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

3.2 The State of Ion Cores

41

or paramagnetic behavior in an applied magnetic field. This phenomenon opens the way to the experimental determination of the electronic state of ion cores.

3.2.1 Hund’s Rules

Consider an atom with some electrons on its outermost, partially filled shell. According to the rules of angular momentum addition, the sum of the orbital and spin angular momenta of individual electrons can lead to many di erent resultant Ls and Ses. Consider, for example, the configuration p2 in which two p-electrons (l = 1) have the same principal quantum number. Denoting the states with magnetic quantum number ml = 1, 0, −1 and spin quantum number ms = ±12 by |ml, ms , the two-electron state can be written as a linear combination of the states |ml1 , ms1 |ml2 , ms2 – where, again, due attention must be paid to the antisymmetrization required by the Pauli exclusion principle. The six one-particle states will then give fifteen two-particle states, which are best grouped according to their total quantum numbers L and S. The total angular orbital momentum of two l = 1 particles is L = 0, 1, or 2. Similarly, the total spin quantum number of two spin-1/2 particles is either S = 0 or S = 1. The singlet state S = 0 is antisymmetric with respect to the interchange of the spin of the two particles. This can appear only in wavefunctions that are symmetric in their spatial variables, that is, the total orbital angular momentum must be L = 0 or L = 2. The triplet configuration S = 1 is symmetric in its spin variables, and thus the corresponding wavefunction must be antisymmetric in its spatial variables. This corresponds to an orbital angular momentum of L = 1. The allowed configurations and their multiplicities – (2L + 1)(2S + 1) – are listed in Table 3.1.

Table 3.1. Allowed values of L and S and multiplicity in states with two p-electrons

L	S	Multiplicity	Notation
0	0	1	1S
1	1	9	3P
2	0	5	1D

In the singlet state S = 0 the spatial parts of the wavefunctions of the two electrons can be identical, while in the S = 1 state they are necessarily di erent. The Coulomb repulsion between the electrons is expected to be weaker in this state, leading to a lower energy than in the singlet state.

Spin–orbit interaction causes further splitting of the ninefold degenerate energy level L = 1, S = 1. In the vector sum J = L + S the allowed values of J range from |L − S| to L + S. Hence for a 3P level the possible values of

42 3 The Building Blocks of Solids

J are 0, 1, and 2. The corresponding multiplicities – given by J(J + 1) – are 1, 3, and 5. Spin–orbit interaction thus splits the ninefold degenerate energy level into three levels, a nondegenerate, a threefold degenerate, and a fivefold degenerate level. Table 3.2 lists the possible configurations for two p-electrons when spin–orbit interaction is taken into account.

Table 3.2. Allowed values of L, S, and J, and multiplicity in states with two p-electrons

L	S	J	Multiplicity	Notation
0	0	0	1	1S0
1	1	0	1	3P0
1	1	1	3	3P1
1	1	2	5	3P2
2	0	2	5	1D2

Experience shows that of the five allowed states listed above, 3P0 (with quantum numbers (L = 1, S = 1, J = 0) is the ground state. Such experimental findings are generalized by Hund’s rules,10 established in the golden age of spectroscopy, which provide a set of rules to determine the quantum numbers of the lowest-energy configuration.

1.) To minimize the Coulomb repulsion due to the overlap of their wavefunctions, electrons tend to occupy states with di erent magnetic quantum numbers ml (as long as this is possible). As we shall see in Chapter 4, even though the Coulomb interaction is spin independent, the indistinguishability of quantum mechanical particles nevertheless gives rise to a spin-dependent interaction, the exchange interaction. As the exchange integral of two orthogonal states is positive, in less than half filled shells it is energetically more favorable to have the spins of all the electrons aligned, which of course requires that their magnetic quantum numbers ml be di erent. When a shell is more than half filled, there will certainly be electrons with identical ml but opposite ms quantum numbers. Nevertheless even in this case the lowest-energy state is found to be the one in which the total spin has the largest possible value. If n of the 2(2l + 1) possible states of the shell with azimuthal quantum number l are filled, the quantum number S of the ground state is

S = 21 2l + 1

− |

2l + 1

−

n

|

=

21 n

1

n

≤ 2l + 1 ,

(3.2.5)

2l + 1

−

2 n

n

≥

2l + 1 .

