Statistical Mechanics of Ideal Paramagnetism

from Statistical Physics using Mathematica
© James J. Kelly, 1996-2002

Introduction

The thermodynamics of ideal paramagnetism is investigated using the canonical ensemble.  We consider a system of N noninteracting spin-j particles subject to an external magnetic field.  We compare the thermodynamic functions predicted by quantum and classical statistical mechanics in considerable detail.  To facilitate this comparison, it is useful to define an alignment variable x which quantum mechanically is the ratio between the magnetic quantum number (with respect to the field axis) and the spin and classically is the cosine of the angle between the magnetic moment and the magnetic field.  Hence, in both approaches x ranges between -1 and +1, but is discrete quantum mechanically and continuous classically.  Thus, the single-particle partition functions for these two theories become:

\(quantum\ theory\ \ ⟹\ \ Z\_1 = ∑\+\(m = \(-j\)\)\%j\((Exp[x\ y] /. {x -> m\/j})\)\)

classical  theory  ⟹  (Subscript[Z, 1]) = 
      2 π (∫_(-1)^1) Exp[
              x y] dx

where y = μ B/(Subscript[k, B])  T.  We evaluate the thermodynamic functions for both approaches below and demonstrate that the quantum theory approaches the classical in the limit of large spin.  We also evaluate the high and low temperature behaviors.  Note that the case of spin-(1/2) is analyzed in spin-half.nb using the microcanonical ensemble.

The present treatment neglects mutual interactions between atoms.  At sufficiently low temperature these mutual interactions promote spontaneous magnetization in the absence of an external magnetic field.  The temperature at which a phase transition with spontaneous magnetization occurs is known as the Curie temperature, (Subscript[T, c]).  Substances with low Curie temperatures are classified as paramagnetic, while a few materials with Curie temperature above room temperature are classified as ferromagnetic.  A mean field model of spontaneous magnetization is presented in Weiss.nb.

Glossary:
	𝒩	= number of spins
	j	= angular momentum
	m	= magnetic quantum number
	x	= alignment of spin with magnetic field
	g	= gyromagnetic ratio
	(Subscript[μ, 0])	= magneton (e.g. Bohr or nuclear)
	μ	= magnetic moment for each spin = g j (Subscript[μ, 0])
	B	= external magnetic field
	(Subscript[k, B])	= Boltzmann constant
	T	= temperature
	β	= 1/(Subscript[k, B]) T 
	y	=  μ B / (Subscript[k, B])  T = magnetic/thermal energies
	τ	= j/y = dimensionless temperature parameter
	(Subscript[Z, 1])	= single-particle partition function
	S	= entropy
	U	= internal energy 
	F	= Helmholtz free energy
	(Subscript[C, B])	= heat capacity
	M	= magnetization
	χ	= magnetic susceptibility

To express the thermodynamic functions in terms of different choices of independent variables with minimal symbolic changes, we employ different arguments.  For simplicity,  we define τ = j/y as a dimensionless temperature parameter.  Thus, S[τ] and S[y]  express the entropy in terms of either the reduced temperature, τ, or the inverse temperature, y, respectively.

Initialization

Defaults, packages, and symbols

ClearAll["Global`*"];
Off[General::spell, General::spell1];

$DefaultFont = {"Times", 12}; 
$TextStyle = {FontFamily->"Times", FontSize->12, FontSlant->"Italic"};

Needs["Miscellaneous`PhysicalConstants`"];
Needs["Miscellaneous`Units`"];
Needs["Utilities`Notation`"];
Needs["Graphics`Master`"]

Symbolize[(Subscript[k, B])];Symbolize[(Subscript[C, B])];Symbolize[(Subscript[μ, 0])];Symbolize[(Subscript[Z, m])_]

Evaluate temperature scales

atomicScale = Convert[BohrMagneton,Joule/Tesla]/BoltzmannConstant

nuclearScale = Convert[NuclearMagneton,Joule/Tesla]/BoltzmannConstant

The temperature scale for nuclear magnetization is much smaller than for atomic magnetization because their elementary magnetic moments are inversely proportional to mass, which is much larger for the nucleon than for the electron.  Therefore, nuclear magnetization is usually achieved by aligning electron spins and then relying on the hyperfine interaction between the electron and the nucleus to transfer that polarization to the nuclear spin system.

