EVALUATION OF THE CALIBRATION SYSTEM

OF AN EBERT-FASTIE SPECTROGRAPH

BENJAMIN R. ARCHER, B.A.

A THESIS

IN

PHYSICS

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of.

MASTER OF SCIENCE

Approved

Director

Accepted^

December, 1970

AtHoiS^

EOS rz 1970 NoJ57 top. 2

ACKNOWLEDGMENTS

I am deeply indebted to Professor Glen A. Mann for his direction of

this thesis and to William H. Almond for his assistance in the prepara­

tion of the instrument.

"V-

11

TABLE OF CONTENTS

LIST OF TABLES iv

LIST OF FIGURES v

I. INTRODUCTION 1

II. INSTRUT'IENTATION 3

Spectrograph 3

Calibration System 6

III. THE 1-0 AND 2-0 INFRARED BANDS OF HCl l6 •

IV. EXPERTtffiNTAL OBSERVATIONS AND RESULTS 29

V. CONCLUSION 35

Sources of Error 35

Recommendations 38

LIST OF REFERENCES ^0

111

LIST OF TABLES

Table I Ca lcu la ted and Observed R Branch Frequencies in cm of

HCl35 Bands 28

Table I I Typical Data for Edser -But le r Band Spacing 32

Table I I I Edse r -But le r Band Reproduc ib i l i t y 33

Table IV Average Uncer ta in ty Per C a l i b r a t i o n Fringe 3^

i v

LIST OF FIGURES

Figure 1 Ebert- Fastie Infrared Spectograph 5

Figure 2 The Calibration System 7

Figure 3 Multiple Beam Interference in a Fabry-Perot Interferometer 12

Figure k Graphs of the Airy Function for Different Values of Reflectance 13

Figure 5 Energy Level Diagram Explaining the Fine Structure of a Rotation-Vibration Band 26

Figure 6 Infrared Absorption Spectrum of the 2-0 Band of HCl • . 27

CHAPTER I

INTRODUCTION

The successful operation of a high resolution infrared spectrograph

requires some suitable means of detemiining the spectral positions with a

high degree of precision. One such method, employed by the infrared spec­

trometry group in the Physics Department at Texas Tech University, in­

volves the use of a Fabry-Perot interferometer. "Calibration fringes"

(Edser-Butler bands or channeled spectra) that result serve to impress a

"wave number scale" on the infrared spectrum to be measured. Ideally,

these calibration fringes should be equally spaced in cm units.

In practice, however, the spacing is not found to be precisely con­

stant. This leads to an uncertainty in the wavelength determination of

spectral lines. In order to determine the accuracy of future infrared

measurements made in this laboratory, the spacing repeatability of the

calibration fringes with particular interferometer plate separations has

been found. Also, the spacings of the fringes for various plate separa­

tions and the associated uncertainties have been determined.

The 1-0 and 2-0 infrared absorption bands of Hydrogen Chloride gas

were chosen as the standards with which to analyze the calibration frin­

ges. HCl was chosen for several reasons. First, both bands occur in the

region of maximum operating efficiency of the spectrograph. At grating

angles less than about 27 or greater than about 33 , adequate calibra­

tion fringes cannot be found without compromising the infrared resolu­

tion of the instrument. Both bands of HCl lie within this region of

1

maximum calibration signal intensity.

Secondly, there are at present two modes of operation in the system

for obtaining infrared data. In the first mode, a transmission filter is

inserted in front of the infrared detector to eliminate all radiation be­

low 2.5 microns. The second mode occurs when the filter is removed. HCl

provides a means of checking fringe consistency for both infrared modes

since the 1-0 band is above and the 2-0 band is below 2.5 microns.

Finally, the infrared spectrum of HCl is easily interpreted and very

precise measurements have been made on both bands under consideration.

CHAPTER II

INSTRUMENTATION

Spectrograph

The data was obtained with an Ebert-Fastie vacuum spectrograph. This

2 instrument is explained in detail in Almond's thesis and is represented

in Figure 1. The infrared source is a carbon arc. A lead sulfide solid

state detector, cooled with liquid nitrogen, is used as the infrared de­

tector. The grating contains 300 lines per millimeter and is blazed for

maximum diffraction of incident radiation of wavelength three microns.

The beam from the carbon arc, continuous infrared radiation, first

passes through the 2 meter path length absorption cell containing HCl.

The infrared spectrum is then scanned by rotation of the grating and is

recorded by one pen of a double-pen recorder. The other pen simultaneous­

ly records the calibration fringes.

The theoretical resolution of the spectrograph at 3 microns is

-1 3

0.05 cm for a single pass of the grating. The resolution is a measure

of the minimum resolvable difference of wavelength at any given wave­

length. In order that two lines be resolved, there must be a dip in the

intensity distribution. A generally accepted criterion for the resolu­

tion of two lines is the Rayleigh criterion. According to this criterion,

two equal lines are considered resolved if the intensity at the saddle

2

poin t i s not g r e a t e r than S/TT t imes the i n t e n s i t y at the two maxima.

The reso lv ing power of a g r a t i ng i s defined by:

R.P. = A/5A ( I l - l )

where 6A is the smallest wavelength difference that produces resolved

images. If the Rayleigh criterion for the resolution is used, the images

must be separated by the angle

66 = A/NdCos0 (II-2)

where N is the number of rulings on the grating, d is the ruling separa­

tion, and 6 is the angle of incident radiation.

