an introduction to the use of neutron scattering...

1 Introduction

11 Neutron scattering as an emerging tool in Earth and Mineral Sciences

Until recently neutron scattering methods had beenapplied only sporadically to problems in the Earth andMineral Sciences The current growing interest in the useof neutron scattering in this field is partly due to the devel-opment of improved neutron sources such as the UK ISISpulsed neutron source and the development of newmethods and instrumentation With regard to instrumenta-tion exciting developments have been facilitated by thepossibility for vastly increase coverage by huge banks ofdetectors with high-throughput data handling capabilitieswhich have been increased by developments in computingdata storage and communications Examples are the GEMgeneral-purpose diffractometer and MAPS magnetic exci-tations spectrometer at ISIS (the best source of informationis the ISIS web site httpwwwisisrlacuk ) Newinstruments give rise to new science possibilities some ofwhich have led to the growing interest in mineral sciencesapplications of neutron scattering These include possibili-ties for diffraction studies of complex minerals with subtlephase transitions high-pressure studies and spectroscopicstudies With the prospects of more advanced neutronsources coming on line over the next decade includingupgrades of ISIS (such as the development of a secondtarget station) the SNS pulsed source at Oak Ridge (USA)and possibly the European ESS pulsed source with the newopportunities these will give it is timely to review thepresent state of the art in applying neutron scattering

methods to solve problems in minerals sciences Some ofthe possibilities for the future have been reviewed byArtioli et al (1996a)

12 The interaction of neutrons with matter

Properties of the neutron

The neutron provides a unique probe of the atomicproperties of matter It carries no charge and has amagnetic moment Because it has no charge and unlikeelectromagnetic radiation and electrons the neutron doesnot interact with the atomic charge distribution Instead theneutron interacts via the strong nuclear force with theatomic nucleus and via its magnetic moment with themagnetisation density fluctuations (spatial and temporal)of the atoms The mass of the neutron (1 amu) is similarto the masses of atomic nuclei (ie from 1ndash240 amu)This means that the momentum and energy of a neutroncan be similar to the momenta and energies of atoms innormal matter Furthermore the wavelength of a beam ofneutrons will be similar to interatomic spacings As aresult neutron beams can be used as probes of both struc-ture and dynamics of materials as will be highlighted inthis review and in other reviews in this collection Thisincludes crystal structure and phonon dynamics which areconcerned with the atoms and magnetic structures andmagnetic excitations

There are two types of processes in which neutronsinteract with matter namely neutron scattering and neutronabsorption The interaction of the neutron with matterwhether through scattering or absorption is usually quite

Eur J Mineral2002 14 203-224

An introduction to the use of neutron scattering methods in mineralsciences

MARTIN T DOVE

Department of Earth Sciences University of Cambridge Downing Street Cambridge CB2 3EQ UK

Abstract We review the current state of the field of neutron scattering as applied to studies of mineral behaviour Neutron scat-tering is a particularly versatile tool because it is able to measure both structure (diffraction) and dynamics (spectroscopy) of mate-rials at an atomic level and because it is able to measure both coherent and incoherent scattering processes As a result there is awide range of phenomena that can be studied using neutron scattering The versatility of the tool is shown to arise within a gen-eral theoretical framework Applications of neutron scattering in the Earth and Mineral Sciences are described

Keywords Neutron scattering inelastic neutron scattering diffuse scattering total scattering diffraction incoherent neutron scat-tering quasi-elastic neutron scattering

0935-1221020014-0203 $ 990copy 2002 E Schweizerbartrsquosche Verlagsbuchhandlung D-70176 StuttgartDOI 1011270935-122120020014-0203

e-mail martinesccamacuk

MT Dove

weak although there are notable (and generally useful)exceptions This is in contrast with other probes such asconventional X-ray diffraction

The strength of the scattering from an atomic nucleus isdefined in either of two ways The first is the formal equiv-alence of the atomic scattering factor which for neutronscattering is called the scattering length (and with units oflength) and it is usually given the symbol b The seconddefinition is as a scattering cross section with symbol s(and with units of area) By convention the cross section islinked to the scattering length by s = 4pb2 The use of thescattering cross section is useful for comparing differenttypes of interaction between the neutron and the nucleus

Neutron scattering processes

The scattering interaction between the neutron andatomic nucleus takes place over extremely short lengthscales of the order of 10ndash15 m By contrast the interactionbetween X-rays and the atomic charge distribution operatesat the length scale of the atom namely around 10ndash10 m(1 Aring) The amplitude of a scattered beam of radiation varieswith scattering vector Q as the inverse of the size of thelength scale involved in the scattering process For X-raydiffraction this means that there is a considerable reductionin the amplitude of scattering over the range of values of Qin a typical experiment (eg 0ndash6 Aringndash1) On the other handthe small distances involved in neutron scattering meanthat there is no appreciable change in the scattering lengthover the same range of Q Together with the high resolutionnow possible with neutron powder diffraction instrumenta-tion the independence of the scattering length on Q hashelped neutron powder diffraction to become particularlyvaluable in determining crystal structures of minerals andthe variation of structure with changing temperature andorpressure This is important because the crystal structures ofminerals can be relatively complicated in comparison withother inorganic crystals Neutron powder diffraction is nowan established tool for the study of subtle changes in struc-ture associated with phase transitions as reviewed byPavese (2002) and Redfern (2002)

Unlike nuclear scattering the scattering of neutrons bythe magnetic moments of atoms does have a significantdependence on Q The magnetisation of an atom arisesfrom the unpaired electrons in the atoms and the magneti-sation density extends across the same length scale as thecharge distribution This means that the scattering lengthfor magnetic scattering varies with Q in a manner similar tothat for X-ray scattering

There are two main types of neutron scatteringprocesses The first is coherent scattering which is thescattering process in which the scattering from each atomof the same type is independent of the nuclear isotope andspin state In effect coherent scattering processes averageover isotope number and spin state The second type ofinteraction is incoherent scattering which accounts fordifferences in scattering from atoms of the same type dueto differences in isotope or spin state Clearly any atomwith a nucleus of zero spin and with only one isotope willhave a zero incoherent cross section The separation of

coherent and incoherent scattering processes is describedin more detail in sect22

Neutron scattering cross sections are actually relativelyweak Unfortunately this is coupled with the fact that eventhe most advanced neutron sources have relatively lowfluxes of neutrons as compared to X-ray laser or electronradiation sources (see Winkler 2002 for a discussion ofsources of neutrons) This means that neutron scatteringexperiments may be time consuming and the high costs ofproducing and detecting a neutron beam (Winkler 2002)mean that neutron scattering experiments are relativelyexpensive However these negative points are more thanoffset by the unique information that can be given byneutron scattering as will be highlighted in this and theother reviews in this collection of papers

Neutron absorption processes

The other type of interaction between neutrons andatomic nuclei is absorption In many cases the crosssection of neutron absorption is quite low However thereare some nuclei that have relatively large absorption crosssections These can occur either as resonances or as anabsorption process that varies as the inverse of the neutronvelocity Two examples of common neutron-absorbingmaterials are boron and cadmium Both of these are oftenused in neutron scattering experiments because of theirabsorbing power usually as components to mask the trajec-tory of the incoming and scattering neutron beam in orderto avoid measurements of neutrons scattering from else-where other than the sample

The low absorption of neutron beams passing throughmost materials means that neutrons can penetrate throughthe walls of sample environment equipment This facili-tates the development of such equipment without needingto compromise the quality of the types of experiment thatcan be performed For example good control at high andlow temperatures is a routine operation in a neutron scat-tering experiment (Pavese 2002 Redfern 2002)Traditionally high pressures have been easier to work withusing X-ray diffraction and diamond anvil cells but the useof time-of-flight neutron scattering techniques (Winkler2002) have facilitated considerable progress in neutronscattering at high pressures (Besson amp Nelmes 1995) It isnow also possible to perform neutron diffraction measure-ments at simultaneous high pressures and temperatures(Zhao et al 1999 2000 Le Godec et al 2001 2002) asreviewed by Redfern (2002)

The usually low absorption of neutron beams alsomeans that neutrons will interact with the bulk of thesample under study In some circumstances there can bemeasurable differences between the behaviour of a materialat its surface and the behaviour in the bulk (displacivephase transitions most often give this effect) In such casesconventional X-ray and neutron diffraction experimentsmay give different results if the results of a conventional X-ray diffraction experiment are dominated by scatteringfrom the surface

Neutron absorption processes are essential for the oper-ation of neutron detectors Common absorbing materials in

204

neutron detectors are lithium and 3He Boron (in thegaseous compound BF3) and gadolinium are also used asneutron absorbing materials in neutron detectors (seeWinkler 2002)

Resonance absorption of neutrons can be exploited inthe design of experiments Fowler amp Taylor (1987)proposed that measurements of the absorption resonanceprofile could be used to measure the temperature of asample through the Doppler broadening of the resonanceline (see also Mayers et al 1989 Frost et al 1989) Thisprocedure has recently been exploited for the measurementof temperature in diffraction experiments at simultaneoushigh pressures and temperatures where thermocoupleswould be difficult to use (Le Godec et al 2001 2002)

Neutron absorption processes can also be exploited formacroscopic imaging and tomographic studies of mineraland melt properties as reviewed by Winkler et al (2002)

Variation with atomic number

Neutron scattering and absorption cross sections do notvary systematically with atomic number (although there aresome general trends albeit with very large fluctuationsaway from these trends) The variation with atomic number

of the coherent scattering lengths incoherent scatteringcross sections and neutron absorption cross sections areillustrated in Fig 1 Several important features emergefrom a consideration of the data in Fig 1

It can be noted that there are interesting contrasts in thecoherent scattering lengths for groups of atoms or ions withsimilar numbers of electrons such as Mg2+ Al3+ and Si4+These will have very similar scattering cross sections forX-ray diffraction but significantly different scatteringlengths for neutron diffraction In fact there are someatoms including 1H Mn and Ti that have negativecoherent scattering lengths In these cases the phase of thescattered neutron beam is reversed compared to the phasewhen scattering from atoms with positive scattering lengthThis will increase the difference between the scatteringlengths of similar atoms even further

In Fig 1 we compare the coherent and incoherent scat-tering cross sections for all atoms It can be seen that formost atoms the coherent scattering cross section is thelarger of the two which is essential for the study of struc-ture through diffraction and for measurements of collectiveexcitations However there are some nuclei for which thecross section for incoherent scattering is particularly large1H is one of these nuclei In a powder diffraction experi-

Neutron scattering methods in mineral sciences 205

- 5

0

5

1 0

1 5

2 0

ndash5 0

0

5 0

1 0 0

1 5 0

1 0 ndash 4

1 0 4

1 0 0

1 0 ndash 2

1 0 2

0 2 0 4 0 6 0 80 100

A t o m i c n um be r

Co

he

ren

t s

ca

tte

rin

g l

en

gth

(fm

)S

ca

tte

rin

g c

ros

s s

ec

tio

n (

ba

rns

)A

bs

orp

tio

n c

ros

s s

ec

tio

n (

ba

rns

)

I n c oh e r e n t x 2 0

C o h er e n t

In c oh e r e n t

Fig 1 Top dependence of coherent neutron scat-tering lengths on atomic number Middle compar-ison of the coherent and incoherent scattering crosssections for the same range of atomic number Forclarity the plot of the coherent scattering crosssections has been lowered by 50 barns and a raisedplot of incoherent scattering cross sections multi-plied by 20 is also included Bottom logarithmic plotof neutron absorption cross sections Among theimportant features of these diagrams are the largeincoherent cross section and negative coherent scat-tering length for hydrogen (atomic number 1) thenear-zero value of the coherent scattering length ofvanadium (atomic number 23) with an appreciablecross section for incoherent scattering the relativelylarge absorption cross sections of boron (atomicnumber 5) cadmium (atomic number 48) andgadolinium (atomic number 64) each of which canbe used to provide neutron shielding for precisionexperiments contrast between the coherent scat-tering lengths of important isoelectronic (or nearlyisoelectronic) atoms such as the MgAlSi andMnFeCoNi series Data are from VF Sears asquoted in Beacutee (1988)

MT Dove

ment on a hydrous mineral the incoherent scattering fromthe hydrogen nucleus can give a large contribution thatdominates the diffraction pattern for all values of Q whicheffectively limits the accuracy of the information that canbe extracted from the diffraction pattern However thedeuterium nucleus has a zero incoherent scattering crosssection and the problem associated with hydrogen can beeliminated if the hydrogen atoms in the sample to bestudied can be completely replaced by deuterium

Figure 1 also shows the wide range of neutron absorp-tion cross sections The relatively large absorption crosssections of some nuclei such as lithium (Z = 3) and boron(Z = 5) can be seen to be several orders of magnitudelarger than for atoms with similar mass Lithium and boronare used in neutron detectors and boron is also used inshielding materials

13 Neutron scattering and the Earth and Mineral Sciences

Rinaldi (2002) has reviewed a wide range of reasonswhy neutron scattering is of direct interest to the Earth andMineral Sciences We have highlighted above the issues ofsensitivity to nuclei of similar atomic number and theability to study magnetic minerals and the possibility tostudy both structure and dynamics

The wide range of particular advantages of neutronscattering listed above and in Rinaldi (2002) can be encap-sulated in the fact that neutron scattering methods have anastonishing versatility This versatility arises from the prop-erties of the neutron In particular the wavelengths andenergies of neutrons are comparable to the length andenergy scales of the atoms in solids and fluids Thiscontrasts with X-rays where the wavelength is similar tothe interatomic spacings but the energy is several orders ofmagnitude higher than the energy of lattice vibrationsLaser light and infrared radiation have energies closer toatomic vibration energies but their wavelengths are muchlarger than interatomic spacings Since neutron scatteringprobes both the length and energy scales of atomicprocesses it is possible to design experiments that focus onboth spatial and dynamic processes at the same time orelse which focus on one or other We will formalise thisversatility below

