Transcript Diffraction Basics II: Intensities

Diffraction Basics II:
Intensities
EPS400-002
Introduction to X-ray Powder Diffraction
Jim Connolly, Spring 2012
Intensity 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 scattering
A “peak” is the result of lots of scattering
 Electron Scattering intensity (Thompson Eq.):

IO
I 2
r
2
 e  1  cos2 (2 )

2 
2
m
c
 e 
2
(Io = intensity of incident beam; r = distance from electron to detector; 2 is
angle between scattering direction and incident beam)

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
Atomic 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 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 Fluorescence

When the incident x-ray energy is sufficient to cause photoelectric xray 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 K1 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 Factor
 B sin 2  
f  f 0 exp 

2




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
Where B, the Debye-Waller
temperature factor is:
B  8 2U 2
U2 is directly related to thermal energy, kT
Structure 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
F (hkl )  ( f N ,  N )
N
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
Extinction
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.


Fhkl
m/2
   f n exp 2i(hx j  ky j  lz j
 j 1

 m/2
h
k
l 
    f n exp2i(hx j   ky j   lz j   
2
2
2

  n1

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:
Fhkl 

m/2
2f
j 1
n

exp 2i(hx j  ky j  lz j

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 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
Reflections: Calculation of the Diffraction
Pattern

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 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 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 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 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
The Intensity Equation
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:
I ( hkl ) 
K e K ( hkl ) v
s
Here Ke is a constant for a particular experimental system:
Io  e 


Ke 
2
64r  me c 
3
2
2
where:
I0 = incident beam intensity
r = distance from the specimen to the detector
 = wavelength of the X-radiation
(e2/mec2)2 is the square of the classical electron radius
s = linear attenuation coefficient of the specimen
v = volume fraction of phase  in the specimen
The Intensity Equation (cont.)
Also, K(hkl) is a constant for each diffraction reflection hkl
from the crystal structure of phase :
K ( hkl )
M hkl
 2 F( hkl )
V
2
 1  cos 2 (2 ) cos 2 (2 m ) 


2
sin  cos 

 hkl
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
2m = the diffraction angle of the monochromator
F(hkl) = the structure factor for reflection hkl including anomalous
scattering and temperature effects
The Intensity Equation (cont.)
Putting it all together we get . . .
I ( hkl )
2
I 0   e  M ( hkl )



F( hkl )
2 
2
64r  me c  V
3
2
2
 1  cos2 (2 ) cos2 (2 m )  v


2
sin  cos

 hkl  s

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 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 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:
K

  cos
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 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  <1m
and not significant for
larger sizes
Residual 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.
Mudmaster

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:
•
•
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)