The grating formula giving the relation between the wavelength and

angle of diffraction is

mA = dSine (II-3)

where the integer m is the order of diffraction. Differentiation of

(II-3) gives

66 = m6A/dCos0 (II-U)

Combining equations (II-2) and (II-U) the resolving power of the grating

spectrograph is obtained:

R.P. = A/6A = mN (II-5)

_A_ For t h i s instrument ^x ~ 62,U00. Thus, i f the s l i t width i s small,

the resolution at 3 microns, as s ta ted should be 0.05 cm for a single

pass of the grat ing. The actual resolu t ion , determined under convenient

operating condit ions, has been found to be approximately 0.06 cm . These

conditions were that a reasonable s ignal- to-noise r a t io was simultaneously

obtained for both the infrared and cal ibra t ion s ignals .

Source

Vacuum Chamber

Figure 1. Ebert-Fastie Infrared Spectrograph

Calibration System

For ca l ibra t ion of the spectrograph, two beams of radiation are sent

through the grat ing optics simultaneously. The beam from the carbon a r c ,

as mentioned above, i s detected by the infrared detector . The second

beam consists of radiat ion from a 300 watt zirconium arc that i s focused

on the entrance s l i t of the monochromator. This resu l t s in nearly mono­

chromatic l igh t emerging frora'the exit s l i t . The emergent beam traverses

the Fabry-Perot interferometer af ter which i t passes through the Wadsworth

prism system and f a l l s on an RCA 1P21 photomultiplier tube. The Edser-

Butler bands tha t resu l t when the interferometer i s swept with white l igh t

that has been diffracted by the grating are detected by the phototube and

recorded by the second pen of the double pen recorder. These cal ibrat ion

fringes are recorded simultaneously with the infrared spectrum.

The ca l ibra t ion system must contain the means of overcoming any e r ­

rors introduced by non-uniform rotat ion of the grat ing, as well as supply

a uniform set of indicator marks which provide a means of determining the

frequency of the spect ra l l i n e s . Both of these functions are performed

effect ively by the use of Edser-Butler bands obtained with the instrumen­

ta t ion shown in Figure 2. Since both the infrared aad cal ibrat ion beams

t r ave l through the same grating optics simultaneously, any s l ight non-uni­

form rotat ion of the grat ing wi l l affect both beams in exactly the same

manner. For example, a minute hes i ta t ion in the rotat ion of the grating

wi l l r esu l t in a s l igh t ly expanded separation of two adjacent infrared

absorption l i n e s . However, t h i s separation is exactly compensated for

u Q)

•H H ft

•H -P I O

o PH

0)

:3

-p o ^ <u

PH 1

^ <u +3 a) S o ^ (U

>>«H ^ S cd f^

^ (U +J C!

o

o u

J=i o o o

o

o •H -P •H CO

o PH

(U ,Q ;3

-P O

o

(U

m >>

CO

O •H +J

•H

O

<u

EH

OJ

•H

P4

a w >>

CD

•H

8

by a similar increase in spacing in the calibration fringe system. Thus,

the calibration fringes do provide a convenient means of calibration as

well as of overcoming anom.alies in the rotation of the grating.

The reliability and reproducibility of the fringes depend primarily

on the characteristics of the Fabry-Perot interferometer. The interfero­

meter used consisted of two partially reflecting circular plates, 1 1/8

inch in diameter. The surfaces of the plates were flat to 1/6 of the

wavelength of green light (56OO angstroms).

The plate spacing was mechanically varied and could be fixed at

1mm, 2mm, 3mm, or ilmm by inserting the proper magnesium spacer in order

to allow no further plate separation than that desired. Once this de­

sirable separation was obtained, it was necessary to make the plates

accurately parallel. To accomplish this, a sodium arc was used to il­

luminate the interferometer. This produced a series of concentric rings

which became readily visible to the eye when the plates were nearly paral­

lel. When the plates become precisely parallel, and if they are exactly

plane, the rings remain the same size as the eye is moved to any point

of the field of view.

Once the plates had been aligned, the sodiimi arc was removed and

the monochromatic light from the exit slit was positioned to strike the

Interferometer at the center of the interference rings. The physical

size of the central ring depended upon the plate separation - the smaller

the separation, the larger the size of the ring.

Before calibration fringes can be obtained, it is necessary to sep­

arate out the multiple diffraction orders that are obtained from the

grating. Each different frequency that passes through the interferome­

ter will, if it is dispersed, produce its own series of equidistant maxi­

ma. However, if the orders are not separated, the resulting output will

consist of the average intensity of all signals combined and will there­

fore be highly periodic in amplitude and spacing. It is thus necessary

to incorporate the Wadsworth prism system in order to separate the orders.

This system consists of.-a 60° prism and a plane mirror introduced

into the beam path after it has traversed the interferometer. The beam

enters the prism at the angle of minimum deviation and is dispersed ac­

cording to wavelength, thus elimj.nating the overlapping of the various

orders. The plane mirror is positioned to reflect the dispersed orders

back on a line parallel to the beam incident on the interferometer.- Each

of the separated images contains the central spot of the interference

pattern produced by the interferometer. This enables the phototube to

be positioned so as to observe the central ring of only one of the dis­

persed orders. This arrangement is shown in Figure 2.

