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Neutron quasielastic scattering from molecular liquids and glasses
PHYSICA
Physica B 182 (1992) 289-301 North-Holland
Neutron quasielastic scattering from molecular liquids and glasses In memory of Professor F. Bermejo-Martinez •
a
•
c
1
F . J . B e r m e j o , A . C h a h i d a, M. G a r c i a - H e r n ~ n d e z a, J . L . M a r t l n e z ' , F . J . M o m p e a n b, W.S. H o w e l l s b a n d E . E n c i s o d alnstituto de Estructura de la Materia, Consejo Superior de lnvestigaciones Cientificas, Madrid, Spain bRutherford Appleton Laboratory, Chilton, Didcot, Oxon 0)(1I OQX, UK Clnstitut Laue Langevin, 156X, F-38042 Grenoble-Cedex, France dDepartamento de Quimica-Fisica 1, Universidad Complutense, E-28040 Madrid, Spain
An overview of the present status of the theory of neutron scattering from disordered molecular systems (liquids and glasses), with the main emphasis on the theoretical modelling of spectral lineshapes, is presented. Applications to several specific eases are described in order to illustrate the capabilities of Quasielastic Neutron Scattering (QENS) for obtaining reliable information about the stochastic dynamics of these systems.
1. Introduction
Although Quasielastic Neutron Scattering ( Q E N S ) , or rather low-energy inelastic scattering, has been employed for nearly three decades for the investigation of the low-frequency dynamics of disordered molecular systems, its capabilities for obtaining reliable information about the characteristic time and length-scales regarding molecular motions have only been fully exploited in recent times• If relatively low counting rates and poor resolution in energy-transfers made up most of the studies until the late seventies, of scarce use for nowadays purposes, the lack of adequate tools for the analysis of lineshapes still represents a hindrance to further progress in this field. If, for monoatomic systems, the field has reached a mature state and fully quantitative Correspondence to: F.J. Bermejo, Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientificas, Serrano 123, E-28006 Madrid, Spain. 1Permanent address: Instituto de Ciencia de Materiales, C.S.I.C., Facultad de Ciencias C-IV, Universidad Aut6noma de Madrid, E-28049 Madrid, Spain.
comparisons between experiment and theory a n d / o r computer simulation results are available [1], the complications introduced by the need to separate rotational from centre-of-mass motions ( C O M ) (and the coupling between them) has precluded the achievement of a comparable degree of refinement in the case of molecular materials. Apart from the difficulties accounting for the spatial dependence of the C O M motions which make the linearized hydrodynamic approaches break down on the scale of wavevectors accessible to Q E N S , the effects of the collisional dynamics on the molecular reorientations have also to be considered in detail in order to extract the rich dynamical information contained in the Q E N S spectra. The use of computer simulations (mainly of static or Monte Carlo types) to help in the analysis of the g(r) pair correlation functions, has been a c o m m o n approach followed by experimentalists since the early days of neutron diffraction. However, the stringent computational requirements needed to carry out molecular simulations regarding time-dependent properties hindered the combined use of both techniques
0921-4526/92/$05.00 ~) 1992- Elsevier Science Publishers B.V. All rights reserved
F.J. Bermejo et al. / QENS.I?om molecular liquMs and glasse,s
290
when the microscopic dynamical properties of this class of fluids were the topic of interest. On the other hand, the advent of powerful desktop computer stations during the last decade has enabled the implementation of simulation codes capable of reproducing the most salient features regarding the thermal and spin dynamics of this kind of systems. As a consequence, several simulational approaches are starting to be considered as indispensable tools for data analysis purposes alongside with the development and testing of theoretical models. The present contribution intends to illustrate the opportunities for further development in this field arising as a consequence of the intricate interrelationship between theory, experiments and computer simulations. An excursion into the present state-of-the-art of theoretical results adequate for the analysis of QENS lineshapes is given in the next section. A survey of experimental results ranging from molecular "simple" liquids to glasses and magnetic molecular systems is then presented and finally some remarks and conclusions are drawn on section 4.
where the single-molecule correlation functions I,,, and I~ and the collective I(Q, t) are given by
I,,,(Q, t) = {exp(iQa,(t)) exp( iQa (t))> , In(Q, t)
1
~ ~, (exp(iQR,(t))exp(-iOR,(0))) '
(2)
, (3)
1
I(Q, t)= ~ ~f~ {exp(iORi(t))exp(-iORi(O)) ) , tl (4) cohjo( Qa,,)] z,
F(Q)=I~b
(5)
where the position vectors of the nuclei belonging to a rigid molecule are given by
r i,,
R + a~t .
(6)
R is the position of the molecular center-of-mass ( C O M ) and a~ is the relative position of each nucleus with respect to the COM. The 1 (Q,t) single-molecule function can be written in terms of a partial-wave expansion which, after a separation of terms corresponding to self- and different particle motions, gives
2. Spectral iineshapes Up to the present moment, no closed form expression exists for the calculation of the S(Q, w) quasielastic response function which does not contain strong simplifying assumptions. Only in the limit of decoupled rotational and center-of-mass motions has it been possible to derive formulae for the calculation of S(Q, w) once the relevant mechanisms for mass-diffusion and rotational relaxation are specified [2,3]. In such a case the molecular dynamics is completely specified by
S(Q, ¢o) = J dt e x p ( - i w t )
+ ~
bu.,.,,,,b .......hlu~(Q,
t)}l~(O, t)
luu(Q,t )
~ (2•+ /
l,,(Q,t)
1)jf(Qa )b~(t),
~(2/+
1 )lT(Qa~)P,(cosO "
J
(1)
)Ft(t ) (s)
where the Fi(t ) are rotational relaxation functions for which information can be made available from computer Molecular Dynamics simulation, optical or N M R spectroscopies etc. Two limiting cases are readily apparent regarding Fl(t ). These correspond to free (inertial) rotation and to infinitesimal rotation steps (i.e. when the rotational motion obeys a diffusion equation). For the case of inertial motion, the expressions for the first two orders in the case of spherical symmetry are 2
I 2r
Fl(r)= 1 2 " e r e x p ( - ½ r e) J + F(Q)(I(Q, t ) - I,(Q, t))]
(7)
0
0
dxexp(xe), (9)
F.J. Bermejo et al. / Q E N S f r o m molecular liquids and glasses 21/2r
F2(7 ) = 1 - ( 3 / 2 ' / 2 ) 7 e x p ( - 2 r 2) f
d x e x p ( x 2).
0
(10)
These are characterized by a rescaled dimensionless time r = (kBT/I)J/2t which is given in terms of the inverse of the free-rotor correlation time. Approximate expressions for its evaluation are given by Sears [2]. For the case of diffusive motion, the relaxation functions are of the simpler form
Fl(t ) = e x p ( - / ( l + 1 ) D r t ) ,
I
exp[/(l +
I)D,
ty (A0~) 1/2= (2D i At) I/2, where D i are the elements of a rotational diffusion tensor represented on a basis which makes it diagonal. The time required for an angular displacement of one radian is usually denoted by r,, so that the collision frequency becomes Z / r o where Z stands for the average number of steps required for a rotation of one radian. For diffusive behaviour it is required that r s ~ r 0. A simple test which gives an indication of the motional regime present is given by the quantity
5 [ k B T ] '/2 X i - lSD, L ~ - , . J
(15)
(11)
where D r is the rotational diffusion constant. The departure from spherical motion requires the introduction of two dynamical constants D 1 and D~ so that
F,U)=
291
+ mZD2]t,
(12)
m= 1
D, = ½(Dxx + Dyy),
(13)
D 2 = D~. - 4(Dxx + Dyy),
(14)
and, as an indication [4], the inertial region is confined to values X < 3, the diffusion regime is approached for X > 5 and an intermediate region is left in between. For cases belonging to the intermediate behaviour, some formulae which interpolate between the two are adequate [5] since both effects are explicitly considered and have been applied to the analysis of some liquids [6]. In particular, the following expression has been found by the authors to give reasonable results when analyzing Q E N S data for liquid CC14:
F,(t)=(r where D , , ii = xx, yy, zz, are the non-zero elements of the rotational diffusion tensor (i.e. expressed on a basis which makes it diagonal). More complicated expressions for the case of asymmetric tops can easily be found although their applicability to Q E N S studies is rather limited due to the number of free parameters involved. In all cases, the constants characterizing the reorientational motion (relaxation times) are given by the integrals r t = fo dt Fl(t ) and these are the relevant quantities to be compared with others obtained from optical, N M R or dielectric relaxation spectroscopies. By their very nature, Brownian dynamics theories assume that the duration of a collision is short in comparison with the period between randomizing events rj. Within such an hydrodynamic view, the angular excursions experienced by a molecule are given by the quanti-
-r+)
l[r_exp(7)
r+exp(~[)l, (16)
r+ = ½(r o' --+2/~), =
(17)
2 -4,o2)"-,
(18)
wj2 = l(l + 1 ) k u T / I ,
(19)
rR = (wZro) 1 ,
(20)
where the two relevant parameters encompassing the dynamical information regarding reorientational motions are r 0 and ¢o~. Since the latter is directly calculable from known constants, only r o is left as adjustable from the experimental intensities. The reorientational motions of real liquids well inside the kinematic region accessible to cold neutron spectroscopy have been shown to
F.J. Bermejo et al. / QENS from molecular liquids and glasses
292
be far more complex than those predicted by hydrodynamic theories. In fact, most of the relaxation functions computed from molecular dynamics simulations seem to follow (on empirical grounds) the formula Fl(t ) = all e x p ( - a 2 , t 2) + a31 e x p ( - a 4 g ) +(1 - a l l - a3t) exp(-a~/t)
(21)
where the Gaussian term dominates at short times (i.e. the regime mostly dominated by binary collisions) and the exponential behaviour characterizes, as expected, the long-time hydrodynamic regime. An example of such behaviour, together with some values for the coefficients, are shown in fig. 1 and table 1 respectively. 1.0 ~t~,~
T = 2 9 3 K (- -) = .,,
0.75
0.5
0,25
0.0
1
2
3
t
As a final remark, it has to be emphasized that, although some deviations are readily apparent between the calculated relaxation functions and those estimated from QENS data, the present day counting statistics as well as the large n u m b e r of free parameters involved in this latter formula preclude its use in fitting lineshapes in terms of physical models. 2. 1. The shape o f I~(Q, t) The use of a simplified hydrodynamic treatment, where the temporal decay of the self (i.e. center-of-mass) correlation function is described in terms of a simple exponential behaviour, is still common practice in the analysis of QENS lineshapes. However, some substantial drawbacks are expected to occur when analyzing data over a large range of momentum transfers. In general terms, an adequate representation of the single-particle scattering law should retain a Gaussian dependence in Q in both the hydrodynamic and free-particle limits. The question of how to interpolate between both is still a matter of different 'schools-of-thought'. In what follows, three cases will be considered: an approach based upon a generalization of hydrodynamics; a result from kinetic theory; a comparison with formulae capable of taking into account the results from molecular dynamics calculations. Some remarks regarding the usage of 'jumpmodels' to analyze the liquid data will complete this section.