The versatility of neutron scattering methods can beillustrated with reference to the hydrogen atom Manyminerals contain hydrogen sometimes in the form ofbound or free hydroxide ions and sometimes in the formof bound water molecules or as free water moleculeswithin cavities in the crystal structure X-rays are veryinsensitive to hydrogen but neutrons are scattered byboth hydrogen 1H and deuterium 2H nuclei Hydrogen1H has an extremely large cross section for incoherentscattering ndash this is the cross section for neutron scatteringthat can be used to study the motions of individualhydrogen atoms (sect4) Slow motions of the hydrogenatoms whether in translational diffusion or reorienta-tional motions of molecules containing hydrogen atomscan be probed by quasi-elastic incoherent scattering(sect42 44) and fast motions of individual atoms as in

lattice or molecular vibrations can be studied by high-energy spectroscopy (sect43) On the other hand sincedeuterium has a reasonable cross section for coherentscattering and no appreciable cross section for inco-herent scattering deuterated samples can be used indiffraction studies for the location of hydrogen sites incrystal structures or in studies of individual collectiveexcitations

The objective of this article is to review theformalism of neutron scattering and to highlight how theformalism leads to the versatility and the great diversityof applications of the methods for studies in mineralsciences We will start with the general formalism andshow how it can be separated into several differentcomponents These include elastic and inelastic scat-tering and coherent and incoherent scattering Thespecific features of these different components will beillustrated using brief examples We will focus mostly onmethods that have already been used for studies ofmineral behaviour and will only briefly comment onmethods (such as measurements of magnetic propertiessmall-angle scattering and surface reflectometry) thathave yet to be used for routine studies

2 General formalism for neutron scattering

21 The neutron scattering law

A typical neutron scattering experiment will involvemeasuring the intensity of the neutrons scattered from abeam incident on a sample The scattering process willinvolve a change in the wave vector of the beam which isdenoted by the scattering vector Q The scattering processmay also involve a change in the energy of the neutrons EThis will be determined by the angular frequency w ofsome dynamic fluctuation in the sample so that

E = w (1)

The intensity measured in an experiment is convertedinto a function of Q and w and is proportional to thedynamical structure factor S(Qw) This function repre-sents the Fourier transforms of the time and spatial fluctu-ations of the atomic density (or spin density for magneticneutron scattering) We present the formalism by consid-ering the scattering from point particles which is appro-priate to the case of neutron scattering because as notedearlier the length scale of the interaction between a neutronand atomic nucleus is around five orders of magnitudesmaller than the wavelength of the neutron beam For agiven particle of label j at position rj at time t the numberdensity can be expressed as the Dirac delta function

rj(rt) = d(r ndash rj(t)) (2)

The spatial Fourier transform of the number density isgiven as

rj(Qt) = ograverj(rt)exp(iQtimesr)dr = exp(iQtimesrj(t)) (3)

206

The fluctuations in time are expressed through thecross-correlation function

fjk(Qt) = ltrj(Qt)rk(ndashQ0)gt (4)

where the angular brackets denote an average over all initialtimes In a neutron scattering process the contribution ofeach atomic nucleus to the amplitude of the scattered beamis weighted by the scattering length bj which reflects thestrength of the interaction between the neutron and thenucleus As a result when we consider the fluctuations ofthe whole sample that will give rise to the neutron scat-tering we weight each contribution accordingly and write

F(Qt) = 1ndashN S

jkltbjbkrj(Qt)rk(ndashQ0)gt

= 1ndashNS

jkbjmdash

bkltrj(Qt)rk(ndashQ0)gt (5)

where N is the number of atoms This function is known asthe intermediate scattering function The overbars repre-sent the average implicit in the time averaging of the cross-correlation functions and can be separated from thefluctuations in the density functions because the scatteringlengths do not depend on time

The dynamical structure factor is given by the timeFourier transform of F(Qt)

S(Qw) = ograveF(Qt)exp(iwt)dt

= 1ndashNograveS

jkbjmdash

bkltrj(Qt)rk(ndashQ0)gtexp(iwt)dt (6)

In many cases this is expressed in terms of energy ratherthan angular frequency using the conversion given in equa-tion (1)

S(QE) = 1ndashNograveS

jkbjmdash

bkltrj(Qt)rk(ndashQ0)gtexp(iEt )dt (7)

The formalism may now appear to be rather compli-cated but we can turn this to our advantage by consideringspecial cases each of which leads to a special techniquethat can give more-or-less unique information about thesystem being studied This includes separation of coherentfrom incoherent scattering processes elastic from inelasticscattering processes (both of which are discussed immedi-ately below) and techniques in which we can integrate overeither energy or scattering vector It is the fact of being ableto carry out the separation of different scattering processesthat leads to the versatility of neutron scattering methodshighlighted in the introduction

22 Separation of coherent and incoherent neutron scattering

In discussions of S(Qw) it is often assumed that theneutron scattering length for any element in a crystal is thesame for each atom of a particular element This impliesthe assumption that the scattering length is independent ofthe nuclear isotope number and the nuclear spin state Inpractice what we have actually been doing is to average outall these effects as we will see below

For many nuclei the spin is zero so that the spin-depen-dent components of the nucleusndashneutron interactions areabsent and even when the nuclei have spin the spin effectsare often relatively weak Moreover for many importantelements only one isotope occurs in significant quantitiesFor example the isotopes of oxygen 15O 16O and 17Ooccur with relative abundances 1 98 and 1 respec-tively Hence the assumption of constant scattering lengthfor the atoms of many elements is quite justified Howeverthere are important exceptions The two major isotopes ofnickel are 58Ni (relative abundance of 683) and 60Ni(relative abundance of 261) with respective scatteringlengths of 144 and 28 fm with neither of these two nucleihaving a non-zero value of the spin On the other handhydrogen (1H) is an example where the spin-dependence ofthe nucleusndashneutron (in this case the protonndashneutron) inter-action is very important and this isotope has a relativeabundance of more than 999 in natural hydrogen Theproton and neutron both have spin values of 12 whichunder the laws of quantum mechanics can be aligned infour ways Three of these have parallel alignment giving atotal spin of S = 1 and z component Sz = ndash1 0 or +1whereas the fourth has antiparallel alignment and S = 0The scattering lengths for parallel and antiparallel align-ment are 104 and ndash474 fm respectively giving an averagevalue of only ndash374 fm Sodium is another nucleus with anincoherent scattering cross section that arises from the spinstates the incoherent scattering length 356 fm is close tothe coherent scattering length 363 fm Vanadium isunusual in that the sum of the coherent contributions isalmost zero with only a significant incoherent crosssection We therefore need to consider how the neutronscattering function can be modified to account for theseeffects and we shall find that there are considerable appli-cations from this extended approach The comparisonsbetween values of the coherent and incoherent scatteringcross sections are illustrated in Fig 1 A compilation ofscattering lengths and cross sections has been reproducedby Beacutee (1988)

In order to account for different scattering lengths foratoms of the same type we separate the intermediate scat-tering factor (equation 5) into two components

F(Qt) = Fcoh(Qt) + Finc(Qt) (8)

where the coherent term is given as

Fcoh(Qt) = 1ndashNograveS

jkbjndash

bkndash ltrj(Qt)rk(ndashQ0)gt (9)

and the incoherent term is given as

Finc(Qt) = 1ndashNS

jk(bjmdash

bk ndash bjndash

bkndash

)ltrj(Qt)rk(ndashQ0)gt (10)

The coherent term has an average value for the scatteringlength and therefore contains the information aboutcoherent (ie correlated) processes such as Bragg andphonon scattering For most applications the coherent termis expanded in terms of instantaneous atomic displace-ments from mean positions and this enables the formalismto be applied to studies of phonon dispersion curves and


MT Dove

other dynamic processes This expansion will be discussedlater

The incoherent term is not yet written in a useful form Wenote however that we do not expect there to be any correla-tion between the particular value of the neutron scatteringlength and the site This means that we have the condition

bjmdash

bk = bjndash

bkndash

for j sup1 k (11)

Hence only the terms for j = k remain so that

Finc(Qt) = 1ndashNS

j(bj

ndash2 ndash bjndash 2)ltrj(Qt)rj(ndashQ0)gt (12)

For reference we can now define two scattering crosssections

sjcoh = 4pbj

ndash 2

sjinc = 4p(bj

ndash2 ndash bjndash 2) (13)

We can write the equation for the incoherent structurefactor in its expanded form

Finc(Qt) =1mdash

4pNSjsj

incltexp(iQtimes[rj(t) ndash rj(0)])gt (14)

It is convenient to define a probability distributionfunction G(r(t)r(0)) which gives the probability offinding a particle at a position r(t) at time t if it had a posi-tion r(0) at time 0 Thus we can write the equation in a newform

Finc(Qt) =1mdash

4pNSjsj

incograveexp(iQtimes[rj(t) ndash rj(0)])

G(r(t)r(0))p(r)dr(t)dr(0) (15)

where p(r) is the probability of finding the atom at positionr at time 0 The significance of writing the intermediatescattering function this way is that for all problems ofinterest the time-dependence of the atomic motions aremost conveniently analysed through the rate equations thatgovern the time-dependence of G(r(t)r(0))

The main point that should be appreciated at this stageis that the incoherent scattering function contains thedynamic information for individual atoms provided theyhave sufficiently large incoherent cross sections Thisdynamic information may include the motions due to allthe phonon modes in the crystal complementary to the

temperature factor but it may also include diffusiondynamics if important This is reviewed briefly below (sect4)a comprehensive treatment has been given by Beacutee (1988)

23 Separation of elastic and inelastic scattering

Elastic scattering arises from the static component of acrystal structure (the idea applies equally well to a glassbut not to a fluid phase) and therefore has w = 0 Howeverit is not identical with S(Qw = 0) as will now beexplained

The function S(Qw = 0) can be written as

S(Qw = 0) = ograveF(Qt)dt (16)

We can separate F(Qt) into two components a constantterm and a term that depends on time and either tends to azero as t reg yen or else oscillates around zero at large t Thuswe can write

F(Qt) = F(Qyen) + Frsquo(Qt) (17)

where the first term is the constant term and the secondcontains all the time dependence The Fourier transform ofthe time-dependent term will give the same energy spec-trum for w sup1 0 as the full F(Qt) The Fourier transform ofthe constant term will yield a d-function at w = 0

ograveF(Qyen)exp(iwt)dt = F(Qyen)d(w) (18)

From equation (5) the function F(Qyen) can be writtenas

F(Qyen) = 1ndashN S

jkltbjbkrj(Qyen)rk(ndashQ0)gt

= 1ndashNS

jkbjmdash

bkltrj(Q)gt ltrk(ndashQ)gt (19)

where we have used the fact that over infinite times thefluctuations in the density function are uncorrelated Forcoherent scattering equation (9) this will reduce to

Fcoh(Qyen) = 1ndashNfrac12S

jbndash

j ltrj(Q)gtfrac122

(20)

The important point to appreciate from this discussionis that the elastic scattering is the Fourier transform of thetime-independent component of the intermediate scatteringfunction The Fourier transform of a constant function will

208

time frequency

F(Q

t)

S(Q

w)

Fouriertransform

constantcomponent

Peak from oscillatingcomponent of F(Qt)

d-function fromtime-independent

component of F(Qt)

Zero-frequency componentfrom time-dependent part

of F(Qt) Fig 2 Illustration of the definition of elasticscattering The scattering function S(Qw) isthe Fourier transform of a time-correlationfunction F(Qt) If F(Qt) does not decay tozero value in the long time limit its Fouriertransform will contain a delta function at zerofrequency It is this delta function and notsimply the scattering at zero frequency thatcorresponds to elastic scattering

give a d-function at zero frequency This is not the same asthe zero-frequency component of the scattered intensityWe illustrate this point in Fig 2 If F(Qt) has a relaxationalcomponent the integral of the time-dependent part whichgives a zero-frequency contribution to S(Qw) is non-zeroHowever it forms part of the continuous distribution ofS(Qw) rather than as a separate d-function

24 Magnetic neutron scattering

The formalism of neutron scattering can be readilyextended for magnetic scattering To illustrate the point weconsider magnetic diffraction using an unpolarised beam ofneutrons The magnetic equivalent of equation (20) is

Fmag(Qyen) = 1ndashNfrac12S

jqjpjltexp(iQtimesrj)gtfrac122

(21)

where the magnetic interaction vector q is given by

q = mm ndash Q(Qtimesm)Q2m (22)

and m is the magnetic moment of an atom or ion If a is theangle between m and q

q =frac12qfrac12 = sina (23)

The magnetic scattering amplitude p is given by

p = e2gmdash2mc2 gJf (24)

where e is the electron charge g is the magnetic momentof the neutron m is the mass of the electron and c is thespeed of light g is the Landeacute factor J is the total orbitalangular momentum number of the atom and gJ gives themagnetic moment of the atom f is the form factor thatdepends on Q in a similar manner to the behaviour of theX-ray atomic scattering factor Over the useful range ofvalues of Q p has values similar to the values of theneutron scattering length for nuclear scattering b (Fig 1)We will not develop the theory of magnetic scattering inthis review but it should be noted that the formalism fornuclear scattering presented here both for elastic andinelastic scattering can be extended for the study of thestructures and dynamics of magnetic materials

3 Formalism for coherent neutron scattering

31 Coherent inelastic neutron scattering

We now develop the analysis of the neutron scatteringfunction for the case when atoms vibrate about their meanpositions as in a crystalline solid (for more details seeAshroft amp Mermin 1976 Dove 1993) The instantaneousposition of atom j can be written as

rj(t) = Rj + uj(t) (25)

where Rj is the average position of the atom and uj(t)represent the instantaneous displacement of the atom from

its average position The intermediate scattering functionequation (9) can therefore be written as

F(Qt) = 1ndashN S

jkbndash

jbndash

k exp(iQtimes[Rj ndash Rk])ltexp(iQtimes[uj(t) ndash uk(0)])gt

(26)