The central ring, as well as all others in the concentric series,

is due to the multiple beam interference utilized in the Fabry-Perot in­

terferometer. The primary beam is partially reflected and partially

transmitted at the first surface. The transmitted part is then reflected

back and forth, as shown in Figure 3, between the two surfaces. If r is

the coefficient of reflection and t the coefficient of transmission,

2 2 2 then the amplitudes of successive transmitted rays are E t , E t r , o o

2 k E t r , where E is the amplitude of the incident beam. The geometric o o

path difference between any two successive transmitted rays is 2dCos9,

10

where d is the separation between reflecting surfaces, and 6 is the angle

either ray makes with the surface normal. The corresponding phase dif­

ference 6 between two successive rays is then given by

6 = 2kdCos0 = dCosG (II-6) A

where k = 27T/A. Taking t h i s phase d i f f e r e n c e i n t o account by i n c l u d i n g

t h e f a c t o r e and adding t h e ampl i tudes of a l l of t h e t r a n s m i t t e d r ays

y i e l d s :

E t ^ ^ ^2 ^ ^2 2 16 ^ ^2 U 2i6 ^ o ,^^ „x

E^ = E t + E t r e + E t r e + . . . = •;--rT ( I I - 7 ) • T o o o _ 2 10

1-r e The i n t e n s i t y of t h e t r a n s m i t t e d l i g h t i s given by

1-r e

I 12 where I = E is the intensity of the primary beam.

o ' o' Since a change of phase can occur on reflection, r is, in general a

k complex number and can be expressed as

r = Irle^'^^/^ (II-9)

where — i s t h e phase change fo r one r e f l e c t i o n . Fu r the rmore , i f t h e r e ­

f l e c t a n c e of one s u r f a c e i s denoted by R and t h e t r a n s m i t t a n c e by T, t h e n

R = | r | ^ , T = | t | ^ (11-10)

Now ( I I - 3 ) may be w r i t t e n as

T =: I _ T ^ ,-^ ( I l - l l ) T o , x2 ^R o- 2/A.

(1-R) 1 + o 2^^ ^9^ (1 -R)^

11

where

A = 6 + 6^ (11-12)

i s t he t o t a l phase d i f fe rence between two success ive t r ansmi t t ed beams.

The i n t e n s i t y , t h e r e f o r e , va r i e s with A according t o t h e funct ion:

ITR ' p A ~ o~A~ (11-13) 1 + - ^ Sin^(f) 1 4- F Sin^(l-)

(1-R)^ •' ^ ^

which is known as the "Airy function.' Here

F = — ^ (II-IM (1-R)^

is a measure of the sharpness of the interference fringes. This does

not refer to the "calibration fringes" recorded as wavelength standards

but only to the concentric ring series produced by a non-varying source .

The Airy function is shown in Figure k for various values of the

reflectance R. These curves show the intensity distribution of the frin­

ges in multiple beam interference. If R is small, the interference frin­

ges are broad and indistinct, whereas if R is close to one, the fringes

are very sharp.

The maximum and minimum values of I are

T^ I r ^ = 1 T(max) o ^_J^)2

= I —^-TT (11-15) T(min) - o ^^^^)2

«

Calibration fringes are produced by what is referred to as the

"scanning method." The rotation of the grating causes the center spot

12

E

Figure 3. Multiple Beam Interference in a Fabry-Perot Interferometer

13

o •H -P O

K

o CO

;=J

-p

c

• H

O

O

•H

a

;:<

•H <

O

m

ft ccJ

(D

w •H

H /q.Tsua:^ui

Ik

to alternate in intensity. This variation at the ring center, recorded

photometrically, is due to the fact that the monochromatic light, inci­

dent on the interferometer at an angle 6, changes by a small amount AA

from A to A±AA as the grating rotates. In order to examine this varia­

tion fully it is necessary to first develop the theory of the interfer­

ence fringes produced by a single wavelength A. When a ray undergoes

two.reflections, it has a ph^se difference of

6 = -- n (2dCos0) (II-I6) A a

from a ray that travels through unreflected, where n is the index of a

refraction of air and is equal to one, 9 is the angle of incidence of the

incident ray, d is the plate separation, and m is the order number. For

constructive interference that results in a bright fringe occurring, 6

must be an integral multiple of 2IT. For an intensity maximum, therefore

mA = 2dSin6 (11-17)

Considering the order number m as constant, d i f ferent ia t ion of

(11-17) y i e lds :

+mAA = T2dSin0A0 (II-I8)

This equation shows that the result of increasing A is a shrinking of

the ring system and that a decrease in A causes an expansion of the rings

When white light is used, A is being continually varied. At those

wavelengths A = ^^^^^ for which m is an integer, maxima will occur. At m

such wavelengths, the intensity of light from the central ring (given in

equation (II-I5)) will be appreciable. This large signal will be regis-

15

tered by the photomultiplier tube and recorded as a calibration fringe

maximum on the chart recorder. As the wavelength continues to increase,

the interference fringe system will collapse, causing a gradual decrease

in intensity of the beam section observed by the phototube until I / . x

given in (11-15) is reached. The next maxima will occur when the wave­

length change results in the order nmnber being changed by one. The

calibration fringes are thus'"a graphical record of the intensity pattern

recorded at the center of the ring system. The distribution is essen-

tially a plot of the Airy function (l + F Sin -)

The spacing of the fringes can now be determined by finding the

difference in cm" corresponding to two adjacent orders of interference.

From the form of the Airy function it is evident that this is given by

setting

A , - A = 27r (11-19) n+1 n

then

A = 6 + 6 (11-20) n n r

where

6 = UTT/A dCose (11-21) n n

but V = 1/A gives the cm"" value of A. After substitution, it is n n '

found that the fringe spacing is given by:

V .. - V n+1 n 2dCose

(11-22)

CHAPTER III

THE 1-0 AND 2-0 INFRARED BANDS OF HCl

In order to attribute justifiable precision to the calibrated wave

number scale it is necessary to have accurate theoretical information on

the individual HCl lines used as standards. Thus, the theoretical infra­

red spectrum must be calculated.

