Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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90 Gero Vogl and Bogdan Sepiol

19. I.S. Anderson, A. Heidemann, J.E. Bonett, D.K. Ross, S.K. Wilson, M. McK-

ergow: J. Less Common Met. 101, 405 (1984)

20. O.G. Randl, B. Sepiol, G. Vogl, R. Feldwisch, K. Schroeder: Phys. Rev. B 49,

8768 (1994)

21. G. Vogl, B. Sepiol: Acta Metall. Mater. 42, 3175 (1994)

22. R. Feldwisch, B. Sepiol, G.Vogl: Acta Metall. Mater. 43, 2033 (1995)

23. H.M. Eggersmann, B. Sepiol, G. Vogl, H. Mehrer: Defect and Diﬀusion Forum

143–147, 339 (1997)

24. M. Eggersmann, H. Mehrer: Phil. Mag. A 80, 1219 (2000)

25. Zs. T¨okei, J. Bernardini, P.Gas, D.L. Beke: Acta Materialia 45, 541 (1997)

26. B. Sepiol, A. Meyer, G. Vogl, R. R¨uﬀer, A.I. Chumakov, A.Q.R. Baron: Phys.

Rev. Lett. 76, 3220 (1996)

27. G.V. Smirnov, V.G. Kohn: Phys. Rev. B 52, 3356 (1995)

28. G. Vogl, B. Sepiol. In: Nuclear Resonant Scattering of Synchrotron Radiation,

ed by G. Langouche, H. de Waard, Hyperﬁne Interactions 123/124, (1999) p

595

29. B. Sepiol, A. Meyer, G. Vogl, H. Franz, R. R¨uﬀer: Phys. Rev. B 57, 10433

(1998)

30. V.G. Kohn, G.V. Smirnov, Phys. Rev. B 57, 5788 (1998)

31. F. Mezei. In: NeutronSpinEcho, ed by F. Mezei (Springer, Berlin Heidelberg

New York 1980)

32. M. Kaisermayr, C. Pappas, B. Sepiol, G. Vogl: Phys. Rev. Lett. 87, 175901

(2001)

33. D.K. Ross, D.L.T. Wilson: Neutron Inelastic Scattering, vol 2,(IAEA, Vienna

1978) p 383

34. T. Springer: Quasielastic Neutron Scattering for the Investigation of Diﬀu-

sive Motions in Solids and Liquids, Springer Tract in Modern Physics, vol 64,

(Springer, Berlin Heidelberg New York 1972)

35. M. Kaisermayr, B. Sepiol, G. Vogl: Physica B 301, 115 (2001)

36. B. Sch¨onfeld: Progress in Mat. Science 44, 435 (1999)

37. M. Kaisermayr, B. Sepiol, H. Thiess, G. Vogl, E.E. Alp, W. Sturhahn: Eur.

Phys. J. B 20, 335 (2001)

38. A.Q.R.Baron,H.Franz,A.Meyer,R.R¨uﬀer,A.I.Chumakov,E.Burkel,W.

Petry: Phys. Rev. Lett. 79, 2823 (1997)

39. D.C. Champeney: Rep. Prog. Phys. 42, 1017 (1979)

40. S. Brauer, B. Stephenson, M. Sutton, R. Br¨uning, E. Dufresne, S.G.J. Mochrie,

G. Gr¨ubel, J. Als-Nielsen, D.L. Abernathy: Phys. Rev. Lett. 74, 2010 (1995)

41. G. Vogl, W. Petry, Th. Flottmann, A. Heiming: Phys. Rev. B 39, 5025 (1989)

42. W. Petry, A. Heiming, J. Trampenau, G. Vogl: Defect and Diﬀusion Forum

66–69, 157 (1989)

43. S. Mantl, W. Petry, K. Schroeder, G. Vogl: Phys. Rev. B 27, 5313 (1983)

44. K. H. Steinmetz, G. Vogl, W. Petry, K. Schroeder: Phys. Rev. B 34

, 107 (1986)