4
(ps)
Fig. 1. Rotational relaxation functions for liquid CCI~ from M D using an LJ potential (dashes and dots) and approximations using eq. (21) (solid lines). The corresponding par a m e t e r s are given in table 1.
2. I. 1. The generalised hydrodynamics approach The so-called 'Gaussian approximation' [1] enables the representation of the incoherent scattering function Is(Q, t) in terms of a series expansion
Table 1 Values for the coefficients corresponding to the curves in rig. 1.
l~(Q,t)
a 1
(/-
(/a
(/4
as
260 K
FI k~
0.050 (1. 192
1.946 2.984
1.312 11.931
0.145 0.360
I). 145 1/.364
293 K
F~ F,
0.089 0.239
1.549 2.905
3.722 2.148
0.287 /).562
1/.325 0.514
exp( ~a(t)Q 2)
x [1 + c~2(t)bQ q
cQ ~ + dQ s + "" "].
(22)
If only the first exponential term is retained, the problem then lies in an adequate choice of a(t). For c~(t)= 2 D J the simple hydrodynamic result is regained whereas making
F.J. Bermejo et al. / Q E N S from molecular liquids and glasses
~(t) = 2D~[(t 2 +
C 2 ) 1/2 - -
C]
for the relaxation time is
with c = m D J k B T gives the approximation due to Egelstaff [7]. Both cases do not satisfy a n u m b e r of frequency moments (sum-rules) and therefore alternatives for overcoming this difficulty have been introduced. In particular, from the short-time expansion of the scattering law, 2
12
2
t4 . . . . .
9
I~(Q, t) = 1 - w(, ~ + w0(3w 0 + ~Q2) 4!
' (23)
where (for a monoatomic liquid) the moments are given by
(~o)=1,
(wz)=~o(2j,
=
293
~"
(26)
1 _ mD~ a 1/20 k B T ~ 2 ""o
and therefore only two parameters D~ and g20 are needed for a complete specification of the response function, since the second frequency moment can be easily evaluated. Such an approximation approaches correctly the lineshapes corresponding to the long-wavelength (Lorentzian shape) and free-particle (Gaussian shape) limits and may be profitably used in the modelling of Q E N S spectra measured for molecular materials, bearing in mind that the results for the higherorder frequency moments should be taken as 'effective' ones since their calculation from first principles still remains a formidable task.
+
2.1.2.
and
~-m
dr ~ dr 0
which is given in terms of the mass-density and the second derivative of the interparticle potential u(r). Hence an approximation which satisfies the m o m e n t relations can be found [8] which is formally analogous to the one developed for the study of spin dynamics in paramagnets. Fourier transformation of eq. (23) above leads [8] to
1 w/3 ~r 1 - e x p ( - h w / 3 )
X [(D'T(O) 2 __ 61 __ 6 2 ) ] 2 -1- (~'0 2 -- 6 1 ) 2
(25)
where the first term is a detailed balance factor and the parameters entering this formula are related to the frequency moments by the expressions 2
v
T
and (61 + 62)61
=
w2 + F~ + D~QQ* - - R e
+ O(Q
'ri'61 62 T
6t = w(l= Q - k B - - , m
S s ( Q , w) ;
Ss(Q, w)= -
The kinetic theory approach
The approaches based upon full kinetic theories have been extensively tested against molecular dynamics results for simple (hard-spheres and LJ fluids) dense fluids and have been used (without further justification) for a wide range of problems (from liquid metals to polymers). Within the De S c h e p p e r - C o h e n version of the m o d e mode coupling theory [9], the incoherent dynamic structure factor is approximated [10] as
2
61+ 67 = oo = 3 6 1 + $ 2 o ,
(034). The usual prescription
G
~/2f(~o/Q2)),
(27)
i.e. in terms of a Fickian contribution with F D s Q 2 and a first-order mode-coupling correction given in terms of the wavenumber Q * = 16"rrl3mnD~, the kinematic viscosity u = ~ / m n , 6 = D~/(D~ + v) and a complex function G ( z ) . Closed-form expressions for the linewidth, the peak height and the product of both have been obtained [10] up to orders of 0(Q3/2): 2 1 G(6 I)Q wQ S~(Q,O)= ~ + DsQ, ,
(28)
F..I. Berrnejo et al. / Q E N S Jrom molecular liquids attd glasses
294
Aoo(Q)/Q ~= D r - D~H(6)Q/Q*
(29)
"rrS,(Q,O) A m ( Q ) = 1 + A(,3)Q/Q*
(3o)
and the functions G, H and A can be evaluated from the parametric expressions given in ref.
[10l. The practical usefulness of such an approach for systems different from rare-gases still has to be demonstrated. It has, however, been applied to the study of the diffusional dynamics of dense hydrogen gas [11] and to liquid deuterium [12].
2.1.3. The approach based on molecular dynamics simulations A limitation which applies to most of the present approaches is the need to account for the first non-Gaussian corrections (i.e. terms in Q4 and above) as the density approaches that characteristic of realistic liquids. As a matter of fact, formulae like the following account for data on liquids composed of particles interacting via simple (LJ plus lower order multipole) potentials:
I~( Q, t) - e x p ( - 1 / 6 Q : ( r 2 ( t ) } ) Q4(r:(t)} ×
1+
7~_
C ( c / t ) e x p ( - t , / t - 1) (31)
where (r:(t)} is the atomic mean-squaredisplacement calculated from the simulated trajectories and C and 6 are constants which characterize the crossover to the free-particle regime. This kind of analysis will undoubtedly help for the separation of mass-diffusion effects from those concerning molecular rotations, thus allowing for a more accurate study of the fine details of the reorientational motions. On the other hand, and whenever possible, information regarding the value for the self-diffusion coefficient obtained in the hydrodynamic limit by means of Pulsed-Field-Gradient NMR will undoubtedly help to separate these two kinds of motion.
sixties [13] wherein the molecular motions were pictured as comprising a damped oscillatory behaviour ('jitter') during a time r() followed by diffusive steps during a time T1 ('jump'). When % "> ~-(), the models become equivalent to hydrodynamic diffusion, while in the opposite limit, assuming that the Debye-Waller term equals unity, one has
l~(Q,t)=exp(-[l+D-~Q2g]t ).
(32)
The usefulness of this kind of models for data analysis purposes comes from the fact that in the specification of the decay rates 1/% and l / r , , some non-Gaussian effects which become vcry relevant at liquid densities are introduced heuristically. However, a detailed analysis of the trajectories calculated from MD using realistic potentials do not show any evidence for the presence of oscillatory behaviour in I~(Q, t). In fact, in all the thoroughly investigated cases which include strongly associated liquids such as liquid water and liquid methanol, the diffusive COM motions arc more continuous than those pictured in this kind of quasi-crystalline motions, and in order to perform a truly discriminating test between the two approaches, the examination of time-scales as short as 0.01 ps becomes necessary [14].
2.2. Beyond the decoupling approximation A number of attempts to approximate the behaviour of the orientation and angular velocity correlation functions have been registered in the last decades. Although some analytical results have been derived for simple molecular shapes [15,16] (linear diatomics, loaded rough spheres and spherocylinders), the problem of how to evaluate the correlation function
( v A t ) v~(O)} (v(t)-v(O)) + ( v ( O ~o(0) × ~(())) + ~'(,,(t). ,(o),o(t) • ,o(o)) ,
(33)
2.1.4. The 'jump model' approach A family of models was developed during the
where v, = v + ½w × v gives the total velocity at
F.J. Bermejo et al. / QENS from molecular liquids and glasses
the ith atomic site in terms of the velocity of the C O M , angular velocity and orientation for an arbitrary molecular shape, remains open and MD simulation is the most common tool to attack the problem. For linear diatomics, however, explicit formulae for the COM velocity ({ v(t) v(O))), angular velocity ( ( w ( t ) o~(0))) and orientation ((~,(t) ~,(0))) autocorrelation functions can be found (ref. [15], p. 362) as well as expressions for the last term in eq. (33). However, the coupling term involves a detailed account of the collisional dynamics and can therefore only be addressed from simulations. On the other hand, some analytical results concerning the long-time tails of these correlation functions have also been obtained [16,17] where the asymptotic behaviour of the angular '~,'2 velocity autocorrelation is shown to follow a t law whereas a t 7/z law is found for its orientational counterpart. Both laws substantially deviate from the hydrodynamic result. Finally, the problem of rotation-translation interaction has also been studied on the basis of simple models [18] where the interaction of a molecule with its surrounding molecules is pictured as resulting from a coupling of the orientational degrees of freedom of a dumbbell with neighbouring oscillators. A simulation based on the solution of a coupled set of Langevin equations shows how the rotational relaxation function of the dumbbell deviates from free or from diffusive behaviour depending upon the choice of the interaction potential. Although this kind of idealized model cannot be sensibly translated to liquid or amorphous materials, the salient features of reorientational motion in real liquids seem to be taken into account.
