Fundamentals of the Physics of Solids / 13-The Experimental Study of Phonons

13.3 Neutron Scattering on a Thermally Vibrating Crystal

439

Neutron emitted by reactors or spallation sources are usually many times more energetic than thermal neutrons. When they are passed through a moderator, a su ciently intense beam of slow (thermal) neutrons is obtained. After collimation and monochromatization via Bragg reflection, a beam with definite energy and momentum emerges. If it is scattered inelastically on the sample, the task is to measure the angular and energy distribution of the scattered beam. Instead of the device used in neutron di raction measurements (Fig. 8.9), a triple-axis spectrometer is now required. This is shown in Fig. 13.10.

Monochromator

Source	Detector

'

Collimator

Analyzer

Sample		&
		&

Fig. 13.10. Schematic sketch of a triple-axis spectrometer, used in inelastic neutron scattering measurements

The beam scattered by the sample falls on an analyzing crystal, which transmits only Bragg reflected neutrons of a particular wavelength (energy) to the detector. The monochromator, the sample holder, and the analyzing crystal can be rotated independently, which explains the origin of the name of the instrument.

13.3.1 Coherent Scattering Cross Section

The general discussion of neutron scattering on crystals with a polyatomic basis is straightforward in principle, however, for notational simplicity, we shall refrain from this, and assume that there is a single atom per primitive cell. We shall denote its position in the mth cell by Rm.

The interaction between the neutron and the nucleus is described by the Fermi pseudopotential

U (r − Rm) =	2πam 2	δ(r − Rm) ,	(13.3.3)
	mn

spectroscopy” (BNB) and “for the development of the neutron di raction technique” (CGS).

440 13 The Experimental Study of Phonons

where am is the scattering length of the atom located at Rm. Even when the sample is chemically pure, the scattering length can be di erent for di erent isotopes of the same element. Based on the isotopic composition, an average scattering length may be introduced, and the sample can be conceived as if atoms with this average scattering length were located at each point of the lattice. This would lead to perfectly coherent scattering. However, in real samples an additional incoherent contribution appears in the scattering pattern because of the random deviations of the scattering length from the average value. The coherent and incoherent components can be fairly well separated experimentally. As we shall see, coherent scattering leads to sharp peaks in the intensity distribution of the scattered beam with respect to energy or scattering angle, while the incoherent part usually gives a featureless, smeared-out background. In our calculations we shall focus exclusively on the coherent contribution.

As demonstrated in the previous chapters on the structure of crystalline and amorphous materials, the intensity of the scattered beam is proportional to the K = k − k component of the Fourier transform of the pair correlation function g(r) that characterizes the spatial correlation of scattering centers. Apart from the contribution of the direct beam, this is just the structure factor

	1
S(K) =				e−iK·Rm eiK·Rn .	(13.3.4)
	N	m,n

For scattering on a thermally vibrating lattice it is plausible to assume that the time-dependent displacements u(m, t) need to be taken into account in the atomic position vectors, and therefore the cross section is related to the Fourier transform of the time-dependent correlation function, the dynamical structure factor

S(K, t) =	1		e−iK·(Rm +u(m,t))eiK·(Rn+u(n) .	(13.3.5)
	N m,n

It will be shown in Appendix E that this is indeed so. Here we just summarize the results derived there.

While neutrons are scattered elastically, without any energy transfer on a static system of atoms, scattering on a vibrating lattice is not necessarily elastic. According to the Van Hove formula, the doubly di erential cross section – in which variations in the energy of the scattered particle are also taken into account – is proportional to the temporal Fourier transform of S(K, t):

∞

S(K, ω) = N1

−∞

∞

= N1

−∞

dt eiωt m,n e−iK·(Rm +u(m,t))eiK·(Rn +u(n))
	(13.3.6)
dt eiωt m,n e−iK·(Rm −Rn ) e−iK·u(m,t)eiK·u(n)	,

13.3 Neutron Scattering on a Thermally Vibrating Crystal

441

where ω and the energy transfer ε are related by

ω = ε =	2k2	−	2k 2	= Ef − Ei .	(13.3.7)
	2mn		2mn

On account of energy conservation, this is the same as the change in the energy of the sample – that is, the energy of the created or annihilated phonon.

