INFRARED LASER ABSORPTION SPECTROSCOPY OF NITRIC OXIDE A DISSERTATION · 2020-01-26 · my...

INFRARED LASER ABSORPTION SPECTROSCOPY OF NITRIC OXIDE

FOR SENSING IN HIGH-ENTHALPY AIR

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Christopher A. Almodovar

August 2019

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/yd623fn5087

Includes supplemental files:

1. AbsorbanceRatioLookupTable (AbsorbanceRatioLookupTable.xlsx)

© 2019 by Christopher Andrew Almodovar. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/yd623fn5087

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Ronald Hanson, Primary Adviser


Mark Cappelli


Christopher Strand,

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Motivated by thermometry in high-enthalpy air, advancements towards the measurement and

modeling of high-pressure laser absorption spectroscopy (LAS) of nitric oxide (NO) are presented.

The primary application of this thermometer is to characterize the stagnation conditions (T = 1000–

2500 K and P = 10–130 atm) in a clean-air hypersonic wind tunnel facility. By characterizing the

thermodynamic conditions upstream of the expansion nozzle, the flow conditions of the expand-

ing air can be determined via enthalpy matching. At high temperatures, the Zeldovich mechanism

describes increasing NO formation in air with increasing temperature, making NO an attractive

species for LAS-based temperature measurements in air. Two optical transitions in the R-branch

of the fundamental rovibrational band of NO are selected and their fundamental spectroscopic pa-

rameters are characterized at high temperatures. The temperature sensor design is demonstrated in

reflected shock wave experiments in a large diameter shock tube at pressures up to 5 atm. Although

the target application’s operating pressure range is well outside the demonstration range, the funda-

mental concept of two-wavelength absorption is still valid. However, at high pressures, the selected

optical transitions begin to blend with their neighboring transitions. Thus, accurate knowledge of

the high-temperature and high-pressure absorption at the selected wavelengths requires knowledge

of the spectroscopic parameters defining the neighboring transitions. To measure the spectroscopic

parameters of the many neighboring transitions, a high-pressure, high-temperature (HPHT) optical

cell (up to 800 K and over 30 atm) is designed and demonstrated for mid-infrared spectroscopy with

usable transmission up to approximately 8 microns. With a functional HPHT optical cell, a detailed,

temperature-dependent study (up to 800 K) of the optical transitions in the NO R-branch near 5.3

v

microns is performed. To extend the study to temperatures relevant for the target sensing applica-

tion, shock tube measurements from 1000 to 2500 K supplement the detailed study. Finally, the

spectrum is studied at high pressures. Static cell measurements reveal deviations from the classical

line shape models used accurately at low pressures. The deviations are attributed to collisional line

mixing that emerges when the line widths of the optical transitions are of similar or greater magni-

tude than the separation of optical transitions. A temperature-dependent line mixing model is built

using statistically-based energy gap fitting laws and the full relaxation matrix expression. A com-

parison with measured data reveals good agreement in the regions where inter-branch coupling can

be neglected. In the end, a thorough treatment of the NO spectrum has provided a temperature- and

pressure-dependent model that can be used to predict the absorption spectra of NO in the R-branch

of the fundamental rovibrational band.

vi

Acknowledgments

The opportunity to attempt and complete a PhD was made possible by so many wonderful

people that have formed a generous community of support to rely upon.

Foremost, I am indebted to my advisor Professor Hanson who graciously granted me the oppor-

tunity to join his lab. I am grateful and humbled to be a part of the world-class research program

he has established. His relentless pursuit of excellence has raised the standard of my own work and

has made me a more confident and articulate engineer.

Throughout the years, past and present members of the Hanson lab have proven to be excellent

mentors, collaborators, and friends. Three specific mentors provided a wealth of advice and a

blueprint for success. First, Chris Goldenstein (now professor at Purdue University) welcomed me

to the lab and taught me the basics of laser absorption spectroscopy and what it takes to successfully

deploy TDLAS sensors in the field. Second, Mitch Spearrin (now professor at UCLA) helped me

get through a difficult start to my experiments. Lastly, Dr. Christopher Strand has been an amazing

resource during the last few years of this journey, and he has fostered a strong community within

the lab that has not gone unnoticed. I am also grateful to the other research scientists in the lab,

Dr. Dave Davidson and Dr. Jay Jeffries, who have done a commendable job of mentoring me and

several generations worth of students in the Hanson lab.

Without my lab mates, much of this dissertation would not exist, and my gratitude goes out

to all of you. Specifically, I would like to thank Wey-Wey Su for assisting me in the construction

of the Stanford High-Pressure, High-Temperature infrared optical cell and the collection of a lot of

nitric oxide (NO) spectroscopy data; Rishav Choudhary and Dr. Jiankun Shao for helping run HPST

experiments; Dr. David Salazar for traveling with me to West Virginia to deploy a sensor to make

vii

measurements; Terry Peng for general helpfulness and expertise in the lab; and Dr. Ritobrata Sur

for his advice and expertise. I am also grateful for other members of the Hanson lab with whom

our relationship has been more social rather than research oriented as you have provided necessary

dimension to my mental and physical activities. Thank you for the memories during IM sports,

rounds of golf, softball games, happy hours, and fun conversation.

Thank you to Professors Mark Cappelli, Reggie Mitchell, and Noah Diffenbaugh for serving on

my dissertation reading and/or oral examination committees.

I would like to express my gratitude for researchers and project sponsors outside the lab. First,

I would like to thank the generosity of Sandisk and Harold and Marcia Wagner who funded my

5-quarter fellowship that ultimately allowed me to attend Stanford. I would like to thank Dr. Mike

Kendra and Dr. Brett Pokines of the Air Force Office of Scientific Research for sponsoring me and

my work throughout most of my PhD. I would also like to thank Professor Christopher Brophy and

his lab at the Naval Postgraduate School for providing the opportunity to engage in two interesting

projects.

My deepest appreciation goes out to the network of friends around the country whose love and

support has been a comfort throughout my life and PhD. My gratitude extends to my family who

were so influential in the formative years of my life. To my parents, Hugo and Marina, you are the

most influential people in my life and have provided unrelenting love and support. Thank you for

instilling the values of hard work, dedication, and perseverance through your humbling example.

To my sister, Audrey, thank you for always having my back and making me smile with just a look.

To my extended family, grandma, aunts, uncles, and cousins, thank you for making every family

reunion an enjoyable and memorable spectacle. I would also like to share my appreciation for my

in-laws, Joe and Rose, who have provided unwavering support and a home away from home.

Finally, I thank my wife, Kate, with the utmost gratitude. She has supported me through the

highs and lows of graduate school all while earning her own PhD. I am undoubtedly grateful for her

understanding and her commitment to weathering a long distance relationship through many years

of graduate school. She encourages me when I am down, humbles me when I am brash, and finds

me when I am lost. Her love and support over many years and many miles were essential to the

completion of this degree.

viii

Contents

Abstract v

Acknowledgments vii

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Laser Absorption Spectroscopy 5

2.1 The Beer-Lambert Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Line Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Line Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Line Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Line Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Two-Color Nitric Oxide Thermometry 12

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Sensor Line Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Spectroscopic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Sensor Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.1 Temperature Measurements in Non-Reacting Shock-Heated Gas . . . . . . 27

ix

3.5.2 Temperature and NO Species Measurements During NO Formation . . . . 30

3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 HPHT Optical Cell 32

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Previous Experimental Facilities and Studies . . . . . . . . . . . . . . . . . . . . . 33

4.3 Design of the High-Pressure, High-Temperature Optical Cell . . . . . . . . . . . . 36

4.4 Characteristics of the HPHT Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4.1 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4.2 Path Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4.3 Temperature Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.4 Pressure Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Experimental Setup and Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5.1 Gas System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5.2 Optical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Measurements and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6.1 Data Reduction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6.2 Room-Temperature Validation . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6.3 High-Temperature, High-Pressure NO Spectra . . . . . . . . . . . . . . . 48


5 Nitric Oxide Line Shapes and Intensities 50

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Measurements and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.1 Multi-Spectral Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.2 Line Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.3 Collision Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.4 Pressure Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3.5 High-Temperature Measurements in a Shock Tube . . . . . . . . . . . . . 67

x

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 High-Pressure NO Spectroscopy 73

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Line Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2.1 The Relaxation Matrix, W . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.2 Constructing W Using Statistically-Based Energy Gap Fitting Laws . . . . 78

6.3 Computing the Spectral Shapes of Interfering Lines . . . . . . . . . . . . . . . . . 81

6.3.1 First-Order Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3.2 Full Relaxation Matrix Expression . . . . . . . . . . . . . . . . . . . . . . 84

6.4 Static Cell Measurements and Analysis . . . . . . . . . . . . . . . . . . . . . . . 85

6.5 Shock Tube Measurements and Analysis . . . . . . . . . . . . . . . . . . . . . . . 93

6.6 High-Pressure NO Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.6.1 Implications of Line Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.6.2 Calculating and Using the Absorbance Ratio for Temperature Measurements 99

6.6.3 Temperature Measurement Results . . . . . . . . . . . . . . . . . . . . . . 103


7 Conclusions and Future Work 107


7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2.1 Utilizing Dynamically-Based Scaling Laws for an Improved Line-Mixing

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2.2 Measurements of Full Nitric Oxide Spectra in the High Pressure Shock Tube 109

7.2.3 Extending the Transmission Range of the High-Pressure, High-Temperature

Optical Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A Uncertainty Analysis of Spectroscopic Measurements 110

Bibliography 114

xi

List of Tables

3.1 Summary and comparison of measured line strengths with HITEMP 2010. Uncer-

tainties are given in parenthesis. Line center frequency and lower state energy are

averages of the hyperfine lambda-doubled pairs and line strengths are the sum of

the hyperfine lambda-doubled pairs. . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Comparison of measured collision broadening parameters for NO with HITEMP

2010. Uncertainties are in parenthesis and represent the statistical uncertainties

from the best-fit power law except for the static cell measurements at 298.7 K. . . . 26

5.1 Summary of temperature-dependent spectroscopic parameters of the v′ ← v′′ =

1 ← 0, X2Π3/2 ← X2Π3/2 band of the NO R-branch measured in static cell

experiments from 294 to 802 K. 2γ(296 K) and n are determined directly from

power law fits to experimental data. δ(296 K) and m are determined from power

law fits to experimental data smoothed by third degree polynomials fits as described

in 5.3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Summary of temperature-dependent spectroscopic parameters of the v′ ← v′′ =

1 ← 0, X2Π1/2 ← X2Π1/2 band of the NO R-branch measured in static cell

experiments from 294 to 802 K. 2γ(296 K) and n are determined directly from

power law fits to experimental data. δ(296 K) and m are determined from power

law fits to experimental data smoothed by third degree polynomials fits as described

in 5.3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xii

5.3 Summary of spectroscopic parameters of the v′ ← v′′ = 2← 1 hot band of the NO

R-branch measured in static cell experiments at 802 K. S(296 K) and 2γ(802 K) are

determined from the multi-spectral fitting routine described in 5.3.1. . . . . . . . . 65

5.4 Measured spectroscopic parameters from high-temperature shock tube experiments 68

6.1 Energy gap fitting laws commonly found in the literature. . . . . . . . . . . . . . . 79

6.2 Steps to determine the relaxation matrix, W , via energy gap fitting laws. . . . . . . 80

6.3 Steps to perform the full relaxation matrix expression calculation. . . . . . . . . . 86

6.4 Modified exponential gap (MEG) law fitting parameters for the NO fundamental

R-branch. The fitting parameters are determined using T0 = 296 K. . . . . . . . . 88

6.5 Steps for measuring temperature using the absorbance ratio surface, R(T, P, ν1, ν2)

for ν1 = 1940.76 cm−1 and ν2 = 1986.55 cm−1. . . . . . . . . . . . . . . . . . . . 100

6.6 Coefficients for the polynomial fit to the absorbance ratio surface in Figure 6.13.

The fifth degree polynomial is defined by Eq. (6.31). Cefficients should only be

used with Eq. (6.31) for NO in N2, T = 1000–2500 K, P = 10–130 atm, ν1 =

1940.76 cm−1, and ν2 = 1986.55 cm−1. . . . . . . . . . . . . . . . . . . . . . . . 100

xiii

List of Figures

2.1 Typical laser absorption spectroscopy experiment. An electromagnetic beam from

a monochromatic light source (e.g. diode laser) with incident intensity I0 is directed

through an absorbing medium and the transmitted intensity I is collected on a pho-

tovoltaic detector. A current waveform drives the wavenumber tuning of the diode

laser as it scans over an absorption transition. . . . . . . . . . . . . . . . . . . . . 6

3.1 Absorption linestrengths (HITEMP 2010) of the infrared NO spectrum at 2000 K

and 1 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Simulated absorbance spectrum (HITEMP 2010) of NO in air at 1 atm, 2000 K, and

XNO fixed at 1%. Water vapor is also simulated at 1 atm, 2000 K and 1000 ppm. . 15

3.3 Absorbance and temperature sensitivity of the selected transitions at fixed pressure

and NO mole fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Experiment Setup. Panel (a) shows the instrumentation and equipment for static

cell and/or shock tube experiments; (b) shows the line of sights during shock tube

experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 The left panel displays Voigt fits to measured line shapes of the Π1/2 R(20.5) and

Π3/2 R(20.5) transitions of nitric oxide’s fundamental band; measurements were

made in a room-temperature static cell. The right panel displays Voigt fits to mea-

sured line shapes of the Π3/2 R(41.5) and Π1/2 R(42.5) transitions; measurements

were made during shock tube experiments. . . . . . . . . . . . . . . . . . . . . . . 21

3.6 Measured collision broadening coefficients of NO in Ar and N2 from 1000 to 3000

K and between 1 and 5 atmospheres. . . . . . . . . . . . . . . . . . . . . . . . . . 24

xiv

3.7 Measured line strength values of the Π3/2 R(41.5) and Π1/2 R(42.5) transitions from

NO in Ar shock tube experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.8 Measurement traces from reflected shock tube experiment of 1.97% NO in N2. T5

= 1550 K and P5 = 4.8 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.9 Demonstration of temperature measurement for fixed concentrations of NO in Ar

and N2. The left panel displays temperature values obtained from the ratio of

R(20.5) line center absorbance and R(41.5) integrated area, and the right panel dis-

plays temperature values obtained from the ratio of R(20.5) line center absorbance

and R(42.5) integrated area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.10 Demonstration of temperature and NO measurements during an NO formation ex-

periment. Conditions at the beginning of Region 5: 1.3% NO2 in argon at 1714 K

and 4.15 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 High-pressure, High-Temperature (HPHT) optical cell assembly schematic. . . . . 36

4.2 Cross-sectional view of HPHT cell 3-D rendering. The components connected to the

tee fitting are plug-gland-collar components (the High Pressure Equipment Com-

pany) that form a high-pressure seal. . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Measured temperature profiles of HPHT system. . . . . . . . . . . . . . . . . . . . 39

4.4 Transmission measurement of a CaF2 crystal at room-temperature. . . . . . . . . . 40

4.5 Experimental setup detailing the laser beam paths through the HPHT optical cell,

reference cell, and optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.6 Data reduction process for high-pressure measurements using the ECQCL. . . . . . 45

4.7 Room-temperature NO spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.8 Measured high-pressure spectra of NO in N2 at 618 and 802 K compared with

HITEMP simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 Experimental setup of the static cell experiments from 294-802 K. . . . . . . . . . 52

xv

5.2 (a) Raw data from an experiment using the ECQCL to interrogate the transitions

near 1897.17 cm−1 at 802 K and several pressures. The top panel displays signals

from the HPHT optical cell, and the bottom panel displays signals from the refer-

ence optical cell. The reference cell signals are used to align the data in relative

frequency. (b) Raw data from an experiment using the DFBQCL to interrogate the

transitions near 1986.75 cm−1 at 802 K and several pressures. . . . . . . . . . . . 54

5.3 Multi-spectral fits near 1897.17 cm−1 of the R(5.5) transitions from the 2Π1/2 and2Π3/2 subbands of the v′ ← v′′ = 1 ← 0 band and the 2Π1/2R(15.5) transition

from the v′ ← v′′ = 2 ← 1 hot band. Measurements were collected at 802 K and

pressures ranging from 0.0267 to 1.0238 atm. . . . . . . . . . . . . . . . . . . . . 56

5.4 Line strength versus temperature for v′ ← v′′ = 1 ← 0 2Π1/2 and 2Π3/2 R(20.5)

transitions. Measurements agree with the values reported by the HITEMP database. 60

5.5 Normalized line strengths of v′ ← v′′ = 1 ← 0 R-branch transitions at several

temperatures. Top: 2Π1/2 subband. Bottom: 2Π3/2 subband. For all temperatures,

the average deviation between measured and HITEMP line strength simulations is

2%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.6 Measured collision broadening coefficients for v′ ← v′′ = 1 ← 0 R-branch transi-

tions at several temperatures. (a) Measured N2 broadening versus J ′′ at four differ-

ent temperatures. Top: 2Π1/2 subband. Bottom: 2Π3/2 subband. The measurements

by Spencer et al. are plotted for comparison with the 294 K data. (b) Measured Ar

broadening versus J ′′ at four different temperatures. Top: 2Π1/2 subband. Bottom:2Π3/2 subband. The measurements by Pope and Wolf are plotted for comparison

with the 294 K data. v′ ← v′′ = 2 ← 1 hot band transitions were measured at 802

K and continue the trend established by the v′ ← v′′ = 1← 0 transitions. . . . . . 62

5.7 Comparison of N2 pressure broadening with air pressure broadening. On average,

measured air broadening is 2.5% less than N2 broadening. . . . . . . . . . . . . . 63

5.8 Pressure broadening power law fit parameters versus J ′′. Error bars represent the

standard error of the best-fit parameters. Solid lines represent third degree polyno-

mials fit to best-fit parameters as a function of J ′′. . . . . . . . . . . . . . . . . . . 64

xvi

5.9 Measured pressure shift coefficients for R-branch transitions at several tempera-

tures. Pressure shifts measured at 296 K by Spencer et al. are shown by the solid

line. Temperature dependence and slight J ′′ dependence of the pressure shifts are

apparent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.10 Line strength measurements of the R(39.5)-R(43.5) transitions. . . . . . . . . . . 69

5.11 Measured pressure broadening coefficients of the R(39.5)-R(43.5) transitions. Open

symbols represent 2γNO−N2 and filled symbols represent 2γNO−Ar. . . . . . . . . 70

5.12 (a) Measured pressure broadening coefficients for the 2Π1/2R(20.5) transition from

room-temperature to 2500 K. (b) Comparison of pressure broadening temperature

exponent determined from measurements in different temperature ranges. . . . . . 71

6.1 Energy level diagram describing the line mixing process between two adjacent op-

tical transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 Comparison of Lorentzian and Rosenkranz (first-order approximation) line shape

profiles. The difference between the two profiles (i.e. the dispersion shape of the

line mixing contribution) is also plotted. . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Energy gap law fits to experimentally determined broadening coefficients at several

temperatures using the modified exponential gap law (MEG). . . . . . . . . . . . . 87

6.4 Off-diagonal relaxation matrix elements (Wkk′) for the 2Π1/2 R(5.5) (left) and

R(15.5) (right) transitions in the Π1/2 subband at 296 K calculated from the MEG

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5 NO spectra measurements in N2 at 294 K and and pressures of 5, 20, and 34 atm.

Simulations using Lorentizian line shapes, Rosenkranz line shapes, and the full re-

laxation matrix expression are plotted for comparison. . . . . . . . . . . . . . . . . 89

6.6 Measured NO spectra in N2 at 618 K and pressures of 5, 20, and 33 atm. Simulations

using Lorentizian line shapes, Rosenkranz line shapes, and the full relaxation matrix

expression are plotted for comparison. . . . . . . . . . . . . . . . . . . . . . . . . 91

xvii

6.7 Measured NO spectra in N2 at 802 K and pressures of 5, 20, and 32 atm. Simulations

using Lorentizian line shapes, Rosenkranz line shapes, and the full relaxation matrix

expression are plotted for comparison. . . . . . . . . . . . . . . . . . . . . . . . . 92

6.8 Measurement traces from fixed-wavelength experiments in a high-pressure shock

tube (HPST). The driven gas is NO in N2 at mole fraction specified in the sub-

caption. Other details of the experiment — T5, P5, ν0 — are also specified in the

subcaption. All of these experiments are monitoring ν = 1940.76 cm−1 near the2Π3/2R(20.5) transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.9 Summary of HPST measurements at low pressures. All simulations use the line

mixing model for absorbance of NO in N2 with the mole fraction specified in the

figure subcaption. For the measurements shown, the temperatures and pressures in

the subcaptions are averages of the nominal experimental conditions. . . . . . . . 96

6.10 Summary of HPST measurements at high pressures. All simulations use the line

mixing model for absorbance of NO in N2 with the NO mole fraction specified in

the figure subcaption. "BL" denotes the use of a simple boundary layer correction.

For the measurements shown, the temperatures and pressures in the subcaptions are

averages of the nominal experimental conditions. . . . . . . . . . . . . . . . . . . 97

6.11 Diagram of the simple boundary layer model used to compare the high-pressure

shock tube absorbance measurements to the line mixing absorbance model. A half

model of the shock tube is shown with region 5 and region 2 being separated by the

reflected shock wave. The boundary layer is assumed to be 1 mm and the thermal

boundary layer is approximated by a linear model with the two ends of the boundary

layer being defined by the shock tube wall temperature, Twall, and the reflected

shock temperature, T5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.12 Implications of line mixing on temperature sensing. (a) The ratio of simulated ab-

sorbance at ν1 = 1940.76 and ν2 = 1986.55 cm−1 from T = 1000–2500 K at P = 90

atm. (b) The estimated temperature measurement error from T = 1000–2500 K at

several pressures. Errors are largest at high number densities when line mixing is

strong. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xviii

6.13 (a) Calculated absorbance ratio surface from 1000-2500 K and 10–130 atm for NO

in N2, ν1 = 1940.76 cm−1, and ν2 = 1986.55 cm−1. Calculations are performed

using the MEG line mixing model presented in previous sections. (b) Residuals of

the fifth degree polynomial fit to the absorbance ratio surface shown in (a). . . . . 102

6.14 (a) The R(T, P, ν1, ν2) surface represented as a heat map and calculated from Eq.

(6.31). (b) Heat map representing the error in temperature due to the imperfect fit of

the polynomial expression for R(T, P, ν1, ν2). Maximum error is ≈ 4% near 2500

K, and errors at other conditions are below 1%. . . . . . . . . . . . . . . . . . . . 103

6.15 Top: pressure traces from two nearly identical shock tube experiments with average

conditions behind the reflected shock of 20.4 amagat, P5 = 87.7 atm, T5 = 1172

K, XNO = 0.0203 in N2. Middle: absorbance traces for ν1 = 1940.76 cm−1 and

ν2 = 1986.55 cm−1. The dashed lines represent the simulated absorbance for the

experimental conditions. Bottom: temperature measurement calculated from the

ratio of the two absorbance traces. The dashed and dot-dashed lines represent the

calculated temperature for each experiment. . . . . . . . . . . . . . . . . . . . . . 104

6.16 Measured temperature versus calculated temperature from reflected shock wave ex-

periments in the Stanford High Pressure Shock Tube (HPST). . . . . . . . . . . . 105

xix

Chapter 1

Introduction

1.1 Background and Motivation

High-velocity aero-propulsion has been a motivating factor of research for several decades dat-

ing back to the cold war. As the boundaries are pushed towards the hypersonic regime (M ≥ 5),

the requirements placed upon the set of tools needed to effectively evaluate new architectures for

hypersonic flight have grown. A number of ground test facilities for hypersonic research exist and

have various benefits depending on the projects with which they are associated [1]. Impulse fa-

cilities such as shock tunnels and expansion tubes provide the high enthalpy conditions needed to

simulate hypersonic flight; however, test duration is limited to a few milliseconds. "Cold" flow facil-

ities provide hypersonic velocities for longer duration (O minutes), but the stagnation temperature

of the flow is considerably less than real flight conditions. Vitiated air facilities utilize combustion

to reach the stagnation temperatures needed to match real flight conditions. However, the com-

bustion process adds combustion product species such as H2O and CO2 to the test flow, and the

presence of additional combustion species in the vitiated air must be accounted for when comparing

experiments with computational results. Furthermore, vitiated air may have a pronounced effect

on the operation of combustion propulsive devices such as scramjets [2]. Clean air facilities mimic

hypersonic flight conditions by matching the free stream enthalpy via sophisticated systems of high-

temperature heaters. The blowdown facilities generally have a set of nozzles designed to reach a

1

2 CHAPTER 1. INTRODUCTION

few well-known flight conditions (e.g. M = 5, M = 7, etc.). A new facility called the Hypersonic

Aero Propulsion Clean Air Testbed (HAPCAT) is designed with a variable Mach nozzle intended

to simulate a variety of flight conditions to provide adaptability during ground testing [3]. Proper

characterization of the simulated flight conditions requires knowledge of the stagnation tempera-

ture (1000–2500 K) and pressure (10–130 atm) upstream of the nozzle. A primary challenge lies

in accurate, time-resolved temperature measurements of the high-enthalpy gas. At such extreme

conditions, conventional diagnostics such as high-temperature thermocouples are susceptible to ox-

idation or melting. This dissertation addresses this problem by developing a thermometry strategy

for high-enthalpy air via laser absorption spectroscopy (LAS) and by investigating the requisite

spectroscopy for a practical and accurate temperature sensor.

Demonstrations of successful LAS sensors for a number of challenging environments can

be found in the literature [4–6]. The benefits of LAS arise from its quantitative, in situ, and

time-resolved nature. Furthermore, the ability to measure the concentrations of specific atoms or

molecules (and their state populations) [7–9], temperature [10, 11], and velocity (via Doppler shift

velocimetry)[12] allows flexibility and utility for numerous applications.

To use an LAS-based temperature sensor in the high-enthalpy environment of HAPCAT or

similar facilities, the target species nitric oxide (NO) was selected due to its increasing formation in

air with increasing temperature. Accurate knowledge of the NO spectral shapes is necessary for a

successful temperature diagnostic. Hence, this dissertation presents a detailed study of the infrared

(IR) spectrum of NO near 5 µm at temperatures and pressures relevant to the present application.

1.2 Overview of Dissertation

The purpose of this dissertation is to provide the fundamental spectroscopy of nitric oxide (NO)

necessary for the implementation of a practical LAS-based temperature sensor in a hypersonic

ground-test facility. The purpose of the remaining chapters is outlined below.

• Chapter 2 introduces the fundamental relations and nomenclature used in this dissertation to

describe absorption spectroscopy. Fundamental concepts such as line position, line strength,

and line shapes are defined.

1.2. OVERVIEW OF DISSERTATION 3

• Chapter 3 outlines the design and demonstration of a two-wavelength thermometry strategy

for high-temperature gases. Wavelength selection is performed by modeling absorption sig-

nals over the temperature range of interest. The spectroscopic parameters of the two absorp-

tion transitions selected are closely studied from 1000 to 3000 K in shock tube experiments

with pressures up to 5 atm. Finally, the sensor’s performance is demonstrated during both

fixed- and transient-concentration experiments.

• Chapter 4 describes the design and demonstration of a high-pressure, high-temperature

(HPHT) optical cell for IR spectroscopy up to ∼ 8 µm . The operating temperature and

pressure ranges were demonstrated to be 294–800 K and 0.025–35 atm, respectively. The

HPHT has a 21.3 cm path length, and the temperature uniformity along the path length was

evaluated with maximum deviations from the mean temperature being no greater than 2.4%

and the standard deviation being less than 1%. IR NO spectra at several temperatures and

pressures were measured to demonstrate the optical cell’s performance. At high gas densities,

deviations from the line shapes described in Chapter 2 are observed. These deviations are

explored further in Chapter 6.

• Chapter 5 focuses on the measurement of fundamental spectroscopic parameters of the R-

branch in the fundamental rovibrational band of NO. A detailed line shape study of over 40

absorption transitions from 294 to 802 K is performed in the HPHT optical cell facility de-

scribed in Chapter 4. Measured transition line strengths agree well with previously published

values and the temperature dependence of collision widths and pressure shifts are determined.

To study line shape temperature dependence beyond the limits of the HPHT optical cell, NO

absorption measurements were made in high-temperature shock tube experiments from 1000

to 2500 K. Evidence of the break down of the typical power law dependence defined in Chap-

ter 5 over a wide temperature range is observed.

• Chapter 6 explores the line shape deviations first encountered in the HPHT optical cell

demonstration in Chapter 4. The primary culprit of the observed deviations is identified as

collisional line mixing. Line mixing arises from inelastic collision processes that result in

population transfers between adjacent states of the absorbing molecule. At sufficiently high

4 CHAPTER 1. INTRODUCTION

gas densities, the isolated transition line shapes described in Chapter 2 are no longer valid and

a different treatment of the line shapes is necessary. Energy gap fitting laws are used to deter-

mine temperature-dependent line mixing parameters and formulate a model for comparison

with high-pressure spectra measured in the HPHT optical cell and the high-pressure shock

tube (HPST).

Chapter 7 summarizes the previous chapters and includes suggestions for future work. Appendix

A details the uncertainty analysis for the line shape parameters measured in Chapters 3 and 5.

Chapter 2

Laser Absorption Spectroscopy

Laser absorption spectroscopy (LAS) is a well-known technique used to study molecular struc-

ture, and it is also utilized in sensors to measure the concentration of particular gaseous species as

well as their temperature, pressure, and even bulk velocity for applications in combustion, propul-

sion, and hypersonics [4, 5]. The main advantages of LAS are its relatively simple operation in its

most basic form and its versatility especially in advanced variants such as wavelength modulation

spectroscopy (WMS) [13–16]. The IR wavelength regions are of particular interest because un-

known molecules can be identified from absorption at different wavelengths correlated to the bonds

and functional groups of the molecules. Furthermore, the strong fundamental absorption bands of

common combustion species (H2O, CO2, CO, and NO) are found in the mid-IR and are becoming

increasingly utilized in sensor designs as mid-IR light sources and optics continue to improve [5].

