Statistical Physics A Final exam one
8 February 2021
Please solve problems and send scans with your own solutions via e-mail to:
byczuk@fuw.edu.pl before 12:45 pm. Write in the subject: SPA EXAM ONE
Each problem is worth 10 points. Explanations and comments in writing, alongside mathema- tical formulation, are necessary and will count to final points. All problems must be solved by yourself and without any help from any other persons. In case of similar solutions both or more persons will have to be interrogated. It is allowed to use your own notes, lecture notes, and books. If you have questions or comments during the test time, please send them via e-mail to either krzysztof.byczuk@fuw.edu.pl or marta.waclawczyk@fuw.edu.pl. They will reply with answers, which will be send to all students via USOS e-mail. Therefore you are recommended to check your e-mails regularly during the test time.
Good luck!
Problem 1. Donor statistics A single impurity in a semiconductor can either be occupied by two electrons or by one electron or be empty. The energy, therefore, depends on the number of electrons and can be expressed as follows:
E(n↑, n↓) = X
σ
nσ + U n↑n↓,
where nσ = 0 or 1 if the state is not occupied or occupied by an electron with spin σ =↑ or
↓, respectively. Assume that > 0 and U > 0. Find: the grand partition function for Nimp
independent impurities, the free energy and the specific heat at constant Nimp and µ. Find and plot the average occupation ¯nσ(ω) at energy ω?
Problem 2. Particles on springs in a crystal A two-dimensional crystal contains Ncryst sites forming a square lattice. In this model, each lattice site is occupied by a non-relativistic particle of mass m. Assume that each particle is enclosed in a circle of radius R and is attached by a spring with a constant κ to the centre of the circle at the respective lattice site. Each particle thus moves in a potential
V (r) = (1
2κr2 for r < R ,
∞ for r> R,
where r = x2 + y2 is the distance from the lattice site. The radius R is less then the distance between the nearest neighbor lattice sites. The system is in a contact with the environment at temperature T .
(a) Which of the ensembles of statistical physics should be used to model this system?
(b) Calculate the partition function for the particles.
(c) Calculate the average total energy and the average potential energy per particle.
(d) What is the average total energy per particle in:
(1) the limit of strong coupling (κR2 kBT )?
(2) the limit of weak coupling (κR2 kBT )?
Compare the two results with the values expected from equipartition of energy.
Problem 3. Three-spin molecules A single molecule is composed of three spins 1/2 on a line
There is associated a magnetic moment 2µsi with each spin i-th, where µ is a positive constant (Bohr magneton) and si = ±1/2 with i = 1, 2, 3. The molecule is placed in an external magnetic field B. In the Ising model, used to represent this system, the Hamiltonian is
H = J s1s2+ J s2s3− 2µB (s1+ s2+ s3) ,
where J is the interaction constant. A gas of such molecules is in equilibrium at the tempera- ture T .
(a) Write down all possible microstates and their corresponding energies for a molecule.
(b) Find the canonical partition function Z1(T, B) for a molecule.
(c) Consider a special case of the absence of the external field (B = 0). What are probabilities of occurence of microstates for a molecule when T → 0? How do they change for kBT J ? (d) Consider a molecule in a weak, but nonvanishing magnetic field (0 < B J/µ). What are probabilities for different microstates in the limit T → 0?
(e) Consider a molecule in a strong field B J/µ. What are probabilities for different micro- states in the limit T → 0?
(f) Consider now an ideal gas of N such molecules. Find the mean energy and the total magnetic moment of the gas in the following cases:
(1) vanishing magnetic field B = 0,
(2) vanishing magnetic field B = 0 in the limit of kT J , (3) weak, non-zero magnetic field B J/µ,
(4) strong magnetic field B J/µ.
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