When atomic displacements are expressed, through (12.1.39), in terms of phonon creation and annihilation operators, the latter appear in the exponents of the structure factor. Because of their operator character, the exponentials cannot be merged; the relationship

eAeB = eA+B

(13.3.8)

holds only when A and B commute. When they do not, but their commutator [A, B] commutes with A and B, then the Baker–Hausdor formula4 applies:

eAeB = eA+B e 21 [A,B] .

(13.3.9)

Since the thermodynamic average of such terms has to be taken, we shall make use of another theorem which asserts that for the linear combination

C =	γq a(q) + γq a†(q)	(13.3.10)

q

of the creation and annihilation operators of harmonic oscillators the thermal average of eiC is given by

eiC = e− 21 C2 .	(13.3.11)
When this is applied to C = A + B,
ei(A+B) = e− 21 A2+AB+BA+B2 .	(13.3.12)
From (13.3.9) we then have
eiAeiB = e− 21 A2−AB− 21 B2 = e− 21 A2 e−AB e− 21 B2 .	(13.3.13)

As suggested by expression (13.3.6) for S(K, ω), we choose A = −K · u(m, t) and B = K · u(n). Since these quantities are the linear combinations of the phonon creation and annihilation operators with complex conjugate coe cients, and their commutator is a c-number, the above relations can be applied. For the first factor in (13.3.13) we introduce the notation

e−W = e− 21 (K·u(m,t))2 .

(13.3.14)

In homogeneous crystals this quantity is independent of time and the equilibrium position Rm of the atom. Therefore the third factor gives the same

4 H. F. Baker, 1905, and F. Hausdorff, 1906.

442 13 The Experimental Study of Phonons

contribution. The combined factor of exp(−2W ) is called the Debye–Waller factor.5 This determines the intensity of the scattered beam. Its temperature dependence will be discussed in the next subsection.

The middle term in (13.3.13) is more interesting. As only those terms contribute to the thermal average in which the same phonon is created and annihilated, their contribution is

exp q,λ	2M N ωλ(q) |K · e(λ)(q)|2		nλ(q) e−i[q·(Rm −Rn)−ωλ (q)t]
	+ ( nλ(q) + 1) ei[q·(Rm −Rn )−ωλ(q)t]			.	(13.3.15)

Collecting every term, we have

S(K, ω) = N		∞
			dt eiωt m,n e−iK·(Rm −Rn )e
1
		−∞					e(λ)(q) 2
× exp						K
				2M N q,λ		|	·ωλ(q) |

+ ( nλ(q) + 1) ei[q·(Rm

−2W

nλ(q) e−i[q·(Rm −Rn )−ωλ (q)t]

−Rn)−ωλ (q)t] . (13.3.16)

The expression in the exponent of the last factor is usually small, therefore the exponential can be expanded in a power series. Regrouping the terms that depend on the coordinates of the ions,

S(K, ω) = N		∞	dt eiωte−2W m,n e−iK·(Rm −Rn )				(13.3.17)
1
		−∞
				e(λ)(q)	2	nλ(q) e−i(K+q)·(Rm −Rn)eiωλ (q)t
+ 2M N mn q,λ |K ·ωλ(q) |

+ ( nλ(q) + 1)e−i(K−q)·(Rm −Rn)e−iωλ (q)t + . . . .

Performing the sum over the lattice points and the integral with respect to time,

S(K, ω) = e−2W G		8δ(K − G)δ(ω)		(13.3.18)
			(λ) 2
+			|K · e (q)| nλ(q) δ(ω + ωλ(q))δ(K + q − G)
	2M N q,λ

ωλ(q)

9

+ ( n(q) + 1)δ(ω − ωλ(q))δ(K − q − G) + . . . .

5 P. Debye, 1914, and I. Waller, 1925.

13.3 Neutron Scattering on a Thermally Vibrating Crystal

443

Because of the presence of the factor δ(ω) the first term corresponds to an elastic scattering with no energy transfer ( ε = 0). Owing to the factor δ(K − G), which arises from the sum over the lattice points, the scattered beam vanishes unless K is a vector of the reciprocal lattice. Thus the Bragg condition for di raction is recovered. The other terms correspond to inelastic processes, in which phonons are emitted or absorbed.

