In most paramagnetic materials, the individual magnetic

Chapter 6, Problem 22P

(choose chapter or problem)

Problem 22P

In most paramagnetic materials, the individual magnetic particles have more than two independent states (orientations). The number of independent states depends on the particle’s angular momentum “quantum number” j, which must be a multiple of 1/ 2. For j = 1/ 2 there are just two independent states, as discussed in the text above and in Section 3.3. More generally, the allowed values of the z component of a particle’s magnetic moment are

where δμ is a constant, equal to the difference in µz between one state and the next. (When the particle’s angular momentum comes entirely from electron spins, δμ equals twice the Bohr magneton. When  orbital angular momentum also contributes, δμ is somewhat different but comparable in magnitude. For an atomic nucleus, δμ is roughly a thousand times smaller.) Thus the number of states is 2j + 1. In the presence of a magnetic field B pointing in the z direction, the particle’s magnetic energy (neglecting interactions between dipoles) is − µzB.

(a) Prove the following identity for the sum of a finite geometric series:

(Hint: Either prove this formula by induction on n, or write the series as a difference between two infinite series and use the result of Problem 1).

(b) Show that the partition function of a single magnetic particle is

where b = βδμB

(c) Show that the total magnetization of a system of N such particles is

Where coth x is the hyperbolic cotangent, equal to cosh x /sin h x. Plot the quantity vs. b, for a few different values of j .

(d) Show that the magnetization has the expected behaviour as T →  0.

(e) Show that the magnetization is proportional to 1/T (Curie’s law) in the limit T → ∞ (Hint: First show that cothwhen x ≪ 1.)

(f) Show hat for j = 1/ 2, the result of part (c) reduces to the formula derived in the text for a two-state paramagnet.

Problem 1:

This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. As in Section 2.2, the allowed energies of each oscillator arc 0, hf, 2hf, and so on.

(a) Prove by long division that

For what values of x does this series have a finite sum?

(b) Evaluate the partition function for a single harmonic oscillator. Use the result of part(a) to simplify your answer as much as possible.

(c) Use formula to find an expression for the average energy of a single oscillator at temperature T. Simplify your answer as much as possible.

(d) What is the total energy of the system of N oscillators at temperature T? Your result should agree with what you found in Problem 1.

(e) If you haven’t already done so in Problem 1, compute the heat capacity of this system aud check that it has the expected limits as T → 0 and T → ∞

Formula:

Problem 2:

In Problem 3 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately.

(a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. Explain why the factors omitted from the formula have no effect on the entropy, when N and q are large.

(b) Use the result of part (a) to calculate the temperature of an Einstein solid as a function of its energy. (The energy is U = qϵ, where ϵ is a constant.) Be sure to simplify your result as much as possible.

(c) Invert the relation you found in part (b) to find the energy as a function of temperature, then differentiate to find a formula for the heat capacity.

(d) Show that, in the limit T → ∞, the heat capacity is C = Nk. (Hint: When x is very small, ex ≈ 1 + x.) Is this the result you would expect? Explain.

(e) Make a graph (possibly using a computer) of the result of part (c). To avoid awkward numerical factors, plot C/Nk vs. the dimensionless variable t = kT/ϵ, for t in the range from 0 to about 2. Discuss your prediction for the heat capacity at low temperature, comparing to the data for lead, aluminium, and diamond shown in Figure Estimate the value of ϵ, in electron-volts, for each of those real solids.

(f) Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x3 in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than (ϵ/kT)2 in the final answer. When the smoke clears, you should find

Problem 3:

Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any large values of N and q, is approximately

The square root in the denominator is merely large, and can often be neglected. However, it is needed in below Problem 4 (Hint: First show that Do not neglect the  in Stirling’s approximation.)

Problem 4:

This problem gives an alternative approach to estimating the width of the peak of the multiplicity function for a system of two large Einstein solids.

a) Consider two identical Einstein solids, each with N oscillators, in thermal contact with each other. Suppose that the total number of energy units in the combined system is exactly 2N. How many different macrostates (that is, possible values for the total energy in the first, solid) are there for this combined system?

b) Use the result of above Problem 2 to find an approximate expression, for the total number of microstates for the combined system. (Hint: Traet the combined system as a single Einstein solid. Do not throw away factors of “large” numbers, since you will eventually be dividing two “very large” numbers that are nearly equal.

c) The most likely macrostate for this system is (of course) the one in which the energy is shared equally between the two solids. Use the result of above Problem 2 to find an approximate expression for the multiplicity of this macrostate.

d)You can get a rough idea of the “sharpness” of the multiplicity function by comparing your answers to parts (b) and (c). Part (c) tells you the Height of the peak, while part (b) tells yon the total area under the entire graph. As a very crude approximation, pretend that the peak’s shape is rectangular. In this case, how wide would it be? Out of all the macrostates, what fraction have reasonably large probabilities? Evaluate this fraction numerically for the case N = 1023.

Figure:

Measured heat capacities at constant pressure (data points) for one mole each of three different elemental solids. The solid curves show the heat capacity at constant volume predicted by the model used in Section 7.5, with the  horizontal scale chosen to best fit the data for each substance. At sufficiently high temperatures, Cv for each material approaches the value 3R predicted by the equipartition theorem. The discrepancies between the data and the sohd curves at high T are mostly due to the differences between Cp and Cv. At T = 0 all degrees of freedom are frozen out, so both Cp and Cv go to zero. Data from Y. S. Touloukian, ed., Thermophysical Properties of Matter (Plenum, New York, 1970).