1549380323-Statistical Mechanics Theory and Molecular Simulation

160 Canonical ensemble

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d dsinψ

ψ

ψ

ki ks

θ θ

r 1

r 2

Fig. 4.4Illustration of Bragg scattering from neighboring planes in a crystal

wheredis the distance between the planes, the condition can be expressedas

2 dsinψ=nλ, (4.6.24)

whereλis the wavelength of the radiation used. However, we can look at thescattering
experiment in another way. Consider two atoms in the crystal at pointsr 1 andr 2 (see
figure), withr 1 −r 2 the relative vector between them. Letkiandksbe the wave
vectors of the incident and scattered radiation, respectively. Since the form of a free
wave is exp(±ik·r), the phase of the incident wave at the pointr 2 is just−ki·r 2 (the
negative sign arising from the fact that the wave is incoming), while the phase atr 1
is−ki·r 1. Thus, the phase difference of the incident wave between the two points is
−ki·(r 1 −r 2 ). Ifθis the angle between−kiandr 1 −r 2 , then this phase difference
can be written as

δφi=−|ki||r 1 −r 2 |cos (π−θ) =|ki||r 1 −r 2 |cosθ=

2 π

λ

dcosθ. (4.6.25)

By a similar analysis, the phase difference of the scattered radiationbetween points
r 1 andr 2 is

δφs=|ks||r 1 −r 2 |cosθ=

2 π

λ

dcosθ. (4.6.26)

The total phase difference is just the sum

δφ=δφi+δφs=q·(r 1 −r 2 ) =

4 π

λ

dcosθ, (4.6.27)

whereq=ks−kiis themomentum transfer. For constructive interference, the total
phase difference must be an 2πtimes an integer, giving the equivalent Bragg condition

4 π

λ

dcosθ= 2πn

2 dcosθ=nλ. (4.6.28)

Sinceθ=π/ 2 −ψ, cosθ= sinψ, and the original Bragg condition is recovered.