Fundamentals of the Physics of Solids / 16-back-matter

648 D Fundamentals of Group Theory

Table D.1. The character table of the irreducible representations of the group Oh

		E 3C42m 6C4m 6C2p 8C3j					I 3σm 6S4m 6σp 8S3j
Γ1	A1g	1	1	1	1	1	1	1	1	1	1
Γ2	A2g	1	1	−1 −1		1	1	1	−1 −1		1
Γ12 Eg		2	2	0	0	−1	2	2	0	0	−1
Γ15	T1g	3	−1	1	−1	0	3	−1	1	−1	0
Γ25	T2g	3	−1 −1		1	0	3	−1	−1 1		0
Γ1	A1u	1	1	1	1	1	−1 −1		−1 −1 −1
Γ2	A2u	1	1	−1 −1		1	−1 −1		1	1	−1
Γ12	Eu	2	2	0	0	−1	−2 −2		0	0	1
Γ15 T1u		3	−1	1	−1	0	−3	1	−1 1		0
Γ25 T2u		3	−1 −1		1	0	−3	1	1	−1	0

of the group O –, and 3 are double-valued. The character table of the double group is given in Table D.2.

Table D.2. Character table of the irreducible representations for the double group O of the point group O

							3C42m									6C2p
			E E							6C4m		6C4m							8C3j 8C3j
									42m
							+3	C								+6C2p
Γ1	A1		1	1			1			1		1				1			1	1
Γ2	A2		1	1			1			−1			−1				−1		1	1
Γ3	E		2	2			2			0		0				0			−1	−1
Γ4	T1		3	3					−1	1		1					−1		0	0
Γ5	T2		3	3					−1	−1			−1			1			0	0
			2				0			√			√			0			1
Γ6	E1			−2						√	2	−√			2					−1
			2				0									0			1
Γ7	E2			−2						− 2		2								−1
Γ8			4	−4						0		0				0			−1	1
	U						0

D.2.2 Group Theory and Quantum Mechanics

The Hamiltonian of a quantum mechanical system may be invariant under certain coordinate transformations R, that is for each point r in space the relation

D.2	Applications of Group Theory	649
H(r ) = H(r) ,	if r = Rr	(D.2.2)

may be satisfied. The coordinate transformations R that meet this requirement form a group G. This is called the symmetry group of the Hamiltonian. Invariance under the elements of the group G also means that the Hamiltonian commutes with the operator O(R) associated with the coordinate transformation R, since

O(R) [H(r)ψ(r)] = H(R−1r)ψ(R−1r) = H(r)O(R)ψ(r) ,

(D.2.3)

and hence

O(R)H(r) = H(r)O(R) .

(D.2.4)

This also implies that if ψ(r) is an eigenfunction of the Hamiltonian with energy ε then O(R)ψ(r) is also an eigenfunction, with the same energy. This is Wigner’s theorem, which was discussed in Chapter 6.

The irreducible representations of the symmetry group of the Hamiltonian and the basis functions of the representations play a privileged role in the solution of the quantum mechanical eigenvalue problem. Below we shall list without proof a couple of theorems which are, to a certain extent, di erent formulations of the same statement.

If the Hamiltonian H is invariant under the transformations of a symmetry group G then the eigenfunctions that belong to the same energy form the basis of a representation of the group G. For an n-fold degenerate energy level there exist n linearly independent eigenfunctions ψi(r); the representation is provided by the matrices D(R) that appear in the equation

O(R)ψi (r) = Dji(R)ψj (r)	(D.2.5)

j

specifying their transformation properties. This representation is usually reducible. If, however, the group G contains every possible symmetry of the Hamiltonian then – barring accidental degeneracies – for each energy level the corresponding eigenfunctions transform according to an irreducible representation of the group G.

In quantum mechanical calculations one often needs to determine the matrix elements

Mij =		φ(μ), Qφ(ν)		(D.2.6)
		i	j
of an operator Q, where φ(μ)	(r) and φ(ν)		(r) are the basis functions of the
i		j

irreducible representations D(μ) and D(ν). If the symmetry of the operator Q is lower than that of the Hamiltonian then the operators O(R) that belong to the elements of the group G take the operator Q into

Qi ≡ O(R)QO−1(R) .

(D.2.7)

The operators Qi obtained this way are transformed into each other according to a representation D that is not the identity representation:

650 D Fundamentals of Group Theory

O(R)QiO−1(R) = Dij (R)Qj .	(D.2.8)

j

It can be shown that the matrix element in (D.2.6) vanishes unless the representation D(μ) appears in the reduction of the direct product of D and D(ν). In other words: the matrix element vanishes unless the identity representation appears in the reduction of the direct product of D(μ) , D, and D(ν). Using these selection rules it is possible to determine whether a quantum mechanical transition occurs.

