Fundamentals of the Physics of Solids / 06-Consequences of Symmetries

6

Consequences of Symmetries

In addition to the spatial symmetries discussed in the previous chapter a physical system may possess further symmetries, for example, it may be invariant under time reversal or gauge symmetries. While they might seem irrelevant to the structure of the crystal, they may nevertheless play an essential role in understanding the physical properties of the system. In general, the existence or lack of a symmetry may have fundamental influence on the system’s behavior.

The connection between symmetries and physical properties is summarized by a fundamental postulate of crystal physics formulated back in the 19th century, known as Neumann’s principle: The symmetry elements of any observable physical property of a crystal must include all the symmetry elements of the point group of the crystal. To put it otherwise: The tensor representing any direction-dependent macroscopic physical property of a crystal must be invariant under the symmetry operations of the point group of the crystal.

For physical quantities given in tensor form, such as resistivity, elastic constants, or optical properties, symmetry operations provide relations among the elements of the tensor. Using symmetry considerations alone it is possible to determine the number of di erent parameters that completely specify the tensor. For example, in the absence of a magnetic field, o -diagonal elements vanish and diagonal ones are identical in the two-index resistance tensor of a cubic crystal. Thus resistivity can be characterized by a single scalar quantity. In crystals with tetragonal symmetry, the resistivity tensor has two independent parameters. As we shall see, the four-index elasticity tensor is specified by three elastic constants in cubic crystals.

Moreover, taking symmetries into account facilitates the determination of the states of the system and provides selection rules for allowed processes. If the Hamiltonian of the system possesses some symmetry that is not observed in the ground state – i.e., if symmetry is spontaneously broken – then further conclusions can be drawn about the excited states of the system and the subgroup of preserved symmetries.

172 6 Consequences of Symmetries

Using simple group theoretical considerations we shall first demonstrate that when symmetries are known, general statements can be made about the degeneracy of electronic states of the crystal, or about lifting this degeneracy. It will be shown that in the presence of an external perturbation, symmetries of the perturbation and of the system imply strict selection rules on the processes that may take place in the solid. Then we shall investigate the consequences of translational symmetry. Finally we shall discuss the implications of symmetry breaking.

6.1 Quantum Mechanical Eigenvalues and Symmetries

If a crystal possesses some symmetry its quantum mechanical Hamiltonian must possess the same symmetry. Knowledge of the symmetries often permits one to make general statements about the quantum mechanical eigenvalue problem without solving the Schrödinger equation explicitly. To this end one has to make recourse to group theory. We assume that the reader is familiar with the basics of group theory – nevertheless we shall summarize the most important theorems and relations for solid-state physics applications in Appendix D. Several theorems are concerned with the relation between the energy spectrum of a system and the irreducible representations of the group of its symmetry operations. Below, after recalling one of Wigner’s theorems,1 we shall show how they can be applied to the study of energy level splitting in solids and to the determination of selection rules.

6.1.1 Wigner’s Theorem

Suppose that a system and its Hamiltonian H(r) are both invariant under the elements Gi of a symmetry group G – that is, operators T (Gi) associated with the symmetry operations take the Hamiltonian into itself:

T (Gi)H(r) = H(Gi−1r) = H(r) .

(6.1.1)

This implies that the operators T (Gi) commute with the Hamiltonian,

T (Gi)H(r) = H(r)T (Gi) ,

(6.1.2)

since by acting on an arbitrary function ψ(r)

) (r)ψ(r) =		H	(G−1r)ψ(G−1r)
T (Gi	H		i	i	(6.1.3)
	=	H	(r)ψ(G−1r)
				i

= H(r)T (Gi)ψ(r) .

Let ψ(r) be an eigenfunction of the Hamiltonian of the system with eigenvalue ε, i.e.,

1 E. P. Wigner, 1927.

6.1 Quantum Mechanical Eigenvalues and Symmetries	173
H(r)ψ(r) = εψ(r) .	(6.1.4)
Operating T (Gi) on both sides and using (6.1.3) gives
T (Gi)H(r)ψ(r) = εT (Gi)ψ(r)	(6.1.5)
and
H(r)T (Gi)ψ(r) = εT (Gi)ψ(r) ,	(6.1.6)

that is, T (Gi)ψ(r) is also an eigenfunction, with the same eigenvalue. This result is summarized by Wigner’s theorem: If ψ(r) is an eigenfunction of the Hamiltonian with an energy eigenvalue ε then for any symmetry operation T (Gi) that leaves the Hamiltonian invariant T (Gi)ψ(r) = ψ(G−i 1r) is also an eigenfunction, with the same energy.