2.) Once the total S is determined, the total orbital angular momentum L has to be chosen as large as possible. If there are n electrons on the shell

10 F. Hund, 1925.

3.2 The State of Ion Cores

43

with azimuthal quantum number l, the total orbital angular momentum L is given by

L = S 2l + 1

−

n

=

21 n(2l + 1

1− n)

2l

1)

n

≤ 2l + 1 ,

(3.2.6)

|

|

2l + 1

−

2 n (n

−

−

n

≥

2l + 1 .

This arrangement is also dictated by the Coulomb energy – although it cannot be illustrated with an intuitive picture.

3.) Owing to the spin–orbit interaction, L and S are no longer conserved, only their sum, the total angular momentum J = L + S is. For fixed values of S and L the allowed values of the quantum number J (i.e., the length of J ) range from |L − S| to L + S. The multiplicities of the split levels indeed add up to (2L + 1)(2S + 1), the multiplicity prior to splitting:

L+S

J(J + 1) = (2L + 1)(2S + 1) .

(3.2.7)

J=|L−S|

The sign of the coupling constant of the spin–orbit interaction determines which of these has a lower energy. It is possible to provide theoretical justification to the empirical rule asserting that if a shell of azimuthal quantum number l is less than half filled (that is, when the number of electrons is smaller than 2l + 1) then the lowest-energy state has J = |L − S|, while if it is more then half filled then it has J = L + S. In case of n electrons

J = S 2l n =			2 n(2l − n			1				≤	(3.2.8)
			1			)			n		2l ,
| − |
				2l + 1	−	2 n (n	−	2l)	n > 2l .
The quantum	numbers S, L, and J of the lowest-energy state satisfying

Hund’s rules are given in Table 3.3 for the case when the 10 states of the d-shell (l = 2) are gradually filled in.

It should be kept in mind that the table shows only schematically how the quantum numbers S and L could be determined for a given filling of the level. The wavefunction itself cannot be inferred directly from it. For this one has to use the angular momentum addition rules given in Appendix F. E.g., if there is a single electron on the 3d level, the wavefunctions of the fourfold degenerate 2D3/2 multiplet are linear combinations of the ten |ml, ms states with orbital and spin quantum numbers l = 2 and s = 1/2. The state with

maximal mJ = 3/2 is

|

2

|

−

−

1

|

5

5

J = 3/2, mJ = 3/2

= √

2,

1/2

√

1, 1/2

.

(3.2.9)

The three other states that correspond to mJ = 1/2, −1/2, and −3/2 can be

obtained from this state by successive applications of the lowering operator

J− = L− + S−.

44 3 The Building Blocks of Solids

Table 3.3. The n-electron ground states of the d-shell, as required by Hund’s rules

n	2	1	0	−1 −2		S	L	J	2S+1LJ
1	↑					1/2	2	3/2	2
									3D3/2
2	↑	↑	↑			1	3	2	4F2
3	↑	↑				3/2	3	3/2	5F3/2
4	↑	↑	↑	↑	↑	2	2	0	6D0
5	↑	↑	↑	↑		5/2	0	5/2	5S5/2
6	↑↓	↑	↑	↑	↑	2	2	4	4D4
7	↑↓	↑↓	↑	↑	↑	3/2	3	9/2	3F9/2
8	↑↓	↑↓	↑↓	↑	↑	1	3	4	2F4
9	↑↓	↑↓	↑↓	↑↓	↑	1/2	2	5/2	1D5/2
10	↑↓	↑↓	↑↓	↑↓	↑↓	0	0	0	S0

The energy splitting between levels with di erent quantum numbers L and S is typically on the order of an electronvolt, since it is due to the Coulomb interaction and its consequence, the exchange interaction. However, the splitting between states with di erent Js is at most 0.1 eV, since this is the result of the spin–orbit interaction. In spectroscopy, energy di erences are customarily given by the wave number (in cm−1 units) of the photon associated with the transition. According to the conversion formulas presented in Appendix A, 1 eV ≈ 8 × 103 cm−1. Thus, the typical energy di erence between multiplets with di erent L and S quantum numbers is 103–104 cm−1, while between states with di erent Js it is usually 102–103 cm−1.