Successful and accurate LAS diagnostics require accurate spectroscopic models of the environment

under investigation. While available spectroscopic databases have generally proven reliable for line

positions and strengths of absorption transitions, their characterization of line shapes beyond low-

temperature applications often proves insufficient. The fundamental relations and corresponding

units describing LAS are detailed below.

5

6 CHAPTER 2. LASER ABSORPTION SPECTROSCOPY

I(t) I0(t) or v(t)

Current Tuning

Figure 2.1: Typical laser absorption spectroscopy experiment. An electromagnetic beam from amonochromatic light source (e.g. diode laser) with incident intensity I0 is directed through an ab-sorbing medium and the transmitted intensity I is collected on a photovoltaic detector. A currentwaveform drives the wavenumber tuning of the diode laser as it scans over an absorption transition.

2.1 The Beer-Lambert Relation

The Beer-Lambert relation

(I

I0

)ν

= e−α(ν) = e−kνL (2.1)

describes the absorption of monochromatic light by an absorbing gas where I is the measured light

intensity, I0 is the incident light intensity, ν is the frequency of light in wavenumber units (cm−1),

α is the spectral absorbance, kν (1/cm−1) is the spectral absorption coefficient, and L (cm) is the

absorption path length. The spectral absorbance is further defined by

α(ν) =∑i

Si(T )φi(ν, T, P,X)PXabsL =∑i

kν,iL (2.2)

where Si (cm−2/atm) is the temperature-dependent line strength of absorption transition i, T (K)

is the absolute gas temperature, φi (cm) is the line shape profile of the absorption transition i, P

(atm) is the total pressure of the gas, X is the mole fraction vector of the gas with Xabs denoting the

absorbing species, and kν,i represents the spectral absorption coefficient for the isolated transition i.

A typical absorption experiment is depicted in Figure 2.1. A monochromatic light source, such

as a laser, emits a collimated beam of electromagnetic radiation that is directed through the absorb-

ing medium of interest (e.g. combustion exhaust gases, atmosphere, high temperature gases, etc.).

2.2. LINE POSITIONS 7

The light intensity reaching the photvoltaic detector, I , represents a fraction of the incident light

intensity, I0, as described by Eq. 2.1.

2.2 Line Positions

The position of absorption transitions within the electromagnetic spectrum depends on the al-

lowed quantum energy states of the absorber (i.e. atom or molecule) of interest and selection rules

dictated by quantum mechanics. These energy states are determined from absorber-specific physi-

cal properties — such as its mass, frequency of vibration, rotational constant, and electron structure

— and absorber-specific selection rules. Consider two energy states of a particular absorber, with

E′′ and E′ representing the lower and upper state energies (cm−1), respectively, such that E′ > E′′.

When the energy of a photon, hcν, is resonant with the difference between the lower and upper

state energy, hcν0 = hc(E′ − E′′), an absorber in the lower state will absorb the photon and the

absorber’s energy state will increase from E′′ to E′. In this dissertation, the spectral and energy

units are given in wavenumber (cm−1) unless otherwise noted.

Fortunately, databases with high accuracy line position tabulations — such as HITRAN [17, 18]

and NIST ASD [19] — exist for a number of absorbers. Aside from these tabulations, line positions

of particular rovibrational or rovibronic bands can be calculated from band origin data, rotational

constants, and selection rules for a particular absorber. For instance, by using the rigid rotor and

simple harmonic oscillator assumptions coupled with the Born-Oppenheimer approximation, the

line positions of the R and P branches of a diatomic linear molecule’s infrared rovibrational band

are given by

R(v′′, J ′′) = ω0 + 2B(J ′′ + 1) (2.3)

P (v′′, J ′′) = ω0 − 2BJ ′′ (2.4)

where v′′ is the lower state vibrational quantum number, J ′′ is the lower state rotational quantum

number, ω0 is the vibrational energy, andB is the rotational constant. Additional parameters and up-

dated expressions that account for non-rigid rotor and anharmonic oscillatory behavior are reported


in the literature.

2.3 Line Strength

The temperature dependence of the pressure-normalized transition line strength in units of

cm−2/atm is defined as [20]

S(T ) = S(T0)Q(T0)T0

Q(T )Texp

[− hcE′′

kB

(1

T− 1

T0

)] [1− exp(− hcν0kBT

)][1− exp

(− hcν0

kBT0

)] (2.5)

with S(T0) (cm−2/atm) representing the transition line strength at reference temperature T0 (typi-

cally T0 = 296 K), Q(T ) is the partition function at temperature T of the absorbing species, h (J-s)

is Planck’s constant, c (cm/s) is the speed of light, kB (J/K) is the Boltzmann constant, E′′ (cm−1)

is the lower state energy of the absorption transitions, and ν0 (cm−1) is the transition line center

frequency. For many molecules, S(T0), Q(T ), E′′, and ν0 are tabulated in the HITRAN database

[17].

2.4 Line Shapes

Understanding transition line shapes and how they change with temperature and pressures is

fundamental to the success of LAS-based sensors. A number of distribution profiles have been used

to account for various effects perturbing transition line shapes. The Lorentzian line shape models

the broadening of line shapes due to energy exchanging collisions that shorten the lifetime of the

absorbing molecule at a given energy level. The Voigt line shape introduces Doppler effects by

accounting for the velocity distribution of the absorbing molecules and their associated Doppler

shifts. The Galatry and Rautian-Sobel’man (or Nelkin-Ghatak) profiles address collisional narrow-

ing (or Dicke narrowing) through the soft and hard collision models, respectively . Finally, speed-

dependent profiles seek to account for the influence of the absorber’s speed or, in other words, its

2.4. LINE SHAPES 9

location within the equilibrium speed-distribution function [21]. Aside from Chapter 6 where col-

lisional line mixing further perturbs the line shapes, the Voigt line shape profile is used throughout

this work as it provides acceptable performance when fitting experimental data and does so with a

minimum number of free parameters.

The Voigt line shape (φV ) is the convolution of Gaussian (φD) and Lorentzian (φL) line shapes

φV (ν) =

∫ ∞−∞

φD(u)φL(ν − u)du (2.6)

Numerical approximations of the Voigt profile [22, 23] require inputs of Doppler (Gaussian) and

collision (Lorentzian) widths, which are given below in Eqs. (2.7) and (2.8).

∆νD = ν0

(8kT ln(2)

mc2

)1/2

= ν0(7.1623× 10−7)

(T

M

)1/2

(2.7)

∆νc = P∑A

XA2γB−A (2.8)

For these expressions, M (g/mol) is molecular weight, XA is the mole fraction of collision partner

A, and 2γB−A (cm−1/atm) is the FWHM collision broadening coefficient of absorbing molecule B

with collision partnerA. An additional input to calculations of Voigt profiles is the pressure-induced

line shift of the absorption transition that results from perturbations of the intermolecular potential

of the collision partners. Similar to Eq. (2.8), Eq. (2.9) describes the pressure shift of absorbing

molecule B as the sum of collision-partner-dependent pressure shift coefficients (δB−A cm−1/atm)

scaled by the partial pressure of collision partner A.

∆νs = P∑A

XAδB−A (2.9)

The temperature dependence of the collision broadening and pressure shift coefficients has generally

been modeled by a power-law with temperature as shown in Eqs. (2.10) and (2.11).

2γ(T ) = 2γ(T0)

(T0

T

)n(2.10)


δ(T ) = δ(T0)

(T0

T

)m(2.11)

Here, n and m are the temperature exponents of the collision broadening and pressure shift co-

efficients, respectively. The power-law model fits experimental data reasonably well over narrow

temperature ranges. However, extrapolation to temperatures outside the studied range, particularly

in the infrared and microwave regions, may lead to significant errors [24–27].

The previous discussion on fundamental absorption spectroscopy of isolated transitions is nec-

essary to understand collisional effects on molecular spectra. As pressure increases, typically ∆νC

follows. At moderate pressures, isolated transitions begin overlapping, and typically a superpo-

sition of transition line shapes sufficiently describes the observed spectrum. At higher pressures,

the spectrum loses its discrete nature in favor of a continuum. Thus, TDLAS sensors designed to

operate at high pressures require accurate knowledge of the spectral parameters for all transitions

with significant contributions to the absorption coefficient at the senor’s optical frequency. How-

ever, additional collision effects such as line mixing and the breakdown of the impact approximation

bring complications beyond the Voigt profile and can introduce significant errors between measured

and simulated absorbance in both the peaks and wings of individual transitions and the entire band

[13, 28, 29]. Facilities such as the high-pressure, high-temperature optical cell described in Chap-

ter 4 are essential to better understand these phenomena as a function of various thermodynamic

conditions.

2.5 Line Mixing

A detailed discussion of collisional line mixing is found in Chapter 6, but a brief discussion

is presented here. Consider two transitions k and k′ whose spacing is |ν0,k − ν0,k′ |. Generally,

line mixing becomes significant when ∆νc is of the same order or greater than the transition spac-

ing [21]. Framing the line mixing problem requires knowledge of the (complex) impact relaxation

matrix, W , that describes the influence of collisions on the spectrum’s shape. This includes the

broadening and pressure shift coefficients of Eqs. 2.10 and 2.11. In fact, the diagonal components

2.5. LINE MIXING 11

of the relaxation matrix are equal to Wk,k = γ − iδ for transition k. As a result, the relaxation ma-

trix can be calculated from fits of empirically determined pressure broadening coefficients through

statistically-based energy gap fitting laws (defined in Chapter 6) or dynamically-based scaling laws.

Once known, the relaxation matrix can be used to calculate line mixing effects via the commonly

used first-order approximation that derives the Rosenkranz line shape profile or via a calculation

using the full relaxation matrix expression. The first-order approximation is convenient because

first-order line mixing coefficients can be easily added to line-by-line databases and the first-order

line mixing coefficients can be determined directly from line shape fitting using the Rosenkranz

profile. However, the approximation is accurate only when the off-diagonal relaxation matrix terms

are much smaller than the transition line spacing (i.e. PWk,k′ << |ν0,k − ν0,k′ | for k 6= k′). For in-

stance, the off-diagonal relaxation matrix terms calculated for NO at room temperature are ≈ 0.005

cm−1/atm, so for a pressure of 20 atm the use of the first-order approximation may be questionable

since PWk,k′ is roughly 3% of the transition spacing. In Chapter 6, line mixing will be discussed

in further detail.

Chapter 3

Two-Color Absorption Spectroscopy for

Nitric Oxide Thermometry

The contents of this chapter have been published in the Journal of Quantitative Spectroscopy

and Radiative Transfer [11]

3.1 Introduction

Nitric oxide (NO) forms in heated air or combustion exhaust gases by the oxidation of nitro-

gen which is described by the Zeldovich mechanism [30]. The extent of its formation is strongly

coupled with temperature as equilibrium calculations of dry air show that the mole fraction of NO

quickly grows from 200 ppm at 1200 K to 4% at 3000 K, at atmospheric pressure. The effects of

high-temperature NO formation from combustion engines can be seen in the photochemical smog

that hovers above urban areas [31]. Despite being unwanted due to environmental concerns, ni-

tric oxide’s increased presence at elevated temperatures can be exploited to measure gas properties

with absorption spectroscopy. This chapter presents a novel sensing strategy for measurement of

temperature and NO mole fraction at high-enthalpy air conditions (1000-3000 K).

The desire to accurately monitor NO and processes that produce it is demonstrated by the body

of work focused on NO absorption spectroscopy, particularly in its fundamental vibration band.

12

3.1. INTRODUCTION 13

2 3 4 5 6 7

10-5

10-3

10-1

Abso

rptio

n Li

nest

reng

th (c

m-2/a

tm)

Wavelength ( m)

6000 4000 2000

2nd Overtone

1st Overtone

Wavenumber (cm-1)

T = 2000 K

Fundamental Band

Figure 3.1: Absorption linestrengths (HITEMP 2010) of the infrared NO spectrum at 2000 K and 1atm [18].

Interest in using mid-IR absorption spectroscopy for remote sensing of NO in the atmosphere has

motivated many spectroscopic studies of the fundamental vibration band at temperatures between

200 and 300 K. These studies, typically performed with Fourier transform absorption spectroscopy,

focused on wide surveys of the fundamental band to improve or provide reassurance for previously

measured or calculated values of spectroscopic parameters such as line positions, line strengths,

collision broadening coefficients, and collision-induced line shifts. With a continually improving

spectroscopic database, atmospheric scientists are able to take meaningful measurements of NO

and better understand its role in atmospheric processes [32–36]. More relevant to the current work,

is the use of NO spectroscopy in high-temperature gas sensing (> 1000 K), which has received

less attention with regards to spectroscopic investigation. In this work, access to the fundamental

vibration band of the NO spectrum is particularly attractive due to strong signals at the conditions

of interest as shown in Figure 3.1.

Heated air, which involves the formation of NO, is utilized in many applications such as power

14 CHAPTER 3. TWO-COLOR NITRIC OXIDE THERMOMETRY

generation from combustion and in propulsion ground test facilities where preheaters heat and pres-

surize air before expanding it to supersonic conditions [37]. Recently, Spearrin et al. developed a

method to measure temperature in high-enthalpy air flows with a continuous wave (CW) external

cavity quantum cascade laser. The method measures the NO concentration via absorption spec-

troscopy at a wavelength near the 2Π3/2R(15.5)(v = 1 ← 0) transition of the fundamental band

and uses the strong temperature dependence of NO concentration in equilibrium air to infer temper-

ature [38]. Furthermore, absorption spectroscopy of the fundamental band of NO has been used to

measure velocity, temperature, and NO concentration in the test section of a high-enthalpy hyper-

sonic wind tunnel [39]. Other high-temperature applications of NO absorption spectroscopy can be

found in combustion and chemical kinetic studies. An early utilization of the fundamental band for

high-temperature NO sensing in shock tubes was demonstrated by Hanson et al. at a fixed wave-

length near the 2Π3/2 R(18.5) transition using a grating-tunable carbon monoxide (CO) laser [40].

Falcone et al. extended high-temperature sensing of NO by studying numerous rovibrational tran-

sitions (P(2.5)–R(14.5)) in the fundamental band using a low-power (< 100µW ), cryogenic diode

laser system. Line shape profiles in nitrogen, argon, and combustion gases were characterized for

use in kinetic shock tube and combustion studies [41, 42]. In a similar manner, von Gersum and

Roth used cryogenically-cooled diode lasers, probing the P(6.5), P(11.5) and R(21.5) transitions

of the 2Π1/2 subband to study the decomposition of NO in argon (Ar) behind shock waves over

a temperature range of 2500–3500 K [43]. More recently, the emergence of room-temperature,

high-power, tunable quantum cascade lasers has allowed field demonstrations of NO sensing in

combustion gases from a coal-fired power plant [44] and in analyzing exhaled human breath for

respiratory issues [45]. Additionally, a number of infrared LAS-based sensors for NO using the first

and second overtone bands have been designed and demonstrated [46, 47].

The current chapter advances the work of Spearrin et al. by developing a sensor for high-

temperature gases that measures temperature under both chemical equilibrium and non-equilibrium

conditions (e.g. situations when residence times are short) when appreciable concentrations of NO

are present. Of note is a new wavelength-pairing selected for sensing in non-equilibrium conditions.

Here, we present the design and development of a new multi-wavelength temperature- and species-

sensing strategy, the measurement of requisite fundamental spectroscopic parameters of NO near 5

3.2. SENSOR LINE SELECTION 15

1920 1940 1960 1980 20000.00

0.05

0.10

0.15

0.20

0.25

R(42.5)R(41.5)

R(20.5)

Absorban

ce

Wavenumber (cm-1)

NO H2O

T = 2000 KP = 1 atmL = 10 cmXNO= 1%XH2O = 0.1%

R(20.5)

1986.5 1987.00.0

0.1

1939 1940 19410.0

0.1

0.2

Figure 3.2: Simulated absorbance spectrum (HITEMP 2010) of NO in air at 1 atm, 2000 K, andXNO fixed at 1%. Water vapor is also simulated at 1 atm, 2000 K and 1000 ppm.

µm, and the demonstration of temperature and NO sensing using the proposed sensor in a shock

tube. Absorption transitions in the fundamental absorption band of NO were selected because of

their strong signals (1 to 2 orders of magnitude stronger than the 1st and 2nd overtone bands; see

Figure 3.1) over the range of conditions studied here and because they can be accessed via commer-

cially available, tunable quantum cascade lasers. Two quantum cascade lasers were used to measure

four rovibrational transitions in a variety of conditions during static cell and shock tube experi-

ments. To the knowledge of the authors, no spectroscopic measurements of the selected transitions

have been made at elevated temperatures as presented here. The spectroscopic parameters were

measured assuming Voigt line shape profiles over a range of temperatures and pressures. Lastly, the

sensor’s capabilities were demonstrated under non-reactive and reactive conditions in a shock tube.


1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8Ab

sorb

ance

Temperature (K)

R(20.5) R(20.5) R(41.5) R(42.5)

(dR/R)/(dT/T)

0

1

2

3

4

5

R(20.5) and R(41.5)

Line Pair Temperature Sensitivity

P = 1 atmL = 10 cmXNO = 2%

Figure 3.3: Absorbance and temperature sensitivity of the selected transitions at fixed pressure andNO mole fraction.

3.2 Sensor Line Selection

Figure 3.2 shows a simulation of the absorbance spectrum of NO in air. The R-branch is sim-

ulated from 1920 to 2000 cm−1 at 1 atm, 2000 K, and 1% NO in air. The water spectrum was

also simulated at 2000 K for a concentration of 1000 ppm. Four transitions were selected to mini-

mize water interference while maintaining a large lower state energy difference (∆E′′) between line

pairs, which is essential for sensitive temperature diagnostics (see temperature sensitivity discussion

below). The insets of Figure 3.2 show simulations of the studied transitions, namely the R(20.5)

transitions of the X2Π1/2 and X2Π3/2 subbands of the electronic ground state and the X2Π3/2

R(41.5) and X2Π1/2 R(42.5) transitions.

Absorbance at transition line centers is simulated versus temperature in Figure 3.3 for fixed

pressure (1 atm) and 2 % NO mole fraction in air. The two R(20.5) transitions show strong variations

in absorbance over the entire temperature range while the R(41.5) and R(42.5) transitions (slightly

overlapped in Figure 3.3) show less drastic changes. The behavior of these transitions with respect

3.3. EXPERIMENTAL SETUP 17

to temperature is advantageous for temperature sensing. Several demonstrations of two-transition

temperature measurements have been reported throughout the literature beginning with [48]. These

methods utilize the integrated absorbance (Ai) of a quantum transition, defined by [20]:

Ai =

∫ ∞−∞

α(ν)dν = PXabsSi(T )L (3.1)

For a single absorbing species, a ratio of integrated absorbances is a function of only line strengths

and thus temperature.

R(T ) =A1

A2=S1(T )

S2(T )(3.2)

Taking the derivative of the ratio of integrated areas with respect to temperature and normalizing

gives the normalized temperature sensitivity

dR/R

dT/T≈(hc

kB

)(E1”− E2”)

T=

(hc

kB

)∆E′′

T(3.3)

Hence when employing this method, it is ideal to select transitions with a large ∆E′′. This technique

requires full or nearly full resolution of the transition’s line shape profile Φi(ν), often achieved via

tunable diode lasers. However, at elevated pressures, individual line shapes are no longer resolvable

which requires further spectroscopic characterization. From Eq. (2.1), a ratio of absorbances is not

only a function of line strengths but also of transition line shape functions. The addition of line shape

profiles to the absorbance ratio requires knowledge of the collision broadening coefficients of the

absorbing species with its collision partners. Therefore, study of collision broadening parameters

over a wide temperature range is essential for development of an accurate LAS-based sensor for

applications where the full line shape profile cannot be resolved (e.g. high-pressure environments).

3.3 Experimental Setup

Two facilities were utilized to measure line strengths and collision widths in a variety of gas

conditions: a room-temperature static cell and a stainless steel shock tube. Pressure in the 21 cm

long static cell was measured with an MKS Baratron capacitance manometer pressure transducer


(resolution to 1 torr). The 15.4 cm inner diameter stainless steel shock tube, with a 3.7 m long driver

section and a 10 m long driven section, was used to achieve a variety of conditions between 1000 –

3000 K and 1 – 5 atm. Before each shock tube experiment, driven and driver sections were separated

by a thin plastic diaphragm, both sections were evacuated, and the driven section was filled with the

test gas (i.e. 1.97% NO in N2 or 2% NO in Ar). The shock wave was generated by filling the driver

section with helium until the diaphragm ruptured. The diaphragm rupture was controlled by a two-

blade, perpendicular cutting device for diaphragms up to 0.020" and by scores in thicker diaphragms

(0.040" and 0.060"). Test conditions were determined by measuring the incident shock speed via a

series of five piezoelectric pressure transducers over the last 1.5 m of the driven section. The five

transducers triggered time-interval counters (Fluke PM6666) from which the incident shock speed

was extrapolated to the end wall where the normal shock jump relations were used to determine

thermodynamic conditions behind the reflected shock. The measurement location was located two

centimeters from the end wall where optical access ports and a Kistler 603B1 pressure gauge were

positioned.

By careful measurement of the pre-shock conditions and incident shock speed, the thermo-

dynamic conditions behind the reflected shock can be known to within ∼ 1% [49]. Furthermore,

Farooq et al. and Spearrin et al. demonstrated the close agreement between measured and calculated

(via the normal shock relations) temperatures behind the reflected shock in non-reactive shock tube

experiments of CO2 balanced in Ar [50, 51]. Also worth considering as a contributor to uncertainty

in LAS temperature measurements in shock tubes is boundary layer development. Since LAS is a

line of sight measurement, cool gas in a relatively thick boundary layer may introduce inaccuracies

to the measurement, especially in small diameter shock tubes and if lowE′′ transitions are involved.

Over the years, several models for boundary layer growth in shock tubes have been developed, be-

ginning with the work by Mirels [52]. For the experimental conditions of interest here, numerical

simulations [53] show that the maximum boundary layer thickness is on the order of 1 mm which is

a small fraction of the 15.24 cm inner diameter of the shock tube.

Two quantum cascade lasers were used to measure the spectral properties of the selected tran-

sitions. A diagram of the experimental setup is shown in Figure 3.4. First, a Daylight Solutions

external cavity quantum cascade laser (ECQCL) probed the transitions near 1940 cm−1. Before

3.3. EXPERIMENTAL SETUP 19

Figure 3.4: Experiment Setup. Panel (a) shows the instrumentation and equipment for static celland/or shock tube experiments; (b) shows the line of sights during shock tube experiments.

entering the test cell, the ECQCL beam was split using a 2-degree wedged CaF2 window. The

reflected beam was diverted to a reference detector that was used to improve signal quality via

common-mode rejection [54]. After passing through the cell, the transmitted beam was collected

by a detection system consisting of a 10 cm focal-length CaF2 plano-convex lens, a baffled 2.54 cm

diameter integrating sphere with 6.35 mm ports, a 5.2 µm band pass filter, and a liquid nitrogen

cooled indium-antimonide detector (2 mm diameter; 1 MHz bandwidth) from Infrared Associates,

Inc. The gold-coated integrating sphere (Labsphere) decreases sensitivity to beam steering due to

density gradients in the shock tube. Light entering the integrating sphere is diffusely reflected,

which results in homogeneous light intensity at the ports [38, 55]. During shock tube experiments,


the ECQCL could not be modulated rapidly enough to fully resolve transition line shape profiles,

so the fixed wavelength was monitored with a Bristol wavelength meter for fixed-wavelength mea-

surements.

The second laser is an Alpes distributed feedback quantum cascade laser (DFBQCL) that mea-

sured the transitions near 1987 cm−1. As a way to spatially filter the TEM10 Hermite-Gaussian

spatial mode of the DFBQCL, the beam was fiber-coupled to a tapered hollow core fiber (from

Opto-Knowledge Systems, Inc.) with an inner diameter varying from 200-to-275 µm [56, 57].

The beam was fiber-coupled to the fiber using a 40 mm focal length infrared anti-reflection coated

plano-convex lens and a 3-axis fiber-coupling stage. Once through the fiber, a 5 cm focal length

plano-convex lens was used to collimate the beam through the shock tube and to the 25.4 mm

focal length collection lens where the light was focused through the 4.9 µm band pass filter and

onto a Vigo thermoelectrically cooled mercury cadmium telluride detector (2mm x 2mm area; 10

MHz bandwidth). For experiments where the output wavelength was modulated, a solid germanium

Fabry-Perot etalon (FSR = 0.0566 cm−1) was used to calibrate the relative output wavelength.

3.4 Spectroscopic Measurements

Important spectroscopic parameters are tabulated in databases such as HITRAN [17] and

HITEMP, however, for high J" quantum numbers and very high temperatures, these databases are

often inadequate due to a lack of experimental investigation and breakdown in the power-law over

a wide-temperature range. For the transitions under investigation, HITRAN and HITEMP tabulate

identical spectroscopic parameter values. The line strength uncertainties are cited to be between

5 and 10 % while the uncertainties in collision broadening coefficients in air at 1500 K for the

R(20.5) transitions propagate to 12.5–25 % and collision broadening coefficients of the R(41.5) and

R(42.5) transitions at elevated temperatures are cited as averages or estimates [18]. Additionally,

for J" > 16.5 the listed collision broadening temperature exponent values are assumed to be 0.6

and 0.7 for Π1/2 and Π3/2 respectively [58]. Quantitative measurements of either temperature or

NO concentration require a more refined spectroscopic model. Thus, measurements in a controlled

environment were made to either validate or improve database values. Furthermore, the HITRAN

3.4. SPECTROSCOPIC MEASUREMENTS 21

0.0

0.2

0.4

Absorbance

P =

1 a

tm P

= 2

atm

Voi

gt F

itT

= 29

8 K

L =

21 c

mX N

O =

1.0

1 %

in N

2

1939

.519

40.0

1940

.519

41.0

-202

Res

idua

ls (%

)

Freq

uenc

y (c

m-1

)

0.0

0.2

0.4

P =

1.3

6 at

m, T

= 1

870

K, X

NO

= 1

.97%

in N

2 P

= 3

.81

atm

, T =

193

7 K,

XN

O =

1.9

7% in

N2

P =

3.8

1 at

m, T

= 2

058

K, X

NO

= 2

.00%

in A

r V

oigt

Fit

L =

15.2

4 cm

Hot

ban

d tra

nsiti

ons

v' =

2

v

" = 1

1986

.419

86.6

1986

.819

87.0

1987

.2-202

Freq

uenc

y (c

m-1

)

Figu

re3.

5:T

hele

ftpa

neld

ispl

ays

Voig

tfits

tom

easu

red

line

shap

esof

the

Π1/2

R(2

0.5)

and

Π3/2

R(2

0.5)

tran

sitio

nsof

nitr

icox

ide’

sfu

ndam

enta

lban

d;m

easu

rem

ents

wer

em

ade

ina

room

-tem

pera

ture

stat

icce

ll.T

heri

ghtp

anel

disp

lays

Voig

tfits

tom

easu

red

line

shap

esof

the

Π3/2

R(4

1.5)

and

Π1/2

R(4

2.5)

tran

sitio

ns;m

easu

rem

ents

wer

em

ade

duri

ngsh

ock

tube

expe

rim

ents

.


Table 3.1: Summary and comparison of measured line strengths with HITEMP 2010. Uncertaintiesare given in parenthesis. Line center frequency and lower state energy are averages of the hyperfinelambda-doubled pairs and line strengths are the sum of the hyperfine lambda-doubled pairs.

ν0 [cm−1] E” [cm−1] Transition Si(296 K) [cm−2/atm]Spin Split v′ ← v” j′ ← j” Measured HITEMP 2010

1939.614 735.468 Π1/2 1← 0 R(20.5) 0.378 (2.5%) 0.373 (5-10%)1940.778 874.735 Π3/2 1← 0 R(20.5) 0.191 (2.5%) 0.189 (5-10%)1986.537 3125.621 Π3/2 1← 0 R(41.5) 6.73E-6 (1%) 6.74E-6 (5-10%)1987.074 3081.772 Π1/2 1← 0 R(42.5) 8.65E-6 (1%) 8.54E-6 (5-10%)

databases do not include collision broadening parameters for Ar as a broadening species, yet it is

commonly used as a bath gas in combustion studies. So, measurements of collision broadening in

Ar were made in addition to measurements in N2. The results of such measurements are presented

here. First, room temperature measurements of NO spectra in N2 were conducted in the static

cell. The results of these measurements are used to determine if the Voigt profile adequately

models the measured spectra and to validate the HITEMP line strength and room-temperature

collision broadening. Next, high-temperature shock tube experiments were used to characterize the

high-temperature spectra of NO in N2 and Ar. Again, the validity of using the Voigt profile was

investigated along with validation of the HITEMP line strengths and the characterization of the

collision broadening coefficients and their temperature dependence from 1000-3000 K, which is

essential for accurate gas sensing over this temperature range.

Static cell experiments were conducted over a range of gas densities to test the Voigt lineshape

model. The room temperature static cell was filled with 1.01% NO in N2, and the ECQCL’s piezo-

electric controller was driven by a 50 Hz sine wave which tuned the laser over the Π1/2 R(20.5) and

Π3/2 R(20.5) transitions to resolve the absorbance profile as shown in the left panel of Figure 3.5.

