Structure Factor Calculation

1. In these set of slides we shall consider:
 scattering from an electron
 scattering from an atom
 structure factor calculations (scattering from an unit cell)
 the relative intensity of ‘reflections’ in power patterns
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
http://home.iitk.ac.in/~anandh/E-book.htm
A Learner’s Guide
Elements of X-Ray Diffraction
B.D. Cullity & S.R. Stock
Prentice Hall, Upper Saddle River (2001)
Click here to directly jump to
structure factor calculations
You may find the first
set of slides a little
difficult and hence

2. Intensity of the Scattered electrons
Electron
Atom
Unit cell (UC)
Scattering by a crystal
A
B
C
Polarization factor
Atomic scattering factor (f)
Structure factor (F)
Click here to jump to structure factor calculations
 To understand scattering from a crystal, we have to
understand scattering from 3 levels: (i) electron, (ii) atom,
(iii) unit cell.
 For crystals the unit cell repeats to give us the crystal and
hence all the information required can be obtained at the
unit cell level.
 Usually materials scientists need to start understanding
from the atomic scattering factor. Most students can jump
directly to structure factor calculations.

3. Scattering by an Electron
)
,
( 0
0 

Sets electron into oscillation
Scattered beams
)
,
( 0
0 

Coherent
(definite phase relationship)
A
 The electric field (E) of the incoming radiation (with wavelength 0 and frequency 0) is the
main cause for the acceleration (& deceleration) of the electron (in an atom) i.e. the
oscillating magnetic field component can be ignored. The effect on the nucleus can be
ignored.
 The oscillating electron is a source of electromagnetic radiation and radiates most strongly in
a direction perpendicular to its motion.
 Hence, the radiation emitted will be polarized along the direction of its motion.
 There is fixed phase relationship between the incoming radiation and the scattered wave.
 The incoming wave may be polarized on un-polarized. We consider the intensity in these two
situations next.
Incoming radiation

4. x
z
r
P

Intensity of the scattered beam due to an electron (I) at a point P such that r >>  is given by:








 2
2
4
2
4
0
r
Sin
c
m
e
I
I

For a wave oscillating in z direction
Case A: intensity of the scattered wave for an polarized wave
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150 180 210 240 270 300 330 360
t
Cos(t)
The reason we are able to neglect
scattering from the protons in the nucleus
The scattered rays are also plane polarized

5. E is the measure of the amplitude of the wave & E2 = Intensity
2
2
2
z
y E
E
E 

z
y I
I
I
0
0
0
2


 
2
4
0 2 4 2
y
Py y
Sin
e
I I
m c r

 
 

 
 
IPy = Intensity at point P due to Ey
IPz = Intensity at point P due to Ez
 
2
4
0 2 4 2
z
Pz z
Sin
e
I I
m c r

 
  
 
 
Total Intensity at point P due to Ey & Ez
   
2 2
4
0 2 4 2
y z
P
Sin Sin
e
I I
m c r
 
 

 

 
 
Case B: intensity of the scattered wave for an un-polarized wave

6.    
2 2
4
0 2 4 2
y z
P
Sin Sin
e
I I
m c r
 
 

 

 
 
           
2 2 2 2 2 2
1 1 2
y z y z y z
Sin Sin Cos Cos Cos Cos
     
     
       
     
     
2 2 2
1
x y z
Cos Cos Cos
  
 
  
  Sum of the squares of the direction cosines =1
       
2 2 2 2
2 2 1 ( ) 1 ( )
y z x x
Cos Cos Cos Cos
   
   
 
      
     
Hence

 
2
4
0 2 4 2
1 ( )
x
P
Cos
e
I I
m c r

 

 

 
 
 
2
4
0 2 4 2
1 (2 )
P
Cos
e
I I
m c r

 

 

 
 
In terms of 2

7. -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150 180 210 240 270 300 330 360
t
Cos(t)
 







 


 2
2
0
0
4
2
4
2
r
Cos
I
I
c
m
e
I
I
I
z
y
Pz
Py
P

 







 
 2
2
4
2
4
0 2
1
2 r
Cos
c
m
e
I
IP
  Scattered beam is not unpolarized
 Forward and backward scattered intensity higher than at 90.
 Scattered intensity minute fraction of the incident intensity.
Very small number
Polarization factor
Comes into being as we used unpolarized beam
0
0.2
0.4
0.6
0.8
1
1.2
0 30 60 90 120 150 180 210 240 270 300 330 360
2t
(1+Cos(2t)^2)/2