45. A.D. Le Claire: J. Nucl.Mater. 69&70, 70 (1978)

46. M.A. Krivoglaz, S.P. Repetskiy: Fiz. Met. Metalloved. 32, 899 (1971), (Phys.

Met. Metallogr. (USSR) 32, 1 (1971))

47. S.K. Chen, M.B. Dutt, A.K. Barua: Phys. Stat. Sol. A 45, 657 (1978); contain-

ing further references on tracer studies of iron diﬀusion in copper

48. G. Vogl, B. Sepiol, C. Czichak, R. R¨uﬀer, R. Weinkamer, P. Fratzl, M. F¨ahnle,

M. Meyer: Mat. Res. Soc. Symp. Proc. 527, 197 (1998)

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 91

49. C.L.Fu,Y.Y.Ye,M.H.Yo,K.M.Ho:Phys.Rev.B48, 6712 (1993)

50. R. Drautz, M. F¨ahnle: Acta Mater. 47, 2437 (1999)

51. S. Divinsky, Chr. Herzig: Intermetallics 8, 1357 (2000)

52. I.V. Belova, M-E. Ivory, G.E. Murch: Phil. Mag. A 72, 871 (1995); Phil. Mag.

A 80, 1481 (2000)

53. M. F¨ahnle, J. Mayer, B. Meyer: Intermetallics 7, 315 (1999)

54. G. Bester, B. Meyer, M. F¨ahnle, C.L. Fu: Mater. Science and Engineering A

323, 478 (2002)

55. M.Kaisermayr,J.Combet,H.Ipser,H.Schicketanz,B.Sepiol,G.Vogl:Phys.

Rev. B 61, 12038 (2000)

56. M.Kaisermayr,J.Combet,H.Ipser,H.Schicketanz,B.Sepiol,G.Vogl:Phys.

Rev. B 63, 054303 (2001)

57. A.T. Donaldson, R.D. Rowlings: Acta Met. 24, 285 (1976)

58. A.G. Balogh, I. Dezsi, J. Pelloth, R.A. Brand, W. Keune, W. Puﬀ: Mat. Sci.

Forum 105–110, 897 (1992)

59. M. Kaisermayr, J. Combet, B. Sepiol, G. Vogl: Defect and Diﬀusion Forum

149–199, 461 (2001)

60. N.A. Stolwijk, M. van Gend, H. Bakker: Phil. Mag. A 42, 783 (1980)

61. A.H. van Ommen, J. de Miranda: Phil. Mag. A 43, 387 (1981)

62. B. Sepiol, G. Vogl: Phys. Rev. Lett. 71, 731 (1993); B. Sepiol, G. Vogl: Hyper-

ﬁne Interact. 59, 149 (1995)

63. A. Gude, H. Mehrer: Phil. Mag. A 76, 1 (1997)

64. O.G. Randl, G. Vogl, W. Petry, B. Hennion, B. Sepiol, K. Nembach: J. Phys.:

Condensed Matter 7, 5983 (1995)

65. E.A. K¨ummerle, K. Badura, B. Sepiol, H. Mehrer, H.-E. Schaefer: Phys. Rev.

B 52, R6947 (1995)

66. I.V.Belova,G.E.Murch:J.Phys.Chem.Solids59, 1 (1998)

67. O.G. Randl, G. Vogl, W. B¨uhrer, M. Kaisermayr, P. Pannetier, W. Petry: J.

Phys.: Condensed Matter 8, 7689 (1996)

68. G. Vogl, M. Kaisermayr, O.G. Randl: J. Phys.: Condensed Matter 8, 4727

(1996)

69. G. Bester, B. Meyer, M. F¨ahnle: Phys. Rev. B 59, 3274 (1999)

70. G.Vogl,O.G.Randl,W.Petry,J.H¨unecke: J. Phys.: Condensed Matter 5,

7215 (1993)

71. M. Sladecek, M. Miglierini, B. Sepiol, H. Ipser, H. Schicketanz, G. Vogl: Defect

and Diﬀusion Forum 149–199, 369 (2001)