Using a non-linear least squares fitting algorithm, the measured absorbance profiles were fit to Voigt

profiles with the line center frequency, ν0, the collision FWHM, ∆νc, and the integrated absorbance,

A, as the best-fit parameters. Despite the evident lambda-doubling of NO spectra at low pressures,

a single Voigt profile fit at moderate pressures accurately captures the measured absorbance profiles

as shown in the peak-normalized residuals (∼ 1%) of Fig 3.5. Using Eqs. (3.1) and (2.8), line

strength and collision broadening coefficients can be calculated from the best-fit parameters ∆νc

and Ai. With room-temperature information, direct comparisons with values in HITEMP can be


made. Measured lines strengths for the R(20.5) transitions differed from HITEMP values by 1.3

and 1%, and the measured collision broadening coefficients differed by 5%. Differences in colli-

sion broadening coefficients can be attributed to the fact that HITEMP reports 2γNO−Air which is

smaller than the present value, likely because 2γNO−O2 is ∼ 17% smaller than broadening in N2

[59]. Moreover, Spencer et al.s measurements of the R(20.5) 2γNO−N2(T = 296K) differ from the

present measurements by less than 1% [34]. Ideally, similar room-temperature experiments would

be performed for the Π3/2 R(41.5) and Π1/2 R(42.5) transitions. However, the high lower-state

energies of these transitions render them unobservable with the experimental path length and room

temperature conditions of our static cell. Therefore, high-temperature shock tube experiments were

used to characterize the spectra of these transitions.

Shock tube experiments were performed over conditions between 1000–3000 K and 1–5 atm

with the test gases being either 1.97 % NO in nitrogen or 2 % NO in argon. Unlike the room-

temperature static cell measurements, the ECQCL cannot be tuned rapidly enough to fully re-

solve transition line shape profiles during shock tube experiments. Therefore, fixed-wavelength

direct absorption (DA) experiments were performed with the ECQCL set near the peak of the Π3/2

R(20.5) transition (1940.778 cm−1). In the fixed-DA experiments, line shape properties were in-

ferred from the measured line center absorbance of NO assuming the Voigt lineshape function.

Given the known temperature and pressure of the shock tube experiments, absorbance simulations

were altered by scaling the broadening coefficient of the Π3/2 R(20.5) transition until the simu-

lated and measured absorbances matched. The left and right panels of Figure 3.6 show results for

experiments in argon and nitrogen, respectively. As mentioned previously, the temperature depen-

dence of the collision broadening coefficients is typically modeled by a power-law function (Eq.

(2.10)). Over the temperature range of 1000 to 3000 K, the best-fit temperature exponents of the

collision broadening coefficient for the Π3/2R(20.5) transition were found to be n = 0.56 and

n = 0.55 for argon and nitrogen, respectively. As expected, broadening due to argon was found to

be less than broadening due to N2. The collision broadening coefficient, 2γB−A, is proportional to

σ2A−B/(µA−B)1/2 where σA−B and µA−B are the optical collision diameter and reduced mass of

molecules A and B, respectively [20]. Thus, broadening due to nitrogen is larger because nitrogen

has a larger effective optical diameter and is lighter than argon. From the power-law fit, the best-fit


1000

1500

2000

2500

3000

0.0

2

0.0

3

0.0

4

0.0

5

Tem

pera

ture

(K)

2γNO−Ar

(cm−1

/atm)

Da

ta R

(20

.5)

Da

ta R

(41

.5)

Da

ta R

(42

.5)

Po

we

r La

w F

it n =

0.5

6P

ow

er L

aw

Fit n

= 0

.38

Po

we

r La

w F

it n =

0.3

7

1000

1500

2000

2500

3000

0.0

2

0.0

3

0.0

4

0.0

5

Tem

pera

ture

(K)

2γNO−N

2

(cm−1

/atm)

Da

ta R

(20

.5)

Da

ta R

(41

.5)

Da

ta R

(42

.5)

Po

we

r La

w F

it n =

0.5

5P

ow

er L

aw

Fit n

= 0

.52

Po

we

r La

w F

it n =

0.5

1

Figure3.6:M

easuredcollision

broadeningcoefficients

ofNO

inA

randN

2from

1000to

3000K

andbetw

een1

and5

atmospheres.


Figure 3.7: Measured line strength values of the Π3/2 R(41.5) and Π1/2 R(42.5) transitions fromNO in Ar shock tube experiments.

2γNO−Ar(T0 = 1000K) = 0.0331 cm−1/atm and 2γNO−N2(T0 = 1000K) = 0.0475 cm−1/atm,

which differs from the HITEMP value by 10%. The collision broadening data was also fit with

T0 = 296K which resulted in 2γNO−N2(T0 = 296K) = 0.0929 cm−1/atm, which underpredicts

both HITEMP 2010 and the measurements made in the room-temperature static cell by 6 and 11

%, respectively. This discrepancy is likely due to inadequacies in Eq. (2.10) over large temperature

ranges that have been discussed in collision broadening studies of CO and CO2 [24, 25, 60]. As a

result, caution should be made when using the data presented here outside of the studied temperature

range or when extrapolating from low-temperature data.

The DFBQCL was operated to perform scanned-DA measurements during the shock tube ex-

periments. The laser is capable of rapid tuning that allowed the entire line shape profiles of the

Π3/2 R(41.5) and Π1/2 R(42.5) transitions to be resolved as shown in the right panel of Figure 3.5.

During these experiments, the current fed to the DFBQCL was modulated at 5 or 10 kHz with either

a sawtooth or triangle wave from a function generator. An example of the output intensity profile

is displayed in Figure 3.8. As in the room-temperature static cell measurements, the resolved line

shape profiles were fit to Voigt profiles with the ν0,i, Ai, and ∆νc,i as best-fit parameters. Addi-

tionally, the Voigt profile fitting routine slightly scaled the incident intensity profiles (0.99-0.999)

to improve the fits and account for any changes in signal due to beam steering or other effects.

Above 1500 K, weaker hot band (v” > 0) transitions became observable (right panel Figure 3.5),


Table 3.2: Comparison of measured collision broadening parameters for NO with HITEMP 2010.Uncertainties are in parenthesis and represent the statistical uncertainties from the best-fit powerlaw except for the static cell measurements at 298.7 K.

ν0 [cm−1] Collision 2γ [cm−1/atm] 2γ(1000K) nPartner Measured (298.7 K) HITEMP (296 K) Best-fit HITEMP Best-fit HITEMP

1939.614 aN2 0.101 (1%) 0.0964 (5-10%) 0.0464 (9-18%) 0.6 (10-20%)1940.778 aN2 0.1045 (1%) 0.0998 (5-10%) 0.0475 (1.1%) 0.0426 (10-20%) 0.55 (3.2%) 0.7 (10-20%)

Ar 0.0331 (1.4%) 0.56 (4.0%)1986.537 aN2 0.0900 (5-10%) 0.0384 (2.1%) 0.0384 (Est.) 0.52 (5.9%) 0.7 (Est.)

Ar 0.0239 (1.6%) 0.38 (5.9%)1987.10 aN2 0.0900 (5-10%) 0.0368 (1.8%) 0.0434 (Est.) 0.51 (4.9%) 0.6 (Est.)

Ar 0.0231 (1.9%) 0.37 (7.3%)

a Listed HITEMP 2010 2γ values are adjusted for air (79% N2, 21% O2)

and when present, Voigt profiles were also fit to these additional transitions. From the best-fit pa-

rameters, the line strength and collision broadening coefficient can be determined from Eqs. (3.1)

and (2.8), respectively. Figure 3.7 displays the measured line strengths of the R(41.5) and R(42.5)

transitions as a function of temperature. Keeping the lower state energy constant, the data was

fit to Eq. (2.5), and the best-fit reference line strengths, Si(T0 = 296K), were found to agree

extremely well with the reference line strengths listed in HITEMP 2010. Figure 3.6 shows the col-

lision broadening measurements as a function of temperature for the nitrogen and argon mixtures,

respectively. The best-fit temperature exponents of the pressure broadening coefficients of the the

Π3/2 R(41.5) and Π1/2 R(42.5) transitions in argon were found to be 0.38 and 0.37, respectively,

with 2γNO−Ar(T0 = 1000K) = 0.0239 and 0.0231 cm−1/atm. In nitrogen, the collision broadening

temperature exponents were found to be 0.52 and 0.51, respectively, with 2γNO−N2(T0 = 1000K)

= 0.0384 and 0.0368 cm−1/atm. As the case for the R(20.5) transition, the collision broadening due

to Ar was observed to be ∼ 37% smaller than N2 induced broadening. For convenience, all mea-

sured spectroscopic parameters are summarized and compared to HITEMP 2010 values in Tables

3.1 and 3.2. A discussion of the uncertainty analysis for these measurements can be found in the

Appendix.

3.5. SENSOR DEMONSTRATION 27

-200 -100 0 100 200 300 400 5000

1

2

3

4

5

6

Pres

sure

(atm

)

Time ( s)

0

1

2

3

Scanned-DA over R(41.5) and R(42.5) Fixed-DA at R(20.5) Line Center

Signal (V)

Figure 3.8: Measurement traces from reflected shock tube experiment of 1.97% NO in N2. T5 =1550 K and P5 = 4.8 atm.

3.5 Sensor Demonstration

3.5.1 Temperature Measurements in Non-Reacting Shock-Heated Gas

The utility of the studied NO transitions is demonstrated through temperature measurements

during shock tube experiments over a wide range of temperatures. Most temperature measurements

were made during experiments with NO mole fraction fixed, but an NO formation experiment that

demonstrated the sensor’s ability to measure temperature during species transients is presented in

the next section. Examples of measured pressure and light transmission data are shown in Figure

3.8 and in the top panel of Figure 3.10. For fixed-NO experiments, the gas mixture was either 1.97%

NO in N2 or 2% NO in Ar. The first step in the pressure trace represents a nearly instantaneous jump

in pressure and temperature of the test gas mixture resulting from the passing of the incident shock

over the measurement location. The increases in temperature and pressure from the incident shock

results in a decrease in the fixed-DA signal from the ECQCL, which corresponds to a decrease in

transmitted light and thus more absorbance. Approximately 50µs later, another rise in pressure is

observed, resulting from the passing of the reflected shock over the measurement location. The


additional jump in pressure and temperature result in an increase of light transmission and thus a

decrease in absorbance. The 10 kHz scanned-DA signal from the DFBQCL with dips in signal due

to the R(41.5) and R(42.5) transitions can also be observed. During experiments with mixtures of

NO in N2, it is evident from the rolling off of the pressure trace that the jump after the reflected

shock is not nearly as instantaneous as the jump after the incident shock. This is the result of shock-

bifurcation that is a phenomenon observed in shock tube experiments containing mostly diatomic

or polyatomic molecules [61, 62]. However, effects of shock-bifurcation are not present during ex-

periments performed with Ar-balanced mixtures. Another carefully considered phenomenon in the

shock tube experiments was vibrational relaxation. Vibrational relaxation is an extensively studied

subject regarding the time-lag required for the vibrational energy states of a gas to reach an equilib-

rium distribution after an instantaneous change in thermodynamic conditions. Works by Taylor et al.

and Wray provide adequate characterization of vibrational relaxation phenomena for our purposes.

Of note is that vibrational relaxation times of mixtures of NO in argon are significantly shorter than

for mixtures of NO in nitrogen [63, 64], so vibrational relaxation was considered for only the shock-

heated NO in nitrogen mixtures. Using the pressure traces, characteristic total relaxation times (e.g.

vibrational relaxation time of 2% NO in N2 at 2000 K and 1 atm∼ 60µs), and fixed-DA absorbance

traces, it was possible to determine when the vibrational energy distribution could be assumed to

be in equilibrium. Furthermore, the temperature measurements presented below provide confidence

that the method used to account for vibrational relaxation was adequate.

Temperature measurements during the fixed-NO experiments are displayed in Figure 3.9. To

convert absorbance data into temperature measurements, the ratio of the Π3/2R(20.5) transition’s

absorbance and the Π3/2R(41.5) (or Π1/2R(42.5)) transition’s integrated absorbance was utilized.

This ratio can be shown to be approximately a function of only temperature and pressure if the

temperature dependence of the line shape is known

R =α(R(20.5))

A(R(41.5 or 42.5))=S1(T )Φ1(ν0,1)

S2(T )≈ f(T, P ) (3.4)

It should be noted that due to the presence of the line shape function in the ratio, there is a slight

dependence on collision partner concentration, but if there is only one primary collision partner, as

3.5. SENSOR DEMONSTRATION 29

1000 1500 2000 2500 3000

1000

1500

2000

2500

3000

Known Temperature (K)

Me

asu

red

Tem

pe

ratu

re (

K)

R =αR(20.5)

AR(41.5)

Slope = 1

NO in ArgonNO in N

2

1000 1500 2000 2500 3000

1000

1500

2000

2500

3000

Known Temperature (K)

R =αR(20.5)

AR(42.5)

Slope = 1

NO in ArgonNO in N

2

Figure 3.9: Demonstration of temperature measurement for fixed concentrations of NO in Ar andN2. The left panel displays temperature values obtained from the ratio of R(20.5) line center ab-sorbance and R(41.5) integrated area, and the right panel displays temperature values obtained fromthe ratio of R(20.5) line center absorbance and R(42.5) integrated area.

is the case for these measurements, then Φi becomes a function of pressure and temperature. Tem-

perature is determined through an iteration loop that converges on a temperature once the simulated

ratio of absorbance and integrated absorbance matches the measured ratio value. Known temper-

atures are determined from the measured shock speed and the normal-shock relations that provide

accurately known conditions within ∼ 1% [49]. The error bars in the measurements receive contri-

butions from the uncertainties in 2γNO−N2/Ar of the R(20.5) transitions measured in the previous

section, line strengths, the measured ratio of absorbance to integrated absorbance, and – at elevated

temperatures – the output wavelength of the ECQCL. At elevated temperatures the line shape pro-

file narrows, increasing the significance of the uncertainty in output wavelength. Furthermore, the

simulations shown in Figure 3.3 indicate that the temperature sensitivity of the sensor is best below

2000 K, which is supported by the reduced scatter in measurements made below 2000 K. Despite

the temperature sensor’s increased uncertainty at temperatures above 2000 K, known and measured

temperatures show good agreement. Differences between measured and known temperatures are

demonstrated to be at most 5% but typically less than 2%. While temperature measurements from

both ratios display similar accuracy and precision, the R(20.5)/R(41.5) scheme is recommended due


0 200 400 600 800 1000 1200 1400 1600 18000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pre

ssu

re (

atm

)

0 200 400 600 800 1000 1200 1400 1600 18000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Sig

na

l (V

olts)

Wavelength tunning of R(41.5) and R(42.5)

Fixed wavelength R(20.5) signal

0 200 400 600 800 1000 1200 1400 1600 1800

400

600

800

1000

1200

1400

1600

1800

2000

Time (µsec)

Te

mp

era

ture

(K

)

0 200 400 600 800 1000 1200 1400 1600 18000

0.25

0.5

0.75

1

1.25

1.5

1.75

2

NO

Mo

le F

ractio

n (

%)

Temperature Measurement

Chemkin Temperature Simulation

XNO

Measurement

Chemkin XNO

Simulation

Figure 3.10: Demonstration of temperature and NO measurements during an NO formation experi-ment. Conditions at the beginning of Region 5: 1.3% NO2 in argon at 1714 K and 4.15 atm.

to a larger ∆E′′.

3.5.2 Temperature and NO Species Measurements During NO Formation

A final demonstration is shown in Figure 3.10 to highlight the capability of the sensor to measure

species at high-bandwidth and temperature in non-equilibrium conditions. Here, NO was formed

from 1.3% NO2 in Ar during a shock tube experiment. The top panel of Figure 3.10 shows the

pressure trace and the recorded signals from the IR-detectors. The bottom panel shows NO mole

fraction and temperature measurements along with the results from a constant-volume chemical

kinetic simulation (Chemkin and GRIMech 3.0 [65]) at the experimental post-reflected shock con-

ditions. Temperature measurements were made using the ratio of Eq. (3.4) as previously explained,

and provide an independent check on the non-reactive data set. From the measured temperature,

NO mole fraction can be calculated from the measured spectroscopic parameters and Eqs. (2.1)

or (3.1). Temperature and NO mole fraction measurements show good agreement with the kinetic

simulation with differences between measured and simulated temperature being less than 3.5% and

3.6. SUMMARY AND CONCLUSIONS 31

differences between measured and simulated mole fraction being less than 4%.

3.6 Summary and Conclusions

Absorption transitions in the fundamental vibration band of nitric oxide were identified for two-

color thermometry and species sensing in high-temperature applications. The R(20.5) transitions

near 1940 cm−1 and the R(41.5) and R(42.5) transitions near 1987 cm−1 were selected for their

large lower state energy difference and relative isolation from water vapor. The R(20.5) transitions

were accessed by an external-cavity quantum cascade laser while the R(41.5) and R(42.5) transitions

were accessed by a distributed-feedback quantum cascade laser. Spectroscopy experiments were

performed in a room-temperature static cell and in shock tube experiments with conditions from

1000 to 3000 K and 1 to 5 atm. Compared to the HITEMP database, our measured transition line

strengths were found to be in good agreement but with reduced uncertainties. Furthermore, room-

temperature collision broadening coefficients of NO in N2 for the R(20.5) transitions were measured

to be within 5% of the reported HITEMP value of pressure broadening in air but within 1% of similar

room-temperature measurements of pressure broadening in N2. The temperature dependence of the

collision broadening for NO in N2 was found to be considerably different from the values reported

in HITEMP which was expected given the lack of high-temperature data for the transitions selected

and the inadequacy of the power-law. Additionally, new spectroscopic data including collision

broadening of the NO transitions in argon was characterized for high temperatures. The sensing

capabilities of the studied transitions were demonstrated by measuring temperature in fixed-NO

shock tube experiments spanning 1000 to 3000 K and by measuring temperature and NO mole

fraction during an NO formation experiment at 1700 K, proving the sensor’s capabilities in non-

equilibrium conditions. The sensor demonstration provided accurate measurements of temperature

from 1000-3000 K and during transient NO conditions.

Chapter 4

High-Pressure, High-Temperature

Optical Cell for Mid-Infrared

Spectroscopy

The contents of this chapter have been published in the Journal of Quantitative Spectroscopy


4.1 Introduction

Laboratory measurements of high-pressure, high-temperature molecular spectra are desirable

for their relevance to combustion sensors [4], planetary atmospheric study [67–69], predicting ra-

diative heat transfer, chemical sensing in industrial processes [7, 70], and sensing for other gas

phase systems [6, 11, 38]. To that end, high-pressure (5-100 atm) and high-temperature (650-2500

K) spectroscopy has been performed in short-duration (∼ms) shock tube experiments [14–16, 38].

In these experiments, typically only a few wavelength regions are studied at a time as traditional

wide spectrum devices (e.g. FTIR) cannot complete a measurement in this time frame. Resolv-

ing full absorption bands or branches in conditions generated behind reflected shock waves in shock

32

4.2. PREVIOUS EXPERIMENTAL FACILITIES AND STUDIES 33

tube experiments has proven difficult until a recent demonstration by Strand et al. using a new rapid-

tuning, broad-scan ECQCL measured the absorption cross sections of ethylene and other molecules

in the 8.5–11.7 µm region [71].

The spectroscopy literature contains numerous descriptions of spectroscopic cells whose achiev-

able temperature or pressure conditions overlap with the range of conditions achieved in shock tube

experiments. However, simultaneously heated and pressurized spectroscopic cells are relatively rare

in the literature, particularly in the mid-infrared (IR) region beyond 5µm. Adding this capability

yields a more convenient, repeatable, and economical means to measure new spectra, test current

spectroscopic models and databases, and to calibrate sophisticated tunable diode laser absorption

spectroscopy (TDLAS) sensors.

In this chapter, the design and construction of a high-pressure, high-temperature optical cell

(HPHT) for mid-infrared spectroscopy via unique calcium fluoride (CaF2) window rods is pre-

sented. The maximum operating conditions successfully demonstrated by the HPHT was simulta-

neous heating and pressurization to 800 K and over 30 atmospheres. Furthermore, the temperature

non-uniformity of the cell is shown to be no greater than 2.4% of the mean absolute temperature

across the entire path length, and the leak rate is insignificant in comparison to the data collection

time of a typical TDLAS system. To demonstrate and validate the reported conditions, high-pressure

and high-temperature laser absorption spectra of nitric oxide (NO) measured in the HPHT optical

cell are presented and compared to simulations calculated from the HITRAN [17] and HITEMP

[18] databases. Evidence of collisional effects beyond Lorentzian-type broadening are observed at

sufficiently high gas densities. These effects further emphasize the need for facilities to perform

high-temperature and high-pressure spectroscopic experiments.

4.2 Previous Experimental Facilities and Studies

A number of experimental facilities have been reported for the study of high-pressure and/or

high-temperature spectroscopy of various gaseous species. For the purposes of improving the

knowledge of NO radiation transport, Green and Tien performed measurements of low-resolution

NO spectra in a static cell capable of reaching temperatures up to 1200 K and pressures up to 4

34 CHAPTER 4. HPHT OPTICAL CELL

atm [72]. The cell body was constructed of a zirconia ceramic tube placed in a high-temperature

furnace, and the windows protruded to the more uniform central zone of the furnace via nitrogen-

cooled stainless steel tubes mounted with sapphire windows. Temperature uniformity was found to

be acceptable as the temperatures measured at the nitrogen-cooled windows and 1.5 cm from the

windows were 20% and 4% lower, respectively, than the center temperature of the furnace [73].

Motivated by TDLAS sensor development for high-pressure combustion, Rieker et al. studied

high-pressure, high-temperature water vapor spectroscopy near 7204 and 7435 cm−1 in a static

cell that demonstrated operation up to 30 atm at 700 K [13]. The window design used a tapered

window housing mounted with a matching tapered sapphire window that was compressed between

two copper seals by a compression nut. The entire cell was placed in a uniform single-zone furnace,

requiring the window seal to survive all operating pressures and temperatures. The Inconel body of

the optical cell presented in this chapter is similar to Rieker’s cell body except with modifications

made for the new window design.

More recently, Stefani et al. studied the spectrum of CO2 from 6000 to 10000 cm−1 (1-1.67 µm)

at pressures and temperatures up to 20 atm and 560 K, respectively [74]. The reported conditions

were obtained in a commercial stainless steel (316) optical cell capable of withstanding pressures

up to 200 bar and temperatures up to 650 K. The 2 cm optical path length cell used zinc sulfide

(ZnS) windows which have good transmission up to 12 µm.

Another example of sapphire’s utilization for high-temperature, high-pressure spectroscopy was

reported by Christiansen et al. who built and demonstrated an optical cell whose operation range

reached simultaneous heating to 1000 K and pressurization to 100 atm [75]. The cell body was made

of concentric, high-grade aluminum oxide ceramic tubes, and the 3 mm thin sapphire windows were

glass-bonded to the internal ceramic tubes to form a seal. A central heating coil heated the 3 cm

optical path length and was supplemented by two more heating coils on either side to form a highly

uniform path length.

A high-temperature, low pressure static cell with excellent temperature uniformity at 1723 K

was reported by Melin and Sanders with sample high-temperature spectra of water vapor pre-

sented [76]. This particular architecture also used concentric alumina ceramic tubes to form the

4.2. PREVIOUS EXPERIMENTAL FACILITIES AND STUDIES 35

cell body. The windows were made from sapphire tubes with optically contacted sapphire win-

dows that formed a seal on the process side. The optical path length of this cell was reported to be

changeable, but for the demonstration presented a path length of 16 cm was used.

In many of the designs, penetration across the temperature gradient from ambient temperatures

to the uniform, high-temperature zone was achieved, in one way or another, by mounting optical

windows to the end of a window mount that protrudes into the center zone of a high-temperature

furnace. For the applications discussed, sapphire’s high strength and resistance to thermal stress

makes it the best solution. However, a limitation of sapphire is that its usable transmission range is

limited to ∼6 µm (depending on the thickness), and to complicate matters the transmission cut-off

wavelength of sapphire shortens as its temperature increases [77]. This limits the number of species

and conditions available for spectroscopic investigation, particularly for hydrocarbons and even NO

if the window is too thick. Other window materials with longer transmission like ZnS, ZnSe, and

the fluoride crystals (CaF2, BaF2, MgF2) have their own design challenges (e.g. brittleness, thermal

stress, lower maximum temperature stability), but are necessary to extend the transmission range of

static cells.

The pressures reached by the optical cells described in this section are representative of com-

bustion engine pressures, yet there is a drive to reach higher temperatures and longer wavelengths.

The need for spectroscopic studies at higher temperatures is clear in that combustion temperatures

can easily exceed 2000 K. The need for measurements of high-temperature and pressure spectra at

longer wavelengths is evident in pyrolysis kinetics. Recent combustion kinetics studies indicate that

fuel fragments from pyrolysis have a pronounced effect on oxidation kinetics [78]. Beyond the C-H

stretch near 3 µm, a number of hydrocarbon (i.e. fuel fragment) identifiers are present in the mid

to far-IR. Accurate spectroscopic databases of pyrolysis fuel fragments can improve time-resolved

species measurements during pyrolysis and combustion processes. Such measurements significantly

improve combustion models. In addition to combustion applications, measurements of the mid-IR

spectra of many molecules over a variety of conditions up to 4000 K and 100 bar are of interest for

the study of exoplanetary atmospheres [67]. The present optical cell provides a stable environment

to study the spectra of many molecules up to ∼8µm in a wide range of relevant thermodynamic

conditions.


4.3 Design of the High-Pressure, High-Temperature Optical Cell

Figure 4.1: High-pressure, High-Temperature (HPHT) optical cell assembly schematic. Locationsof critical components such as the water-chilled copper coil collars and band heaters are highlighted.

The High-Pressure, High-Temperature (HPHT) optical cell was initially designed for infrared

spectroscopy of gases at high pressures and temperatures up to 900 K. In previous works, such con-

ditions were reached using shock tube or static cell facilities with windows made with robust optical

materials such as sapphire or fused quartz. The diminishing transmission of longer wavelength light

through these materials is problematic for infrared spectroscopy as sapphire’s transmission begins to

cut off near 4.5 µm and fused quartz’s near 3.5 µm. Infrared spectroscopy at longer wavelengths re-

quires use of window materials with superior transmission properties. At room temperature, use of

materials such as zinc selenide, calcium fluoride, barium fluoride, or germanium for infrared optics

is common. Unfortunately, many of these infrared optical materials are much less robust than sap-

phire or fused quartz and are more susceptible to a variety of failure modes at elevated temperatures

such as thermal shock, oxidation, and opacity [79–81]. Although a compromise in transmission

wavelength range (up to∼8µm) compared to other IR materials, calcium fluoride (CaF2) presents a

stable alternative at high-temperatures when care is used to prevent damage due to thermal stress or

exposure to moisture [80, 81]. Additionally, CaF2’s low index of refraction mitigates back-reflection

problems other high index materials may cause. Thus, CaF2 was selected as the window material

for the HPHT.

The body of the HPHT is made from Inconel 625 high-pressure fittings (High Pressure Equip-

ment Co.); its geometry and dimensions are shown in Figure 4.1 and 4.2. SAE ports were machined

4.3. DESIGN OF THE HIGH-PRESSURE, HIGH-TEMPERATURE OPTICAL CELL 37

Figure 4.2: Cross-sectional view of HPHT cell 3-D rendering. The components connected to thetee fitting are plug-gland-collar components (the High Pressure Equipment Company) that form ahigh-pressure seal.

into the boss fittings at either end of the HPHT and are where the window assembly and o-ring sit to

form a vacuum tight seal. Inlet and outlet tubes at opposite end of the optical cell provide flexibility

in setting up a flow configuration. The HPHT body sits in a 40 cm wide single-zone tube furnace

(Barnstead|Thermolyne 21100 Tube Furnace) that provides the primary heating load and insulation.

Additional high-temperature insulation is added to cover the exposed ends of the metal body. Be-

fore being placed in the tube furnace, the HPHT was fitted with a stainless steel sheath that is used

to translate a long K-type thermocouple along the HPHT to measure the temperature at different

points along the optical axis. In this configuration, the HPHT temperature distribution was found

to be unacceptably non-uniform for spectroscopy measurements (see Figure 4.3). To overcome the

non-uniform path length, high-temperature band heaters (Watlow) were secured around the center

tube of the cell body (see Figure 4.1). Independent control was provided by temperature controllers

(Omega CN7533) with feedback from type-K thermocouples placed at the band heater locations.

Figure 4.3 shows the improved temperature uniformity of the HPHT with the band heaters.

The window design consists of three components; namely, the window housing, the window,

and the sealing adhesive. To minimize coefficient of thermal expansion (CTE) mismatch with CaF2,

aluminum SAE plugs were selected for the threaded window housings that mated with the SAE ports

on the cell body. The plugs were given a clear aperture of 1 cm for optical access. Although the band

heaters dramatically improved the uniformity of a 22 cm region in the center of the furnace, a large

temperature gradient exists between the band heater and the window housing locations. To penetrate


the temperature gradient, 16 cm long, 1.168 cm diameter CaF2 rods are used as the window material,

creating a gas path length of 21.3 cm between the internal faces of the CaF2 rods. The internal

faces of the CaF2 rods are wedged to 1◦ to mitigate etalons due to constructive and deconstructive

interference of the laser beam intensity. Two adhesive materials were examined to bond the CaF2

rods to the aluminum window housings. A high-temperature epoxy selected for its close CTE match

with CaF2 and aluminum proved to be unsuitable as it cured significantly harder and stronger than

the CaF2 crystal. After curing and upon installation into the HPHT cell, the CaF2 rod fractured near

the bonding location when applying torque to the SAE window fitting. A successful alternative was

found in a low-outgassing silicone adhesive (ACC Adhesives AS1724) with a maximum working

temperature of 200 C. Despite a larger difference in CTE compared to CaF2, the softer, more elastic

silicone adhesive has continued to seal under many heating and pressurization cycles. To further

combat thermal stress at the bond location, the boss ends of the HPHT were cooled by water-chilled

copper coil collars (see Figure 4.1). With the HPHT temperature set to 900 K, the temperature at

the bond location was measured to be less than 80 C, well within the adhesive’s limits.