• _

For a diatomic molecule-'there are two modes of motion that are of

primary importance in determining its infrared spectrum. First, the

molecule can rotate as a whole about an axis passing through the center

of gravity and perpendicular to the internuclear axis, and second the

atoms can vibrate relative to each other along the internuclear axis.

To investigate what type of spectrum would be expected for such a

vibrating or rotating system, on the basis of quantum theory, one begins

with the so-called rigid rotator model. Consider the two atoms to be a

distance r apart, separated by a massless rigid rod.

In order to determine the possible energy states of such a rigid

rotator the Schrodinger equation must be solved.

2

1 ' (Sine -^) . - V — ^ - E^ = 0 (III-I) Sine 88 ' dB' „. 2. ..2 ,2 r

Sin 6 d4» h

where I is the moment of inertia of the molecule. The solution of this

7 equation, carried out in Pauling and Wilson, is found to be

^ = N P,(">(Cose)ei"* (III-2)

^r r J

where ip represents the rigid rotator eigenfunctions and N^ is a nor­

malization constant. The energy eigenvalues are

16

17

y2 E = - ^ I • J(J+1) (III-3)

87T

where the rotational quantum number J can take the integral values

0,1,2,. . .

The absorption of a light quantum of the proper frequency produces

a transition from a lower to a higher rotational level. The wave number

of the absorbed radiation is

V = E'/hc - E"/hc (Ill-ii)

where E' and E" are rotational energies in the upper and lower states,

respectively. Define F(j) as the rotational energy such that

F(J) = E/hc = —~—J(J+1) = BJ(J+1) (III-5) 87T cl

where

B = ^ 87T^Cl

Thus,

V = F(J') - F(J") = BJ'(J'+1) - BJ"(J"+1) (III-6)

In order to calculate the frequencies that are actually absorbed,

it is necessary to know the selection rules for the quantum number J.

These rules are found by evaluating the m.atrix elements of the dipole

moment. If M is the constant dipole moment of the rotator, its com-o

ponents are

M = M SineCosd) X o ^

M = M SineSin* (III-7) y o ^ M^ = M^Cose .

18

The matrix elements to be evaluated are of the form

^ J'M'J"M" K. 1

*/'"' M^,/""" d. (III-8)

where i = x,y,z.

o

Evaluation of these elements show that they vanish except when

AJ = J'-J" = ±1.

Hence

V = F(J"+1) - F(J") = B(j"+l)(j"+2) - B(J"+1)

= 2B(J+1) J = J" = 0,1,2,. . . (III-9)

Thus, the spectrum of the simple r ig id ro ta tor consists of a ser ies

of equidistant l i n e s , the f i r s t at 2B (J=0). The separation of succes­

sive l i ne s i s also 2B.

The simplest possible assumption about the form of the vibrat ions

in a diatomic molecule i s that each atom executes simple harmonic motion.

If the masses of the atoms are replaced by y = TiLm/m+m^, the reduced

mass, and the displacement of each atom from i t s equilibrium pos i t ion ,

TQ and rgp, by the d i s to r t ion of the distance between the two pa r t i c l e s

from the equilibrium bond length, then the vibration of the two pa r t i c l e

system may be t r ea t ed as a single vibrat ing p a r t i c l e . The Schrodinger

equation for the vibrat ions of a diatomic molecule with reduced mass y

i s

7^f^^I^\-^\ (iii-io) OlT y dX

19

the solut ion functions can be writ ten as

o —l/2ax

K " V H^(v^) V = 0 ,1 ,2 , . . . ( I l l - l l )

where v i s the v ibra t ional quantum number, N^ i s a normalizing constant ,

H^ i s the Hermite polynomial of degree v, and

a = 27T/yk/h

The allowed energies are given by the expression^

E^ = (v+l) hv^AT

o^ E^ = (v+l") ho) V = 0,1,2, . . .(III-12)

Since ^ = "2:^ /k/y i s the c lass ica l frequency of vibrat ion. In

order to express the resul t of (III-12) in cm" , the energy values are

transformed into "term values" (G(v)). Thus,

G(v) = E(v)/h = a)(v+|) ( III-13)

where cu i s measured in cm

The o s c i l l a t o r can be set in vibration by absorption of l ight of

the proper frequency. This frequency i s given by

V = G(v') - G(v") (III-IU)

In order to find the par t i cu la r t r ans i t ions that may occur, the

matrix elements of the dipole moment must be evaluated, t h i s time using

the vibrat ional wave fun ct ions. I t is found that the selection rule for

the v ibra t ional quantum number of the harmonic osc i l l a to r i s Av = +1.

20

Av = v' - v" = +1 for absorption

so

- V = G(v + 1) - G(v) =00 (III-15)

Thus, all of the allowed transitions give rise to the same frequency.

The frequency of radiated light is equal to the frequency of the oscilla­

tor, regardless of what the Value of the initial state may be.

If the theoretical results obtained so far, are compared with the ob­

served spectra of HCl, the following interpretation results. The spectrum

in the far infrared, since it consists of nearly equidistant lines, is a

rotation spectrum. The spectrum in the near infrared, since it consists

essentially of a single very intense line, is a vibration spectrum. On

this assumption the occurrence of bands with nearly two and three times

the frequency would be connected with deviations from the harmonic oscil­

lator.

