Part 8' Diffraction: Intensity

1
Part 8. Diffraction Intensity (From Chapter 4 of
Textbook 2 and Chapter 9 of Textbook 1)
? Scattering by an electron An electron
which has been set into oscillation by an
X-ray beam is continuously accelerating and
decelerating during its motion and therefore
emits an EM wave ? being said to scatter
x-rays. The emitted X-ray has the same
frequency (wavelength) as the excited X-ray
beam ? i.e. coherent. (the phase change on
scattering from an electron is ?/2) The
intensity of the scattered beam is angular
dependent. J.J. Thomson the intensity I
of the beam scattered by a single electron
of charge e (C) and mass m (kg), at a
distance r (meters) from the electron, is
given by
P
r
?
2
?0 4??10-7 mkgC-2
z
Incident beam wavevector along x-axis, the
electric field lies on the yz plane. Electron at
O. Diffracted beam at P on xz plane, i.e.
diffraction plane is xz. Every E can be
resolved into components Ey and Ez.
P
r
2?
y
O
Random polarized
x
On average, Ey Ez, since E is perfectly random.
The y component accelerates the electron in
the direction Oy ? the intensity of the scattered
beam at P is
3
? ?yOP ?/2
The z component accelerates the electron in
the direction Oz ? the intensity of the scattered
beam at P is
? ?zOP ?/2 -2?
The total scattered intensity at P is
Polarization factor
r ? ? I ? ? ? ? I?
If a monochromator is used with the Bragg angle
?M ? the polarization factor for powdered
crystalline sample becomes
It depends on geometry and sample condition
4
Another kind of scattering Compton
scattering when X-ray encounter loosely bound or
free electrons incoherent scattering,
wavelength changed, no fixed phase
relation belongs to scattering
loss (attenuation). X-ray before impact h?1 (?1),
after Compton scattering h?2 (?2).
h?1
e
h?2
2?
e
? Atomic scattering (or form) factor
Consider the physics of coherent scattering by
atoms. The atomic scattering factor is the
ratio of the scattering from an atom to that
from a single bound electron.
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Assume that each electron has a spherical charge
density ?(r) around the atomic nucleus and that
the X-ray wavelength ? is not too close to the
absorption edge. The amplitude scattered to the
plane P from a small volume of the charge density
of one electron, dV, is given by ?(r)dV. s0, s
unit vectors presenting the incident and
scattered wave The path different between
the wave scattered at O and at dV is R-(x1 x2).
x2
x1
dV
r
O
s
s0
2?
R
The differential atomic scattering factor is
defines by
Ee the magnitude of the wave from a bound
electron
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Spherical integration, the integration element is
dV 2?r2sin? d? dr
dV
?
S-S0
r
S
S0
Evaluate (S - S0)?r S - S0rcos? (S -
S0)/2 sin?.
S
S-S0
S0
2?
Let
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For n electrons in an atom
Tabulated
For ? 0, only k 0 ? sinkr/kr 1.
Number of electrons in the atom
equal to 1 bound electrons
If one consider the absorption of the X-ray, the
atomic form factor is expressed as a complex
number. The calculation above is the real part of
the factor. The imaginary part (absorption) is
also tabulated for different materials and X-ray
source.
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? Scattering by a unit cell Next, we have to
consider the diffracted beam from a group of
atoms that can make up the crystal. Atoms
representing unit cell ? crystal Diffracted
beam from the cell ? diffracted beam from the
crystal Find out the phase difference from
the group of atoms representing a unit cell.
1
1?
3
3?
The path difference between 11? and 22? is NCM.
A
?
2
2?
S
R
B
a
C
N
M
(h00)
The path difference between 11? and 33? is SBR.
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? the phase difference between 11? and 33? is
If the position of atom B is specified by the
fractional coordinate (normalized to the
magnitude of the base vector), u x/a.
Extending the argument to other two direction,
the atom B has actual coordinates x y z or
fractional coordinates x/a y/b z/c ( u v w),
respectively. The phase difference between the
wave scattered by atom B and that scattered By
atoms A at the origin for the hkl reflection
If u v w is integer, no phase difference.
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Waves differing in amplitude and phase may also
be added by representing them as vectors.
A3
?2
A2
A1
?1
?3
The analytical expression for a
vector representing a wave
i
A
Any scattered wave can be expressed in the
complex exponential form
?
real
The resultant wave scattered by all the atoms of
the unit cell is called the structure factor (F).
