Quantum States for an Ideal Localized Spin System

Consider a solid paramagnetic system of N particles, each with spin angular momentum tiS and associated magnetic dipole moment ft = -gpB S, given in terms of the g factor and the Bohr magneton //B = eh/lm. This expression applies to electron spins, and a similar relation may be written for nuclear spins. For electrons in our ideal spin system, we take g = 2. The minus sign in the expression that relates//to S shows that the magnetic dipole vector is antiparallel to the spin angular momentum vector for negatively charged electrons.

We assume that the spins are located on fixed lattice sites in a solid, and there are negligible interactions between the localized moments. Localization of the spins is important, as we shall see. For a single spin in applied magnetic field of induction В along the z-direction, the Hamiltonian operator is

where В is the local field seen by the spin. This is the operator form of Equation 2.27. For a paramagnetic system, the local field is very nearly the same as the applied field В = ц0Н along z, and we have

The energy eigenvalues of this Hamiltonian are simply multiples of the eigenvalues of Sz, with where ms = 5, 5-1, ..., -S. For S = there are two eigenstates, I and I- ^}, with energies

These states are depicted in Figure 4.4 for g = 2. The energy gap ДE increases linearly with B.

For nuclei, it is usual to represent the spin operator by /, and the nuclear magnetic dipole moment is given by // = у til. where y. the magnetogyric ratio for the nucleus of interest, can be positive or negative. For the present discussion, the form Equation 4.7 for the Hamiltonian and Equation 4.9 for the eigenvalues will be adopted.

Energy levels for a single electron spin — in a magnetic field B. The energy gap between the

FIGURE 4.4 Energy levels for a single electron spin — in a magnetic field B. The energy gap between the

2

two levels is 2ц,B.

For N non interacting dipoles, the total energy is the sum of the individual energies. The state of the system may be specified by giving the set of quantum numbers, (m„ m2,..., m„) or in Dirac notation l/n„ m2,..., m„). The subscript S has been dropped to simplify the notation. If the energy of the system E is fixed corresponding to fixed В and N, only certain combinations of quantum numbers are allowed subject to the constraint

where n+ denotes the number of spins in the lower energy --0 state and n_ the number in the higher

energy +^- state. (It is convenient from now on in our general discussion to denote the number of

spins in the lower energy state, with their dipole moment/i oriented along B, as n+.) For weak interactions between the spins, the individual quantum numbers can change with time through mutual spin energy exchange. However, the total numbers in each state n+ and n_ will stay constant because of the fixed total energy constraint.

The Number of Accessible Quantum States

The labeled quantum states for our two model systems permit us to count states in a particular energy range. The number of accessible states is denoted by Q.(E), and this is interpreted as the number of states in the energy range E to E + SE, where SE is a small interval that allows for some uncertainty in the fixed total energy E of the system. In terms of the energy density of state p(E), the number of accessible states is

In the calculation of p(E), allowance must be made for any degeneracy of states with the same total energy.

a. The Density of States for a Single Particle in a Box. In Section 4.3, the quantum states for N particles in a box of volume V are considered. The single particle states are from Equation 4.5 given by e„ = (n2b2/2m V2/3)(tr? + n2 + n|), with nx, ny, n, = 1, 2, 3, .... It is convenient to introduce a geometrical representation of the states in a space spanned by the values of the quantum numbers nx, ny, and nz, and this representation is shown in Figure 4.5.

The number of states N(e) in the range from 0 to e is easily obtained. Each state corresponds to a cell of volume unity in the quantum number representation. The total number

Representation of particle in a box states in quantum number space specified by nx, ny, nz= 1,2, 3,.... The octant of the sphere shown encloses states with particular values of n + n +

FIGURE 4.5 Representation of particle in a box states in quantum number space specified by nx, ny, nz= 1,2, 3,.... The octant of the sphere shown encloses states with particular values of n + ny +

of states is therefore given by the volume of the octant of a sphere as shown in Figure 4.5,

/ ~ л ч 1/2

with radius R = yn; + n~ + n; J , where the selected set of quantum numbers (inx, np and n.)

generates the spherical surface corresponding to energy e. This enumeration procedure gives the total number of states as

From Equation 4.5, we obtain (n2 +n; + n2) in terms of e and other quantities, and substitution in Equation 4.12 gives

The density of states follows directly by differentiation of N(e) with respect to e

This is an important result, which shows that p(e) ~e'n for three-dimensional single particle states. The number of states in a shell of thickness Se at e is

In Chapter 5, we make use of this expression for the number of accessible states.

