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Summary of lecture from 09.03.2018 Structure factors, systematic abscences, and XRD plots

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(1)

Summary of lecture from 09.03.2018

Structure factors, systematic abscences, and XRD plots

(2)

Form factor and structure factor

I

Structure factor

S

k

= X

atomj∈unit cell

f

j

(k) e

ik·rj

Proportional to peak intensity: |S

k

|

2

∝ I

k I

Atomic form factor f

j

(k)

"Strength of scattering from atom j "

f

j

(k) = Z

e

ik·r

V

j

(r ) dr

Fourier transform of scattering potential V

j

(r )

(3)

Monoatomic bases

S

k

= j · f (k) X

j

e

ik·rj

= j · f (k) X

j

e

i2π

(

hxj+kyj+lzj

) = j · f · S

klattice

with S

klattice

= P

j

e

ik·rj

the structure factor for the lattice and f (k) the form factor

lattice points S

klattice

systematic absence sc [0, 0, 0] 1 all h, k, l allowed bcc [0, 0, 0] 2 h + k + l must be even

1

2

,

12

,

12

fcc [0, 0, 0] 4 h, k , l all even or all odd

1

2

,

12

, 0

1

2

, 0,

12

(4)

Systematic absences

I

For some combinations of h, k, l : S

k

= 0

I

(hkl ) not allowed: destructive interference

I

absence of peak

(5)

Polyatomic bases

S

k

= S

klattice

· S

kbasis

example crystal basis S

lattice

S

basis

CsCl sc Cs: [0, 0, 0] 1 f

Cs

+ f

Cl

(−1)

h+k+l

Cl:

1

2

,

12

,

12

NaCl fcc Na: [0, 0, 0] 4 f

Na

+ f

Cl

(−1)

h+k+l

Cl:

1

2

,

12

,

12

If f

A

6= f

B

systematic absences same as for S

klattice

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