8. B Scattering by an Atom
Scattering by an atom  [Atomic number, (path
difference suffered by scattering from each e−, )]
Scattering by an atom  [Z, (, )]
 The scattering by an atom is the sum (vectoral sum, including phase) of the waves scatted by
the electrons. Angular differences involved in scattering leads to path differences.
 In the forward direction all scattered waves are in phase.
 The scattering from an atom is captured in the atomic scattering factor (f), which is the ratio of
the amplitude of the wave scattered by the atom to that scattered by a single electron. It shows
the amplification obtained by the presence of multiple electrons in an atom.
 Scattering by an atom is proportional to the atomic number (the number of electrons). The
natural coordinates to plot the variation of ‘f’ is not  or 2, but Sin()/. At 0 the curve starts
at the atomic number as in the schematic below.
electron
an
by
scattered
wave
of
Amplitude
atom
an
by
scattered
wave
of
Amplitude
Factor
Scattering
Atomic
f


f
→

)
(
Sin
(Å−1) →
0.2 0.4 0.6 0.8 1.0
10
20
30
Schematic showing variation in
atomic scattering factor with
Sin()/

)
(
Sin
Atomic Scattering Factor or Form Factor (f)
Equals number of electrons, say for
f = 29, it should Cu or Zn+
Coherent
scattering
Incoherent (Compton)
scattering
Z   
Sin() /    
As θ↓, path difference ↓  constructive interference
As λ↓, path difference ↑  destructive interference

9. B Scattering by an Atom
BRUSH-UP
 The conventional UC has lattice points only at the vertices/corners.
 Additional lattice points may be present within the unit cell, if the unit cell is non-
primitive.
 Needless to say, lattice in itself cannot scatter and atoms are required.
 Motif is associated with lattice and motif may consist of one or more atoms, ions,
molecules (or combinations).
 There may or may not be atoms located at the lattice points.
 The shape of the UC is a parallelepiped (Greek parallēlepipedon) in 3D.
 There may be additional atoms in the UC due to two reasons:
 The chosen UC is non-primitive (hence additional lattice points exist within the cell, which are associated with atoms).
 The additional atoms may be part of the motif.
 We use the Bragg’s viewpoint to derive the general structure factor equation.

10. C Scattering by the Unit cell (uc)
 We use the following points to derive the scattering from a unit cell (which is captured in the
structure factor (F)): (i) the scattering is coherent, (ii) Unit Cell (UC) is representative of the
crystal structure, (iii) scattered waves from various atoms in the UC interfere to create the
diffraction pattern.
 The essential feature of the need for structure factor calculation can be understood from the
figure below. Let us assume a BCC crystal. If the waves are in phase for the ‘reflection’ from
the green planes, then the presence of the additional atom in the orange plane will lead to
scattering of waves, which are exactly out of phase with that scattered from the green planes
and hence the 100 reflection will ‘go missing’. Note that the atomic density of both the planes
(green and orange) is the same (one atom per a2) and hence the scattering power of these
planes is the same.
 If we consider a simple cubic crystal with a two atom motif with atoms as: A at (0,0,0) and B
at (½, ½, ½), then the atomic scattering factor of the two atoms will not be the same (i.e. fA 
fB) and hence there will be a ‘residual’ intensity from the 100 planes (i.e. 100 reflection will
‘not go totally missing’, but will be weak called a superlattice reflection).
The wave scattered from the middle plane is out of phase
with the ones scattered from top and bottom planes. I.e. if
the green rays are in phase (path difference of ) then the
red ray will be exactly out of phase with the green rays
(path difference of /2).

11. d(h00)
B 
Ray 1 = R1
Ray 2 = R2
Ray 3 = R3
Unit Cell
x
M
C
N
R
B
S
A
'
1
R
'
2
R
'
3
R
(h00) plane
a
 Let us consider scattering of waves (or “rays”) R1 and R2 from planes P1 and P2 (belonging to
the (h00) set of planes. The scattered rays are denoted by ‘primes’.
 Let there be an additional atom located at a distance ‘x’ along x-axis (on plane P3).
 The rays R2 and R3 travel longer distances than R1 and hence accrue path () & phase
differences ().
P1
P2
P3

12. h
a
d
AC h 
 00

::
:: AC
MCN
x
AB
RBS ::
::
h
a
x
x
AC
AB



1 2 00
2 ( )
  