Exercise 4.2

Estimate the number of single particle states N(e), with energies in the range from 0 to e = kBT,

for helium atoms in a box of volume 1 L at a temperature of 300 K. Compare the number of states N(e) with Avogadro's number NA.

We have shown in Equation 4.13 that the number of single particle states in the range from 0 to

e is given by the expression fV(e) = ^-(v// n2h3)(2mefn =(4n/3)(V/VQ). (Note that the number

N(e) increases as V increases, for constant quantum volume Vq.) The upper energy limit is, for conservative estimate purposes, based on the equipartition of energy theorem. For helium, the atomic mass is m = 4.002x1.660x10 27kg. For these values for e and m, we get N(e) 6x1030. This is much larger than Avogadro's number NA = 6.02x1023moM. The large ratio N(e)/NA supports the claim made in Section 4.2 that the single particle states are sparsely populated in the classical high- temperature, low-density limit.

b. The Density of States for N Noninteracting Particles in a Box. In the determination of p(E) for N particles, with total energy E, confined in a box of volume V, great care must be exercised. The particles are indistinguishable, and it is therefore physically impossible to count states as distinct that simply involve particle interchange or, in simple terms, it is not possible to label particles. A further complication is that the particles may be either fermions or bosons. Fermions obey the Pauli exclusion principle whereas bosons do not. However, in the classical limit of low densities and high temperatures, where the states are sparsely populated, it is legitimate to ignore the quantum statistics features, as we show in detail in Chapter 12.

For N particles, the total number of states with energy in the range from 0 to E, if we ignore indistinguishability for the moment, is estimated as N (E) ~ [MT)],V where e ~ EIN. This estimate combines all states for particle 1 with all the states for particle 2 and so on. Because of the indistinguishability of particles, the estimate is not correct and overcounts states by a large factor. To a good approximation, the indistinguishability problem can be overcome by multiplying by 1AV! to allow for those permutations of particles that give rise to indistinguishable states of the system. Our revised estimate of the number of states is N(E) ~ (l/N)[N(e)]N. This correction factor is introduced in a natural way in the classical limit of the quantum distribution functions, as discussed in Chapter 12.

With N(e) from Equation 4.13, we get

and using dc/d£ ~ 1 IN, we obtain the density of states from Equation 4.16 as

or

This gives the density of states to a good approximation as

The number of states in the range from £ to £ + <5£ follows immediately as

For future reference, it is convenient to obtain ln£2, (£) with use of an extremely useful approximation known as Stirling’s formula: In ZV!~ N N-N as given in Appendix A. With use of this result, we obtain

3 о

As a good approximation, the term In—8E is ignored in comparison with the other retained

terms that involve N ~1023 and are therefore very large.

A number of observations may be made concerning Q.(E). First, Q(E) corresponds to a very large number of states. In Q.(E) is of the order of N, and if for our estimate purposes we choose N of order Avogadro’s number, this implies that fl(E) ~ e10 , an extremely large number. Second, Q.(E) is a very rapidly increasing function of £ because Q(£)oc£3A72 These two features play an important role in the development of the statistical theory. The expression for In Q.(E) obtained above involves a number of approximations. However, a reliable expression on the basis of results obtained in Chapter 16 for a classical ideal gas leads to the very similar result

or

In Equation 4.19, VA = V/N is the volume per particle, and VQ is the quantum volume introduced in Section 4.3. We take (e) = E/N in introducing the quantum volume. In the classical limit, we have VA » VQ, and In Q(E) is clearly very large, of the order of the number of particles N, as expected. The expression for In Q(£) will prove useful in a later discussion of the entropy of an ideal gas in Chapter 5.

c. Accessible States for a System of N Noninteracting Spins. In Section 4.4, it is shown that a

spin ^ particle (electron or nucleus) with magnetic moment p in a magnetic field В has two

energy levels with separation 2/r.B. (To simplify the notation, we drop the subscript on pin the following discussion.) For N spins, the possible energies are shown in Figure 4.6, where the convention for specifying the numbers of spins in a state as given in Section 4.4 has been adopted.

