Concise Physical Chemistry

c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

CRYSTALS 173

θ

θ θ

FIGURE 11.9 Bragg’s law for constructive reflection.

serves as well, but one that is some uneven multiple ofλwill interfere destructively.

The condition demanded of wavelength for the emerging radiation to be in phase is

( 1 , 2 , 3 ,...,)λ=nλ, wherenis an integer.

From this we can extract a triangle with a hypotenuse that is one-half the wave-

length of incoming radiation of wavelengthλ(Fig. 11.9). The opposite side of the

triangle so extracted isd, the distance between adjacent lines of atoms in the two-

dimensional model. We measureθ, the angle of constructive reflection. All other

angles give destructive interference and we do not see the radiation reflected. The

sine ofθis its hypotenuse divided by its opposite side, which isd,sowehave

sinθ=

nλ

2

(

1

d

)

or

d=

nλ

2

(

1

sinθ

)

which isBragg’s law.

11.4.1 X-Ray Diffraction: Determination of Interplanar Distances

The earliest and by far most common diffraction studies were on reflection and

diffraction of X-radiation. When radiation is reflected from two parallel planes of

a three-dimensional crystal, the reflected beams may be in phase, out of phase, or

something in between. Ifθis varied until the Bragg equation is satisfied, a sharp

maximum in reflected radiation is recorded by photographic or other means. The

(small) integernin the Bragg equation is called theorderof the reflection, anddis

the distance between reflecting planes.

On the basis of the distance between planes found at various different incident

angles, a repeating pattern is observed; from that, a repeatingunitof crystal structure

in three dimensions, theunit cell, is assigned. For example, ifdis found to be the

same in all three orthogonal directions, the cell is a simple cube; but ifdvalues differ,