72. G. Vogl: Hyperﬁne Interactions 53, 197 (1990)

73. B. Sepiol: Defect and Diﬀusion Forum 125–126, 1 (1995)

74. G. Vogl. In:

Neutron Scattering from Hydrogen in Materials, ed by A. Furrer

(World Scientiﬁc, Singapore 1994)

75. G. Vogl, W. Mansel, P.H. Dederich:, Phys. Rev. Lett. 36, 1497 (1976)

3 Diﬀusion Studies of Solids by Quasielastic

Neutron Scattering

Tasso Springer and Ruep E. Lechner

3.1 Introduction

The diﬀusion in solids is usually investigated on a macroscopic scale, for

instance following the interpenetration of two atomic species across an inter-

face which is related to the chemical diﬀusion coeﬃcient. On the other hand,

the self-diﬀusion coeﬃcient is measured by labelling atoms with radioactive

isotopes. These techniques have been presented in Chap. 1. Labelling can

also be achieved by Larmor precession in a magnetic ﬁeld gradient where the

corresponding dephasing depends on the diﬀusive migration of the precess-

ing particles (Chaps. 10 and 17). In contrast to such “macroscopic” methods

the diﬀusion process can be studied for atomistic time scales by the spin-

lattice relaxation in nuclear magnetic resonance spectroscopy (Chap. 9). The

method of quasielastic neutron scattering (QENS)

, which focuses on scat-

tering processes at small energy transfers, penetrates into the region of 10

−13

to 10

−8

s. In addition, quasielastic neutron scattering explores the diﬀusive

motion in space, for a range comparable with the wavelengths of the neu-

trons used in spectroscopy. Typical correlation lengths, diﬀusive paths, jump

distances, and vibrational displacements can be evaluated between 10

−6

and

−9

cm. Inelastic neutron scattering on such diﬀusive processes leads to a

spectrum of energy transfers

ω = E

− E

, (3.1)

in a range from 10

−3

to 10

−7

eV, where E

and E

are the energies before and

after scattering, respectively. The corresponding scattering vector in such a

process is

Q = k

− k

, (3.2)

where k

and k

are the neutron wave vectors before and after scattering,

respectively. The values of Q =(4π/λ)sin(ϕ/2) are in the region of typi-

cally 0.1to5

−1

(λ = neutron wavelength, ϕ = scattering angle) such that

1/Q corresponds to interatomic distances. The scattered intensity in such a

In the following, we use “QENS” for quasielastic neutron scattering in general,

and “QINS”, if the quasielastic scattering is purely incoherent, in order to dis-

tinguish this case from the coherent one.

94 Tasso Springer and Ruep E. Lechner

process is proportional to the so-called scattering function or dynamic struc-

ture factor S(Q,ω), which can be calculated for typical diﬀusive processes;

the calculation and determination of this function is the subject of the fol-

lowing sections. To anticipate brieﬂy what will be explained in detail, the

widths of the Lorentzian shaped “quasielastic line” is given by Γ = Q

for small values of Q,whereD

is the self-diﬀusion coeﬃcient. For larger

scattering vectors, Γ depends on the details of the jump process. Obviously,

for a particle at rest, Γ =0andS(Q,ω) is a sharp line δ(ω)atω =0.

For resonance absorption of gamma rays this corresponds to the well-known

M¨oßbauer line. It results from absorption processes where the gamma quan-

tum transmits momentum k to the sample as a whole, without any mea-

surable energy transfer. Furthermore, for M¨oßbauer absorption spectroscopy,

a diﬀusing particle produces a corresponding broadening of this line and it

is described by the same concepts as used for incoherent neutron scattering

(see Chap. 2 for nuclear resonance gamma ray absorption and scattering).

The interpretation of the dynamic structure factor S(Q,ω)intermsof

the diﬀusive process is uncomplicated because

– the diﬀusion can be described by classical physics; in general, quantum

eﬀects can be completely neglected, and

– the scattering is treated in ﬁrst Born approximation. This allows an eval-

uation and interpretation of S(Q,ω) by pair correlation functions for the

scattering nuclei in space and time.