2.3. Inclusion of coherent effects Although the possibility of separating the coherent and incoherent contributions to the Q E N S response of molecular liquids by means of polarization analysis has opened up a new era within this field, most of the available tools used for data analysis still rely on the 'convolution approximation' or ad hoc modifications of it. The problem seems to be solvable analytically only in
295
the case where the anisotropic contribution to the total structure factor (i.e. all those contributions which cannot be represented in terms of eq. (1)) is very small [19]. A recent application to a molecular liquid [6] used an extension of the formalism developed by Singwi [20] for the analysis of the coherent response in liquid argon, since attempts to model this contribution in terms of the abovementioned approximations lead to rather poor results. The model, which belongs to those of the 'quasicrystalline' family, represents the interference contribution by
Q=kBT Scoh(O, w)~ I~mVT L(R, O)Z(w)
(34)
where v r represents the mean isothermal sound velocity, Z(o~) is the generalized vibrational density of states of the liquid and L(R, Q) is a function given in terms of a 'coherence length' R and the structure factor for molecular centres Sere(Q):
L(R, Q)~
R3
4(Qv)~, =
f q, dq'[Scm(q')
1]
0
x e x p ( - 4R2 -
(O_ q,)2)
x exp( R~2( Q - q')2-1) .
(35)
We have found that this expression represents a fair approximation to the experimental data, although a comparison between the measured spectra and those calculated by means of eq. (1) as given in fig. 2, evidences that the present tools are still incapable of reproducing the QENS spectra for wavevectors close to Qp (i.e. the Q value corresponding to the maximum of S(Q)). Some effort on the development of lattice-gas models, aiming at a more accurate description of the diffusion of light atoms in metal hosts, has been reported in recent times [22]. In particular, the width of the coherent quasielastic component at low Q is given by AWcoh = Dch,.mQ=, i.e. in terms of a 'chemical' diffusion coefficient D~h.... which is related to the self-diffusion (tracer) coefficient by
F.J. Bermejo et al. / QENS.fi'om molecular liquids amt glasse,s
296
veals a d r a m a t i c slowing-down effect since D,~.....
0.4 Q = 1.06 A-~
D {T-
T.)/T,
w h e r e a s D~ maintains a regular behaviour. Alt h o u g h this kind of models is not directly applicable to liquid state problems, it represents a very promising tool for studies regarding critical phenomena.
0.3 3
u~ 0.2
3.
o.1
o.G
0.0
2.5
5.0 ~fi ¢~ (meV)
7.5
10.0
Some
recent
examples
In what follows, a brief review of recent studies of several molecular liquids, which epit o m i z e a whole range of different p h e n o m e n a , is given. T h e interested reader should consult the original references for m o r e details. 3.1.
A m o l e c u l a r s i m p l e liquid
1.5 Q=
1.25
3
1.37A
I1
l.O 0.75
0.5
0,25
0.0
I: "l I]ill:I :l /:1
•
/,-/ ,/'"
\ 0.0
2.5
5.0
7.5
10.0
"h ~ (meV) Fig. 2. A comparison between the experimental S( Q, to) anti that calculated by MD of liquid CCI~ for two wavevectors {Qp 1.34,~ ~ in this system). The solid line represents the measurement and the dashes the approximation given by eq. (1). In the lower frame the two different approximations represent distinct choices of F(Q) (see cq. (1)).
Liquid c a r b o n tetrachloride constitutes one of the most simple molecular liquids as well as being p e r h a p s the most heavily studied one tog e t h e r with water. D u e to its high molecular s y m m e t r y the interaction potential, the accuracy of which has been extensively tested, is mostly d o m i n a t e d by LJ terms, which makes it a m o p ecular a n a l o g u e to liquified classical rare gases [23]. R e c e n t Q E N S [21] and inelastic [24] studies c o m b i n e d with an M D simulation have enabled a fairly detailed analysis of the dynamics of this liquid. In o r d e r to exemplify the use of simulation data as a valuable tool for data analysis purposes s o m e c o m m e n t s regarding the use of the m e m o r y - f u n c t i o n formalism [19] follow. In principle, the advantages of the use of the K ( t ) m e m o r y functions associated with a given a u t o c o r r e l a t i o n ¢tj(t) stem from the simpler structure of K ( t ) (less oscillatory behaviour) in comparison with qJ(t) which will result in an easier modelling. T h e definition of these quantities c o m e s f r o m the set of equations
Dch .... = D J ( 1 - c)
w h e r e c stands for the average o c c u p a n c y of the available sites. N e a r the critical point D~ ..... re-
dO{O d~-
K{t - "r)dJ('r) d'c , {}
{36}
F.J. Bermejo et al. / QENS from molecular liquids and glasses
OK . I OT
f
K,,(t - ~-)K._I(': ) d-r,
which define the memory function K(t) as the kernel of an integral equation which is then given in terms of a hierarchy of K . ( t ) functions. The autocorrelation functions can be easily expressed afterwards in terms of the kernels K(t) if the Laplace transforms d~(s) are introduced. In order to approximate t~(s) the continued fraction
(37)
o
O(s)
-
-
+ •..
-
s + Ko(s )/(s + K, (s))
s + K(s)
(38)
T = 293
1"0I|
= 0.75[i
297
K
(---)
T = 260 K C.)
o.251
O.C 0.0
I
L0
I
I
I
t
I
I
I
0.5
[
I
I
I
I
1.0 t(ps) I
I
I
1.5
I
t
I
I
2.0
t
1
t
i
T = 260 K
i
;
l
I
:
t
t
I
I
I
f
k
T = 293 K
0.75 -
k~ ,.j 0.5> 0.25-
i
0.0
;
i
0.5
i
1.0 t ( ps )
i
J
I
1.5
i
i
i
i
J
2.0
i
I
i
I
0.5
i
i
i
i
i
l.O
i
i
i
E
F
1.5
i
i
i
i
2.0
t ( ps
Fig. 3. T h e first m e m o r y f u n c t i o n s f o r t h e l i n e a r v e l o c i t y c o r r e l a t i o n f u n c t i o n s in liquid CCI 4 f r o m M D . T h e g r a p h s b e l o w s h o w a c o m p a r i s o n o f t h e s i m u l a t e d $ ( t ) a n d t h e i r a p p r o x i m a t i o n u s i n g K,,(t) = K,(O)y(t), w h e r e y(t) is a s u m o f t w o G a u s s i a n s .
298
F..1. Berrmjo et al.
QENSfrom molecular liquids and glasse,s
has to be truncated at some step. The usual practice is to approximate K , ( t ) = K,,(O)y(t), where y(t) is a simple model-function and K(0) can be estimated from the frequency m o m e n t s of d,(w). An evaluation of different model functions which can describe adequately the m e m o r y functions associated with the linear and angular velocity correlation functions is given in ref. [21], and a comparison between the shape of the m e m o r y functions and the relevant autocorrelation functions as derived from MD simulations is shown in fig. 3. Some important discrepancies are seen between the computed function and the chosen model. Further improvements are obviously possible at the expense of the introduction of additional parameters. Once the m e m o r y functions are approximated (usually only the zeroth and first orders arc required) the evaluation of the Laplace transform of the Q E N S response is simply given by an expression analogous to eq. (38), with K 0 and K t substituted by the relevant expressions. Such a procedure thus enables the specification of the intermediate scattering functions in terms of relatively simple expressions whose validity is not limited to the hydrodynamic domain. Finally, as it was already shown in fig. 2, the M D results have provided a means to test the validity of the decoupling approximation given by eq. (1) on a real liquid, thus evidencing the need of far more refined theoretical models if an understanding of the microscopic dynamics at the same level of refinement as the one achieved on present day Q E N S spectroscopy is required.
ortho-D~ a mode-coupling analysis of the Iow-Q incoherent contribution of liquid deuterium was found to reproduce the wavevector-dependences of the linewidths and peak-heights up to a m o m e n t u m - t r a n s f e r of 1 ,~ r. The coherent Q E N S linewidths were rather well approximated by the expressions given above in section 2.3, possibly due to the low densities and viscosities involved in comparison with liquid C C I 4. However, systematic discrepancies were found for wavevectors about the main diffraction peak where some intensity around energy-transfers of about 2 meV could not be accounted for I12]. Although these systems have been investigated from the early days of neutron scattering, a n u m b e r of questions remains open and fully deserve a new re-examination using present-day techniques. In particular, it would bc worth exploring the following: • If some broadening is apparent in the quasielastic line of the solid. Predictions of about 1 IxeV for such a broadening have been available since the mid-sixties and such a measurement will constitute a stringent test for calculations based on simple quadrupolar interactions. • The mixtures of ortho- and para-components with a controlled amount of each species. These systems are considered to be an optimal model for the oricntational glass state [27] and a wealth of N M R and optical data are already available. • The t e m p e r a t u r e - d e p e n d e n c e of the coherent Q E N S spectra near critical points, since these systems are a natural testbed for models of coherent interference effects.
3.2. Quantum molecular liquids Due to the light masses and the simplicity of the intermolecular potential found in the liquid hydrogens, noticeable quantum features are readily apparent, which manifest themselves in the quantized nature of the rotational motion. In a recent set of Q E N S experiments [12,251 both the coherent and incoherent contributions to the Q E N S spectra were studied and in recent inelastic neutron scattering experiments [12,26] the higher-energy collective dynamics were probed. Due to the spherical symmetry of para-H z and
3.3. A complex (and vitrifiable ) liquid: Methanol Strong effects on the microscopic dynamics caused by hydrogen-bond ( H B ) formation have been found by the concurrent use of Q E N S and MD simulations [28,29]. Furthermore, by a comparison with an isomorphous material (methanethiol) where there is no evidence for the existence of any noticeable HB network, a quantitative explanation of the intermolecular effects on the microscopic dynamics has been obtained
F.J. Bermejo et al. / QENS from molecular liquids and glasses
[30], thus enabling a separation of dynamic effects arising from a purely thermal origin from those caused by the presence of strongly anisotropic intermolecular potentials. Since the vitrification of these materials is relatively easy and leads to glasses that are stable for many hours, a comparative study on the dynamics of liquid and glassy states has been carried out [31]. The density of states, Z(~o), provides evidence for some features which are characteristic of the glassy state [32] and most of the quasielastic intensity is transferred to a lowenergy (~-2 meV) inelastic peak which seems to be one of the most significant signatures of the presence of low-energy excitations in the glass not present in its crystalline counterpart.