Thus, the main characteristics of the infrared spectra are explained

by the models of the rigid rotator and the harmonic oscillator. However,

to obtain the finer details of the spectra, the models must be refined.

The rotator must be treated as nonrigid and the oscillator as anharmonic.

According to equation (III-IO), an harmonic oscillator is character­

ized by a parabolic potential curve. In the actual molecule, however,

when the atoms are a great distance apart, the attractive force is zero,

and the potential energy is constant. This is not consistent with the

parabolic potential function in which the potential energy increases in­

definitely with increasing distance from the equilibrium position.

21

As a f i r s t approximation to the actual potent ia l energy function

of the molecule a cubic term i s added to the quadratic function of the

harmonic o s c i l l a t o r . Thus,

U = f ( r - r ^ ) ^ - g ( r - r^ )^ ( I I I - I6 )

Here, the coefficient g is very much smaller than f and r is the equi­

librium separation.

If (III-I6) is substituted into the Schrodinger equation as the po­

tential function, the energy values of the anharmonic oscillator are

given by

E^ = hcw^Cv-t^) - hca)^x^(v+|-)^ + hca)^y^(v+|-) + . . .(III-I7)

and correspondingly the term values are

% = ^e^"""^^ " %\^^^^^ '' '^e^e^'^'i^^ + . . . ( I I I - I8 )

1 / ^ where the constant w x <<a) and oj y <<co x . Here oj = 7 and w x e e e e e e e 27TC y e e

and 0) y are anharmonicity constants. e' e "

Equation (III-I8) shows that the energy levels of the anharmonic

oscillator are not equidistant like those of the harmonic oscillator,

but that their separation decreases slowly with increasing v. However,

in order to consider the spectrum more fully, the selection rules must

be considered. They are

Av = 1,2,3

22

The t r ans i t i ons with Av=l are expected to give r i se to the most intense

t r a n s i t i o n s . I t can be seen that the t rans i t ions with Av=2,3,. . .

have approximately but not exact ly, two, three . . . times the frequency

of the t r ans i t i on Av=l. In spi te of the s l ight deviation from the c las ­

s i ca l r e s u l t , the 1-0, 2-0, 3-0, . . . are referred to as the fundamental,

f i r s t overtone, second overtone, and so on. The formula that gives the

posi t ions of the v ibra t ional - levels above the lowest vibrat ional level i s

V = G(v) - G(o) = w V - 0) X V + OJ X v- -(111-19) o o o o o

Thus far the models of the rigid rotator and the harmonic (or an­

harmonic) oscillator have been used independently of each other. How­

ever, it is obvious that the molecule cannot be a strictly rigid rotator

where it has the capability of vibrating. Therefore, a new model, in

which the atoms are joined by a massless spring instead of a rigid rod

will be considered. This model is called the nonrigid rotator.

In this system, an increase in angular speed will increase the cen­

trifugal force which in turn will increase the internuclear distance and

consequently the moment of inertia. To a good approximation, the rota­

tional terms of the nonrigid rotator are given by

F(J} = BJ(J+1) - DJ^(J+1)^ (III-20)

where D is the centrifugal distortion constant and D<<B. The selection

rule derived previously AJ = ±1 is valid whether or not the rotator is

rigid. Using (III-20) the spacing of lines is given by

23

V = F(J+1) - F(J) = 2B(J+1) - 4 D ( J + 1 ) ^ ( I I I - 2 1 )

The l i n e s are no longer e q u i d i s t a n t , as for a r i g i d r o t a t o r , but t h e i r

s epa ra t ion decreases s l i g h t l y with inc reas ing J .

To t h i s po in t t h e v i b r a t i o n and r o t a t i o n of t he molecule have been

regarded as be ing t o t a l l y independent. However, observed data show

s t rong i n d i c a t i o n s t h a t r o t a t i o n and v ib ra t ion take place simultaneously.

Hence, i t i s necessary t o consider t he v i b r a t i n g r o t a t o r .

During t h e v i b r a t i o n the i n t e r n u c l e a r dis tance and consequently the

moment of i n e r t i a and the rota . t ional constant B are changing. The r o t a ­

t i o n a l constant B i n t he v i b r a t i o n a l s t a t e v i s given by V

B = B - a (v-i^) + . . . ( I I I -22 ) V e e 2

where a i s a constant which i s small compared to B . Analogously, D , Q C V

representing the influence of centrifugal forces, is given by

D = D + 3 (v4-) + . . . (III-23) V e e ^ 2'

IfR 3 Here 3 <<D = , which r e f e r s t o t he completely v ib r a t i on l e s s s t a t e e e ^

We The r o t a t i o n a l terms in a given v i b r a t i o n a l l e v e l are

F ( J ) = B^J(J+1) - D^J^(J+1)^+ . . . (III-2I1)

The term values of a v i b r a t i n g r o t a t o r , obtained by t ak ing in to account

t he i n t e r a c t i o n of v i b r a t i o n and r o t a t i o n , are

2U

T = G(v) + F (J)

= 0) , ( v 4 ) - a)^xjv-f |)2 + . e e

+ B^J(J+1) - D^J^(J+1)^+. . .

( I I I - 2 5 )

Since the eigenfunct ions of the v i b r a t i n g r o t a t o r are e s s e n t i a l l y

the products of r o t a t o r and o s c i l l a t o r func t ions , the same se lec t ion

r u l e s hold as for these systems i n d i v i d u a l l y . Thus, v can change by any

i n t e g r a l amount although Av=+1 gives by far the most in tense t r a n s i t i o n .

The value of J can only change by un i ty .