If the atoms move with harmonic motions a standardresult gives

ltexp(iQtimes[uj(t) ndash uk(0)])gt = exp ndash 1ndash2 lt(Qtimes[uj(t) ndash uk(0)])2 gt

= exp ndash 1ndash2 lt[Qtimesuj(t)]2 gt ndash

1ndash2 lt[Qtimesuk(0)]2gt + lt[Qtimesuj(t)][Qtimesuk(0)]gt

(27)

The first two terms in the expanded exponent corre-spond to the normal temperature factors (see Willis ampPryor 1975 Ashcroft amp Mermin 1976 Castellano ampMain 1985)

exp ndash 1ndash2 lt[Qtimesuj(t)]2 gt = exp(ndashWj)

exp ndash 1ndash2 lt[Qtimesuk(0)]2 gt = exp(ndashWk) (28)

The third term can be expanded as a power series

exp(lt[Qtimesuj(t)][Qtimesuk(0)]gt) = yen

Sm=0

1mdashmlt[Qtimesuj(t)][Qtimesuk(0)]gtm (29)

The first term in the series m = 0 corresponds to elasticscattering which we will discuss below The second termm = 1 turns out to be the most interesting of the other termsin the series Within the harmonic theory of latticedynamics the instantaneous displacement can be written as

uj(t) = (Nmmdash1mdash

j)12Skn

ej(kn)exp(iktimesrj)Q(kn) (30)

where Q(kn) is the normal mode coordinate for a phononof wave vector k and branch n (the number of branches ina crystal containing z atoms per unit cell is equal to 3z) andej(kn) is the normalised vector that gives the relativedisplacements of each atom Treating the normal modecoordinate as a quantum operator it can be shown that them = 1 term contributes the following term to the dynamicalscattering factor

S1(Qw) = 1ndashN S

jkbndash

jbndash

k exp(iQtimes[Rj ndash Rk])exp(ndashWj ndash Wk)

ogravelt[Qtimesuj(t)][Qtimesuk(0)]gtexp(ndashiwt)dt

= 1ndashN S

n 2w(mdash mdash

kn)frac12Fn(Q)frac122

([1 + n(w)]d(w + w(kn)) + n(w)d(w ndash w(kn)))(31)

where the phonon structure factor component is given as

Fn(Q) = Sj

bndash

jmdashmj

exp(ndashWj)exp(iQtimesRj)Qtimese(kn) (32)


MT Dove

The formula for the one-phonon dynamical scatteringfactor requires some unpacking The Dirac delta functionsrepresent the fact that scattering from a phonon of angularfrequency w(kn) only occurs for changes in the energy ofthe neutron beam of plusmn w(kn) The probability of phononscattering is determined by the thermal population ofphonons which is determined by the Bose-Einstein factorn(w)

n(w) = expmdash

( w mdash1mdash

kBT)ndash1mdash (33)

The case where the scattering mean has lost energycorresponds to the situation where the neutron beam hasscattered following the creation of a phonon and thisprocess occurs with probability [1 + n(w)] The case wherethe scattered beam gains energy corresponds to the situa-tion where the neutron beam absorbs a phonon and proba-bility of this process is given by the number of phononsn(w) At high temperatures the intensity from a singlephonon is proportional to

S1(Qw)microkBTmdashw2d(w plusmn w(kn)) (34)

In both cases the intensity of scattering is determined bythe phonon structure factor which accounts for the relativepositions of atoms and the relative orientations of the scat-tering vector Q and the displacements of the atoms Thesefactors give selection rules for the one-phonon neutronscattering process which are not as restrictive as the selec-tion rules in Raman or infrared spectroscopy and whichcan be useful in separating measurements when there are alarge number of phonons for any wave vector k

The terms for m gt 1 involve multiphonon processesThese processes do not give enough structure in measuredspectra to be useful and understanding the behaviour ofthese processes is mostly motivated by the need to subtractthem from absolute measurements of intensities in bothinelastic and total scattering measurements

The one-phonon coherent inelastic processes describedabove are often used for measurements of phonon disper-sion curves with single crystal samples (Chaplot et al2002) These studies are not trivial and as a result it isoften the case that measurements will only focus on a smallsubset of all phonon modes (particularly the lower energymodes) Examples of measurements of relatively completesets of dispersion curves of minerals include calcite(Cowley amp Pant 1970) sapphire (Schober et al 1993) andquartz (Strauch amp Dorner 1993) The measured dispersioncurves for quartz are shown in Fig 3 It should be appreci-ated that the measurements of the high-frequency modesare particularly difficult ndash it can be seen from equation (34)that the one-phonon intensity scales as wndash2 Since thesemodes can be better accessed by other spectroscopicprobes and indeed their variation across range of wavevector is often slight it is often not worth attempting tomeasure complete sets of dispersion curves Moreover theexperimental difficulties are increased with more complexminerals Examples of partial sets of dispersion curvesinclude the olivines forsterite (Rao et al 1988) and fayalite(Ghose et al 1991) pyrope (Artioli et al 1996b)andalusite (Winkler amp Buumlhrer 1990) and zircon (Mittal etal 2000a)

Because of the difficulties in obtaining full dispersioncurves inelastic coherent scattering measurements usingsingle crystals may focus on selected modes An exampleis a detailed study of an anomalous inelastic scatteringprocess associated with a particular zone boundary modein calcite (Dove et al 1992 Harris et al 1998) The rel-evant phonon dispersion relation is shown in Fig 4 Thefrequency of the lowest mode dips considerably at the zoneboundary and softens on increasing temperature This isshown in inelastic neutron scattering spectra for differenttemperatures in Fig 5 This mode is actually the soft modefor the high-pressure displacive phase transition in calciteIn addition to the phonon peak there is a broad band of

210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)

Reduced wave vector component x(x00) (00x)

Fig 3 Phonon dispersion curves of quartz obtained by inelasticneutron scattering measurements using a triple-axis spectrometerat the ILL reactor neutron source (Strauch amp Dorner 1993)

00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector

Fig 4 Phonon dispersion curves for calcite along the direction(000) to (120ndash2) in reciprocal space measured using a triple-axisspectrometer at the Chalk River reactor neutron source The impor-tant point is the dip in the lowest-frequency branch at the zoneboundary wave vector (Dove et al 1992)

inelastic scattering that is localised (in Q) at the zoneboundary This anomalous inelastic scattering can be seenas a weak feature barely above the background level atenergies lower than the phonon peak at ambient tempera-ture in Fig 5 The intensity of this scattering increasesrapidly on heating eventually dominating the spectra athigh temperatures The spectra have been analysed byusing a model in which the phonon is coupled to a relax-ation process in the crystal although there is no clear iden-tification of the nature of this relaxation process There isan order-disorder phase transition at 1260 K (which hasalso been studied by neutron powder diffraction Dove ampPowell 1989) Phonon instabilities associated with phasetransitions have also been studied by inelastic neutronscattering in quartz (Boysen et al 1980 Bethke et al1987 1992 Dolino et al 1992) and leucite (Boysen1990)

It is also possible to perform inelastic coherent scat-tering measurements on polycrystalline samples either

performing measurements as a function of the modulus ofthe scattering vector Q or integrating over all Q to obtaina weighted phonon density of states The latter type ofmeasurement was used to obtain the first experimentalevidence for a significant enhancement of low-energyrigid-unit modes in the high-temperature phase of cristo-balite (Swainson amp Dove 1993) Measured phonon densi-ties of states have been reported for complex minerals suchas almandine (Mittal et al 2000b) pyrope (Pavese et al1998) sillimanite and kyanite (Rao et al 1999) orthoen-statite (Choudhury et al 1998) and fayalite (Price et al1991)

With spectrometers such as the MARI instrument atISIS it is now possible to perform inelastic scatteringmeasurements from powdered samples as functions of QThe value of these types of measurements is that theyenable comparisons to be made between the excitationspectra of different phases Fig 6 shows measurements forthe two phases of cristobalite (Dove et al 2000a) It isclear that there is considerable additional scattering at lowenergies (below 5 meV) in the high-temperature disorderedphase This scattering can be associated with the rigid-unitmodes discussed by Swainson amp Dove (1993) Hammondset al (1996) and Dove et al (2000b and c) It has also beenshown by these measurements that the inelastic neutronscattering spectra of silica glass is closely related to thespectra of the two phases of cristobalite (Dove et al 2000aand b)

There has been a considerable effort in the physics andchemistry communities to use inelastic neutron scatteringmethods to study magnetic dynamics which can often bedescribed as spin waves Measurements of spin wavedispersion curves can provide information about the inter-actions between atomic magnetic moments the so-calledexchange interactions There have been comparatively fewinelastic neutron scattering measurements on magneticminerals Spin wave dispersion curves have been reportedfor hematite (Samuelson amp Shirane 1970) and crystalfield magnetic transitions in Co-bearing cordierite andspinel phases have been studied by inelastic neutron scat-tering (Winkler et al 1997)


Fig 5 Inelastic neutron scattering spectra from calcite at the(120ndash2) zone boundary point for several temperatures Theimportant points are the phonon peak that softens on heating andthe additional scattering at lower frequencies that increases dramat-ically on heating (Harris et al 1998)

Fig 6 Inelastic neutron scattering data forthe two phases of cristobalite obtained asfunctions of both scattering vector andenergy transfer High intensities are shownas lighter shading The important point ofcomparison is the additional scattering seenat low energies in the high-temperature bphase The data were obtained on the MARIchopper spectrometer at the ISIS spallationneutron source

MT Dove

32 Coherent elastic scattering and Bragg scattering

The case m = 0 in equation (29) gives the result

F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash

jexp(iQtimesRj)exp(ndashWj)frac122

(35)

This result is independent of time As a result the timeFourier transform gives a delta function

S0(Qw) = ograveF0(Qt)exp(ndashiwt)dt

= 1ndashNfrac12Sj

bndash

j exp(iQtimesRj)exp(ndashWj)frac122d(w) (36)

This result is equivalent to the elastic scattering ie thew = 0 limit of S(Qw) obtained earlier equations (18) and(19) This can be demonstrated by starting from the generalequation for the intermediate structure factor for coherentneutron scattering

Sj

bndash

j ltexp(iQtimesRj)gt = Sj

bndash

jexp(iQtimesRj)ltexp(iQtimesuj)gt= S

jbndash

j exp(iQtimesRj)exp(ndashlt[Qtimesuj]2gt) (37)

This is the normal Bragg diffraction structure factorbut can be generalised to include glasses For glasses itsapplication relies on the fact that the atoms vibrate aroundmean positions that do not change over the time scale of ameasurement This is not the case for fluids and as a resultthere is no elastic scattering for a liquid

Bragg diffraction is commonly used in two modes Themost common is powder diffraction with data analysed usingthe Rietveld refinement method This method is now quitefast on a high-intensity neutron source and resolution canalso be high An example of a high-resolution measurement isshown in Fig 7 Powder diffraction can be used for studyingthe response of a structure to changes in temperature or press-ure particularly if there is a displacive or orderdisorder

phase transition Examples of temperature-induced displacivephase transitions studied by neutron powder diffractioninclude cristobalite (Schmahl et al 1992) leucite (Palmer etal 1997) calcite and the analogue sodium nitrate (Swainsonet al 1997) and lawsonite (Myer et al 2001) Examples ofthe use of neutron powder diffraction for the study of cationordering include the aringkermanitendashgehlenite solid solution(Swainson et al 1992) MnMg ordering in cummingtonite(Reece et al 2000) and cation ordering in other amphiboles(Welch amp Knight 1999) cation ordering in theilmenitendashhematite solid solution (Harrison et al 2000)cation ordering in various spinel phases and spinel solid solu-tions (Harrison et al 1998 1999 Pavese et al 1999ab and2000a Redfern et al 1999) and cation partioning in olivinephases (Henderson et al 1996 Redfern et al 1998 2000)and phengite (Pavese et al 1997 1999c 2000b) Powderdiffraction is an excellent tool for the variation of structurewith temperature for properties such as thermal expansioneg crocoite (Knight 2000) or for studies of dehydrationeg gypsum (Schofield et al 1997) and fluorapophyllite(Stahl et al 1987) As noted in sect12 neutron scattering ismuch more sensitive to scattering from hydrogen ordeuterium than X-rays and because of this neutron powderdiffraction has often been used to study the behaviour ofhydrogen in minerals Examples include analcime (Line etal 1996) laumontite (Stahl amp Artioli 1993) lawsonite(Myer et al 2001) gypsum (Schofield et al 2000) and theamphiboles (Welch amp Knight 1999 Reece et al 2000)Recent exciting developments in high-pressure neutron scat-tering are most advanced in powder diffraction Examples ofsome of the studies that have been carried out include severalstudies of brucite (Parise et al 1993 Catti et al 1995 LeGodec et al 2001 2002) FeSi (Wood et al 1997) hydro-garnet (Lager amp von Dreele 1996) cristobalite (Dove et al2000) muscovite (Catti et al 1994) and Phase A (Kagi et al

212

10 20d-spacing (Aring)

Inte

nsity

Fig 7 High-resolution powder diffraction measurement fromorthoenstatite obtained on the HRPD diffractometer at the ISISspallation neutron source (courtesy of SAT Redfern)

Fig 8 Map of diffuse scattering measured in b-quartz using thePRISMA indirect-geometry spectrometer at the ISIS spallationneutron source (Tucker et al 2001a) The most intense scatteringis shown as the darker regions (such as the scattering from thepoints in the reciprocal lattice) The most important features are thestreaks of diffuse scattering which correspond to low-energy rigidunit phonon modes (Hammonds et al 1996)

2000) It is now possible to perform neutron powder diffrac-tion meaurements at simultaneous high pressures and tempera-tures (Zhao et al 1999 2000 Le Godec et al 2001 2002)