13.3.2 Temperature Dependence of the Intensity of Bragg Peaks

In spite of thermal vibrations, Bragg peaks are not smeared out in the harmonic approximation. Their intensity is nonetheless reduced, and the Debye– Waller factor appears in their temperature dependence. Expressing once again atomic displacements, through (12.1.39), in terms of phonon creation and annihilation operators, and making use of the property that once again only those terms contribute to the thermal average in which the same phonon is created and annihilated,

2W =

=

=

(K · u(

m))2

(λ)

(q)

2

aλ(q)aλ† (q) + aλ† (q)aλ

(q)

2M N ωλ(q) K · e

q,λ

2

M N ωλ(q)

K · e(λ)(q)

nλ(q)

+

21 .

(13.3.19)

q,λ

In the isotropic case the branches that correspond to the three polarization directions are degenerate, and the polarization vectors of the longitudinal and two transverse branches can be taken as the basis vectors of a Cartesian coordinate system. Then summation over the polarization index λ of the branches leads to the expression

2W =	K2	+ 1 .	(13.3.20)
	n(q)
	M N ω(q)	2
q

Note that via (12.3.2) a very simple relation can be established between this quantity and the mean square displacement of ions:

2W =	K2	u2(m) .	(13.3.21)
	3

When the phonon spectrum is taken in the Debye approximation, the mean square displacement – and with it, the Debye–Waller factor – can be evaluated. Using the formulas given in (12.3.10) and (12.3.12),

	3		2K	2	1			π2		T		2
2W =							1 +	2				+ . . .	(13.3.22)
	2		2M			kBΘD		3		ΘD

444 13 The Experimental Study of Phonons
at low temperatures, and
	2K2	1			T	+	1		Θ	+ . . .
2W = 6									D		(13.3.23)
	2M		kBΘD		ΘD		36		T

at high temperatures. Since W appears in the exponent, the intensity of the Bragg peaks decreases with increasing temperature. The decrease in the scattering intensity occurs because neutrons emit virtual phonons and then (after an extremely short time) reabsorb them, or absorb phonons and after a short time re-emit them. Energy and momentum conservation is imposed only on the initial and final states but not on the intermediate virtual state – however, such processes modify the transition probability, and through it the scattered intensity. The probability of the process in which the phonon is first absorbed and then re-emitted is proportional to the thermal population of phonon states. The other process is proportional to n(q) + 1 because of the possibility of stimulated emission. This explains the factor 2 n(q) + 1 in W . Since an infinite succession of such processes can occur, W appears in the exponent in the Debye–Waller factor.

13.3.3 Inelastic Phonon Peaks

Having analyzed the first term in the expansion (13.3.18), which corresponds to elastic scattering (di raction), let us now turn to the study of higher-order terms. These describe how additional peaks – which are due to processes with nonvanishing energy transfer – appear in the scattering pattern, at shifted positions relative to the Bragg peaks. Owing to the factor δ(K + q − G), the second term corresponds to a process in which the momentum of the scattered neutron is k = k + q − G. The energy transfer in the scattering process can be written as Ef = Ei − ωλ(q) because of the restriction ε = Ef − Ei = − ωλ(q). This means that a phonon is absorbed in the scattering process. In Fig. 13.11 the phonon dispersion relation and εn(k + q) − εn(k) are plotted against the component of q along the direction of k. To account for the reciprocal-lattice vector that appears in the equation for crystal-momentum conservation, the phonon dispersion relation is repeated periodically, at equivalent wave vectors q.

As the figure shows there are at least two intersection points, i.e., there are two qs that satisfy both conditions simultaneously. Scattered neutrons must occur at the corresponding values of energy transfer. The intensity of this process is proportional to the thermal occupation n(q) of the phonon state that has to be annihilated. As (13.3.18) shows, the temperature-dependent Debye– Waller factor modifies the scattered intensity independently of the momentum and energy transfer – that is, the intensity of phonon peaks decreases with increasing temperature in the same way as the intensity of Bragg peaks.