If the operator Q possesses the full symmetry of the Hamiltonian then the matrix element is finite provided the identity representation appears in the reduction of the direct product D(μ) D(ν). As a special case consider the matrix elements of the Hamiltonian between the basis functions of irreducible representations. If D(μ)(R) and D(ν)(R) are two irreducible unitary represen-

tations of the symmetry group of the Hamiltonian, and the associated basis functions are φ(1μ)(r), φ(2μ)(r), . . . and φ(1ν)(r), φ(2ν)(r), . . . then

φi(μ), Hφj(ν)	= ε(μ)δμν δij ,	(D.2.9)

that is, the Hamiltonian has only diagonal matrix elements in this basis, and the matrix element is the same for each basis function of the μth irreducible representation. This means that if the Hamiltonian H is invariant under the transformations of a group G then the eigenfunctions that transform according to the same irreducible representation are of the same energy.

As a consequence of this, consider a Hamiltonian that may be written in the form H = H0 + H1, where H1 can be taken as a perturbation. Assume further that the symmetry group of H0 is G0, while H1 is invariant only under the transformations of a subgroup G of G0. Then under the perturbation H1 any energy level ε0 of H0 will split so that when the (usually reducible) representation of the group G over the eigenfunctions of the level ε0 is reduced only those states will necessarily be of the same energy that belong to the same irreducible representation. The energy level ε0 will be split into at most as many levels of di erent energy as the number of irreducible representations that appear in the reduction (and that are compatible with the representation of the level ε0). If the representation of the group G over the eigenfunctions of the level ε0 is irreducible then the perturbation only shifts but does not split the level.

References

1.J. E. Cornwell, Group Theory in Physics, Vol. 1, Fourth Printing, Academic Press, London (1989).

2.A. P. Cracknell, Group Theory in Solid State Physics, Taylor & Francis Ltd., London (1975).

D.2 Applications of Group Theory

651

3.M. Hamermesh, Group Theory and Its Applications to Physical Problems, Addison-Wesley/Pergamon, New York (1962).

4.R. L. Libo , Primer for Point and Space Groups, Undergraduate Texts in Contemporary Physics, Springer-Verlag, Berlin (2004).

5.M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill, New York (1964).

6.E. P. Wigner, Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York (1959).

E

Scattering of Particles by Solids

A frequently used method for the investigation of solids is the scattering of particles (electrons, photons, neutrons, etc.) by the solid. In an ideal setup well-collimated monoenergetic beams are used in which the energy and the wave vector of the incident particles are known. By measuring the energy of the particles scattered in various directions (i.e., the energy distribution), the changes in energy and momentum permit us to determine the dispersion relation of the excitations created or absorbed in the sample, and through it to study the internal dynamics of the system.

By assuming potential scattering and magnetic interactions, we shall derive the general formula for the scattering cross section expressed in terms of some correlation function of the system. For potential scattering this will be the density–density correlation function, while for magnetic interactions it will be the correlation function for magnetic density.

E.1 The Scattering Cross Section

For simplicity, assume that the incoming and scattered beams can be both approximated by plane waves, of wave vectors k and k , respectively. The properly normalized wavefunctions are

k	=	1		eik·r ,	k		=	1		eik ·r ,	(E.1.1)
|		√			|			√
			V						V

where V is the total volume, not only that of the sample. Assuming that the interaction with the scattered particle gives rise to the creation or absorption of an elementary excitation in the sample, and using the consequences of discrete translational symmetries derived in Chapter 6, the wave vector of the

elementary excitation created in the scattering process between states |k and

|k is

q = k − k + G .

(E.1.2)

654 E Scattering of Particles by Solids

In the absorption process an elementary excitation of wave vector −q is absorbed. G is a vector of the reciprocal lattice.

Simultaneously, the crystal makes a transition from the initial state |i of energy Ei to a final state |f of energy Ef. The energy of the elementary excitation is the di erence of these two. On account of energy conservation, the change in the energy of the sample can be expressed with the change in the particle energy. For the scattering of particles of mass mn with a quadratic dispersion relation, the energy transferred to the sample in the scattering process is

ε =	2k2	−	2k 2	= Ef − Ei .	(E.1.3)
	2mn		2mn

If a single elementary excitation is created or annihilated in the scattering process, then an inelastic peak appears in the distribution of the scattered particles. Its direction is determined by the momentum of the excitation, while its energy is specified by the energy transfer in the process. If the changes in the momentum and energy of the scattered particle can be measured simultaneously, then the dispersion relation for the elementary excitation created or annihilated in the process can be determined.