299
T=88K
/
1.5
f
E
/
0
f""
0.5
%
I
1.5
i / E 0
0.5
•
/
0
-
3.4. Magnetic liquids Although the macroscopic behaviour of magnetic liquids, such as oxygen, resembles that characteristic of a normal paramagnet, polarized neutron diffraction studies have revealed the existence of magnetic short-range order [35]. A Q E N S study of the liquid and plastic crystal phases [35] has evidenced the dynamical nature of the observed magnetic correlations. An MD simulation of the structure-related effects [36] has enabled an approximate separation of magnetic and nuclear effects, thus allowing for a lineshape analysis of the magnetic response. Such a separation can be justified on the basis that the potential used has been shown to reproduce the structural features in both the liquid and plastic-crystal phases where the coupling between magnetic and structural degrees of freedom is weak enough. As a result, the evidence of an exchange-coupled paramagnetic state was inferred from the wavevector-dependence of the linewidths as well as from the spectral moments which have been estimated by means of a model for the Smagn(Q, co) contribution to the total structure factor based upon an exchange coupled Heisenberg model. The most significant results are depicted in fig. 4 and estimates for the coupling mechanism, based on current hydrodynamic approaches [33,34], have
0
i
i
i
r
0.5
1
1.5
2
2.5
Q (A "1)
5
"a 10
K
0.5
L0
1.5
2.0
Q cA~)
Fig. 4. The upper part shows the Q-dependence of the magnetic linewidths of liquid oxygen at two temperatures. The solid line represents the best approximation in terms of antiferromagnetic coupling and the dashed lines the ferromagnetic case. The I~, values calculated from the experiment are shown as full circles. The lower part shows the second frequency moment of the spectral weight function and its approximation by means of a model for nearest-neighbours interactions in a cubic Heisenberg paramagnet. The solid line corresponds to the case of ferromagnetic coupling (J 22 K) and the dashed line to the antiferromagnetic case giving a similar value for the coupling parameter.
300
F.J. Bermejo et al. / Q E N S /?ore molecular liquids and glasses
been inferred from them. A number of significant questions do however remain open: • The precise form of the coupling interaction remains to be determined. In principle the exchange constant should exhibit significant distance and orientation dependences; the modelling of it in computationally bearable terms is still awaiting. • How to model a Heisenberg paramagnet in the liquid state? • The neutron results concerning the comparison of the magnetic dynamics of the plastic crystal and liquid phases are in an apparent contradiction with recent I*SR studies [37]. Additional neutron measurements under wlrying magnetic field strengths may clarify this point. 3.5. Conclusion
Due to the difficulties in analyzing QENS lineshapes, most of the reported work has relied on coarse approximations (linearized hydrodynamics, jump-diffusion models etc.). In particular, the concurrent use of QENS spectroscopy with computer simulations has: • separated the COM motions from the total Q E N S intensity, thus allowing for the determination of reliable information regarding the reorientational dynamics; • analysed the rotational component with far greater detail than that achievable using simplified models whose validity is only ensured in limiting cases such as free rotation or diffusion; • explored the validity of the approximations which neglect the coupling between rotational and translational motions; • quantified the predictive capability of models developed to account for coherent interference effects; • enabled an approximate separation between structural and spin effects in magnetic liquids. Finally, future developments within the realm of fully microscopic theories can also be contrasted with the existing experimental and simulated data, so that a level of understanding comparable to the one nowadays existing for simple monoatomics (liquified rare gases and metals) will become possible.
Acknowledgements This work has been supported in part by D G I C Y T grant No. PB89-0037-C03. The help given by Dr. A.J. Dianoux of the Institut Laue Langevin, Dr. C.J. Carlile and Mr. M. Adams from R.A.L. during the inelastic neutron scattering experiments is warmly acknowledged.
References [1] J.P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill, New York, 1982). [2] V.F. Scars, Canad. J. Phys. 44 (1966) 1279, 1299. [3[ G. Venkataram, B.A. Dasannaracharya and K.R. Rao, Phys. Rev. 61 (1967) 1. [4] K.T. Gillen and J.H. Noggle, J. (~'hem. Phys. 53 (1970) S01. [5] W.G. Rothschild, Dynamics of Molecular Liquids (Wiley, New York, 1984). [6] F.J. Bermejo, M. Alvarez, M. Garc/a-Hernandez, F.J. Mompean, R.P. White, W.S. Howells, C.J. Cariile, E. Enciso and F. Batallan, J. Phys.: Cond. Matter 3 (1991) 851. [7] P.A. Egelstaff, An Introduction to the Liquid State (Academic Press, New York, 1967). [8] S.W. Lovesey, Theory of Neutron Scattering from Condensed Manor, Vol. 2 (Oxford Science Publications, 1986): see also S.W. Lovesey, J. Phys. C 7 (1974) 20118. [9] I.M. de Schepper and M.H. Ernst, Physica A 98 (1979) 189. [10] P. Verkerk, J.H. Builtjes and I.M. de Schepper, Phys. Rev. A 31 (10851 1731. [ 1 I] P. Vcrkerk. U. Baffle, B. Farago and F. Mezei, Physica B 168 (1991) I. [12] F.J. Bermejo, I:.J. Mompean, J.L. Martinez, M. Garcia-Hernfindez, A. Chahid, G. Senger and M.L. Ristig, Phys. Rev. B, submitted. 3] A series of papers on these topics have appeared. The lirst one is due to K.S. Singwi and A. Sj61ander Phys. Rev. 1211 (1960) 1(193, and the second by V. Ardente, G.F. Nardelli and L. Reatto, Phys. Rev. 148 ( 19661 124 should be consulted for references regarding these models. 14] An attempt to test the jump-diffusion models can be found in M. Sakamoto, B.N. Brockhouse. R.G. Johnson and N.K. Popc, J. Phys. Soe. Japan Suppl. B 17 (1062) 370. [15] M. Evans, G.J. Evans, W.T. Coffey and P. Grigolini, Molecular Dynamics and Thcory of Broadband Spectroscopy (Wiley, New York, 1982). [16] F. Garisto and R. Kapral, Phys. Rev. A 11)(1974) 309. [17] S,K, Deb, Chem. Phys. 12(1 (1988) 225. [18] A. Rau and R.W. Gerling, Z. Phys. B 78 (19901 275.
F.J. Bermejo et al. / QENS from molecular liquids and glasses [19] J.P. Hansen and I.R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). [20] K.S. Singwi, Phys. Rev. A 4 (1964) 969; Physica 31 (1965) 1275. [21] A. Chahid, F.J. Bermejo, E. Enciso, M. Garc/aHern~indez and J.L. Martinez, J. Phys.: Cond. Matter 4 (1992) 1213. [22] S.K. Sinha and D.K. Ross, Physica B 149 (1988) 51. [23] F.J. Bermejo, E. Enciso, J. Alonso, N. Garc/a and W.S. Howells, Molec. Phys. 64 (1988) 1169. [24] M. Garc/a-Hernfindez, A. Chahid, F.J. Bermejo, E. Enciso and J.L. Martinez, J. Chem. Phys. 96 (1992) 8477. [25] F.J. Bermejo, A. Chahid, J.L. Martinez, F.J. Mompean and M. Garcfa-Hern~indez, Physica B 180&181 (1992) 845. [26] F.J. Bermejo, J.L. Martinez, D. Martin, J.J. Mompean, M. Garc/a-Hernfindez and A. Chahid, Phys. Lett. A 158 ( 1991 ) 253. [27] A. Brooks-Harris and H. Meyer, Can. J. Phys. 63 (1985) 3. [28] F.J. Bermejo, F. Batallan, E. Enciso, R. White, A.J. Dianoux and W.S. Howells, J. Phys.: Cond. Matter 2 (199(I) 1301. [29] J. Alonso, F.J. Bermejo, M. Garc/a-Hernfindez, J.L. Martinez and W.S. Howells, J. Mol. Struc. 250 (1991) 147.
301
[30] F.J. Bermejo, F. Batallan, W.S. Howells, C..I. Carlile, E. Enciso, M. Garcia-Hern~indez, M. Alvarez and J. Alonso, J. Phys.: Cond. Matter 2 (199(I) 5005. [31] F.J. Bermejo, D. Martin, J.L. Martinez, F. Batallan, M. Garc/a-HernS_ndez and F.J. Mompean, Phys. Lett. A 150 (19901 201; F.J. Bermejo, J.L. Martinez, M. Garc/a-Hermindez, D. Martin, F.J. Mompean, J. Alonso and W.S. Howells, Europhys. Lett. 15 (1991) 5//9. [32] F.J. Bermejo, J. Alonso, A. Criado, F.J. Mompean, J.L. Martinez, J. Garc/a-Hernfindez and A. Chahid, Phys. Rev. B, in press. [33] D. Foster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Addison-Wesley, New York, 1990). [34] A.Z. Patashinskii and V.L. Pokrovskii, Fluctuation Theory of Phase Transitions (Pergamon Press, Oxford, 1979) ch. 4. [35] J.L. Martinez, F.J. Bermejo, M. Garcia-Herntindez and F.J. Mompean, J. Phys.: Cond. Matter 3 (1991) 3849; A. Chahid, F.J. Bermejo, J.L. Martinez and F.J. Mompean, Physica B 180&181 (1992) 843. [36] A. Chahid, F.J. Bermejo, J.L. Martinez, M. GarciaHerntindez, E. Enciso and F.J. Mompean, Europhys. Lett., in press. [37] V.G. Storchak et al., I.V. Kurchatov Inst. Alom. Energ. Preprint No. IAE-5333/9, 1991.
