Midterm 1

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

Problem 1

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

Problem 2

Give some examples of spontaneous symmetry breaking. How does it depend on the system size? Explain how it can happen, in apparent violation of the ergodicity hypothesis of Boltzmann. According to Boltzmann, what determines the probability of some microscopic configuration (10 points)

Problem 3

What determines the Curie temperature of a ferromagnet? How does it depend on the coordination number of the corresponding crystal lattice? Sketch the magnetization as a function of temperature in zero and in finite magnetic field. How does the field affect the transition? (10 points)

Problem 4

The mean-field equation of state for a ferromagnet is given by the expression

$m=\tanh \left(\beta Jzm+\beta h\right)$.

Consider the system at high temperature, where nonzero magnetization is found only in presence of an external magnetic field. Assume that both the field and the magnetization are small, and determine (linearizing the RHS in m and h) the magnetization as a function of temperature, in presence of a weak field. Calculate the magnetic susceptibility and sketch its inverse as a function of T. Where does it diverge? (30 points)

Problem 5

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)

Problem 6

Consider a molecule consisting of two different atoms. Assume that the energy difference between the ground states of the two atoms is $\Delta$, and that the corresponding hopping element between the two atoms is $t$. Calculate the energy of the bonding and the anti-bonding molecular state for this molecule. What is the energy needed to break the molecule, the energy of the covalent bond that binds the atoms in the molecule? (30 points)