The lowest energy state corresponds to all moments being aligned parallel to the field (or up) with n+ = N and the highest energy state, in turn, to all moments being aligned antiparallel to the field (or down) so that n_ = N. The density of states is constant for the N spin system and is given by

Energy levels for /V electron spins -i in a magnetic field B. The energy states are specified by

FIGURE 4.6 Energy levels for /V electron spins -i in a magnetic field B. The energy states are specified by

the spin quantum number sets as shown. The number of up moments aligned parallel to В is denoted by n+, with n_ = N- nt. Successive energy levels correspond to turning over a single spin with a change in n+, and consequently n_, by plus or minus 1, respectively.

which corresponds to just one state in the energy range 2pB. For localized moments in a rigid lattice, the spins are, in principle, distinguishable. Using clever imaging techniques involving particle beams, for example, it is feasible that we can determine the orientation of any given spin. Although this might be extremely difficult in practice, the fact that it is possible in a gedanken experiment means that the localized spins must be considered in a different way to the delocalized particles in a gas. The factor 1/7V! introduced to overcome overcounting of states in the case of the gas is not needed for localized, distinguishable spins.

The number of states in the range from E to E + 8E is Q.(E)= g(N) p(E) SE, with g(N) a degeneracy factor that allows for the permutation of up and down spins without altering n+ or n_. Note that

The factor g(N) is easily obtained and is simply the number of ways of arranging N objects, where n are of one kind with moment up (or spin down) and (N-n) are of another kind with moment down. The result is

( N

where denotes the binomial coefficient. The binomial distribution, which involves the

l " )

binomial coefficient, is discussed in Chapter 10 and Appendix B. Using Equation 4.22 for g(N) and Equation 4.20 for p(E) gives

For future reference, it is useful to obtain an expression for In Q(E), that is, In Q.(E) = In TV! — In (N- n)!-lnn!-ln(6E/2//#), and using Stirling’s formula, we get to a good approximation

The term ln(<5£/2//B) has been omitted because (SE/2/гВ) is of order unity, and the logarithm of this quantity will be negligible compared with the other terms, which are of order N. The right- hand side of Equation 4.24 does not contain E explicitly. From Equation 4.21, it is readily seen that

n = ^[N-(E/ цВ)] and (W-/t) = ^[Af + (£7jUfi)].

Substituting for n and (N-n) in Equation 4.24 leads to

Exercise 4.3

Differentiating the expression for In £2(£), given in Equation 4.25, with respect to E, and equating the result to zero show that an extremum (maximum) occurs at E = 0.

From Equation 4.25, d In £2(£)/d£ = ln[(/V/2-£/2/rS)/(/V/2 + E/2fiB)] = 0, and solving for E gives E/2yB = 0. The extremum occurs at E = 0 as predicted. By differentiating again, it is readily found that the extremum at E = 0 is a maximum.

A plot of In £2(£) as a function of E is given in Figure 4.7.

For E = 0, the function In Q.(E) has a maximum value of N In 2, as may be seen from Equation 4.25 or Equation 4.24 with n = N12. Q(E) increases, from a value of 1 (In Q(E) = 0) when n = N to a value of 2N for n = N12 and then decreases to 1 at the upper energy bound, NfiB. which corresponds to n = 0 with all moments down (spins up). For N~ NA. the maximum value for Q(£) is very large. It is clear that Q.(E) increases extremely rapidly with E for E < 0 and decreases again extremely rapidly for E > 0. This is an important characteristic that is used in establishing a connection between the entropy S and In LUE). Figure 4.8 gives a schematic representation of the variation of £2(£) with £.

The curve, shown in Figure 4.8 for N = 100, tends to Gaussian form and peaks more and more sharply at £ = 0 as N is increased. The sharply peaked feature of the distribution is of central importance in the discussion of accessible states given in Chapter 5.

Plot of In Q(E) versus E for a system of N spins —

FIGURE 4.7 Plot of In Q(E) versus E for a system of N spins —.

Plot of Q.(£) versus E for a spin system. The rapid increase near E = 0 is due to the explosive increase in the degeneracy factor g(n) given in Equation 4.22

FIGURE 4.8 Plot of Q.(£) versus E for a spin system. The rapid increase near E = 0 is due to the explosive increase in the degeneracy factor g(n) given in Equation 4.22.

 
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