Establish  transformation rules

Simplification with assumptions

MyAssumptions = {j > 0, (Subscript[k, B]) > 0, T > 0, 
          y > 0, (Subscript[μ, 0]) ∈ Reals, B ∈ Reals, 
          g ∈ Reals};
MySimplify = 
        Simplify[#, MyAssumptions, 
            Trig -> False] &;
MyFullSimplify = 
        FullSimplify[#, MyAssumptions, Trig -> False] &;

Rules for changing variables

yToT = {y -> ((μ B)/((Subscript[k, B]) T)), μ -> 
            j g (Subscript[μ, 0])};
TtoY = {T -> ((μ B)/((Subscript[k, B]) y)), (Subscript[μ, 0]) -> (μ/(j g))};
yToTau = {y -> (j/τ)};

dydT = -(y/T);
dydB = (y/B);

Quantum theory

Derive thermodynamic functions using canonical ensemble

The thermodynamic functions are derived using the canonical ensemble and are expressed as functions of y, the ratio between the basic units of magnetic and thermal energy.

Partition function and Helmholtz free energy

The single-particle partition function is obtained by summation with respect to the magnetic quantum number, represented here by the alignment variable x.  The sum is simply a restricted geometric series and is easily converted to a compact expression in terms of hyperbolic trigonometric functions, but Mathematica needs some guidance in order to obtain the most aesthetically pleasing expression.  The Helmholtz free energy is then obtained from the relationship F = -N (Subscript[k, B]) T Log[(Subscript[Z, 1])] for a collection of identical independent elements.  We then obtain the entropy, internal energy, and other thermodynamic functions directly.

(Subscript[Z, 1]) = (∑_(m = -j)^j)(Exp[
                  x y] /. {x -> (m/j)}) /. {(E^x)_ - 
                1 -> 2 (E^((x/2))) Sinh[(x/2)]} // 
        MySimplify

F[y] = -𝒩 (Subscript[k, B]) T Log[(Subscript[Z, 1])] /. TtoY // 
        MySimplify

F[T] = F[y] //.yToT//MySimplify

Magnetization and magnetic susceptibility

The magnetization is obtained by differentiating the free energy with respect to B.  The relative magnetization, M/N μ, is often called the Brillouin function, (Subscript[B, j])[y].  Here we label this function as Brillouin[j, y] to avoid any confusion with the magnetic field, B.

M[T]  =  -(∂_B) F[T] // MySimplify

M[y]  =  M[T]  /. TtoY

Brillouin[j_, y_] = M[y]/(𝒩 μ)

Plot[ Evaluate[Table[Brillouin[j, y], {j, (1/2), (9/2)}]], {y, 0, 10}, 
        PlotRange -> {{0, 10}, {0, 1}}, 
        PlotLabel -> "Brillouin Function", Frame -> True,  
        FrameLabel -> {"y",  "B[j,y]"}, 
        DisplayFunction -> $DisplayFunction];

The Brillouin function saturates earliest for spin-(1/2) and for larger spin approaches a limiting curve which we will find is identical with the classical model.  The limit of large y represents either strong magnetic fields or low temperatures where the spins become aligned with the external field.  For given values of g and (Subscript[μ, 0]), stronger fields are needed to align large spins because the magnetic moments μ  =  g j (Subscript[μ, 0]) are larger.  For high temperatures or small fields, corresponding to small y, the magnetization is linear in y.  Thus, we obtain Curie's law, M ∝  B/T.

χ[y] = (∂_y) M[y] dydB  // Simplify

χ[T] = χ[y] //.yToT //Simplify

Entropy

S[T] = -(∂_T) F[T]  // MySimplify

S[y] = S[T]/.TtoY//MySimplify

Internal energy and heat capacity

U[T] = F[T] + T S[T] // MySimplify

U[y] = U[T] /. TtoY

(Subscript[C, B])[T] = (∂_T) U[T]  // MySimplify

(Subscript[C, B])[y] = (Subscript[C, B])[T] /. TtoY

Special case: spin-(1/2)

In this section we verify that the familiar results for the special case of spin-(1/2) are recovered.  Recall that the spin-(1/2) case was analyzed in the notebook spin-half.nb using the microcanonical ensemble.

