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Problem Set 3

1. The document is a problem set from an MIT physics course on the physics of solids. It contains 5 problems related to crystal lattices, ion interactions, and determining phase diagrams. 2. Problem 2 asks students to generalize the calculation of van der Waals attraction between oscillators to the case where the oscillators have different spring constants. 3. Problem 5 asks students to calculate a phase diagram as a function of parameters f and n, determining what ranges of those parameters result in crystals with CsCl, NaCl, or cubic ZnS structures.

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130 views1 page

Problem Set 3

1. The document is a problem set from an MIT physics course on the physics of solids. It contains 5 problems related to crystal lattices, ion interactions, and determining phase diagrams. 2. Problem 2 asks students to generalize the calculation of van der Waals attraction between oscillators to the case where the oscillators have different spring constants. 3. Problem 5 asks students to calculate a phase diagram as a function of parameters f and n, determining what ranges of those parameters result in crystals with CsCl, NaCl, or cubic ZnS structures.

Uploaded by

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Physics Department

8.231, Physics of Solids I Due on Wed., Sept. 27.

Problem set #3

1. Consider an orthorhombic C lattice whose conventional unit cell has a size a × b × c. Deter­
mine the reciprocal lattice by giving the fundamental translation vectors �b1 , �b2 , and �b3 , and
specifying the Bravais lattice type.

2. In the book (and in the lecture), the Van der Waals attraction is calculated for two harmonic
oscillators of same mass m and same spring constant K. Generalize that result to the case
where the spring constants of the two oscillators are different and given by K1 and K2 . (The
mass is still assumed to be the same.)

3. Assume that the interaction between two ions of charge qi and qj is given by
� �12
qi qj e2 a
Uij = + (1)
r a r

To see how good (or bad) is the above form of interaction, determine the values of a for NaF
and RbCl using the nearest neighbor separations in table 7 of the Kittel book on page 66. Then
calculate the lattice energy compared to free ions per NaF or RbCl. (Note e2 /1A ˚ = 14.39eV.)
Compare your result with the data in the table 7.
The values for γ6 and γ12 are given by

lattice: FCC SC BCC diamond

γ6 : 14.5 8.40 12.2 5.12

γ12 : 12.1 6.20 9.1 4.04

4. Problem 7 on page 86 in Kittel book.

5. (Due on Wed. Oct. 4)


A simplified interaction between two ions of charge qi and qj is given by
�� � � �6 �
11
qi qj e2 a a
Uij = +f − (2)
r a r r

which include both the Coulomb and a modified Van der Waals interaction (the exponent 12
is changed to 11). Let the charges of the two kinds of ions are given by ne and −ne. Here
we treat n as a continuous parameter. Depending on different values of f and n, the two
kinds of ions may form several kinds of crystals, which include CsCl, NaCl, and the cubic ZnS
structures. In this project, we like to calculate the phase diagram of the system. Determine
for what ranges of f and n that the ions form the CsCl, NaCl, or the cubic ZnS structures.
Present your result in a 2D phase diagram in f -n plane. (You might need to write a small
�
program to calculate γ11 = �j 1/pij 11 . Note that γ
11 is slightly larger then γ12 .)

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