Statistical Thermodynamics of a Paramagnetic

The properties of paramagnetic crystals are essential interest at extremely low T of a few Kelvin or less. A number of simplifying assumptions will be made, but the procedure is the same as in more complicated cases.

2.5 Application of Quantum Statistics 59

Paramagnetism results from the tendency of unpaired magnetic dipole moments associated with the electron orbital angular momentum, the elec-tron spin, or the both, to align themselves parallel to the applied field. Para-magnetism occurs in compounds containing transition metal ions that have either incomplete d shells (the iron, palladium and platinum groups) or in-complete f shells (the lanthanide and actinide groups). Both of the net spin and orbital magnetic moments may contribute to paramagnetism, but for an ion with an incomplete d shell, the effective orbital moment may be quenched by electrostatic interactions with its neighboring ions, leading to a predom-inant spin moment. Paramagnetism is due mainly to the spin angular mo-ments of the electrons. Every electron in an atom has not only an electric charge but also a magnetic moment μB of 1 Bohr magneton, being equal to 9.27× 10⁻²⁴A· m², as if the electron were a tiny sphere of electric charge spinning about an axis.

It was shown in Sec. 1.6 that the thermodynamic properties of a para-magnetic crystal could be calculated from a knowledge of G^ = H^− T S.

Using the methods of statistics, G^ can be derived in terms of T and the parameters that determine the energy levels of the atoms in the crystal. In a paramagnetic material, because the atoms can be labeled according to the positions they occupy in a crystal lattice, the system obeys MB statistics. As usual, the first step is to calculate Z, defined as Z =

jΔg_jexp(−εj/kT ).

Because of their oscillatory motion, the molecules have the same set of vibrational energy levels as those of any solid, and the total vibrational en-ergy constitutes the internal enen-ergy Uvib. In addition, the small interaction between the magnetic ions, and their interactions with the electric field set up by the remainder of the lattice, gives rise to an additional internal energy (of the ions only) Uint. Each magnetic ion in a paramagnetic crystal is a small permanent magnet and is equivalent to a tiny current loop as shown in Fig.

2.2.

Fig. 2.2 Magnetic ion of the magnetic moment μv equivalent to a small current loop.

The ion has a magnetic moment μv, which can be represented by a vector perpendicular to the plane of the loop. If the paramagnetic crystal is placed in a magnetic field B, where the moment vector makes an angle θ with the direction of B, a torque τ of magnitude μ^vB sin θ is exerted on the loop, in such a direction as to align μv in the same direction as B. Finally, the

60 Chapter 2 Statistical Thermodynamics

ions have a magnetic potential energy which, like the gravitational potential energy of particles in a gravitational field, is a joint property of the ions and the source of the field and cannot be considered as internal energy. The total magnetic potential energy is Emp.

The vibrational energy levels associated with internal magnetic and elec-trical interactions, and the potential energy levels are all independent. Z can be expressed as the product of independent partition functions Zvib, Zint, and Z_Hmag. Thus, Z = ZvibZintZ_Hmag. The magnetic ions constitute a subassem-bly, characterized by the partition functions Zint and Z_Hmag only, and they can be considered independent of the remainder of the lattice, and simply as a container of the subassembly.

The paramagnetic salts most widely used contain paramagnetic ions sur-rounded by large number of nonmagnetic particles. A typical example is Cr2(SO4)3·K2SO4·24H2O (chromium potassium alum). Its magnetic prop-erties are due solely to the chromium ions existing in the crystal. Cr⁺⁺⁺

has three unpaired electron spins and therefore a magnetic moment of 3μB. Besides the two chromium ions there are four sulphur ions, two potassium ions, forty oxygen ions, and forty-eight hydrogen ions. Hence there are a total of ninety-four nonmagnetic particles. The magnetic ions are so widely sepa-rated in the molecules that the interaction between them is negligibly small.

At the same time, the effect of the orbital motions of the valence electrons is quenched by the fields of neighboring ions. What remains is a net electron spin. Although Uintand Zint play important roles in the complete theory, we shall neglect them and consider that the total energy of the subassembly is its potential energy Ep only. Thus we solely consider Z_Hmag.

Empis the work that must be done to rotate the magnetic dipole from its zero energy position π/2 to θm,

For simplicity, only a subassembly of ions having a magnetic moment of 1 Bohr magneton μB is considered. The principles of quantum mechanics restrict the possible values of θm, for such an ion, to either 0^◦ or 180^◦, so that the magnetic moment is either parallel or antiparallel to the field (other angles are permitted if the magnetic moment is greater than μB). As an example, for an electron with a single net electron spin, there would be two possible energy levels, with spin quantum number ms = 1/2 (spin parallel to B) and ms = −1/2 (spin antiparallel to B). The corresponding values of cos θm are then +1 and −1, and the possible energy levels are −μBB and +μBB. The energy levels are nondegenerate. There is only one state in each level, but there is no restriction on the number of ions per state. Z_Hmag therefore reduces also to the sum of two terms,

Z_Hmag = exp

2.5 Application of Quantum Statistics 61

since by definition the hyperbolic cosine is given by cosh x = (1/2)[exp(x) + exp(−x)].