Neutron Bragg diffraction can also be used with singlecrystals (Artioli 2002) Single-crystal diffraction is capableof giving structural information with higher precision thanwith powder diffraction although at the cost of a greatlyincreased time necessary for a single measurement thatprecludes studies at many temperatures or pressuresExamples of single-crystal neutron diffraction measure-ments include anorthite (Ghose et al 1993) beryl (Artioliet al 1995a) pyrope (Artioli et al 1997) diopside(Prencipe et al 2000) enstatite (Ghose et al 1986) cationordering in olivine (Artioli et al 1995b Rinaldi et al2000) and hydrogen bonding in schultenite (Wilson 1994)

In addition to the use of Bragg diffraction for determi-nation and refinement of crystal structures Bragg diffrac-tion can be used to give information about the internal statesof stress within a material This is reviewed by Schaumlfer(2002) A few magnetic neutron diffraction measurementshave been reported These include ilvaite (Ghose et al1984 1990) hedenbergite (Coey amp Ghose 1985) Li-aegerine (Redhammer et al 2001) and the ilmenite-hematite solid solution (Harrison amp Redfern 2001)

33 Coherent diffuse scattering

The coherent diffuse scattering from a crystalline mate-rial is defined as the total diffraction with the Bragg scat-

tering subtracted The diffuse scattering corresponds to ameasurement of S(Qw) at a fixed value of Q integratedover all w

S(Q) = ograveS(Qw)dw (38)

Since F(Qt) is the reverse Fourier transform of S(Qw)the equation for S(Q) is simply the zero-time component ofthe reverse Fourier transform

S(Q) = F(Q0) = 1ndashN S

jkbndash

j bndash

k ltexp(iQtimesrj)exp(ndashiQtimesrk)gt (39)

Expressing this in terms of the density functions andsubtracting the Bragg scattering to give the diffuse scat-tering we obtain

Sdiffuse(Q) = 1ndashN S

jkbndash

j bndash

k (ltrj(Q)rk(ndashQ)gt ndash ltrj(Q)gt ltrk(ndashQ)gt (40)

The diffuse scattering can therefore be seen as arisingfrom instantaneous fluctuations of the atomic density fromthe average density Many aspects of the technique ofneutron diffuse scattering have recently been discussed byNield amp Keen (2001)

There have been far fewer studies of minerals usingdiffuse scattering than diffraction studies An example of ameasurement of diffuse scattering from a single crystal ofquartz heating into the high-temperature b phase is shownin Fig 8 (Tucker et al 2001a) The important features arethe streaks of diffuse scattering which correspond to scat-


893 K

761 K

756 K

01ndash01 00 00 01ndash01 02ndash02

40

41

39

40

41

39

40

41

39

Wave vector (h00) Wave vector (h04)

Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200

300Fig 9 Left maps of scattering from Na2CO3 atvarious temperatures showing the Bragg peaks andthe diffuse scattering associated with the Braggpeaks Right cuts through the Bragg peaks showingthe decline of the intensity in the Bragg peaks oncooling towards the phase transition and the growthof diffuse scattering around the Bragg peaks Thedata have been fitted by a model of this type offerroelastic phase transition The measurements wereobtained by Harris et al (1996) on the PRISMA indi-rect-geometry spectrometer at the ISIS spallationneutron source

MT Dove

tering from low-frequency rigid unit phonon modes(Hammonds et al 1996) and which are less prominent indiffuse scattering measurements from the a phase Themain problem with such large-scale diffuse scatteringmeasurements lies in the interpretation It is very difficultto develop data-based models and such measurements areusually used for comparing with independent model calcu-lations

Diffuse scattering measurements have more success forthe study of specific features as a function of temperaturesuch as critical scattering associated with a phase transi-tion An example is the study of the critical scattering asso-ciated with the phase transition in NaNO3 (Payne et al1997) An interesting example is the diffuse scattering inNa2CO3 associated with the displacive hexagonalndashmono-clinic phase transition at 760 K (Harris et al 1996) aneutron powder diffraction study of the phase transition hasbeen reported by Harris et al (1993) and Swainson et al(1995) Maps of the single-crystal diffuse scattering areshown for several temperatures in Fig 9 together withdetailed cuts through the peaks of diffuse scattering Thisphase transition is an unusual example of a second-orderferroelastic phase transition in which the acoustic softeningtakes place over a plane of wave vectors (the acoustic soft-ening is seen macroscopically as the softening of the c44elastic constant which is isotropic within the plane normalto [001]) It has been shown on general grounds that theferroelastic instability in such as case will lead to a diver-gence of the temperature factors similar to the behaviourin two-dimensional melting (Mayer amp Cowley 1988) Thiswill lead to the Bragg peaks vanishing being replaced bybroad diffuse scattering This effect is clearly seen in thedata shown in Fig 9

34 Coherent total scattering

When dealing with isotropic samples we have toaverage the scattering function over all relative orientationsof Q and r We write rjk = |rj ndash rk| and obtain the orienta-tional average (Q = |Q|)

ltexp(iQtimes[rj ndash rk])gt =1mdash

4p ograve2p

0djograve

p

0sinqdqexp(iQrjkcosq)

= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)

Using this average we obtain

S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash

k sin(Qrjk)Qrjk (42)

where we separate the terms involving the same atoms (theself terms) and those involving interference betweendifferent atoms

Rather than perform a summation over all atoms pairswe can express the equation using pair distribution func-tions First we define gmn(r)dr as the probability of findinga pair of atoms of types m and n with separation betweenr and r + dr This function will have peaks correspondingto specific sets of interatomic distances For example insilica (SiO2) there will be a peak corresponding to theSindashO bond at ~16 Aring a peak corresponding to the OndashObond at ~23 Aring and a third peak corresponding to thenearest-neighbour SihellipSi distance at ~32 Aring g(r) will bezero for all r below the shortest interatomic distance and

214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7

Fig 10 Total scattering dataQi(Q) for a polycrystallinesample of berlinite atambient temperature withdata from different banks ofdetectors being shown sepa-rately The curves throughthe data are fits using aninverse Monte Carlo method(Tucker et al 2002) Thedata were obtained on theGEM diffractometer at theISIS spallation neutronsource

will tend to a value of 1 at large r Thus we can rewriteS(Q) as

S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen

04pr2(gmn(r) ndash1)sinmdash

QrmdashQr

dr (44)

where cm and cn the proportions of atoms of type m and nrespectively and r0 is the number density S0 is determinedby the average density and gives scattering only in the exper-imentally inaccessible limit Q reg 0 It is convenient to writethe pair distribution functions in either of two overall forms

G(r) = Smn

cmcnbndash

mbndash

n(gmn(r) ndash1) = Sn

ij=1cicjb

ndashibndash

j(gij(r) ndash1) (45)

D(r) = 4prr0G(r) (46)

Thus we can write the scattering equations and associ-ated Fourier transforms as

i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)

G(r) = (2p)mdash1mdash

3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)

D(r) = 2ndashp ograve

yen

0Qi(Q)sinQrdQ (50)

This formalism is discussed in more detail by Keen(2001) and Dove et al (2002)

The technique of total scattering is described in more detailelsewhere in this special collection (Dove et al 2002) and byDove amp Keen (1999) To illustrate the point we show total scat-tering data Qi(Q) for berlinite AlPO4 in Fig 10 obtained usingseveral different banks of detectors in the modern instrumentGEM at ISIS Each bank will measure the total scattering signalover a different range of values of Q The low-Q data showBragg peaks superimposed on a background of diffuse scat-tering The high-Q data show only oscillatory diffuse scatteringwhich can be related to the shortest interatomic spacings Theresultant pair distribution function D(r) is shown in Fig 11

The important point about the data shown in Fig 11 isthat it has been possible to resolve separately the peaks in thepair distribution functions associated with the AlO4 and PO4tetrahedra To achieve this resolution it is essential to performtotal scattering measurements to large values of Q (see thediscussion of Dove et al 2002) This is something that canbe achieved using spallation neutron sources such as ISIS

Neutron total scattering experiments have been used instudies of phase transitions in various silica polymorphs(Dove et al 1997 2000b and c Keen amp Dove 1999Tucker et al 2000a 2001a and b) for a comparison of thestructures of crystalline and amorphous silica (Keen ampDove 1999 2000) and for a direct determination of thethermal expansion of the SindashO bond (Tucker et al 2000b)Further examples are given by Dove et al (2002)


ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)

PndashOPndashPAlndashAlOndashOP

OndashOAl

AlndashO

Fig 11 The D(r) pair distribution function for berliniteobtained as the Fourier transform of the total scatteringdata show in Fig 10 using an inverse transform method(Tucker et al 2002) The features in D(r) are associatedwith interatomic distances The resolution on themeasurement is good enough to differentiate between theinteratomic distances associated with the AlO4 and PO4

tetrahedra

MT Dove

35 Small-angle coherent scattering

Most of this article focuses on structural features on theatomic scale but important processes (such as exsolution)have rather longer length scales (of order 10ndash1000 Aring)These produce density fluctuations over a length scalecovering many atoms which lead to interference effects inthe scattered neutron beams Because the length scales arerelatively large the scattering processes occur for smallvalues of Q and hence they are called small-angle scat-tering Typically the signature of small-angle scattering is apeak centred on Q = 0 with a width that is related to thecharacteristic length scale of the long-range density fluctu-ations The strength of the small-angle scattering is deter-mined by the contrast between the mean scattering lengthsof different regions in the sample For this reason small-angle scattering in solids is most easily seen for exsolutionprocesses where the variation in the density across thesample is associated with clustering of different types ofatoms (similarly one important application of small-anglescattering is to study clustering of molecules in solutions)

The simplest analysis of small-angle scattering isthrough the Guinier formula which is appropriate for theexsolution of small particles within a matrix In this limitthe intensity of small angle scattering is

I = I0 exp(ndashQ2R2g 3) (51)

where Rg is the radius of gyration of the particles Forspherical particles of radius R

R2g = 3R 5 (52)

Other formulations of the small-angle scattering lawhave been developed for other mechanisms of density fluc-tuations including cases that cannot be described as parti-cles within a matrix At the time of writing small-anglescattering is a tool that has yet to be routinely exploitedwithin Earth and Mineral Sciences beyond a few casesincluding a study of opals by Graetsch amp Ibel (1997)

4 Formalism for incoherent scattering

41 General considerations

Hydrogen is the ideal atom for the incoherent scatteringof neutrons since the cross section for incoherent scat-tering is large relative to its own coherent scattering crosssection and the cross sections of most atoms likely to beencountered in natural materials Hydrogen often occurs aspart of a molecule such as water or methane Consider firsta single hydrogen atom in a molecule Its instantaneousabsolute position r(t) can be written as

r(t) = R(t) + d(t) + u(t) (53)

where R(t) is the instantaneous position of the centre-of-mass of the molecule (not to be confused with the meanposition for which we used the same symbol earlier) d(t)

is the instantaneous position of the hydrogen atom withrespect to the molecular centre-of-mass assumed to haveconstant magnitude but variable direction and u(t) is thevibrational displacement of the atom from its lsquomeanrsquo posi-tion (magnitudes d raquo u) Note that we are taking account ofthe fact that the motions associated with the vibrationaldisplacement u(t) are much faster than the rotationalmotion of the molecule which is why we can treat thebehaviour of the bond vector d(t) separately from the vibra-tional displacement u(t) equation (14)

We are fundamentally interested in the quantityltexp(iQtimes[r(t) ndash r(0)])gt (equation 14) This can be written as

ltexp(iQtimes[r(t) ndash r(0)])gt =

ltexp(iQtimes[R(t) ndash R(0)]) acute exp(iQtimes[d(t) ndash d(0)])

acute exp(i(Qtimes[u(t) ndash u(0)])gt (54)

If the translational rotational and vibrational motionsare not coupled we can perform the averages on the threeexponential terms separately obtaining the total function(with self-explanatory notation)

Ftotinc(Qt) = Ftrans

inc (Qt) acute Frotinc (Qt) acute Fvib

inc (Qt) (55)

When we take the Fourier transforms we have theconvolution of the incoherent inelastic scattering functionsfor the three types of motion

Stotinc(Qw) = Strans

inc (Qw) Auml Srotinc (Qw) Auml Svib

inc (Qw) (56)

In practice the different types of motion within crystalsoccur on sufficiently different time scales that the effects ofthe convolution do not cause any major problems in theanalysis of incoherent neutron scattering data ndash these prob-lems mostly occur in the study of molecular fluids

42 Atomic diffusion

Incoherent neutron scattering can be used to provideinformation about the motions of individual atoms as theydiffuse through a crystalline or liquid medium which aredescribed by the time dependence of R One example is thediffusion of hydrogen atoms in metals such as palladiumAtomic hydrogen enters the metal in interstitial sites and isable to jump from site to site rather quickly Anotherexample is diffusion within molecular fluids For thesecases incoherent neutron scattering can provide informa-tion about the time constants for the diffusion and the diffu-sion pathways

Consider first the case of isotropic diffusion as in asimple liquid or as a first approximation to a crystal In thiscase there is no potential to worry about The probabilitydistribution function G(r(t)r(0)) in equation (15) whichwe rewrite as G(rt) r = r(t) ndashr(0) is determined by thestandard rate equation for diffusion (Fickrsquos second law)

paraGmdashparat= DNtilde2G (57)

216

where D is the diffusion constant This has the solution

G(rt) = (4pDt)ndash32 exp(ndashr2 4Dt) (58)

where r is the modulus of r This solution is consistent withthe boundary conditions

G(r0) = d(r)

ograveG(rt)dr = 1 for all t (59)

Substituting the solution into the equation for the inter-mediate scattering function (noting that for the present casethe probability p(r) is constant for all values of r) leads tothe result

Finc(Qt) = exp(ndashDQ2t) (60)

where Q is the modulus of Q and N is the number ofatoms Fourier transformation gives the final incoherentscattering function for our example