By operating all elements of the symmetry group G on the wavefunction ψ(r), a set of eigenfunctions with the same eigenvalue is obtained. One may then construct a representation of the group on the space of these functions. If the representation is irreducible then the degeneracy of the energy levels is not accidental but the consequence of symmetry. When, on the other hand, the representation is not irreducible, then after its reduction to irreducible representations symmetry operations will mix only those wavefunctions that belong to the same representation, and only their eigenvalues will be necessarily equal. Higher degeneracy may appear only accidentally. It is plausible that by choosing the potential (interaction term) in the Hamiltonian in the most general form consistent with symmetry accidental degeneracy of the energy levels is lifted, and only the degeneracy imposed by symmetries survives. In such cases the representation of the symmetry group is irreducible on the wavefunctions of the selected energy value.

Consequently, energy levels can be characterized by the irreducible representation according to which the wavefunctions transform, and the degree of degeneracy is equal to the dimension of the irreducible representation associated with the energy level. This also means that when the symmetries of the system and the irreducible representations of the symmetry group are known, one can determine the degree of degeneracy of the energy levels, as well as the way they are split in the presence of a lower-symmetry external perturbing field.

6.1.2 Splitting of Atomic Levels in Crystals

The state of atomic electrons in a spherical potential is characterized by the principal quantum number n = 1, 2, 3, . . . , the azimuthal quantum number l = 0, 1, 2, . . . (that specifies the orbital angular momentum of the states s, p, d, . . . ), the magnetic quantum number ml = −l, −l + 1, . . . , l − 1, l, and the spin quantum number ms = ±1/2. To determine energy levels, in addition to the nucleus one must also take into account the e ects of the other electrons. Approximating the latter by a spherically symmetric e ective

174 6 Consequences of Symmetries

potential, the potential field is invariant under all rotations around any axis through the origin, irrespective of the angle. The group of these symmetry operations is equivalent to the special orthogonal group SO(3). Its irreducible representations can be characterized by the eigenvalues of the orbital angular momentum operator L. As the dimension of the irreducible representation associated with quantum number l is 2l + 1, the corresponding energy level is 2l + 1-fold degenerate. Owing to the radial part of the wavefunction, energy obviously depends on the principal quantum number n as well, however, it is independent of ml.

Instead of a single isolated atom consider one at a lattice site of a regular crystal, where the e ects of its neighbors may no longer be neglected. Even when electrons are localized on atoms and e ects of overlapping between electronic states can be ignored, the electrostatic field produced by neighboring atoms, the crystal field may change the electronic states. This potential possesses the local symmetry of the site in question, that is, the symmetry in the arrangement of the neighboring atoms. This is usually lower than perfect spherical symmetry, therefore, within a crystal, the free ion’s originally 2l + 1-fold degenerate energy level of quantum number l is split into several sublevels. This phenomenon is called crystal-field splitting.

The amount of splitting depends on the strength of the potential at the atom, nevertheless the character of splitting – i.e., the number and degeneracy of the sublevels into which the atomic level is split – are determined solely by the symmetries of the atomic position. Without knowing the exact form of the crystal field, these can be specified from the symmetry properties of the crystal. This is of particular importance as it allows for the determination of atomic positions.

As an example, consider an atom in a cubic environment for which the group of local symmetries is Oh. This is the case for an atom at the center of the Bravais cell in a face-centered cubic crystal. As we shall see in the next chapter, this position is called an octahedral site. Then the crystal field Vcr field(r, θ, ϕ) acting on the electrons orbiting the atom possesses cubic symmetry, i.e., it is invariant under the rotations and reflections listed in Tables 5.1 and 5.4. The simplest such potential contains the lowest-order polynomial that shows cubic symmetry but is not invariant under an arbitrary

	rotation:
	Vcr field = A(r) + D(x4 + y4 + z4			−	3	r4) + . . .
					5				(6.1.7)
	= A(r) + Dr4 1 −	5	sin4	θ sin2 2ϕ −			5	sin2 2θ .
		4					4

The solutions to the Schrödinger equation with this potential must transform according to the irreducible representations of the symmetry group of the cubic crystal. If the 2l + 1 states of a free atom with angular momentum l are used as a set of basis functions, the representation of the symmetry group of the cubic crystal (which contains a finite number of rotations through 90◦,

6.1 Quantum Mechanical Eigenvalues and Symmetries

175

120◦, and 180◦) is usually reducible on it. Thus the splitting of the state with orbital angular momentum l depends on how this reducible representation is reduced to the sum of the irreducible representations of the cubic symmetry group. The remaining degeneracy is determined by the dimensions of these irreducible representations. This reduction is readily calculated using the characters of the representations.