As thermal energy reaches 0.1 eV at temperatures around 103 K, higher multiplets are usually not excited thermally because of the large energy difference. Therefore in the overwhelming majority of cases it is su cient to consider only levels whose quantum numbers S and L are determined by Hund’s first and second rules. Moreover, it is often enough to consider only the ground-state multiplet whose quantum number J is determined from Hund’s third rule.

3.2.2 Angular Momentum and Magnetic Moment

It is known from the Zeeman11 splitting of energy levels that the angular momentum L of an ion is always accompanied by a magnetic moment

μ = −μBL ,

(3.2.10)

where μB = e /(2me) is the Bohr magneton. We shall shortly see how this is derived from the Schrödinger equation of an electron in a magnetic field. Using the physical angular momentum L instead of the dimensionless L, the previous formula is alternatively written as

11 See footnote on page 2.

3.2	The State of Ion Cores	45
μ = −γe L ,	(3.2.11)

where γe is the gyromagnetic (magnetomechanical) ratio of the electron. Its

	value is	e	= 1.760 860 × 1011 s−1 T−1 .
	γe =			(3.2.12)
		2me

Owing to its orbital motion, an electron in a state characterized by the quantum numbers n, l, and ml has a magnetic moment of magnitude

μl = μB l(l + 1) , (3.2.13)

while the z component of this magnetic moment is

μlz = −μBml .

(3.2.14)

The electron’s intrinsic angular momentum, its spin s is accompanied by an intrinsic magnetic moment μs. Just like orbital angular momentum and magnetic moment, spin and intrinsic magnetic moment are also oppositely directed. However, as the Einstein–de Haas experiment 12 revealed, the gyromagnetic ratio is now di erent from that obtained for the orbital angular momentum. The same conclusion is drawn from the Dirac equation. As mentioned earlier, the following relation holds between the dimensionless spin and the intrinsic magnetic moment in the nonrelativistic limit:

μs = −2μBs .

(3.2.15)

This relation is slightly modified by the radiation corrections of quantum electrodynamics:

μs = geμBs ,

(3.2.16)

where calculations and measurements give the same value for ge, the electron g-factor:

ge = −2	1 +	α	+ O(α2 ) = −2.002 319 .	(3.2.17)
		2π

As before, α is the fine-structure constant. To a very good approximation, ge is equal to −2.13

Thus the z component of the magnetic moment in a state with spin quantum number ms = ±21 takes the value

μsz = ±21 geμB ≈ μB .

(3.2.18)

12A. Einstein and J. W. de Haas, 1915. Albert Einstein (1879–1955) was awarded the Nobel Prize in 1921, “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric e ect”.

13Note that while in its rigorous definition ge is negative, most solid-state physics references nevertheless use its absolute value. For easy comparison, we shall often use the notation |ge| below.

46 3 The Building Blocks of Solids

3.2.3 The Magnetic Hamiltonian of Atomic Electrons

To determine the magnetic properties of atomic electrons, consider an electron moving in a potential U (r) and an applied magnetic field B(r). If the latter is represented by a vector potential A(r), the canonical momentum p is replaced by the kinetic momentum p + eA in the electron’s Hamiltonian. When the interaction between the magnetic field and the magnetic moment due to the electron spin is taken into account, the behavior of a single electron is governed by the Hamiltonian (3.1.28),

H =	1	p + eA(r)	2 + U (r) + |ge|μBB · s .	(3.2.19)
	2me

When expanding the quadratic expression of the kinetic energy, it should be remembered that the order of the operators p and A is not immaterial. It can be seen from the form of the momentum operator,

					∂		∂		∂
p =		=				,		,		(3.2.20)
	i		i		∂x		∂y		∂z

that p and A do not commute in general:

[p, A] =		div A .	(3.2.21)
	i

To simplify calculations it is useful to choose the Coulomb gauge (also known as radiation gauge or transverse gauge) div A = 0. When the magnetic field is uniform, one possible choice for the vector potential is

A = 21 [B × r] .

(3.2.22)

This is the so-called symmetric gauge. It is easily seen that both B = curl A and the condition div A = 0 are satisfied.

Substituting this form of the vector potential into the first term of the Hamiltonian (3.2.19), the following form is obtained:

1

p −

e

[r × B]

2

p2

e

e2

[r × B]2

=

−

p · [r × B] +

2me

2

2me

2me

8me

(3.2.23)

p2

e

e2

2

=

+

[r × p] · B +

[r × B] .

