Fundamentals of the Physics of Solids / 03-The Building Blocks of Solids

3.2 The State of Ion Cores

51

the same order are found for a great number of elements as well. Experimental data obtained at room temperature for some of them are listed in Table 3.5.

Table 3.5. Molar and specific magnetic susceptibility values (in SI units) for diamagnetic elements at room temperatures. For carbon, diamond data are given

		χmol [SI]		χmass [SI]
		10−12 m3/mol		10−9 m3/kg
Cu		−68.6		−1.08
Ag		−245		−2.27
Ga		−271		−3.9
C		−74.1		−6.17
Si		−39.2		−1.8
Ge		−146		−1.33
Sn		−470		−3.3
Sb		−1244		−10
Bi		−3520		−16.8
Se		−314		−4.0
Te		−478		−3.9

Alert readers have no doubt noticed that among diamagnetic materials there are not only covalently bonded semiconductors and insulators with closed electron shells but also semimetals, like bismuth, and a few metals, such as noble metals. In the latter electrons outside the closed d-shell can be considered practically free: they are responsible for electric conduction. While they give a positive, paramagnetic contribution to susceptibility,15 the total susceptibility receives a more important diamagnetic contribution from d-electrons.

3.2.6 Atomic Paramagnetism

When outer shells are not completely filled and thus J = 0, the atom has a permanent magnetic moment. If an external magnetic field is applied, this magnetic moment will be more likely to line up with the field than against it. Consequently a magnetization proportional to the magnetic field appears. This is called paramagnetic behavior. The value of paramagnetic susceptibility can be easily determined classically. In a magnetic field of induction B the energy of a classical permanent magnetic moment μ is −μ ·B. Using classical Boltzmann statistics, the probability that the magnetic moment points in the elementary solid angle dΩ around the direction given by the polar angles θ and ϕ is

15The susceptibility due to the electrons not bound to the ion core (free electrons) will be determined in Volume 2.

52	3 The Building Blocks of Solids
	P (θ, ϕ) dΩ =	1	eμ·B/kBT dΩ ,	(3.2.47)
		Z

where μ ·B = μB cos θ, and Z is some normalization factor. With this weight

factor, the thermal average of the magnetic moment is found to be			(3.2.48)
μ = Z		μ eμ·B/kBT dΩ .
1

If there are N magnetic moments in volume V , then magnetization is given by the density of the total magnetic moment, M = μ N/V , and susceptibility by its derivative with respect to magnetic field H – for which the approximation H = B/μ0 can be used. The components of the susceptibility tensor are thus

χαβ = V kBT				Z μαμβ eμ·B/kBT dΩ
	N μ0			1
								μβ eμ·B/kBT dΩ
				−Z2 μαeμ·B/kBT dΩ
						1
=	N μ0			μαμβ − μα μβ ! .				(3.2.49)
	V kBT

Thermal fluctuations of the magnetic moment are recognized on the righthand side. This is an example of the fluctuation–dissipation theorem which states that the linear response of the system to an external perturbation can be expressed in terms of the fluctuations of some quantity in thermal equilibrium.

The initial susceptibility is obtained by determining the thermal averages in (3.2.49) in thermal equilibrium, in the absence of any applied field. As free moments can point in any direction, μα = 0, and the various components are uncorrelated – that is, o -diagonal elements vanish in μαμβ . Equivalence of the three spatial directions implies that the diagonal elements are equal to one-third of the squared magnitude of the momentum,

μαμβ = 31 μ2δαβ .

(3.2.50)

Thus the magnetic susceptibility of a system made up of atoms with classical magnetic moments of magnitude μ is

χ =	N		μ0μ2	.	(3.2.51)
	V 3k T
			B

It was first observed by P. Curie16 (1895) that the susceptibility of paramagnetic materials varies inversely with temperature. This temperature dependence is called Curie’s law, and susceptibilities proportional to 1/T are

16Pierre Curie (1859–1906) shared the Nobel Prize with his wife, Marie Curie née Sklodowska (1867–1934), and Henri Becquerel (1852–1908); the Curies received the accolade “in recognition of the extraordinary services they have rendered by their joint researches on the radiation phenomena discovered by Professor Henri Becquerel”.

3.2 The State of Ion Cores

53

in general known as Curie susceptibilities. The above formula was, however, first derived by P. Langevin (1905), hence the name Langevin susceptibility is also used.