M[y] //. j -> (1/2)  // Simplify

χ[y] //. j -> (1/2)  // Simplify

U[y] //. j -> (1/2)  // Simplify

(Subscript[C, B])[y] //. j -> (1/2)  // Simplify

S[y] //. j -> (1/2)   // Simplify

F[y] //. j -> (1/2)  // Simplify

Examine thermodynamic functions of y

plotMvsY[k_] := 
      Plot[Evaluate[
          M[y]/(𝒩 μ) /. {j -> 
                k}], {y, 0.01, 5}, PlotLabel -> "Magnetization", 
        PlotRange -> {{0, 5}, {0, 1}}, GridLines -> None, 
        Frame -> True, FrameLabel -> {"y", "M/(Nμ)"}, 
        DisplayFunction -> Identity]

\(plotUvsY[k_] := 
      Plot[Evaluate[
          U[y]\/\(𝒩\ μ\ B\) /. {j -> k}], {y, 0.01, 5}, PlotLabel -> "Energy", 
        PlotRange -> {{0, 5}, {\(-1\), 0}}, GridLines -> None, 
        Frame -> True, FrameLabel -> {"y", "U/(NμB)"}, 
        DisplayFunction -> Identity]\)

\(plotCvsY[k_] := 
      Plot[Evaluate[
          C\_B[y]\/\(𝒩\ k\_B\) /. {j -> 
                k}], {y, 0.01, 5}, PlotLabel -> "Heat Capacity", 
        PlotRange -> {{0, 5}, {0, 1}}, GridLines -> None, 
        Frame -> True, 
        FrameLabel -> {"y", \*""\!\(C\_B\)/(\!\(Nk\_B\))""}, 
        DisplayFunction -> Identity]\)

\(plotCHIvsY[k_] := 
      Plot[Evaluate[\(χ[y]\ B\)\/\(𝒩\ μ\) /. {j -> k}], {y, 0.01, 5}, 
        PlotLabel -> "Magnetic Susceptibility", 
        PlotRange -> {{0, 5}, {0, 0.5}}, GridLines -> None, 
        Frame -> True, 
        FrameLabel -> {"y", "χB/(Nμ)"}, 
        DisplayFunction -> Identity]\)

\(plotSvsY[k_] := 
      Plot[Evaluate[
          S[y]\/\(𝒩\ k\_B\) /. {j -> 
                k}], {y, 0.01, 5}, PlotLabel -> "Entropy", 
        PlotRange -> {{0, 5}, {0, 2}}, GridLines -> None, 
        Frame -> True, 
        FrameLabel -> {"y", \*""S/(\!\(Nk\_B\))""}, 
        DisplayFunction -> Identity]\)

\(plotFvsY[k_] := 
      Plot[Evaluate[
          F[y]\/\(𝒩\ μ\ B\) /. {j -> k}], {y, 0.01, 5}, PlotLabel -> "Free Energy", 
        PlotRange -> {{0, 5}, {\(-10\), 0}}, GridLines -> None, 
        Frame -> True, 
        FrameLabel -> {"y", "F/(N μ B)"}, 
        DisplayFunction -> Identity]\)

yfig[k_, opts___Rule] := 
      GraphicsArray[{{plotMvsY[k], plotSvsY[k]}, {plotCHIvsY[k], 
            plotCvsY[k]}, {plotUvsY[k], plotFvsY[k]}}, opts]

Show[
        yfig[Range[(1/2), (5/2)], 
          GraphicsSpacing -> {-0.1, 0.1}]];

We find that the magnetization saturates at smaller values of y for small spins j.  Similarly, the peak of the magnetic susceptibility is obtained at smaller B/T for smaller spins for a given value g (Subscript[μ, 0]).  All paramagnetic systems exhibit a Schottky anomaly in their heat capacities, where the Schottky anomaly refers to the peak at finite temperature.  The heat capacity vanishes for large y because the entropy approaches zero when all spins are aligned and also for small y because the entropy is saturated; hence, the heat capacity must reach a maximum for an intermediate value of y that depends upon j.  The greater the spin, the smaller the temperature or greater the field at which the peak occurs for given g (Subscript[μ, 0]).

Examine thermodynamic functions of T

Here the thermodynamic functions are expressed in terms of reduced temperature, defined as τ = j/y = (Subscript[k, B]) 
            T/g (Subscript[μ, 0]) B.