The situation is often not so simple in scattering experiments, therefore evaluating the scattering cross section, which determines the strength of the scattering process is of the utmost importance.

Suppose that scattering takes the sample from a well-defined initial state |i to an equally well defined final state |f . The joint state of the sample and the particle is denoted by |i |k before scattering, and by |f |k after it. Assume, furthermore, that the Hamiltonian Hint of the interaction between the particle and the crystal is known.

Owing to the weakness of the interaction, the Born approximation su ces to determine the scattering cross section in the majority of cases. According to quantum mechanics, the transition probability between the initial state and a specified final state is

		2π		k | f |Hint|i |k 2 δ(ε − Ef + Ei) ,				(E.1.4)
Wi,k→f,k =
where the factor δ(ε
	−	E		+ E )	ensures the conservation of	energy.
		f		i			1/2	arising
We shall follow the convention that separates the factor 1/V

from the normalization of the incident and scattered wavefunctions, i.e., uses

the notations |k

= e

ik·r

k

= eik ·r

for the expressions without this

and |

normalization factor. Therefore

Wi,k→f,k

2π 1

k | f |Hint|i |k 2

δ(ε − Ef + Ei) .

(E.1.5)

= V

2

The energy and propagation direction of the scattered particle can be measured only with a certain precision. Therefore we have to sum the contributions of those states with wave vector k whose energy is in the interval

			E.1	The Scattering Cross Section		655
	dε =	dε	dk =	2k	dk	(E.1.6)
		dk		mn

and whose direction is in the element of solid angle dΩ = sin θ dθ dϕ. The number of such states is given by

	ρ(ε)dε dΩ =	V	k 2 dk sin θ dθ dϕ = V	k mn	dε dΩ .	(E.1.7)
		(2π)3		(2π)3 2

The intensity of the beam scattered into the selected solid angle element and energy range is obtained by multiplying the transition probability by this density of final states and the number Nk of incoming particles in state k:

ΔIk = NkV	k mn		2π 1			k | f |Hint|i |k		2	δ(ε − Ef + Ei) dε dΩ . (E.1.8)
	(2π)3 2		V 2

The doubly di erential cross section is then obtained through division by the incoming flux Φ. This flux – that is, the number of particles passing through a unit surface in unit time – is just the product of the density of particles in state k and their velocity:

Φ =	Nk		k	.	(E.1.9)
	V mn

Thus the volume factors eventually cancel out, and the cross section is

d2σ	=	k		mn	2	k | f |Hint|i |k		2	δ(ε − Ef + Ei) .	(E.1.10)
dΩ dε		k		2π 2

Very similar expressions are derived for photon scattering, only the prefactors of the matrix element are slightly di erent: owing to the di erent dispersion relation, neither the density of final states nor the incident flux are the same as before. Using the angular frequency instead of the energy, the density of final states, derived from

ρ(ω)dω dΩ =

V

2

dk sin θ dθ dϕ ,

k

(E.1.11)

(2π)3

is now

V

dk

V

n

ρ(ω) =

k 2

=

k 2

,

(E.1.12)

(2π)3

dω

(2π)3

c

where n is the refractive index of the sample. Writing the incoming photon

	flux as			c
	Φ =	v	=		,	(E.1.13)
		V		nV

and expressing it in terms of the change in frequency rather than the change in energy, the cross section is now

dΩ dω	=		2π c		2	k | f |Hint|i |k 2 δ(ω − (Ef − Ei)/ ) . (E.1.14)
d2σ			nk

656 E Scattering of Particles by Solids

Below we shall derive the formulas for neutron scattering, however, our results will apply – up to a multiplicative factor – to light scattering as well.

The expression studied above is for fixed initial and final states of the sample. However, neither is known from measurements, which provide data only about the incident and scattered particles. Assuming that the sample is initially in thermal equilibrium, the probability of the initial state |i is

pi =	1	e−βEi .	(E.1.15)
	Z

This probability has to be used as the weighting factor of the initial states in the averaging procedure. Finally, the final states |f for which the transition matrix element is finite must be summed over. The doubly di erential cross section is then

d2σ

=

k

mn

2

if

pi

k | f |Hint|i |k

2

δ(ε − Ef + Ei) .