R R h
MCN d Sin
  
1 3
  
R R
AB x
RBS
a
AC
h
  




2

1 3
2
2

  

R R
x x
h
a a
h
  x
coordinate
fractional
a
x



1 3
2
 
R R hx

Extending to 3D 2 ( )
h x k y l z
    
   Independent of the shape of UC
Note: R1 is from corner atoms and R3 is from atoms in additional positions in UC



 
2
To satisfy Bragg’s equation the path
difference between rays R1 & R2 has to be 
Correspondingly the path difference between
rays R1 and R3 will be
Using fractional coordinates (x’) the
phase difference becomes
The phase
difference
can be
written as

13.  If atom B is different from atom A  the amplitudes must be weighed by the respective
atomic scattering factors (f).
 The resultant amplitude of all the waves scattered by all the atoms (say ‘n’atoms) in the UC
gives the scattering factor for the unit cell .
 The unit cell scattering factor is called the Structure Factor (F).
 The intensity of the ‘diffracted’wave is proportional to the square of F.
 Once we have the structure factor then the intensity obtained from a set of planes (hkl) is
known.
Scattering by an unit cell = f(position of the atoms, atomic scattering factors)
electron
an
by
scattered
wave
of
Amplitude
uc
in
atoms
all
by
scattered
wave
of
Amplitude
Factor
Structure
F 

[2 ( )]
i i h x k y l z
E Ae fe
    
 
 
2 ( )
h x k y l z
    
  
The phase difference gives rise to the amplitude (which can be written in complex notation as)
2
F
I 
[2 ( )]
1 1
j j j j
n n
i i h x k y l z
hkl
n j j
j j
F f e f e
    
 
 
   
 
   
 
Structure factor is independent of the shape and size of the unit cell!
F → Fhkl
For n atoms in the UC
If the UC distorts so do the planes in it!!
hkl
n
F

14. n
ni
e )
1
(


)
(
2



Cos
e
e i
i

 

 ni
ni
e
e 

1
)
(



i
n
odd
e
1
)
(



i
n
even
e
( 1)
n i n
e 
 
Note:
n i n i
e e
 


n is an integer
Some useful relations
[2 ( )]
1 1
j j j j
n n
i i h x k y l z
hkl
n j j
j j
F f e f e
    
 
 
 
 
 The summation ‘j’ is done over the ‘n’ atoms in the UC.
 In a primiteve unit cell (e.g. in the SC crystal) there is just one atom
(in the UC).
 x’(y’& z’) refer(s) to the fractional coordinates of each atom.
 The fj is the atomic scattering factor for each atom. If there is just
one type of atom (say Po then fj = fPo). If there is more than one type
of atom, the fractional coordinates should match with the atom type.
 The intensity is an actual observable (if sampled in an experiment) and hence
structure factor is a real quantity.
 A set of useful equations to evaluate the F is given on the right side.
Evaluation of the equation

15. How to understand the structure factor? Is there a simpler way so as to avoid calculations?
Funda Check
 Structure factor (F) tells us that which planes in the crystal scatter from an unit cell and what
is the resultant intensities.
 I.e. which hkl reflections are present and which ones are absent?
‘Which ‘hkl’ has what intensities?’ (which have higher intensities and which have lower
intensities).
 The ‘F’ has each term weighed by the atomic scattering factor (f) and hence (we will see soon) that
intensities are weighed by some function of the ‘f’ of atoms in the unit cell.
 Using the ‘F’ calculations we can construct the reciprocal crystal. And we will see later that diffraction
patterns are nothing but selective sampling of this reciprocal crystal (displayed/plotted in 2 axis or 1/x axis).
 The structure factor calculation is performed by evaluating the sum for all “non-equivalent”
(this term will become clear from the examples considered soon) atoms in the unit cell.
 What is the source of these additional atoms*?
Ans:
(i) The unit cell may be non-primitive and hence there might be additional lattice points
associated with additional atoms.
(ii) The motif may have many atoms.
* “Atoms” is used in a general sense here, it includes neutral atoms, ions, molecules etc.

16.  A set of planes (hkl) in real space becomes a point in reciprocal space (labeled without the
brackets as hkl ).
 These points define the reciprocal lattice (a purely geometrical construction).
 Structure factor calculations give us the intensity associated with each of these points. We
use a sphere to designate the intensity, with the diameter of the sphere proportional to the
intensity.
 On decorating the reciprocal lattice with intensities, we get the reciprocal crystal.
Plotting Structure Factor calculations Please refer to topic on Reciprocal Lattice to know all the details
Real Space Reciprocal Space
Lattice decorated with Motif Reciprocal lattice decorated with Intensities as motif
Planes
Points
(planes in real space become points in reciprocal space)
Motif
Intensities
(motif in real space determines the intensities in reciprocal space)
Needless to point out the reciprocal of the reciprocal lattice is the real lattice.