I f a t r a n s i t i o n from v' t o v" i s considered, the loca t ions of the

r e s u l t i n g l i n e s are given by

V = v^ + B , J ' ( J ' + 1 ) - D , J ' 2 ( J ' + 1 ) 2 _ B „J"(J"+1)

+ D „J"^(J"+1)2 ( I I I -26 )

where v = G ( V ' ) - G(v") , the frequency of the pure v ib r a t i ona l t r a n s i ­

t i o n . I f t he AJ=+1 t r a n s i t i o n i s defined as the R branch and the AJ=-1

as t he P branch the r e s u l t i n g l i n e s are

>R = ^^ + (2B ,-kl) , ) + (3B ,-B „-12D , ) J + (B ,-B „-13D ,+D „ ) j ' n O V V V V V V V V V

- ( 6 D , + 2 D „ )J -^ - (D ,+D „)J V V V V

( I I I -27 )

vp = v^ + (B ,-B „)J + (B ,-B ,,-D ,-D „)J^ + (-2D ,+2D „)J^ r O ^ V V V V V V V V

- (D ,+D „ ) J V V

(III-28)

25

A single formula can be used to represent the two branches - namely

V = v^ + (B^,+B^.,)m + (B^,-B^„-D^,+D^.,)m2

-2(D^,+D^„)m2 - (D^,-D^„)m^ (III-29)

where m is an integral running number that takes the values 1,2, . . .

for the R branch (that is, m=J+l) and the values -1,-2, . . . for the P

branch (m=-J). Thus, there is a single series of lines for which a line

is missing at m=0. The missing line at v=v is called the band origin.

o ^

F i g u r e 5 i s an energy l e v e l diagram t h a t e x p l a i n s t h e f i n e s t r u c t u r e

of a. r o t a t i o n - v i b r a t i o n band. In g e n e r a l , t h e s e p a r a t i o n of two v i b r a ­

t i o n a l l e v e l s i s much g r e a t e r t h a n t h e spac ing of t h e r o t a t i o n a l l e v e l s

shown i n t h e f i g u r e . The schemat ic spec t rogram below t h e f i g u r e r e p r e ­

s e n t s t h e e x p e c t e d v i b r a t i o n - r o t a t i o n s p e c t r a . F igure 6 shows t h e a c t u a l

i n f r a r e d s p e c t r a of HCl fo r t h e 2-0 band. The s h o r t e r peaks r e p r e s e n t 37 35 .

t h e i s o t o p e HCl . The e v a l u a t i o n , however , was done only wi th HCl

The c a l c u l a t i o n of t h e s e l i n e p o s i t i o n s as w e l l as a c c u r a t e e x p e r i ­

men ta l measurements , have been c a r r i e d out and p u b l i s h e d by D. H. Rank.

His r e s u l t s a re d i s p l a y e d i n Table 1 fo r v a l u e s of J up t o and i n c l u d i n g

J=10 .

26

v=2

v=0

J'

5 -

k -

1 -0 -

J"

5

1 0

r 'T

CM O H

PH

OJ

PH

oo P

. ^

PH

LTN

P-.

h

..A..—-—a.

I 1 ji • »-jfc-y

V.

Figure 5 Energy Level Diagram Explaining the Fine Structure of a Rotation-Vibration Band

o

o

pq

o I

CM

(U

- P

O

- P o (U

ft CO

o •H -p P^ U O CO

(U

u

M

27

>

5 rz:> n _ z>

13-___ i_

L_

U

•H

-i-

28

TABLE I

CALCULATED AND OBSERVED R BRANCH FREQUENCIES IN CM " OF HCl" ^ BANDS

1-0 Band

J (Ca lc )

R(J) (Obs)

Calc - Obs

X 10"^

0

1

2

3

h

5

6

7

8

9

10

2-0 Band

0

1

2

3

k

5

6

7

8

9

10

2906.2I179

2925.8977

29i|I | .9ili6

2963.2866

2981.0015

2998.0)476

30lU.U83^

3030.0876

30U5.0592

3059.3171

3072.8509

5687 .651^

5706.0951

5723.3033

5739.26^3

5753.9665

5767.3989

5779.5510

5790.U121

5799.9725

5806.2225

5815.1527

.2521

.8950

.915^

.2865

.0013

.0I438

.lilli^

.0862

.0569

.3179

.8U90

.61+98

.0932

.302U

.2651+

.9673

.3980

.51^98

.U1U7

.9688

.2205

. l i^8l

-U.2

+2 .7

- 0 . 8

+ 0 . 1

+0.2

+ 3 .8

+2.0

+l.k

+ 2 . 3

- 0 . 8

+1.9

+1.6

+1.9

+0.9

- 1 . 1

- 0 . 3

+0.9

+1.2

- 2 . 6

+3.7

+2.0

+I1.6

CHAPTER IV

EXPERIMENTAL OBSERVATIONS AND RESULTS

In analyzing the data, great care was taken to justify the degree

of accuracy that is presented. At least four independent spectra were

made with each spacer. The results for each spacer were determined us­

ing both the 1-0 and 2-0 bands of HCl as standards. A comprehensive

error analysis, as described in Errors of Observation and Their Treat­

ment , was then made. A brief indication of this analysis follows.

If X , Xp, . . • J x^ represent the averages of fringe spacings be­

tween adjacent HCl absorption peaks in a single spectrum, the most pro­

bable value is the arithmetic mean.