2me

2me

8me

The angular momentum r × p = l is immediately recognized in the middle term, and the Bohr magneton in its coe cient. The total Hamiltonian can therefore be written as

	p2		e2	2
H =		+ μBl · B +		[r × B] + U (r) + |ge|μBB · s
	2me		8me				(3.2.24)
	p2				e2
=		+ U (r) + μB(l + |ge|s) · B +				[r × B]2 .
	2me				8me

3.2 The State of Ion Cores

47

Using r , the component of the electron’s position vector perpendicular to the magnetic field, the vector product in the last term can be rewritten as |r × B| = |B| · |r |, leading to

H =	p2	+ U (r) + μB(l + |ge|s) · B +	e2
				B2r2 .	(3.2.25)
	2me		8me

This expression is readily generalized for the case when the atom has Z electrons. Labeling the electrons by i, one starts o with the counterpart of the Hamilton (3.2.19),

Z		2me (pi + eA(ri))2		+ U (ri) + |ge|μBB · si .	(3.2.26)
H = i=1
		1

Following the same steps as above, the final result is now

	Z	p2				e2			Z
H =		i	+ μB (L +	|ge|S) · B +				2
									2
							B		ri ,
	i=1	2me				8me
									i=1
where			Z		Z
			L =	li =	(ri × pi)
			i=1		i=1

(3.2.27)

(3.2.28)

is the total orbital angular momentum due to the orbital motion of the electrons, and

Z
S = si	(3.2.29)
i=1

is the total spin of the system of electrons.

3.2.4 Magnetization and Susceptibility

The Hamiltonian of the electrons contains terms that are linear and quadratic in the magnetic field. The linear term is present when the total magnetic moment −μB(L + |ge|S) is finite (nonzero). In this case this moment tends to be aligned with the field. This paramagnetic contribution is dominant unless the total moment vanishes for some reason. Before analyzing the conditions for this, we have to say a few words about the determination of magnetization and susceptibility.

If the energies Ei of the states of the system are shifted under the influence of a magnetic field B, the free energy also depends on the field. Using the partition function

	(3.2.30)
Z = e−Ei (B)/kBT ,

i

the Helmholtz free energy can be written as

48 3 The Building Blocks of Solids

F = −kBT ln Z = −kBT ln	e−Ei (B)/kBT .	(3.2.31)

i

Magnetization is the negative partial derivative of the free energy density f = F/V with respect to magnetic induction:

M (B, T ) = −

∂f

= −

1 ∂F

(3.2.32)

.

∂B

V

∂B

This can equally be written as

M (B, T ) =

i

M i(B)e−Ei (B)/kBT

(3.2.33)

i e−Ei(B)/kB T

,

where

1 ∂Ei(B)

M i(B) = −

(3.2.34)

V

∂B

is the contribution of the ith state to magnetization.

The customary definition of magnetic susceptibility – the response of the system to a magnetic field – is not based on the straightforward choice

		∂M
χ = ∂B				B=0	(3.2.35)
but rather on
	∂M				(3.2.36)
χ = ∂H H =0 .

The derivative can also be taken at finite values of the field, leading to a field-dependent susceptibility. Susceptibility, with its ordinary definition, is the weak-field limit of the latter. In certain systems nonlinear contributions to the magnetization are also used to characterize magnetic properties.

Susceptibility defined in this manner is usually a tensor quantity,

∂Mα

χαβ = . (3.2.37)

∂Hβ H =0

However, in the isotropic case o -diagonal elements vanish and diagonal ones are identical – so susceptibility can be considered a scalar. The dimensionless scalar susceptibility is related to magnetic permeability via

μ = μrμ0 = (1 + χ)μ0 .

(3.2.38)

Apart from some magnetically ordered materials (to be discussed in Chapter 14), the magnetization of solids is usually rather small (χ 1). As χ of (3.2.26) is much more easy to determine theoretically, the relation B ≈ μ0H permits one to obtain the dimensionless susceptibility χ from

χ = μ0χ .