M[τ] = M[y] /. yToTau;
U[τ] = U[y] /. yToTau;
S[τ] = S[y] /. yToTau;
F[τ] = F[y] /. yToTau;
(Subscript[C, B])[τ] = (Subscript[C, B])[y] /. yToTau;
χ[τ] = χ[y] /. yToTau;

\(\(plotMvsT[k_] := 
        Plot[Evaluate[
            M[τ]\/\(𝒩\ μ\) /. {j -> k}], {τ, 0.01, 5}, PlotLabel -> "Magnetization", 
          PlotRange -> {{0, 5}, {0, 1}}, GridLines -> None, 
          Frame -> True, 
          FrameLabel -> {"τ", "M/(Nμ)"}, 
          DisplayFunction -> Identity];\)\)

\(\(plotUvsT[k_] := 
        Plot[Evaluate[
            U[τ]\/\(𝒩\ μ\ B\) /. {j -> k}], {τ, 0.01, 5}, PlotLabel -> "Energy", 
          PlotRange -> {{0, 5}, {\(-1\), 0}}, GridLines -> None, 
          Frame -> True, 
          FrameLabel -> {"τ", "U/(NμB)"}, 
          DisplayFunction -> Identity];\)\)

\(\(plotCvsT[k_] := 
        Plot[Evaluate[
            C\_B[τ]\/\(𝒩\ k\_B\) /. {j -> k}], {τ, 0.01, 5}, PlotLabel -> "Heat Capacity", 
          PlotRange -> {{0, 5}, {0, 1}}, GridLines -> None, 
          Frame -> True, 
          FrameLabel -> {"τ", \*""(Subscript[C, B])/((Subscript[Nk, B]))""}, DisplayFunction -> Identity];\)\)

\(\(plotFvsT[k_] := 
        Plot[Evaluate[
            F[τ]\/\(𝒩\ μ\ B\) /. {j -> k}], {τ, 0.01, 5}, PlotLabel -> "Free Energy", 
          PlotRange -> {{0, 5}, {\(-10\), 0}}, GridLines -> None, 
          Frame -> True, 
          FrameLabel -> {"τ", "F/(NμB)"}, 
          DisplayFunction -> Identity];\)\)

\(\(plotSvsT[k_] := 
        Plot[Evaluate[
            S[τ]\/\(𝒩\ k\_B\) /. {j -> k}], {τ, 0.01, 5}, PlotLabel -> "Entropy", 
          PlotRange -> {{0, 5}, {0, 2}}, GridLines -> None, 
          Frame -> True, 
          FrameLabel -> {"τ", \*""S/((Subscript[Nk, B]))""}, 
          DisplayFunction -> Identity];\)\)

\(\(plotCHIvsT[k_] := 
        Plot[Evaluate[\(χ[τ]\ B\)\/\(𝒩\ μ\) /. {j -> k}], {τ, 0.01, 5}, 
          PlotLabel -> "Magnetic Susceptibility", 
          PlotRange -> {{0, 5}, {0, 0.5}}, GridLines -> None, 
          Frame -> True, 
          FrameLabel -> {"τ", "χB/(Nμ)"}, 
          DisplayFunction -> Identity];\)\)

Tfig[k_, opts___Rule] := 
      GraphicsArray[{{plotMvsT[k], plotSvsT[k]}, {plotCHIvsT[k], 
            plotCvsT[k]}, {plotUvsT[k], plotFvsT[k]}}, opts]

Show[
        Tfig[Range[(1/2), (5/2)], 
          GraphicsSpacing -> {-0.1, 0.1}]];

At high temperatures, the entropy approaches a maximum value N (Subscript[k, B]) Log[2  j + 1] determined by the total number of microstates for the collection of N independent spin-j atoms with 2  j + 1 magnetic substates each.  The greater the spin the higher the temperature or the smaller the field needed to approach maximum entropy for given g (Subscript[μ, 0]).  Thus, the location of the Schottky peak increases with j.

Limiting Behaviors

High Temperature

The Curie law of paramagnetism states that at high temperature the magnetic susceptibility is inversely proportional to temperature.  Hence, we define curie to be the constant of proportionality.  Notice that this constant is proportional to the ensemble average of (μ^2) and the quantum mechanical expectation value for (j^2) is, in fact, j(j + 1).

curie = 
      T Normal[Series[χ[y], {y, 0, 2}]] //. yToT  // 
        Simplify

Normal[Series[M[y], {y, 0, 2}]]/curie //. yToT // Simplify

Normal[Series[U[y], {y, 0, 2}]]/curie //. yToT // Simplify

Normal[Series[(Subscript[C, V])[y], {y, 0, 2}]]/curie //. yToT // 
      Simplify

The asymptotic value of the entropy should be proportional to the logarithm of the total number of states, N Log[2j+1].  Furthermore, since the internal energy is small, the free energy is negative and is dominated by entropy at high temperature.

