5-XRD.pdf

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Diffraction I: X-ray Diffractometry (XRD)
Elastic Scattering (NO ENERGY LOST):
• High resolution measurements are feasible of:
• distances between atomic planes
• Planes present or missing (symmetry)
• Atoms regularly present or missing in the
planes
• Uniformity (sample quality)
• Even particle or grain size
• Note: All measurements with XRD determine
average atomic positions.

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Spectrum (point detector)
•Peak positions, intensities, and breadth act as a fingerprint to the crystal, its
symmetry, the atomic spacing, the atoms involved, and relative fractions.
•Most systems have been studied by someone else, and catalogued, so one
can simply compare data to these standards to identify an unknown.
n = 2d(sin)
Theta/2 Theta Diffraction Spectrum

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Diffraction
Can be thought of/described in 3 ways:
• Graphical: Constructive interference
• Mathematical: Elastic scattering
• Intuitive: Simple graphical

takenfrombdhuey
B. D. Huey
Outline
• Diffraction Review
– Graphical diffraction derivation
• Diffraction Conditions
– d spacings
– Reflexions and forbidden reflexions
• Mathematical diffraction derivation
– Structure factor
– exp(2*pi*i) equation
– Lattice+basis approach
• Diffraction equation terms
• Diffraction peak analysis

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B. D. Huey
Graphical Approach
• X-rays generally considered to behave as waves (not
particles).
– The phase is shifted due to the path length.
• Diffraction is detectable when there is constructive
interference between parallel waves ‘bouncing off’ of
consecutive planes of atoms along a given sample
orientation.
– Full 360° phase shifting satisfies diffraction conditions (‘allowed
reflexion’).
– Half phase shifting (180°) causes zero diffraction (‘forbidden
reflexion’).
• How to identify circumstances for constructive
interference?

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Constructive Interference
n = 2d(sin)
Bragg’s Law:
 
 
)
sin(
2
)
(
sin
2
)
sin(
/
)
(
sin
2
1
1
))
(
sin
2
1
(
)
sin(
/
))
(
sin
2
1
(
)
2
cos(
2
2
2
2












d
n
d
n
x
n
x
x
y
x
so
d
x
and
x
x
y
y
x
n
















• For diffraction
we need the
angle,
wavelength,
and d spacing
so that
multiple
reflected
beams
constructively
interfere.
• As drawn, will
there be any
diffraction?
– out of phase,
not in phase,
so NO! Note: n=integer

takenfrombdhuey
B. D. Huey
Bragg’s law
For a theta/2theta experiment:
• We usually know wavelength
• We usually measure theta for specific planes that diffract
– Angle between incident beam and atomic planes in a sample at
which diffraction occurs.
• This can thus reveal the ‘d-spacings’ (distances between
atomic planes)
– Every repeating plane in a crystal has it’s own d-spacing.
– Different crystal symmetries have different possible planes.
– Thus provides a fingerprint of the composition of our sample.
n = 2d(sin)

takenfrombdhuey
Generalized d-spacing calculation
1
d2

1
V2
(S11h2
S22k2
S33l2
2S12hk 2S23kl 2S31
lh)
V  abc 1cos2
 cos2
 cos2
 2coscoscos
S11 b2
c2
sin2

S22 c2
a2
sin2

S33  a2
b2
sin2

S12  abc2
coscos cos
 
S23  a2
bc coscos cos
 
S31  ab2
c cos cos cos
 
To determine the d-spacing between arbitrary atomic planes in an
arbitrary crystal, use the following (not used in this class, but good to
know for some of you):

takenfrombdhuey
Simple Lattice Parameters and d Spacings
Triclinic: 1
d2

1
V2
S11h2
S22k2
S33l2
 2S12hk  2S23kl S13hl


 


Cubic: 1
d2

h2
k2
l2
a2
Tetragonal: 1
d2

h2
k2
a2

l2
c2
Orthorhombic: 1
d2

h2
a2

k2
b2

l2
c2
Hexagonal: 1
d2

4
3
h2
hk k2
a2









l2
c2
Monoclinic: 1
d2

1
sin2

h2
a2

k2
sin2

b2

l2
c2

2hl cos
ac








As complex as we will get in this
class
Note: hkl are indices of a given plane, eg. (001) (110) etc, and MUST be integers.
Also note: abc are lattice parameters, αβγ are angles, S and V are defined next.