17. Structure factor calculations
1 Atom at (0,0,0) and equivalent positions Simple Cubic Crystal
 Now we perform calculation of structure factor calculations for some simple crystal structure
unit cells.
 We basically perform the summation of the function f exp(i) for the different atoms ions in
the unit cell and work out the restrictions on the allowed values of hkl (and the corresponding
intensities).
a
UC of reciprocal
lattice (primitive)
a
a
Let us start with the simple cubic crystal. The first step is to construct the reciprocal lattice from the real
lattice. The real lattice is simple cubic and so is the reciprocal lattice.
1/a
1/a
1/a
 Each of the points in the reciprocal lattice corresponds to a set of planes in real space.
 The dimensions are reciprocal to that in real space. Hence, lattice parameter ‘a’ in real space becomes
‘1/a’ in reciprocal space.
 Now we have to perform structure factor calculations to determine the intensities which decorate this
lattice.
Underlying
lattice: SC
UC of SC lattice

18. [2 ( )]
j j j j
i i h x k y l z
hkl
j j
F f e f e
    
 
   
 
   
[2 ( 0 0 0)] 0
i h k l
F f e f e f
     
  
2
2
f
F   F is independent of the scattering plane (h k l)
 All reflections are
present
We note that in simple cubic crystals
there is no restrictions on the allowed
values of hkl (i.e. for all values of hkl
reflections are present).
1/a
1/a
1/a
The diameter of the
sphere scales with the
intensity of the ‘spot’.
UC of reciprocal crystal
(intensities decorating the reciprocal lattice)
(also primitive)
In the reciprocal crystal, intensities decorate the lattice points (shown in orange colour). The magnitude of
the intensity (as determined from the structure factor calculation) is f2.

19. 2 Atom at (0,0,0) & (½, ½, 0) and equivalent positions
[2 ( )]
j j j j
i i h x k y l z
j j
F f e f e
    
 
 
]
1
[ )
(
)
(
)
(
)]
2
(
2
[
)]
2
(
2
[
)]
2
(
2
[
)]
0
(
2
[
h
l
i
l
k
i
k
h
i
h
l
i
l
k
i
k
h
i
i
e
e
e
f
e
e
e
e
f
F



























Face Centred Cubic (CCP crystal)
Real
f
F 4

0

F
2
2
16 f
F 
0
2

F
(h, k, l) unmixed
(h, k, l) mixed
111, 200, 220, 333, 420
100, 211; 210, 032, 033
(½, ½, 0), (½, 0, ½), (0, ½, ½)
]
1
[ )
(
)
(
)
( h
l
i
l
k
i
k
h
i
e
e
e
f
F 





 


Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
h,k,l → all even or all odd
CCP (“FCC crystal”)
becomes “BCC crystal” in
reciprocal space
Continued…
UC of reciprocal crystal: non-primitive

20. Mixed indices CASE h k l
A o o e
B o e e
( ) ( ) ( )
CASE A: [1 ] [1 1 1 1] 0
i e i o i o
e e e
  
       
( ) ( ) ( )
CASE B: [1 ] [1 1 1 1] 0
i o i e i o
e e e
  
       
0

F 0
2

F
(h, k, l) mixed e.g. 100, 211; 210, 032, 033.
Mixed indices Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
Unmixed indices CASE h k l
A o o o
B e e e
Unmixed indices
f
F 4
 2
2
16 f
F 
(h, k, l) unmixed
e.g. 111, 200, 220, 333, 420.
All odd (e.g. 111); all even (e.g. 222)
( ) ( ) ( )
CASE A: [1 ] [1 1 1 1] 4
i e i e i e
e e e
  
       
( ) ( ) ( )
CASE B: [1 ] [1 1 1 1] 4
i e i e i e
e e e
  
       
 This implies that in FCC only h,k,l
‘unmixed’ reflections are present.
Face Centred Cubic

21. 3 Atom at (0,0,0) & (½, ½, ½) and equivalent positions
[2 ( )]
j j j j
i i h x k y l z
j j
F f e f e
    
 
 