Physica B 182 (1992) 289-301 North-Holland
Neutron quasielastic scattering from molecular liquids and glasses In memory of Professor F. Bermejo-Martinez •
a
•
c
1
F . J . B e r m e j o , A . C h a h i d a, M. G a r c i a - H e r n ~ n d e z a, J . L . M a r t l n e z ' , F . J . M o m p e a n b, W.S. H o w e l l s b a n d E . E n c i s o d alnstituto de Estructura de la Materia, Consejo Superior de lnvestigaciones Cientificas, Madrid, Spain bRutherford Appleton Laboratory, Chilton, Didcot, Oxon 0)(1I OQX, UK Clnstitut Laue Langevin, 156X, F-38042 Grenoble-Cedex, France dDepartamento de Quimica-Fisica 1, Universidad Complutense, E-28040 Madrid, Spain
An overview of the present status of the theory of neutron scattering from disordered molecular systems (liquids and glasses), with the main emphasis on the theoretical modelling of spectral lineshapes, is presented. Applications to several specific eases are described in order to illustrate the capabilities of Quasielastic Neutron Scattering (QENS) for obtaining reliable information about the stochastic dynamics of these systems.
1. Introduction
Although Quasielastic Neutron Scattering ( Q E N S ) , or rather low-energy inelastic scattering, has been employed for nearly three decades for the investigation of the low-frequency dynamics of disordered molecular systems, its capabilities for obtaining reliable information about the characteristic time and length-scales regarding molecular motions have only been fully exploited in recent times• If relatively low counting rates and poor resolution in energy-transfers made up most of the studies until the late seventies, of scarce use for nowadays purposes, the lack of adequate tools for the analysis of lineshapes still represents a hindrance to further progress in this field. If, for monoatomic systems, the field has reached a mature state and fully quantitative Correspondence to: F.J. Bermejo, Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientificas, Serrano 123, E-28006 Madrid, Spain. 1Permanent address: Instituto de Ciencia de Materiales, C.S.I.C., Facultad de Ciencias C-IV, Universidad Aut6noma de Madrid, E-28049 Madrid, Spain.
comparisons between experiment and theory a n d / o r computer simulation results are available [1], the complications introduced by the need to separate rotational from centre-of-mass motions ( C O M ) (and the coupling between them) has precluded the achievement of a comparable degree of refinement in the case of molecular materials. Apart from the difficulties accounting for the spatial dependence of the C O M motions which make the linearized hydrodynamic approaches break down on the scale of wavevectors accessible to Q E N S , the effects of the collisional dynamics on the molecular reorientations have also to be considered in detail in order to extract the rich dynamical information contained in the Q E N S spectra. The use of computer simulations (mainly of static or Monte Carlo types) to help in the analysis of the g(r) pair correlation functions, has been a c o m m o n approach followed by experimentalists since the early days of neutron diffraction. However, the stringent computational requirements needed to carry out molecular simulations regarding time-dependent properties hindered the combined use of both techniques
0921-4526/92/$05.00 ~) 1992- Elsevier Science Publishers B.V. All rights reserved
F.J. Bermejo et al. / QENS.I?om molecular liquMs and glasse,s
290
when the microscopic dynamical properties of this class of fluids were the topic of interest. On the other hand, the advent of powerful desktop computer stations during the last decade has enabled the implementation of simulation codes capable of reproducing the most salient features regarding the thermal and spin dynamics of this kind of systems. As a consequence, several simulational approaches are starting to be considered as indispensable tools for data analysis purposes alongside with the development and testing of theoretical models. The present contribution intends to illustrate the opportunities for further development in this field arising as a consequence of the intricate interrelationship between theory, experiments and computer simulations. An excursion into the present state-of-the-art of theoretical results adequate for the analysis of QENS lineshapes is given in the next section. A survey of experimental results ranging from molecular "simple" liquids to glasses and magnetic molecular systems is then presented and finally some remarks and conclusions are drawn on section 4.
where the single-molecule correlation functions I,,, and I~ and the collective I(Q, t) are given by
I,,,(Q, t) = {exp(iQa,(t)) exp( iQa (t))> , In(Q, t)
1
~ ~, (exp(iQR,(t))exp(-iOR,(0))) '
(2)
, (3)
1
I(Q, t)= ~ ~f~ {exp(iORi(t))exp(-iORi(O)) ) , tl (4) cohjo( Qa,,)] z,
F(Q)=I~b
(5)
where the position vectors of the nuclei belonging to a rigid molecule are given by
r i,,
R + a~t .
(6)
R is the position of the molecular center-of-mass ( C O M ) and a~ is the relative position of each nucleus with respect to the COM. The 1 (Q,t) single-molecule function can be written in terms of a partial-wave expansion which, after a separation of terms corresponding to self- and different particle motions, gives
2. Spectral iineshapes Up to the present moment, no closed form expression exists for the calculation of the S(Q, w) quasielastic response function which does not contain strong simplifying assumptions. Only in the limit of decoupled rotational and center-of-mass motions has it been possible to derive formulae for the calculation of S(Q, w) once the relevant mechanisms for mass-diffusion and rotational relaxation are specified [2,3]. In such a case the molecular dynamics is completely specified by
S(Q, ¢o) = J dt e x p ( - i w t )
+ ~
bu.,.,,,,b .......hlu~(Q,
t)}l~(O, t)
luu(Q,t )
~ (2•+ /
l,,(Q,t)
1)jf(Qa )b~(t),
~(2/+
1 )lT(Qa~)P,(cosO "
J
(1)
)Ft(t ) (s)
where the Fi(t ) are rotational relaxation functions for which information can be made available from computer Molecular Dynamics simulation, optical or N M R spectroscopies etc. Two limiting cases are readily apparent regarding Fl(t ). These correspond to free (inertial) rotation and to infinitesimal rotation steps (i.e. when the rotational motion obeys a diffusion equation). For the case of inertial motion, the expressions for the first two orders in the case of spherical symmetry are 2
I 2r
Fl(r)= 1 2 " e r e x p ( - ½ r e) J + F(Q)(I(Q, t ) - I,(Q, t))]
(7)
0
0
dxexp(xe), (9)
F.J. Bermejo et al. / Q E N S f r o m molecular liquids and glasses 21/2r
F2(7 ) = 1 - ( 3 / 2 ' / 2 ) 7 e x p ( - 2 r 2) f
d x e x p ( x 2).
0
(10)
These are characterized by a rescaled dimensionless time r = (kBT/I)J/2t which is given in terms of the inverse of the free-rotor correlation time. Approximate expressions for its evaluation are given by Sears [2]. For the case of diffusive motion, the relaxation functions are of the simpler form
Fl(t ) = e x p ( - / ( l + 1 ) D r t ) ,
I
exp[/(l +
I)D,
ty (A0~) 1/2= (2D i At) I/2, where D i are the elements of a rotational diffusion tensor represented on a basis which makes it diagonal. The time required for an angular displacement of one radian is usually denoted by r,, so that the collision frequency becomes Z / r o where Z stands for the average number of steps required for a rotation of one radian. For diffusive behaviour it is required that r s ~ r 0. A simple test which gives an indication of the motional regime present is given by the quantity
5 [ k B T ] '/2 X i - lSD, L ~ - , . J
(15)
(11)
where D r is the rotational diffusion constant. The departure from spherical motion requires the introduction of two dynamical constants D 1 and D~ so that
F,U)=
291
+ mZD2]t,
(12)
m= 1
D, = ½(Dxx + Dyy),
(13)
D 2 = D~. - 4(Dxx + Dyy),
(14)
and, as an indication [4], the inertial region is confined to values X < 3, the diffusion regime is approached for X > 5 and an intermediate region is left in between. For cases belonging to the intermediate behaviour, some formulae which interpolate between the two are adequate [5] since both effects are explicitly considered and have been applied to the analysis of some liquids [6]. In particular, the following expression has been found by the authors to give reasonable results when analyzing Q E N S data for liquid CC14:
F,(t)=(r where D , , ii = xx, yy, zz, are the non-zero elements of the rotational diffusion tensor (i.e. expressed on a basis which makes it diagonal). More complicated expressions for the case of asymmetric tops can easily be found although their applicability to Q E N S studies is rather limited due to the number of free parameters involved. In all cases, the constants characterizing the reorientational motion (relaxation times) are given by the integrals r t = fo dt Fl(t ) and these are the relevant quantities to be compared with others obtained from optical, N M R or dielectric relaxation spectroscopies. By their very nature, Brownian dynamics theories assume that the duration of a collision is short in comparison with the period between randomizing events rj. Within such an hydrodynamic view, the angular excursions experienced by a molecule are given by the quanti-
-r+)
l[r_exp(7)
r+exp(~[)l, (16)
r+ = ½(r o' --+2/~), =
(17)
2 -4,o2)"-,
(18)
wj2 = l(l + 1 ) k u T / I ,
(19)
rR = (wZro) 1 ,
(20)
where the two relevant parameters encompassing the dynamical information regarding reorientational motions are r 0 and ¢o~. Since the latter is directly calculable from known constants, only r o is left as adjustable from the experimental intensities. The reorientational motions of real liquids well inside the kinematic region accessible to cold neutron spectroscopy have been shown to
F.J. Bermejo et al. / QENS from molecular liquids and glasses
292
be far more complex than those predicted by hydrodynamic theories. In fact, most of the relaxation functions computed from molecular dynamics simulations seem to follow (on empirical grounds) the formula Fl(t ) = all e x p ( - a 2 , t 2) + a31 e x p ( - a 4 g ) +(1 - a l l - a3t) exp(-a~/t)
(21)
where the Gaussian term dominates at short times (i.e. the regime mostly dominated by binary collisions) and the exponential behaviour characterizes, as expected, the long-time hydrodynamic regime. An example of such behaviour, together with some values for the coefficients, are shown in fig. 1 and table 1 respectively. 1.0 ~t~,~
T = 2 9 3 K (- -) = .,,
0.75
0.5
0,25
0.0
1
2
3
t
As a final remark, it has to be emphasized that, although some deviations are readily apparent between the calculated relaxation functions and those estimated from QENS data, the present day counting statistics as well as the large n u m b e r of free parameters involved in this latter formula preclude its use in fitting lineshapes in terms of physical models. 2. 1. The shape o f I~(Q, t) The use of a simplified hydrodynamic treatment, where the temporal decay of the self (i.e. center-of-mass) correlation function is described in terms of a simple exponential behaviour, is still common practice in the analysis of QENS lineshapes. However, some substantial drawbacks are expected to occur when analyzing data over a large range of momentum transfers. In general terms, an adequate representation of the single-particle scattering law should retain a Gaussian dependence in Q in both the hydrodynamic and free-particle limits. The question of how to interpolate between both is still a matter of different 'schools-of-thought'. In what follows, three cases will be considered: an approach based upon a generalization of hydrodynamics; a result from kinetic theory; a comparison with formulae capable of taking into account the results from molecular dynamics calculations. Some remarks regarding the usage of 'jumpmodels' to analyze the liquid data will complete this section.
