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

the cross section is written as

Looking back at (E.2.5), the previous expression is factorized into three terms:

k mn 2, which is specific to the scattered particle alone, |UK |2 which de- k 2π 2

scribes the interaction, and S(K, ε/ ), the dynamical structure factor, which depends only on the internal dynamics of the scattering system. The above expression for the cross section is called the Van Hove formula. It is very generally valid, since only the applicability of the Born approximation has been assumed. Whether the correlation function of nuclear positions, electronic charge density, or spin density appears in the cross section is determined by the dominant interaction between the incoming particles and the sample.

In nonmagnetic scattering neutrons interact with nuclei at positions rm. Using the Fermi pseudopotential

for characterizing the interaction, the point-like character of the scattering center makes the Fourier transform independent of K:

For identical atoms the factor AK takes the K-independent form (k/k )|am|2, and the Fourier transform of the correlation function for the density of the nuclei appears in the cross section. From this formula information can be obtained about nuclear positions, about the equilibrium atomic positions in the crystal, and, through the motion of the nuclei, about phonons. This is discussed in more detail in Chapter 13.

Expression (E.2.10) for the cross section is valid also for the scattering of electrons by crystals. Through electronic position coordinates the spatial distribution of electrons bound to atoms appear in this case. If the electronic wavefunction is φi (r) in the initial and φf (r) in the final state, the transition matrix element is

k, i |Hint| f, k = drn drφi (r)φf (r)e−i(k−k )·rn U (r − rn) . (E.2.22)

When the relative coordinate is used, then, besides the Fourier transform of the potential, an additional factor

F (K) = drφi (r)φf (r)e−iK·r (E.2.23)

For r = 0 the first term yields the well-known dipole–dipole interaction:

An extra contribution arises at r = 0; this is clearly seen when the relation

is used in the second term. However, a similar singular contribution arises from the first term, too, as it was shown in Section 3.3.1. By repeating the derivation presented there, the interaction Hamiltonian can be given as the sum of two terms:

where, in addition to the usual dipole–dipole interaction term, the Fermi contact interaction

appears. This term is related to the overlap of the two moments, and plays a fundamental role in hyperfine interactions.

In magnetic scattering the spin direction can change for neutrons as well as for electrons bound to atoms. Therefore, besides the wave vector of the neutron and the spatial part of the electron wavefunction, the spin states of the neutron and electron must also be explicitly included in the wavefunction of the initial and final states. Denoting the spatial part of the electron wavefunction before

To evaluate this expression, the form (E.2.28) of the interaction is best used. Integration by parts of the first term gives

Expressed in terms of the unit vector e along the direction of K, the matrix element can be rewritten as

If zm electrons of the mth atom participate in the magnetic scattering process, and their common wavefunction is Ψm(r1, r2, . . . rzm ), then the cross section contains the magnetic form factor

which characterizes the spatial distribution of “magnetic” electrons. Here Sm is the total spin of the mth atom; the associated magnetic moment μm is given by

This total atomic moment appears in the matrix element, too,

k, i |Hint|f, k = −μ0 sn|μn|sn · σj | (e × [μm × e]) |σj Fm(K) . (E.2.41)

In expressions (E.2.37) and (E.2.39) for the atomic form factor rj and rν denote the absolute position of the electron. Changing to the relative coordinate

allows for the separation of a phase factor due to the actual atomic position, and the obtained Fm(K) is the true atomic magnetic form factor.

The cross section is obtained by summing over all atoms. The calculations below are for the simple case when the incident neutron beam is unpolarized and the polarization of the scattered beam is not measured, either. By characterizing the probabilities of the possible spin states of the neutron by the density matrix ρn, summing over final-state spins, and using the density matrix ρ for specifying the initial spin state of the atoms, the cross section is given by

If spin–phonon interactions are neglected, the averaging procedures can be performed separately for spins and atomic positions. When calculating the contribution of magnetic scattering, atoms are assumed to occupy their equilibrium positions, and so

The spatial distribution of the moment is characterized by the atomic magnetic form factor, while the spin correlation function describes spin dynamics.

References

1.E. Balcar and S. W. Lovesey, Theory of Magnetic Neutron and Photon Scattering, Oxford University Press, Oxford (1989).

2.S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vol. I: Nuclear Scattering, Clarendon Press, Oxford (1986).

3.S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vol. II: Polarization E ects and Magnetic Scattering, Clarendon Press, Oxford (1986).

4.W. Marshall and S. W. Lovesey, Theory of Thermal Neutron Scattering, Clarendon Press, Oxford (1971).

Using the corresponding expression for the operator of the rotation around the z -axis,

is finally obtained.

A possible representation of the rotation matrices can be easily obtained from determining their action on the function ψ(r) = r. For a rotation around the z-axis through angle φ,

Likewise, for a rotation around the y-axis through angle ϑ,

while for a rotation around the z-axis through angle ψ,

Owing to the transformations of the axes, rotations appear in the reverse order in (F.1.15):

(F.1.19) Thus we have obtained the representation of the rotation group by 3 × 3 orthogonal matrices of unit determinant, as R T = R −1. Spatial rotations in three dimensions are indeed isomorphic to the group SO(3).

F.1.2 The Irreducible Representations of the Rotation Group

To determine further representations of the rotation group, we shall exploit the fact that J 2 and Jz commute, and therefore have a common set of eigenfunctions. Denoting the eigenvalues of J 2 by j(j + 1) and those of Jz by m,

668 F The Algebra of Angular-Momentum and Spin Operators

it follows from the properties of the operators that j can take only positive, while m any integer or half-integer1 value,

j = 0, 1/2, 1, 3/2, 2, . . .,

The angular-momentum operator can be given in the space of states with quantum numbers j, m – denoted as |j, m – by (2j + 1) ×(2j + 1) matrices. To determine the matrix element, the raising and lowering operators J± = Jx±iJy are introduced; they increase and decrease the component Jz by unity:

It is straightforward to show that a suitable unitary transformation can take these matrices into (D.1.72), that is, the two forms are equivalent.

Expressing the rotation matrices in terms of these generators, as in

1By half-integer or half-odd-integer numbers we mean numbers of the form (2n + 1)/2, where n is integer.