To obtain a more precise, quantum mechanical description of paramagnetic behavior, one has to focus on the second term in the Hamiltonian (3.2.27),

μB (L + |ge|S) · B .

(3.2.52)

Assuming that the applied magnetic field is weak, the contribution of this term is calculated in the first nonvanishing order of perturbation theory. For this one needs to know the matrix elements of L and S in the atomic state.

Because of spin–orbit interaction, L and S are not conserved separately: the state is determined by the magnitude of the resultant J = L + S and its component along the magnetic field. When the z-axis is chosen along the magnetic field direction, states are characterized by the quantum numbers L, S, J, and mJ – the last being the eigenvalue of Jz . The matrix elements necessary for determining the energy are obtained using the Wigner–Eckart theorem.17 Before formulating this theorem, the concept of vector operators has to be introduced first.

An operator V is called a vector operator if it is transformed as a vector under rotations – or, equivalently, if its commutation relations with the components of the total angular momentum are

[Jx, Vx] = 0 ,
[Jx, Vy ] = i Vz ,	(3.2.53)
[Jx, Vz ] = −i Vy ,

and other similar expressions obtained by cyclic permutation of the indices. Making use of the equality J = L + S, it is easy to show that both L and S satisfy the above conditions, i.e., they are vector operators.

The Wigner–Eckart theorem asserts that for fixed L and S, in the (2L + 1)(2S + 1)-dimensional subspace of the eigenfunctions of the total angular momentum operator, the matrix elements of any vector operator are proportional to the matrix elements of the total angular momentum operator, and the constant of proportionality is independent of the value of mJ :

L, S, J, mJ |V |L, S, J, mJ = α(L, S, J) L, S, J, mJ |J |L, S, J, mJ . (3.2.54)

In other words: within this subspace the vector operator V is equivalent to

the quantity αJ :
V ≡ α(L, S, J)J .	(3.2.55)

17C. Eckart (1930), E. P. Wigner (1931). Eugene Paul Wigner (1902–1995) was awarded the Nobel Prize in 1963 “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles”.

54 3 The Building Blocks of Solids

Let us now introduce the operator
P (L, S, J) =		(3.2.56)
	|L, S, J, mJ L, S, J, mJ | .

mJ

Exploiting the mutual orthogonality of angular momentum eigenfunctions it is readily seen that P 2(L, S, J) = P (L, S, J), and also that the completeness relation

P (L, S, J) =	|L, S, J, mJ L, S, J, mJ | = 1	(3.2.57)
J	J mJ

holds, thus for given L and S the operator P (L, S, J) projects onto the subspace of quantum number J. Moreover, this projection operator commutes will all three components of J . When (3.2.54) is multiplied by |L, S, J, mJ from the left and by L, S, J, mJ | from the right, summation yields

P (L, S, J)V P (L, S, J) = α(L, S, J)P (L, S, J)J P (L, S, J) .

(3.2.58)

To determine the coe cient α(L, S, J), consider the following matrix element of the scalar V · J :

	|		·	|	, mJ		(3.2.59)
	L, S, J, mJ	V		J L, S, J		.

Making use of the completeness relation, the fact that only those matrix elements of J are nonzero that belong to the same quantum number J, and the Wigner–Eckart theorem we have

L, S, J, mJ |V · J |L, S, J , mJ = α(L, S, J) L, S, J, mJ |J 2|L, S, J , mJ .

(3.2.60)

When α(L, S, J) is expressed, we arrive at

V =	V · J	J .	(3.2.61)
	J 2

in the space of the angular momentum eigenfunctions. Thus one can draw the intuitive conclusion that within this subspace only the J -projection of the vector operator is important.

For the vector operator L + |ge|S this implies

L + |ge|S = gJ J

(3.2.62)

in the relevant subspace. The value of gJ , called the Landé g-factor 18 can be determined from

gJ =	(L + |ge|S) · J	.	(3.2.63)
	J 2

18 A. Landé, 1923.

3.2 The State of Ion Cores 55

One only has to make use of

L · S = 21 [J 2 − L2 − S2] ,

(3.2.64)

and thus

L · J = L · (L + S) = L2 + 21 [J 2 − L2 − S2] ,

(3.2.65)

as well as

S · J = S · (L + S) = S2 + 21 [J 2 − L2 − S2] .

