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. 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 the temperature 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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(k) = \frac{\hbar^2 k^2}{2m}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = \frac{2\pi m}{L}\;} ; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 0, \pm 1, \pm 2,...}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = \frac{N}{L} = \frac{2}{L} \sum_{k < |k_F |} = \frac{2}{2 \pi } \sum_{k < |k_F |}\frac{2 \pi}{L} = \frac{2}{2 \pi} 2\int_0^{k_F} dk }

Here the extra factor of 2 is due to spin

The other factor of two comes from the addition of the negative values for k

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2k_F}{\pi} \Rightarrow k_F = \frac{n \pi}{2}}

Now plugging this value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F} into our original energy function gives:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(k_F) = \frac{\hbar^2 k_F^2}{2m}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow E_F = \frac{\hbar^2 \pi^2 n^2}{8m}.}

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

The energy density (per unit length) is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_F = \frac{2}{L}\sum_{k < |k_F |} \frac{\hbar^2 k^2}{2m} = \frac{2}{\pi}\int_o^{k_F} \frac{\hbar^2 k^2}{2m} dk = \frac{\hbar^2 k_F^3}{3m} = \frac{\pi^2 \hbar^2}{24m} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega (k) = 2\sqrt{\tfrac{K}{M}} sin (\tfrac{ka}{2})}

The highest frequency corresponds to the edge of the Brillouin zone Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{max} = \tfrac{\pi}{a}} . This gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{max} = 2\sqrt{\tfrac{K}{M}} } .

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

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_n (k) \sim e^{ikna}\;} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\tfrac{\pi}{a} \;} . Therefore

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \rightarrow 0} , we find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega (k) \approx \left( a\sqrt{\tfrac{K}{M}} \right) k,} or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = a \sqrt{\tfrac{K}{M}} } .

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = \frac{N}{L} = \frac{1}{2\pi} \sum_{k < |k_D |} = \frac{1}{\pi}\int_o^{k_D} dk = \frac{k_D}{\pi}.}

kFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _B T_D = \hbar \omega_D = \hbar c k_D = \hbar c n \pi = \hbar\pi a \sqrt{\tfrac{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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon (T) = \frac{1}{2\pi} \sum_{k < |k_D |} \frac{\hbar c k}{e^{\beta\hbar c k}-1}}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_V \sim T.\; }

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

At high temperatures Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (k_B T >> \hbar c k_D)} , the expression of the energy density reduces to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon (T) \approx n k_B T} ,

so the specific heat is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_V = \frac{\partial \varepsilon (T)}{\partial T} = n k_B}

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