According to the formulas given in Appendix C, the character of a rotation through angle ϕ is

		l		sin(l +	1 )ϕ
	χl(ϕ) =		eimϕ =			(6.1.8)
					2
				sin 1	ϕ
		m=−l	2

in the representation of index l of the rotation group.

Taking this expression for integer values of l at angles ϕ = 0, π, π/2, and

2π/3, which appear among the symmetries of the cubic system,
χl(0)	=	2l + 1 ,				(6.1.9-a)
χl(π) = (−1)l ,						(6.1.9-b)
χl(π/2)		(	1)l/2		for l even ,	(6.1.9-c)
	= (−1)(l−1)/2				for l odd ,
			−
			1	for l = 3k ,
χl(2π/3)			1	for l = 3k + 2 .		(6.1.9-d)
	=		0	for l = 3k + 1 ,

−

The characters for the 48 elements of the cubic symmetry group Oh are given in Table 6.1. When calculating them one must take into account that the elements of the cubic group can be divided into ten classes (see Appendix D). The first class contains the identity element; the second – rotations through 180◦ around the three equivalent directions 100 (C42m, m = x, y, z); the third – rotations through 90◦ and 270◦ around the directions 100 (C4m and C43m); the fourth – rotations through 180◦ around the six equivalent directions110 (C2p, p = a, b, c, d, e, f ); the fifth – rotations through 120◦ and 240◦ around the directions 111 (C3j and C32j , j = a, b, c, d). The characters for these classes can be read o from the formulas in (6.1.9). Elements of the other five groups are obtained by multiplying the above by the inversion operation. A basis function is odd (even) under inversion when the azimuthal quantum number is odd (even).

To proceed with the reduction process one must also know the characters of the irreducible representations of the group Oh. These are listed in Table D.1, in Appendix D. The general rules of reduction are also given there. Some straightforward algebra leads to the results in Table 6.2.

176 6 Consequences of Symmetries

Table 6.1. Character table for the representation of the point group Oh using the spherical harmonics Ylm(θ, ϕ) as basis functions. Only the first few values of the quantum number l are listed

l	E 3C42m		6C4m 6C2p 8C3j			I 3σm 6S4m 6σp 8S3j
0	1	1	1	1	1	1	1	1	1	1
1	3	−1	1	−1	0	−3	1	−1	1	0
2	5	1	−1	1	−1	5	1	−1	1	−1
3	7	−1	−1	−1	1	−7	1	1	1	−1
4	9	1	1	1	0	9	1	1	1	0
5	11	−1	1	−1	−1	−11	1	−1	1	1
6	13	1	−1	1	1	13	1	−1	1	1

Table 6.2. Reduction of representation Dl of the rotation group to the irreducible representations of the group Oh. Only the first few values of the quantum number l are listed

l	Reduction	Dimension
0	D0 = Γ1	1
1	D1 = Γ15	3
2	D2 = Γ12 Γ25	5 = 2 + 3
3	D3 = Γ2 Γ15 Γ25	7 = 1 + 3 + 3
4	D4 = Γ1 Γ12 Γ15 Γ25	9 = 1 + 2 + 3 + 3

As an application, consider an atom with a 3d-electron placed in a cubic environment, and examine the possible states of the electron. According to Hund’s first and second rules, S = 1/2, L = 2 in ground state – thus the state is 2D. In a free atom this tenfold degenerate state is split by the spin–orbit interaction into a fourfold degenerate J = 3/2 state and a sixfold degenerate J = 5/2 state. According to Hund’s third rule, the state with angular momentum J = 3/2 is the ground state. However, this is not necessarily the case for atoms in a crystal. The reason for this is that in transition metals – especially the elements of the iron group – with partially filled 3d, 4d, or 5d shells the crystal-field splitting for d electron levels (that are considered to belong to the core) is on the order of Ecr field 103 cm−1 0.1 eV, which is larger than the energy correction due to spin–orbit interaction, Es–o 102 cm−1 0.01 eV. Thus to determine the state of the electron, the e ects of crystal field in the configuration specified by Hund’s first and second rules have to be taken into account first, and spin–orbit interactions second.