Series[S[y], {y, 0, 2}]  // Normal /. yToTau // 
      Simplify

Series[F[y], {y, 0, 2}]  // Normal /. yToTau  // 
      Simplify

The accuracy of Curie's law can be improved by replacing T -> T - (Subscript[T, c]), where (Subscript[T, c]) is the Curie temperature marking the onset of spontaneous magnetization.  A model of spontaneous magnetization is discussed in Weiss.nb.

Low Temperature

The essential singularity in Exp[(1/τ)] at τ=0 is handled by representing the exponentials in terms of a new variable, d=Exp[-(1/τ)], which is small at low temperatures and hence can be used for Taylor expansions for positive temperature.

\(\(lowTrules = {Exp[\(-\(1\/τ\)\)] -> d, 
          Exp[1\/τ] -> 1\/d, Exp[x_. \/τ] -> d\^\(-x\), 
          Log[d] -> \(-\(1\/τ\)\), 
          Log[1 + a_. \ d\^n_. ] -> a\ d\^n + 1\/2\ \((a\ d\^n)\)\^2, 
          Sinh[x_] -> 1\/2\ Exp[x]\ \((1 - Exp[\(-2\)\ x])\), 
          Cosh[x_] -> 1\/2\ Exp[x]\ \((1 + Exp[\(-2\)\ x])\), 
          Tanh[x_] -> 1 - 2\ Exp[\(-2\)\ x], 
          Coth[x_] -> 1 + 2\ Exp[\(-2\)\ x], 
          Sech[x_] -> 2\ Exp[\(-x\)]\ \((1 - Exp[\(-2\)\ x])\), 
          Csch[x_] -> 
            2\ Exp[\(-x\)]\ \((1 + Exp[\(-2\)\ x])\)};\)\)

The magnetization and internal energy become saturated at low temperature.

M[τ] //. lowTrules  // Simplify

U[τ] //. lowTrules  // Simplify

(F[τ] //. lowTrules  // PowerExpand // Simplify) //. 
        lowTrules  // Simplify

(S[τ] //. lowTrules  // PowerExpand // Simplify) //. 
        lowTrules  // Simplify

Large spin

The limit of large spin, where we expect classical behavior, is easily obtained using built-in rules for evaluating limits.

Limit[M[y], j -> ∞]  // Simplify

Limit[χ[y], j -> ∞]  // Simplify

Limit[U[y], j -> ∞]  // Simplify

Limit[(Subscript[C, B])[y], j -> ∞]  // Simplify

Classical theory

These classical results should be compared to the quantum theory in the limit of large spin for fixed magnetic moment μ.  The simplest way to obtain this limit is to express thermodynamic functions in terms of y and j and then to neglect the dependence of y upon j.  Alternatively, we can replace g -> 1/j to obtain a classical magnetic moment that is independent of quantum-mechanical spin.

Derive thermodynamic functions using canonical ensemble

Partition function and Helmholtz free energy

Classically, the spin can point in any direction and hence the partition function is obtained by integration of the Boltzmann factor over the unit sphere.

Z1Classical = 
      2 π (∫_(-1)^1) Exp[x y] \[DifferentialD]x   // 
        ExpToTrig

fClassical[y] = -𝒩 (Subscript[k, B]) T Log[Z1Classical] /. 
        TtoY

fClassical[
        T] = fClassical[y] /. yToT /. {g -> 1, 
          j -> 1, (Subscript[μ, 0]) -> μ}

Entropy

sClassical[y] = -(∂_y) fClassical[y] dydT //. TtoY // 
        Simplify

sClassical[T] = sClassical[y] /. yToT

Magnetization

mClassical[T] = -(∂_B) fClassical[T]  // 
        Simplify

mClassical[y] = mClassical[T] /. TtoY

chiClassical[y] = (∂_y) mClassical[y] dydB  // 
        Simplify

The relative magnetization in the classical model is called the Langevin function and is compared below with the Brillouin function for large spin.

Langevin[y_] = 
      mClassical[y]/(𝒩 μ)  // Simplify

Limit[Brillouin[j, y], j -> ∞] == Langevin[y]  // 
      Simplify

Internal energy

uClassical[T] = -(T^2) (∂_T) fClassical[T]/T  // 
        Simplify

cbClassical[T] = (∂_T) uClassical[T]  // Simplify

cbClassical[y] = cbClassical[T]//.TtoY

Therefore, all of the thermodynamic functions obtained from the quantum theory approach the corresponding classical results in the limit j → ∞, as expected.