1 1 1
[2 ( )]
[2 ( 0 0 0)] 2 2 2
[ 2 ( )]
0 ( )
2
[1 ]
i h k l
i h k l
h k l
i
i h k l
F f e f e
f e f e f e




    
    
 
 
 
   
Body centred Orthorhombic
Real
]
1
[ )
( l
k
h
i
e
f
F 


 
f
F 2

0

F
2
2
4 f
F 
0
2

F
e.g. 110, 200, 211; 220, 022, 310.
e.g. 100, 001, 111; 210, 032, 133.
 This implies that (h+k+l) even reflections are only present.
 The situation is identical in BCC crystals as well.
Continued…
UC in real
space (crystal)

22. UC of reciprocal crystal
UC in real
space (crystal)
UC of the
reciprocal lattice
(primitive)
Underlying
lattice: BCO
UC of reciprocal crystal: non-primitive

23. 4 Atoms at (0,0,0) & (½, ½, 0) and equivalent positions
[2 ( )]
j j j j
i i h x k y l z
hkl
j j
F f e f e
    
 
 
C- centred Orthorhombic Crystal
 For this case there is one additional lattice point with an associated atom. Hence, there will be
two terms in the summation for the structure factor.
UC of the
reciprocal lattice
(primitive)
Ball & stick model of UC in real space
UC in real
space (crystal)
 This is a UC of the reciprocal lattice. The UC
corresponds to a simple orthorhombic lattice
in reciprocal space.
 Note the basis vectors.
 The magnitude of the basis vectors are 1/a,
1/b & 1/c.
 However, the reciprocal crystal has a
different unit cell.
 Unit cell of the reciprocal crystal (as we shall see)
is a c-centred orthorhombic UC (i.e. UC usually
chosen for the c-centred orthorhombic lattice).
*
1
1
| |
b
a

*
2
1
| |
b
b

*
3
1
| |
b
c


24. 1 1
[2 ( 0)]
[2 ( 0 0 0)] 2 2
[ 2 ( )]
0 ( )
2
[1 ]
i h k l
hkl i h k l
h k
i
i h k
F f e f e
f e f e f e


 
    
    


 
   
 F is independent of the ‘l’index
Real
]
1
[ )
( k
h
i
e
f
F 

 
f
F 2

0

F
2
2
4 f
F 
0
2

F
e.g. 001, 110, 112; 021, 022, 023.
e.g. 100, 101, 102; 031, 032, 033.
Important note:
 The 100, 101, 210, etc. points in
the reciprocal lattice exist (as the
corresponding real lattice planes
exist), however the intensity
decorating these points is zero.
1/a 1/b
1/c
Missing reflections Unit cell of reciprocal
lattice (primitive)
C-centred orthorhombic UC
UC of reciprocal crystal: non-primitive

25.  If the blue planes are scattering in phase then on C-
centering the red planes will scatter out of phase (with
the blue planes- as they bisect them) and hence the
(210) reflection will become extinct
 This analysis is consistent with the extinction rules: (h
+ k) odd is absent
 The result derived (i.e. the effect of lattice centring) can be understood by a simple geometric consideration. This is
illustrated for the C-centred OR crystal considered before, but is valid for all crystals.
 Let us view the [001] projection of the crystal and consider two sets of planes: (210) and (310) planes.
 On introducing a centring a new set of lattice planes between the original (210) planes have to be introduced, which
scatter exactly out of phase with the original planes and hence the 210 reflection goes missing on introducing a C-
centring. In the case of the (310) planes no new planes need to be introduced and this reflection survives.
 In case of the (310) planes no new translationally
equivalent planes are added on lattice centering  this
reflection cannot go missing.
 This analysis is consistent with the extinction rules: (h
+ k) even is present

26. How come the unit cell for the reciprocal lattice is different from that of the
reciprocal crystal (for the case of the c-centred orthorhombic crystal*)?
Funda Check
 The reciprocal lattice is a pure geometric construct from the real lattice; wherein, planes are
transformed into points.
 The physics is introduced into the problem via the presence of atoms, which may scatter in
phase according to Laue’s or Braggs’ equations. This results in intensities decorating the
reciprocal lattice.
 Due to presence of additional atoms in the UC (arising from additional lattice points in a
non-primitive UC or as a part of the motif) some of these intensities are zero.
 Thus the UC (according to the picture: crystal = lattice + motif) is altered.
* I.e. c-centred orthorhombic lattice decorated with a single ‘atom’motif.
UC of crystal in
reciprocal space
Lattice: C-centred orthorhombic
Motif: Intensity (4f2)
UC of reciprocal
lattice (primitive)
UC of the reciprocal
crystal
UC of reciprocal crystal: non-primitive

