MENA3100

Structure factors

The structure factors for x-ray, neutron and electron diffraction are similar. For neutrons and electrons we need only to replace by fj

(n) or fj(e) .

N

j

xjhklg fFF

1

)( 2exp( ))( jjj lwkvhui X-ray:

The coordinate of atom j within the crystal unit cell is given rj=uja+vjb+wjc. h, k and l are the miller indices of the Bragg reflection g. N is the number of atoms within the crystal unit cell. fj(n) is the x-ray scattering factor, or x-ray scattering amplitude, for atom j.

rj

ujaa b

x

z

c

y

vjb

wjc

The intensity of a reflection is proportional to:

ggFF

MENA3100

Example: Cu, fcc• eiφ = cosφ + isinφ • enπi = (-1)n

• eix + e-ix = 2cosx

N

jjhklg fFF

1

2exp( ))( jjj lwkvhui

Atomic positions in the unit cell: [000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ]

Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l))

If h, k, l are all odd then:Fhkl= f(1+1+1+1)=4f

If h, k, l are mixed integers (exs 112) thenFhkl=f(1+1-1-1)=0 (forbidden)

What is the general condition for reflections for fcc?

What is the general condition for reflections for bcc?