Page 31 - Modern physical chemistry

20 Structure in Solids

1.3 Show that in a given crystal the density of atoms in a reflecting plane is proportional to the
spacing between successive planes in its set.
1.4 If a (123) plane intersects the y axis at b, where does it intersect the z and x axes?
1.5 If equivalent solid spheres are packed to form the diamond lattice, what fraction of the total
volume is occupied by the spheres?
1.6 X rays 1.540 A in wavelength were reflected by powdered copper at the following O's:
21.65°,25.21°,37.06°,44.96°,47.58°.
What kind of cubic lattice does the metal possess? What is the length of an edge of its unit cell?
1. 7 At room temperature, sodium assumes a body-cantered cubic structure with unit cell edges
4.2906 A in length. When cooled, it changes form. At -195° C, the unit cell edges are still
mutually perpendicular, but each has become 5.350 A long. The density has increased by
4.0%. Determine the lattice which the metal possesses at the low temperature.
1.8 Consider a sodium chloride lattice in which the cations touch neighboring anions but the
anions do not touch each other(a stable situation). Suppose that the radius of each anion
is increased while everything else is kept constant. Calculate the radius ratio at which the
anions contact each other.
1.9 If a crystal exhibited periodic fivefold symmetry about parallel axes, the smallest nonzero
translation a in one direction could be rotated by ±2n / 5 radians to get other possible tran-
sitions. Show that such translations can add to give a nonzero resultant shorter than a. What
does one conclude?
1.10 Show how a face-cantered cubic lattice can be described. as a rhombohedral lattice and cal-
culate the angle a between the pertinent rhombohedral axes. Vector algebra may be employed.

1.11 A grating with 320 lines per centimeter was oriented so that th~ lines were perpendicular
to an X-ray beam while the angle between the plane of the lines and the beam was 6' 0". If
first- order diffraction appeared at a reflection angle of 12' 21", what was the wavelength
of the X rays?
1.12 In figure 1.4, consider the distance between atoms along each dashed line to be c and the
angle of reflection to be ¢>, rather than 9, the angle of incidence. Alter equations (1.4) and
(1.1) to fit this situation. For what set of crystal planes are the diffracted rays the result of
specular reflection satisfying the Bragg condition?
1.13 To the points in a simple cubic lattice is added a point at the middle of the base of each
cube. What is the symmetry of the resulting lattice?
1.14 X rays of given wavelength reflect from NaF at a Bragg angle of 8° 47', while the corre-
sponding reflection from KF is at 7° 40'. Calculate the ratio of the molar volume of KF to
the molar volume of NaF.
1.15 If uniform solid spheres are packed to form the simple hexagonal lattice, what fraction of
the total volume is occupied?
1.16 On entering a crystal, X rays are refracted slightly. The refractive index is given by
s: _ cos 0 _ A
I -u-----,
cosO' A'
where 0 is the angle of incidence and 9' the angle of refraction measured from the crystal
surface, while A is the external and A' the internal wavelength. Correct the Bragg equation
for this effect.
1.17 X rays 1.540 A in wavelength were reflected most strongly by powdered molybdenum at the
following angles 9:

What kind of cubic lattice does the metal possess? What is the length of an edge of its unit cell?
1.18 If a-iron has a body-cantered cubic lattice with unit cell width 2.8605 A and density 7.865 g
cm-3, what is Avogadro's number?
1.19 Determine the radius ratio at which the cation is just small enough so the surrounding anions
begin to touch each other in the zinc blende structure.