27. F SC, Al at (0, 0, 0), Ni at (½, ½, ½) NiAl: Simple Cubic (B2- ordered structure)
SC
1 1 1
[2 ( )]
[2 ( 0 0 0)] 2 2 2
[ 2 ( )]
0 [ ]
2
i h k l
i h k l
Al Ni
h k l
i
i h k l
Al Ni Al Ni
F f e f e
f e f e f f e




    
    
 
 
 
   
Real
Al Ni
F f f
 
e.g. 110, 200, 211, 220, 310.
e.g. 100, 111, 210, 032, 133.
[ ]
i h k l
Al Ni
F f f e   
 
2 2
( )
Al Ni
F f f
 
Al Ni
F f f
  2 2
( )
Al Ni
F f f
 
Click here to know more about ordered structures
 When the central atom is identical to the corner ones we have the BCC case.
 This implies that (h+k+l) even reflections are only present in BCC.
This term is zero for BCC

28. Reciprocal lattice/crystal of NiAl
2
( )
Al Ni
I f f
 
2
( )
Al Ni
I f f
 
e.g. 110, 200, 211, 220, 310.
e.g. 100, 111, 210, 032, 133.
Click here to know more about
* Coordinates in reciprocal space.
& B2 gives NaCl in reciprocal space !!
UC in reciprocal
space&
Reciprocal
Crystal
Lattice: FCC.
Motif: fundamental reflection at (0,0,0) & superlattice reflection at (½,0,0).
UC of reciprocal crystal: non-primitive

29. G FCC, C at (0,0,0) & (¼, ¼, ¼)
[2 ( )]
[2 ( 0)] 4
( ) ( ) ( )
[ ]


  
h k l
i
i
C C
i h k i k l i l h
Al Ni
F f e f e
f f e e e
 
  
 
 
   
 
 
   
Diamond Cubic
Real
(h, k, l) unmixed
(h, k, l) mixed
(111), (200), (220), (333), (420)
(100), (211); (210), (032), (033)
Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
Ni
Al
( ) ( ) ( )
[ ]
i h k i k l i l h
Al Ni
F f f e e e
  
  
   
3
Al Ni
F f f
  2 2
( 3 )
Al Ni
F f f
 
Al Ni
F f f
  2 2
( )
Al Ni
F f f
 
h,k,l → all even or all odd

30. G SC, Al at (0,0,0), Ni at (½, ½, 0) and equivalent positions
[2 ( )] [2 ( )] [2 ( )]
[2 ( 0)] 2 2 2
( ) ( ) ( )
[ ]
h k k l l h
i i i
i
Al Ni
i h k i k l i l h
Al Ni
F f e f e e e
f f e e e
  

  
  
  
 
 
   
 
 
 
   
Simple Cubic (L12 ordered structure)
Real
(h, k, l) unmixed
(h, k, l) mixed
(111), (200), (220), (333), (420)
(100), (211); (210), (032), (033)
(½, ½, 0), (½, 0, ½), (0, ½, ½)
Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
Ni
Al
( ) ( ) ( )
[ ]
i h k i k l i l h
Al Ni
F f f e e e
  
  
   
3
Al Ni
F f f
  2 2
( 3 )
Al Ni
F f f
 
Al Ni
F f f
  2 2
( )
Al Ni
F f f
 
h,k,l → all even or all odd
Click here to know more about ordered structures

31. e.g. (111), (200), (220), (333), (420)
e.g. (100), (211); (210), (032), (033).
2 2
( 3 )
Al Ni
F f f
 
2 2
( )
Al Ni
F f f
 
Reciprocal lattice/crystal of Ni3Al Click here to know more about
UC of reciprocal crystal: non-primitive

32. E Na+ at (0,0,0) + Face Centering Translations  (½, ½, 0), (½, 0, ½), (0, ½, ½)
Cl− at (½, 0, 0) + FCT  (0, ½, 0), (0, 0, ½), (½, ½, ½)



























)]
2
(
2
[
)]
2
(
2
[
)]
2
(
2
[
)]
2
(
2
[
)]
2
(
2
[
)]
2
(
2
[
)]
2
(
2
[
)]
0
(
2
[
l
k
h
i
l
i
k
i
h
i
Cl
h
l
i
l
k
i
k
h
i
i
Na
e
e
e
e
f
e
e
e
e
f
F