(E.1.16)

dΩ dε

k

2π 2

E.2 The Van Hove Formula for Cross Section

As it was pointed out by L. Van Hove in 1954, the above form of the scattering cross section can be expressed with some correlation function of the scattering centers, the exact form of which depends on the type of the interaction between the scattered particles and the sample.

E.2.1 Potential Scattering

The previous general expression will be reformulated by assuming that the interaction between the scattered particle and the electrons or ions of the crystal can be written as

Hint = U (r − rm) ,	(E.2.1)

m

where rm stands for the position coordinates of the mth scattering center in the crystal. Substituting this expression into the transition matrix element, and choosing the wavefunction of the scattered particles as a plane wave,

k| i|Hint|f |k	=	dr		i e−i(k−k )·r					m	U (r − rm) f
	= UK		i		e−	iK	rm	f			(E.2.2)
				m		·

is obtained, where K = k − k and K is the momentum transferred in the scattering process, while

E.2 The Van Hove Formula for Cross Section		657
UK =	dr e−iK·r U (r)	(E.2.3)

is the Fourier transform of the interaction at the wave number that corresponds to the above momentum transfer.

Substituting this into the cross section formula (E.1.16),

d2σ

=

k

mn

2

|UK |

2

i,f m,n pi

i

e−iK·rm f

(E.2.4)

dΩ dε

k

2π 2

×f eiK·rn i δ(ε − Ef + Ei) .

Using the notation

k

mn

2

2

AK =

|UK |

,

(E.2.5)

k

2π 2

the cross section is written as

d2σ

dΩ dε = AK

pi

i

e−iK·rm

f f

eiK·rn

i

δ(ε − Ef + Ei) . (E.2.6)

i,f m,n

Making use of the integral representation (C.3.1-d) of the Dirac delta function,

∞

dt ei(ε−Ef+Ei)t/ .

(E.2.7)

δ(ε − Ef + Ei) = 2π

1

−∞

A suitable rearrangement of the terms in the exponent leads to

d2σ

1

∞

= AK

dt

i,f m,n pi

i eiEit/ e−iK·rm e−iEft/

f

dΩ dε

2π

(E.2.8)

−∞

×f eiK·rn i eiεt/ .

Expressing the time-dependent factors of the first matrix element in terms of the Hamiltonian H0 of the sample, and using the usual form of the timedependent operators,

d2σ

1

∞

dt eiεt/

i,f m,n pi

i

eiH0t/ e−iK·rm e−iH0t/

f

dΩ dε

= AK 2π

−∞

× f

eiK

rn

∞

·

i

(E.2.9)

dt e

i

i,f m,n pi

i e−

iK

rm (t)

f

f

e

iK

rn

i .

= AK 2π

·

·

1

εt/

−∞

Making use of the completeness relation for the states, the sum over final states can be evaluated. Using the notation of thermal averaging for denoting summation over the initial states multiplied by the weight factors pi,

dr dr e−iK·r

658 E Scattering of Particles by Solids
	d2σ		1		∞
		= AK				dt eiεt/ m,n		e−iK·rm (t)eiK·rn .	(E.2.10)
	dΩ dε			2π
					−∞
Using the identity
e−iK·rm (t) eiK·rn			= 1			dr e−iK·r δ(r − rm(t)) dr eiK·r			δ(r − rn)2
									(E.2.11)
and the density
					ρ(r, t) = δ(r − rm(t))				(E.2.12)
						m

of the scattering system, the thermal average in the cross section can be rewritten as

m,n	e−iK·rm (t) eiK·rn	=	dr dr e−iK·(r−r ) ρ(r, t)ρ(r , 0) . (E.2.13)

This formula contains the density–density correlation function of scattering centers.

In line with the normalization introduced in Chapter 2,
Γ (r, r , t) =	V	ρ(r, t)ρ(r , 0) ,	(E.2.14)
	N

where N is the number of scattering centers. For spatially homogeneous systems Γ (r, r , t) is a function of r − r alone, therefore it is customary to use

the notation			V
	Γ (r, t) =			ρ(r + r , t)ρ(r , 0) ,	(E.2.15)
			N
which gives		=		ρ(r + r , t)ρ(r , 0) .
m,n	e−iK·rm (t) eiK·rn				(E.2.16)

Since the integrand is in fact independent of r , integration with respect to r gives the volume of the sample, so

m,n	e−iK·rm (t) eiK·rn	= N	dr e−iK·r Γ (r, t) .	(E.2.17)

Introducing the spatial and temporal Fourier transform of the density–density correlation function Γ (r, t) by

∞
S(K, ε/ ) =	dt eiεt/ dr e−iK·r Γ (r, t) ,	(E.2.18)

−∞