The thermodynamic properties of a two-level system exhibit the same features as those of more complex systems and are easier to calculate. Let N ↑ and N ↓ represent respectively the number of ions whose moments are aligned parallel and antiparallel to B. The corresponding energy is ε ↑= −μBB and ε↓= +μBB. The average occupation numbers in the two directions are then

N ↑= N

The excess of those ions in the parallel, over those in the antiparallel align-ment, is

The net magnetic moment M of the crystal is the product of μB of each ion and the excess number of ions aligned parallel to B. Thus, M =

N ↑ −N ↓

μB = N μB tanh (μBB/kT ). This is the magnetic equation of state of the crystal, expressing M as a function of B and T . Note that M depends only on the ratio B/T .

Since the limits of small and large x are tanh(x) = (e^x − e^−x)/(e^x+ e^−x)−−−→

x→ 0(1−1)/(1+1) = 0 and tanh (x) = (e^x−e^−x)/(e^x+e^−x)−−−−→x→ ∞e^x/e^x

= 1, in the case of μBB >> kT, tanh (μBB >> kT )→ 1 and M = Nμ^B. This is simply the saturation magnetic moment Msat, which would result if all ionic magnets were parallel toB.

At the other extremes of weak B and high T, (μ^BB/kT ) << 1, tanh (μBB/kT ) → μBB/kT, and M = (N μ²_B/k)(B/T ). This is just the exper-imentally observed Curie law, stating that M ∝ B/T or M = CCurieB/T in weak B and at high T where CCurie is the Curie constant. The meth-ods of statistics therefore not only lead to the Curie law, but also provide a theoretical value of CCurie= N μ²_B/k.

We now calculate other thermodynamic properties of the system. Emp

can then be found from

The energy is−Nμ^BB at T = 0 and approaches zero asymptotically at high T . Here higher T produces increased randomization of the dipole moments and Empgoes to zero.

62 Chapter 2 Statistical Thermodynamics

The magnetic contribution to the heat capacity also has distinctive char-acteristics, which is given by

CB=

∂Emp

∂T



B,N

= N k

μBB kT

2

sech²

μBB kT



. (2.63)

Figure 2.3 shows graphs of Empand CB(both divided by N k) as a function of kT /(μBB). The curves differ from the corresponding curves for the internal energy and heat capacity of an assembly of harmonic oscillators because there are only two permitted energy levels and the energy of the subassembly can-not increase indefinitely with increasing T. CBhas a fairly sharp peak known as a Schottky anomaly. It is called anomalous because the heat capacity usu-ally increases with increasing T , or stays constant. The anomaly is useful for determining energy level splitting of ions in rare-earth and transition-group metals.

Fig. 2.3 The specific potential energy and specific heat capacity at constant mag-netic intensity, both divided by Nk, for a paramagnetic crystal as a function of kT /(μBB).

Now we compare CB of the magnetic subassembly with C_V of the entire crystal. Letting T = 1 K and B = 1 T (a comparatively strong laboratory magnetic field), we have (kT /μBB) ≈ 1.5, sech²(μBB/kT ) = 0.66, and by Eq. (2.63), CB ≈ Nk(1.5)⁻² × 0.66 ≈ 0.29Nk. Assuming that there are 50 nonmagnetic particles for every magnetic ion, and taking a ΘD = 300 K as a typical value, we have from the Debye T³ law, C_V ≈ Nk(50) × (12π⁴/5)(1/300)³≈ 0.5 × 10⁻⁵N k. Hence, CB at T = 1 K is about 100000 times C_V. Much more energy is required to orient the ionic magnets than to increase the vibrational energy of the molecules of the lattice. It is this energy of the orientation which allows the cooling of the lattice during the process of adiabatic demagnetization described in Sec. 3.1.2.

The most important thermodynamic property of two-level system is the

2.5 Application of Quantum Statistics 63

entropy. For MB statistics, S = Emp/T + N k ln Z. Substituting Eqs. (2.61) and (2.62) into this expression, S = N k

 ln



2 coshμBB kT



−μBB

kT tanhμBB kT

 . Figure 2.4 is a graph of S/(N k), plotted as a function of kT /μBB. At low T, ln[2 cosh(μBB/kT )] = ln[exp(μBB/kT ) + exp(−μBB/kT )] → ln[exp(μBB/kT )] = μBB/kT and (μBB/kT ) tanh(μBB/kT ) → μBB/kT . Thus, S → 0 as T → 0. As T → 0 all the dipoles are in the lowest energy state pointing to a direction parallel to the applied magnetic field. There is only one possible microstate. As a result, W = 1 and S = k ln 1 = 0. At high T , the second term in brackets approaches zero, cosh(μBB/kT )→ 1, and S → Nk ln 2. This is exactly what we would expect. At the upper tem-perature limit, W = 2^N, the number of equally probable microstates, and S = N k ln 2. This corresponds to a pattern of random dipole orientations, involving equal numbers of parallel and antiparallel magnets in any chosen direction. In this disordered state S as a function of B/T only reaches the maximum. In a reversible adiabatic demagnetization, S and hence B/T re-main constants. Thus as B drops, T must decrease too, in agreement with the thermodynamic result.

Fig. 2.4 The entropy of a paramagnetic crystal.