4.4 Characteristics of the HPHT Cell

4.4.1 Transmission

Depending on the sample thickness and data source, the reported optical transmission of CaF2

crosses 10% between roughly 8-13 µm (900-1250 cm−1) [79, 82]. Since the CaF2 rods used for the

HPHT are much longer than the specimens typically measured in the literature, the authors felt it

was worthwhile to measure the transmission spectrum to better understand the limits of the facility

and to plan future experiments. The transmission spectrum of a 12 cm long CaF2 rod from the same

manufacturer of the 16 cm long rods used in the HPHT was measured at room temperature with a

Nicolet 6700 FTIR spectrometer. The measured spectrum is shown in Figure 4.4 from 2.5-8.3 µm

with transmission above 90% over much of the measured range until the transmission transitions

to 10% between 6-8 µm. Theoretical transmission through the rod was calculated from the Fresnel

equations and the wavelength-dependent index of refraction and absorption coefficient of CaF2

[83]. The theoretical simulation, plotted in Figure 4.4, agrees with measurements at all measured

4.4. CHARACTERISTICS OF THE HPHT CELL 39

-10 -5 0 5 10

Axial Position (cm)

400

600

800

1000

Te

mp

era

ture

(K

) Tmean

= 453.3 K Tmean

= 617.9 K Tmean

= 802.1 K

-10 -5 0 5 10500

550

600

650

700T

em

pe

ratu

re (

K)

HPHT Path Length = 21.3 cm

Band heaters on

Band heaters off

Figure 4.3: Measured temperature profiles of HPHT system. Top panel: a comparison betweenHPHT temperature distributions (for the same desired mean temperature) with the band heatersturned on (circles) and with the band heaters turned off (squares). Bottom panel: measured temper-ature at 1 cm increments along the axis of the HPHT cell for three different temperature operatingpoints.

wavelengths.

With increasing temperature, the transmission edge of many infrared optical materials becomes

shorter [77, 79]. CaF2 is no exception and its infrared transmission edge has been studied from

cryogenic temperatures to the elevated temperatures relevant here [82, 84, 85]. Namjoshi et al. and

Lipson et al. measured the absorption coefficient of CaF2 and other fluoride crystals at tempera-

tures between 295 and 800 K and found the temperature dependence of the absorption coefficient

in the infrared transmission edge to be described by a multiphonon absorption process. During

elevated temperature operation, the temperature gradient across the window rods would limit the

useful transmission range of the HPHT cell to ∼6.5-7 µm. Additional transmission data found in

studies of the contamination of fluoride crystals after heat treatment in oxidative and wet environ-

ments suggest great care must be taken to limit the exposure to moisture and oxidative environments

when performing experiments above 600 C [80, 81, 83]. Furthermore, published thermal disper-

sion coefficients for CaF2 indicate the index of refraction exhibits little change with temperature


3 4 5 6 7 8

Wavelength (µm)

0

0.2

0.4

0.6

0.8

1

Tra

nsm

issio

n

Measurement

Simulation

Figure 4.4: Transmission measurement of a similar CaF2 crystal (L = 12 cm) at room-temperature.A simulated transmission spectrum of the CaF2 rod is also displayed and agrees with the measuredspectrum.

(dn/dT ∼ −10−5 1C ) [86]. Thus, the temperature gradient across the window rods during elevated

temperature operation of the HPHT is not expected to significantly alter the refractive properties

observed at room temperature.

4.4.2 Path Length

The path length of the HPHT cell was calculated to be 21.3 cm based on the precisely measured

geometry of the various components used in the cell’s construction. Additionally, room-temperature

absorption measurements of NO at known pressure and concentration confirmed the path length to

within 2% (See Figure 4.7(a)). Since the cell is a metal body, changes in path length due to thermal

expansion must be considered. At the maximum operating condition, the change in path length due

to thermal expansion was estimated to be less than 1% of the nominal length due mostly to the

expansion of the Inconel metal body (CTE = ∼ 14 × 10−6 1/m) being countered by the expansion

of the CaF2 rods (CTE = ∼ 20× 10−6 1/m) .

4.4. CHARACTERISTICS OF THE HPHT CELL 41

4.4.3 Temperature Uniformity

To measure the HPHT cell’s temperature distribution along the optical path length, a type-

K thermocouple with 1 cm graduations was translated through the stainless steel thermocouple

sheath. The temperature measurements of the outer wall are equal to the inner gas temperature

as shown in the analysis of a similar gas cell by Schwarm et al. [87]. The measured temperature

distributions at different conditions are shown in Figure 4.3. Without supplementary heating from

the band heaters, the heat load supplied by the single-zone furnace provided only a short region of

uniform temperature (∼5 cm). Furthermore, the inherent distribution of the furnace was found to

be biased to one side. To improve and optimize the temperature uniformity, the band heater settings

were iteratively adjusted until the results were satisfactory. A 22 cm region bounding the 21.3 cm

optical path length was measured at 1 cm increments for each of the profiles shown. At 802 K,

the temperature profile saw a maximum deviation of 19 K or 2.37% from the mean. However,

this maximum deviation represents only a small fraction of the total path length. Perhaps a more

indicative metric of the temperature uncertainty along the path length would be a standard deviation

which is less than 1% for each condition shown.

4.4.4 Pressure Stability

Pressure transducers (Setra) rated to 67 and 1.7 atm were used to measure pressures above and

below 1.7 atm, respectively. Pressure loss was evaluated at different temperature and pressure con-

ditions. However, no significant change in leak rate was observed at elevated temperatures. Under

vacuum conditions, the pressure was found to be very stable at all temperatures with a pressure

change of 1 mTorr/s which is negligible for the duration of a TDLAS measurement. At pressures

above 1 atm, the rate of pressure change per minute was on the order of 0.01% of the nominal pres-

sure. For instance, the drop in pressure at 30 atm was observed to be < 0.02 atm after two minutes

at any temperature condition. During the measurements discussed in the following sections, the

resolution of the 67 atm pressure transducer could not resolve any pressure changes throughout the

duration of the external cavity quantum cascade laser measurement that lasted roughly 6 seconds.

Thus, the pressure stability was found to be adequate for high-pressure spectroscopy.


4.5 Experimental Setup and Procedure

Figure 4.5: Experimental setup detailing the laser beam paths through the HPHT optical cell, refer-ence cell, and optics.

4.5.1 Gas System

The HPHT was connected to a high-pressure valve manifold that controls the flow of gases

to the HPHT facility, room-temperature reference cell, gas cylinders, mixing tank, vacuum pump,

and exhaust ventilation. At several strategic locations in the manifold matrix, pressure transducers

monitor the system pressure. Before making a measurement, the desired gas mixture of dilute

nitric oxide in nitrogen was allowed to slowly flow through both optical cells for several minutes to

allow saturation of NO wall adsorption. Next, the room-temperature reference cell was filled with

the gas mixture to a low pressure (< 0.1 atm), providing an absolute wavelength marker during

experiments. After isolating the reference cell, the HPHT was slowly filled to the desired pressure

with the NO mixture.

4.5. EXPERIMENTAL SETUP AND PROCEDURE 43

4.5.2 Optical System

An external cavity quantum cascade laser (ECQCL) (Daylight Solutions), with a maximum

continuous wave output power of 50 mW, was used to probe the R-branch of the NO fundamental

band. The laser’s output frequency range (1880-1943 cm−1) spans most of the R-branch and can

be tuned over that range via the rotation of its diffraction grating. However, its mode-hop-free

region spans only about one third of the total frequency range from 1924-1943 cm−1. Out of the

laser package, the beam displays excellent beam quality and good collimation with little divergence

observed through the many laser paths. Its line width is specified to be less than 10 MHz which

qualifies as monochromatic light for the purposes of this work.

Figure 4.5 shows the optical components and laser beam paths through the HPHT and reference

cells. After exiting the laser cavity, the laser beam was split with a zinc selenide (ZnSe) window

wedge. One of the split beams was directed through the HPHT and focused onto a thermoelectrically

cooled mercury-cadmium-telluride (MCT) detector (Vigo Systems) with a parabolic mirror. Before

reaching the parabolic mirror, the remaining optical power (∼70% after transmission through ZnSe)

was attenuated by collecting the reflection off a CaF2 flat to ensure a maximum laser power to

thermal emission ratio. The second beam path from the ZnSe beam splitter was split a second time

to a reference detector (IR Associates) and a third beam splitter. The third beam splitter directs one

beam through a solid germanium Fabry-Perot etalon (FSR = 0.0163 cm−1) before it is focused onto

a thermoelectrically cooled MCT detector (Vigo Systems) with a CaF2 lens. Lastly, the final beam is

directed through the reference cell after which the beam is focused onto a thermoelectrically cooled

MCT detector (Vigo Systems) with another CaF2 lens.

The grating tuning was controlled by the ECQCL controller with no external inputs to the con-

troller or laser head. Timing signals from the ECQCL controller and signals from the four detectors

were recorded at 100 kHz by the multi-channel DAQ system (National Instruments PXI-6115 boards

in a PXIe-1062Q chassis). This sampling rate provided more than adequate resolution as full scans

from 1880-1945 cm−1 were complete in roughly 6 s. For each measurement, a dark signal, a trans-

mission background signal, and a transmission signal were needed for proper data processing. The

dark signals capture any systematic detector signal offsets from the system without the laser passing


through. The transmission background signals represent the I0 in Eq. (2.1) and are recorded when

no absorbing gas species is present in the HPHT cell. Finally, the transmission signals are recorded

with an absorbing gas species present in the HPHT cell. The data processing steps will be discussed

in the next section.

4.6 Measurements and Results

4.6.1 Data Reduction Process

Proper data reduction required the 4 detector signals and 3 measurement types described in the

previous section. The ECQCL exhibits slight scan-to-scan intensity variations, and this is mitigated

through use of a common-mode rejection strategy. In common-mode rejection, a ratio between the

reference detector and gas cell detectors is measured as a function of optical frequency during the

transmission background measurement. Thus, any non-absorption intensity variations measured be-

tween I0 and It are removed. The etalon signal was used as a high-resolution wavelength calibration

to ensure alignment between the I0 and It measurement sets. To calibrate absolute wavelength, all

measurements sets were collected with a low pressure mixture of dilute NO in N2 in the reference

cell. Figure 4.6 displays transmission spectra after implementing common mode rejection and an

example of the wavelength calibration procedure.

After implementing common-mode rejection and properly calibrating wavelengths, the ab-

sorbance baseline was measured from two background transmission measurement sets. Many

optical materials under mechanical stress exhibit birefringence. Although the extent of the stress-

induced birefringence in the long CaF2 rods was expected to be low, this effect was investigated at

several pressures. The top panel of Figure 4.6 shows the measured absorbance baseline when the

HPHT cell was filled with 20 atm of oxygen. The oxygen ensured any remaining NO in the gas

system was oxidized, resulting in a true absorbance baseline measurement. No significant changes

in transmitted light intensity were observed, and the measured baseline noise was found to be

≈ ±0.01. Observed temperature effects were negligible since all measurements are recorded with

the same heating load applied to the CaF2 rods.

4.6. MEASUREMENTS AND RESULTS 45

1925 1930 1935 1940

0

0.5

1

De

tecto

r S

ign

al (a

.u.)

or

Ab

so

rba

nce

1924 1924.2 1924.4 1924.6 1924.8 1925

Wavenumber (cm-1

)

0

0.5

1

De

tecto

r S

ign

al (a

.u.)

or

Tra

nsm

issio

n

FSR

HPHT Detector

Ref. Cell

Detector

Absorbance Baseline

Alignment of

etalon signals

Alignment of measured and simulated

absorption transitions

Ref. Cell DetectorHITRAN Simulation

Figure 4.6: Top panel: Detector signals from the HPHT cell and reference cell detectors. The solidlines represent the signals recorded during a background measurement (I0) when the HPHT cell isunder vacuum and the reference cell is filled with a low pressure NO-N2 mixture. The dashed linesrepresent the detector signals during a transmission measurement (It) when the HPHT cell is filledwith 20 atm of pure oxygen. The resulting baseline absorbance for 20 atm is also plotted. Bottompanel: Visualization of the alignment of I0 and It signals. A HITRAN simulation of the referencecell conditions is used to align I0 and It and to assign absolute wavenumbers.

4.6.2 Room-Temperature Validation

Operation of the cell and performance of the ECQCL were first evaluated in room-temperature

(294 K here) experiments at several pressures and compared to simulations using the HITRAN

database. Following the experimental procedure outlined in the previous section, a mixture of 2.03%

(Praxair certified grade with XNO uncertainty of ±2%) NO in nitrogen was added to the HPHT

and reference cells. Figure 4.7(a) displays the validation measurement at 1 atm superimposed by

a HITRAN simulation at the same conditions. Transition peak absorbances from the HITRAN


1925 1930 1935 1940Wavenumber (cm

-1)

0

0.5

1

1.5

2

2.5

3

3.5

Ab

so

rba

nce

Measurement

HITRAN

HITRAN, 1.04×γ

T = 294 K

P = 1 atm

XNO

= 0.0203 in N2

(a)

1890 1900 1910 1920 1930 1940

Wavenumber (cm-1

)

0

1

2

3

4

Ab

so

rba

nce

XNO

= 0.0035 in N2

T = 294.1 K

P = 34 atm

P = 20 atm

P = 10 atm

P = 5 atm

P = 1 atm

HITRAN

Super

Lorentzian

Sub

Lorentzian

(b)

Figure 4.7: Room-temperature NO spectra. (a) Measured 1 atm spectra compared with the simula-tions using the HITRAN database and the HITRAN database with a modified γ to account for thedifferences between air and N2 broadening. (b) Measured R-branch spectra of the NO fundamentalband at 294 K for various pressures up to 34 atm. HITRAN simulations (1.04 × γ) at the sameconditions are superimposed for comparison. Sub-Lorentzian and super-Lorentzian regions of thespectra are evident at high pressures.


simulation are about 4% higher than the measured absorbance because the collision-broadening

coefficients (γ) reported by HITRAN are adjusted for air whereas the current measurements are in

nitrogen [58]. Measurements of the oxygen broadening coefficient [59] indicate that γNO−O2 is

about 20% smaller than γNO−N2 leading to γNO−air reported by HITRAN to be about 4% less than

γNO−N2 . With this adjustment made, the measured and simulated spectra agree more favorably

with peak absorbances differing by less than 0.5%.

Room-temperature measurements at higher pressures up to 34 atm are shown in Figure 4.7(b).

To remain below the optically thick limit, a mixture of 3500 ppm NO in nitrogen was made for

these measurements. Since the uncertainty in this mixture is much higher than that of the 2.03%

NO mixture, Voigt line shapes were fit to a 1 atm measurement to more accurately determine the

concentration of NO. The resulting best-fit concentration was then used in high-pressure simulations

for a better comparison with the measurements.

Observed deviations from the HITRAN simulations grow with pressure. Super-Lorentzian ab-

sorbance (defined here as stronger absorbance than a super-position of Voigt line shape profiles) is

observed at the lower frequencies until a crossover to sub-Lorentzian absorbance occurs near 1936

cm−1. These observations are consistent with trends found in line mixing studies [29, 88, 89] and

in particular with experiments and calculations by Abels and DeBall [90] and Hirono and Ichikawa

[28, 91], who reported a band-correction function for the absorption coefficient in the troughs be-

tween adjacent transitions at 296 K and below 1 atm in NO-N2 mixtures. The band-correction

function, K(ν), is defined in Eq. (4.1) as the ratio of the observed absorption coefficient with line

mixing and the absorption coefficient due to Lorentz-type broadening.

K(ν) =kνkν,L

(4.1)

Measured and calculated band correction functions in [28, 91] reported peak values of about 1.1

near the R-branch center which are consistent with the deviations observed here between 1.1 and

1.2 in the troughs between absorption peaks. Additionally, the reported super-to-sub Lorentzian

transition frequency is near 1936 cm−1 as observed in this work.


4.6.3 High-Temperature, High-Pressure NO Spectra

1890 1900 1910 1920 1930 1940

Wavenumber (cm-1

)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ab

so

rba

nce

XNO

= 0.0034 in N2

P = 33 atm, T = 618 K

P = 32 atm, T = 802 K

HITEMP

Figure 4.8: Measured high-pressure spectra of NO in N2 at 618 and 802 K compared with HITEMPsimulations. Super-Lorentzian absorbance is evident, yet the crossover to sub-Lorentzian is notobserved. Instead, it is expected to exist at higher frequencies. Comparisons between measured lineshapes and the simulated line shapes indicate that improvements to collision broadening parametersin HITRAN can be made.

Using the methods outlined above, the spectrum of NO in N2 was measured at 618 and 802 K at

several pressures up to 33 atm. These measurements are shown in Figure 4.8. For comparable pres-

sures, deviations from the Voigt model using the HITEMP database are less pronounced in high-

temperature measurements than in room-temperature measurements. Studies have demonstrated

that line-mixing and the breakdown of the impact approximation are primarily effects of gas density

rather than simply gas pressure [13, 89]. Observed deviations trend similarly at comparable gas

densities with the room-temperature measurements, yet the transition from super-to-sub Lorentzian

is no longer observed near 1936 cm−1 as in the room-temperature case. To the authors’ knowledge

band-correction functions for NO as a function of temperature, pressure, and collision partner have

not been reported. Line mixing studies of other small molecules reporting band-correction functions

at several temperatures indicate that as temperature increases the super-to-sub Lorentzian transition


frequency shifts, like the rovibrational intensity distribution, to frequencies near higher J” transi-

tions [88, 89]. Therefore, a super-to-sub Lorentzian transition is expected to exist outside the tuning

range of the ECQCL used in this study. Additional observed deviations between the measured and

simulated spectra reveal that improvements to the HITEMP database collision broadening param-

eters, particularly for high J” transitions can be made. Furthermore, the inconsistencies between

simulated and measured line shape broadening makes determination of the super-to-sub Lorentzian

transition frequency difficult. The need to improve existing databases to properly account for col-

lisional effects at high-temperatures is evident from the high-pressure, high-temperature spectra

presented here.


The design, build, and operation of an optical cell for high-temperature, high-pressure mid-IR

spectroscopy was presented. For optical access from ∼0.15-8 µm, the optical cell uses 16 cm CaF2

rods to penetrate the temperature gradient imposed by the single-zone furnace used as the primary

heating load. Penetration by the CaF2 rods allows the optical cell’s gas-tight seal to be maintained at

a much lower temperature during operation at high-temperature set points. Temperature uniformity

across the optical path length of 21.3 cm is maintained by the temperature-controlled single-zone

furnace and temperature-controlled band heaters attached to the body of the optical cell. Good

temperature uniformity up to 800 K and pressure stability above 30 atm were demonstrated.

Spectra of the R-branch in the fundamental rovibrational band of nitric oxide at several tempera-

tures and pressures up to 800 K and 34 atm, respectively, were measured by an ECQCL for the opti-

cal cell’s characterization and demonstration of utility. Deviations between the measurements and a

Voigt-based model using the HITRAN/HITEMP databases were observed with increasing gas den-

sity. These deviations are attributed primarily to line-mixing, but at elevated temperatures, incon-

sistencies between the observed pressure broadening and that of the HITEMP/HITRAN databases

is evident. These findings emphasize the need for high-pressure and high-temperature spectroscopy

experiments to characterize high gas density phenomena, and this facility provides a practical means

to study the spectra of molecules into the mid-IR.

Chapter 5

High-Resolution Line Shapes and

Intensities of Nitric Oxide Near 5.3 µm

The contents of this chapter have been submitted for publication in the Journal of Quantitative

Spectroscopy and Radiative Transfer [92]

5.1 Introduction

The absorption and emission spectra of nitric oxide (NO) have been used for quantitative diag-

nostics of gases in a variety of applications including atmospheric sensing [93], combustion [94–96],

high-temperature gases [11, 38, 97], and compressible flows [39, 98–100]. Due to NO formation

described by the Zeldovich mechanism, a substantial amount of NO is present in chemically equili-

brated air at high temperatures (> 1000 K) [30], making it an attractive optical target for absorption-

based temperature measurements. Furthermore, the emergence of commercially available, tunable

quantum cascade lasers (QCLs) provides access to the strongly absorbing (and emitting) fundamen-

tal band of NO near 1900 cm−1 (5.3 µm). Thus, accurate knowledge of NO line intensities and

temperature-dependent line shape parameters is needed for accurate implementation of optical gas

sensors.

Many of the previous spectroscopic measurements targeting the fundamental absorption band

50

5.2. EXPERIMENTAL DETAILS 51

of NO have focused on the spectrum at atmospheric relevant conditions (i.e. 180-300 K) [32–35,

42, 59, 101, 102]. Additionally, Falcone et al. measured the line strength and collision widths of

the R(10.5) transitions at temperatures up to 2500 K in shock tube experiments [42]. These works

made significant contributions to the NO component of the HITRAN and HITEMP spectroscopic

databases [17, 18, 58]. More recently, Spearrin et al. and Almodovar et al. performed additional

high-temperature measurements of the NO fundamental band to motivate strategies for single- and

two-wavelength thermometry in high-temperature gases [11, 38].

The focus of this chapter is to improve the accuracy of NO spectroscopic parameters at high

temperatures and to provide validation data for theoretical model improvements at the studied con-

ditions. To that end, over 40 transitions within the R-branch of the NO fundamental absorption

band are studied with two QCLs over a range of temperatures and pressures from 300 to 2500 K

and 0.025 to 3.2 atm, respectively. A static cell is used to study NO line shapes between 300 and 800

K in gas mixtures with nitrogen, argon, and air (79% N2 and 21% O2) serving as the primary col-

lision partners. Line intensities, pressure broadening coefficients, and pressure-induced lines shift

coefficients are determined from multi-spectral Voigt line shape fitting. To supplement the static

cell measurements and extend the temperature range studied, mixtures of NO and collision partners

(i.e. nitrogen and argon) were shock-heated up to 2500 K in reflected shock wave experiments.

5.2 Experimental Details

A high-pressure, high-temperature (HPHT) optical cell was used to generate the desired ther-

modynamic conditions for spectroscopic investigation. Specifications of the cell can be found in

Chapter 4, but a brief summary is provided here for completeness. The HPHT cell is composed of

an inconel cell body and 16 cm long calcium fluoride (CaF2) window rods mounted in aluminum

window housings. The CaF2 window rods provide access to longer mid-IR wavelengths than mate-

rials typically used in high-temperature optical cells such as fused silica or sapphire. Additionally,

the window rods penetrate the temperature gradient between the heated optical path and ambient

conditions. The heating load is supplied mainly by a temperature-controlled single-zone furnace

52 CHAPTER 5. NITRIC OXIDE LINE SHAPES AND INTENSITIES

DAQ Computer

HPHT Optical Cell

Beam

Splitter

ECQCL

Photovoltaic

Detector

Beam Dump

Ref. Cell

Germanium

Etalon

CaF2

Lens

Laser

Controllers

DFBQCL

Piezo

Driver

Hollow Core Fiber

Function

Generator

Manifold

P P

P

Gas Mixtures Vacuum/

Vent

Figure 5.1: Experimental setup of the static cell experiments from 294-802 K.

and is supplemented by two temperature-controlled band heaters to improve uniformity. The tem-

perature controllers for the heating elements are each controlled by dedicated type-K thermocouples.

Furthermore, an independent thermocouple can be placed at different axial locations along the op-

tical path length to verify a uniform temperature distribution. Together, the cell components form a

uniform 21.3 cm path length that can be simultaneously heated and pressurized to 900 K and over

30 atm. The maximum deviation from the mean absolute temperature was demonstrated to be no

greater than 2.4%, and the standard deviation across the path length was found to be less than 1%.

In addition to the HPHT optical cell, a room-temperature optical cell with an aluminum body and

sapphire windows was used with low-pressure, room-temperature spectra as a reference cell. The

two optical cells are connected via a stainless steel valve manifold and tubing. Three pressure trans-

ducers – two Setra capacitive transducers with 68 and 1.7 atm full scale range (±0.11% full scale

5.2. EXPERIMENTAL DETAILS 53

accuracy) and one MKS Baratron with 100 torr full scale range – are connected to the manifold to

monitor the pressure of the two cells.

Two QCLs were used to probe absorption transitions in the NO fundamental rovibrational band.

The first laser was a tunable external cavity QCL (ECQCL) from Daylight Solutions with a us-

able tuning range of 1880-1945 cm−1 and peak continuous wave (CW) output power of 50 mW.

Under CW operation, the monochromatic line width of the ECQCL is specified to be <10 MHz.

The frequency output was changed by rotating a grating in the laser cavity via two methods: 1) a

stepper motor for coarse tuning across the whole tuning range and 2) a piezoelectric motor for fine

local tuning (about 1 cm−1). For coarse adjustment, the ECQCL’s dedicated laser controller from

Daylight Solutions changed the output wavelength. For fine adjustments, an external piezoelectric

controller (ThorLabs MDT694) drove the piezoelectric motor. For fine tuning across ∼ 1 cm−1,

a 50 Hz sinusoid from a function generator was supplied to the piezoelectric driver that drove the

piezoelectric motor at 50 Hz. The second laser was a distributed feedback QCL (DFBQCL) from

Alpes Lasers mounted in a high heat load (HHL) collimation housing. The collimated output emits

up to 20 mW of CW optical power. The current and voltage supplied to the DFBQCL and the in-

tegrated thermoelectric cooler (TEC) were controlled by a combined laser driver and temperature

controller (Arroyo Instruments ComboSource 6310QCL). In contrast to the ECQCL’s wide tuning

range the DFBQCL output frequency range spans 1980-1990 cm−1. To tune ∼ 1 cm−1, the laser

driver was supplied a 100 Hz sawtooth wave. As described in [11], this laser was fiber-coupled to

improve the spatial mode of the output beam.

As shown in Figure 5.1, the QCL output beams were routed through the HPHT and reference

optical cells by a series of mirrors and beam splitters until being focused onto one of four photo-

voltaic detectors serving different purposes. One mercury-cadmium-telluride (MCT) detector (Vigo

systems) collected light that passed through the HPHT optical cell. A similar MCT detector cap-

tured light that passed through the reference cell. A third MCT detector collected light passing

through a solid germanium Fabry-Perot etalon (FSR = 0.0162 cm−1) that monitored the relative fre-

quency of light as the lasers tuned. Finally, the fourth detector was a liquid-nitrogen-cooled indium

antimonide (InSb) photovoltaic detector (IR Associates) used to monitor and cancel out the scan-

to-scan intensity variations of the laser systems via common-mode rejection. Signals from all four


photvoltaic detectors were sampled and recorded by the DAQ system (PXI-6115 in a PXIe-1062Q

chassis) at 1 MHz.

0.006 0.008 0.01 0.012 0.014 0.016

Time (s)

0

0.5

1

1.5

De

tecto

r V

olta

ge

I0

0.682 atm

0.305 atm

0.098 atm

0.006 0.008 0.01 0.012 0.014 0.016

Time (s)

0.2

0.3

0.4

0.5

0.6

0.7

De

tecto

r V

olta

ge

I0

0.1 atm

0.1 atm

0.1 atm

HPHT optical cell detecor

Reference optical cell detector

(a) ECQCL Experiment

0.006 0.008 0.01 0.012 0.014 0.016

Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

Dete

cto

r V

oltage

I0

1.037 atm

0.636 atm

0.201 atm

(b) DFBQCL Experiment

Figure 5.2: (a) Raw data from an experiment using the ECQCL to interrogate the transitions near1897.17 cm−1 at 802 K and several pressures. The top panel displays signals from the HPHT opticalcell, and the bottom panel displays signals from the reference optical cell. The reference cell signalsare used to align the data in relative frequency. (b) Raw data from an experiment using the DFBQCLto interrogate the transitions near 1986.75 cm−1 at 802 K and several pressures.

Prior to each static cell experiment, the HPHT and reference cells were evacuated by a vacuum

pump and the transmitted intensity, I0, was recorded. The reference cell was then filled with ≤ 0.1

atm of an NO mixture dilute in either nitrogen (N2) or argon (Ar). Next, the HPHT cell was filled

with the test mixture to approximately 1 atm. At this point, incident intensity, I , measurements

were recorded at different pressures until the pressure was reduced to approximately 0.025 atm.

Ultimately, a set of measurements at the desired test temperature spanning a pressure range of

0.025–1 atm was recorded for each interrogated NO rovibrational transition. Examples of detector

signals from experiments using the ECQCL and DFBQCL are presented in Figure 5.2. For the

measurements performed with the DFBQCL, it should be noted that the accessible transitions are

inactive at room-temperature making the reference cell unusable. However, the DFBQCL is more


stable than the ECQCL in terms of wavelength and intensity drift, so alignment of the scans is

repeatable enough for the multi-spectral fitting routine described in the next section.

Aside from in-house mixtures, the test mixtures used were certified standard mixtures (XNO ±

2%) from Praxair with NO concentrations ranging from 608 ppm to 2.03% by mole fraction and

balanced by either N2 or Ar. For certain experiments, in-house mixtures of NO in N2 were made

by diluting the 2.03% NO in N2 using a jet-stirred mixing tank and partial pressure relations. To

measure pressure broadening due to air, two digital flow controllers (Alicat MCS Series) were used

to mix the gases inline. Inline mixing is necessary because at room-temperature, NO exposed to O2

rapidly oxidizes into NO2 through the reaction 2NO + O2 ↔ 2NO2 [103]. At high temperatures,

this reaction slows down significantly, however mixture preparation and delivery of the the gases

at sufficiently high temperatures was impractical. A mixture of NO dilute in N2 was flowed at 770

(±26) mL/min with 200 (± 2) mL/min of pure O2 to form the inline mixture.