4
(ps)
Fig. 1. Rotational relaxation functions for liquid CCI~ from M D using an LJ potential (dashes and dots) and approximations using eq. (21) (solid lines). The corresponding par a m e t e r s are given in table 1.
2. I. 1. The generalised hydrodynamics approach The so-called 'Gaussian approximation' [1] enables the representation of the incoherent scattering function Is(Q, t) in terms of a series expansion
Table 1 Values for the coefficients corresponding to the curves in rig. 1.
l~(Q,t)
a 1
(/-
(/a
(/4
as
260 K
FI k~
0.050 (1. 192
1.946 2.984
1.312 11.931
0.145 0.360
I). 145 1/.364
293 K
F~ F,
0.089 0.239
1.549 2.905
3.722 2.148
0.287 /).562
1/.325 0.514
exp( ~a(t)Q 2)
x [1 + c~2(t)bQ q
cQ ~ + dQ s + "" "].
(22)
If only the first exponential term is retained, the problem then lies in an adequate choice of a(t). For c~(t)= 2 D J the simple hydrodynamic result is regained whereas making
F.J. Bermejo et al. / Q E N S from molecular liquids and glasses
~(t) = 2D~[(t 2 +
C 2 ) 1/2 - -
C]
for the relaxation time is
with c = m D J k B T gives the approximation due to Egelstaff [7]. Both cases do not satisfy a n u m b e r of frequency moments (sum-rules) and therefore alternatives for overcoming this difficulty have been introduced. In particular, from the short-time expansion of the scattering law, 2
12
2
t4 . . . . .
9
I~(Q, t) = 1 - w(, ~ + w0(3w 0 + ~Q2) 4!
' (23)
where (for a monoatomic liquid) the moments are given by
(~o)=1,
(wz)=~o(2j,
=
293
~"
(26)
1 _ mD~ a 1/20 k B T ~ 2 ""o
and therefore only two parameters D~ and g20 are needed for a complete specification of the response function, since the second frequency moment can be easily evaluated. Such an approximation approaches correctly the lineshapes corresponding to the long-wavelength (Lorentzian shape) and free-particle (Gaussian shape) limits and may be profitably used in the modelling of Q E N S spectra measured for molecular materials, bearing in mind that the results for the higherorder frequency moments should be taken as 'effective' ones since their calculation from first principles still remains a formidable task.
+
2.1.2.
and
~-m
dr ~ dr 0
which is given in terms of the mass-density and the second derivative of the interparticle potential u(r). Hence an approximation which satisfies the m o m e n t relations can be found [8] which is formally analogous to the one developed for the study of spin dynamics in paramagnets. Fourier transformation of eq. (23) above leads [8] to
1 w/3 ~r 1 - e x p ( - h w / 3 )
X [(D'T(O) 2 __ 61 __ 6 2 ) ] 2 -1- (~'0 2 -- 6 1 ) 2
(25)
where the first term is a detailed balance factor and the parameters entering this formula are related to the frequency moments by the expressions 2
v
T
and (61 + 62)61
=
w2 + F~ + D~QQ* - - R e
+ O(Q
'ri'61 62 T
6t = w(l= Q - k B - - , m
S s ( Q , w) ;
Ss(Q, w)= -
The kinetic theory approach
The approaches based upon full kinetic theories have been extensively tested against molecular dynamics results for simple (hard-spheres and LJ fluids) dense fluids and have been used (without further justification) for a wide range of problems (from liquid metals to polymers). Within the De S c h e p p e r - C o h e n version of the m o d e mode coupling theory [9], the incoherent dynamic structure factor is approximated [10] as
2
61+ 67 = oo = 3 6 1 + $ 2 o ,
(034). The usual prescription
G
~/2f(~o/Q2)),
(27)
i.e. in terms of a Fickian contribution with F D s Q 2 and a first-order mode-coupling correction given in terms of the wavenumber Q * = 16"rrl3mnD~, the kinematic viscosity u = ~ / m n , 6 = D~/(D~ + v) and a complex function G ( z ) . Closed-form expressions for the linewidth, the peak height and the product of both have been obtained [10] up to orders of 0(Q3/2): 2 1 G(6 I)Q wQ S~(Q,O)= ~ + DsQ, ,
(28)
F..I. Berrnejo et al. / Q E N S Jrom molecular liquids attd glasses
294
Aoo(Q)/Q ~= D r - D~H(6)Q/Q*
(29)
"rrS,(Q,O) A m ( Q ) = 1 + A(,3)Q/Q*
(3o)
and the functions G, H and A can be evaluated from the parametric expressions given in ref.
[10l. The practical usefulness of such an approach for systems different from rare-gases still has to be demonstrated. It has, however, been applied to the study of the diffusional dynamics of dense hydrogen gas [11] and to liquid deuterium [12].
2.1.3. The approach based on molecular dynamics simulations A limitation which applies to most of the present approaches is the need to account for the first non-Gaussian corrections (i.e. terms in Q4 and above) as the density approaches that characteristic of realistic liquids. As a matter of fact, formulae like the following account for data on liquids composed of particles interacting via simple (LJ plus lower order multipole) potentials:
I~( Q, t) - e x p ( - 1 / 6 Q : ( r 2 ( t ) } ) Q4(r:(t)} ×
1+
7~_
C ( c / t ) e x p ( - t , / t - 1) (31)
where (r:(t)} is the atomic mean-squaredisplacement calculated from the simulated trajectories and C and 6 are constants which characterize the crossover to the free-particle regime. This kind of analysis will undoubtedly help for the separation of mass-diffusion effects from those concerning molecular rotations, thus allowing for a more accurate study of the fine details of the reorientational motions. On the other hand, and whenever possible, information regarding the value for the self-diffusion coefficient obtained in the hydrodynamic limit by means of Pulsed-Field-Gradient NMR will undoubtedly help to separate these two kinds of motion.
sixties [13] wherein the molecular motions were pictured as comprising a damped oscillatory behaviour ('jitter') during a time r() followed by diffusive steps during a time T1 ('jump'). When % "> ~-(), the models become equivalent to hydrodynamic diffusion, while in the opposite limit, assuming that the Debye-Waller term equals unity, one has
l~(Q,t)=exp(-[l+D-~Q2g]t ).
(32)
The usefulness of this kind of models for data analysis purposes comes from the fact that in the specification of the decay rates 1/% and l / r , , some non-Gaussian effects which become vcry relevant at liquid densities are introduced heuristically. However, a detailed analysis of the trajectories calculated from MD using realistic potentials do not show any evidence for the presence of oscillatory behaviour in I~(Q, t). In fact, in all the thoroughly investigated cases which include strongly associated liquids such as liquid water and liquid methanol, the diffusive COM motions arc more continuous than those pictured in this kind of quasi-crystalline motions, and in order to perform a truly discriminating test between the two approaches, the examination of time-scales as short as 0.01 ps becomes necessary [14].
2.2. Beyond the decoupling approximation A number of attempts to approximate the behaviour of the orientation and angular velocity correlation functions have been registered in the last decades. Although some analytical results have been derived for simple molecular shapes [15,16] (linear diatomics, loaded rough spheres and spherocylinders), the problem of how to evaluate the correlation function
( v A t ) v~(O)} (v(t)-v(O)) + ( v ( O ~o(0) × ~(())) + ~'(,,(t). ,(o),o(t) • ,o(o)) ,
(33)
2.1.4. The 'jump model' approach A family of models was developed during the
where v, = v + ½w × v gives the total velocity at
F.J. Bermejo et al. / QENS from molecular liquids and glasses
the ith atomic site in terms of the velocity of the C O M , angular velocity and orientation for an arbitrary molecular shape, remains open and MD simulation is the most common tool to attack the problem. For linear diatomics, however, explicit formulae for the COM velocity ({ v(t) v(O))), angular velocity ( ( w ( t ) o~(0))) and orientation ((~,(t) ~,(0))) autocorrelation functions can be found (ref. [15], p. 362) as well as expressions for the last term in eq. (33). However, the coupling term involves a detailed account of the collisional dynamics and can therefore only be addressed from simulations. On the other hand, some analytical results concerning the long-time tails of these correlation functions have also been obtained [16,17] where the asymptotic behaviour of the angular '~,'2 velocity autocorrelation is shown to follow a t law whereas a t 7/z law is found for its orientational counterpart. Both laws substantially deviate from the hydrodynamic result. Finally, the problem of rotation-translation interaction has also been studied on the basis of simple models [18] where the interaction of a molecule with its surrounding molecules is pictured as resulting from a coupling of the orientational degrees of freedom of a dumbbell with neighbouring oscillators. A simulation based on the solution of a coupled set of Langevin equations shows how the rotational relaxation function of the dumbbell deviates from free or from diffusive behaviour depending upon the choice of the interaction potential. Although this kind of idealized model cannot be sensibly translated to liquid or amorphous materials, the salient features of reorientational motion in real liquids seem to be taken into account.
