Diffraction Basics II:Diffraction Basics II:IntensitiesIntensities

EPS400-002

Introduction to X-ray Powder Diffraction

Jim Connolly, Spring 2012

Intensity variations in Diffraction DataIntensity variations in Diffraction Data

Locations of diffraction peaks are related to d-spacings of families of lattice planes

Relative intensities of diffraction peaks can yield information about the arrangement of atoms in the crystal structure

Intensities may be measured as– Peak heights– Areas under peaks (minus background)

By convention, the strongest peak (in a pattern or of a particular phase in a pattern) is assigned an intensity of 100, and all others are reported in proportion to it.

Intensity = the vector sum of all scatteringIntensity = the vector sum of all scattering

A “peak” is the result of lots of scattering Electron Scattering intensity (Thompson Eq.):

2

)2(cos1 22

2

2

2

cm

e

r

II

e

O

Shows inverse square relationship to intensity Is classical electron radius term Is the polarization factor indicating that the Incident

beam is polarized by the scattering process

(Io = intensity of incident beam; r = distance from electron to detector; 2 is angle between scattering direction and incident beam)

Atomic ScatteringAtomic Scattering Scattering from an atom

will be the sum of all electron scattering, defining the atomic scattering factor, fo

Scattering at 0° will be the sum of electron scattering from all electrons

At higher scattering angles, quantum approximations are very complex

fo is calculated for all atoms in the “International Tables for Crystallography” Vol C.

A Graph of Atomic Scattering FactorsA Graph of Atomic Scattering Factors Plot of fo for several

elements at right

Plot shows atomic scattering factor vs. Sin/

Denser atoms scatter with greater intensity (with some variation related to oxidation state)

Plots show how intensity decreases as the scattering angle increases

Anomalous Scattering and FluorescenceAnomalous Scattering and Fluorescence

When the incident x-ray energy is sufficient to cause photoelectric x-ray production in a target atom, that atom is said to fluoresce.

The photoelectric x-rays produce scattered x-rays slightly out of phase with other scattered x-rays resulting in a reduced intensity.

Result is the “absorption edge” phenomenon in which peak intensity is attenuated with certain x-radiation for particular elements.

Target (Anode) Element

of K1 in

Angstroms

Elements with strong fluoresence

Cr 2.2909 Ti, Sc, Ca

Fe 1.9373 Cr, V, Ti

Co 1.7902 Mn, Cr, V

Cu 1.5418 Co, Fe, Mn

Mo 0.7107 Y, Sr, Rb

Thermal FactorThermal Factor

2

2

0

sinexp

B

ff228 UB Where B, the Debye-Waller

temperature factor is:

Qualitatively, as T increases, B increases and scattering intensity decreases

Shown for Cu at right

Calculations are very complex and “standard” values not well agreed upon

U2 is directly related to thermal energy, kT

Structure FactorStructure Factor The structure factor is a means of grouping the atoms

in the unit cell into planar elements, developing the diffraction intensities from each of those elements and integrating the results into the total diffraction intensity from each dhkl plane in the structure.

It describes total scattering intensity

),()( NNNfhklF

Where F(hkl) is the structure factor for the hkl reflection of the unit cell, f is the atomic scattering factor for each of N atomic planes and is the repeat distance between atomic planes measured from a common origin (called the phase factor)

It is most easily visualized as an addition of vectors (next slide)

In simplified 2D, scattering of dhkl is the vector sum of scattering from atomic planes containing atoms P, R & Q

Note how the vector sum is always less than the sum of the scalar values

The actual 3D calculations are extremely complex

ExtinctionExtinction In certain lattice types, the arrangement and spacing of lattice planes

produces diffractions from certain classes of planes in the structure that are always exactly 180º out of phase producing a phenomenon called extinction.

For a body-centered cubic cell, for each atom located at x, y, z there will be an identical atom located at x+½, y+½ , z+½. The structure factor Fhkl is represented by the following equation.

222(2exp(2exp

2/

1

2/

1

llz

kky

hhxiflzkyhxif jjj

m

nnjjjn

m

jhkl F

If h + k + l is even, the second term will contain an integer. An integral

number of 2’s will have no effect on the value of this term and Fhkl is: jjjn

m

jhkl lzkyhxif

(2exp22/

1

F

If h + k + l is odd, the second term will contain an term including 2 (n/2)

and the scattering vector will be 180º out of phase, and Fhkl = 0.