High temperature experiments were performed in a large diameter (15.24 cm) shock tube. Prior

to a shock tube experiment, a low pressure mixture of the test (driven) gas is separated from the

high pressure driver section by a thin plastic diaphragm. Once the pressure differential between the

driver and driven section is large enough, the diaphragm ruptures and a shock wave forms. The gas

behind the incident shock wave is nearly instantaneously heated and pressurized. As the incident

shock wave reflects off the shock tube end wall, the driven gas is further heated and pressurized.

Measurements were made behind the reflected shock 2 cm from the end wall. The shock tube

experimental setup and procedures used in this chapter are very similar to those used in Chapter 3.

5.3 Measurements and Results

5.3.1 Multi-Spectral Fitting

For the study of R-branch absorption transitions, line shapes were measured using the proce-

dure described in the previous section. A multi-spectral least squares fitting routine similar to the

one used by [104] was employed to determine line shape parameters of the measured absorption

transitions. In multi-spectral fitting, several line shapes of the transition(s) of interest are measured

over a range of well-known conditions, usually pressure and/or temperature, and simultaneously fit


-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.10

0.5

1

1.5

Ab

so

rba

nce

1.0238 atm

0.6818

0.3911

0.2035

0.0981

0.0267

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

Relative Frequency (cm-1

)

-5

0

5

Pe

ak-N

orm

aliz

ed

Re

sid

ua

ls (

%)

v'←v"=2←1

Π1/2

R(15.5)

v'←v"=1←0

Π3/2

R(5.5)

v'←v"=1←0

Π1/2

R(5.5)

T = 802 K

XNO

= 2.03% in N2

L = 21.3 cm

Figure 5.3: Multi-spectral fits near 1897.17 cm−1 of the R(5.5) transitions from the 2Π1/2 and 2Π3/2

subbands of the v′ ← v′′ = 1← 0 band and the 2Π1/2R(15.5) transition from the v′ ← v′′ = 2← 1hot band. Measurements were collected at 802 K and pressures ranging from 0.0267 to 1.0238 atm.

to a set of best-fit parameters that define the line shape model being used (e.g. Voigt). The mer-

its of employing a multi-spectral fitting routine onto NO absorption spectra are twofold. First, the

e/f components of the Λ-doublet transitions characteristic of NO can be fit independently over a

wide range of pressures, ranging from when the doublets are easily distinguishable to when they

are blended. Second, the line shape parameters (i.e. S(T ), 2γNO−A(T ), and δNO−A(T )) can be

determined directly rather than requiring additional linear fits of integrated areas or collision widths

with pressure as required by single line shape fitting.

At a fixed temperature and mixture of NO and N2 or Ar, a set of ≈ 10 line shapes at pressures

ranging from 0.025 to 1 atm were measured for each transition. Each line shape measured in the

HPHT cell has a corresponding constant pressure, room-temperature line shape measured in the


reference cell like those shown in the bottom panel of Figure 5.2(a). The reference line shapes were

used to align all line shapes in frequency space. For each set of measured line shapes, Voigt line

shapes were simultaneously fit by a least squares method to extract the free parameters ν0,rel, δ,

2γ, and S. Here, ν0,rel is the relative line center frequency while δ, 2γ, and S hold their meanings

as described in Chapter 2. To prevent the fitting routine from biasing the free parameters towards

line shapes with larger absorbance, each line shape was weighted by 1 over the square root of its

peak absorbance. A few constraints were employed by the fitting routine to better handle NO’s

characteristic Λ-doubled transitions. First, the line strengths of the doublet’s e/f-components were

set equal to one another. Second, a single pressure shift coefficient for the doublet was used. Third,

the transition line center spacing reported by the HITEMP database was used as a guide when

selecting a first guess for ν0,rel, but ν0,rel for each e/f-component was allowed to float. Lastly,

transitions from the 2Π3/2 spin-split subband were fit primarily as single transitions rather than

doublets due to their closer component spacing. However, the doublet spacing for the 2Π3/2 subband

grows with J ′′ while the spacing for the 2Π1/2 decreases with J ′′. For transitions in the 2Π3/2

subband with J ′′ ≥ (15.5), a single Voigt profile does not adequately fit the transition at low

pressures, so two Voigt profiles were used for these transitions. Since the doublet transitions remain

indistinguishable, the doublet spacing reported by HITEMP was used to constrain the spacing.

In certain regions of the fundamental band, multiple doublet transitions were measured in a

single ≈ 1 cm−1 wide scan as shown in Figure 5.3, where the two R(5.5) transitions from the2Π1/2 and 2Π3/2 subbands of the v′ ← v′′ = 1 ← 0 band and the 2Π1/2R(15.5) transition of the

v′ ← v′′ = 2← 1 hot band are visible in a single scan. As J ′′ increases, so does the spacing between

corresponding transitions of the spin-split subbands. Thus, beyond the J ′′ = 10.5 transitions, a

single doublet transition was measured per scan except at high-temperatures when v′ ← v′′ = 2←

1 hot band transitions were visible. When present with sufficient signal-to-noise ratio, hot band

transitions were fit according to the rules outlined in the previous paragraph.

5.3.2 Line Strengths

Transition line strengths at four different temperatures were inferred from the measured line

shapes using the multi-spectral fitting routine described in Section 5.3.1. For comparison with the


Table5.1:

Summ

aryof

temperature-dependentspectroscopic

parameters

ofthe

v′←

v′′

=1←

0,X

2Π3/2←

X2Π

3/2

bandof

theN

OR

-branchm

easuredin

staticcellexperim

entsfrom

294to

802K

.2γ(296

K)and

nare

determined

directlyfrom

powerlaw

fitsto

experimentaldata.

δ(296K

)and

mare

determined

frompow

erlaw

fitsto

experimentaldata

smoothed

bythird

degreepolynom

ialsfits

asdescribed

in5.3.4.

X2Π

3/2←X

2Π3/2

Nitrogen

Argon

Transitionνa0

SHT

(296K

) bS

(296K

)2γ(296

K)

nδ(296

K)

m2γ(296

K)

nδ(296

K)

mcm−

1cm−

2/atmv

cm−

2/atmcm−

1/atmcm−

1/atmcm−

1/atmcm−

1/atmR

(2.5)1887.635

0.90060.9335(4.7%

)0.1317(3.3%

)0.72(7.1%

)-0.0029(31.3%

)0.92(51.9%

)0.1046(4.0%

)0.71(8.7%

)-0.0016(23.2%

)0.71(49.6%

)R

(3.5)1890.912

1.19141.1655(4.4%

)0.1284(1.0%

)0.70(2.2%

)-0.0027(28.2%

)0.92(46.5%

)0.1001(1.1%

)0.70(2.5%

)-0.0017(19.9%

)0.76(39.9%

)R

(4.5)1894.150

1.40921.3689(4.1%

)0.1250(3.5%

)0.69(7.7%

)-0.0026(24.6%

)0.93(40.4%

)0.0980(1.3%

)0.70(2.8%

)-0.0018(17.1%

)0.79(32.9%

)R

(5.5)1897.353

1.55651.5221(4.3%

)0.1239(5.4%

)0.70(11.9%

)-0.0024(20.6%

)0.93(33.7%

)0.0956(2.5%

)0.70(5.5%

)-0.0019(14.8%

)0.82(27.6%

)R

(6.5)1900.517

1.63671.5971(4.1%

)0.1216(4.0%

)0.70(8.8%

)-0.0023(16.2%

)0.93(26.6%

)0.0944(2.6%

)0.73(5.4%

)-0.0019(12.9%

)0.84(23.5%

)R

(7.5)1903.644

1.65431.5801(5.1%

)0.1214(2.8%

)0.73(6.1%

)-0.0023(11.7%

)0.93(19.2%

)0.0911(0.7%

)0.70(1.7%

)-0.0020(11.2%

)0.85(20.2%

)R

(8.5)1906.731

1.61741.6025(4.5%

)0.1197(2.8%

)0.71(6.0%

)-0.0022(7.6%

)0.93(12.4%

)0.0902(3.6%

)0.71(7.6%

)-0.0021(9.8%

)0.85(17.5%

)R

(9.5)1909.785

1.53451.5238(3.3%

)0.1206(3.3%

)0.73(6.9%

)-0.0022(5.5%

)0.93(9.0%

)0.0882(4.8%

)0.71(10.2%

)-0.0021(8.7%

)0.86(15.4%

)R

(10.5)1912.795

1.41761.4011(2.9%

)0.1178(3.3%

)0.72(7.1%

)-0.0021(7.2%

)0.93(11.9%

)0.0854(2.0%

)0.69(4.5%

)-0.0021(7.7%

)0.86(13.6%

)R

(11.5)1915.768

1.27721.2576(5.1%

)0.1166(1.9%

)0.72(4.1%

)-0.0021(10.5%

)0.92(17.4%

)0.0849(1.2%

)0.70(2.6%

)-0.0022(6.9%

)0.86(12.2%

)R

(12.5)1918.703

1.12371.1149(4.5%

)0.1158(3.4%

)0.72(7.3%

)-0.0021(13.7%

)0.92(22.8%

)0.0827(1.4%

)0.67(3.1%

)-0.0022(6.2%

)0.87(10.9%

)R

(13.5)1921.599

0.96650.9604(3.5%

)0.1145(3.0%

)0.71(6.4%

)-0.0022(16.1%

)0.91(26.9%

)0.0809(0.8%

)0.67(1.8%

)-0.0022(5.5%

)0.87(9.6%

)R

(14.5)1924.457

0.81350.8164(2.2%

)0.1129(4.3%

)0.69(9.5%

)-0.0022(17.5%

)0.91(29.4%

)0.0801(1.5%

)0.66(3.6%

)-0.0023(4.7%

)0.88(8.2%

)R

(15.5)1927.275

0.67040.6743(2.7%

)0.1105(3.0%

)0.71(6.5%

)-0.0022(17.7%

)0.90(29.9%

)0.0774(3.4%

)0.67(7.8%

)-0.0023(3.8%

)0.88(6.6%

)R

(16.5)1930.054

0.54130.5434(3.1%

)0.1113(0.7%

)0.70(1.6%

)-0.0022(16.5%

)0.90(28.0%

)0.0766(3.2%

)0.64(7.6%

)-0.0024(2.9%

)0.90(4.9%

)R

(17.5)1932.794

0.42830.4306(2.3%

)0.1105(1.4%

)0.70(3.0%

)-0.0022(13.9%

)0.90(23.7%

)0.0759(4.0%

)0.66(9.2%

)-0.0024(2.6%

)0.91(4.4%

)R

(18.5)1935.495

0.33230.3336(2.2%

)0.1088(0.9%

)0.69(2.1%

)-0.0022(10.1%

)0.89(17.2%

)0.0739(0.5%

)0.64(1.2%

)-0.0025(3.8%

)0.94(6.2%

)R

(19.5)1938.156

0.25290.2533(2.9%

)0.1077(1.0%

)0.69(3.0%

)-0.0022(6.8%

)0.89(11.7%

)0.0729(1.8%

)0.65(4.4%

)-0.0026(6.2%

)0.96(9.9%

)R

(20.5)1940.778

0.18890.1891(2.3%

)0.1063(1.6%

)0.68(3.7%

)-0.0022(10.9%

)0.89(18.7%

)0.0712(3.3%

)0.63(8.0%

)-0.0027(9.5%

)1.00(14.5%

)R

(21.5)1943.360

0.13850.1388(2.9%

)0.1046(1.4%

)0.67(3.1%

)-0.0022(21.7%

)0.89(37.2%

)0.0695(3.1%

)0.63(7.5%

)-0.0028(13.4%

)1.04(19.7%

)a

Weighted

averagesofthe

Λ-doubletand

hyperfineline

positionsfrom

theH

ITE

MP

2010database.

bFrom

theH

ITE

MP

2010database.

ForS

(296K

),parenthesesrepresentuncertainty

dueto

fits(Fig.5.4)and

systematic

effectssuch

asN

Oconcentration

andpath

length.For2

γ,n,δ,and

m,parentheses

represent95%confidence

intervalsfrom

powerlaw

fits.


Tabl

e5.

2:Su

mm

ary

ofte

mpe

ratu

re-d

epen

dent

spec

tros

copi

cpa

ram

eter

sof

thev′←

v′′

=1←

0,X

2Π

1/2←

X2Π

1/2

band

ofth

eN

OR

-bra

nch

mea

sure

din

stat

icce

llex

peri

men

tsfr

om29

4to

802

K.2γ

(296

K)a

ndn

are

dete

rmin

eddi

rect

lyfr

ompo

wer

law

fits

toex

peri

men

tald

ata.δ(

296

K)

andm

are

dete

rmin

edfr

ompo

wer

law

fits

toex

peri

men

tald

ata

smoo

thed

byth

ird

degr

eepo

lyno

mia

lsfit

sas

desc

ribe

din

5.3.

4.

X2Π

1/2←X

2Π

1/2

Nitr

ogen

Arg

onTr

ansi

tion

νa 0

SHT

(296

K)b

S(2

96K

)2γ

nδ(

296K

)m

2γ

nδ(

296K

)m

cm−

1cm−

2/a

tmcm−

2/a

tmcm−

1/a

tmcm−

1/a

tmcm−

1/a

tmcm−

1/a

tme

fe

fe

fe

fR

(2.5

)18

87.5

201.

9377

1.89

40(2

.7%

)0.

1253

(8.7

%)

0.12

73(4

.0%

)0.

72(1

8.4%

)0.

71(8

.7%

)-0

.000

7(57

.8%

)0.

79(1

12.3

%)

0.09

67(2

.6%

)0.

0991

(7.9

%)

0.71

(5.6

%)

0.72

(5.7

%)

-0.0

010(

43.2

%)

0.41

(161

.9%

)R

(3.5

)18

90.7

182.

3775

2.33

74(3

.3%

)0.

1226

(10.

2%)

0.12

67(4

.9%

)0.

69(2

2.6%

)0.

71(1

0.6%

)-0

.000

9(39

.4%

)0.

87(6

8.9%

)0.

0915

(3.0

%)

0.09

24(7

.8%

)0.

68(6

.7%

)0.

68(6

.1%

)-0

.001

2(29

.6%

)0.

55(8

2.9%

)R

(4.5

)18

93.8

702.

7169

2.66

50(2

.5%

)0.

1178

(2.2

%)

0.12

07(2

.9%

)0.

69(5

.0%

)0.

69(6

.4%

)-0

.001

1(29

.5%

)0.

91(4

9.4%

)0.

0897

(0.6

%)

0.09

04(2

.8%

)0.

69(1

.3%

)0.

69(2

.1%

)-0

.001

4(20

.4%

)0.

64(4

8.6%

)R

(5.5

)18

96.9

902.

9482

2.88

17(2

.5%

)0.

1170

(0.9

%)

0.11

75(0

.4%

)0.

70(2

.0%

)0.

69(1

.0%

)-0

.001

3(23

.2%

)0.

93(3

8.1%

)0.

0879

(1.7

%)

0.08

84(6

.5%

)0.

69(3

.8%

)0.

70(4

.9%

)-0

.001

6(14

.0%

)0.

71(2

9.9%

)R

(6.5

)19

00.0

753.

0712

2.96

31(2

.2%

)0.

1135

(2.4

%)

0.11

54(3

.4%

)0.

71(5

.2%

)0.

70(7

.4%

)-0

.001

4(18

.7%

)0.

93(3

0.5%

)0.

0843

(1.7

%)

0.08

52(1

.9%

)0.

69(3

.8%

)0.

69(1

.5%

)-0

.001

8(9.

7%)

0.77

(19.

2%)

R(7

.5)

1903

.128

3.08

932.

9258

(4.8

%)

0.11

23(4

.5%

)0.

1154

(4.8

%)

0.71

(10.

1%)

0.71

(10.

9%)

-0.0

015(

15.1

%)

0.93

(24.

7%)

0.08

36(2

.4%

)0.

0848

(10.

9%)

0.69

(5.6

%)

0.68

(4.0

%)

-0.0

019(

7.2%

)0.

82(1

3.5%

)R

(8.5

)19

06.1

453.

0141

3.01

33(4

.7%

)0.

1122

(1.9

%)

0.11

51(0

.9%

)0.

71(4

.0%

)0.

70(2

.0%

)-0

.001

6(12

.2%

)0.

93(2

0.1%

)0.

0821

(1.4

%)

0.08

38(5

.3%

)0.

70(3

.0%

)0.

69(4

.0%

)-0

.002

0(6.

3%)

0.86

(11.

2%)

R(9

.5)

1909

.128

2.86

082.

7704

(4.4

%)

0.10

87(2

.4%

)0.

1130

(2.3

%)

0.68

(5.4

%)

0.69

(5.0

%)

-0.0

017(

9.8%

)0.

92(1

6.2%

)0.

0813

(3.2

%)

0.08

30(6

.7%

)0.

70(7

.0%

)0.

69(5

.1%

)-0

.002

2(6.

2%)

0.89

(10.

6%)

R(1

0.5)

1912

.075

2.64

712.

5805

(3.7

%)

0.10

85(4

.9%

)0.

1139

(4.4

%)

0.69

(11.

0%)

0.71

(9.5

%)

-0.0

017(

7.7%

)0.

92(1

2.9%

)0.

0786

(0.7

%)

0.08

06(9

.4%

)0.

67(1

.5%

)0.

67(7

.4%

)-0

.002

2(6.

2%)

0.93

(10.

2%)

R(1

1.5)

1914

.990

2.39

102.

3788

(2.2

%)

0.10

67(3

.8%

)0.

1127

(5.0

%)

0.69

(8.3

%)

0.70

(10.

8%)

-0.0

018(

6.2%

)0.

91(1

0.3%

)0.

0764

(3.1

%)

0.08

01(4

.9%

)0.

67(7

.2%

)0.

67(3

.8%

)-0

.002

3(5.

8%)

0.96

(9.3

%)

R(1

2.5)

1917

.868

2.11

13–

––

––

-0.0

018(

4.9%

)0.

91(8

.3%

)–

––

–-0

.002

4(4.

9%)

0.99

(7.5

%)

R(1

3.5)

1920

.711

1.82

421.

8204

(2.7

%)

0.10

67(8

.4%

)0.

1102

(4.1

%)

0.70

(18.

3%)

0.69

(9.0

%)

-0.0

018(

4.1%

)0.

91(6

.8%

)0.

0744

(3.6

%)

0.07

84(6

.8%

)0.

65(8

.4%

)0.

66(5

.4%

)-0

.002

5(3.

3%)

1.02

(5.0

%)

R(1

4.5)

1923

.518

1.54

261.

5318

(3.6

%)

0.10

52(5

.2%

)0.

1091

(3.5

%)

0.69

(11.

5%)

0.67

(8.0

%)

-0.0

019(

3.3%

)0.

92(5

.5%

)0.

0735

(3.2

%)

0.07

73(8

.5%

)0.

65(7

.5%

)0.

65(6

.9%

)-0

.002

5(1.

5%)

1.05

(2.2

%)

R(1

5.5)

1926

.290

1.27

871.

2722

(3.2

%)

0.10

52(1

.8%

)0.

1097

(1.8

%)

0.71

(3.9

%)

0.70

(3.8

%)

-0.0

019(

2.4%

)0.

94(3

.9%

)0.

0719

(2.4

%)

0.07

60(1

1.0%

)0.

65(5

.5%

)0.

65(8

.9%

)-0

.002

6(3.

0%)

1.07

(4.3

%)

R(1

6.5)

1929

.026

1.03

871.

0413

(3.3

%)

0.10

24(1

0.1%

)0.

1068

(7.6

%)

0.68

(22.

4%)

0.67

(17.

2%)

-0.0

020(

1.5%

)0.

96(2

.3%

)0.

0699

(1.6

%)

0.07

42(4

.7%

)0.

62(3

.9%

)0.

63(3

.9%

)-0

.002

7(7.

0%)

1.10

(9.7

%)

R(1

7.5)

1931

.728

0.82

750.

8306

(3.3

%)

0.10

08(4

.5%

)0.

1057

(3.6

%)

0.67

(10.

1%)

0.67

(8.4

%)

-0.0

020(

2.0%

)0.

99(3

.1%

)0.

0691

(1.4

%)

0.07

33(1

.0%

)0.

63(3

.3%

)0.

64(0

.8%

)-0

.002

7(12

.1%

)1.

13(1

6.4%

)R

(18.

5)19

34.3

920.

6466

0.64

33(2

.5%

)0.

1011

(3.2

%)

0.10

50(4

.4%

)0.

70(6

.9%

)0.

67(1

0.0%

)-0

.002

1(4.

7%)

1.03

(7.0

%)

0.06

82(5

.0%

)0.

0722

(17.

5%)

0.64

(11.

9%)

0.64

(14.

2%)

-0.0

028(

18.3

%)

1.16

(24.

2%)

R(1

9.5)

1937

.021

0.49

580.

4928

(3.2

%)

0.10

02(3

.9%

)0.

1037

(2.9

%)

0.70

(8.6

%)

0.66

(6.7

%)

-0.0

022(

8.6%

)1.

08(1

2.1%

)0.

0672

(5.4

%)

0.07

09(1

5.4%

)0.

65(1

2.7%

)0.

64(1

2.6%

)-0

.002

8(25

.5%

)1.

18(3

2.9%

)R

(20.

5)19

39.6

140.

3732

0.37

06(3

.0%

)0.

0978

(3.1

%)

0.10

13(1

.8%

)0.

67(7

.0%

)0.

64(4

.2%

)-0

.002

4(13

.6%

)1.

14(1

8.2%

)0.

0646

(3.5

%)

0.06

83(9

.8%

)0.

61(8

.7%

)0.

60(8

.6%

)-0

.002

9(33

.7%

)1.

21(4

2.5%

)R

(21.

5)19

42.1

690.

2758

––

––

–-0

.002

6(19

.8%

)1.

22(2

4.9%

)–

––

–-0

.003

0(43

.0%

)1.

24(5

2.9%

)R

(22.

5)19

44.6

890.

2001

0.19

66(2

.8%

)0.

0966

(1.2

%)

0.09

89(3

.7%

)0.

66(2

.8%

)0.

61(9

.2%

)-0

.002

8(27

.1%

)1.

29(3

2.0%

)0.

0621

(1.9

%)

0.06

57(1

5.5%

)0.

60(4

.9%

)0.

57(1

4.3%

)-0

.003

0(53

.2%

)1.

27(6

3.9%

)

aW

eigh

ted

aver

ages

ofth

eΛ

-dou

blet

and

hype

rfine

line

posi

tions

from

the

HIT

EM

P20

10da

taba

se.

bFr

omth

eH

ITE

MP

2010

data

base

.Fo

rS(2

96K

),pa

rent

hese

sre

pres

entu

ncer

tain

tydu

eto

fits

(Fig

.5.4

)and

syst

emat

icef

fect

ssu

chas

NO

conc

entr

atio

nan

dpa

thle

ngth

.Fo

r2γ

,n,δ

,andm

,par

enth

eses

repr

esen

t95%

confi

denc

ein

terv

als

from

pow

erla

wfit

s.


200 400 600 800 1000

Temperature (K)

0

0.1

0.2

0.3

0.4

0.5

0.6

Lin

e S

trength

(cm

-2/a

tm)

Measurement 21/2

R(20.5)

Measurement 23/2

R(20.5)

Best Fit

HITEMP2010

Figure 5.4: Line strength versus temperature for v′ ← v′′ = 1 ← 0 2Π1/2 and 2Π3/2 R(20.5)transitions. Measurements agree with the values reported by the HITEMP database.

HITEMP database, the line strength at the reference temperature, i.e. S(T0), was determined by

fitting Eq. 2.5 to the measured line strengths and floating S(T0). Examples of line strength mea-

surements versus temperature for the 2Π1/2 and 2Π3/2 R(20.5) transitions are displayed in Figure

5.4. The best fit and HITEMP line strength curves are plotted along with the measured data. For

these particular transitions, the best fit and HITEMP S(T0) agree very well. For the other transi-

tions measured, all S(T0) agree within the uncertainty provided by HITEMP and the experimental

uncertainty of the present measurements. Figure 5.5 summarizes the line strength measurements in

the static cell experiments. For the purpose of presenting on a single plot, line strengths for each

temperature case are normalized by the transition with the maximum line strength calculated from

Eq. 2.5 using the HITEMP parameters. Across all temperatures, the average deviation from the

HITEMP database is 2%. The best fit S(T0) for the measured transitions are reported in Tables 5.1

and 5.2.


0 10 20 30 40

J"

0

0.2

0.4

0.6

0.8

1

Norm

aliz

ed L

ine S

trength

21/2

Subband

294 K

453 K

618 K

802 K

HITEMP

0 10 20 30 40

J"

0

0.2

0.4

0.6

0.8

1

Norm

aliz

ed L

ine S

trength

23/2

Subband

294 K

453 K

618 K

802 K

HITEMP

Figure 5.5: Normalized line strengths of v′ ← v′′ = 1 ← 0 R-branch transitions at several temper-atures. Top: 2Π1/2 subband. Bottom: 2Π3/2 subband. For all temperatures, the average deviationbetween measured and HITEMP line strength simulations is 2%.

5.3.3 Collision Broadening

The collision broadening coefficients, 2γ, for NO transitions with N2 and Ar collision partners

at four different temperatures were determined via the multi-spectral fitting routine described in

section 5.3.1. Figure 5.6 shows 2γ versus J ′′ for each spin-split subband broadened by N2 and

Ar at the four temperatures studied in the static cell. For the 2Π1/2 transitions, broadening for

both e/f-components are displayed. The f-component broadening was found to be larger than the

e-component broadening. At all temperatures, the average pressure broadening due to Ar is less

than that of N2 by ≈ 27%. However, the difference grows with J ′′. For the R(2.5) transitions,


0 10 20 30 40

J"

0.04

0.06

0.08

0.1

0.12

0.14

2N

O-N

2

(cm

-1/a

tm)

21/2

Subband

e-component

f-component

Average component

2 1 hot band

Spencer (1994)

0 10 20 30 40

J"

0.04

0.06

0.08

0.1

0.12

0.14

2N

O-N

2

(cm

-1/a

tm)

23/2

Subband

e-component

f-component

Average component

2 1 hot band

Spencer (1994)

294 K

453 K

618 K

802 K

294 K

453 K

618 K

802 K

(a) Nitrogen Broadening

0 10 20 30 40

J"

0.02

0.04

0.06

0.08

0.1

0.12

2N

O-A

r (cm

-1/a

tm)

21/2

Subband

e-component

f-component

Average component

2 1 hot band

Pope and Wolf (2001)

0 10 20 30 40

J"

0.02

0.04

0.06

0.08

0.1

0.12

2N

O-A

r (cm

-1/a

tm)

23/2

Subband

e-component

f-component

Average component

2 1 hot band

Pope and Wolf (2001)

294 K

453 K

618 K802 K

294 K

453 K

618 K802 K

(b) Argon Broadening

Figure 5.6: Measured collision broadening coefficients for v′ ← v′′ = 1 ← 0 R-branch transitionsat several temperatures. (a) Measured N2 broadening versus J ′′ at four different temperatures. Top:2Π1/2 subband. Bottom: 2Π3/2 subband. The measurements by Spencer et al. [34] are plotted forcomparison with the 294 K data. (b) Measured Ar broadening versus J ′′ at four different tempera-tures. Top: 2Π1/2 subband. Bottom: 2Π3/2 subband. The measurements by Pope and Wolf [36] areplotted for comparison with the 294 K data. v′ ← v′′ = 2← 1 hot band transitions were measuredat 802 K and continue the trend established by the v′ ← v′′ = 1← 0 transitions.

Ar broadening is less than N2 broadening by ≈ 21% for all temperatures while the difference is ≈

40% for the R(40.5) transitions. The general trend of N2 broadening exceeding that of Ar can be

explained physically because ∆νc ∝ σ2AB/√µAB where σAB and µAB are respectively the optical

collision diameter and reduced mass of the colliding molecules A and B. N2 is lighter than Ar and

generally has a larger effective optical collision diameter than Ar. However, the reader interested in

theoretical insights of the J ′′ dependence of the N2 to Ar broadening ratios is encouraged to refer

to [21] and the sources within. Comparable room-temperature (near 296 K) N2 and Ar broadening

data for the fundamental band of NO found in the literature are plotted with the data in Figure 5.6.


0 5 10 15 20 25 30

J"

0.04

0.06

0.08

0.1

0.12

0.14

2N

O-A

(cm

-1/a

tm)

21/2

Subband

A = N2

A = 79% N2 + 21% O

2

0 5 10 15 20 25 30

J"

0.04

0.06

0.08

0.1

0.12

0.14

2N

O-A

(cm

-1/a

tm)

23/2

Subband

294 K

618 K

802 K

294 K

618 K

802 K

Figure 5.7: Comparison of N2 pressure broadening with air pressure broadening. On average,measured air broadening is 2.5% less than N2 broadening.

For N2 broadening, measurements by Spencer et al. [34] agree well with both spin-split subbands.

The most recent and complete J ′′-dependent study of Ar broadening available is that of Pope and

Wolf [36]. However, the values reported by Pope and Wolf are systematically high on average by ≈

20% and 13%, respectively, in the 2Π1/2 and 2Π3/2 subbands. Though less extensive, older studies

of Ar broadening show better agreement with the present measurements [42, 105].

