PHZ3400 Midterm Solution

PHZ 3400 – Midterm Exam (with solution) – March 20, 2009

Problem 1

Give some examples of spontaneous symmetry breaking. How large must the system be to display spontaneous symmetry breaking? (10 points)

Examples of spontaneous symmetry breaking include ferromagnetism, antiferromagnetism, superconductivity, superfluidity, etc. Spontaneous symmetry breaking can only occur in an infinite system.

Problem 2

Describe the difference between a first order and a second order phase transition. Give examples of each type. (10 points)

A first order phase transition displays a jump in all quantities, and is characterized by a latent heat (e.g. of melting, evaporation, etc.). Examples are melting, evaporation, etc. A second order phase transition is continuous and usually corresponds to some form of spontaneous symmetry breaking. For examples see above.

Problem 3

What determines the Curie temperature of a ferromagnet? How does it depend on the coordination number of the corresponding crystal lattice? (10 points)

Within mean-field theory, the Curie temperature of a ferromagnet is proportional the magnetic exchange interaction and the coordination number of the lattice in question.

Problem 4

What is the physical principle behind the fact that atoms act as hard spheres, i.e. they have a “core” impenetrable to other atoms? (10 points)

The core electrons in an atom are prevented from all occupying the lowest electronic state by the Pauli exclusion principle. The same mechanism also prevents the electrons from another atom to penetrate the atomic core, which thus acts essentially as an impenetrable sphere.

Problem 5

Consider a one dimensional metal with electronic density n.

a. Determine the relation between the Fermi energy and the density in this case. (15 points)

$E(k)={\frac {\hbar ^{2}k^{2}}{2m}}$

$k=2\pi m/L\;$; $m=0,\pm 1,\pm 2,...$

$n={\frac {N}{L}}2\sum _{k<|k_{F}|}={\frac {1}{\pi }}\int _{o}^{k_{F}}dk={\frac {k_{F}}{\pi }}.$

Here the extra factor of 2 is due to spin

$E_{F}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2m}}.$

b. Determine the ground state kinetic energy as a function of density. (15 points)

The energy density (per unit length) is:

$\varepsilon ={\frac {2}{L}}\sum _{k<|k_{F}|}{\frac {\hbar ^{2}k^{2}}{2m}}={\frac {1}{\pi }}\int _{o}^{k_{F}}{\frac {\hbar ^{2}k^{2}}{2m}}dk={\frac {\pi ^{2}\hbar ^{2}}{6m}}n^{3}$.

Problem 6

Consider a one dimensional vibrational system consisting with atoms of mass m connected by harmonic springs with spring constant K.

a. Determine the highest possible vibrational frequency of this system. (5 points)

The sound mode dispersion for this system is given by

$\omega (k)=2{\sqrt {K/M}}sin(ka/2)$

The highest frequency corresponds to the edge of the Brillouin zone $k_{max}=\pi /a$. This gives

$\omega _{max}=2{\sqrt {K/M}}$.

b. Sketch the motion of atoms corresponding to that mode. (5 points)

Here $u_{n}(k)\sim e^{ikna}\;$, with $k=\pi /a\;$. Therefore

$u_{n}(k)\sim e^{i\pi n}=(-1)^{n}\;$.

The neighboring atoms thus move out-of-phase (opposite direction from each other).

c. Determine the low frequency speed of sound. (5 points)

At $k\rightarrow 0$, we find $\omega (k)\approx \left(a{\sqrt {K/M}}\right)k,$ or

$c=a{\sqrt {K/M}}$.

d. Determine the Debye temperature for this system. (10 points)

$n={\frac {N}{L}}={\frac {1}{2\pi }}\sum _{k<|k_{D}|}={\frac {1}{\pi }}\int _{o}^{k_{D}}dk={\frac {k_{D}}{\pi }}.$

k$_{B}T_{D}=\hbar \omega _{D}=\hbar ck_{D}=\hbar cn\pi =\hbar \pi a{\sqrt {K/M}}n$.

e. Compute the temperature dependence of the specific heat at low temperatures. (5 points)

The energy density (per unit length) for this system is given by

$\varepsilon (T)={\frac {1}{2\pi }}\sum _{k<|k_{D}|}{\frac {\hbar ck}{e^{\beta \hbar ck}-1}}$

For low temperatures, we can conver the sum into an integral and extend the upper integration limit to infinity. We get:

$\varepsilon (T)\approx {\frac {k_{B}^{2}T^{2}}{\pi \hbar c}}\int _{o}^{+\infty }dx{\frac {x}{e^{x}<-1}}\sim T$.

We conclude that in this one-dimensional system

$C_{V}\sim T.\;$

f. Compute the temperature dependence of the specific heat at high temperatures. (5 points)

At high temperatures $(k_{B}T>>\hbar ck_{D})$, the expression of the energy density reduces to

$\varepsilon (T)\approx nk_{B}T$,

so the specific heat is

$C_{V}={\frac {\partial \varepsilon (T)}{\partial T}}=nk_{B}$

This is a one-dimensional version of the Dulong-Petit Law.