Within this concept, the following sections deal with diﬀerent kinds of

diﬀusion. First of all, hydrogen diﬀusion in metals will be explained. Be-

cause of the dominant incoherent scattering of the proton, the interpretation

of the results is straightforward. Then we deal with other cases such as va-

cancy induced diﬀusion in metals, diﬀusion in the presence of impurities, ionic

diﬀusion, and, brieﬂy, collective eﬀects as investigated in substances where

coherent scattering dominates; this is more complicated, and coherent qua-

sielastic scattering has been exploited to a lesser extent than its incoherent

counterpart. It is clear that we cannot cover the whole ﬁeld. For instance, ex-

periments concerning diﬀusion in polymers (see Chaps. 13 and 16) are omit-

ted. The methodical aspect, namely high resolution neutron spectroscopy,

will be explained to the extent this is needed for the presented experimental

studies.

3.2 The Dynamic Structure Factor S(Q, ω)

Atomistic information is obtained by analyzing the intensity of neutrons from

a monochromatic beam with a current density I

scattered into a solid angle

element ∆Ω and an energy interval ∆ω,onasamplewithavolumeV and

an atomic number density n

(see Fig. 3.1). This intensity is given by

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 95

Fig. 3.1. Neutron scattering experi-

ment. The intensity scattered into a

solid angle element dΩ andanenergy

element dω is proportional to the dou-

ble diﬀerential scattering cross-section

σ/dΩdω in (3.4).

∆I

= I



dΩ dω



∆Ω ∆ω, (3.3)

where the double diﬀerential scattering cross-section is

dΩ dω

4π

S(Q,ω) . (3.4)

This cross-section is factorized in three components: i) The ratio of the wave

numbers k

and k

characterizing the scattering process, ii) the cross-section

σ for a rigidly bound scattering nucleus (where σ =4πb

and b is the cor-

responding scattering length of the nucleus), and iii) the dynamic structure

factor S(Q,ω). The latter depends on the scattering vector Q and the energy

transfer ω as deﬁned by (3.2) and (3.1), respectively. The structural and dy-

namical properties of the scattering sample are described by S(Q,ω)which

does not depend on the neutron-nuclear interaction. In the following, we will

explain this function. For a detailed derivation and discussion we refer to the

corresponding literature [1–3]. To start with, we formulate the well-known

static structure factor S(Q)forthesimplestcaseofN identical atoms at po-

sitions r

, r

...r

. The scattering (“diﬀraction”) intensity is proportional

to the square of the amplitude sum, taking into account the mutual phase

diﬀerences of the outgoing waves for pairs of particles. The structure factor

per atom is then

I(Q)=N

−1



−iQ(r

−r

)

= N

−1





i=1

iQr



(3.5)

Instead, for moving nuclei, corresponding to lattice vibrations and diﬀusion,

one gets the so-called intermediate dynamic structure factor (or intermediate

scattering function) with time dependent space coordinates r(t)

I(Q,t)=N

−1





−iQr

(0)

iQr

(t)



. (3.6)

Actually, r

are operators, ···is a thermal average of the expectation value

for the product enclosed in the brackets. Below we explain that, under the

96 Tasso Springer and Ruep E. Lechner

given conditions, the quantities r are vectors in space and not operators,

and ···is just the thermal average. For the following we need mainly the

self-terms in (3.6) which leads to

(Q,t)=



−iQr(0)

iQr(t)



. (3.7)

The index i was left out and the sum



is replaced by N. From these rela-

tions the Van Hove theory [4] derives an elegant formulation of the dynamic

structure factor S(Q,ω), namely

S(Q,ω)=(2π)

−1



I(Q,t)e

−iωt

dt, (3.8)

and for its self-part, namely the so-called “incoherent” contribution (see be-

low),

inc

(Q,ω)=(2π)

−1



(Q,t)e

−iωt

dt. (3.9)