]
[
]
1
[
)
(
)
(
)
(
)
(
)
(
)
(
)
(
l
k
h
i
l
i
k
i
h
i
Cl
h
l
i
l
k
i
k
h
i
Na
e
e
e
e
f
e
e
e
f
F






















]
1
[
]
1
[
)
(
)
(
)
(
)
(
)
(
)
(
)
(





















k
h
i
h
l
i
l
k
i
l
k
h
i
Cl
h
l
i
l
k
i
k
h
i
Na
e
e
e
e
f
e
e
e
f
F







]
1
][
[ )
(
)
(
)
(
)
( h
l
i
l
k
i
k
h
i
l
k
h
i
Cl
Na
e
e
e
e
f
f
F 








 





NaCl: FCC lattice

33. ]
1
][
[ )
(
)
(
)
(
)
( h
l
i
l
k
i
k
h
i
l
k
h
i
Cl
Na
e
e
e
e
f
f
F 








 





Zero for mixed indices
Mixed indices CASE h k l
A o o e
B o e e
]
2
][
1
[ 

 Term
Term
F
( ) ( ) ( )
CASEA: 2 [1 ] [1 1 1 1] 0
i e i o i o
Term e e e
  
         
0
]
1
1
1
1
[
]
1
[
2
:
B
CASE )
(
)
(
)
(









 o
i
e
i
o
i
e
e
e
Term 


0

F 0
2

F
(h, k, l) mixed 100, 211; 210, 032, 033
Mixed indices
NaCl: FCC lattice
Unmixed indices CASE h k l
A o o o
B e e e
4
]
1
1
1
1
[
]
1
[
2
:
A
CASE )
(
)
(
)
(









 e
i
e
i
e
i
e
e
e
Term 


4
]
1
1
1
1
[
]
1
[
2
:
B
CASE )
(
)
(
)
(









 e
i
e
i
e
i
e
e
e
Term 


Unmixed indices
Continued…

34. (h, k, l) unmixed ]
[
4 )
( l
k
h
i
Cl
Na
e
f
f
F 


 
 
]
[
4 
 
 Cl
Na
f
f
F If (h + k + l) is even
2 2
16[ ]
Na Cl
f I
F f
 

  
]
[
4 
 
 Cl
Na
f
f
F If (h + k + l) is odd
2 2
16[ ]
Na Cl
f I
F f
 

  
111, 222; 133, 244
222, 244
111, 133
h,k,l → all even or all odd
NaCl: FCC lattice
UC in reciprocal
space*
* NaCl gives B2 in reciprocal space !!
UC of reciprocal crystal: non-primitive

35. Bravais Lattice
Reflections which may be
present
Reflections necessarily
absent
Simple all None
Body centred (h + k + l) even (h + k + l) odd
Face centred h, k and l unmixed h, k and l mixed
End centred
h and k unmixed
C centred
h and k mixed
C centred
Bravais Lattice Allowed Reflections
SC All
BCC (h + k + l) even
FCC (CCP) h, k and l unmixed
DC
(Rules for CCP +
additional condition
h, k and l are all odd
Or
all are even
& (h + k + l) divisible by 4
Selection
/
Extinction
Rules
 Presence of additional atoms/ions/molecules in the UC can alter the
intensities of some of the reflections

36. h2 + k2 + l2 SC FCC BCC DC
1 100
2 110 110
3 111 111 111
4 200 200 200
5 210
6 211 211
7
8 220 220 220 220
9 300, 221
10 310 310
11 311 311 311
12 222 222 222
13 320
14 321 321
15
16 400 400 400 400
17 410, 322
18 411, 330 411, 330
19 331 331 331
List of observed reflections
SC, FCC, BCC & DC
crystals
Cannot be expressed as a sum
of the square of 3 integers

37.  We have already noted that absolute value of intensity of a peak (which is the area under a
given peak) has no significance w.r.t structure identification.
 The relative value of intensities of the peak gives information about the motif.
 One factor which determines the intensity of a hkl reflection is the structure factor.
 In powder patterns (& other experimental conditions) many other factors come into the
picture as in the next slide. Some of these have origin in crystallography (like the
multiplicity factor), some have origin in the powder diffraction geometry (like the Lorentz
factor), some are related to the radiation (polarization factor), etc.
 The multiplicity factor relates to the fact that we have 8 {111} planes giving rise to single
peak, while there are only 6 {100} planes (and so forth). Hence, by this very fact the
intensity of the {111} planes should be more than that of the {100} planes.
 A brief consideration of some these factors are in upcoming slides. The reader may consult
Cullity’s book for more details.
Relative intensity of peaks in powder patterns