2.3. Inclusion of coherent effects Although the possibility of separating the coherent and incoherent contributions to the Q E N S response of molecular liquids by means of polarization analysis has opened up a new era within this field, most of the available tools used for data analysis still rely on the 'convolution approximation' or ad hoc modifications of it. The problem seems to be solvable analytically only in
295
the case where the anisotropic contribution to the total structure factor (i.e. all those contributions which cannot be represented in terms of eq. (1)) is very small [19]. A recent application to a molecular liquid [6] used an extension of the formalism developed by Singwi [20] for the analysis of the coherent response in liquid argon, since attempts to model this contribution in terms of the abovementioned approximations lead to rather poor results. The model, which belongs to those of the 'quasicrystalline' family, represents the interference contribution by
Q=kBT Scoh(O, w)~ I~mVT L(R, O)Z(w)
(34)
where v r represents the mean isothermal sound velocity, Z(o~) is the generalized vibrational density of states of the liquid and L(R, Q) is a function given in terms of a 'coherence length' R and the structure factor for molecular centres Sere(Q):
L(R, Q)~
R3
4(Qv)~, =
f q, dq'[Scm(q')
1]
0
x e x p ( - 4R2 -
(O_ q,)2)
x exp( R~2( Q - q')2-1) .
(35)
We have found that this expression represents a fair approximation to the experimental data, although a comparison between the measured spectra and those calculated by means of eq. (1) as given in fig. 2, evidences that the present tools are still incapable of reproducing the QENS spectra for wavevectors close to Qp (i.e. the Q value corresponding to the maximum of S(Q)). Some effort on the development of lattice-gas models, aiming at a more accurate description of the diffusion of light atoms in metal hosts, has been reported in recent times [22]. In particular, the width of the coherent quasielastic component at low Q is given by AWcoh = Dch,.mQ=, i.e. in terms of a 'chemical' diffusion coefficient D~h.... which is related to the self-diffusion (tracer) coefficient by
F.J. Bermejo et al. / QENS.fi'om molecular liquids amt glasse,s
296
veals a d r a m a t i c slowing-down effect since D,~.....
0.4 Q = 1.06 A-~
D {T-
T.)/T,
w h e r e a s D~ maintains a regular behaviour. Alt h o u g h this kind of models is not directly applicable to liquid state problems, it represents a very promising tool for studies regarding critical phenomena.
0.3 3
u~ 0.2
3.
o.1
o.G
0.0
2.5
5.0 ~fi ¢~ (meV)
7.5
10.0
Some
recent
examples
In what follows, a brief review of recent studies of several molecular liquids, which epit o m i z e a whole range of different p h e n o m e n a , is given. T h e interested reader should consult the original references for m o r e details. 3.1.
A m o l e c u l a r s i m p l e liquid
1.5 Q=
1.25
3
1.37A
I1
l.O 0.75
0.5
0,25
0.0
I: "l I]ill:I :l /:1
•
/,-/ ,/'"
\ 0.0
2.5
5.0
7.5
10.0
"h ~ (meV) Fig. 2. A comparison between the experimental S( Q, to) anti that calculated by MD of liquid CCI~ for two wavevectors {Qp 1.34,~ ~ in this system). The solid line represents the measurement and the dashes the approximation given by eq. (1). In the lower frame the two different approximations represent distinct choices of F(Q) (see cq. (1)).
Liquid c a r b o n tetrachloride constitutes one of the most simple molecular liquids as well as being p e r h a p s the most heavily studied one tog e t h e r with water. D u e to its high molecular s y m m e t r y the interaction potential, the accuracy of which has been extensively tested, is mostly d o m i n a t e d by LJ terms, which makes it a m o p ecular a n a l o g u e to liquified classical rare gases [23]. R e c e n t Q E N S [21] and inelastic [24] studies c o m b i n e d with an M D simulation have enabled a fairly detailed analysis of the dynamics of this liquid. In o r d e r to exemplify the use of simulation data as a valuable tool for data analysis purposes s o m e c o m m e n t s regarding the use of the m e m o r y - f u n c t i o n formalism [19] follow. In principle, the advantages of the use of the K ( t ) m e m o r y functions associated with a given a u t o c o r r e l a t i o n ¢tj(t) stem from the simpler structure of K ( t ) (less oscillatory behaviour) in comparison with qJ(t) which will result in an easier modelling. T h e definition of these quantities c o m e s f r o m the set of equations
Dch .... = D J ( 1 - c)
w h e r e c stands for the average o c c u p a n c y of the available sites. N e a r the critical point D~ ..... re-
dO{O d~-
K{t - "r)dJ('r) d'c , {}
{36}
F.J. Bermejo et al. / QENS from molecular liquids and glasses
OK . I OT
f
K,,(t - ~-)K._I(': ) d-r,
which define the memory function K(t) as the kernel of an integral equation which is then given in terms of a hierarchy of K . ( t ) functions. The autocorrelation functions can be easily expressed afterwards in terms of the kernels K(t) if the Laplace transforms d~(s) are introduced. In order to approximate t~(s) the continued fraction
(37)
o
O(s)
-
-
+ •..
-
s + Ko(s )/(s + K, (s))
s + K(s)
(38)
T = 293
1"0I|
= 0.75[i
297
K
(---)
T = 260 K C.)
o.251
O.C 0.0
I
L0
I
I
I
t
I
I
I
0.5
[
I
I
I
I
1.0 t(ps) I
I
I
1.5
I
t
I
I
2.0
t
1
t
i
T = 260 K
i
;
l
I
:
t
t
I
I
I
f
k
T = 293 K
0.75 -
k~ ,.j 0.5> 0.25-
i
0.0
;
i
0.5
i
1.0 t ( ps )
i
J
I
1.5
i
i
i
i
J
2.0
i
I
i
I
0.5
i
i
i
i
i
l.O
i
i
i
E
F
1.5
i
i
i
i
2.0
t ( ps
Fig. 3. T h e first m e m o r y f u n c t i o n s f o r t h e l i n e a r v e l o c i t y c o r r e l a t i o n f u n c t i o n s in liquid CCI 4 f r o m M D . T h e g r a p h s b e l o w s h o w a c o m p a r i s o n o f t h e s i m u l a t e d $ ( t ) a n d t h e i r a p p r o x i m a t i o n u s i n g K,,(t) = K,(O)y(t), w h e r e y(t) is a s u m o f t w o G a u s s i a n s .
298
F..1. Berrmjo et al.
QENSfrom molecular liquids and glasse,s
has to be truncated at some step. The usual practice is to approximate K , ( t ) = K,,(O)y(t), where y(t) is a simple model-function and K(0) can be estimated from the frequency m o m e n t s of d,(w). An evaluation of different model functions which can describe adequately the m e m o r y functions associated with the linear and angular velocity correlation functions is given in ref. [21], and a comparison between the shape of the m e m o r y functions and the relevant autocorrelation functions as derived from MD simulations is shown in fig. 3. Some important discrepancies are seen between the computed function and the chosen model. Further improvements are obviously possible at the expense of the introduction of additional parameters. Once the m e m o r y functions are approximated (usually only the zeroth and first orders arc required) the evaluation of the Laplace transform of the Q E N S response is simply given by an expression analogous to eq. (38), with K 0 and K t substituted by the relevant expressions. Such a procedure thus enables the specification of the intermediate scattering functions in terms of relatively simple expressions whose validity is not limited to the hydrodynamic domain. Finally, as it was already shown in fig. 2, the M D results have provided a means to test the validity of the decoupling approximation given by eq. (1) on a real liquid, thus evidencing the need of far more refined theoretical models if an understanding of the microscopic dynamics at the same level of refinement as the one achieved on present day Q E N S spectroscopy is required.
ortho-D~ a mode-coupling analysis of the Iow-Q incoherent contribution of liquid deuterium was found to reproduce the wavevector-dependences of the linewidths and peak-heights up to a m o m e n t u m - t r a n s f e r of 1 ,~ r. The coherent Q E N S linewidths were rather well approximated by the expressions given above in section 2.3, possibly due to the low densities and viscosities involved in comparison with liquid C C I 4. However, systematic discrepancies were found for wavevectors about the main diffraction peak where some intensity around energy-transfers of about 2 meV could not be accounted for I12]. Although these systems have been investigated from the early days of neutron scattering, a n u m b e r of questions remains open and fully deserve a new re-examination using present-day techniques. In particular, it would bc worth exploring the following: • If some broadening is apparent in the quasielastic line of the solid. Predictions of about 1 IxeV for such a broadening have been available since the mid-sixties and such a measurement will constitute a stringent test for calculations based on simple quadrupolar interactions. • The mixtures of ortho- and para-components with a controlled amount of each species. These systems are considered to be an optimal model for the oricntational glass state [27] and a wealth of N M R and optical data are already available. • The t e m p e r a t u r e - d e p e n d e n c e of the coherent Q E N S spectra near critical points, since these systems are a natural testbed for models of coherent interference effects.
3.2. Quantum molecular liquids Due to the light masses and the simplicity of the intermolecular potential found in the liquid hydrogens, noticeable quantum features are readily apparent, which manifest themselves in the quantized nature of the rotational motion. In a recent set of Q E N S experiments [12,251 both the coherent and incoherent contributions to the Q E N S spectra were studied and in recent inelastic neutron scattering experiments [12,26] the higher-energy collective dynamics were probed. Due to the spherical symmetry of para-H z and
3.3. A complex (and vitrifiable ) liquid: Methanol Strong effects on the microscopic dynamics caused by hydrogen-bond ( H B ) formation have been found by the concurrent use of Q E N S and MD simulations [28,29]. Furthermore, by a comparison with an isomorphous material (methanethiol) where there is no evidence for the existence of any noticeable HB network, a quantitative explanation of the intermolecular effects on the microscopic dynamics has been obtained
F.J. Bermejo et al. / QENS from molecular liquids and glasses
[30], thus enabling a separation of dynamic effects arising from a purely thermal origin from those caused by the presence of strongly anisotropic intermolecular potentials. Since the vitrification of these materials is relatively easy and leads to glasses that are stable for many hours, a comparative study on the dynamics of liquid and glassy states has been carried out [31]. The density of states, Z(~o), provides evidence for some features which are characteristic of the glassy state [32] and most of the quasielastic intensity is transferred to a lowenergy (~-2 meV) inelastic peak which seems to be one of the most significant signatures of the presence of low-energy excitations in the glass not present in its crystalline counterpart.