In addition to N2 and Ar broadening, air broadening was also measured at three temperatures for

a few transitions in each subband. Comparisons of air and N2 broadening are presented in Figure

5.7, and for simplicity, the average of the e/f-component broadening coefficients for the 2Π1/2 sub-

band are shown in Figure 5.7. At all temperatures, air broadening coefficients were found to be ≈

2.5% less than N2 broadening. This is consistent with measurements of O2 broadening by Chacke-

rian et al. who found that O2 broadening in NO’s fundamental rovibrational band at 299 K was about


0 5 10 15 20 25

J"

0.55

0.6

0.65

0.7

0.75

0.8

0.85

n

Nitrogen 1/2,e

Nitrogen 1/2,f

Nitrogen 3/2

0 5 10 15 20 25

J"

0.09

0.1

0.11

0.12

0.13

0.14

2N

O-N

2

(To =

296 K

) (c

m-1

/atm

)

Nitrogen 1/2,e

Nitrogen 1/2,f

Nitrogen 3/2

(a) Nitrogen broadening

0 5 10 15 20 25

J"

0.55

0.6

0.65

0.7

0.75

0.8

0.85

n

Argon 1/2,e

Argon 1/2,f

Argon 3/2

0 5 10 15 20 25

J"

0.06

0.07

0.08

0.09

0.1

0.11

2N

O-A

r(To =

296 K

) (c

m-1

/atm

)

Argon 1/2,e

Argon 1/2,f

Argon 3/2

(b) Argon broadening

Figure 5.8: Pressure broadening power law fit parameters versus J ′′. Error bars represent the stan-dard error of the best-fit parameters. Solid lines represent third degree polynomials fit to best-fitparameters as a function of J ′′.

17% less than comparable N2 broadening [59]. The HITRAN and HITEMP databases use Chack-

erian’s O2 broadening values in their reported values of γNO−air = 0.79γNO−N2 + 0.21γNO−O2

[58] that would lead to average air broadening coefficients to be ≈ 3.6% less than comparable N2

broadening. Furthermore, there was no detectable difference in the observed temperature depen-

dence between air and N2 broadening. To the authors’ knowledge, the air broadening coefficients

at elevated temperatures presented here are the first of their kind for the fundamental rovibrational

band of NO.


For each J ′′, the power law presented in Eq. (2.10) was fit to the data presented in Figure 5.6.

The fit parameters, 2γ(T0 = 296K) and n, for different collision partners are presented in Tables

5.1 and 5.2 and Figure 5.8 for the different collision partners. For the temperature range studied, the

temperature exponents have a slight J ′′ dependence, and the 2Π3/2 subband has a larger temperature

exponent than the 2Π1/2 temperature exponents. Furthermore, the temperature exponent for N2

collisions is on average slightly larger than for Ar collisions. Overall, these measurements agree

favorably with temperature exponents presented in other works with the temperature dependence

exponents being near 0.6–0.7 [33, 101, 102].

Table 5.3: Summary of spectroscopic parameters of the v′ ← v′′ = 2 ← 1 hot band of the NOR-branch measured in static cell experiments at 802 K. S(296 K) and 2γ(802 K) are determinedfrom the multi-spectral fitting routine described in 5.3.1.

Transition νa SHT (802 K)a S(802 K) 2γNO−N2 (802 K) 2γNO−Ar(802 K)cm−1 cm−2/atm cm−2/atm cm−1/atm cm−1/atm

2Π3/2R(11.5) 1887.218 0.0290 0.0283(2.0%) 0.0572(0.3%) 0.0430(0.4%)2Π1/2R(15.5) 1897.627 0.0352 0.0336(3.0%) 0.0533(1.8%) 0.0393(2.1%)2Π1/2R(16.5) 1900.328 0.0339 0.0324(5.2%) 0.0515(5.9%) 0.0379(3.4%)2Π3/2R(18.5) 1906.688 0.0239 0.0247(11.0%) 0.0526(8.4%) 0.0399(6.0%)2Π3/2R(19.5) 1909.312 0.0224 0.0229(13.4%) 0.0589(17.4%) 0.0398(18.4%)2Π1/2R(25.5) 1922.997 0.0166 0.0158(18.4%) 0.0484(7.6%) 0.0344(13.0%)2Π3/2R(25.5) 1924.224 0.0127 0.0123(14.1%) 0.0491(10.4%) 0.0336(6.3%)2Π3/2R(26.5) 1926.569 0.0112 0.0098(35.3%) 0.0455(14.6%) 0.0314(13.0%)2Π1/2R(27.5) 1927.627 0.0130 0.0124(15.2%) 0.0475(9.7%) 0.0323(6.6%)2Π3/2R(27.5) 1928.874 0.0099 0.0098(15.9%) 0.0506(14.3%) 0.0369(20.6%)2Π1/2R(28.5) 1929.885 0.0114 0.0118(24.7%) 0.0514(20.2%) 0.0362(14.2%)2Π1/2R(29.5) 1932.104 0.0099 0.0091(49.1%) 0.0453(13.3%) 0.0292(9.6%)2Π3/2R(29.5) 1933.364 0.0075 0.0071(9.1%) 0.0468(5.7%) 0.0332(6.2%)2Π1/2R(30.5) 1934.286 0.0085 0.0086(60.0%) 0.0517(28.8%) 0.0505(17.4%)2Π3/2R(30.5) 1935.547 0.0064 0.0068(12.7%) 0.0627(16.7%) 0.0299(19.3%)2Π3/2R(32.5) 1939.793 0.0046 0.0041(22.0%) 0.0394(16.3%) 0.0229(24.2%)2Π1/2R(33.5) 1940.598 0.0053 0.0053(11.0%) 0.0477(11.2%) 0.0329(11.7%)

To indicate the precision of the measurements, 95% confidence intervals for the best-fit parametersare reported in parentheses.a Values determined from the HITEMP 2010 database.

At the highest temperature (802 K) studied in the HPHT static cell, hot band transitions were

included in the multi-spectral Voigt fits as shown in Figure 5.3. The resulting v′ ← v′′ = 2← 1 hot

band broadening parameters can be seen in Figure 5.6 and are reported in Table 5.3. For some values

of J ′′, 2γ was measured for both v′ ← v′′ = 1← 0 and v′ ← v′′ = 2← 1 bands, and these values


are equivalent within experimental uncertainty. Furthermore, the measured v′ ← v′′ = 2 ← 1

transitions for J ′′ > 22.5 follow the trend established by the J ′′ = 2.5-22.5 and J ′′ = 39.5-42.5

transitions in the v′ ← v′′ = 1 ← 0, providing evidence that there is insignificant vibrational level

dependence on 2γ.

5.3.4 Pressure Shift

0 5 10 15 20 25

J"

-3

-2

-1

0

1

2

NO

-N2

(cm

-1/a

tm)

10-3

21/2

Subband

294 K

453 K

618 K

802 K

Spencer (1994)

0 5 10 15 20 25

J"

-3

-2

-1

0

1

2

NO

-N2

(cm

-1/a

tm)

10-3

23/2

Subband

(a) Nitrogen Pressure Shift

0 5 10 15 20 25

J"

-3

-2

-1

0

1

2

NO

-Ar (

cm

-1/a

tm)

10-3

2

1/2 Subband

294 K

453 K

618 K

802 K

0 5 10 15 20 25

J"

-3

-2

-1

0

1

2

NO

-Ar (

cm

-1/a

tm)

10-3

2

3/2 Subband

(b) Argon Pressure Shift

Figure 5.9: Measured pressure shift coefficients for R-branch transitions at several temperatures.Pressure shifts measured at 296 K by Spencer et al. [34] are shown by the solid line. Temperaturedependence and slight J ′′ dependence of the pressure shifts are apparent.

Pressure shift coefficients, δ, were also determined at several temperatures from the multi-

spectral Voigt fitting discussed in Section 5.3.1 and are presented in Figure 5.9. Currently, few

pressure shift coefficients for rovibrational NO bands are presented in the literature. Determination

of NO pressure shift is difficult due to Λ-doublet transitions and small shift magnitudes. Spencer et


al. measured pressure shift coefficients for many of the transitions in the fundamental rovibrational

band at 296 K [34], and Pine et al. measured pressure shift coefficients in the first overtone band

[106]. The pressure shift coefficients of transitions in the overtone bands have been shown to be

larger than those of the fundamental band [107], which is is the case for Spencer et al. and Pine et

al. However, the magnitudes of the pressure shift coefficients at 294 K measured in this work fall

between the measurements by Spencer et al. and Pine et al. For comparison with the present mea-

surements, the pressure shift coefficients as a function of J ′′ reported by Spencer et al. are plotted

in Figure 5.9 (a) for both subbands. The observed trends with J ′′ of Spencer et al. and this work are

similar with the shifts in this work being systematically larger than those of Spencer et al. by about

0.001 cm−1.

A visible temperature dependence is present in the pressure shift measurements presented in Fig-

ure 5.9. The decreasing pressure shift magnitude with increasing temperature supports the power

law temperature dependence of Eq. (2.11). Fitting the data to Eq. (2.11) resulted in scattered results

for the temperature dependence coefficients, m, with values varying between 0.3 to 1.5. To reduce

the scatter, the data was smoothed at each temperature by fitting third degree polynomials to the

measured pressure shift coefficients over the range of J ′′ studied. The smoothed data was then fit

to Eq. (2.11), and the resulting best-fit values are presented in Tables 5.1 and 5.2. For the 2Π1/2

subband, averages for the fit parameters are as follows: mN2 = 0.97, δNO−N2(T0) = −0.0018

cm−1/atm, mAr = 0.95, and δNO−Ar(T0) = −0.0023 cm−1/atm. For the 2Π3/2 subband, av-

erages for the fit parameters are as follows: mN2 = 0.91, δNO−N2(T0) = −0.0023 cm−1/atm,

mAr = 0.87, and δNO−Ar(T0) = −0.0022 cm−1/atm. Measured air pressure shifts do not differ

significantly from measured N2 pressure shifts.

5.3.5 High-Temperature Measurements in a Shock Tube

To extend the temperature range of this study to 2500 K, absorption measurements were made

during shock tube experiments. Conditions behind the reflected shock ranged from 1000-2500

K and 1.5-3.2 atm. The test mixtures were either 2% NO in Ar or 2.03% NO in N2. Scanned-

direct-absorption measurements of the 2Π1/2R(40.5)–R(43.5) and 2Π3/2R(39.5)–R(42.5) line

shapes were collected with the DFBQCL. The ECQCL is unable to wavelength tune rapidly


Table5.4:M

easuredspectroscopic

parameters

fromhigh-tem

peratureshock

tubeexperim

ents

Nitrogen

Argon

Transitionνa

SHT

(296K

) aS

(296K

)2γ(296

K)

nb

2γ(1000

K)

nc

2γ(296

K)

nb

2γ(1000

K)

nc

cm−

1cm−

2/atmcm−

2/atmcm−

1/atm(296–2500

K)

cm−

1/atm(1000–2500

K)

cm−

1/atm(296–2500

K)

cm−

1/atm(1000–2500

K)

2Π1/2 R

(5.5)1896.990

––

0.1130(4.6%)

0.6074(6.3%)

0.0546(5.0%)

0.6112(14.3%)

0.0852(6.2%)

0.62(7.4%)

0.0398(3.9%)

0.58(15.3%)

2Π3/2 R

(7.5)1903.644

––

0.1216(2.8%)

0.7216(5.0%)

0.0515(9.3%)

0.7549(23.1%)

0.0934(3.3%)

0.77(5.5%)

0.0369(9.9%)

0.79(22.9%)

2Π1/2 R

(10.5)1912.075

––

0.1068(5.3%)

0.6228(7.1%)

0.0481(5.7%)

0.5395(18.0%)

0.0792(2.0%)

0.67(2.4%)

0.0343(3.8%)

0.63(10.3%)

2Π3/2 R

(10.5)1912.795

––

0.1152(2.9%)

0.6717(4.5%)

0.0518(6.4%)

0.6915(16.6%)

0.0872(0.7%)

0.71(0.9%)

0.0370(2.0%)

0.72(4.9%)

2Π1/2 R

(15.5)1926.290

––

0.1039(3.9%)

0.6298(5.2%)

0.0482(4.1%)

0.6140(11.7%)

0.0729(2.8%)

0.62(3.0%)

0.0339(4.2%)

0.59(12.4%)

2Π3/2 R

(15.5)1927.275

––

0.1067(3.7%)

0.6329(5.1%)

0.0490(0.4%)

0.6059(1.3%)

0.0755(3.5%)

0.61(4.3%)

0.0360(3.2%)

0.62(9.3%)

2Π1/2 R

(20.5)1939.614

––

0.0975(3.0%)

0.6077(3.8%)

0.0467(6.0%)

0.6051(16.2%)

0.0656(4.3%)

0.58(4.0%)

0.0327(9.8%)

0.59(28.6%)

2Π3/2 R

(20.5)1940.778

––

0.1026(4.7%)

0.6098(5.8%)

0.0475(1.1%)

0.5500(3.2%)

0.0696(2.9%)

0.60(3.6%)

0.0331(1.4%)

0.56(4.0%)

2Π3/2 R

(39.5)1982.955

2.47e-052.48e-05(3.0%

)0.0694

d–

0.0386(2.8%)

0.4824(11.2%)

0.0373d

–0.0240(1.8%

)0.36(9.0%

)2Π

1/2 R

(40.5)1983.543

3.16e-053.26e-05(3.0%

)0.0695

d–

0.0378(2.9%)

0.5007(20.2%)

0.0377d

–0.0240(1.7%

)0.37(16.7%

)2Π

3/2 R

(40.5)1984.767

1.30e-051.33e-05(3.0%

)0.0659

d–

0.0379(2.6%)

0.4538(10.6%)

0.0373d

–0.0241(2.3%

)0.36(10.8%

)2Π

1/2 R

(41.5)1985.329

1.66e-051.67e-05(3.0%

)0.0687

d–

0.0376(1.6%)

0.4947(5.9%)

0.0375d

–0.0234(1.7%

)0.39(7.7%

)2Π

3/2 R

(41.5) e1986.537

6.74e-066.73e-06(1.0%

)0.0723

d–

0.0384(2.0%)

0.5200(6.0%)

0.0380d

–0.0239(2.0%

)0.38(6.0%

)2Π

1/2 R

(42.5) e1987.074

8.54e-068.65e-06(1.0%

)0.0685

d–

0.0368(2.0%)

0.5100(5.0%)

0.0362d

–0.0231(2.0%

)0.37(7.0%

)2Π

3/2 R

(42.5)1988.265

3.43e-063.55e-06(3.0%

)0.0631

d–

0.0375(2.6%)

0.4271(11.7%)

0.0359d

–0.0236(2.7%

)0.34(15.9%

)2Π

1/2 R

(43.5)1988.779

4.33e-064.44e-06(3.0%

)0.0596

d–

0.0361(2.5%)

0.4114(11.6%)

0.0381d

–0.0229(2.4%

)0.42(11.7%

)

95%confidence

intervalsare

reportedin

parenthesesforpow

erlawfitting

parameters.

aV

aluesfrom

theH

ITE

MP

2010database.

bR

epresentstem

peratureexponentcalculated

usingboth

thestatic

cellandshock

tubem

easurements.

cR

epresentstem

peratureexponentcalculated

usingonly

shocktube

measurem

ents.d

Extrapolated

fromhigh

temperature

experiments.

eE

xperimentalvalues

inthis

rowfrom

[11].


enough to measure line shapes at the time scales of a shock tube experiment. Thus, fixed-direct-

absorption measurements of absorption peaks for the 2Π1/2R(5.5), R(10.5), R(15.5), R(20.5) and2Π3/2R(7.5), R(10.5), R(15.5), R(20.5) transitions were collected with the ECQCL.

Measured line strengths of the high J ′′ transitions at temperatures up to 2500 K are presented

in Figure 5.10. Like the line strengths measured in the HPHT optical cell, the line strengths mea-

sured in shock tube experiments agree with the HITEMP line strengths within uncertainties. For the

v = 1 ← 0 X2Π1/2R(40.5) transition, a hot band transition not resolved by the wavelength scan

interferes with the fitting routine and biases the measured integrated area. For a better comparison

with HITEMP, the measured line strength is compared with the summation of the 2Π1/2R(40.5)

transition and hot band transition in Figure 5.10. The other v = 1← 0 transitions were distinguish-

able from neighboring hot band transitions.

500 1000 1500 2000 2500 3000

Temperature (K)

0

0.01

0.02

0.03

0.04

0.05

Lin

e S

trength

(cm

-2/a

tm)

2

1/2 Subband

ST NO in N2

ST NO in Ar

HPHT

HITEMP

Best Fit

500 1000 1500 2000 2500 3000

Temperature (K)

0

0.01

0.02

0.03

0.04

0.05

Lin

e S

trength

(cm

-2/a

tm)

2

3/2 Subband

ST NO in N2

ST NO in Ar

HPHT

HITEMP

Best Fit

R(39.5)

R(42.5)

R(40.5)R(41.5)

R(43.5)

R(40.5) + (2 1, 1/2

R(63.5))

Figure 5.10: Line strength measurements of the R(39.5)-R(43.5) transitions.

Pressure broadening coefficients measured in the shock tube for the high J ′′ transitions are

presented in Figure 5.11. The power law of Eq. 2.10 was fit to the data to determine the tempera-

ture dependence exponent and collision width at the reference temperature. Since these transitions

are weak at temperatures below 800 K, T0 = 1000 K was used in Eq. 2.10. Best fit values are

summarized in Table 5.4 . Present measurements agree with 2γ(T0) and n of 2Π3/2R(41.5) and2Π1/2R(42.5) reported by [11].


1000 1500 2000 2500 3000

Temperature (K)

0.01

0.02

0.03

0.04

0.05

2 (

cm

-1/a

tm)

21/2

Subband

R(40.5)

R(41.5)

R(43.5)

Power Law Fit

1000 1500 2000 2500 3000

Temperature (K)

0.01

0.02

0.03

0.04

0.05

2 (

cm

-1/a

tm)

23/2

Subband

R(39.5)

R(40.5)

R(42.5)

Power Law Fit

Figure 5.11: Measured pressure broadening coefficients of the R(39.5)-R(43.5) transitions. Opensymbols represent 2γNO−N2 and filled symbols represent 2γNO−Ar.

Pressure broadening coefficients of the low J ′′ transitions were inferred from line center absorp-

tion measurements as described in [11]. Briefly, the pressure broadening coefficient is iteratively

scaled until the measured absorbance at line center matches a simulated absorbance at the transi-

tion’s line center within the desired tolerance. Comparisons with static cell measurements reveal

the breakdown of the power law (Eq. 2.10) over this wide temperature range (294-2500 K). Figure

5.12 (a) plots measured 2γ versus temperature for the 2Π1/2R(20.5) transition from both the HPHT

static cell and the shock tube. Best fit power law curves for two situations are also plotted. First, the

best fit determined from only the HPHT measurements is plotted and extrapolated to 2500 K. Sec-

ond, the best fit determined from both the HPHT and shock tube measurements are plotted. Based

on this data, extrapolating the best fit 2γ from relatively low temperature static cell data to higher

temperatures would induce errors in peak absorbance by ≈ 7%. Similar calculations for the other

transitions studied in the shock tube were made, and the resulting temperature dependence exponent

for N2 are plotted in Figure 5.12 (b). For the studied transitions, the temperature exponent changes

significantly with the extended temperature range. Several previous studies have investigated the

insufficiencies of the power law and its limitations beyond the temperature range for which it was

determined [25–27]. For the purpose of modeling the temperature dependence of NO line shapes,

the authors recommend using the values presented in Tables 5.1 and 5.2 for applications between

296 and 802 K and the 2γ(296 K) and n(296–2500 K) values in Table 5.4 for applications between

5.4. SUMMARY 71

1000 and 2500 K.

500 1000 1500 2000 2500

Temperature (K)

0.02

0.04

0.06

0.08

0.1

2N

O-N

2 (

cm

-1/a

tm)

HPHT Data

ST Data

Best Fit HPHT

Best Fit ST+HPHT

n = 0.61

21/2

R(20.5) transition

n=0.65

(a)

5 10 15 20

J"

0.5

0.6

0.7

0.8

0.9

n

2

1/2 (294-802 K)

2

1/2 (294-2500 K)

2

3/2 (294-802 K)

2

3/2 (294-2500 K)

(b)

Figure 5.12: (a) Measured pressure broadening coefficients for the 2Π1/2R(20.5) transition fromroom-temperature to 2500 K. (b) Comparison of pressure broadening temperature exponent deter-mined from measurements in different temperature ranges.

5.4 Summary

Spectroscopic parameters of over 40 rovibrational transitions in the fundamental NO absorption

band near 1900 cm−1 were determined from transition line shapes measured in a static cell from

294 to 802 K and in a shock tube from 1000 to 2500 K with two quantum cascade lasers. Measured

line strengths did not differ significantly from those tabulated by the HITRAN/HITEMP databases

or other measurements found in the literature. Collision broadening coefficients, 2γ, were deter-

mined for collisions with N2, Ar, and air. At room temperature, measured N2 broadening agreed

well with previous works found in the literature. For room-temperature Ar broadening, compara-

ble published data is more sparse, yet available data found moderate agreement with this work. The

HITRAN/HITEMP databases report γNO−air rather than γNO−N2 , and room-temperature measure-

ments of 2γNO−air presented here also agree well with those found in the databases. Pressure shift

coefficients were also determined and compared to other room-temperature measurements in the

literature with a systematic offset of 0.001 cm−1 found between the most extensive study.


The primary contribution of this chapter is found in the determination of the temperature depen-

dence of the collision broadening and pressure shift coefficients and the measurement of high J ′′

transitions. A power law adequately describes the temperature dependence of both collision broad-

ening and pressure shift phenomena studied here. However, high-temperature shock tube measure-

ments of broadening coefficients deviated from the temperature behavior predicted by static cell

measurements. This highlights insufficiencies in the power law outside of the studied temperature

range. To the authors’ knowledge, the temperature dependence of the pressure shift for NO transi-

tions in the fundamental rovibrational band was measured for the first time. New measurements of

high J ′′ transitions line strength and broadening coefficients are presented from 800 to 2500 K. Ad-

ditionally, v = 2← 1 hot band transitions were measured at 800 K and their broadening coefficients

were in agreement with the trends established by the v = 1← 0 transitions.

Chapter 6

High-Pressure Spectroscopy of Nitric

Oxide Near 5.3 µm

6.1 Introduction

To accurately model the absorption spectra of molecules in extreme, high-pressure environ-

ments, an understanding of the collisional processes that affect the spectra is necessary for quan-

titative sensing applications. Two collisional effects relevant at high pressures are line mixing and

the breakdown of the impact approximation. This chapter will focus primarily on line mixing as the

experimental results presented here indicate that line mixing is the primary contributor to deviations

from classical line shape models. However, the break down of the impact approximation has been

studied for commonly measured molecules such as H2O [13, 21]. To be clear, the present discussion

of line mixing is formulated within the impact approximation. In situations where the breakdown of

the impact approximation is significant, the frequency dependence of line mixing parameters must

be included.

Line mixing has been studied extensively in absorption and Raman Q branches where the tran-

sitions are more closely spaced [108–113]. Line mixing generally becomes relevant when the colli-

sional linewidth of the transition, ∆νc, is of similar or greater magnitude to the transition spacing,

so Q branch transitions and other closely spaced transition manifolds may be susceptible to line

73

74 CHAPTER 6. HIGH-PRESSURE NO SPECTROSCOPY

mixing effects at even moderate pressures (∼ 1 atm). Modeling and quantification of line mixing

has been approached in a variety of ways from theoretical quantum calculations to more empirical

approaches such as the band correction function defined in Chapter 4.

This chapter focuses on the energy gap fitting law approach to build an empirically based model

of the line mixing phenomena observed in NO spectra measured at high pressures. To that end, first,

introductory line mixing theory and concepts, such as the impact relaxation matrix, are presented.

Second, the energy gap fitting law approach used here is discussed in depth. Third, computational

approaches are introduced and a temperature-dependent model is built from experimentally deter-

mined line broadening parameters. Finally, the model is compared with high-pressure absorption

spectra measurements from high-pressure, high-temperature optical cell and high-pressure shock

tube experiments. The end of this chapter explores using the line mixing model in temperature

sensing applications.

6.2 Line Mixing

Collisional line mixing (also called collisional line interference or coupling) is a collision-

induced phenomenon that significantly affects spectral signals when transitions are closely spaced

and sufficiently pressure broadened. When there is significant overlap between interfering tran-

sitions, line mixing most notably affects the core regions of the line shape profiles. Although a

collision induced phenomenon, line mixing can still be important in low pressure situations. For in-

stance if the interfering transitions are closely spaced as in Q branches and near the band head of an

R-branch, line mixing may be observed below 1 atm [109, 111–114]. Additionally, at relatively low

pressures line mixing can play a critical role in measurements of the low-absorbing microwindows

(regions between transitions) [28, 91, 115, 116]. In the context of this work and the NO R-branch,

the focus is on situations where the core regions of absorption transitions significantly overlap due

to pressure broadening.

The line shape distortion due to line mixing is caused by rotationally inelastic collision processes

resulting in population transfers between the energy states defining an optical transition. Consider

a system of two absorption transitions with lower and upper state energies represented by Ei and

6.2. LINE MIXING 75

𝐸𝐸𝑘𝑘′ = 𝐸𝐸𝑓𝑓

𝐸𝐸𝑘𝑘′′ = 𝐸𝐸𝑖𝑖

𝜈𝜈0,𝑙𝑙 = 𝐸𝐸𝑓𝑓 − 𝐸𝐸𝑖𝑖

𝒌𝒌 represents the optical transition 𝒇𝒇 ← 𝒊𝒊 𝒌𝒌′ represents the optical transition 𝒇𝒇′ ← 𝒊𝒊′

𝐸𝐸𝑘𝑘′′ = 𝐸𝐸𝑓𝑓′

𝐸𝐸𝑘𝑘′′′ = 𝐸𝐸𝑖𝑖′

𝜈𝜈0,𝑙𝑙′ = 𝐸𝐸𝑓𝑓′ − 𝐸𝐸𝑖𝑖′

Figure 6.1: Energy level diagram describing the line mixing process between two adjacent opticaltransitions.

Ef for transition k and and Ei′ and Ef ′ for transition k′ (see Figure 6.1). Collisions induce energy

(population) transfers. If collisional population transfers between i, i′ and f, f ′ are allowed, an

absorber in state i can follow two paths to state f . First, there is the usual absorption from i to

f . Second, collisions can cause i to i′ transfers, then photon absorption from i′ to f ′, and finally

relaxation from f ′ to f . As a result, absorbers initially in state i can contribute to the absorption

transition from i′ to f ′. A similar argument can be made for absorbers initially in state i′ and their

contribution to absorption in the i to f transition [21].

The extent of the processes described above depends on the allowability of collision population

transfers and how efficient the process is between states. It should be noted that line mixing between

different molecules does not occur. Generally, line mixing effects are efficient when the collision


width (∆νc) is of the same order or greater than the transition spacing (νif − νi′f ′). However,

certain situations affect this general criteria. For instance, despite potentially close spectroscopic

positioning, coupling between transitions in different bands or branches is generally less efficient

because the difference between states (e.g. Ei′,v=1−Ei,v=0 ≈ 100 cm−1) is much greater resulting

in less efficient collision population transfers. Another example specific to this work is line mixing

between the spin split sub states of NO. Mixing between the 2Π3/2 and 2Π1/2 subbands can be

considered inefficent because the ineleastic collision transition probabilities for 2Π3/2 ←2 Π1/2 or2Π1/2 ←2 Π3/2 transitions are much smaller than for 2Π1/2 ←2 Π1/2 or 2Π3/2 ←2 Π3/2 transitions

[117–119].

Despite the effects line mixing imposes on spectroscopic line shapes, the equilibrium Maxwell-

Boltzmann population distribution remains preserved by collisions. Detailed balancing of the pro-

cesses (Eq. (6.1)) leads to an important characteristic of line mixing.

ρkRk′←k = ρk′Rk←k′ (6.1)

In Eq. (6.1), ρ represents the relative equilibrium population of a given state and R represents the

population transfer rate between two states. Thus, for the two sides to balance, the rate from the

least populated state must have a more efficient transfer rate. For example, ρk > ρk′ necessitates

Rk←k′ > Rk′←k. As a result, the effect line mixing has on the line shape is to favor transfer to

stronger absorbing regions. For instance, the center of an absorption band will absorb more and the

wings will absorb less than predicted by a model that neglects line mixing (e.g. Lorentzian or Voigt)

as demonstrated by high-pressure infrared absorption spectra measurements in [29, 66].

6.2.1 The Relaxation Matrix, W

Generally, line mixing modeling requires the construction of the impact relaxation matrix, W ,

which describes the influence of collisions on the spectrum’s shape. As discussed in [21, 110,

114], the theory of overlapping and interfering lines is historically attributed to Baranger [120] and

Kolb and Griem [121] before the relaxation matrix formalism was introduced by Fano [122]. In

the typical line shape modeling described in Chapter 2, parameters such as pressure broadening

6.2. LINE MIXING 77

coefficients, γ, and pressure shift coefficients δ are components of the relaxation matrix. It is a

complex matrix with its diagonal components given by

Wk,k = γk − iδk (6.2)

where k represents the kth transition. The off-diagonal components, Wk,k′ (short for Wk←k′),

represent the line mixing interference terms between transitions k′ and k where k 6= k′. The inverse

processes, i.e. Wk′,k, can be determined and vice versa by the detail balance relation [21, 28, 110,

114, 123]

ρkWk,k′ = ρk′Wk′,k (6.3)

More specifically, the line mixing terms represent collision-induced state population transfers. Un-

der conditions where line mixing can be justifiably neglected, the off-diagonal terms of the relax-

ation matrix are zero, and the relaxation matrix can be used to recover the Lorentzian line shape

profile.