X = Cx^ + x^ + . . • x^)

N (lV-1)

The s tandard e r r o r of the mean i s — , where a i s the s tandard devia-

t i o n for the se t of measurements. The bes t es t imate of a i s found t o be

- (x^-x) + (x^-x) + . . . (x^-x) a2 = _ 1 ^ 1 (lV-2)

N-1

Hence, the measured spacing is reported as " x + ".

Table 2 shows typical data obtained for the 1-0 and 2-0 bands.

The data presented was obtained with the Umm spacer. The first column

indicates the absorption peak number of the HCl. The second gives the

separation of adjacent peaks in fringe numbers. The separation of

these peaks in cm""'' is listed in the next column. In the final column,

the fringe spacings in cm" are tabulated. The most probable value of

29

30

fringe spacing and the standard error of the mean for each band is also

reported.

Table 3 shows the reproducibility of fringe separation. The follow­

ing procedure was used to determine this reproducibility; (l) the spac­

er \mder investigation, was inserted and the interferometer plates made

parallel, (2) the spectrum was run, (3) the spacer was removed, (U) the

interferometer plates were closed down and then reopened to accomodate

the spacer again, (5) the plates were realigned and the next spectrum

was made. It was found that after replacing the spacer the interfero­

meter plates did not assume the original parallel position. In some

cases, when using the larger plate separations, the interference pattern

as viewed with the sodium arc had entirely disappeared. The adjustment

necessary to reacquire the pattern was small, however, as was the inher­

ent error in the reproducibility. It was also found that the interfero­

meter plates could not be adjusted to precise parallelness. The probable

reasons for and the effect of this will be discussed in the following

chapter.

The ratio of the average spacings of the fringes in the 1-0 band

with respect to the 2-0 band are listed in column three of Table 3.

These ratios are approximately constant. A value in the neighborhood of

.58 will encompass all the ratios if the inherent uncertainty in each is

considered. This indicates that the separation in cm of a pair of

fringes used with the infrared region above 2.5 microns will represent

about .58 of the separation in cm" of the same pair of fringes used

with the region below 2.5 microns. The 2.5 micron division is due, as

31

previously explained, to the fact that the 1-0 band was examined using

the 2.5 micron infrared filter, while the 2-0 band was examined without

it.

Table k lists the average uncertainty per fringe for the various

spacings. These were compiled by computing the fringe spacing uncerta.in-

ty in each spectrum and then finding the average of these uncertainties

for*each spacer.

32

TABLE I I

TYPICAL DATA FOR EDSER-BUTLER BAND SPACING

{hzam Spacer )

1-0 Band

Pe'ak

No.

R(0)

R ( l )

R(2)

R(3)

R(l+)

R(5)

R(6)

R(7)

R(8)

HCl

F r i r

S e p a r a t i o n s in

ige Numbers

97 .52

9^^.66

9 1 . 3 1

88.22

Qk.86

8 1 . lU

77 .98

7^^.32

HCl S e p a r a t i o n s in

cim*~-'-

19.6ii98

19.0169

18.3720

17.71I+9

17.0U61

16.3658

15.67I42

l i [ . 9 7 l 6

F r inge Spacing i n

cm"-'-

.2015

.2009

.2012

.2008

.2009

.2017

.2010

.20lii

Most P robab le Value: .2012 + .0002

2-0

R(o;

R( l -

R(2;

R(3 .

Bik,

R(5;

R(6;

R(7;

R(8;

Band

) 53.15

) 1+9.75

) i+6.12

) 1+2.73

) 38 .73

) • 35.02

I 31 .38

) 27.1+2

18.1+1+37

17.2082

15.9610

IU.7022

I3.I+32I+

12 .1521

10 .8611

9.560I+

t P robab l e Value :

.31+71

.31+59

.31+61

.31+1+1

.31+68

.31+70

.3961

.3U87

.31+65 ± .0006

33

TABLE III

EDSER-BUTLER BAND REPRODUCIBILITY

Spacer

1 mm

2 mm

3 mm

k mm

Band . *

1-0

.582 ± .051+

.367 ± .007

.2588 ± .0008

.2006 ± .0001+

2-0

1.050 ± .Ol+l

.650 ± .006

.1+1+91 ± .0006

.31+62 ± .0002

Ratio ^ ^

552 ± .069

56I+ ± .016

5762 ± .0026

579^ ± .0015

TABLE IV

AVERAGE UNCERTAINTY PER CALIBRATION FRINGE

Band Spacer

1-0 2-0

31+

1mm .0016 .0031

2mm .0006 .0011

3mm .0005 .0009

i+mm .0003 .0008

CHAPTER V

CONCLUSION

Sources of Error

The data displayed in Tables III and IV indicates that the best

possible accuracy that may be obtained with the present instrumentation

results when the 1+ mm spacer is used. The accuracy drops off so consi­

derably for separation of 1 mm and 2 mm that these spacers do not ap­

pear to be suitable for infrared calibration. This increase in uncer­

tainty occurs because the large widely-spaced calibration fringes that

result when the 1 and 2 mm spacers are used, are not suited to the pre­

cise evaluation that high resolution analysis requires. The exact cen­

ter line, for example, of the large, rounded peaks produced is difficult

to find. Whereas, the centers of the sharper, pointed peaks obtained

with k mm separation are readily ascertained. Furthermore, a small par­

titioning error in a fringe produced with small separation leads to much

greater uncertainty than the same error produced with large separations.

This is a result of the fact that there are many more calibration fringes

between the same HCl peaks, when larger separations are used.

There are other factors that contribute to the uncertainties also.