Systematic ExtinctionSystematic Extinction Systematic extinction is

as a consequence of lattice type

At right is table of systematic extinctions for translational (i.e., non-rotational) symmetry elements

Other extinctions can occur as a consequence of screw axis and glide plane translations

Accidental Extinctions may occur resulting from mutual interference of other scattering vectors

Symmetry Extinction Conditions

P none

C hkl; h + k = odd

B hkl; h + l = odd

A hkl; k + l = odd

I hkl; h + k + l = odd

F hkl; h, k, l mixed even and odd

21 ║ b 0k0: k = odd

C b h0l: l = odd

Key: P = primitive lattice; C, B, A = side-centered on c-, b-, a-face; I = body centered; F = face centered (001)

Factors affecting intensities of Bragg Factors affecting intensities of Bragg Reflections: Calculation of the Diffraction Reflections: Calculation of the Diffraction

PatternPattern

Consideration of all factors contributing to relative intensities of Bragg Reflections allows calculation of theoretical diffraction pattern for phases whose structures are well characterized

Calculated patterns in the ICDD Powder Diffraction File are created from this information

A qualitative understanding of these factors is very useful in interpreting your experimental results

The Plane Multiplicity FactorThe Plane Multiplicity Factor

The number of identically spaced planes cutting a unit cell in a particular hkl family is called the plane multiplicity factor. – For high symmetry systems, the factor can be very high– For low symmetry systems, it is low

As an example, each cubic crystal face has a diagonal (110) and an equivalent plane. With six faces, there are 12 crystallographic orientations. The (100) will similarly have 6 orientations. Thus, the (110) family will have twice the intensity of the (100) family because of the multiplicity factor.

The Lorentz FactorThe Lorentz Factor As each lattice point on the

reciprocal lattice intersects the diffractometer circle, a diffraction related to the plane represented will occur.

As angles increase, the intersection approaches a tangent to the circle; thus at higher angles, more time is spent in the diffracting condition.

This increased time will be a function of the diffraction angle and may be corrected by inserting the term I/(sin2 cos) into the expression for calculating diffraction intensities.

This term is called the Lorentz factor. In practice, this is usually combined with the atomic scattering polarization term (Thompson equation) and called the Lorentz polarization (Lp) correction.

Extinction from Strong ReflectionsExtinction from Strong Reflections A phase-shifted reflection can occur from the

underside of very strongly reflecting planes. This will be directed towards the incident beam but 180 out of phase with it.

The net effect is to reduce the intensity of the incident beam, and therefore the intensity of the diffraction from that plane.

A similar phenomenon will reduce the penetration of the beam into strongly diffracting planes by reducing the primary beam energy.

Corrections are very difficult to calculate or estimate. Careful sample preparation and a uniform small

crystallite size (~ 1 m) will reduce this effect. This effect can still reduce the experimental

intensities of the strongest reflecting peaks by up to 25%.

Other Absorption EffectsOther Absorption Effects Absorption

– Related to Depth of Penetration: In a diffractometer, at low 2 values a larger area of sample is irradiated with less depth of penetration. At higher 2 values, the irradiated area is smaller, but depth of penetration greater. In general, these tend to be offsetting effects as related to diffracted intensity over the angular range of the data collection.

– Linear Absorption: Calculated intensities include a term for 1/s where s is the linear absorption coefficient of the specimen.

Microabsorption

– Occurs in polyphase specimens when large crystals preferentially interact with the beam causing both anomalous absorption and intensities not representative of the proportions of the phases.

– The effect is minimized in diffraction experiments by decreasing the crystallite size in the specimen

s

hklehkl

vKKI

)(

)(

2

2

23

64

cm

e

r

IK

e

oe

The Intensity of diffraction peak from a flat rectangular sample of phase in a diffractometer with a fixed receiving slit (neglecting air absorption), may be described as:

Here Ke is a constant for a particular experimental system:

where:I0 = incident beam intensityr = distance from the specimen to the detector = wavelength of the X-radiation(e2/mec

2)2 is the square of the classical electron radiuss = linear attenuation coefficient of the specimenv = volume fraction of phase in the specimen

The Intensity EquationThe Intensity Equation

The Intensity Equation (cont.)The Intensity Equation (cont.)