With these relations, the so-called Van Hove correlation functions can be

deﬁned, namely

G(r,t)=(2π)

−3



−i(Qr−ωt)

S(Q,ω)dQ dω, (3.10)

and for the self-part

(r,t)=(2π)

−3



−i(Qr−ωt)

inc

(Q,ω)dQ dω, (3.11)

Inversion leads to the dynamic structure factors as a Fourier transform of

these correlation functions, namely

S(Q,ω)=(2π)

−1



G(r,t)e

i(Qr−ωt)

dr dt, (3.12)

inc

(Q,ω)=(2π)

−1



(r,t)e

i(Qr−ωt)

dr dt. (3.13)

In the classical approximation, (3.6) and (3.7) can be interpreted in the fol-

lowing sense. The cross terms with i = j in I(Q,t) correspond to a superposi-

tion of scattered neutron waves originating from pairs of diﬀerent particles at

positions r

and r

at diﬀerent times 0 and t. I

(Q,t), however, is obviously

caused by scattered waves originating from the same nucleus at diﬀerent po-

sitions for diﬀerent times. Correspondingly, one can show that the Van Hove

correlation functions have the following classical meaning (Fig. 3.2): G(r,t)

is the probability per unit volume to ﬁnd a particle (nucleus) at a position r

for a given time t if this or another particle has been at a position r =0for

aprevioustimet =0.Inthesameway,G

(r,t) is the probability of ﬁnding

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 97

Fig. 3.2. The Van Hove correlation function:

G(r

− r

,t) is the conditional probability to ﬁnd

an atom at r

for a time t, if this atom or another

one was at r

, with a distance vector r = r

− r

for t = 0. For the self-correlation function G

(r,t),

and r

locate the same particle.

aparticleatr for a time t if the same particle was at the origin r =0for

t =0.

In particular this implies

(r,t=0)=δ(r)andG(r,t=0)=δ(r)+g(r) , (3.14)

where g(r) is the usual instantaneous pair correlation function, and one gets

S(Q)=

+∞



−∞

S(Q,ω)dω =1+



g(r)e

iQr

dr . (3.15)

Furthermore, we note that G

(0,t) is the probability that a certain nucleus

which was at r =0fort = 0, is still (or again) at r =0foratimet.

For a particle diﬀusing in a space, which is large as compared to the

atomic scale, the self-correlation function G

(r,t) vanishes, if t goes to inﬁn-

ity, whereas for a particle bound to a ﬁnite volume (e. g. as part of a rotating

molecule ﬁxed in a crystal), G

(r,t) approaches a ﬁnite value G

(r, ∞)for

r varying in the interior of this volume. In fact, following Van Hove [4],

Turchin [5] has pointed out that very generally, the self-correlation function

can be split into its asymptotic value in the long-time limit and a time-

dependent term G



(r,t), according to G

(r,t)=G

(r, ∞)+G



(r,t). The

Fourier transform of this expression reads:

inc

(Q,ω)=(2π)

−1



i(Qr−ωt)

(r, ∞)+G



(r,t)] dr dt. (3.16)

which gives

inc

(Q,ω)=S

inc

(Q)δ(ω)+S

inc

(Q,ω) (3.17)

It is seen that the incoherent scattering function is decomposed into a purely

elastic line, S

inc

(Q)δ(ω), with the integrated intensity S

inc

(Q), and a nonelas-

tic component, S

inc

(Q,ω). The elastic line is the result of diﬀraction of the

neutron on the “inﬁnite time” distribution in space of a single nucleus spread

over a ﬁnite volume by its motion, as already pointed out by Stiller [6]. There-

fore we can derive information about the structure in a very direct way from

the incoherent scattering [7]. This is clearly a rather important result of the

98 Tasso Springer and Ruep E. Lechner

Van Hove theory. We now turn to its application which will be further dis-

cussed later, in the context of practical examples in Sects. 3.4, 3.5, 3.9, 3.10

and 3.12.