There are variations between Rank's calculated and observed values for

the HCl absorption lines. Table I shows that the average difference is

about .002 cm" . This indicates that the uncertainty is not entirely due

to fringe irregularities but is partly a result of instrumental inaccura­

cies that are to be expected when observing the positions of the infrared

of HCl. 35

36

Another source of error mentioned in the preceding chapter is that

the plates could not be adjusted to exact parallelness. This problem

appears to be due to the surface defects of the Fabry-Perot plates,

which according to Chabbal,^^ can significantly change the properties of

the interferometer. The ordinary optical flat of 1/1+ wavelength flat­

ness is not good enough for precise work. A flatness of the order of

1/20 to 1/100 wavelength is strongly recommended. As stated previously,

the plates used in obtaining the data presented had a listed flatness of

1/6 wavelength - clearly a minimal value for interferometer plates.

A final, and probably the smallest source of error is due to the

fact that the temperature of the interferometer plates and spacer was

not regulated. During periods in which the data was taken, both were ex­

posed to the temperature of the room. The magnesium spacers used in the

-6

interferometer had a coefficient of linear expansion of about 26 x lO" 1/C

The maximum possible change in temperature that was likely to occur dur­

ing a data run was about 5°. This amounts to a maximum change in length

of the spacer of about .005 mm. On the basis of many observations of the

behavior of the interference pattern with small displacements of the

plates, it seems reasonable to conclude that distortion due to tempera­

ture expansion is small compared to the other errors mentioned.

The ratio of fringe spacing (from Table III) , as discussed in the

prior section, was found to be about O.58. This can be attributed to a

change in the angular dispersion of the grating for different orders of

diffraction. The theoretical angular separation between two spectral

37

lines differing in wavelength by AX is

^^ " dCose

where n is the order number and d is the grating spacing. Obviously,

therefore, the dispersion of the 2-0 band should be about twice that of

the 1-0 band. The fact that the ratio is not exactly 0.50 indicates

that the dispersion in the second order falls off from the expected

value.

38

Recommendations

Despite the above listed sources of error in the present calibra­

tion system, its accuracy is nearly equivalent to or surpasses many of

the present techniques employed for wavelength calibration. It is, how­

ever, very far from the accuracy claimed by Professor T. H. Edwards at

13 Michigan State University. Using fused quartz plates aluminized to

ahout 80^ reflectivity and a fused quartz spacer, fringe spacing was de­

termined to ±0.000001 cm""''.

It is interesting to note some of the characteristics of this in­

terferometer. Fused quartz is used because it has one of the lowest

-6 n coefficients of linear expansion, .1+2 x 10 l/C . The importance of

the high reflectivity was shown in Figure 1+, the graph of the Airy func­

tion. As can be seen, the fringes become sharper for large values of R.

It is doubtful that the cost involved in this kind of accuracy is

necessary in the Ebert-Fastie system discussed in this thesis. However,

if the resolution is improved, which can be accomplished by the multiple

II+

pass system described in Almond's thesis, an improvement in the cali­

bration system would seem desirable. This could be accomplished by con­

struction of a Fabry-Perot interferometer of high reflectance using high

quality plates and fused-quartz spacers as described in Born and Wolf.

For general infrared calibration, a spacer of about 3-5 mm would be de­

sirable. This would allow good accuracy as well as permit excellent frin­

ges to be readily obtained — a combination that is difficult to achieve

with 1+ mm separation.

39

Since t h i s study was completed, high, opt ical quali ty plates have

been obtained and the construction of a Fabry-Perot e talon, with the

spacing recommended in t h i s work, wi l l soon be undertaken.

LIST OF REFERENCES

1. D. H. Rank, B. S. Rao, and T. A. Wiggins, "Molecular Constants of HCl-^^ J. Mol. Spectry., IJ (1965) , pp. 122-130.

2. W. H. Almond, "The Construction and Calibration of an Ebert-Fastie Vacuum Spectrograph," (Texas Technological College, 1966, Unpub­lished M.S. Thesis).

3. Ibid. , p. 22.

h. G. R. Fowles , I n t r o d u c t i o n t o Modern Op t i c s ( H o l t , R i n e h a r t , and Wins ton , I n c . , New York, I 9 6 8 ) , pp . 8 6 - 9 0 .

5 . M. Born and E. Wolf, P r i n c i p l e s of Op t i c s (The Macmillan Company, New York, I96I+), p . 326.

6 . K. W. M e i s s n e r , " I n t e r f e r e n c e S p e c t r o s c o p y , " J . Opt. Soc. Am., 13 (19I+I) , pp . I+O5-I+O9.

7 . L. P a u l i n g and E. B. Wi l son , I n t r o d u c t i o n t o Quantum Mechanics (McGraw-Hill Book Company, I n c . , New York, 1 9 3 5 ) , pp . 263-27I+.

8 . G. H e r z b e r g , S p e c t r a of Diatomic Molecules (D. Van Nost rand Company I n c . , T o r o n t o , 1 9 5 0 ) , p . 7 2 .

9 . I b i d . , p . 7 5 .

10. Rank, Rao, and Wiggins, op. cit.

11. J. Topping, Errors of Observation and Their Treatment (Chapman and Hall, London, I961).

12. R. Chabbal, "Finesse Limite D'un Fabray-Perot Forme de Lames Impar-fartes," J. Phys. Rad. , 1£ (1958), pp. 295-300.

13. D. H. Rank, K. N. Rao, C. J. Humphreys •> Wavelength Standards in the Infrared (Academic Press, New York, I966) , p. 16O.

II+. Almond, op. cit. , pp. 22-21+.

15. Born and Wolf, op. cit. , p. 330.