(3.2.66)

In the subspace of the eigenvectors of J = L + S

L · J = L(L + 1) + 21 [J(J + 1) − L(L + 1) − S(S + 1)] ,

(3.2.67)

S · J = S(S + 1) + 21 [J(J + 1) − L(L + 1) − S(S + 1)] .

This finally yields

g

= 1

( g

+ 1) + 1

( g

1)

S(S + 1) − L(L + 1)

J(J + 1)

J

2

| e|

2

| e| −

(3.2.68)

J(J + 1) + S(S + 1) − L(L + 1)

= 1 + ( g

e| −

1)

.

|

2J(J + 1)

The importance of this result lies in the fact that gJ determines the total magnetic moment

μ = −μB(L + |ge|S) = −gJ μBJ

(3.2.69)

of an ion core, and therefore the energy level splitting in an applied magnetic field.

The interaction of the magnetic moment with an applied magnetic field is

included in the Hamiltonian as the perturbation term
Hext = −μ · B = gJ μB J · B .	(3.2.70)

This completely lifts the 2J + 1-fold degeneracy of the original state, splitting it into 2J + 1 nondegenerate Zeeman levels. When the quantization axis is chosen along the applied field direction, the energy shift of the individual levels is given by

E = gJ μBmJ B ,

(3.2.71)

where mJ = −J, −J + 1, . . . , J.

A simple physical interpretation can be given to the foregoing by picturing the moments L and S as classical vectors of length L(L + 1) and

S(S + 1)				precess about a
. Because of the spin–orbit interaction, these will
third classical vector, J , of length			J(J + 1)
			, as shown in Fig. 3.1.
Owing to the precession, all	components perpendicular to the direction of

Jare averaged out, and only the projection of L and S on the direction of

Jsurvive. As the gyromagnetic ratio is di erent for spin and orbital angular

56 3 The Building Blocks of Solids

B

	S	gJ	J
	L |ge|S
	J
		S

L

L

Fig. 3.1. Precession of the orbital angular momentum and the spin about the total angular momentum

momentum, the total magnetic moment – that is, the resultant of the intrinsic magnetic moment and the magnetic moment associated with the orbital angular momentum – does not point in the same direction as J but rather precesses about it, and only its component parallel to J is conserved. This component is given in terms of the lengths of the vectors μL and μS ,

|μL| = μB

L(L + 1)

, |μS | = |ge|μB

S(S + 1)

(3.2.72)

and the angles θL and θS that J makes with L and S as

|μJ | = |μL| cos θL + |μS | cos θS .

(3.2.73)

By reading o the angles from the figure and making use of the relation

|μJ | = gJ J(J + 1)μB (3.2.74)

the previous result for the Landé g-factor is recovered. When a magnetic field is turned on, this average magnetic moment (of magnitude |μJ | and direction J ) starts to precess about the field direction, and only its component along the field contributes to energy. The classical picture cannot account for the fact that the component along the field direction is quantized, i.e., it can take only discrete values.

To specify the Curie or Langevin susceptibility, we shall assume that the energy di erence between the ground state (as determined by Hund’s rules) and higher-lying states is larger than the thermal energy kBT at experimentally relevant temperatures, and consequently the thermal occupation of the latter states can be neglected. If this assumption is justified, only a single level with quantum number J has to be considered for each atom – which, however, splits into 2J + 1 levels in a magnetic field. A well-known theorem in statistical mechanics states that the partition function of a system made

3.2 The State of Ion Cores

57

up of independent subsystems is the product of the partition functions of the subsystems. In a first approximation, the state of individual atoms will be considered independent of each another. The partition function Za of all possible states of a single atom is given by

Za = e−βEn =	J
	exp(−βE0) exp(−βgJ μBmJ B) .

nmJ =−J

Summation yields

Za = exp(−βE0) exp(−βgJ μBJB) exp[βgJ μB(2J + 1)B] − 1 exp(βgJ μBB) − 1

= e−βE0 exp[βgJ μB(J + 1/2)B] − exp[−βgJ μB(J + 1/2)B] exp(βgJ μBB/2) − exp(−βgJ μBB/2)

= e−βE0 sinh[βgJ μB(J + 1/2)B] . sinh[βgJ μBB/2]

(3.2.75)

(3.2.76)

Because of the independence of individual atoms, the partition function of the entire system with N atoms is just the N th power of the partition function of a single atom,

Z = ZaN .