Sinc(Qw) = 1ndashp (D

mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)

The final scattering function for the case of isotropicdiffusion is thus a single Lorentzian function centred onzero frequency with a frequency width that varies with Q2

and amplitude that varies as Qndash2 The width is also propor-tional to the diffusion constant D and hence the timeconstant associated with the diffusion In general this peakis narrow (by some orders of magnitude) compared tophonon frequencies and the term quasi-elastic scatteringhas been coined to describe such scattering Incoherentquasi-elastic neutron scattering (IQNS) probes time scalesthat are so much slower than the time scales for phononmotion that special instruments have to be designed forthese experiments The typical resolution of an IQNSinstrument will range from 108 to 1010 Hz which is muchfiner than the typical value of 1011 Hz for a triple-axis spec-trometer

For crystalline media the basic model needs to bemodified in two ways Firstly we need to take account ofthe position-dependent potential and secondly we alsoneed to take account of the fact that the diffusion constantwill in general be a tensor rather than a single valuereflecting the fact that the diffusion will be easier in somedirections than in others The mathematics gets far beyondthe scope of this book at this point but three points shouldbe noted Firstly the general form of the Q2-dependentLorentzian term is usually recovered from more compli-cated treatments although the general solution may containa superposition of several Lorentzian functions Howeverby analysing data obtained over a range of values of Qparticularly as Q tends to small values the behaviour of theindividual components (particularly the lowest-ordercomponent) can be deduced Secondly because the diffu-sion constant is a tensor and hence anisotropic the scat-tering function will depend on the orientation of Q ratherthan only upon its magnitude Work with powders willtherefore give an average over all orientations which may

not matter if it is known that diffusion is only significantalong one direction in the crystal Thirdly the existence ofa periodic potential (as found in a crystal) leads to the exis-tence of elastic incoherent scattering when Q is equal to areciprocal lattice vector This elastic scattering containsinformation about the strength of the periodic potential Inmany cases the effect of the potential is to cause thediffusing atoms to be able to exist only on well-definedsites within the crystal and the diffusion proceeds by theatoms jumping between these sites In the limit where thetime taken to jump is much less than the average time thatan atom remains in a site standard jump diffusion modelscan be used in the interpretation of the quasi-elastic scat-tering

The incoherent scattering method has not yet been usedfor studies of diffusion in mineral sciences The main appli-cations could be for the study of hydrogen diffusion inminerals and fluids for which there have been extensivemeasurements on metals Beacutee (1988) describes many of theexperimental and theoretical details including the develop-ment of the formalism beyond the lsquotrivialrsquo case describedabove

43 Collective vibrations and vibrational spectroscopy

Having remarked that the frequency regime for theobservation of translational diffusion is much lower thanfor phonons we should also note that the time dependenceof both R and u contains a contribution from the phononsWe can broadly interpret the time-dependence of R asarising from lattice vibrations in which the molecules moveas rigid bodies giving the lower-frequency modes and thetime-dependence of u as arising from the higher-frequencyinternal vibrations of the molecules These motions can bemeasured if the incoherent spectrum is recorded over theappropriate frequency range The resultant form of Sinc(w)when averaged over Q gives the phonon density of statesweighted by the relevant incoherent neutron cross sectionsFor systems containing hydrogen the spectrum will bedominated by the motions of the hydrogen atoms

Vanadium is an unusual system in that the scatteringlengths for the two spin states are of opposite sign andnearly equal magnitude leading to a very small coherentscattering length (as a result of which vanadium is verysuitable for making sample holders for neutron scatteringas it will not contribute any Bragg scattering to themeasured spectrum) but to a reasonably large incoherentscattering cross section (see Fig 1) The incoherent scat-tering from vanadium is often used to calibrate neutronscattering instruments Vanadium has therefore been usedfor the measurement of the phonon density of states usingincoherent neutron scattering

The fact that hydrogen has a large incoherent crosssection means that incoherent neutron scattering can beused for vibrational spectroscopy We have previouslynoted that there is not a significant wave vector dependencefor the high frequency modes which means that thephonon modes measured as a density of states will havesharp peaks and will not require a good average over scat-tering vectors The advantage of incoherent neutron scat-


MT Dove

tering as a tool for vibrational spectroscopy over light scat-tering methods is that there are no selection rules that causemodes to be unobservable

Although the phonon modes have a much higherfrequency than the energies associated with translationaldiffusion and rotational motions the quasi-elastic scat-tering is convoluted with the vibrational spectrum In prac-tice this convolution gives rise to a DebyendashWaller prefactorto the correct expression for the quasi-elastic intensityexactly as in the case of coherent neutron scattering

An example of the information that can be obtainedusing vibrational incoherent neutron spectroscopy is shownin Fig 12 which gives the spectra for analcime in variousstates of hydration and a comparison with the closelyrelated structure leucite The fact that the intensities of thespectra are reduced as the hydrogen content of the samplesis lowered indicates that the spectra are dominated by theincoherent scattering from the hydrogen atoms The spectrahave two main features The first is a peak around 15 meVwhich is due to whole molecular librations The secondfeature is a broader peak around 50 meV This arises fromthe coupling of the water molecules to the vibrations of thenetwork of SiO4 and AlO4 tetrahedra Similar studies havebeen carried out on bassanite gypsum and cordierite(Winkler amp Hennion 1994 Winkler 1996) and small-pored zeolites (Line amp Kearley 2000)

44 Rotational motions

As a molecule rotates about its centre-of-mass the rela-tive positions of the hydrogen atoms d(t) move over thesurface of a sphere of constant radius d This motion iscalled rotational diffusion as the end point of the vector ddiffuses over the surface of the sphere We will firstconsider the simple illustrative case of isotropic rotational

diffusion of an atom ndash this is a good model for a molecularliquid and is also a good first-approximation for systemssuch as crystalline methane just below the meltingtemperature where experiments indicate that although thepositions of the molecules are ordered on a lattice themolecules are freely rotating We can define the orientationof d by the polar angle W = (qf) The relevant probabilitydistribution function G for equation (15) is now a functionof W(t) and W(0) Formally G(W(t)W(0) is the probabilityof finding the bond orientation at W(t) at time t if it has theorientation W(0) at time 0 and it follows a similar rate lawto the case of translation diffusion

paraGmdashparat= DRNtilde2

WG (62)

where DR is the rotational diffusion constant and itsinverse gives the time constant for reorientational motionsThe differential operator in spherical coordinates is givenas

Ntilde2W =

sinmdash1mdashq

paramdashparaq sin q

paramdashparaq +

sinmdash1mdash

2 qpara2mdashparaf2 (63)

As for the case of translational diffusion the apparentsimplicity of the rate equation belies the complexity of thesolution And as before we can only quote the solution(Beacutee 1988)

G(W(t)W(0)) = 4pSyen

=0exp(ndashDR ( + 1)t)S

+

m=ndashYm(W(t))Y

m (W(0))

(64)

where the functions Ym(W) are the normal sphericalharmonics (the indicates the complex conjugates)Substitution of this expression generates the intermediatescattering function

Finc(Qt)microj20(Qd) + 1ndashpS

yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)

where the functions j are the regular Bessel functions TheFourier transform gives the incoherent scattering function

Sinc(Qw)micro j20(Qd)d(w)

+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)

where d(w) is the Dirac delta function This equation hastwo components The first ( = 0) is an elastic peak centredon w = 0 which will be considered in more detail belowThe second is a superposition of Lorentzian peaks withwidths that are independent of Q The lowest order term inthis second component ( = 1) has a width equal to 2DRand higher order terms have larger widths On the otherhand the amplitudes of the quasi-elastic components aredependent on Q through the Bessel functions

The theory needs to be modified to take account oforientational crystal potentials For example the ammo-nium molecular ion NH+

4 is known to undergo frequentreorientations in the CsCl form of ammonium chlorideThe electrostatic potentials are sufficiently anisotropic toensure that the NndashH bonds lie along the aacute111ntilde directions ndash

218

0 50 100

Mas

s-no

rmal

ised

inte

nsity

Energy transfer (meV)

Fully-hydrated

Dehydrated

Leucite

Partially-deuterated

Fig 12 Incoherent inelastic neutron scattering from samples ofanalcime with various degrees of hydration and compared withscattering from leucite The data were obtained from the TFXAcrystal-analyser spectrometer at the ISIS spallation neutron source(unpublished data of CMB Line and MT Dove reported in Line1995)

there are effectively two independent orientations of theammonium ion on this site This results in a more compli-cated scattering function but the essential details ndash namelythe elastic and the quasielastic components ndash are preservedA complete description of the theory for a range ofcomplex situations is given by Beacutee (1988)

It is often found that the temperature dependence of thewidth of the quasi-elastic component follows an Arrheniusrelation

tndash1 = D = D0exp(ndashERRT) (67)

where ER is the activation energy associated with the rota-tional motion

An example of a quasi-elastic spectrum associated withrotations of water molecules in analcime is shown in Fig13 (Line et al 1994) This spectrum has had the elasticpeak subtracted out The important point from the Fig isthe energy range of the quasi-elastic spectrum which ismuch lower than the energy range of vibrational excitationsseen in Fig 12 It is clear that high resolution is requiredfor quasi-elastic scatterning experiments ndash the varioustypes of technology are reviewed by Beacutee (1988)

The widths of the quasi-elastic spectra for analcimehave been shown to be constant with Q and to increase onheating These features are consistent with rotationalmotions that become faster with increasing temperatureThe Arrhenius plot of the relaxation time (the inverse of thewidth of the quasi-elastic spectrum) is shown in Fig 14The fit to the data gave an activation energy ofERR = 780 plusmn 200 K (Line et al 1994)

45 Elastic incoherent scattering function

The quasi-elastic incoherent scattering gives a means ofmeasuring the time constants for rotational diffusion Theelastic component on the other hand gives informationconcerning the positions of the hydrogen atoms

The elastic component is used to define the elastic inco-herent structure factor (EISF) IEISF(Q)

IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe

(Q)mdash(Q)mdash (68)

where Sel(Q) is the intensity of the incoherent elastic peakand Sqe(Q) is the intensity of the quasi-elastic peak inte-grated over all energies The integration of S(Qw) over allfrequencies is equal to F(Q0) Thus the elastic componentSel(Q) is given by the = 0 term and the quasi-elastic termSQE(Q) is given by the terms for sup3 1 Given that j (x) tendsto the value 0 when x = 0 for all values of sup1 0 we see thatIEISF(Q) has a maximum value of 1 at Q = 0 We also notethat the form of IEISF(Q) is determined only from geometricproperties ndash there is no dependence on the time constantsand hence no dependence on temperature apart from anydependence of the structure on temperature

For isotropic rotational motions of molecules the EISFhas the simple form

IEISF(Q) = sinmdash

QrmdashQr

(69)

where r is the radius of the molecule The model ofisotropic rotational diffusion is usually not realistic formost types of rotational diffusion Usually the rotationaldiffusion is influenced by the rotational potential experi-enced by the molecules which will have the samesymmetry as the local site The development of a model forthe rotational diffusion from which the form of IEISF(Q) canbe calculated is usually extremely difficult and analyticalsolutions only occur for idealised cases such as uniaxialrotation in an n-fold potential or for jump motions betweena small number of well-defined orientations Inevitably it isnecessary to use a model that is an oversimplification ofthe real situation but given that even a simple model willcapture the essential dynamic and geometric aspects of themotion this may not be too much of a problem In the inter-pretation of measurements of IEISF(Q) it is common tocompare the experimental data with the predicted functionobtained from the model in order to assess whether themodel is correct In many cases the calculated form of


ndash100 0 100

Inte

nsity


Fig 13 Quasi-elastic scattering spectrum for analcime The curveis a Lorentzian of halfwidth 30 microeV convoluted with the resolutionfunction The data were obtained from the high-resolution back-scattering IRIS spectrometer at the ISIS spallation neutron source(Line et al 1994)

3030

40

40

35

35

45

20 251000T

ln(t

)

Fig 14 Arrhenius plot of the relaxation time for water reorienta-tions in analcime obtained from the quasi-elastic spectra (Line etal 1994)

MT Dove

IEISF(Q) does not have any free parameters ndash the parametersin the model are quantities such as the bond lengths whichare known in advance

The biggest problem in extracting IEISF(Q) from experi-mental data is accounting for multiple scattering Becausethe incoherent cross section for hydrogen is so large thereis always a high probability that any neutron will be scat-tered more than once whilst travelling through the sampleThe effect of this will be to reduce the size of the elasticcomponent and hence of IEISF(Q) The existence ofmultiple scattering will be revealed if the extracted form ofIEISF(Q) tends towards a limiting value lower than 1 forsmall values of Q Complicated methods are available foraccounting for multiple scattering but a better approach isto reduce the size of multiple scattering in the experimentby the use of thin samples and to measure spectra only fortransmission with short path lengths or reflection It is theproblem of multiple scattering that has discouraged the useof single crystal samples given that the best single crystalsamples will ideally be in the form of a thin flat plate withthe large faces being coincident with a crystal plane ofinterest It is clear that there should be a higher informationcontent if single crystals are used in that for rotationaldiffusion in an anisotropic potential it is expected that theelastic incoherent structure factor will have a dependenceon the direction of the scattering vector Q as well as on itsmagnitude Despite the experimental difficulties somemeasurements of IEISF(Q) have been obtained from singlecrystals

The measured EISF for analcime is shown in Fig 15(Line et al 1994) The data are fitted by two types ofcurves The first is a standard isotropic rotational EISF asgiven by equation (69) with the effective radius beingfitted at each temperature and reproduced in the figure Thesecond type of curve is again based on the standardisotropic rotational EISF with a molecular radius of 088 Aringbut allowing for the possibility that only a fraction a of thewater molecules participate in the rotational motions Thedata cannot give a judgement as to the most appropriatemodel For small molecules such as water it is necessary to

extend the measurements to higher values of Q but thereare then difficulties with overlapping Bragg peaks