38. Structure Factor (F)
Multiplicity factor (p)
Polarization factor
Lorentz factor
Relative Intensity of diffraction lines in a powder pattern
Absorption factor
Temperature factor
Scattering from UC (Atomic Scattering Factor is included
in this term)
Number of equivalent scattering planes
Effect of wave polarization
Combination of 3 geometric factors
Specimen absorption
Thermal diffuse scattering
  














 2
1
2
1
Sin
Cos
Sin
factor
Lorentz
 
 

2
1 2
Cos
IP 


39. Multiplicity factor
Lattice Index Multiplicity Planes
Cubic
(with highest
symmetry)
(100) 6 [(100) (010) (001)] ( 2 for negatives)
(110) 12 [(110) (101) (011), (110) (101) (011)] ( 2 for negatives)
(111) 12 [(111) (111) (111) (111)] ( 2 for negatives)
(210) 24* (210)  3! Ways, (210)  3! Ways,
(210)  3! Ways, (210)  3! Ways
(211) 24
(211)  3 ways, (211)  3! ways,
(211)  3 ways
(321) 48*
Tetragonal
(with highest
symmetry)
(100) 4 [(100) (010)] ( 2 for negatives)
(110) 4 [(110) (110)] ( 2 for negatives)
(111) 8 [(111) (111) (111) (111)] ( 2 for negatives)
(210) 8* (210) = 2 Ways, (210) = 2 Ways,
(210) = 2 Ways, (210) = 2 Ways
(211) 16 [Same as for (210) = 8]  2 (as l can be +1 or 1)
(321) 16* Same as above (as last digit is anyhow not permuted)
* Altered in crystals with lower symmetry
Actually only 3 planes ! (as (hkl)  (h k l)

40. Cubic
hkl hhl hk0 hh0 hhh h00
48* 24 24* 12 8 6
Hexagonal
hk.l hh.l h0.l hk.0 hh.0 h0.0 00.l
24* 12* 12* 12* 6 6 2
Tetragonal
hkl hhl h0l hk0 hh0 h00 00l
16* 8 8 8* 4 4 2
Orthorhombic
hkl hk0 h0l 0kl h00 0k0 00l
8 4 4 4 2 2 2
Monoclinic
hkl h0l 0k0
4 2 2
Triclinic
hkl
2
* Altered in crystals with lower symmetry (of the same crystal class)
Multiplicity factor

41. All peaks present
Look at general trend line!
0
5
10
15
20
25
30
0 20 40 60 80
Bragg Angle (, degrees)
Lorentz-Polarization
factor
Polarization factor
  














 2
1
2
1
Sin
Cos
Sin
factor
Lorentz
 
 

2
1 2
Cos
IP 

 







 




Cos
Sin
Cos
factor
on
Polarizati
Lorentz 2
2
2
1
XRD pattern from Polonium
Click here for details
Example of effect of Polarization factor on
power pattern
 The polarization factor and the Lorentz factor both have ‘’ dependence (only) can be combined into the
L-P factor. Note that the LP factor has a dip in middle 2 values (with large values at low and high
angles). [Fig.1].
 The signature of this factor is evident in the powder diffraction pattern from Po (simple cubic) as shown
in Fig.2.
Fig.2
Fig.1
Lorentz factor

42. Intensity of powder pattern lines (ignoring Temperature & Absorption factors)







 




Cos
Sin
Cos
p
F
I 2
2
2 2
1
 Valid for Debye-Scherrer geometry
 I → Relative Integrated “Intensity”
 F → Structure factor
 p → Multiplicity factor
 POINTS
 As one is interested in relative (integrated) intensities of the lines constant factors are
omitted:
 Volume of specimen  me , e  (1/dectector radius)
 We have assumed random orientation of crystals  in a material with Texture relative
intensities are modified. So if a powder pattern has to be compared (both in intensity and
angle of peaks) with standard JCPDS/ICDD files, then the effect of texture has to be
removed first. Texture can be removed by: (i) making a fine powder and then annealing the
same (if required) or (ii) rotating the sample to all orientations (practically how this can be
done is not explained here!).
 I is really diffracted energy (as Intensity is Energy/area/time).
 Ignoring Temperature & Absorption factors is valid for lines close-by in pattern.