Also, K(hkl) is a constant for each diffraction reflection hkl

from the crystal structure of phase :

hkl

mhkl

hklhkl F

V

MK

cossin

)2(cos)2(cos12

222

)(2)(

where:

Mhkl – multiplicity for reflection hkl of phase

V = volume of the unit cell of phase the fraction in parentheses equals the Lorentz and polarization corrections for the diffractometer (Lp)hkl, including a correction for the diffracted beam monochromator

2m = the diffraction angle of the monochromator

F(hkl) = the structure factor for reflection hkl including anomalous

scattering and temperature effects

Putting it all together we get . . .

The Intensity Equation (cont.)The Intensity Equation (cont.)

shkl

mhkl

hkl

ehkl

vF

V

M

cm

e

r

II

cossin

)2(cos)2(cos1

64 2

222

)(2

)(

2

2

230

)(

This equation may be used to calculate the diffraction pattern for an “ideal” powder.

Calculations may be done relatively easily for simple structures. A computer (with specialized software) is required for anything else.

Most real-world variations from ideal intensities are related to anisotropic effects in your specimen . . .

Small disordered cells are referred to as amorphous

Crystallites show sufficient extent of repeated unit cells to display diffraction.

Random orientation of crystallites produce the ideal diffraction pattern

Preferred orientation of crystallites will produce a significant distortion of diffraction intensities from the ideal.

Preferred OrientationPreferred Orientation Many materials exhibit preferred orientation as a characteristic

property of the material.

These include many types of ceramic magnets, extruded wires, most pressed powders and many engineered films and polymers. Study of these materials usually require use of a special pole-figure diffractometer to measure a particular single diffraction.

Preferred orientation of powders is the most common source of deviation from the “ideal” pattern in your diffraction data

Special specimen preparation techniques may be utilized to minimize effects in powders

Recognition of the effects may be used to successfully identify oriented phases in a powder pattern

Pattern fitting techniques (i.e., Reitveld) may be used to compensate in quantitative analysis

Specialized analytical methods (i.e., for clays) make use of preferred orientation for specialized structure analysis

Crystallite Size EffectsCrystallite Size Effects Extremely small crystallites will produce incoherent

scattering from the edges of the crystals. For crystallites containing large numbers of unit cells, these edge effects are minimized, and the diffraction pattern approaches the “ideal”

The incoherent scatter can result in broadening of the resultant peak, described by the Scherrer equation:

cos

K

Where is the mean crystallite dimension, K is the shape factor (typically about 0.9), is the wavelength, and is the line broadening (equal to the B – b, B being the breadth of the observed diffraction line at its half-intensity maximum, and b the instrumental broadening)

Crystallite Size vs. Instrument BroadeningCrystallite Size vs. Instrument Broadening Instrument broadening

is the “baseline” line width shown by any diffraction peak

It is a characteristic of the instrument and not related to the specimen

Crystallite or particle size broadening is related to deviation of crystallite size from ideality

Broadening related to crystallite size is measurable at <1m and not significant for larger sizes

Residual Stress and StrainResidual Stress and Strain Strain in the crystal lattice will

produce a distortion of the diffraction line

Macrostrain causes the lattice parameters to change resulting in a peak shift. Glycolation or heating of clay minerals are examples of induced macrostrains.

Tensile and compressive forces can produce Microstrains resulting in a broadening of the diffraction peaks or peak asymetry in some cases.

Dislocations, vacancies, shear planes, etc can produce Microstress. The result can be a distribution of peaks around the unstressed peak location, appearing like a crude broadening of the peak.

MudmasterMudmaster An excellent tool for understanding particle size and

strain effects (and calculating them) in the diffraction pattern is “Mudmaster” from Dennis Eberl

Reference: “Mudmaster: A Program for Calculating Crystallite Size from Distributions and Strain from the Shapes of the Diffraction Peaks” by Dennis Eberl, et al., U.S. Geological Survey Open-File Report 96-171, 1996.

Program uses Microsoft Excel (with Solver and Statistical add-on tools installed) to do calculations

Available by free download via FTP at: ftp://brrcrftp.cr.usgs.gov/pub/ddeberl/MudMaster/ or ftp://eps.unm.edu/pub/xrd/MudMast.zip

Next time:Next time:

• Clay Minerals, mineralogy and X-ray diffraction (Guest lecture by Dr. Dewey Moore)

• Review Q&A Session prior to hour exam (Exam two weeks from today)