F is obtained by simply adding together all the
waves scattered by the individual atoms. Assume
the unit cell contains atoms 1, 2, 3, , N, with
fractional coordinates u1v1w1, u2v2w2, , uNvNwN.
?
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F is, in general, a complex number. F
amplitude of the resultant wave in terms of the
amplitude of the wave scattered by a single
electron.
? Structure factor calculations
How to choose the groups of atoms to represent a
unit cell of a structure? First, determine the
number of atoms in the unit cell. Second, choose
the representative atoms for a cell properly
(ranks of equipoints).
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? Example 1 Simple cubic There are 1 atoms
per unit cell, 000 and 100, 010, 001, 110,
101, 011, 111 are all equipoints of rank 1.
Choose any one will have the same result.
for all hkl
? Example 2 Body centered cubic There are 2
atoms per unit cell, 000 and 100, 010, 001,
110, 101, 011, 111 are all equipoints of rank 1.
Another equipoints of rank 1 is ½ ½ ½. Two
points to choose are 000 and ½ ½ ½.
when hkl is even
when hkl is odd
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? Example 3 Face centered cubic There are 4
atoms per unit cell, 000 and 100, 010, 001,
110, 101, 011, 111 are all equipoints of rank 1.
Another equipoints of rank 3 is ½ ½ 0, ½ 0 ½,
0 ½ ½, ½ ½ 1, ½ 1 ½, 1 ½ ½. Four atoms chosen
are 000, ½ ½ 0, ½ 0 ½, 0 ½ ½.
when h, k, l is unmixed (all evens or all odds)
when h, k, l is mixed
? Example 4 Diamond Cubic There are 8 atoms
per unit cell, 000 and 100, 010, 001, 110,
101, 011, 111 are all equipoints of rank 1.
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Another equipoints of rank 3 is ½ ½ 0, ½ 0 ½,
0 ½ ½, ½ ½ 1, ½ 1 ½, 1 ½ ½. The other
equipoints of rank 4 is ¼ ¼ ¼, ¾ ¾ ¼, ¾ ¼ ¾, ¼
¾ ¾. Eight atoms chosen are 000, ½ ½ 0, ½ 0 ½,
0 ½ ½ (the same as FCC), ¼ ¼ ¼, ¾ ¾ ¼, ¾ ¼ ¾,
¼ ¾ ¾.
FCC structure factor
Two FCC shifted by ¼ ¼ ¼
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when h, k, l are all odd
when h, k, l are all even and h k l 4n
when h, k, l are all even and
h k l ? 4n
when h, k, l are mixed
? Example 5 Close packed hexagonal cell
There are two atoms in a unit cell The eight
atoms in the corner are equipoints of rank 1.
Choose 000 to represent it. The other point
is 1/3 2/3 1/2.
( 1/3 2/3 1/2)
(001)
(000)
(010)
(100)
(110)
Set h 2k/3 l/2 g
equipoints
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h 2k
l
3m 3m 3m?1 3m?1
even odd even odd
1 0 0.25 0.75
4f 2 0 f 2 3f 2
? Example 6 NaCl (two kinds of atoms), can
be considered as 4 Na at 000 FCC structure 4
Cl at ½ ½ ½ FCC structure
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The structure factor
FCC structure factor
when h, k, l mixed
when h, k, l all even
when h, k, l all odd
You can also think of the structure as simple
FCC and the atom is replaced by a pair of
atoms (1 Na 1 Cl) with the new structure
factor
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? Multiplicity Factor The following factors
discussed are related to the diffraction from
polycrystalline sample. The multiplicity
factor, p, for hkl planes may be defined as
the number of permutations of position and sign
of ?h, ?k, ?l. It represents the number of
equivalent planes to yield the same
diffraction cone. E.g. d100, d010, d001,
d-100, d0-10, d00-1 form part of the same
diffraction cone for cubic crystal ? p 6.
If the crystal is tetragonal, d100, d010,
d-100, d0-10, form part of the same diffraction
cone ? p 4 d001, d00-1 ? p 2
? Lorentz factor Certain trigonometrical
factors, including Angular distribution of
the diffraction (finite spreading of the
intensity peak, fraction of crystal contributing
to a
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diffraction peak, and the intensity
spreading in a cone. ? Integrated
intensity (finite spreading) from
constructive interference at Bragg angle
to complete destructive interference ?
transition region
2?
?
Imax
Imax/2
Intensity
B
Integrated Intensity
?2
2?B
2?