The only other study of quasi-elastic scattering fromrotations of water molecules in minerals was carried out byWinkler et al (1994) on hydrous cordierite There isconsiderable scope for developing this line of work withminerals that contain molecular water Quasielastic scat-tering experiments provide unique information about thedynamics of water molecules both in respect to the timescales for motion and the geometric changes associatedwith these motions

6 Summary

The main point that I have sought to convey in thisreview article is that neutron scattering has a tremendousversatility that can be exploited for studies of mineralbehaviour This versatility is being constantly extended asnew neutron sources and new instrumentation come intooperation The focus of this review has been on how thisversatility arises from the fact that the characteristic lengthand energy scales of neutron beams are closely matched tothose of the atoms in condensed matter enabling neutronscattering to provide information about structure anddynamics These two aspects are encapsulated within thegeneral formulation of the neutron scattering law Theversatility of neutron scattering is further enhanced by theunique possibility to separate out both coherent and inco-herent scattering

We have described in detail a selection of the neutronscattering techniques that are routinely used in the widerscience communities and which have been more-or-lesspicked up by the Earth and Mineral Sciences communityThese include coherent inelastic scattering for single crys-tals and polycrystalline samples diffraction on powdersand single crystals single-crystal diffuse scattering totalscattering from polycrystalline materials vibrational inco-herent scattering and incoherent quasi-elastic scatteringThe review also touches on magnetic scattering (diffraction

220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring

r = 043 Aringr = 041 Aring

r = 032 Aring

a = 019

a = 026

a = 032 Fig 15 The EISF for analcime obtained atthree temperatures (squares are 260 K filledcircles are 320 K and open circles are 373 K)The curves marked with a value for the molecu-lar radius r are fits with the model EISF forisotropic rotation with a variable effectiveradius for rotation The curves marked with avalue of a are for a model EISF with fixed rota-tion radius (088 Aring) but with varying fraction aof the molecules that participate in the rota-tional motions (Line et al 1994)

and inelastic) and small-angle scattering both of which areripe for exploitation This review has not touched on someother applications such as surface reflectometry and deepinelastic scattering (otherwise known as neutron Comptonscattering) which again are ripe for exploitation

Acknowledgements I am pleased to acknowledge collab-oration in neutron scattering experiments with many peopleover the past few years These include Brian Powell and IanSwainson (Chalk River) David Keen and Mark Harris(ISIS) Mark Hagen (Keele) Bjoumlrn Winkler (Kiel)Christina Line Matthew Tucker and Simon Redfern(Cambridge) I am grateful for financial support fromEPSRC to support these collaborations

References

Artioli G (2002) Single crystal neutron diffraction Eur JMineral 14 233ndash239

Artioli G Rinaldi R Wilson CC Zanazzi PF (1995a) Single-crystal pulsed-neutron diffraction of a highly hydrous berylActa Cryst B 51 733ndash737

mdash mdash mdash mdash (1995b) Highndashtemperature FendashMg cation parti-tioning in olivine in situ single-crystal neutron-diffractionstudy Am Mineral 80 197ndash200

Artioli G Besson JM Dove MT Geiger CA Rinaldi RSchaefer W (1996a) Earth Sciences in ldquoScientific Prospectsfor Neutron Scattering with Present and Future Sources (ESFFramework Studies into Large Research Facilities)rdquo ESF(Strasbourg) 94ndash108

Artioli G Pavese A Moze O (1996b) Dispersion relations ofacoustic phonons in pyrope garnet Relationship between vibra-tional properties and elastic constants Am Mineral 81 19ndash25

Artioli G Pavese A Stahl K McMullan RK (1997) Single-crystal neutron-diffraction study of pyrope in the temperaturerange 30ndash1173 K Can Mineral 35 1009ndash1019

Ashcroft NW amp Mermin ND (1976) Solid State Physics HoltRinehart and Winston New York 826 p

Beacutee M (1988) Quasielastic neutron scattering principles andapplications in solid state chemistry biology and materialsscience Adam Hilger Bristol 437 p

Besson JM amp Nelmes RJ (1995) New developments in neutron-scattering methods under high pressure with the Paris-Edinburgh cells Physica B 213 31ndash36

Bethke J Dolino G Eckold G Berge B Vallade M ZeyenCME Hahn T Arnold H Moussa F (1987) Phonondispersion and mode coupling in highndashquartz near the incom-mensurate phasendashtransition Europhys Lett 3 207ndash212

Bethke J Eckold G Hahn T (1992) The phonon dispersion andlattice dynamics of a-AlPO4 an inelastic neutronndashscatteringstudy J Phys Condensed Matter 4 5537ndash5550

Boysen H (1990) Neutron scattering and phase transitions inleucite in ldquoPhase transitions in ferroelastic and co-elastic crys-talsrdquo Cambridge University Press Cambridge 334ndash349

Boysen H Dorner B Frey F Grimm H (1980) Dynamic struc-ture determination for two interacting modes at the M point ina- and b-quartz by inelastic neutron scattering J Phys C SolidState Phys 13 6127ndash6146

Castellano EE amp Main P (1985) On the classical interpretationof thermal probability ellipsoids and the DebyendashWaller factorActa Cryst A 41 156ndash157

Catti M Ferraris G Hull S Pavese A (1994) Powder neutron-diffraction study of 2M1 muscovite at room pressure and at 2GPa Eur J Mineral 6 171ndash178

mdash mdash mdash mdash (1995) Static compression and H-disorder in bruciteMg(OH)2 to 11 GPa ndash a powder neutron-diffraction studyPhys Chem Minerals 22 200ndash206

Chaplot SL Choudhury N Ghose S Rao MN Mittal RGoel P (2002) Inelastic neutron scattering and latticedynamics of minerals Eur J Mineral 14 291ndash329

Choudhury N Ghose S Chowdhury CP Loong CK ChaplotSL (1998) Lattice dynamics Raman spectroscopy andinelastic neutron scattering of orthoenstatite Mg2Si2O6 PhysRev B 58 756ndash765

Coey JMD amp Ghose S (1985) Magnetic order in hedenbergiteCaFeSi2O6 Solid State Comm 53 143ndash145

Cowley ER amp Pant AK (1970) Lattice dynamics of calcitePhys Rev B8 4795ndash4800

Dolino G Berge B Vallade M Moussa F (1992) Origin of theincommensurate phase of quartz 1 Inelastic neutron-scatteringstudy of the high-temperature b-phase of quartz J Phys I 21461ndash1480

Dove MT (1993) Introduction to lattice dynamics CambridgeUniversity Press Cambridge 258 p

Dove MT amp Keen DA (1999) Atomic structure of disorderedmaterials in ldquoMicroscopic properties and processes inmineralsrdquo (ed CRA Catlow and K Wright) 371ndash387

Dove MT amp Powell BM (1989) Neutron diffraction study of thetricritical orientational orderdisorder phase transition in calciteat 1260 K Phys Chem Minerals 16 503ndash507

Dove MT Hagen ME Harris MJ Powell BMSteigenberger U Winkler B (1992) Anomalous inelasticneutron scattering from calcite J Phys Condensed Matter 42761ndash2774

Dove MT Keen DA Hannon AC Swainson IP (1997)Direct measurement of the SindashO bond length and orienta-tional disorder in b-cristobalite Phys Chem Minerals 24311ndash317

Dove MT Craig MS Keen DA Marshall WG RedfernSAT Trachenko KO Tucker MG (2000a) Crystal struc-ture of the high-pressure monoclinic phase-II of cristobaliteSiO2 Min Mag 64 569ndash576

Dove MT Hammonds KD Harris MJ V Heine Keen DAPryde AKA Trachenko K Warren MC (2000b)Amorphous silica from the Rigid Unit Mode approach MinMag 64 377ndash388

Dove MT Tucker MG Keen DA (2002) Neutron total scat-tering method simultaneous determination of long-range andshort-range order in disordered materials Eur J Mineral 14331ndash348

Dove MT Trachenko KO Tucker MG Keen DA (2000c)Rigid Unit Modes in framework structures theory experimentand applications Rev Mineral Geochem 39 1ndash33

Fowler PH amp Taylor AD (1987) Temperature imaging usingepithermal neutrons Rutherford Appleton Laboratory ReportRAL-87-056

Frost JC Meehan P Morris SR Ward RC Mayers J (1989)Non-intrusive temperature-measurement of the components ofa working catalyst by neutron resonance radiography CatalysisLetters 2 97ndash104

Ghose S Hewat AW Marezio M (1984) A neutron powderdiffraction study of the crystal and magneticndashstructures ofilvaite from 305 K to 5 K a mixed-valence iron silicate with anelectronicndashtransition Phys Chem Minerals 11 67ndash74


MT Dove

Ghose S Schomaker V Mcmullan RK (1986) EnstatiteMg2Si2O6 a neutron diffraction refinement of the crystal struc-ture and a rigid-body analysis of the thermal vibration ZeitKristall 176 159ndash175

Ghose S Hewat AW Pinkney M (1990) A powder neutrondiffraction study of magnetic phasendashtransitions and spin frus-tration in ilvaite a mixed-valence iron silicate showing a semi-conductorndashinsulator transition Solid State Comm 74413ndash418

Ghose S Hastings JM Choudhury N Chaplot SL Rao KR(1991) Phonon dispersion relation in fayalite Fe2SiO4 PhysicaB 174 83ndash86

Ghose S Mcmullan RK Weber HP (1993) Neutron diffractionstudies of the P1

ndashndashI1

ndashtransition in anorthite CaAl2Si2O8 and the

crystal structure of the body-centered phase at 514 K ZeitKristall 204 215ndash237

Graetsch H amp Ibel K (1997) Small angle neutron scattering byopals Phys Chem Minerals 24 102ndash108

Hammonds KD Dove MT Giddy AP Heine V WinklerB (1996) Rigid unit phonon modes and structural phasetransitions in framework silicates Am Mineral 81 1057ndash1079

Harris MJ Cowley RA Swainson IP Dove MT (1993)Observation of lattice melting at the ferroelastic phase transi-tion in Na2CO3 Phys Rev Lett 71 2939ndash2942

Harris MJ Dove MT Godfrey KW (1996) A single crystalneutron scattering study of lattice melting in ferroelasticNa2CO3 J Phys Condensed Matter 8 7073ndash7084

Harris MJ Dove MT Swainson IP Hagen ME (1998a)Anomalous dynamical effects in calcite CaCO3 J PhysCondensed Matter 10 L423ndash429

Harrison RJ amp Redfern SAT (2001) Short- and long-rangeordering in the ilmenite-hematite solid solution Phys ChemMinerals 28 399ndash412

Harrison RJ Redfern SAT OrsquoNeill HSC (1998) Thetemperature dependence of the cation distribution in synthetichercynite (FeAl2O4) from in situ neutron structure refinementsAm Mineral 83 1092ndash1099

Harrison RJ Dove MT Knight KS Putnis A (1999) In situneutron diffraction study of nonndashconvergent cation ordering inthe (Fe3O4)1ndashx(MgAl2O4)x spinel solid solution Am Mineral84 555ndash563

Harrison RJ Redfern SAT Smith RI (2000) Inndashsitu study ofthe R3

ndashto R3

ndashc phase transition in the ilmenitendashhematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194ndash205

Henderson CMB Knight KS Redfern SAT Wood BJ(1996) High-temperature study of octahedral cation exchangein olivine by neutron powder diffraction Science 2711713ndash1715

Kagi H Parise JB Cho H Rossman GR Loveday JS(2000) Hydrogen bonding interactions in phase A[Mg7Si2O8(OH)6] at ambient and high pressure Phys ChemMinerals 27 225ndash233

Keen DA (2001) A comparison of various commonly used corre-lation functions for describing total scattering J Appl Cryst34 172ndash177

Keen DA amp Dove MT (1999) Comparing the local structures ofamorphous and crystalline polymorphs of silica J PhysCondensed Matter 11 9263ndash9273

mdash mdash (2000) Total scattering studies of silica polymorphs simi-larities in glass and disordered crystalline local structure MinMag 64 447ndash457

Knight KS (2000) A high temperature structural phase transitionin crocoite (PbCrO4) at 1068 K crystal structure refinement at1073 K and thermal expansion tensor determination at 1000 KMin Mag 64 291ndash300

Lager GA amp von Dreele RB (1996) Neutron powder diffractionstudy of hydrogarnet to 90 GPa Am Mineral 81 1097ndash1104

Le Godec Y Dove MT Francis DJ Kohn SC Marshall WGPawley AR Price GD Redfern SAT Rhodes N RossNL Schofield PF Schooneveld E Syfosse G TuckerMG Welch MD (2001) Neutron diffraction at simultaneoushigh temperatures and pressures with measurement of temper-ature by neutron radiography Min Mag 65 749ndash760

Le Godec Y Dove MT Redfern SAT Marshall WG TuckerMG Syfosse G Besson JM (2002) A new high P-T cell forneutron diffraction up to 7 GPa and 2000 K with measurementof temperature by neutron radiography High PressureResearch 65 737-748

Line CMB amp Kearley GJ (2000) An inelastic incoherentneutron scattering study of water in smallndashpored zeolites andother water-bearing minerals J Chem Phys 112 9058ndash9067

Line CMB Winkler B Dove MT (1994) Quasielastic inco-herent neutron scattering study of the rotational dynamics of thewater molecules in analcime Phys Chem Minerals 21451ndash459

Line CMB Dove MT Knight KS Winkler B (1996) Thelow-temperature behaviour of analcime I High-resolutionneutron powder diffraction Min Mag 60 499ndash507

Mayer AP amp Cowley RA (1988) The continuous melting transi-tion of a 3-dimensional crystal at a planar elastic instability JPhys C Solid State Physics 21 4827ndash4834

Mayers J Baciocco G Hannon AC (1989) Temperature-measurement by neutron resonance radiography Nucl InstrMethods Phys Research A 275 453ndash459

