Chapter 2, Atomic Cohesion And Crystal Binding Video Solutions, Elements of Solid State Physics

Problem 1

Briefly answer the following questions.
(a) If positive and negative ions attract each other, why does the structure not collapse?
(b) Why is it that the cohesive energy of an ionic crystal is almost equal to the attractive Coulomb energy?
(c) What is the origin of cohesion in metals? Can it be described by interatomic potentials or covalent or any other bonds?

Emily Himsel

Numerade Educator

00:51

Problem 2

Consider a system of two argon atoms with position of the second atom with respect to the location of the first atom being denoted by vector $\mathbf{r}$ at certain instant of time. If for a fraction of second the centre of positive charges is shifted from the centre of negative charges (electrons) in the first atom, show that this would lead to an attractive potential of the system expressible as
$$
U=-\frac{\alpha_{\mathrm{e}} p^{2}}{\left(4 \pi \epsilon_{0}\right)^{2} r^{6}}\left(1+3 \cos ^{2} \theta\right)
$$
where
$\alpha_{\mathrm{c}}$ is the electronic polarizability of the argon atom
$p$ is the instantaneous electric dipole moment of the first atom $\theta$ is the angle between $\mathbf{p}$ and $\mathbf{r}$.

Prem Bijarniya

Numerade Educator

04:01

Problem 3

Show that the Madelung constant for one-dimensional array of ions of alternating sign with a distance $a$ between successive ions is given by $2 \ln 2$.

Suzanne W.

Numerade Educator

02:14

Problem 4

Calculate the cohesive energy per ion-pair of LiCl and KCl crystals using the following data:
$\begin{array}{lll} & \mathrm{LiCl} & \mathrm{KCl} \\ \text { Madelung constant } & 1.748 & 1.748 \\ \mathrm{Li}^{+}-\mathrm{Cl}^{-} / \mathrm{K}^{+}-\mathrm{Cl}^{-} \text {spacing } & 2.57 \AA & 3.14 \mathrm{~A} \\ \text { lonization energy of } & 5.4 \mathrm{eV} & 4.34 \mathrm{eV} \\ \mathrm{Li} / \mathrm{K} & & \end{array}$ Assume that the repulsive potential energy is negligibly small.

Suzanne W.

Numerade Educator

02:31

Problem 5

Assume that the repulsive potential energy of an ion-pair in an ionic crystal arises because of the action of the Pauli exclusion principle and that it can be expressed as $$
U_{\mathrm{rep} .}=\frac{B}{r^{n}}
$$
where $r$ is the separation between the ions of the opposite sign, $n$ is a large number and $B$ is a constant of the crystal. Show that the total potential energy per ion-pair for equilibrium separation $r_{0}$ is
$$
U=-\frac{\alpha \mathrm{e}^{2}}{4 \pi \epsilon_{0} r_{0}}\left(1-\frac{1}{n}\right)
$$
The observed cohesive energy of LiCl crystal is $6.8 \mathrm{eV}$ per ion-pair. Using the data given in Problem 2.4, evaluate the value of $n$ in the repulsive potential energy term (refer to the above problem).

Suzanne W.

Numerade Educator

01:25

Problem 6

Calculate the cohesive energy of hydrogen in $\mathrm{kJ}$ per mole using the Lennard-Jones parameters
$$
\varepsilon=5 \times 10^{-22} \mathrm{~J} \text { and } \sigma=2.96 \AA
$$
Assume that hydrogen molecules are spherical and crystallize into an FCC structure. Compare your result with the measured value ( $0.751 \mathrm{~kJ}$ per mole) and comment.

David Collins

Numerade Educator

01:08

Problem 7

In a crude model of alkali metals, the charge of each electron is treated as uniformly distributed over the volume of a sphere of radius $r_{\mathrm{s}}$ centred at each ion. Prove that the electrostatic energy per electron is then given by
$$
\begin{aligned}
U_{\text {coul }} &=-\frac{9 a_{0}}{5 r_{s}} \text { rydberg } \\
&=\frac{24.49}{r_{s} / a_{0}} \mathrm{eV}
\end{aligned}
$$
where $a_{0}$ is the Bohr radius.

Adriano Chikande

Numerade Educator

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Problem 8

Symbols $r^{>}$ and $r^{<}$ represent the radius of the bigger and smaller atom, respectively, in a crystal of diatomic basis. Show that the critical ratio,
$\frac{r^{>}}{r^{<}}=\frac{\sqrt{3}+1}{2}$ for the $\mathrm{CsCl}$ structure
$=2+\sqrt{6}$ for the zinc blende structure

Vipin Singh

Numerade Educator

15:25

Problem 9

Show that the longitudinal and shear wave velocities in the [111] direction in a cubic crystal are, respectively, given by
$$
\begin{array}{l}
v_{l}=\left[\frac{C_{11}+2 C_{12}+4 C_{44}}{3 \rho}\right]^{1 / 2} \\
v_{s}=\left[\frac{C_{11}-C_{12}+C_{44}}{3 \rho}\right]^{1 / 2}
\end{array}
$$

Dr. Rajveer Singh

Numerade Educator

07:28

Problem 10

Show that in a cubic crystal, the effective elastic constant for a shear across the (110) plane in the $[1 \overline{1} 0]$ direction is equal to $\frac{\left(C_{11}-C_{12}\right)}{2}$.

Dr. Rajveer Singh

Numerade Educator