The third term describes processes in which phonons are created. The presence of the factor δ(K −q −G) implies that the momentum of the neutron

13.3 Neutron Scattering on a Thermally Vibrating Crystal

445

!(q)	n(k+q) n(k)
!(q2)	!(q1)
	q
n(k)	n(k)
k	q1
q2

Fig. 13.11. Phonon and neutron dispersion relations for phonon absorption processes. The points of intersection indicate the solutions for which energy and quasimomentum conservation are both satisfied

is changed to k = k − q − G, while the restriction ε = Ef −Ei = ωλ(q) shows that the energy of the scattering system is increased by the amount of the phonon energy: Ef = Ei + ωλ(q). As this is a stimulated emission process, its intensity is proportional to n(q) + 1 . The conditions for phonon emission processes are shown in Fig. 13.12. It can be demonstrated that emission occurs only above a certain threshold energy, which corresponds to the requirement that the neutron should possess more energy than the phonon that is to be created.

!(q)
n(k)	!(q1)
k	q
	2k
n(k) n(k-q)	q1
	q2

Fig. 13.12. Phonon and neutron dispersion relations for phonon emission processes. The points of intersection indicate the solutions for which energy and quasimomentum conservation are both satisfied

The equations for energy and crystal-momentum conservation impose severe restrictions on the energy and direction of the scattered beam. When neutrons of energy εn and wave vector k are incident on the sample, the above conditions are satisfied only at certain values of the energy for a given direction of the scattered beam. Using a triple-axis spectrometer, one can change the energy of the incident beam and its direction relative to the crystallographic axes of the sample. By measuring the energy of the scattered beam in

446 13 The Experimental Study of Phonons

several directions, the changes in energy and crystal momentum permit the determination of the phonon spectrum.

In higher orders of the expansion (13.3.17) multiphonon processes are obtained with the corresponding restrictions of energy and momentum conservation. These usually give rise to a wide, smeared-out background.

13.3.4 The Finite Width of Phonon Peaks

In contrast to the infinitely narrow δ(ω ωλ(q))-type peaks of the previous description, peaks of finite width are observed in experiments, as shown in Fig. 13.13.

Fig. 13.13. The intensity of scattered neutrons as a function of energy for the onephonon absorption peak in aluminum [K.-E. Larsson et al., Proc. of the Symp. on Inelastic Scattering of Neutrons in Solids and Liquids, Vienna, p. 587 (1960)]

The finite resolution of detectors is just one of the reasons. More important is that the phonon states obtained in the harmonic approximation are not exact eigenstates of the system. As we have seen, anharmonicity gives rise to phonon decay, leading to a finite phonon lifetime. Assume that the probability of finding the phonon decreases exponentially because of decay processes. Denoting the inverse lifetime by Γλ(q), the time dependence of the phonon creation and annihilation operators are expected to be given by

aλ	(q, t) = aλ	(q)e−iωλ(q)t−Γλ (q)|t|,
aλ†	(q, t) = aλ†	(13.3.24)
		(q)eiωλ(q)t−Γλ (q)|t|

instead of (12.1.37). Therefore S(K, ω) in the cross section will have a form analogous to (13.3.17), however, an exponentially decreasing factor appears in the time integral because of the finite lifetime. Expression

13.3 Neutron Scattering on a Thermally Vibrating Crystal

447

∞	dt eiωte±iωλ(q)t−Γλ (q)|t| = 2 (ω ± ωλ(q))2 + Γλ2		(q)	(13.3.25)
		Γλ(q)

−∞

shows that the inelastic peaks that correspond to phonon creation and annihilation are no longer sharp deltas: their intensity distribution is a Lorentzian curve. The smaller the parameter Γ (i.e, the longer the lifetime) the sharper the phonon peak. From the peak width the phonon lifetime can be determined quantitatively.

Further Reading

1.P. Brüesch, Phonons: Theory and Experiments II, Experiments and Interpretation of Experimental Results, Springer-Verlag, Berlin (1986).

2.W. Marshall and S. W. Lovesey, Theory of Thermal Neutron Scattering: The Use of Neutrons for the Investigation of Condensed Matter, Clarendon Press, Oxford (1971).

3.S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vol. I: Nuclear Scattering, Clarendon Press, Oxford (1984).