In order to exploit the theoretical fact expressed by (3.17) in neutron scat-

tering experiments on orientationally disordered (plastic and liquid) crystals,

the concept of the elastic incoherent structure factor (EISF) was formulated

by Lechner [8]. First applications of this concept were published, as soon as

the progress of instrument development at the Institute Laue-Langevin (ILL)

in Grenoble permitted to carry out corresponding experiments [9–13]. First of

all, an experimental separation of the elastic component according to (3.17)

was required. Novel high-resolution quasielastic spectrometers which could

achieve this were built at the start-up of the high-ﬂux reactor of the ILL.

Secondly, for the analysis of the data a method was needed that would allow

the isolation of the scattering eﬀect of speciﬁc motions, such as the stochas-

tic molecular reorientations known to occur in these dynamically disordered

crystals. The EISF concept provides such a method. It permits the extrac-

tion of structural information on such localized single-particle motions by

the determination of the elastic fraction of the measured spectral intensity.

The idea is simple: First, by employing suﬃciently high energy-resolution,

the measured integrals of elastic (I

) and nonelastic (I

) components of the

scattering function in (3.17) - after trivial corrections for the factor

(see

(3.4)), the sample self-attenuation and the energy-dependent detector eﬃ-

ciency - are determined separately. Then an intensity ratio involving the two

integrals can be deﬁned,

EISF = I

/(I

+ I

)=AS

inc

(Q)/[A



inc

(Q,ω)dω] (3.18)

where A is a normalization factor proportional to experiment parameters

such as the incident neutron ﬂux, the sample size, the detector eﬃciency, the

duration of the measurement, etc. Obviously, the diﬃculty of an absolute

intensity calibration is avoided in the determination of the EISF by (3.18):

The normalization factor cancels, the integral of the incoherent scattering

function is equal to 1, and we simply have EISF = S

inc

(Q). Here, we have

used (3.17) and (3.18) as a starting point for obtaining a deﬁnition of the

EISF. The coeﬃcient S

inc

(Q) in these two equations is the EISF in its most

general form, since it includes all the motions of the scattering atom.

However, the determination of this “global” EISF is generally not the

immediate aim of an experiment, for the following reasons. First of all, one

is often more interested in speciﬁc types of motions than in all of them. Sec-

ondly, an unambiguous measurement of a global EISF is not easily achieved

in just one single experiment. Every measurement has a well-deﬁned energy

resolution connected with an eﬀective energy transfer window (see the cor-

responding discussion later in this section). Essentially only the dynamics of

the speciﬁc motions occurring in this energy range are visible, because much

slower motions are hidden within the energy resolution function, whereas

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 99

much faster motions appear only as a ﬂat background. This can be used to

experimentally isolate the eﬀect of a speciﬁc motion. It is therefore much

more interesting to apply the EISF concept to the elastic component of each

speciﬁc type of motion, e. g. to molecular rotation, rather than to the com-

bined eﬀect of all atomic motions. The expression to be used for the analysis

of such speciﬁc motions is formally the same as (3.18). However, we have to

replace the nonelastic integral (I

) by the corresponding quasielastic integral

), and keep in mind that the incoherent scattering function, S

inc

(Q,ω),

must now be replaced by a partial scattering function consisting merely of

an elastic term (measured integral: I

) and a quasielastic term (measured

integral: I

) corresponding to the speciﬁc motion under study. The eﬀect

of faster motions has been subtracted as a ﬂat “inelastic” background and

only appears as an attenuating Debye-Waller factor bound to be cancelled,

because it is included in the normalization factor A. The feasibility conditions

are that i) the energy resolution is adapted to the time scale of the motion

of interest, ii) this motion is suﬃciently well separated on the energy-scale

from other motions of the same atom, and iii) the assumption of dynamical

independence of the diﬀerent modes from each other, e. g. rotations and vi-

brations, represents an acceptable approximation (see (3.64), Sect. 3.9). It is

this possibility of isolating the EISF of speciﬁc modes of motions that has

proved to be of most practical importance for the development of the tech-

nique. The success of a series of prototype experiments is documented in the

original literature [9–13].