Writing the spatial part of the wavefunction of the electrons of azimuthal quantum number l in a free atom as

ψnlm	(r, θ, ϕ) = Rnl(r)Y ml (θ, ϕ) = Rnl(r)P ml (cos θ)eiml ϕ ,		(6.1.10)
l	l	l

6.1 Quantum Mechanical Eigenvalues and Symmetries

177

the associated Legendre polynomials for the d-state (l = 2) are

1 sin2 θ

3

P ml (cos θ) =	1 sin θ cos θ
2	3

1 (3 cos2 θ − 1)

2

ml = ±2,

ml = ±1,

(6.1.11)

ml = 0.

These functions transform according to the representation D2 of the rotation group. In a cubic environment, this fivefold (or, when spin is also taken into account, tenfold) degenerate level is split into two levels that transform according to Γ12 and Γ25. A twoand a threefold (or, with spin, a fourand a sixfold) degenerate energy level are thus obtained.

It is readily shown that the basis functions of the irreducible representations are the symmetric and antisymmetric combinations of the functions with ml = ±2 and ml = ±1, plus the function with ml = 0:

ψxy = Rn2	(r) √−2		Y22 − Y2−2	=		16π		1/2
								Rn2(r) sin2 θ sin 2ϕ
		i				15

1/2

=15 Rn2(r)xy/r2 ,

4π

ψx2−y2 = Rn2(r) √2

Y22 + Y2−2

=

1/2

16π

Rn2(r) sin2 θ cos 2ϕ

1

15

=

1/2

16π

Rn2(r)(x2 − y2)/r2 ,

15

ψxz = Rn2(r) √−2

Y21 − Y2−1

=

4π

1/2

Rn2(r) sin θ cos θ cos ϕ

1

15

=

4π

1/2

(6.1.12)

Rn2(r)xz/r2 ,

15

ψyz = Rn2(r) √2

Y21 + Y2−1

=

4π

1/2

Rn2(r) sin θ cos θ sin ϕ

i

15

=

4π

1/2

Rn2(r)yz/r2 ,

15

ψz2 = Rn2(r)Y20(θ, ϕ) =

16π

1/2

Rn2(r)(3 cos2 θ − 1)

5

=

1/2

Rn2(r)(3z2 − r2)/r2 .

16π

5

These wavefunctions are real. The corresponding charge distributions are schematically shown in Fig. 6.1.

Wavefunctions ψxy , ψyz , and ψxz are equivalent in a cubic environment and transform according to the irreducible representation Γ25 of the cubic

178	6 Consequences of Symmetries
	z			z		z			z
		y		y				y		y
				x
		x						x		x
	dz2			dx2-y2		dxy			dyz

	z
		y
		x
	dxz

Fig. 6.1. Schematic representation of d-state wavefunctions

point group. In chemical nomenclature, this representation is also denoted by T2g – which is why these states are often called t2g-states.

Functions ψx2 −y2 and ψz2 transform according to the irreducible representation Γ12, and they, too, have the same energy in a cubic environment. Chemical nomenclature uses the symbol Eg for this representation and the name eg-state for the corresponding states.

As t2g- and eg-states belong to di erent irreducible representations, their energies are usually di erent. Which of them is lower depends on the spatial arrangement of the surrounding atoms. When nearest neighbors are located along crystallographic axes forming an octahedral environment, t2g-states tend to be of lower energy, as wavefunctions overlap to a smaller extent and the Coulomb repulsion is weaker. The pattern of energy-level splitting is similar when a single 3d-electron is missing – however, the order of the levels is then reversed.

Using the wavefunctions given in (6.1.12), the expectation value for the z component of the angular momentum contains either the integral

2π			∂
0
	sin(mlϕ)								sin(mlϕ) dϕ	(6.1.13)
			i				∂ϕ
or
2π		∂
0
	cos(mlϕ)							cos(mlϕ) dϕ .		(6.1.14)
		i				∂ϕ
Both of them vanish, therefore
	Lz = 0 .									(6.1.15)

The vanishing of the two other components can be shown along similar lines. The orbital angular momentum is thus said to be quenched. This explains the curiosity presented in Chapter 3 stating that one should ignore orbital angular momentum contributions to the e ective magneton number of transitionmetal ions. The hitherto neglected spin–orbit interaction may slightly polarize states in a crystal field, giving rise to a slight change in the e ective magneton number.