(3.2.77)

As free energy is the logarithm of the partition function, the component of magnetization along the magnetic field is

M = −	1 ∂F			=	N
						gJ μB JBJ (βgJ μBJB) ,	(3.2.78)
	V		∂B		V

where BJ (x), the Brillouin function can be written as

BJ (x) =	2J + 1	coth	2J + 1	x −	1	coth	1	x .	(3.2.79)
	2J		2J		2J		2J

For J = 1/2, the Brillouin function takes the particularly simple form

B1/2(x) = tanh x .

(3.2.80)

In the J 1 limit the component of the magnetic moment along the magnetic field is found to be

μz = μ L(μB/kBT ) ,

(3.2.81)

in perfect agreement with the result that would be obtained from the classical equation (3.2.48). Here

L(x) = coth x − 1/x

(3.2.82)

is the Langevin function.

58 3 The Building Blocks of Solids

In strong fields, where the energy due to the coupling to the magnetic field is much larger than the thermal energy (gJ μBJB kBT ), the projection of the spins will be maximal along the field direction, and so magnetization reaches saturation. Apart from extremely low temperatures, μBB kBT in fields customarily applied in susceptibility measurements, therefore the argument of the Brillouin function is small. Using the small-x expansion of the coth function,

1

+

1

1

coth x =

x −

x3 + . . . ,

(3.2.83)

x

3

45

we have

J + 1 x

(2J + 1)4 − 1

x3

B

(x) =

−

+ . . . .

(3.2.84)

J 3

J

(2J)4 45

Substitution into (3.2.78) gives the leading-order term of the susceptibility:

χ =	N μ0			(gJ μB)2J(J + 1)		(3.2.85)
					.
	V			3kBT

To compare this formula with the expression in (3.2.51) valid for classical magnetic moments, the e ective magnetic moment μe and the e ective magneton number p are customarily introduced via

χ =

N

μ0μe2

=

N

μ0μB2 p2

.

(3.2.86)

V 3k T

V 3k T

B

B

Their theoretical values are μth = gJ

μB and pth = gJ

.

J(J + 1)

J(J + 1)

e

Table 3.6 lists the values of

p calculated from Hund’s rules for trivalent rare-

earth metals (pth) as well as determined from experimental data on the susceptibility of paramagnetic salts containing such ions (pexp).

As the data in the table show, the agreement between calculated and measured values is excellent – except for samarium and europium. The reason for the discrepancy in the latter is that our earlier assumption – that it is su cient to consider only the lowest 2J + 1 energy levels associated with the value of J given by Hund’s rules – is not justified now. In this case somewhat higher-lying states must also be included in calculations of the susceptibility, as their occupation can no longer be neglected. We shall see a remarkable example in the following subsection.

A far worse agreement is found for 3d transition metals when the calculated magnetic moment of the configuration given by Hund’s rules is compared to the e ective moment determined from measured data of the paramagnetic susceptibility in solids containing transition-metal ions. This is illustrated in Table 3.7.

The table clearly shows that an agreement can be found only when the e ective magneton number due to the electrons’ intrinsic magnetic moment

3.2 The State of Ion Cores

59

Table 3.6. Ground-state electron configuration, Landé g-factor, as well as calculated and measured e ective magneton numbers for trivalent rare-earth ions

Ion	4fn	L	L	J	Ground	gJ	pth	pexp
					state
La3+	0	0	0	0	1S0	0	0	0
Ce3+	1	1/2	3	5/2	2F5/2	6/7	2.535	2.4–2.7
Pr3+	2	1	5	4	3H4	4/5	3.578	3.4–3.6
Nd3+	3	3/2	6	9/2	4I9/2	8/11	3.618	3.4–3.7
Pm3+	4	2	6	4	5I4	3/5	2.683	–
Sm3+	5	5/2	5	5/2	6H5/2	2/7	0.845	1.3–1.6
Eu3+	6	3	3	0	7F0	0	0	3.2–3.4
Gd3+	7	7/2	0	7/2	8S7/2	2	7.937	7.9–8.0
Tb3+	8	3	3	6	7F6	3/2	9.721	9.4–9.8
Dy3+	9	5/2	5	15/2	6H15/2	4/3	10.646	10.5–10.7
Ho3+	10	2	6	8	5I8	5/4	10.607	10.3–10.6
Er3+	11	3/2	6	15/2	4I15/2	6/5	9.581	9.4–9.6
Tm3+	12	1	5	6	3H6	7/6	7.561	7.3–7.6
Yb3+	13	1/2	3	7/2	2F7/2	8/7	4.536	4.4–4.6
Lu3+	14	0	0	0	1S0	0	0	0