299
T=88K
/
1.5
f
E
/
0
f""
0.5
%
I
1.5
i / E 0
0.5
•
/
0
-
3.4. Magnetic liquids Although the macroscopic behaviour of magnetic liquids, such as oxygen, resembles that characteristic of a normal paramagnet, polarized neutron diffraction studies have revealed the existence of magnetic short-range order [35]. A Q E N S study of the liquid and plastic crystal phases [35] has evidenced the dynamical nature of the observed magnetic correlations. An MD simulation of the structure-related effects [36] has enabled an approximate separation of magnetic and nuclear effects, thus allowing for a lineshape analysis of the magnetic response. Such a separation can be justified on the basis that the potential used has been shown to reproduce the structural features in both the liquid and plastic-crystal phases where the coupling between magnetic and structural degrees of freedom is weak enough. As a result, the evidence of an exchange-coupled paramagnetic state was inferred from the wavevector-dependence of the linewidths as well as from the spectral moments which have been estimated by means of a model for the Smagn(Q, co) contribution to the total structure factor based upon an exchange coupled Heisenberg model. The most significant results are depicted in fig. 4 and estimates for the coupling mechanism, based on current hydrodynamic approaches [33,34], have
0
i
i
i
r
0.5
1
1.5
2
2.5
Q (A "1)
5
"a 10
K
0.5
L0
1.5
2.0
Q cA~)
Fig. 4. The upper part shows the Q-dependence of the magnetic linewidths of liquid oxygen at two temperatures. The solid line represents the best approximation in terms of antiferromagnetic coupling and the dashed lines the ferromagnetic case. The I~, values calculated from the experiment are shown as full circles. The lower part shows the second frequency moment of the spectral weight function and its approximation by means of a model for nearest-neighbours interactions in a cubic Heisenberg paramagnet. The solid line corresponds to the case of ferromagnetic coupling (J 22 K) and the dashed line to the antiferromagnetic case giving a similar value for the coupling parameter.
300
F.J. Bermejo et al. / Q E N S /?ore molecular liquids and glasses
been inferred from them. A number of significant questions do however remain open: • The precise form of the coupling interaction remains to be determined. In principle the exchange constant should exhibit significant distance and orientation dependences; the modelling of it in computationally bearable terms is still awaiting. • How to model a Heisenberg paramagnet in the liquid state? • The neutron results concerning the comparison of the magnetic dynamics of the plastic crystal and liquid phases are in an apparent contradiction with recent I*SR studies [37]. Additional neutron measurements under wlrying magnetic field strengths may clarify this point. 3.5. Conclusion
Due to the difficulties in analyzing QENS lineshapes, most of the reported work has relied on coarse approximations (linearized hydrodynamics, jump-diffusion models etc.). In particular, the concurrent use of QENS spectroscopy with computer simulations has: • separated the COM motions from the total Q E N S intensity, thus allowing for the determination of reliable information regarding the reorientational dynamics; • analysed the rotational component with far greater detail than that achievable using simplified models whose validity is only ensured in limiting cases such as free rotation or diffusion; • explored the validity of the approximations which neglect the coupling between rotational and translational motions; • quantified the predictive capability of models developed to account for coherent interference effects; • enabled an approximate separation between structural and spin effects in magnetic liquids. Finally, future developments within the realm of fully microscopic theories can also be contrasted with the existing experimental and simulated data, so that a level of understanding comparable to the one nowadays existing for simple monoatomics (liquified rare gases and metals) will become possible.
Acknowledgements This work has been supported in part by D G I C Y T grant No. PB89-0037-C03. The help given by Dr. A.J. Dianoux of the Institut Laue Langevin, Dr. C.J. Carlile and Mr. M. Adams from R.A.L. during the inelastic neutron scattering experiments is warmly acknowledged.
References [1] J.P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill, New York, 1982). [2] V.F. Scars, Canad. J. Phys. 44 (1966) 1279, 1299. [3[ G. Venkataram, B.A. Dasannaracharya and K.R. Rao, Phys. Rev. 61 (1967) 1. [4] K.T. Gillen and J.H. Noggle, J. (~'hem. Phys. 53 (1970) S01. [5] W.G. Rothschild, Dynamics of Molecular Liquids (Wiley, New York, 1984). [6] F.J. Bermejo, M. Alvarez, M. Garc/a-Hernandez, F.J. Mompean, R.P. White, W.S. Howells, C.J. Cariile, E. Enciso and F. Batallan, J. Phys.: Cond. Matter 3 (1991) 851. [7] P.A. Egelstaff, An Introduction to the Liquid State (Academic Press, New York, 1967). [8] S.W. Lovesey, Theory of Neutron Scattering from Condensed Manor, Vol. 2 (Oxford Science Publications, 1986): see also S.W. Lovesey, J. Phys. C 7 (1974) 20118. [9] I.M. de Schepper and M.H. Ernst, Physica A 98 (1979) 189. [10] P. Verkerk, J.H. Builtjes and I.M. de Schepper, Phys. Rev. A 31 (10851 1731. [ 1 I] P. Vcrkerk. U. Baffle, B. Farago and F. Mezei, Physica B 168 (1991) I. [12] F.J. Bermejo, I:.J. Mompean, J.L. Martinez, M. Garcia-Hernfindez, A. Chahid, G. Senger and M.L. Ristig, Phys. Rev. B, submitted. 3] A series of papers on these topics have appeared. The lirst one is due to K.S. Singwi and A. Sj61ander Phys. Rev. 1211 (1960) 1(193, and the second by V. Ardente, G.F. Nardelli and L. Reatto, Phys. Rev. 148 ( 19661 124 should be consulted for references regarding these models. 14] An attempt to test the jump-diffusion models can be found in M. Sakamoto, B.N. Brockhouse. R.G. Johnson and N.K. Popc, J. Phys. Soe. Japan Suppl. B 17 (1062) 370. [15] M. Evans, G.J. Evans, W.T. Coffey and P. Grigolini, Molecular Dynamics and Thcory of Broadband Spectroscopy (Wiley, New York, 1982). [16] F. Garisto and R. Kapral, Phys. Rev. A 11)(1974) 309. [17] S,K, Deb, Chem. Phys. 12(1 (1988) 225. [18] A. Rau and R.W. Gerling, Z. Phys. B 78 (19901 275.
F.J. Bermejo et al. / QENS from molecular liquids and glasses [19] J.P. Hansen and I.R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). [20] K.S. Singwi, Phys. Rev. A 4 (1964) 969; Physica 31 (1965) 1275. [21] A. Chahid, F.J. Bermejo, E. Enciso, M. Garc/aHern~indez and J.L. Martinez, J. Phys.: Cond. Matter 4 (1992) 1213. [22] S.K. Sinha and D.K. Ross, Physica B 149 (1988) 51. [23] F.J. Bermejo, E. Enciso, J. Alonso, N. Garc/a and W.S. Howells, Molec. Phys. 64 (1988) 1169. [24] M. Garc/a-Hernfindez, A. Chahid, F.J. Bermejo, E. Enciso and J.L. Martinez, J. Chem. Phys. 96 (1992) 8477. [25] F.J. Bermejo, A. Chahid, J.L. Martinez, F.J. Mompean and M. Garcfa-Hern~indez, Physica B 180&181 (1992) 845. [26] F.J. Bermejo, J.L. Martinez, D. Martin, J.J. Mompean, M. Garc/a-Hernfindez and A. Chahid, Phys. Lett. A 158 ( 1991 ) 253. [27] A. Brooks-Harris and H. Meyer, Can. J. Phys. 63 (1985) 3. [28] F.J. Bermejo, F. Batallan, E. Enciso, R. White, A.J. Dianoux and W.S. Howells, J. Phys.: Cond. Matter 2 (199(I) 1301. [29] J. Alonso, F.J. Bermejo, M. Garc/a-Hernfindez, J.L. Martinez and W.S. Howells, J. Mol. Struc. 250 (1991) 147.
301
[30] F.J. Bermejo, F. Batallan, W.S. Howells, C..I. Carlile, E. Enciso, M. Garcia-Hern~indez, M. Alvarez and J. Alonso, J. Phys.: Cond. Matter 2 (199(I) 5005. [31] F.J. Bermejo, D. Martin, J.L. Martinez, F. Batallan, M. Garc/a-HernS_ndez and F.J. Mompean, Phys. Lett. A 150 (19901 201; F.J. Bermejo, J.L. Martinez, M. Garc/a-Hermindez, D. Martin, F.J. Mompean, J. Alonso and W.S. Howells, Europhys. Lett. 15 (1991) 5//9. [32] F.J. Bermejo, J. Alonso, A. Criado, F.J. Mompean, J.L. Martinez, J. Garc/a-Hernfindez and A. Chahid, Phys. Rev. B, in press. [33] D. Foster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Addison-Wesley, New York, 1990). [34] A.Z. Patashinskii and V.L. Pokrovskii, Fluctuation Theory of Phase Transitions (Pergamon Press, Oxford, 1979) ch. 4. [35] J.L. Martinez, F.J. Bermejo, M. Garcia-Herntindez and F.J. Mompean, J. Phys.: Cond. Matter 3 (1991) 3849; A. Chahid, F.J. Bermejo, J.L. Martinez and F.J. Mompean, Physica B 180&181 (1992) 843. [36] A. Chahid, F.J. Bermejo, J.L. Martinez, M. GarciaHerntindez, E. Enciso and F.J. Mompean, Europhys. Lett., in press. [37] V.G. Storchak et al., I.V. Kurchatov Inst. Alom. Energ. Preprint No. IAE-5333/9, 1991.