6.1 Quantum Mechanical Eigenvalues and Symmetries

179

6.1.3 Spin Contributions to Splitting

Rare earth metals with a partially filled 4f shell – lanthanoids (lanthanides)2

– show a markedly di erent behavior from transition metals. Because of the screening e ect of electrons on the 5s and 5p shells, splitting caused by the crystal field due to neighboring atoms is typically on the order of just Ecr field 102 cm−1 0.01 eV, whereas splitting caused by the spin–orbit interaction is an order of magnitude larger, Es–o 103 cm−1 0.1 eV. For lanthanoids – as well as actinoids – the e ect of the crystal field is thus smaller than that of the spin–orbit interaction; therefore the ground-state electron configuration has to be specified using Hund’s three rules, and then the e ects of the crystal field have to be treated as a perturbation, with J

fixed.

For example, when there is a single electron on the 4f shell, then, according to Hund’s rules the quantum numbers of the ground state are S = 1/2, L = 3, J = L − S = 5/2 (2F5/2 state), and the state with the same L and S but J = 7/2 is the first excited state. To determine the crystal-field splitting of the sixfold degenerate ground state, the behavior of the wavefunction under rotations should be known not only for integer values of the angular momentum but also half-integer values, which will be denoted by j. Relations (F.1.29) and (6.1.8) are still valid:

	χj (ϕ) =	sin(j + 1 )ϕ			(6.1.16)
			2	.
		sin 1	ϕ
	2
	This implies
	χj (ϕ + 2π) = −χj (ϕ) ,				(6.1.17)

that is, because of the spinor character of spin, rotation through 2π is not an identity transformation, only rotation through 4π is. Thus, in the spirit of the discussion in Appendix D, the group of symmetries has to be extended to the double group in which rotation through 2π (denoted by E) and its products with ordinary rotations also appear.

The characters for rotations compatible with cubic symmetry are

χj (0)	= 2j + 1 ,	(6.1.18-a)
χj (π)	= 0 ,	(6.1.18-b)

2Instead of the commonly used names lanthanides and actinides, the International Union of Pure and Applied Chemistry (IUPAC) recommends the usage of lanthanoids and actinoids for the groups of atoms from La to Lu and from Ac to Lr.

1806 Consequences of Symmetries

√

			2		for	j = 1/2, 9/2, . . . ,
χj (π/2) =			0		for	j = 3/2, 7/2, 11/2, . . . ,	(6.1.18-c)
			√2		for	j = 5/2, 13/2, . . . ,
		−
χj (2π/3) =			1		for	j = 3/2, 9/2, . . . ,	(6.1.18-d)
			1		for	j = 1/2, 7/2, . . . ,
		−	0		for	j = 5/2, 11/2, . . . .

From these relations, one may determine the characters for the elements of the double group O of the octahedral group for various values of j. These are listed in Table 6.3.

Table 6.3. Characters for the elements of the double group O on the basis of wavefunctions with half-integer angular momentum, for various values of j

					3C42m										6C2p
	j E E							6C4m 6C4m										8C3j 8C3j
							42m
					+ 3	C									+ 6C2p
	1/2 2 −2				0			√			√				0			1	−1
								2		− 2
	3/2	4	−4		0			√	0		√		0		0			−1	1
	5/2	6	−6		0			− 2		2					0			0	0
	7/2	8	−8		0			√	0		√		0		0			1	−1
	9/2	10	−10		0			2		− 2					0			−1	1

Using the character table of the irreducible representations of the double group O – Table D.2 of Appendix D –, the reductions listed in Table 6.4 are readily established.

Turning back to the case when the f (l = 3) shell contains a single electron, the total angular momentum may be either j = 5/2 or j = 7/2. The wavefunctions of the j = 5/2 level that transform according to the representation D5/2 will, in a cubic crystal field, go over into functions that transform

Table 6.4. Reduction of the representations Dj with half-integer j of the continuous rotation group to the irreducible representations of the double group O

j		Reduction			Dimension
1/2		D1/2	= Γ6		2
3/2		D3/2	= Γ8		4
5/2	D5/2 = Γ7 Γ8				6 = 2 + 4
7/2	D7/2	= Γ6 Γ7		Γ8	8 = 2 + 2 + 4
9/2	D9/2	= Γ6 Γ8		Γ8	10 = 2 + 4 + 4