Table 3.7. Electron configuration, quantum numbers of the ground state (as specified by Hund’s rules), calculated and measured values of the e ective magneton number p, and spin the theoretically determined contribution to the e ective magneton number ps for some transition-metal ions

Ion	3dn	S	L	J	Ground	gJ	pth	pexp	psth
					state
Ti3+	1	1/2	2	3/2	2D3/2	4/5	1.549	1.6–1.8	1.732
V4+	1	1/2	2	3/2	2D3/2	4/5	1.549	1.6–1.8	1.732
V3+	2	1	3	2	3F2	2/3	1.633	2.5–2.8	2.828
V2+	3	3/2	3	3/2	4F3/2	2/5	0.775	3.7–4.0	3.873
Cr3+	3	3/2	3	3/2	4F3/2	2/5	0.775	3.7–4.0	3.873
Mn4+	3	3/2	3	3/2	4F3/2	2/5	0.775	3.7–4.0	3.873
Cr2+	4	2	2	0	5D0	–	0	4.8–5.0	4.900
Mn3+	4	2	2	0	5D0	–	0	4.8–5.0	4.900
Mn2+	5	5/2	0	5/2	6S5/2	2	5.916	5.9	5.916
Fe3+	5	5/2	0	5/2	6S5/2	2	5.916	5.9	5.916
Fe2+	6	2	2	4	5D4	3/2	6.708	5.2–5.4	4.900
Co3+	6	2	2	4	5D4	3/2	6.708	5.2–5.4	4.900
Co2+	7	3/2	3	9/2	4F9/2	4/3	6.633	4.8–4.9	3.873
Ni2+	8	1	3	4	3F4	5/4	5.590	3.1–3.2	2.828
Cu2+	9	1/2	2	5/2	2D5/2	6/5	3.550	1.9	1.732
Zn2+	10	0	0	0	1S0	0	0	0	0

60 3 The Building Blocks of Solids

ps = |ge| S(S + 1) is considered, i.e., the contribution of the orbital angular momentum is neglected. It seems as if the electrons’ orbit were rigidly fixed

– quenched – and una ected by the magnetic field. As we shall see in Chapter 6, this happens because inside the solid the transition-metal ions feel the relatively strong electrostatic field of the neighboring ions. Therefore ions can no longer be considered to be independent of each other, and so Hund’s rules may become invalid. For iron, cobalt, and nickel ions the agreement is not very good even for ps when the e ective moment is taken from room-temperature measurements. The reason for this is that for such ions thermal energies are on the order of the energy di erence between the ground state and higher-lying energy levels, and thus the contribution of the latter to susceptibility cannot be neglected.

3.2.7 Van Vleck Paramagnetism

In this subsection we shall return to the problem of europium, in which the outermost electron shell is not closed although according to Hund’s rules J = 0 in the ground state. The same situation occurs when two electrons are present in the p-shell or four in the d-shell (or, in general, n = 2l electrons in the shell with azimuthal quantum number l). In such cases the first nonvanishing contribution to paramagnetic susceptibility is obtained in the second order of perturbation theory, and so higher-lying energy levels must also be taken into account as intermediate states. The second-order correction to the ground-state energy is given by

E0	=	| 0|μBB(Lz + |ge|Sz )|n |2 .		(3.2.87)
	n=0	E0 − En

This energy shift is always negative as the energy denominator is negative in each term. The resulting magnetization is always along the magnetic field direction, and the susceptibility is positive:

χ = 2	N	μ0	μ2	| 0|(Lz + |ge|Sz )|n |2	.	(3.2.88)
				−
	V		B	En E0
			n

This phenomenon is called Van Vleck paramagnetism, and χ the Van Vleck susceptibility.19 If the excitation energy of the state |n is not too large, the small energy denominator may make this correction more important than the diamagnetic contribution. When this Van Vleck contribution is taken into account, the calculated paramagnetic susceptibility of substances containing trivalent europium ions is in good agreement with measured data.

In the ground state of the samarium ion J takes a nonzero value, therefore Sm3+ has a nonvanishing paramagnetic moment. Nevertheless the lowest-lying excited states are close enough for that the Van Vleck second-order correction

19 J. H. Van Vleck, 1932. See footnote on page 6.