?B
Diffraction Angle 2?
?1
1
2
1?
2?
path difference for 11?-22? AD
CB acos?2 - acos?1 acos(?B-??) - cos
(?B??) 2asin(??)sin?B 2a?? sin?B.
D
C
?1
?2
a
A
B
1?
N?
2Na?? sin?B ? ? completely
cancellation (1- N/2, 2- (N/21) )
Na
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Maximum angular range of the peak
Imax ? ?? ? 1/sin?B, Half maximum B ? 1/cos?B
(will be shown later) ? integrated intensity
of the reflection ? ImaxB ? (1/sin?B)(1/cos?B)
? 1/sin2?B.
? The second geometrical factor arises from
the fact that the integrated intensity
depends on the number of crystals
orientated at or near the Bragg angle. ?
estimated the fraction of the crystal around the
angle. See Fig. 4-16.
??
Fraction of crystal
crystal plane
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In accessing relative intensity, one should
compare the total diffracted energies in
different cone. See, Fig. 4-17. For different
cones, the radius of the diffracted energy
intersecting the film (Hull/Debye- Scherrer
film) is different. Assume the total diffracted
energy is equally distributed in the cone
(2?Rsin2?B) ? the relative intensity per unit
length ? 1/sin?B. ? Lorentz factor
? Lorentzpolarization factor (omitting
constant)
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? Absorption factor The factor accounts for
the X-ray being absorbed during its in and
out of the sample. Two geometries are
considered. ? Geometry for Hull/Debye-Scherrer
Camera difficult to calculate, different
path different absorption, see Fig. 4-19.
In general, the absorption factor is written as
A(?). A(?) varies with ?, qualitatively,
A(?) ? as ??. ? Geometry for Diffractometer
dID
1cm
I0
?
Incident I0, is 1cm2 in cross section, and
is incident on the powder plate at an angle ?.
The beam incident on the plate is
.
?
C
A
x
dx
B
l
2?
? linear absorption coefficient
Let a volume fraction of the
specimen containing particles having
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the correct orientation for diffraction of
the incident beam b the fraction of the
incident energy which is diffracted by one
unit volume. ? volume l ? dx ? 1cm
ldx. ? actual diffracted volume abldx ?
diffracted beam ?
diffracted beam absorbed before escaping from
the sample
If ? ? ? ?
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a varies with ?, being accounted for in
the Lorentz factor already. The case
discussed is for infinite thickness
(defined in the textbook, dID(x 0)/dID(x t)
1000 and ? ? ?).
? Temperature factor Also known as Debye
Waller factor. Atoms in lattice vibrate
(Debye model). Temperature ? ? (1) lattice
constants ? 2? ? (2) Intensity of diffracted
lines ? (3) Intensity of the background
scattering ?.
u
Lattice vibration is more than important at
high ?B (u/d). High ?B decrease more!
Introducing the temperature factor e-2M.
d
1
low ?B
u
e-2M
d
high ?B
sin? / ?
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? Formally, the factor is included in f as
Because F f 2 ? factor e-2M shows up
? What is M?
Mean square displacement in the direction
normal to the diffraction plane, difficult to
calculate, Debye has given the following
expression
h Planks constant T absolute temperature m
mass of vibrating atom k Boltzmanns constant
? Debye temperature of the substance x ?/T
?(x) tabulated function (appendix 15)
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Put all numbers in and change m to atomic weight
(A)
? Temperature diffuse scattering general
coherent scattering in all directions by
the displacement (u) ? contributing to the
general background of the pattern ? as ? ?
? Temperature effect on the diffraction
intensity, see Fig. 4-22. The diffraction
peak width B ? slightly as T ?
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? Summary of the intensities of diffraction peaks
from polycrystalline samples ?
Hull/Debye-Scherrer Camera
? Diffractometer
A(?) independent of ?.
? Effects that make the above intensity
equation invalid (1) Preferred
orientation like films, wire, not true
random orientation of the crystal ? Lorentz
factor depends on how the crystals
orientated (2) Extinction for crystals
with small sizes (10-4-10-5 cm)?
kinematical theory intensity is the sum of all
diffraction planes
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For large perfect crystal ?
dynamical theory is required takes
into account the multiple reflections
between diffracted planes (absorption also
increases)
Actually to match the intensity calculation
with the measured intensity is difficult.
But, at least the magnitude (very strong,
strong, weak) is qualitatively matched.