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B. D. Huey
What is the d spacing for these simple examples
(assume a primitive cubic specimen)?
(hkl):
• (100) cubic
• (110) cubic
• (111) cubic
• (200) cubic
• (100) tetragonal
• (001) tetragonal
st
Tetragonal: 1
d2

h2
k2
a2

l2
c2

takenfrombdhuey
Primitive (Simple Cubic) Reflexions
{400}
-
{321}
{320}
{222}
{311}
{310}
{221}/{300}
{220}
-
{211}
{210}
{200}
{111}
{110}
{100}
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
h2+k2+l2 Primitive hkl
Simple Cubic
• Diffraction is possible for any atomic
planes (d-spacings) in a crystal
• Known as a “Reflexion”
• Doesn’t mean diffraction occurs at
every angle—it cannot by the eq.
2θ
θ
incident
diffracted
1/d
n = 2d(sin)
d spacing diffraction angle
1
d2

h2
k2
l2
a2

takenfrombdhuey
Forbidden Reflexions
•“Forbidden Reflexions” with zero
intensity occur when a plane of atoms
is separated from its equal by another
plane of the same atoms (usually
uniformly shifted in x, y, and/or z).
•Nothing forbidden for simple
cubic.
•But for BCC, there is a plane
through the cube center that is
equivalent to cube faces (just
shifted).
•The reflexion for (100) is thus
“forbidden”
Nothing
forbidden for
simple cubic.
Some
forbidden for
BCC.

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B. D. Huey
Forbidden Reflexions for BCC
THUS no {100} “reflexions” for BCC (if blue=green atoms).
Diffraction from (100) is out of phase with diffrac from (200).
So, even if properly aligned at θ100, destructive interference
occurs for EVERY diffracted photon (on average).
Cube body centers (same atom
as cube corner, but shifted ½
unit cell above/below the {100}
planes)
Cube corners
{100}
± ± ±
± ± ±
± ± ±

takenfrombdhuey
400
-
321
320
222
311
310
221/300
220
-
211
210
200
111
110
100
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
h2+k2+l2 BCC
Primitive
400
-
321
-
222
-
310
220
-
211
200
-
110
-
-
-
h+k+l = even
Allowed/Forbidden Reflexions
Recognizing these patterns on diffraction data allows easy
identification of a structure.
Obviously {100} are
forbidden, but so are
{111}, {311}, etc. Not so
easy to visualize? Later
learn to calculate…

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B. D. Huey
Simple theta/2theta measurement
Monochromatic incident beam.
forbidden
forbidden
forbidden
Since several possible
peaks are missing,
following a particular
pattern, this material
MUST be something
else. BCC in this case.
There SHOULD be peaks at each of the red peaks, but
also at the blue arrows for a simple cubic system.

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B. D. Huey
400
-
321
320
222
311
310
221/300
220
-
211
210
200
111
110
100
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
h2+k2+l2 BCC
Primitive
400
-
321
-
222
-
310
220
-
211
200
-
110
-
-
-
h+k+l = even
FCC
st
-
-
• Is 100 allowed or
Forbidden for FCC?
• What about 110?

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Forbidden Reflexions
Selection Rules:
• Listed for each space group in the International
Tables for X-Ray Crystallography – Vol. A
• Simple arithmetic criteria for identifying allowed
and forbidden reflexions can thus be looked up.
• Or, they can easily be calculated.
• How does one know which reflexions are allowed
and which are not?
• Visual inspection tells you a few, but that won’t
help for a complicated crystal or higher order
planes.

takenfrombdhuey
Scattering by Unit Cells
• Each atom scatters an amplitude A, which is a
function of the atom itself (atomic number) and the
incident angle (). Aexpi  A cos i sin
 
• For diffraction from any given plane (hkl), the
contribution to the scattered amplitude from each
atom in the unit cell (located at relative position
[uvw]) is given by:
 
 
lw
kv
hu
i
f
i
A atom
atom 

 
 2
exp
exp
where hkl are the indices of the diffraction plane, and u,v,w is the
atomic position within the cell relative to the origin (thus ≤ [111])
For example: uvw=(½, ½, ½) for the body center
uvw=(0,0,0) for a corner atom [note: you
only need to identify 1 corner due to symmetry]
uvw=(0, ¼, 0.234) for an arbitrary position

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The Structure Factor
To determine the total diffraction intensity per unit cell,
simply add the contributions to the scattered amplitude by
each symmetrically unique atom in the unit cell.
This yields the Structure Factor F.
For a single element cell containing N atoms (all same type):
 
 
 


N
n
hkl lw
kv
hu
i
f
F
1
2
exp 
•What are the symmetrically
unique positions uvw for BCC?
•(0,0,0) AND (½,½,½)
•The term ‘f’ depends on the
atom itself.

takenfrombdhuey
B. D. Huey
Summary
• Graphical diffraction derivation
• Diffraction Conditions
– d spacings
– Reflexions and forbidden reflexions
• Mathematical diffraction derivation
– Structure factor
– exp(2*pi*i) equation
• Reading: Rest of Diffraction Chapter
• HW1 due Tuesday

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