Compare classical and quantum results

Curie law

Both the classical and quantum theories predict that the magnetization is inversely proportional to temperature at high temperature, but the proportionality constants are different.  The classical Curie constant depends only upon the magnetic moment, whereas  in the quantum theory the proportionality constant also depends upon spin.  Although, agreement is obtained in the limit of large spin, the classical constant is only one-third the quantum result for spin-(1/2).

curieClassical =  T Normal[Series[chiClassical[y], {y, 0, 2}]] //. 
          yToT /. {g -> 1/j, (Subscript[μ, 0]) -> μ}

curieRatio = 
      (curieClassical/curie) /. {μ -> g j (Subscript[μ, 0])}

The classical and quantum Curie constants differ most for small spins because there the classical and quantum expectation values for (j^2) differ most, being (j^2) classically or j(j+1) quantum mechanically.

curieRatio /. {j -> (1/2)}

To illustrate the approach to the classical limit, we compare the classical result (solid), with quantum results for j=(1/2)  and j=10.

\(\(Plot[
        Evaluate[{mClassical[y], M[y] /. {j -> 1\/2}, 
              M[y] /. {j -> 
                    10}} /. {𝒩 -> 1, μ -> 
                1}], {y, 0.01, 20}, PlotLabel -> "Magnetization", 
        PlotRange -> {{0, 20}, {0, 1.2}}, GridLines -> None, 
        Frame -> True, FrameLabel -> {"y", "M/(Nμ)"}, 
        PlotStyle -> {{}, Dashing[{0.02, 0.02}], 
            Dashing[{0.04, 0.01, 0.02, 0.01}]}, 
        PlotLegend -> {"Classical", \*""j=(1/2)"", "j=10"}, LegendPosition -> {0.3, \(-0.2\)}, 
        LegendSize -> {0.5, 0.4}, 
        DisplayFunction -> $DisplayFunction];\)\)

Heat capacity

The classical heat capacity approaches a constant for low temperature (large y), but in the quantum theory the heat capacity vanishes at low temperatures because absorption of heat is inhibited by the energy gap between the lowest two single-particle states.  The difference between the two theories is greatest for small spin, where the discrepancy occurs for small y.

At high temperature (small y), the heat capacities for both theories approach zero as the entropy approaches its maximum value asymptotically.  Thus, both series below show that the entropies have slopes ∂(S/∂)T ∝ 
        (T^(-3)) as T -> ∞ so that the heat capacity vanishes.

Series[S[y], {y, 0, 2}]  // Simplify // 
      Collect[#, {N,  (Subscript[k, B])}] &

Series[sClassical[y], {y, 0, 2}]  // Simplify  // 
      Collect[#, {N,  (Subscript[k, B])}] &

To illustrate the approach to the classical limit, we compare the classical result (solid), with quantum results for j = (1/2)and j = 10.

\(\(Plot[
        Evaluate[{cbClassical[y], 
              C\_B[y] /. {j -> 1\/2}, 
              C\_B[y] /. {j -> 
                    10}} /. {𝒩 -> 1, 
              k\_B -> 1}], {y, 0.01, 20}, 
        PlotLabel -> "Heat Capacity", 
        PlotRange -> {{0, 20}, {0, 1.2}}, GridLines -> None, 
        Frame -> True, 
        FrameLabel -> {"y", \*""(Subscript[C, B])/((Subscript[Nk, B]))""}, 
        PlotStyle -> {{}, Dashing[{0.02, 0.02}], 
            Dashing[{0.04, 0.01, 0.02, 0.01}]}, 
        PlotLegend -> {"Classical", \*""j=(1/2)"", "j=10"}, LegendPosition -> {0.0, \(-0.3\)}, 
        LegendSize -> {0.5, 0.4}, 
        DisplayFunction -> $DisplayFunction];\)\)

The Schottky anomaly does not occur in the classical theory because the energy spectrum is continuous in the classical limit; hence, in the quantum theory the maximum in the heat capacity occurs only as y → ∞ when j → ∞.   Of course, both theories predict that the magnetic contribution to the heat capacity vanishes as T → ∞ because the entropy approaches an asymptotic maximum at high temperatures.

Problems