Mittal R Chaplot SL Parthasarathy R Bull MJ Harris MJ(2000a) Lattice dynamics calculations and phonon dispersionmeasurements of zircon ZrSiO4 Phys Rev B 6212089ndash12094

Mittal R Chaplot SL Choudhury N Loong CK (2000b)Inelastic neutron scattering and lattice dynamics studies ofalmandine Fe3Al2Si3O12 Phys Rev B 61 3983ndash3988

Myer H-W Marion S Sondergeld P Carpenter MA KnightKS Redfern SAT Dove MT (2001) Displacive compo-nents of the phase transitions in lawsonite Am Mineral 86566ndash577

Nield VM amp Keen DA (2001) Diffuse neutron scattering fromcrystalline materials Oxford University Press Oxford 317 p

Palmer DC Dove MT Ibberson RM Powell BM (1997)Structural behavior crystal chemistry and phase transitions insubstituted leucites Am Mineral 82 16ndash30

Parise JB Leinenweber K Weidner DJ Tan K von DreeleRB (1993) Pressure-induced H-bonding ndash neutron-diffractionstudy of brucite Mg(OD)2 to 93 GPa Am Mineral 79193ndash196

Pavese A (2002) Neutron powder diffraction and Rietveld anal-ysis applications to crystal chemical studies of minerals at non-ambient conditions Eur J Mineral 14 241ndash249

Pavese A Ferraris G Prencipe M Ibberson R (1997) Cationsite ordering in phengite 3T from the DorandashMaira massif(western Alps) a variable-temperature neutron powder diffrac-tion study Eur J Mineral 9 1183ndash1190

Pavese A Artioli G Moze O (1998) Inelastic neutron scatteringfrom pyrope powder experimental data and theoretical calcula-tions Eur J Mineral 10 59ndash69

222

Pavese A Artioli G Hull S (1999a) In situ powder neutrondiffraction of cation partitioning vs pressure in Mg094Al204O4

synthetic spinel Am Mineral 84 905ndash912Pavese A Artioli G Russo U Hoser A (1999b) Cation parti-

tioning versus temperature in (Mg070Fe023)Al197O4 syntheticspinel by in situ neutron powder diffraction Phys ChemMinerals 26 242ndash250

Pavese A Ferraris G Pischedda V Ibberson R (1999c)Tetrahedral order in phengite 2M(1) upon heating frompowder neutron diffraction and thermodynamic consequencesEur J Mineral 11 309ndash320

Pavese A Artioli G Hoser A (2000a) MgAl2O4 syntheticspinel cation and vacancy distribution as a function of temper-ature from in situ neutron powder diffraction Zeit Kristall215 406ndash412

Pavese A Ferraris G Pischedda V Radaelli P (2000b) Furtherstudy of the cation ordering in phengite 3T by neutron powderdiffraction Min Mag 64 11ndash18

Payne SJ Harris MJ Hagen ME Dove MT (1997) Aneutron diffraction study of the orderndashdisorder phase transi-tion in sodium nitrate J Phys Condensed Matter 92423ndash2432

Prencipe M Tribaudino M Pavese A Hoser A Reehuis M(2000) Single-crystal neutron diffraction investigation of diop-side at 10 K Can Mineral 38 183ndash189

Price DL Ghose S Choudhury N Chaplot SL Rao KR(1991) Phonon density of states in fayalite Fe2SiO4 PhysicaB 174 87ndash90

Rao KR Chaplot SL Choudhur y N Ghose S Hastings JMCorliss LM Price DL (1988) Latticendashdynamics andinelastic neutronndashscattering from forsterite Mg2SiO4 phonondispersion relation density of states and specific heat PhysChem Minerals 16 83ndash97

Rao MN Chaplot SL Choudhury N Rao KR Azuah RTMontfrooij WT Bennington SM (1999) Lattice dynamicsand inelastic neutron scattering from sillimanite and kyaniteAl2SiO5 Phys Rev B 60 12061ndash12068

Redfern SAT (2002) Neutron powder diffraction of minerals athigh pressures and temperatures some recent technical devel-opments and scientific applications Eur J Mineral 14251ndash261

Redfern SAT Knight KS Henderson CMB Wood BJ(1998) FendashMn cation ordering in fayalitendashtephroite(FexMn1ndashx)2SiO4 olivines a neutron diffraction study MinMag 62 607ndash615

Redfern SAT Harrison RJ OrsquoNeill HSC Wood DRR(1999) Thermodynamics and kinetics of cation ordering inMgAl2O4 spinel up to 1600degC from in situ neutron diffractionAm Mineral 84 299ndash310

Redfern SAT Artioli G Rinaldi R Henderson CMBKnight KS Wood BJ (2000) Octahedral cation ordering inolivine at high temperature II an in situ neutron powderdiffraction study on synthetic MgFeSiO4 (Fa50) Phys ChemMinerals 27 630ndash637

Redhammer GJ Roth G Paulus W Andre G Lottermoser WAmthauer G Treutmann W Koppelhuber-Bitschnau B(2001) The crystal and magnetic structure of Li-aegirineLiFe3+Si2O6 a temperature-dependent study Phys ChemMinerals 28 337ndash346

Reece JJ Redfern SAT Welch MD Henderson CMB(2000) MnndashMg disordering in cummingtonite a high-temper-ature neutron powder diffraction study Min Mag 64255ndash266

Rinaldi R (2002) Neutron scattering in mineral sciences prefaceEur J Mineral 14 195ndash202

Rinaldi R Artioli G Wilson CC McIntyre G (2000)Octahedral cation ordering in olivine at high temperature I insitu neutron singlendashcrystal diffraction studies on naturalmantle olivines (Fa12 and Fa10) Phys Chem Minerals 27623ndash629

Samuelson EJ amp Shirane G (1970) Inelastic neutron scatteringinvestigation of spine waves and magnetic interactions in a-Fe2O3 Physica Status Solidi 42 241ndash256

Schaumlfer W (2002) Neutron diffraction applied to geologicaltexture and stress analysis Eur J Mineral 14 263ndash289

Schmahl WW Swainson IP Dove MT Graeme-Barber A(1992) Landau free energy and order parameter behaviour ofthe andashb phase transition in cristobalite Zeit Kristall 201125ndash145

Schober H Strauch D Dorner B (1993) Lattice dynamics ofsapphire (Al2O3) Zeit Physik B Condensed Matter 92273ndash283

Schofield PF Stretton IC Knight KS Hull S (1997) Powderneutron diffraction studies of the thermal expansion compress-ibility and dehydration of deuterated gypsum Physica B 234942ndash944

Schofield PF Wilson CC Knight KS Stretton IC (2000)Temperature related structural variation of the hydrous compo-nents in gypsum Zeit Kristall 215 707ndash710

Staringhl K amp Artioli G (1993) A neutron powder diffraction studyof fully deuterated laumontite Eur J Mineral 5 851ndash856

Staringhl K Kvick A Ghose S (1987) A neutron diffraction andthermogravimetric study of the hydrogen bonding and dehydra-tion behavior in fluorapophyllite KCa4(Si8O20)F8H2O and itspartially dehydrated form Acta Cryst B 43 517ndash523

Strauch D amp Dorner B (1993) Lattice dynamics of a-quartz 1Experiment J Phys Condensed Matter 5 6149ndash6154

Swainson IP amp Dove MT (1993) Low-frequency floppy modesin b-cristobalite Phys Rev Lett 71 193ndash196

Swainson IP Dove MT Schmahl WW Putnis A (1992)Neutron powder diffraction study of the AringkermanitendashGehlenitesolid solution series Phys Chem Minerals 19 185ndash195

Swainson IP Dove MT Harris MJ (1995) Neutron powderdiffraction study of the ferroelastic phase transition in sodiumcarbonate J Phys Condensed Matter 7 4395ndash4417

mdash mdash mdash (1997) The phase transitions in calcite and sodiumnitrate Physica B 241 397ndash399

Tucker MG Dove MT Keen DA (2000a) Simultaneous anal-yses of changes in long-range and short-range structural orderat the displacive phase transition in quartz J Phys CondensedMatter 12 L723ndashL730

mdash mdash mdash (2000b) Direct measurement of the thermal expansionof the SindashO bond by neutron total scattering J PhysCondensed Matter 12 L425ndashL430

Tucker MG Keen DA Dove MT (2001a) A detailed struc-tural characterisation of quartz on heating through the andashbphase transition Min Mag 65 489ndash507

Tucker MG Squires MD Dove MT Keen DA (2001b)Dynamic structural disorder in cristobalite Neutron total scat-tering measurement and Reverse Monte Carlo modelling JPhys Condensed Matter 13 403ndash423

Welch MD amp Knight KS (1999) A neutron powder diffractionstudy of cation ordering in highndashtemperature synthetic amphi-boles Eur J Mineral 11 321ndash331

Willis BTM amp Pryor AW (1975) Thermal vibrations in crystal-lography Cambridge University Press Cambridge


MT Dove

Wilson CC (1994) Structural studies of schultenite in the temper-ature range 125ndash324 K by pulsed single-crystal neutron diffrac-tion hydrogen ordering and structural distortions Min Mag58 629ndash634

Winkler B (1996) The dynamics of H2O in minerals Phys ChemMinerals 23 310ndash318

mdash (2002) Neutron sources and instrumentation Eur J Mineral14 225ndash232

Winkler B amp Buehrer M (1990) The lattice dynamics ofandalusite prediction and experiment Phys Chem Minerals17 453ndash461

Winkler B amp Hennion B (1994) Low-temperature dynamics ofmolecular H2O in bassanite gypsum and cordierite investigatedby high-resolution incoherent inelastic neutronndashscatteringPhys Chem Minerals 21 539ndash545

Winkler B Coddens G Hennion B (1994) Movement ofchannel H2O in cordierite observed with quasindashelasticneutronndashscattering Am Mineral 79 801ndash808

Winkler B Harris MJ Eccleston RS Knorr K Hennion B(1997) Crystal field transitions in Co2[Al4Si5]O18 cordierite

and CoAl2O4 spinel determined by neutron spectroscopy PhysChem Minerals 25 79ndash82

Winkler B Knorr K Kahle A Vontobel P Hennion BLehmann E Bayon G (2002) Neutron imaging and neutrontomography as non-destructive tools to study bulk rocksamples Eur J Mineral 14 349ndash354

Wood IG David WIF Hull S Price GD (1996) A high-pres-sure study of e-FeSi between 0 and 85 GPa by time-of-flightneutron powder diffraction J Appl Cryst 29 215ndash218

Zhao YS von Dreele RB Morgan JG (1999) A high P-T cellassembly for neutron diffraction up to 10 GPa and 1500 KHigh Pressure Research 16 161ndash177

Zhao YS Lawson AC Zhang JZ Bennett BI von DreeleRB (2000) Thermoelastic equation of state of molybdenumPhys Rev B 62 8766ndash8776

Received 29 July 2001Modified version received 12 September 2001Accepted 6 November 2001

224

MT Dove

weak although there are notable (and generally useful)exceptions This is in contrast with other probes such asconventional X-ray diffraction

The strength of the scattering from an atomic nucleus isdefined in either of two ways The first is the formal equiv-alence of the atomic scattering factor which for neutronscattering is called the scattering length (and with units oflength) and it is usually given the symbol b The seconddefinition is as a scattering cross section with symbol s(and with units of area) By convention the cross section islinked to the scattering length by s = 4pb2 The use of thescattering cross section is useful for comparing differenttypes of interaction between the neutron and the nucleus

Neutron scattering processes

The scattering interaction between the neutron andatomic nucleus takes place over extremely short lengthscales of the order of 10ndash15 m By contrast the interactionbetween X-rays and the atomic charge distribution operatesat the length scale of the atom namely around 10ndash10 m(1 Aring) The amplitude of a scattered beam of radiation varieswith scattering vector Q as the inverse of the size of thelength scale involved in the scattering process For X-raydiffraction this means that there is a considerable reductionin the amplitude of scattering over the range of values of Qin a typical experiment (eg 0ndash6 Aringndash1) On the other handthe small distances involved in neutron scattering meanthat there is no appreciable change in the scattering lengthover the same range of Q Together with the high resolutionnow possible with neutron powder diffraction instrumenta-tion the independence of the scattering length on Q hashelped neutron powder diffraction to become particularlyvaluable in determining crystal structures of minerals andthe variation of structure with changing temperature andorpressure This is important because the crystal structures ofminerals can be relatively complicated in comparison withother inorganic crystals Neutron powder diffraction is nowan established tool for the study of subtle changes in struc-ture associated with phase transitions as reviewed byPavese (2002) and Redfern (2002)

Unlike nuclear scattering the scattering of neutrons bythe magnetic moments of atoms does have a significantdependence on Q The magnetisation of an atom arisesfrom the unpaired electrons in the atoms and the magneti-sation density extends across the same length scale as thecharge distribution This means that the scattering lengthfor magnetic scattering varies with Q in a manner similar tothat for X-ray scattering

There are two main types of neutron scatteringprocesses The first is coherent scattering which is thescattering process in which the scattering from each atomof the same type is independent of the nuclear isotope andspin state In effect coherent scattering processes averageover isotope number and spin state The second type ofinteraction is incoherent scattering which accounts fordifferences in scattering from atoms of the same type dueto differences in isotope or spin state Clearly any atomwith a nucleus of zero spin and with only one isotope willhave a zero incoherent cross section The separation of

coherent and incoherent scattering processes is describedin more detail in sect22

Neutron scattering cross sections are actually relativelyweak Unfortunately this is coupled with the fact that eventhe most advanced neutron sources have relatively lowfluxes of neutrons as compared to X-ray laser or electronradiation sources (see Winkler 2002 for a discussion ofsources of neutrons) This means that neutron scatteringexperiments may be time consuming and the high costs ofproducing and detecting a neutron beam (Winkler 2002)mean that neutron scattering experiments are relativelyexpensive However these negative points are more thanoffset by the unique information that can be given byneutron scattering as will be highlighted in this and theother reviews in this collection of papers