(3.2.39)

3.2 The State of Ion Cores

49

When this approximation is justified, susceptibility can be calculated from the free-energy expression as

μ0 ∂2F (B)
χ = − V ∂B2 .	(3.2.40)

As we started o with the free-energy density, χ above is frequently referred to as the volume susceptibility or bulk susceptibility.

Note that instead of the dimensionless susceptibility often χmass, the mass or specific susceptibility, or χmol, the molar susceptibility is given as experimental data. The former can be obtained by dividing χ by the density, χmass = χ/ρ, while the latter by multiplying χ by the molar volume,

χmol = χVmol.

3.2.5 Langevin or Larmor Diamagnetism

If the outermost electron shell of an atom or ion is completely filled, as in noble gases (He, Ne, Ar, etc.) or ionic crystals (Na+, Cl−, etc.), the ground-state quantum numbers are

L = S = J = 0 .

(3.2.41)

When the energy correction due to the magnetic field is calculated to the lowest order in perturbation theory, the second term in (3.2.27) does not contribute. The only surviving term is proportional to B2. For a single atom with Z electrons, the change in energy is

		e2		Z
	E =			0 i=1 ri2 0 .		(3.2.42)
		8me B2

For completely filled electron shells the electron distribution is spherically symmetric. It is an elementary exercise to show that in such cases the mean square of the position vector component perpendicular to the magnetic field is equal to two-thirds of the mean square of the magnitude of the position vector. Thus,

		e2		Z
	E =			0 i=1 ri2	0 .	(3.2.43)
		12me B2

In systems containing N atoms in volume V , this shift of the ground-state energy gives rise to a susceptibility on the order of

	N ∂2 E				e2μ0 N			Z
				= −				0 i=1 ri2	0	,	(3.2.44)
χ = −μ0 V ∂B2					6me V

as at T = 0 susceptibility should be calculated from the energy instead of the free energy. As the negative sign shows, ions with closed shells behave diamagnetically. This is the Langevin or Larmor diamagnetism. The latter name

50 3 The Building Blocks of Solids

reflects that the phenomenon lends itself to classical interpretation based on Larmor’s theorem on the motion of electrons in a magnetic field14 – however, the interpretation is due to P. Langevin (1905). This asserts that when an electron moving in a closed orbit around a nucleus is placed in a magnetic field, the angular frequency of the orbiting electron will be shifted by the Larmor frequency ωL = eB/2me, inducing a magnetic moment that is proportional to the applied magnetic field. According to Lenz’s law, the induced moment will oppose the applied field. This is the root of diamagnetism.

To obtain an order-of-magnitude estimate for the diamagnetic energy correction, we shall approximate the radius of atomic orbits by the Bohr radius a0 = 4π 0 2/mee2 = 2/mee˜2. The energy shift per electron is then

e2

(3.2.45)

Edia ≈

B2a02 .

12me

In terms of the Larmor frequency,

e2

2

1

eB

2 a0

1

e˜2

−1

Edia ≈

B2a0

=

2

=

( ωL)2

.

12me

mee˜2

3

2me

e˜2

3

a0

(3.2.46)

While e˜2/a0 is on the order of atomic energies, the magnetic energy ωL is much smaller. Even for fields as strong as B 1 tesla the di erence is five orders of magnitude. In SI units, the dimensionless susceptibility χ is on the order of 10−9 for noble gases. This corresponds to a molar susceptibility of 10−10–10−11 m3/mol, and a specific susceptibility on the order of 10−9 m3/kg. Molar and specific susceptibilities for noble gases are listed in Table 3.4. When working with CGS units, the numerical value of the susceptibility di ers by a factor of 4π: χCGS = χSI/4π.

Table 3.4. Molar and specific susceptibilies of noble gases at room temperature

		χmol [SI]		χmol [CGS]		χmass [SI]
		10−12 m3/mol		10−6 cm3/mol		10−9 m3/kg
He		−25.4		−2.02		−5.9
Ne		−87.5		−6.96		−4.2
Ar		−243		−19.3		−6.16
Kr		−364		−29.0		−4.32
Xe		−572		−45.5		−4.20

In ionic crystals, the dimensionless susceptibility is on the order of 10−4 to 10−6 – e.g., for NaCl χ = −13.9 × 10−6, which corresponds to a molar susceptibility of χmol = −3.75 ×10−10 m3/mol. Diamagnetic susceptibilities of

14 J. Larmor, 1897.