The EISF method was developed as a strategy for ﬁnding the appropriate

diﬀerential equations and their boundary conditions for the dynamical mech-

anisms of localized atomic motions in condensed systems. These early and

subsequent experiments have been reviewed extensively [14–16]. In [16] the

importance of the dynamic independence approximation for the deﬁnition

of the EISF of speciﬁc motions, the relation between Debye-Waller factor,

Lamb-M¨oßbauer factor and EISF, as well as the observation-time depen-

dence of the latter are discussed in detail. More general discussions of Van

Hove’s theory and its application can be found in text books; see, e. g. [3,17].

It is interesting to note, that the EISF concept also plays a role in pulsed-

ﬁeld gradient (PFG) NMR studies of diﬀusion in conﬁned volumes, where

at suﬃciently large observation time the spin-echo attenuation approaches

a time-independent value conveying geometrical information on the spatial

restriction (see (10.18) of Chap. 10).

In our explanation we have interpreted the nuclear motion in terms of

classical time-dependent coordinates. What does this mean? The average de

Broglie wavelength of the scattering nucleus with a mass M is given by

≈ h(2Mk

T )

−1/2

, (3.19)

if this particle moves with the average speed v

within a sample at temper-

ature T . A classical description of the nuclear motion is certainly valid for

100 Tasso Springer and Ruep E. Lechner

times during which the particle has passed a distance which is larger than its

de Broglie wavelength: tv

>λ

. For instance, at 300

◦

C (3.19) yields

t ≥

≈ 10

−13

s . (3.20)

This holds in all cases to be discussed in the following, i.e. for typical dif-

fusive jump rates < 10

−1

or diﬀusion coeﬃcients D

< 10

−8

/s. On

the other hand, the criterium fails for the atomic vibrations in solids with

frequencies above 10

−1

. In this case, the quantum mechanical meaning of

the Van Hove correlation functions or the usual matrix element description

is preferable.

Let us now turn to the problem of experimental resolution. (3.12) and

(3.13) represent a Fourier analysis of the neutron scattering functions S(Q,ω)

and S

inc

(Q,ω) , respectively, with the Van Hove correlation functions in space

and time as coeﬃcients. We draw the attention to the fact that the scattering

functions, i. e. the dynamic structure factors S and S

inc

deﬁned in (3.4)

to (3.13), as well as the corresponding correlation functions G, G

,andthe

intermediate scattering functions I, I

, cannot be determined experimentally

in their pure forms: The measured scattering functions are broadened due

to convolution with the experimental resolution functions R(Q,ω)inthe

four-dimensional (Q,ω)-space. In the case of incoherent scattering, the Q-

spread of the resolution can often be neglected when the studied functions

are only slowly varying with Q. Then it is suﬃcient to merely deconvolute the

measured spectra from the energy resolution function R(ω). The latter may

have a shape close to a Gaussian or a Lorentzian, with an energy width ∆(ω)

deﬁned as the half-width at half maximum (HWHM). Note that this width

is connected, by the uncertainty relation, with the experimental observation

time ∆t which is the decay time of the observation function R

∗

(t)inthe

Fourier time domain [15–18]:

∆(ω)∆t

∼

 (3.21)

While the resolution function R(ω) has the eﬀect of broadening the neutron

scattering function along the energy transfer coordinate of the experiment,

the observation function R

∗

(t) is a factor, which increasingly attenuates the

corresponding correlation function with increasing Fourier time: R

∗

(t)isthe

Fourier transform of R(ω) and is therefore in most practical cases a func-

tion essentially decaying with increasing time. The net eﬀect is, that the

correlation functions are observed in a Fourier time window, with an upper

limit determined by the decay time constant of the observation function. The

low-time limit of this window has a diﬀerent origin: it is a consequence of the

(always limited) statistical accuracy of the measurement, because quasielastic

intensities typically decrease with increasing energy transfer, and therefore

counting statistics become the poorer the larger the energy transfers are as

compared to the energy-resolution width.