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Problem Set 3 for Solid State II: Magnetic Susceptibilities and Ising Model, Assignments of Solid State Physics

University of Florida (UF)Solid State Physics

A problem set for a university course in solid state ii, focusing on magnetic susceptibilities and the ising model. Students are required to solve various problems using the provided reading materials and software from the sss package. Topics include diamagnetic and paramagnetic susceptibilities, magnetic field dependence of paramagnetic specific heat, and the ising model simulation.

Typology: Assignments

Pre 2010

Uploaded on 09/17/2009

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PHZ7427–Solid State II
Spring 2001
Problem Set 3
Feb. 7, 2001
Due: Feb. 21, 2001
Reading: Chs. 31-33, Ashcroft & Mermin; Ch. 4, Kittel
1. Diamagnetic/Paramagnetic susceptibilities. Problem 31-7, Ashcroft &
Mermin.
2. Magnetic field dependence of paramagnetic specific heat. Problem 31-
10, Ashcroft & Mermin.
3. Ising Model. This problem requires the use of the ”ising” software from the
SSS package. The Ising model is
H=−1
2JX
ij n.n.
SiSj−X
j
SiH,
where the first sum is over nearest neighbor sites (”n.n.”), and His an external
field.
a) Using PRESET 3, change the INITIAL CONDITIONS to M=1,andRUN
briefly at T=1.4J(T and H are measured in units of J - the exchange
energy). You see the occasional reversed spin resulting from the occasional
toss of the loaded coin that puts the spin in the less favorable energy state. It
is these few reversed spins that give a magnetization less than the saturation
value of 1.
b) Increasing the temperature in steps of .1J,goupto1.9J, collecting data for
M(T). Remember after each run to click the RESET button in the upper left
side of the window. If desired the speed of simulation could be adjusted to
20 or more so that the time per simulation decreases. It is a good ideea to
run several sweeps for each temperature and take the average of the readings
as the value for M.
c) Next you need to collect data for the interval 2 - 2.3 J at the suggested tem-
peratures 2, 2.1, 2.2, 2.25, 2.3 J. As the temperature increases approaching
Tc(Tc=2.27Jfor an infinite two-dimensional Ising system) the fluctuations
of M in time become larger and can affect the resuts. Therefore the ”number
of sweeps” cursor in the middle should be adjusted to 5.00 ×10−3to have
a longer recording time. You should also increase the speed to 100. Espe-
cially for 2.25J and 2.3J temperatures you will observe that the trace will
be relatively stable for a while, and then have a large fluctuation or even
change sign before stabilizing again for a little while. Try estimating the
”stable” values from the graph rather than using the <M >readout(take
the absolute values for M).
1
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Download Problem Set 3 for Solid State II: Magnetic Susceptibilities and Ising Model and more Assignments Solid State Physics in PDF only on Docsity!

PHZ7427–Solid State II Spring 2001 Problem Set 3 Feb. 7, 2001

Due: Feb. 21, 2001 Reading: Chs. 31-33, Ashcroft & Mermin; Ch. 4, Kittel

  1. Diamagnetic/Paramagnetic susceptibilities. Problem 31-7, Ashcroft & Mermin.
  2. Magnetic field dependence of paramagnetic specific heat. Problem 31- 10, Ashcroft & Mermin.
  3. Ising Model. This problem requires the use of the ”ising” software from the SSS package. The Ising model is

H = −

J

∑

ij n.n.

SiSj −

∑

j

SiH ,

where the first sum is over nearest neighbor sites (”n.n.”), and H is an external field.

a ) Using PRESET 3, change the INITIAL CONDITIONS to M = 1, and RUN briefly at T = 1. 4 J (T and H are measured in units of J - the exchange energy). You see the occasional reversed spin resulting from the occasional toss of the loaded coin that puts the spin in the less favorable energy state. It is these few reversed spins that give a magnetization less than the saturation value of 1. b ) Increasing the temperature in steps of. 1 J, go up to 1. 9 J, collecting data for M(T). Remember after each run to click the RESET button in the upper left side of the window. If desired the speed of simulation could be adjusted to 20 or more so that the time per simulation decreases. It is a good ideea to run several sweeps for each temperature and take the average of the readings as the value for M. c ) Next you need to collect data for the interval 2 - 2.3 J at the suggested tem- peratures 2, 2.1, 2.2, 2.25, 2.3 J. As the temperature increases approaching Tc (Tc = 2. 27 J for an infinite two-dimensional Ising system) the fluctuations of M in time become larger and can affect the resuts. Therefore the ”number of sweeps” cursor in the middle should be adjusted to 5. 00 × 10 −^3 to have a longer recording time. You should also increase the speed to 100. Espe- cially for 2.25J and 2.3J temperatures you will observe that the trace will be relatively stable for a while, and then have a large fluctuation or even change sign before stabilizing again for a little while. Try estimating the ”stable” values from the graph rather than using the < M > readout(take the absolute values for M).

d ) Using the results above make a graph for M(T) and demonstrate graphically that the deviation of M from its saturation value has a dependence e−A/T with A constant.

e ) Using the mean field theory derive a result for the dependence of (1-M) in the limit M ≈ 1. For a quantitative comparison add the prediction of the mean field theory to the plot from d). f ) From the data of point c) make a linear plot of M versus T and estimate a value for Tc. Is that a reasonable value?

g ) Use the mean field theory to determine the temperature dependence of M near Tc , or more precisely, the limiting behaviour of M(T) as T → 0. (Hint: use M = tanh

( H+4M T

) and expand the tanh function keeping the two lowest orders in its argument.) Plot the result on the graph of the preceding point. Did you obtain the expected behaviour? Could you give example of a reason why you didn’t get the correct behaviour?

  1. Paramagnetism of free Fermi gas. Using the Kubo formula, calculate the magnetization 〈μˆz〉 of a free Fermi gas to an external Zeeman term in the Hamiltonian H′^ = −

∫ dr′^ μˆz(r′)Bz (r′, t), (1)

where the magnetic moment operator is

μˆz(r) = μ 0 (ˆn↑(r) − nˆ↓(r)). (2)

Following the calculation of the Lindhard function in the notes and/or elsewhere, calculate the spin susceptibility χμz μz (q, ω) and show it is proportional to the same Lindhard function as the charge susceptibility, up to an overall factor. Show the susceptibility reduces to the Pauli susceptibility, χSz Sz (q → 0 , ω = 0 , T → 0) = μ^20 N 0.