Neutron absorption processes

The other type of interaction between neutrons andatomic nuclei is absorption In many cases the crosssection of neutron absorption is quite low However thereare some nuclei that have relatively large absorption crosssections These can occur either as resonances or as anabsorption process that varies as the inverse of the neutronvelocity Two examples of common neutron-absorbingmaterials are boron and cadmium Both of these are oftenused in neutron scattering experiments because of theirabsorbing power usually as components to mask the trajec-tory of the incoming and scattering neutron beam in orderto avoid measurements of neutrons scattering from else-where other than the sample

The low absorption of neutron beams passing throughmost materials means that neutrons can penetrate throughthe walls of sample environment equipment This facili-tates the development of such equipment without needingto compromise the quality of the types of experiment thatcan be performed For example good control at high andlow temperatures is a routine operation in a neutron scat-tering experiment (Pavese 2002 Redfern 2002)Traditionally high pressures have been easier to work withusing X-ray diffraction and diamond anvil cells but the useof time-of-flight neutron scattering techniques (Winkler2002) have facilitated considerable progress in neutronscattering at high pressures (Besson amp Nelmes 1995) It isnow also possible to perform neutron diffraction measure-ments at simultaneous high pressures and temperatures(Zhao et al 1999 2000 Le Godec et al 2001 2002) asreviewed by Redfern (2002)

The usually low absorption of neutron beams alsomeans that neutrons will interact with the bulk of thesample under study In some circumstances there can bemeasurable differences between the behaviour of a materialat its surface and the behaviour in the bulk (displacivephase transitions most often give this effect) In such casesconventional X-ray and neutron diffraction experimentsmay give different results if the results of a conventional X-ray diffraction experiment are dominated by scatteringfrom the surface

Neutron absorption processes are essential for the oper-ation of neutron detectors Common absorbing materials in

204










- 5

0

5

1 0

1 5

2 0

ndash5 0

0

5 0

1 0 0

1 5 0

1 0 ndash 4

1 0 4

1 0 0

1 0 ndash 2

1 0 2

0 2 0 4 0 6 0 80 100


Co

he

ren

t s

ca

tte

rin

g l

en

gth

(fm

)S

ca

tte

rin

g c

ros

s s

ec

tio

n (

ba

rns

)A

bs

orp

tio

n c

ros

s s

ec

tio

n (

ba

rns

)


C o h er e n t

In c oh e r e n t


MT Dove












E = w (1)





206




F(Qt) = 1ndashN S


= 1ndashNS

jkbjmdash





= 1ndashNograveS

jkbjmdash




jkbjmdash










jkbjndash



Finc(Qt) = 1ndashNS

jk(bjmdash

bk ndash bjndash

bkndash




MT Dove



bjmdash

bk = bjndash

bkndash

for j sup1 k (11)


Finc(Qt) = 1ndashNS

j(bj



sjcoh = 4pbj

ndash 2

sjinc = 4p(bj



Finc(Qt) =1mdash

4pNSjsj



Finc(Qt) =1mdash

4pNSjsj


G(r(t)r(0))p(r)dr(t)dr(0) (15)













F(Qyen) = 1ndashN S


= 1ndashNS

jkbjmdash




jbndash

j ltrj(Q)gtfrac122

(20)


208

time frequency

F(Q

t)

S(Q

w)

Fouriertransform

constantcomponent



component of F(Qt)








(21)











rj(t) = Rj + uj(t) (25)



F(Qt) = 1ndashN S

jkbndash

jbndash


(26)





(27)






Sm=0




j)12Skn



S1(Qw) = 1ndashN S

jkbndash

jbndash



= 1ndashN S

n 2w(mdash mdash




Fn(Q) = Sj

bndash

jmdashmj



MT Dove


n(w) = expmdash

( w mdash1mdash








210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)



00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector










MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224










- 5

0

5

1 0

1 5

2 0

ndash5 0

0

5 0

1 0 0

1 5 0

1 0 ndash 4

1 0 4

1 0 0

1 0 ndash 2

1 0 2

0 2 0 4 0 6 0 80 100


Co

he

ren

t s

ca

tte

rin

g l

en

gth

(fm

)S

ca

tte

rin

g c

ros

s s

ec

tio

n (

ba

rns

)A

bs

orp

tio

n c

ros

s s

ec

tio

n (

ba

rns

)


C o h er e n t

In c oh e r e n t


MT Dove












E = w (1)





206




F(Qt) = 1ndashN S


= 1ndashNS

jkbjmdash





= 1ndashNograveS

jkbjmdash




jkbjmdash










jkbjndash



Finc(Qt) = 1ndashNS

jk(bjmdash

bk ndash bjndash

bkndash




MT Dove



bjmdash

bk = bjndash

bkndash

for j sup1 k (11)


Finc(Qt) = 1ndashNS

j(bj



sjcoh = 4pbj

ndash 2

sjinc = 4p(bj



Finc(Qt) =1mdash

4pNSjsj



Finc(Qt) =1mdash

4pNSjsj


G(r(t)r(0))p(r)dr(t)dr(0) (15)













F(Qyen) = 1ndashN S


= 1ndashNS

jkbjmdash




jbndash

j ltrj(Q)gtfrac122

(20)


208

time frequency

F(Q

t)

S(Q

w)

Fouriertransform

constantcomponent



component of F(Qt)








(21)











rj(t) = Rj + uj(t) (25)



F(Qt) = 1ndashN S

jkbndash

jbndash


(26)





(27)






Sm=0




j)12Skn



S1(Qw) = 1ndashN S

jkbndash

jbndash



= 1ndashN S

n 2w(mdash mdash




Fn(Q) = Sj

bndash

jmdashmj



MT Dove


n(w) = expmdash

( w mdash1mdash








210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)



00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector










MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove












E = w (1)





206




F(Qt) = 1ndashN S


= 1ndashNS

jkbjmdash





= 1ndashNograveS

jkbjmdash




jkbjmdash










jkbjndash



Finc(Qt) = 1ndashNS

jk(bjmdash

bk ndash bjndash

bkndash




MT Dove



bjmdash

bk = bjndash

bkndash

for j sup1 k (11)


Finc(Qt) = 1ndashNS

j(bj



sjcoh = 4pbj

ndash 2

sjinc = 4p(bj



Finc(Qt) =1mdash

4pNSjsj



Finc(Qt) =1mdash

4pNSjsj


G(r(t)r(0))p(r)dr(t)dr(0) (15)













F(Qyen) = 1ndashN S


= 1ndashNS

jkbjmdash




jbndash

j ltrj(Q)gtfrac122

(20)


208

time frequency

F(Q

t)

S(Q

w)

Fouriertransform

constantcomponent



component of F(Qt)








(21)











rj(t) = Rj + uj(t) (25)



F(Qt) = 1ndashN S

jkbndash

jbndash


(26)





(27)






Sm=0




j)12Skn



S1(Qw) = 1ndashN S

jkbndash

jbndash



= 1ndashN S

n 2w(mdash mdash




Fn(Q) = Sj

bndash

jmdashmj



MT Dove


n(w) = expmdash

( w mdash1mdash








210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)



00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector










MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224




F(Qt) = 1ndashN S


= 1ndashNS

jkbjmdash





= 1ndashNograveS

jkbjmdash




jkbjmdash










jkbjndash



Finc(Qt) = 1ndashNS

jk(bjmdash

bk ndash bjndash

bkndash




MT Dove



bjmdash

bk = bjndash

bkndash

for j sup1 k (11)


Finc(Qt) = 1ndashNS

j(bj



sjcoh = 4pbj

ndash 2

sjinc = 4p(bj



Finc(Qt) =1mdash

4pNSjsj



Finc(Qt) =1mdash

4pNSjsj


G(r(t)r(0))p(r)dr(t)dr(0) (15)













F(Qyen) = 1ndashN S


= 1ndashNS

jkbjmdash




jbndash

j ltrj(Q)gtfrac122

(20)


208

time frequency

F(Q

t)

S(Q

w)

Fouriertransform

constantcomponent



component of F(Qt)








(21)











rj(t) = Rj + uj(t) (25)



F(Qt) = 1ndashN S

jkbndash

jbndash


(26)





(27)






Sm=0




j)12Skn



S1(Qw) = 1ndashN S

jkbndash

jbndash



= 1ndashN S

n 2w(mdash mdash




Fn(Q) = Sj

bndash

jmdashmj



MT Dove


n(w) = expmdash

( w mdash1mdash








210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)



00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector










MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove



bjmdash

bk = bjndash

bkndash

for j sup1 k (11)


Finc(Qt) = 1ndashNS

j(bj



sjcoh = 4pbj

ndash 2

sjinc = 4p(bj



Finc(Qt) =1mdash

4pNSjsj



Finc(Qt) =1mdash

4pNSjsj


G(r(t)r(0))p(r)dr(t)dr(0) (15)













F(Qyen) = 1ndashN S


= 1ndashNS

jkbjmdash




jbndash

j ltrj(Q)gtfrac122

(20)


208

time frequency

F(Q

t)

S(Q

w)

Fouriertransform

constantcomponent



component of F(Qt)








(21)











rj(t) = Rj + uj(t) (25)



F(Qt) = 1ndashN S

jkbndash

jbndash


(26)





(27)






Sm=0




j)12Skn



S1(Qw) = 1ndashN S

jkbndash

jbndash



= 1ndashN S

n 2w(mdash mdash




Fn(Q) = Sj

bndash

jmdashmj



MT Dove


n(w) = expmdash

( w mdash1mdash








210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)



00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector










MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224






(21)











rj(t) = Rj + uj(t) (25)



F(Qt) = 1ndashN S

jkbndash

jbndash


(26)





(27)






Sm=0




j)12Skn



S1(Qw) = 1ndashN S

jkbndash

jbndash



= 1ndashN S

n 2w(mdash mdash




Fn(Q) = Sj

bndash

jmdashmj



MT Dove


n(w) = expmdash

( w mdash1mdash








210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)



00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector










MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove


n(w) = expmdash

( w mdash1mdash








210

0

10

20

30

40

05 05 0500 0001020304

Fre

quen

cy (

TH

z)



00

1

2

3

4

05

Freq

uenc

y (T

Hz)

Reduced wave vector










MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224









MT Dove



F0(Qt) = 1ndashN S

jkbndash

j bndash


= 1ndashNfrac12Sj

bndash


(35)



= 1ndashNfrac12Sj

bndash



Sj

bndash


bndash


jbndash





212


Inte

nsity












jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224










jkbndash

j bndash




jkbndash

j bndash





893 K

761 K

756 K


40

41

39

40

41

39

40

41

39


Wav

e ve

ctor

(00

l)

893 K

773 K

761 K

0

100

200

300

0

100

200

300

0

100

200


MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove






4p ograve2p

0djograve

p


= 1ndash2 ograve

+1

ndash1exp(iQrjkx)dx

= sin(mdash

QrjkmdashQrjk) (41)


S(Q) = 1ndashN S

jkbndash

jbndash

k sin(Qrjk)Qrjk

= 1ndashN S

jbndash

j2 +

1ndashN S

jsup1kbndash

j bndash




214

0

10

20

0

10

20

0

10

20

0

10

20

0 5 10 15 20 30 40 50

0

3

ndash3

0

3

ndash3

0

3

ndash3

0

3

ndash3

Qi(Q) Qi(Q)

Q (Aring)

Bank 2

Bank 4

Bank 5

Bank 7



S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224


S(Q) = Sm

cmbndash2

m + i(Q) + S0 (43)

i(Q) = r0Smn

cmcnbndash

mbndash

n ograveyen


QrmdashQr

dr (44)


G(r) = Smn

cmcnbndash

mbndash


ij=1cicjb

ndashibndash


D(r) = 4prr0G(r) (46)


i(Q) = r0 ograveyen

04pr2G(r)

sinmdashQrmdashQr

dr (47)

Qi(Q) = ograveyen

0D(r) sinQr dr (48)


3r0ograveyen

04pQ2i(Q)

sinmdashQrmdashQr

dQ (49)


yen

0Qi(Q)sinQrdQ (50)






ndash1

0

1

2

0 5 10 20 3015 25

ndash1

0

1

2

0 51 2 3 4

r (Aring)

D(r)


OndashOAl

AlndashO


tetrahedra

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove






R2g = 3R 5 (52)





r(t) = R(t) + d(t) + u(t) (53)










inc (Qt) (55)




inc (Qw) (56)


42 Atomic diffusion




216




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224




G(r0) = d(r)






mdashQ2mdashDQ2

mdash)2 +

mdashw2 (61)











MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove








WG (62)


Ntilde2W =

sinmdash1mdashq


paramdashparaq +

sinmdash1mdash





+

m=ndashYm(W(t))Y

m (W(0))

(64)



yen

=1(2 + 1) j2(Qd)exp(ndashDR ( + 1)t (65)



+ 1ndashpS

yen

=1(2 + 1) j2(Qd)mdash

(DRmdashDRmdash

( + 1mdash

))2mdash( + 1)mdash

+w2 (66)




218

0 50 100

Mas

s-no

rmal

ised

inte

nsity


Fully-hydrated

Dehydrated

Leucite












IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224










IEISF(Q) = Selmdash

(Q)mdashSel

+ Sqe




IEISF(Q) = sinmdash

QrmdashQr

(69)



ndash100 0 100

Inte

nsity



3030

40

40

35

35

45

20 251000T

ln(t

)


MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove






6 Summary



220

00 05

09

08

0710

10

15 20

Q (Aringndash1)

EIS

F

r = 10 Aring


r = 032 Aring

a = 019

a = 026




References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224



References



































MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224

MT Dove





ndashndashI1












ndashto R3


























222








































MT Dove














224








































MT Dove














224

an introduction to the use of neutron scattering...

Documents