Another important characteristic of the relaxation matrix is satisfaction of the sum rule [21, 28,

88] ∑k′

dk′Wk,k′ = 0 (6.4)

with dk′ being the transition dipole moment matrix element of transition k′. The sum rule is accurate

only when the rigid rotor assumption is valid (i.e. no vibrational coupling, etc.) but is still a useful

approximation for modeling. When the rigid rotor assumption breaks down, the sum in Eq. (6.4) is

no longer zero. The diagonal, Wk,k, and off-diagonal, Wk,k′ , terms are related by combining Eqs.

(6.4) and (6.2):

− 1

dk

∑k′ 6=k

dk′Wk,k′ = γk − iδk (6.5)

The utility of this expression lies in relating the real component of the off-diagonal elements,

Re(Wk,k′), to the pressure broadening coefficients, γk, that can be readily determined from ex-

perimental data. Thus, the real component of the relaxation matrix can be constructed by fitting

experimentally determined γ to statistically-based energy gap fitting laws or dynamically-based


scaling laws. The energy gap fitting laws relate the inelastic collision rate between states i and i′

to their lower state energy difference, and the scaling laws are more physical models that include

both spectroscopic and dynamical effects [21]. The use of statistically-based energy gap fitting laws

is discussed in detail in the following section. As for the imaginary part of the relaxation matrix

elements, the pressure shift coefficients, δk, for the diagonal elements can be determined through

experiments like those found in Chapter 5. However, determination of the imaginary off-diagonal

components is difficult, yet these values are expected to be small [112]. Thus, the imaginary off-

diagonal parameters are set to zero and the imaginary part reduces to the pressure shift coefficients.

6.2.2 Constructing W Using Statistically-Based Energy Gap Fitting Laws

The use of statistically-based energy gap fitting laws is an empirical approach to determine

the relaxation matrix, W . The real component of the off-diagonal relaxation matrix elements are

proportional to the rotational state-to-state population transfer rates (or inelastic collision rates)

[21, 28, 110, 114]

Re(Wk′,k) ∝ −Rk′←k, k′ 6= k (6.6)

For isotropic Raman Q branches, the proportionality constant of Eq. (6.6) is exactly 1. However,

for infrared and multibranch spectra the relationship between the population transfer rates and the

relaxation matrix is more complex. In practice, ad hoc proportionality constants are often used

Wk′,k = −AXYRk′←k, k′ 6= k (6.7)

In Eq. (6.7), AXY scales the amount of coupling between branches X and Y and is typically deter-

mined from measured spectra [21, 110, 114].

These rates are modeled by an analytical, temperature-dependent function of the energy gap

between states, ∆Ei′,i′ = |Ei − Ei′ |,

Rk′←k = f(∆Ei′,i, T ) (6.8)

Commonly used energy gap functions are the power gap law (PGL), the exponential gap law (EGL),

6.2. LINE MIXING 79

Table 6.1: Energy gap fitting laws commonly found in the literature.

Name Formula Fit ParametersPower Gap Law (PGL) f = a1(∆Ei′,i/kBT )−a2 a1, a2

Exponential Gap Law (EGL) f = a1exp(−a2∆Ei′,i/kBT ) a1, a2

Modified Exponential Gap Law (MEG) f = a1(1+1.5Ei/a2kBT1+1.5Ei/kBT

)2exp(−a3∆Ei′,i

kBT) a1, a2, a3

Power Exponential Gap Law (PEG)a f = a1(∆Ei′,iB )−a2exp(

−a3∆Ei′,ikBT

) a1, a2, a3

aB in the PEG model is the rotational constant in cm−1.

the modified exponential gap law (MEG), and the power exponential gap law (PEG) [21]. These

fitting laws are defined in Table 6.1 and contain two to three fitting parameters, ai (i = 1, 2, 3). With

the energy gap laws, f(∆Ei′,i, T ), defining the off-diagonal components of the relaxation matrix,

the real diagonal components (i.e. the pressure broadening coefficients, γ) are related to the inelastic

collision rates through a sum rule similar to Eq. (6.5)

γk(T ) =∑k′ 6=k

Rk′←k (6.9)

Thus, the off-diagonal relaxation matrix elements can be determined through fits to pressure broad-

ening coefficients using Eqs. (6.6)–(6.9). It should be noted, however, that Eq. 6.9 is exact only

for spectroscopically unperturbed isotropic Raman Q-branches [21] and is approximate for in-

frared spectra. However, the fitting laws have been applied to infrared spectra with reasonable

success [28, 110, 114]. Additionally, extrapolation and interpolation of rotational-dependent colli-

sion broadening coefficients with fitting laws typically leads to more reliable results than polynomial

fits (see Figure 6.3).

The summation of the inelastic collision rates (Eq. 6.9) captures the lifetime of the lower and

upper states of transition k, i and f . Through the Heisenberg Uncertainty Principle, the uncertainty

in energy is limited by the lifetime of the energy level (∆Ei ≥ h/(2πcτi)). Hence, the linewidth of


Table 6.2: Steps to determine the relaxation matrix, W , via energy gap fitting laws.

1. Assemble a database of broadening coefficients, γ, for transitions in the vibrational bandand branch of interest from either experiments, databases, or the literature. The energy gapmodel will be fit to this data.

2. Assemble a database of lower state energy for all transitions in the band. These are usedto calculate the energy gaps, ∆E = E′i − E′′i , between transitions. Include the lower stateenergy of transitions whose broadening coefficients are unavailable because the energy gapsbetween all transitions are included in the model. For molecular spectra, the transition lowerstate energies can be found in the HITRAN databases [17, 18].

3. Calculate the upward transfer rates, Rk′←k for k′ > k, from the selected energy gap fittinglaw defined in Table 6.1.

4. Use the detail balance expression, Eq. (6.1), to calculate the downward rates, Rk′←k fork′ < k.

5. Sum over all depopulation rates from transition, k, to calculate γk using Eq. (6.9). The γkcalculations are compared to the database assembled in step 1. until reaching convergencevia the selected fitting method.

6. Once converged, the resulting Rk′←k are used in Eq. (6.7) to determine W .

the transition k (neglecting Doppler effects) is

∆ν =1

2πc

(1

τi+

1

τf

)(6.10)

where τi and τf are the lifetimes of the lower and upper levels of transition k, respectively. When

the collision frequency is high enough, energy level lifetimes are dominated by collisions and Eq.

(6.10) can be written in terms of the total collision frequency, Z,

∆νcP

= 2γ ∝ Z

P∝ σi′←iv (6.11)

where σ is the inelastic collision cross section and v is the mean relative speed. Therefore, the

summation of all depopulation rates from state i of transition k is a direct indicator of the state’s

lifetime and thus transition k’s linewidth.

6.3. COMPUTING THE SPECTRAL SHAPES OF INTERFERING LINES 81

The fit parameters, ai, of f(∆Ei′,i, T ) are temperature-dependent parameters and can be deter-

mined from fits to experimental values of pressure broadening coefficients at a specific temperature.

An additional way to model the temperature dependence is with a power law [124]. Since the energy

gap fitting laws can be used to calculate γ, it is reasonable to formulate the temperature dependence

of the fitting laws as a power law

Rk′←k =

(T0

T

)nf(∆Ei′,i, ai) (6.12)

where T0 is the reference temperature (typically 296 K), T is the temperature, and n is an experi-

mentally determined parameter. Thus, temperature independent ai and n can be determined from

fits to experimental broadening coefficients. The steps to determine the relaxation matrix, W , by

applying energy gap fitting laws to pressure broadening coefficient are outlined in Table 6.2.

6.3 Computing the Spectral Shapes of Interfering Lines

The full representation of the absorption coefficient, kν (cm−1), in terms of the relaxation matrix

is given in Eq. (6.13) [21, 112]. Note that this expression is valid within the impact approximation

and neglects Doppler effects.

kν =8π2

3hcν

[1− exp

(− hcν

kBT

)]PXa(

7.34× 1021

T)

×∑k

∑k′

ρkdkdk′Im{〈〈k′|[Σ− La − iPWa,b(T )]−1|k〉〉

}(6.13)

Here, ν (cm−1) is the current wavenumber , T (K) is temperature, P (atm) is the pressure, Xa is the

mole fraction of absorber a, ρk is the relative equilibrium population of the lower state of transition

k, and dk (Debye) is the transition dipole moment matrix element of transition k. Finally, Σ, La,

and Wa,b are operators in the Liouville or line space. Wa,b is the relaxation matrix for absorber a

perturbed by collision partner b. For clarity, the meaning of these parameters and how to obtain

them is explained in the next few paragraphs.

The transition dipole moment has many forms that has led to much confusion over the years


[125, 126]. Here, dk is defined with respect to the Einstein A coefficient, Ak (s−1), as

gfAk =16π3

3hε0ν3kd

2k (6.14)

with gf representing the upper state degeneracy, ε0 representing the permitivity of vacuum, and νk

(cm−1) being the frequency at line center. In Eq. (6.13), dk is in cgs units (i.e. Debye), so for proper

unit conversion, ε0 should be replaced with (4π)−1. For many IR applications, the parameters used

in Eq. (6.14) to calculate dk are available in the HITRAN databases [17, 18]. Additional details for

utilizing the HITRAN databases for this purpose can be found in [126].

The Σ and La Liouville operators are diagonal matrices and related to the current wavenumber

of the computation and the wavenumber of transition line centers, respectively. More precisely,

Σ = νIN (6.15)

and

La(k, k′) = δk,k′ν0,k (6.16)

where IN is the identity matrix withN equal to the total number of transitions, δk,k′ is the Kronecker

delta, and ν0,k is the line position of transition k.

6.3.1 First-Order Approximation

At low pressures when PWk′,k << |ν0,k′−ν0,k| for k 6= k′ is satisfied, line mixing is considered

weak and a perturbation approximation of Eq. (6.13) can be made [21]. If the terms up to first order

are kept, the resulting expression for the absorption coefficient is

k1stν =

1

πPXa

∑k

Sk(T )Im

{1 + iPYk(T )

ν − ν0,k − Pδk(T )− iPγk(T )

}(6.17)

Yk(T ) = 2∑k′ 6=k

dk′

dk

Wk′,k(T )

ν0,k − ν0,k′(6.18)

6.3. COMPUTING THE SPECTRAL SHAPES OF INTERFERING LINES 83

-2 -1 0 1 2

Relative Wavenumber (cm-1

)

0

0.5

1

1.5

Norm

aliz

ed A

bsorb

ance

Lorentzian

Rosenkranz

Lorentz - Rosenkranz

Figure 6.2: Comparison of Lorentzian and Rosenkranz (first-order approximation) line shape pro-files. The difference between the two profiles (i.e. the dispersion shape of the line mixing contribu-tion) is also plotted.

where Sk (cm−2/atm) and Yk (atm−1) are the line strength and first-order line mixing coupling

coefficient (Eq. (6.18)) of transition, k. With respect to the transition dipole moment and Eq. (6.13),

the line strength in pressure normalized units (cm−2/atm) is defined as

Sk(T ) =8π3

3hcν0,k

[1− exp

(−hcν0,k

kBT

)]ρk(T )d2

k

(7.34× 1021

T

)(6.19)

The spectral profile defined by Eq. (6.17) is known as the Rosenkranz profile [127]. A com-

parison between Lorentzian and Rosenkranz line shapes is displayed in Figure 6.2. The difference

between the two profiles is a dispersion shape that is scaled by Yk.

Since the first-order approximation is most useful at sufficiently low pressures, Doppler effects

may be significant depending on the application. A convolution of the complex Lorentzian with a

Gaussian (Doppler) profile leads to [112]

k1stν =

1

πPXa

∑k

Sk(T )2

∆νD,k

√ln(2)

πRe{

(1− iPYk(T ))W(w + ia)}

(6.20)


where ∆νD (cm−1) is the Doppler FWHM andW is the complex probability function defined by

W = e−(w+ia)2erfc(−i(w + ia)) (6.21)

with erfc(x) being the complementary error function and a and w being the usual Voigt line shape

parameters.

a =

√ln(2)∆νC

∆νD(6.22)

w =2√ln(2)(ν − ν0 + Pδ(T ))

∆νD(6.23)

Computationally efficient algorithms forW are available in the literature [22, 116].

The first-order approximation has been used in many line mixing studies [28, 91, 115, 116, 123,

128] and is particularly useful for applications such as atmospheric remote sensing where the micro-

windows between absorption transitions become important. Additionally, the HITRAN database

recently began adding first-order line mixing parameters for some molecules and transitions [17]. It

is also possible to determine Yk directly by fitting measured line shapes to the Rosenkranz profile.

6.3.2 Full Relaxation Matrix Expression

When the relaxation matrix is available, calculations of the spectral line shape with line mixing

can be performed relatively straight forward. Eq. (6.13) can be reformulated into matrix form.

kν =8π2

3hcν

[1− exp

(− hcν

kBT

)]PXa(

7.34× 1021

T)

×∑k

∑k′

ρkdkdk′Im{(

Σ− La − iPW (T ))−1

k,k′

}(6.24)

Rather than performing a matrix inversion for the absorption coefficient at each wavenumber,

Eq. (6.24) can be rewritten for computational efficiency [21, 112, 129]. The complex matrix [La +

iPW ] can be diagonalized such that

[La + iPW ] = V (P, T )D(P, T )V (P, T )−1 (6.25)

6.4. STATIC CELL MEASUREMENTS AND ANALYSIS 85

where D is the diagonal matrix of eigenvalues and V is the corresponding matrix of eigenvectors.

Now, the elements of the inverted matrix in Eq. (6.24) can be rewritten as

(Σ− La − iPW (T )

)−1

k,k′=∑m

V (P, T )k′,m1

ν −D(P, T )m,mV (P, T )−1

m,k (6.26)

Incorporating Eq. (6.26) into Eq. (6.24), the matrix expression for the absorption coefficient be-

comes

kν =8π2

3hcν

[1− exp

(− hcν

kBT

)]PXa(

7.34× 1021

T)× Im

{∑m

Bm(P, T )

ν −D(P, T )m,m

}(6.27)

with

Bm(P, T ) =∑k

∑k′

ρkdkdk′V (P, T )k′,mV (P, T )−1m,k (6.28)

If including Doppler effects is necessary, Eq. (6.20) can be used with equivalent line parameters

defined in [21] and [112]. However, the analysis in this chapter ignores Doppler effects when

comparing experiments to both the line mixing and non-line mixing models (i.e. Lorentzian line

shapes). The omission of Doppler effects is justified for the experimental conditions studied in the

following sections because the ratio of Lorentzian to Doppler widths (L/D ratio) is over 20, which

leads to an error of less than 0.5% between the Voigt and Lorentzian line shape models. Table 6.3

lists the necessary steps to perform the full relaxation matrix expression calculation.

6.4 Static Cell Measurements and Analysis

The data used to build and evaluate the line mixing model discussed in this section include

measurements of collision broadening coefficients (detailed in Chapters 3 and 5) and high-pressure,

high-temperature (HPHT) static cell measurements (detailed in Chapter 4). To construct the re-

laxation matrix for the NO-N2 and NO-Ar systems, measured broadening coefficients were fit by

Eqs. (6.9) and (6.12) with the MEG energy gap fitting law described in Table 6.1. For simplicity,

a single MEG law with a power-law temperature dependence is used. The results of the fitting

are presented in Figure 6.3 and resulting fitting parameters are presented in Table 6.4. The fitting


Table 6.3: Steps to perform the full relaxation matrix expression calculation.

1. Gather the necessary parameters for all transitions affecting the absorption coefficient atthe desired wavenumber: ν0,k, γk, δk, Ak, E′′k , gf , ρk, and dk (if available). For manymolecules, the necessary parameters can be found in the HITRAN databases [17, 18].

2. Assemble the relaxation matrix, W (T ), for the desired temperature and gas mixture. Formulti-component gas mixtures, the relaxation matrix can be written as

PW =∑b

PbWa,b (6.29)

with Pb representing the partial pressure of collision partner b. For a given ab-sorber/perturber combination, the diagonal of the relaxation matrix, Wk,k, is made upof the pressure broadening and pressure shift coefficients and shown in Eq. (6.2). In thiswork, the off-diagonal components are determined by fitting statistically-based energygap fitting laws to experimental collision broadening data as described in 6.2.2. Anothercommon method for determining the off-diagonal components of the relaxation matrix isthe fitting dynamically-based scaling laws to measured pressure broadening coefficients[21, 91, 110, 123].

3. Calculate the transition dipole moments using Eq. (6.14). This step is necessary whenusing HITRAN 2004 [130] or later as the Einstein A coefficients are reported rather thanthe transition dipole moments.

4. Create the other Liouville space operators (matrices) — Σ for the current wavenumber andLa for the transition line centers — using Eqs. (6.15) and (6.16).

5. Apply the computationally efficient version of the line mixing formulation using Eqs.(6.25)–(6.28).

parameters determined for the MEG law are applied to a wide temperature range with acceptable

results. It is clear that the model deviates from the measurements at high temperatures and high J ′′.

As discussed in Chapter 5, the power law is often insufficient over a wide temperature range and the

pressure broadening temperature exponent is also J ′′ dependent. While individual fits to every set

of temperatures can be used instead, no obvious temperature dependence for the fitting parameters

was observed, making a temperature-dependent model of the relaxation matrix built from MEG law

fits at individual temperatures intractable.


0 10 20 30 40 50 60

J"

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

γN

2

(cm

-1/a

tm)

2Π

1/2 Subband MEG Fit

294 K

453 K

618 K

802 K

1000 K

1500 K

2000 K

2500 K

(a)

0 10 20 30 40 50 60

J"

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

γN

2

(cm

-1/a

tm)

2Π

3/2 Subband MEG Fit

294 K

453 K

618 K

802 K

1000 K

1500 K

2000 K

2500 K

(b)

0 10 20 30 40 50 60

J"

0

0.01

0.02

0.03

0.04

0.05

0.06

γA

r (cm

-1/a

tm)

2Π

1/2 Subband

MEG Fit

294 K

453 K

618 K

802 K

1000 K

1500 K

2000 K

2500 K

(c)

0 10 20 30 40 50 60

J"

0

0.01

0.02

0.03

0.04

0.05

0.06

γA

r (cm

-1/a

tm)

2Π

3/2 Subband

MEG Fit

294 K

453 K

618 K

802 K

1000 K

1500 K

2000 K

2500 K

(d)

Figure 6.3: Energy gap law fits to experimentally determined broadening coefficients at severaltemperatures using the modified exponential gap law (MEG).

Relaxation matrices are then constructed for each subband of NO (i.e. 2Π1/2 and 2Π3/2). Inter-

ference between spin-split subbands (i.e. 2Π1/2 →2 Π3/2) is expected to be negligible because their

inelastic collision cross-sections are much smaller than spin-split conserving inelastic collisions as

suggested by [117–119]. Furthermore, for high J ′′, interbranch line mixing between R-P and R-Q

branches is expected to be much smaller than R-R line mixing as shown by scaling law calcula-

tions in [91, 123]. Examples of calculated off-diagonal elements of the relaxation matrix for R-R

line mixing at 296 K are shown in Figure 6.4 with elements for 2Π1/2R(J ′′) ←2 Π1/2R(5.5) and2Π1/2R(J ′′)←2 Π1/2R(15.5).


0 5 10 15 20

J"

0

0.002

0.004

0.006

0.008

0.01

WN

O-X

(cm

-1/a

tm)

X = N2

X = Ar

5 10 15 20 25

J"

0

0.002

0.004

0.006

0.008

0.01

WN

O-X

(cm

-1/a

tm)

X = N2

X = Ar

Figure 6.4: Off-diagonal relaxation matrix elements (Wkk′) for the 2Π1/2 R(5.5) (left) and R(15.5)(right) transitions in the Π1/2 subband at 296 K calculated from the MEG model.

Table 6.4: Modified exponential gap (MEG) law fitting parameters for the NO fundamental R-branch. The fitting parameters are determined using T0 = 296 K.

Subband Collision Partner a1 a2 a3 n2Π1/2 N2 0.0097 1.2869 1.3368 1.25382Π3/2 N2 0.0137 1.4534 1.3705 1.34852Π1/2 Ar 0.0077 1.3798 1.3926 1.27982Π3/2 Ar 0.0123 1.7007 1.4202 1.4148

Measured absorption spectra of NO in N2 are presented in Figures 6.5 through 6.7. At tempera-

tures of 294, 618, and 802 K, the spectra was measured at approximately 5, 20, and 35 atm. For com-

parison with the measurements, absorbance is simulated using Lorentzian line shapes, Rosenkranz

(first-order approximation) line shapes with Yk(T ) calculated from the relaxation matrix via Eq.

(6.18), and the full relaxation matrix expression (Eq. (6.24)). The broadening coefficients for each

simulation are calculated from the temperature-dependent MEG expression and the best-fit param-

eters.

For the spectra at 294 K (Figure 6.5), the most pronounced effects of line mixing are observed as

the number density of molecules is highest and the pressure broadening coefficients are also greatest.

At 5 atm, the difference between model simulations is virtually indistinguishable compared to mea-

surement noise. As pressure increases to 20 and 30 atm, differences between the simulations become

more obvious. At the peak of the R-branch near 1905 cm−1 a simple superposition of Lorentzian


(a) 4.6 amagat, P = 5.0 atm, T = 294 K, XNO = 0.0035, L = 21.3 cm

0

0.5

1

1.5

Ab

so

rba

nce

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.020

0.020.040.06

De

via

tio

n

Lorentzian Rosenkranz Full Calculation

(b) 18.8 amagat, P = 20.1 atm, T = 294 K, XNO = 0.0054, L = 21.3 cm

0

1

2

3

Absorb

ance

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.2

-0.1

0

Devia

tion


(c) 31.7 amagat, P = 34.0 atm, T = 294 K, XNO = 0.0035, L = 21.3 cm

0

1

2

3

4

Absorb

ance

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.8-0.6-0.4-0.2

0

Devia

tion


Figure 6.5: NO spectra measurements in N2 at 294 K and and pressures of 5, 20, and 34 atm.Simulations using Lorentizian line shapes, Rosenkranz line shapes, and the full relaxation matrixexpression are plotted for comparison.


line shapes is no longer adequate to accurately model the observed absorbance. Moreover, the

first-order approximation is also a worse predictor of the observed spectra compared to the full cal-

culation. Recalling the requirements of the first-order approximation — |ν0,k − ν0,k′ | >> PWk′,k

— and the off-diagonal elements of W , we see that requirement becomes more difficult to justify

with increasing pressure. In the wing from 1930 to 1940 cm−1, deviations from the Lorentzian be-

gin to decrease until the Lorentzian begins to overpredict the measured absorbance near 1936 cm−1.

This transition is a characteristic of line mixing that has been observed and calculated for the IR P

and R branches of linear molecules [28, 88, 89, 91, 123, 131] and is the manifestation of line mix-

ing favoring higher absorbing regions. On the other side of the R-branch near the band center near

1880 cm−1, the first-order approximation and the full calculation begin to significantly underpre-

dict the measured absorbance. In this region, justification of the first-order approximation is more

difficult given the closer spacing between transitions at low J ′′. However, other factors leading to

disagreement between the measurements and both the first-order approximation and the full calcu-

lation in this region are the omission of interbranch coupling and the fact that the energy gap fitting

approach inherently neglects elastic collision contributions to the broadening coefficient [21]. A

more physically based approach is the dynamically-based scaling law approach that includes upper

state quanta and angular momentum dynamics of the radiator in its formulation [21, 91, 123]. The

use of dynamically-based scaling laws is an avenue worth exploring; however, sufficiently far from

the band center the energy gap fitting approach incorporated here provides significant improvements

over a superposition of Lorentzian or Voigt line shapes.

The general trends from the room-temperature spectra remain true for the high-temperature

spectra shown in Figures 6.6 and 6.7. For a given number density (represented in amagat in the

captions of Figures 6.5–6.7), we see similar deviations between the Lorentzian model and the data.

For instance, the differences between models are virtually indistinguishable in the 20 atm, 802 K

case (6.8 amagat) as they were for the 5 atm, 294 K case (4.6 amagat). Across all cases, deviations

between the Lorentzian model and the measured spectra become increasingly noticeable above≈ 10

amagat. Another observed distinction between room-temperature and elevated temperatures is the

absence of the characteristic transition from under- to overprediction of the Lorentzian model in

the frequency range studied here. At high temperatures, one would expect this transition point to



0

0.2

0.4

0.6

0.8

Absorb

an

ce

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.02

0

0.02

De

via

tio

n



0

0.5

1

Absorb

an

ce

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.05

0

0.05

Devia

tion



0

0.5

1

1.5

Ab

so

rban

ce

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.1

-0.05

0

Devia

tio

n


Figure 6.6: Measured NO spectra in N2 at 618 K and pressures of 5, 20, and 33 atm. Simulationsusing Lorentizian line shapes, Rosenkranz line shapes, and the full relaxation matrix expression areplotted for comparison.



0

0.1

0.2

0.3

0.4

Ab

so

rba

nce

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.02

0

0.02

0.04

De

via

tio

n



0

0.2

0.4

0.6

Ab

so

rba

nce

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.020

0.020.04

Devia

tion



0

0.5

1

Ab

so

rbance

Measurement

Lorentzian

Rosenkranz

Full Calculation

1880 1890 1900 1910 1920 1930 1940

Wavenumber (cm-1)

-0.05

0

0.05

De

via

tio

n


Figure 6.7: Measured NO spectra in N2 at 802 K and pressures of 5, 20, and 32 atm. Simulationsusing Lorentizian line shapes, Rosenkranz line shapes, and the full relaxation matrix expression areplotted for comparison.

6.5. SHOCK TUBE MEASUREMENTS AND ANALYSIS 93

exist at higher wavenumbers because the population distribution will favor higher energy levels. The

present model also predicts this with the transitions points of the R-branch being near 1956 and 1965

cm−1 for 618 and 802 K, respectively. This argument is further supported by (low-temperature)

temperature-dependent studies of line mixing in infrared CO and CO2 [88, 115].

6.5 Shock Tube Measurements and Analysis

To further evaluate high-pressure spectra and the line mixing model, reflected shock wave ex-

periments were performed in the Stanford High Pressure Shock Tube (HPST) at temperatures and

pressures from 1000 to 2500 K and 10 to 120 atm. The HPST driver section is 3 m long with a

7.62 cm inner diameter, and the driven section is 5 m long with a 5 cm inner diameter. The two

sections are separated by either an aluminum or steel diaphragm that is scored for rupture control.

The driven section is filled with a relatively low-pressure mixtures of NO in N2 prior to filling the

driver section with helium until the diaphragm bursts and generates a shock wave. The incident

shock wave propagates down the length of the driven section, heating and pressurizing the driven

gas. The driven gas is further heated and pressurized after the shock wave reflects off the end wall.

The shock speed is monitored by pressure transducers (PCB Pl 13A) spaced along the length of

the driven section. Conditions behind the reflected shock are calculated from the measured shock

speed and the normal-shock relations. For optical access, window ports with sapphire windows

are located 1.13 cm from the end wall. Additionally, the pressure at this location is monitored by

a Kistler 603B1 pressure gauge. The optical configuration used here is virtually identical to that

described in Chapter 3 except applied to a different shock tube system.

Examples of fixed-wavelength absorption and pressure measurements from high-pressure shock

tube experiments are shown in Figure 6.8. In these experiments, an external cavity quantum cascade

laser was set to 1940.76 cm−1. Simulations using both the Lorentzian and line mixing models are

overlayed for comparison. In Figure 6.8(a), the line mixing and Lorentzian models both agree well

with the measured absorbance, which is expected given that line mixing is weak at the relatively

low pressure and high temperature (1.7 amagat). In (b), the absorbance at a significantly higher

pressure (20.2 amagat) is measured at 1203 K. The Lorentzian and line mixing models are 12%


0

5

10

15

Pre

ssure

(atm

)

0

0.5

1

Absorb

ance

Measurement

Line Mixing Model

Lorentzian Model

-100 0 100 200 300

Time ( s)

-0.1

0

0.1

Devia

tion

Line Mixing Model Lorentzian Model

(a) 1.7 amagat, T5 = 1759 K, P5 = 11.2 atmXNO = 0.0509, ν = 1940.76 cm−1

0

50

100

Pre

ssure

(atm

)

0

0.5

1

1.5

2

Absorb

ance

Measurement

Line Mixing Model

Lorentzian Model

-100 0 100 200 300

Time (µs)

-0.2

0

0.2

Devia

tion


(b) 20.2 amagat, T5 = 1203 K, P5 = 88.9 atmXNO = 0.0203, ν = 1940.76 cm−1

0

50

100

150

Pre

ssure

(atm

)

0

0.5

1

1.5

Absorb

ance

Measurement

Line Mixing Model

Lorentzian Model

-100 0 100 200 300

Time (µs)

-0.2

0

0.2

Devia

tion


(c) 26.5 amagat, T5 = 1217 K, P5 = 118 atmXNO = 0.0104, ν = 1940.76 cm−1

0

50

100

Pre

ssure

(atm

)

0

0.5

1

Absorb

ance

Measurement

Line Mixing Model

Lorentzian Model

-100 0 100 200 300

Time (µs)

-0.1

0

0.1

Devia

tion


(d) 8.9 amagat, T5 = 2480 K, P5 = 80.7 atmXNO = 0.0203, ν = 1940.76 cm−1

Figure 6.8: Measurement traces from fixed-wavelength experiments in a high-pressure shock tube(HPST). The driven gas is NO in N2 at mole fraction specified in the subcaption. Other details ofthe experiment — T5, P5, ν0 — are also specified in the subcaption. All of these experiments aremonitoring ν = 1940.76 cm−1 near the 2Π3/2R(20.5) transition.


and 5.8% less than the measured absorbance, respectively. In (c), the experiment produces a similar

temperature but even higher pressure (26.5 amagat). The Lorentzian and line mixing models are

16% and 7.5% less than the measured absorbance, respectively. Finally, (d) shows an experiment

at 2480 K and 80.7 atm (8.9 amagat) high-temperature experiment. After the bifurcated region [62]

of the shock wave passes near 100 µs, the Lorentzian and line mixing models are 13% and 10%

less than the measured absorbance, respectively. At similar temperatures, the deviations between

measurement and model are consistent across pressures ((b) and (c)). However, in (d) the deviation

(in percentage) is significantly larger despite being at a lower pressure and number density.

