A proton has tWo posslble spins States, spin up With energies e+ and Spin down with...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
an evaporation-cry .9. A simple model of a ferromagnet supposes that it consists of a macroscopic number N of so-called "spins." Each spin can have one of two orientations, either up or down. Assume here that the spins do not interact with one another; that is, their energies are independent of the up or down state of their neighbors Instead, for every up spin, there is an energy of -h and for every down one an energy of th (a)...
Question 1 A system of N identical non-interacting magnetic ions of spin Y%, has energy u tHo for each spin. μο is the magnetic moment in a crystal at absolute temperature T in a magnetic field B. For this system calculate: a) The partition function, Z. b) Free energy, F. c) The entropy. S d) The average energy, U e) The average magnetic moment, M
To first approximation, a neutron star may be treated as a non-interacting gas of neutrons (spin-1/2 fermions). ( a) Find an expression for the degeneracy (zero point) energy of the star as a function its mass M and radius R. (Don’t worry that our calculation for degenerate Fermi systems was done for particles in a cubical box, the same results apply to a sphere of the same volume.) (b) Write down an expression for the gravitational potential energy of the...
4. When an external magnetic field B is applied, a "spin-1" ion has 3 magnetic states with energies given Em=aBm, m=-1,0,1, where a is a constant of order a few times the Bohr magneton up = en/(2m). (Note: the notation here is quite different from that of Kittel & Kroemer who use "m" for the elementary magnetic moment which we have denoted a. In our terminology, m=ms is an integer quantum number: m=-1 labels the "spin down" state, m=0 labels...
Question 10 Statistical thermodynamics may be used to find the radiation pressure P for cavity (or black body) radiation in terms of the energy per unit volume u. (a) An ideal quantum gas comprises non-interacting identical particles with discrete quantum states labelled 1, 2, ...,r ,....The partition function is given by Z (T,V,N)- > exp(-B(n,&, + п,&, +...)} пп. (i) Define the symbols n1, n2,...,n,...and 81, 82, ..., Er,... (iiExplain why, for photons, the partition function may be expressed as:...
Consider a molecule that has two energy levels separated by e, where the ground state has a degeneracy of 2 and the excited state has a degeneracy of 3. (a) What is the expression for the partition function at temperature T? (b) What are the fractional populations of the two states at temperature T? (c) What is the internal energy per particle at temperature T?
The partition function at constant V, T is the sum of Boltzmann factors where the sum is over independent states, . Week #10's Lesson showed that Z is related to two state functions, U and F F=-kT lnZ Using these two relations, derive the relationships between Z and the following state functions and C (a) Entropy: S k In Z + kT (oln Z/oT)v (b) Pressure: P- kT (oln ZƠVr (d) Gibbs Free Energy: G-kTInZ + kTV(OlnZ/aV)T (e) Heat Capacity:...
Question 1 (8 marks in total) The deuteron is a bound state of a proton and a neutron. Treating nucleons as identical particles with spin and isospin degrees of freedom, the total state of the deuteron can be writ- ten space Ψ spin Ψ isospin. The deuteron has a total angular momentum quantum number J - 1 and a total spin S -1. Our goal is to determine the parity of the deuteron Q1-1 (1 mark) Show that the possible...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...