THE UNIVERSITY OF BRITISH COLUMBIA

PHYSICS 455

Lecture # 5 :

Wed. 15 Jan. 1997

The Boltzmann Distribution

LOGISTICS: Assignment 1 is nominally due today, but almost no one has turned it in yet. Since there are no solutions (or ``right answers'') to this first assignment, I can accept them any time right up to the last day of classes (there must be some limit!) - but of course it is not in your interest to let assigments pile up....

What do people think about having a take-home final exam? Several options are possible; if you can reach a consensus on the P455 WebCT Bulletin Board (which see!) in time for me to make plans, I will try to implement your choice.

HANDOUTS: Assignment 2 - due next Wed.

READING: You should be finished with Ch. 1 and Ch. 2 and starting on Ch. 3 now.

I. Recapitulate:

II. ADIABATIC DEMAGNETIZATION:

• For a large system of N independent spin-½ particles, each of magnetic moment m, in a magnetic field B, the equilibrium fractional magnetization was shown to have the value M/mN = mB/ in the limit of high temperature (it can never actually get larger than 100% spin polarization, of course). This corresponds to a net spin energy of U = Nm²B²/ so U² = N²m⁴B⁴ /². If we plug this back into = _o - U²/2m²B²N and pretend that the thermal equilibrium value of U is actually the value of U, we get

  = _o - N m² B² / 2 ² .

• Now suppose we prepare such a system in a field B and let it come to thermal equilibrium with a large heat reservoir (usually known as ``the lattice'' in recognition that most spin systems are incorporated into crystalline solids). This may take a while for the systems we are interested in, since the next step is to reduce the magnetic field so rapidly that the spins cannot ``relax'' to their new thermal equilibrium polarization during the change. That is, the microscopic state of the spin system does not change during this process, nor, therefore, does the entropy. Such a process is called isentropic or (equivalently) adiabatic.

• In the formula above, the entropy is a function only of the ratio of B to . Therefore if stays constant while B is reduced by some factor, then gets reduced by the same factor. In such a process the temperature of the spin system is abruptly decreased by the same factor as the magnetic field.

• If one then waits for the spin system to come into thermal equilibrium with the lattice, the lattice temperature will drop as the spin system warms up. This method is used to cool from a few milliKelvin (mK) - the practical lower limit of most ³He/⁴He dilution regrigerators (about which we will learn more later) - to a fraction of a mK, using paramagnetic salts.

• Because our high temperature approximation breaks down when mB approaches (the spins cannot be more than 100% polarized) the limit on the starting temperature _initial depends upon the magnetic moment m and the field B. For paramagnetic moments m µ_B = 9.27410^-21 erg/G, while k_B = 1.38110^-16 erg/K, so T_initial[K] B_initial[G]/10⁴. That is, for a 10,000 G [1 T] initial field one can cool from no lower than about 1 K.

  To start from even lower _initial one can use adiabatic nuclear demagnetization, for which typical magnetic moments are several thousand times weaker. Then one may start from 1 mK at 3 T and cool into the microKelvin (µK) range. The coldest temperatures ever achieved in laboratories on Earth use huge refrigerators several stories high (with elaborate vibration-free mountings) with a final stage based on adiabatic nuclear demagnetization.

III. Boltzmann Distribution and Partition Function:

1. (See text.) A large thermal reservoir is placed in thermal contact with a small system in a fully specified microscopic quantum state | > of energy . The total energy of the two combined systems is U and so the energy of the reservoir is U_R = U - . The probability P of such a configuration is proportional to the product of the multiplicities for the two systems: g_R for the reservoir and 1 for the fully specified state. Thus P g_R(U-) = exp(_R(U) - [_R /U_R] ) exp(-/) . (Recall the definition of ) This is called the Boltzmann factor.

2. We would like to normalize the Boltzmann factor to turn it into the actual probability P of finding the small system in that particular microscopic quantum state. To do so, we much divide by the sum of all such factors: P = exp(-/)/Z, where

   Z exp(-/)

   This is called the partition function for the small system.

3. A little algebra converts the formula for the thermal average energy of the small system from its obvious definition

   < U > exp(-/)

   into the somewhat more compact form

   < U > = ² (log Z)/ .

   Subsequently we shall tend to ignore the distinction between the actual energy U of such a small system and its thermal average energy < U >.

4. The same logic used to derive the Boltzmann factor can be used to explain the Overhauser effect (see text, Problem 5 in Ch. 3) is which, by contriving to add evergy to the reservoir every time it gives up an energy to the small system, a device can cause higher energy states of a system to be more likely than lower energy states even though the system is nominally in thermal equilibrium with a large reservoir.

5. In another problem (K&K Ch. 3, Problem 4) we see how to use the obvious definition of the rms deviation of the energy from its mean value to derive a formula for the variance of the energy at a given temperature:

   var U < (U - < U > )² > = ² U/ .