Additional shock tube measurements at other wavelengths are summarized in Figures 6.9 and

6.10. The lower pressure experiments in Figure 6.9 agree well with the model at all temperatures,

yet at higher pressures in Figure 6.10, systematic deviations beyond line mixing are apparent. A

possible reason for this is the presence of a non-uniform temperature profile in the boundary layer.

Behind the incident shock a boundary layer grows. It has been shown [53, 132] that during high-

pressure experiments the flow behind the incident shock will transition from laminar to turbulent

resulting in significant growth of the viscous boundary layer. The transition from laminar to tur-

bulent occurs more quickly during higher pressure experiments. Thus, the boundary layer will

affect the measured absorbance more significantly for cases when the reflected shock conditions are

greater than ≈ 20 atm.

Using the results from [53], it is reasonable to assume that the boundary layer isO(1 mm) which

is ≈ 4% of the 5 cm path length. Assuming the thermal boundary layer behind the reflected shock

is of similar magnitude, there may be a significant increase in the absorbance measured across the

shock tube due to the spectral structure and temperature-dependence of the NO fundamental band.

If a simple linear boundary layer model with a thermal boundary layer of 1 mm (see Figure 6.11) is

added to the absorbance simulation, the simulated absorbance agrees more favorably with the mea-

sured absorbance at high pressures (Figure 6.10). At low pressures, the boundary layer is expected

to remain laminar and much smaller at the measurement location, so additional absorbance from

the boundary layer is expected to be negligible. This is consistent with the observations presented

in Figure 6.9 where the line mixing model without a boundary layer corrections agrees with the

measurements.


0

0.5

1

1.5

Ab

so

rba

nce

Line Mixing

Measurements

1880 1900 1920 1940 1960 1980 2000

Wavenumber (cm-1

)

-0.1

-0.05

0

De

via

tio

n

(a) 3.1 amagat, T5 = 1065 K,P5 = 12.1 atm, XNO = 0.0509

0

0.2

0.4

0.6

0.8

Ab

so

rba

nce

Line Mixing

Measurements

1880 1900 1920 1940 1960 1980 2000

Wavenumber (cm-1

)

-0.1

-0.05

0

0.05

De

via

tio

n

(b) 1.9 amagat, T5 = 1800 K,P5 = 12.6 atm, XNO = 0.0509

0

0.1

0.2

0.3

0.4

0.5

Ab

so

rba

nce

Line Mixing

Measurements

1880 1900 1920 1940 1960 1980 2000

Wavenumber (cm-1

)

-0.05

0

0.05

0.1

De

via

tio

n

(c) 1.6 amagat, T5 = 2500 K,P5 = 14.5 atm, XNO = 0.0509

Figure 6.9: Summary of HPST measurements at low pressures. All simulations use the line mixingmodel for absorbance of NO in N2 with the mole fraction specified in the figure subcaption. For themeasurements shown, the temperatures and pressures in the subcaptions are averages of the nominalexperimental conditions.


0

0.5

1

1.5

2

2.5

Ab

so

rba

nce

Line Mixing w\ BL

Line Mixing No BL

Measurements

1880 1900 1920 1940 1960 1980 2000

Wavenumber (cm-1

)

-0.2

-0.1

0

De

via

tio

n

Line Mixing w\ BL

Line Mixing No BL

(a) 21.1 amagat, T5 = 1150 KP5 = 89 atm, XNO = 0.0203

0

0.5

1

1.5

2

Ab

so

rba

nce

Line Mixing w\ BL

Line Mixing No BL

Measurements

1880 1900 1920 1940 1960 1980 2000

Wavenumber (cm-1

)

-0.2

0

0.2

De

via

tio

n Line Mixing w\ BL

Line Mixing No BL

(b) 9.4 amagat, T5 = 1750 KP5 = 60 atm, XNO = 0.0509

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ab

so

rba

nce

Line Mixing w\ BL

Line Mixing No BL

Measurements

1880 1900 1920 1940 1960 1980 2000

Wavenumber (cm-1

)

-0.2

-0.1

0

0.1

De

via

tio

n

Line Mixing w\ BL

Line Mixing No BL

(c) 3.93 amagat, T5 = 2500 KP5 = 36 atm, XNO = 0.0509

Figure 6.10: Summary of HPST measurements at high pressures. All simulations use the line mix-ing model for absorbance of NO in N2 with the NO mole fraction specified in the figure subcaption."BL" denotes the use of a simple boundary layer correction. For the measurements shown, thetemperatures and pressures in the subcaptions are averages of the nominal experimental conditions.


25

Twall

T5

1 mm 2.5 cm

𝟏 mm

2.5 cm

Figure 6.11: Diagram of the simple boundary layer model used to compare the high-pressure shocktube absorbance measurements to the line mixing absorbance model. A half model of the shock tubeis shown with region 5 and region 2 being separated by the reflected shock wave. The boundarylayer is assumed to be 1 mm and the thermal boundary layer is approximated by a linear model withthe two ends of the boundary layer being defined by the shock tube wall temperature, Twall, and thereflected shock temperature, T5.

6.6 High-Pressure NO Thermometry

6.6.1 Implications of Line Mixing

This chapter demonstrates that at sufficiently high gas densities line mixing effects can have

a significant impact on the absorbance of NO. As discussed in Chapter 3, the measured ratio of

absorbance at two distinct wavelengths can be a sensitive thermometer.

R(T, P, ν1, ν2) =α(ν1, T, P )

α(ν2, T, P )=kν1(T, P )L

kν2(T, P )L=kν1(T, P )

kν2(T, P )(6.30)

To investigate the implications of line mixing on laser absorption based temperature measurements,

the absorbance ratio in Eq. (6.30) is calculated for NO in N2 for both the Lorentzian and line mixing

models at wavenumbers near the transitions selected in Chapter 3 (1940.76 and 1986.55 cm−1). In

6.6. HIGH-PRESSURE NO THERMOMETRY 99

Figure 6.12(a), the Lorentzian and line mixing absorption ratios are compared for 90 atm. At all

temperatures, the ratios calculated using the line mixing model are larger than the purely Lorentzian

model. As demonstrated in the absorption measurements presented in Sections 6.4 and 6.5, the

deviations between the Lorentzian and line mixing models are largest at high density, and this is

also true for the calculated absorbance ratios. Thus, the ratios converge and the deviation between

models vanishes with increasing temperature (i.e. decreasing number density). In Figure 6.12(b),

the estimated temperature measurement error is plotted versus temperature for several pressures.

Below 2000 K, the measurement error can be substantial, ranging from 2–10% depending on the

pressure. However, at low gas densities (amagat), the error is more manageable. Nevertheless,

line mixing can play a significant role in the accuracy of temperature measurements utilizing laser

absorption spectroscopy.

1000 1500 2000 2500

Temperature

0

2

4

6

8

10

Ab

so

rba

nce

Ra

tio

Line Mixing

Lorentzian

P = 90 atm

1 = 1940.76 cm-1

2 = 1986.55 cm-1

(a)

1000 1500 2000 2500

Temperature

-12

-10

-8

-6

-4

-2

0

Me

asu

rem

en

t E

rro

r (%

)

10 atm

50 atm

90 atm

130 atm

(b)

Figure 6.12: Implications of line mixing on temperature sensing. (a) The ratio of simulated ab-sorbance at ν1 = 1940.76 and ν2 = 1986.55 cm−1 from T = 1000–2500 K at P = 90 atm. (b) Theestimated temperature measurement error from T = 1000–2500 K at several pressures. Errors arelargest at high number densities when line mixing is strong.

6.6.2 Calculating and Using the Absorbance Ratio for Temperature Measurements

For practical temperature sensing applications, it is desirable to reduce the information presented

to this point into a more compact and easily transferable form. Consider the three-dimensional

surface of the absorbance ratio of Eq. 6.30 versus temperature and pressure for NO in N2, ν1 =


Table 6.5: Steps for measuring temperature using the absorbance ratio surface, R(T, P, ν1, ν2) forν1 = 1940.76 cm−1 and ν2 = 1986.55 cm−1.

1. Calculate the R(T, P ) surface using Eq. (6.31) and Table 6.6 or the line mixing absorbancemodel. This needs to be calculated only once and can be stored for future use.

2. Measure the absorbance ratio, Rmeas = α(ν1)/α(ν2).

3. Measure or estimate pressure in the sensing environment, Pmeas.

4. Isolate the constant pressure ratio curve, R(T, Pmeas, ν1, ν2), corresponding to the mea-sured or estimated pressure, Pmeas, of the sensing environment.

5. Find the temperature, Tmeas, that satisfies R(Tmeas, Pmeas, ν1, ν2) = Rmeas where Rmeasis the measured absorbance ratio.

Table 6.6: Coefficients for the polynomial fit to the absorbance ratio surface in Figure 6.13. Thefifth degree polynomial is defined by Eq. (6.31). Cefficients should only be used with Eq. (6.31) forNO in N2, T = 1000–2500 K, P = 10–130 atm, ν1 = 1940.76 cm−1, and ν2 = 1986.55 cm−1.

Coefficients for Eq. (6.31)p00 p10 p01 p20 p11 p02

2.757 -0.9562 0.04666 0.3123 -0.05422 -0.02151p30 p21 p12 p03 p40 p31

-0.1190 0.003276 0.08150 -0.02944 0.1873 -0.01382p22 p13 p04 p50 p41 p32

-0.005975 0.03401 0.02865 -0.08504 0.009804 0.005927p23 p14 p05

-0.007182 -0.03856 -0.004391


1940.76 cm−1, and ν2 = 1986.55 cm−1. Over the thermodynamic operating space of T = 1000–

2500 K and P = 10–130 atm, this absorbance ratio was calculated using the MEG line mixing model

and the resulting surface is presented in Figure 6.13 (a). Ideally, a lookup table of this surface can be

used for rapid determination of temperature via interpolation and the process described in Table 6.5.

A digital lookup table of the absorbance ratio surface will be made available with this dissertation.

However, if the lookup table is inaccessible to the reader, another method to compute the absorbance

ratio surface is provided in the following paragraphs.

An alternative approach for practically storing and transferring absorbance ratio surface in-

formation is to empirically fit the calculated surface with a polynomial. Doing so reduces future

computations of the absorbance ratio from a complex spectroscopic calculation to one using a few

polynomial coefficients. The absorbance ratio surface in Figure 6.13 (a) was fit by a fifth degree

polynomial. The residuals of the fit are shown in Figure 6.13 (b) and are no greater than 5% with the

largest residuals near the upper limit of the temperature range. The fitting polynomial is presented

in Eqs. (6.31) and (6.32) and the coefficients of the polynomial are presented in Table 6.6.

R(T, P, 1940.76 cm−1, 1986.55 cm−1) = p00 + p10T + p01P + p20T2 + p11T P + p02P

2

+p30T3 + p21T

2P + p12T P2 + p03P

3 + p40T4 + p31T

3P

+p22T2P 2 + p13T P

3 + p04P4 + p50T

5 + p41T4P + p32T

3P 2

+p23T2P 3 + p14T P

4 + p05P5

(6.31)

In Eq. (6.31), T and P represent the normalized temperature and pressure defined in Eq. (6.32). The

temperature and pressure were normalized to better condition the fitting equation. Eqs. (6.31) and

(6.32) and Table 6.6 should only be used for T = 1000–2500 K and P = 10–130 atm.

T =T − 1750

433.3

P =P − 70

35.21

(6.32)

The steps necessary to measure temperature using the polynomial expression for the absorbance


ratio are outlined in Table 6.5. It should be noted that these steps may also be performed by calcu-

lating R(T, P ) with the absorbance line mixing model rather than the polynomial expression of Eq.

(6.31).

(a) (b)

Figure 6.13: (a) Calculated absorbance ratio surface from 1000-2500 K and 10–130 atm for NO inN2, ν1 = 1940.76 cm−1, and ν2 = 1986.55 cm−1. Calculations are performed using the MEG linemixing model presented in previous sections. (b) Residuals of the fifth degree polynomial fit to theabsorbance ratio surface shown in (a).

Before confidently using Eq. (6.31) to calculate the absorbance ratio needed for temperature

measurements, it is important to evaluate how the residuals (see Figure 6.13 (b)) of the fit propagate

error into the temperature measurement. For such an analysis, the steps in Table 6.5 are followed

by replacing Rmeas with the simulated ratio using the line mixing model, Rsim(Tsim, Psim), and

setting Pmeas = Psim. Then, an estimate of the error can be made by comparing the difference

between the output temperature from Table 6.5, Tmeas, and Tsim. Figure 6.14 (b) presents this

comparison as a heat map of the difference between the two temperatures. The maximum deviation

is ≈ 4% near 2500 K and below ≈ 1% at most other conditions. For completeness, Figure 6.14 (a)

shows the R(T, P ) surface calculated from Eq. (6.31). If available, using a lookup table of R(T, P )

generated from the line mixing model is recommended, but the fifth degree polynomial defined by

Eqs. (6.31) provides a convenient alternative without significant sacrifices in accuracy.


(a) (b)

Figure 6.14: (a) The R(T, P, ν1, ν2) surface represented as a heat map and calculated from Eq.(6.31). (b) Heat map representing the error in temperature due to the imperfect fit of the polynomialexpression for R(T, P, ν1, ν2). Maximum error is≈ 4% near 2500 K, and errors at other conditionsare below 1%.

6.6.3 Temperature Measurement Results

As a final demonstration, the high pressure shock tube absorbance measurements presented in

6.5 were used to measure temperature. Experimental traces from two nearly identical reflected

shock wave experiments is shown in Figure 6.15. The average conditions behind the reflected shock

are T = 1172 K, P = 87.7 atm resulting in an average gas density of 20.4 amagat. The top panel

shows the experimental pressure traces with the difference between reflected shock pressures being

2.3 atm. The middle panel shows absorbance traces at ν1 and ν2. To account for the difference

in pressure, the absorbance traces are scaled by the ratio of the average pressure to each respec-

tive reflected shock pressure. Absorbance simulations for ν1 and ν2 are included for comparison.

Finally, the lower panel shows the resulting temperature measurement determined from a look up ta-

ble and the process outline in Table 6.5. For comparison, temperatures calculated from shock speed

measurements are included. After about 100 µs, the average measured temperature is about 60 K

greater than the average calculated temperature. The reason for the difference is apparent in that the

measured absorbances are larger than the simulated values. However, despite the disagreement in

absorbance the measured and calculated temperatures still agree within ≈5%.


-100 0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

Pre

ssu

re (

atm

)

P Run 1

P Run 2

-100 0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

Absorb

ance 1

(1940.76 cm-1)

2 (1986.55 cm-1)

Simulation

-100 0 100 200 300 400 500 600 700 800 900 1000

Time ( s)

800

1000

1200

1400

1600

Tem

pera

ture

(K

)

Measurement

Calc. T Run 1

Calc. T Run 2

Figure 6.15: Top: pressure traces from two nearly identical shock tube experiments with averageconditions behind the reflected shock of 20.4 amagat, P5 = 87.7 atm, T5 = 1172 K, XNO = 0.0203in N2. Middle: absorbance traces for ν1 = 1940.76 cm−1 and ν2 = 1986.55 cm−1. The dashedlines represent the simulated absorbance for the experimental conditions. Bottom: temperaturemeasurement calculated from the ratio of the two absorbance traces. The dashed and dot-dashedlines represent the calculated temperature for each experiment.


In total, six laser absorption based temperature measurements were made in reflected shock

temperatures and pressures between 1000 and 2500 K and 11 and 90 atm, respectively. These mea-

surements are shown in Figure 6.16 and display reasonable agreement with temperature calculations

from shock speed measurements.

500 1000 1500 2000 2500 3000

Calculated Temperature (K)

500

1000

1500

2000

2500

3000

Me

asu

red

Te

mp

era

ture

(K

)

Measurement

Slope = 1

Figure 6.16: Measured temperature versus calculated temperature from reflected shock wave exper-iments in the Stanford High Pressure Shock Tube (HPST).


In Chapter 4, high-pressure measurements of NO absorbance were found to deviate significantly

from simulations using the superposition of individual Voigt line shapes. These deviations were

identified to be the result of collisional line mixing between transitions. In this chapter, the concepts

and theory of line mixing were introduced before a discussion of the impact relaxation matrix and

how to construct it from empirically-determined collision broadening coefficients using energy gap

fitting laws. With a constructed relaxation matrix, two computational strategies — the first-order

approximation and the full relaxation matrix expression — were used to simulate the spectral shape

of the R-branch in the fundamental rovibrational band of NO influenced by line mixing. Comparison

between high-pressure static cell measurements and simulations at three temperatures demonstrated


that line mixing models predicted the measured absorbance more accurately than a Lorentzian-

based model, with the full relaxation matrix expression giving better results than the first-order

approximation. To extend the range of conditions to higher temperatures and pressures, fixed-

wavelength measurements of NO absorbance were made in high-pressure shock tube experiments.

At low pressures, the measurements agreed well with both the Lorentzian and line mixing models as

expected given the low number density. At higher pressures, the measured absorbance significantly

disagreed with the Lorentzian model. However, to a lesser extent, it also disagreed with the line

mixing model. These deviations are likely due to the presence of a thermal boundary layer along

the measurement line of sight in addition to other noise sources in such challenging environments.

With line mixing modeling improvements, NO thermometry at high pressures was evaluated.

Simulations suggest that using a typical superposition of Lorentzian (or Voigt) line shape profiles

when line mixing is present results in estimated errors of up to 10% depending on the thermody-

namic conditions. To provide the reader with the capability to make temperature measurements

using line mixing results, the absorbance ratio surface of NO in N2 for ν1 = 1940.76 cm−1 and ν2

= 1986.55 cm−1 was calculated and a digital lookup table of the calculations is provided with this

dissertation. Additionally, a polynomial model was fit to the surface and the best-fit coefficients

were provided; errors due to the imperfection of the fit were also evaluated. Since the pressure

broadening of pure N2 and air differ by ≈ 3% (as shown in Chapter 5), the present calculations

of the absorbance ratio surface for NO in N2 should be adequate for use in high-temperature air

environments. However, full line mixing calculations for the case with air would slightly improve

the accuracy. Finally, a demonstration of the thermometry technique was presented for a few exper-

iments of NO in N2 in a high pressure shock tube. The measurement results agree favorably with

temperatures calculated from measured shock speeds.

Chapter 7

Conclusions and Future Work


The work presented in this dissertation represents significant progress towards the goal of de-

veloping an accurate temperature sensor for high-enthalpy air. To that end, the dissertation can be

broken up into five primary contributions:

1. In Chapter 3, a novel temperature sensing strategy based on nitric oxide laser absorption was

designed and demonstrated at high temperatures in shock tube experiments.

2. In Chapter 4, as a means to further investigate the infrared spectrum of NO, a new high-

pressure, high-temperature optical cell capable of transmission in the mid-IR up to 8 µm was

designed and demonstrated.

3. In Chapter 5, a thorough analysis of the temperature-dependence of the R-branch in the fun-

damental rovibrational band of nitric oxide near 5.3 µm was performed in the new optical cell

and shock tube experiments.

4. In Chapter 6, high-pressure NO spectra were measured and deviations from conventional line

shape profiles were observed in the form of collisional line mixing. A temperature-dependent

model of the NO spectra was built using measured line shape parameters and statistically-

based energy gap fitting laws. The line mixing model was evaluated by comparing simulations

107

108 CHAPTER 7. CONCLUSIONS AND FUTURE WORK

to high-pressure NO spectra measured in static cell and shock tube experiments. Overall, the

line mixing model found favorable agreement with experiments.

5. Chapter 6 also provides an approach for thermometry in high-pressure air applications. The

construction of the absorbance ratio surface necessary for temperature measurements is pre-

sented and made available to the reader in the form of a digital lookup table paired with this

dissertation and a polynomial model.

The tools used and demonstrated here provide the capability to predict the absorption spectra of NO

over a wider range of thermodynamic conditions. Moreover, the tools applied here can be adapted

to other sensing applications in extreme environments where line mixing contributes to the spectral

shape. Detailed summaries of the preceding chapters are available at the end of each respective

chapter.

7.2 Future Work

7.2.1 Utilizing Dynamically-Based Scaling Laws for an Improved Line-Mixing

Model

A clear limitation of the NO line mixing model presented in Chapter 6 is accurate modeling

of the absorption spectrum near the band center of the NO fundamental band. The inaccuracies

are likely due to the omission of line mixing between R- and P-branches and R- and Q-branches.

Fortunately, the present model remains useful for the purposes of this dissertation because the wave-

lengths selected for the sensing application are sufficiently removed from the band center. However,

a logical next step would be the introduction of dynamically-based scaling laws such as the Infinite

Order Sudden (IOS) and Energy Corrected Sudden (ECS) approximations to account for the molec-

ular dynamics during collisions as described in [21, 91, 123].

7.2. FUTURE WORK 109

7.2.2 Measurements of Full Nitric Oxide Spectra in the High Pressure Shock Tube

In Chapter 6, high-pressure shock tube measurements of NO absorbance at fixed wavelengths

were collected and compared with the line mixing model developed therein. While the model gener-

ally agrees reasonably well with the data, without the spectral structure characteristic of line mixing,

it is difficult to build sufficient confidence in the model based on only a few sparse data points. How-

ever, recent developments in mid-infrared laser technology in the form of rapid-tuning, broad-scan

ECQCLs [71] allow the potential to measure significant portions of the NO fundamental band during

short-duration reflected shock wave experiments at extreme conditions. This additional information

will provide a means to either confirm or modify the line mixing model at high temperatures and

pressures.

7.2.3 Extending the Transmission Range of the High-Pressure, High-Temperature

Optical Cell

The High-Pressure, High-Temperature optical cell described in Chapter 4 is a valuable asset for

studying high-temperature and/or high-pressure spectra in the infrared. As mentioned previously,

the use of CaF2 as the window material allows transmission beyond the limitations of fused silica

or sapphire, particularly at high temperatures. However, optical crystals suffer from multiphonon

absorption processes that limit their transmission range, and these limitations are exacerbated with

increasing temperature. However, other infrared optical materials exist that may be able to extend

the transmission range of the cell at high-temperatures (BaF2, MgF2, ZnSe, etc.). Of course, the

availability of proper materials coupled with creative engineering is necessary.

Appendix A

Uncertainty Analysis of Spectroscopic

Measurements

The contents of this Appendix have been published in the Journal of Quantitative Spectroscopy


Uncertainty analysis was performed for individual data points and for the temperature dependent

data, S and 2γ, via regression analysis. The error bars of individual data points in Figures 3.6 and 3.7

are calculated by addition in quadrature of the uncertainty contributors listed below. Uncertainties

of room-temperature static cell measurements of 2γ and S listed in Tables 3.1 and 3.2 were also

calculated by addition in quadrature. The reported uncertainties of the values in Tables 3.1 and

3.2 for best-fit R(41.5) and R(42.5) S(T0) and best-fit 2γ(1000K) and n were determined from

regression analysis of their respective data sets over the 1000 – 3000 K temperature range.

For the scanned-DA experiments, contributors to uncertainty of individual S and 2γ data points

include uncertainties in:

1. Temperature, T :

(a) Room-temperature static cell experiments: The measurement uncertainty in the sample

temperature was 1 K which contributed 1% to uncertainty in line strength at 296 K.

110

111

(b) Shock tube experiments: For the temperature dependent data, uncertainty in tempera-

ture affects the uncertainty of the fitting parameters (i.e. S(T0), 2γ(T0), and n). As

previously mentioned, thermodynamic conditions behind the reflected shock are known

to within ∼ 1%.

2. Pressure, P :

(a) Room-temperature static cell experiments: The uncertainty in the measured pressure

was 0.5% which resulted in contributions of 0.5% to uncertainties in both S and 2γ.

(b) Shock tube experiments: The pressure uncertainty (in %) from each shock tube exper-

iment contributes the same percent uncertainty to S and 2γ. As previously mentioned,

thermodynamic conditions behind the reflected shock are known to within ∼ 1%.

3. NO mole fraction, XNO: The 2% uncertainty in NO mole fraction contributed 2% and <

0.1% uncertainty in S and 2γ, respectively.

4. Path length, L:

(a) Room-temperature static cell experiments: The path length uncertainty of the room-

temperature optical cell is 1% which contributed 1% uncertainty in S.

(b) Shock tube experiments: The 15.24 cm path length of the shock tube is well-known

and documented in other works using the same facility. Uncertainty in the path length

contributed 1% uncertainty to S.

5. Non-linear regression confidence intervals of the best-fit parameters from the Voigt profile

fits, A and ∆νc:

(a) Room-temperature static cell experiments: The confidence intervals of the best-fit pa-

rameters contributed 0.06 and 0.09% to uncertainty in S and 2γ, respectively.

(b) Shock tube experiments: Fitting parameter confidence intervals for the shock tube ex-

periments were much larger due to the increased residuals of the fit as shown in Fig. 3.5.

Uncertainties in A and ∆νc contributed ∼ 2% uncertainty to S and 2γ, respectively.

112 APPENDIX A. UNCERTAINTY ANALYSIS OF SPECTROSCOPIC MEASUREMENTS

6. NO self-broadening, 2γNO−NO:

(a) Room-temperature static cell experiments: The HITEMP 2010 room-temperature

2γNO−NO values for the R(20.5) transitions have reported uncertainties of 10-20%.

For these measurements, however, uncertainty in 2γNO−NO is a small contributor

to uncertainty in 2γNO−N2 (0.25%) because the mixture is only 1.01% NO by mole

fraction.

(b) Shock tube experiments: NO self-broadening at elevated temperatures is not known

for the R(41.5) or R(42.5) transitions, but since the mixtures are only 2% NO, the

contribution to uncertainty in 2γNO−N2 or 2γNO−Ar from uncertainty in 2γNO−NO

is expected to be low. This was checked by using 2γNO−NO(296K) reported in the

HITEMP database and assuming a temperature exponent nself = 0.75. Furthermore,

the uncertainty of both 2γNO−NO(296K) and nself were assumed to be 20%. The un-

certainty analysis for 2γNO−N2 or 2γNO−Ar resulted in uncertainty contributions from

2γNO−NO of < 1%.

For the fixed-DA measurements of the R(20.5) transition in the shock tube, contributors to

uncertainty of inferred 2γ data points were evaluated via perturbation analysis. These contributors

included uncertainties in:

1. Temperature, T : As mentioned previously, conditions behind the reflected shock are known to

within ∼ 1%, but with increasing temperature the nominal uncertainty (in Kelvin) increases,

leading to increased uncertainty at elevated temperatures. The contribution to uncertainty of

inferred 2γ data points due to temperature uncertainty ranges from 1-3%.

2. Pressure, P : As mentioned previously, conditions behind the reflected shock are known to

within ∼ 1%. The contribution to uncertainty of inferred 2γ data points due to pressure

uncertainty ranges from 0.1-0.3%.

3. Absorbance, α: Uncertainty in the measured absorbance is quantified as the standard devi-

ation of the measurement. As the absorbance changes over the temperature range (see Fig.

113

3.3), the percent uncertainty in absorbance increases resulting in a range of 0.2 – 2% uncer-

tainty contribution to the inferred 2γ data points.

4. NO mole fraction, XNO: The mixtures used in these experiments have 2% uncertainty in

XNO which results in a 2–2.5% contribution to uncertainty of the inferred 2γ data points.

5. Path length, L: The 15.24 cm path length of the shock tube is well-known and documented

in other works using the same facility. The uncertainty contribution to the inferred 2γ data

points due to uncertainty in L (1%) is around 0.8% for all experiments.

6. Laser output frequency, ν: The uncertainty of the wavelength meter cited by the manufacturer

(Bristol) is < 0.001 cm−1 at 5µm. However, the ECQCL output frequency was found to be

slightly unstable. The uncertainty in the frequency was quantified by observing the room

temperature absorbance signal prior to experiments. Since the line shape and line strength at

room temperature are well-known for the R(20.5) transitions, the absorbance signal observed

directly translated to laser frequency uncertainty. This uncertainty was found to be 0.0075

cm−1 which resulted in contributions to uncertainty of 0.4-0.8% for the inferred 2γ data

points.

7. Line strength, S: The uncertainty of the R(20.5) line strengths measured during the static cell

experiment was found to be 2.5%. This resulted in contributions to uncertainty of 4.2-4.8%

for the inferred 2γ data points.

8. NO self-broadening, 2γNO−NO: NO self-broadening at elevated temperatures is not known

for the R(20.5) transitions, but since the mixtures are only 2% NO, the uncertainty in

2γNO−N2 or 2γNO−Ar due to uncertainty in 2γNO−NO is expected to be low. This was

checked by using 2γNO−NO(296K) reported in the HITEMP database and assuming a

temperature exponent nself = 0.75. Furthermore, the uncertainty of both 2γNO−NO(296K)

and nself were assumed to be 20%. The resulting perturbation analysis for 2γNO−N2 or

2γNO−Ar resulted in uncertainty contributions of ∼ 0.1% for the inferred 2γ data points.

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