Investigation of Metal-Hydrogen Systems by Means of Neutron Scattering

Zeitschrift für Physikalische Chemie Neue Folge, Bd. 115, S. 141-163 (1979)© by Akademische Verlagsgesellschaft, Wiesbaden 1979

Investigation of Metal-Hydrogen Systemsby Means of Neutron Scattering

ByT. Springer

Institut Laue-Langevin, F-38042 Grenoble, Cedex 156X, Franceand Institut für Festkörperforschung der KFA Jülich, W. Germany

(Received March 7, 1979)

Neutron scattering I Diffusion of hydrogen / Vibrations

Slow neutron scattering is a standard method to investigate (i) the diffusion of hydrogendissolved in metals, (ii) the localized vibrations of the hydrogen atoms, and (iii) their influence on

the host lattice phonons. Furthermore, integrated quasielastic coherent scattering (at D atoms)yields information on the mutual interaction of the dissolved atoms, and on their interactionwith the host lattice as well. Selected experiments dealing with these subjects will be presented.

Die Streuung langsamer Neutronen ist eine Standardmethode zur Untersuchung 1. derDiffusion von in Metallen gelöstem Wasserstoff, 2. der lokalen Schwingungen der Wasserstoff-atome und 3. ihres Einflusses auf die Phononen des Wirtsgitters. Darüber hinaus liefert dieintegrierte quasielastische kohärente Streuung (an D-Atomen) Information über die gegenseitigeWechselwirkung der gelösten Atome und ebenso über ihre Wechselwirkung mit dem Wirtsgitter.Eine Reihe von ausgewählten Experimenten, die sich mit solchen Problemen befassen, wirdvorgestellt.

1. Introduction

The dynamical behaviour of a hydrogen or deuterium atom in a metal canbe understood in the following way : the dissolved atom alternates betweenoscillatory motions at certain interstitial sites (e.g. tetrahedral in bcc hostlattices), and rapid transitions or jumps between such sites which lead todiffusion. Scattering of slow neutrons is the standard, and most successfullmethod to study these motions. This has two reasons : first, the investigationof the scattered neutron intensity as a function of energy and momentumtransfer is directly related to the atomistic behaviour of this motion in timeand in space. Secondly, hydrogen scattering is particularly strong and can beeasily separated from the host lattice contribution which makes theevaluation of the experiments very simple.

This review presents investigations where the neutrons have been appliedto study the dynamical behaviour of hydrogen. Chapters 2 and 3 are dealing

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with the determination of the diffusion constant and with the investigation ofthe diffusive step itself, with emphasis on the carefully studied bcc hydrides.Chapter 4 treats the problem of diffusion in a metal with impurities acting as

traps for the diffusing hydrogen atoms. Chapter 5 deals with the hydrogenvibrations, and with the influence of dissolved hydrogen on the host latticephonons, in particular referring to the interesting aspect ofcoupling betweendiffusive motions and transversal phonons. The last chapter describes alsothe investigation of static effects, namely the interaction of the deuteriumatoms with the host lattice, and their mutual interaction, as studied bycoherent scattering. The paper presents only examples of such studies. Fordetailed reviews we refer to the literature [1,2].

2. Neutron Scattering and the Determinationof the Hydrogen Diffusion Constant

One of the special features of H in certain metals and alloys is its highmobility, leading to selfdiffusion constants as large as D = 10"5

—

10~4cm2/s,which are many orders of magnitude higher than the diffusion constants forheavy impurities in metals. On the other hand, these values are as high as theself diffusion constant of an atom in a liquid metal, or of an ion in a solidelectrolyte. Such diffusion constants are close to the maximum possible valuewhich should be D = s2v(h/3 where s is of the order of the lattice parameter,and vth is the thermal velocity of the diffusing atom. In addition to diffusion,the dissolved hydrogen atom carries out vibrations about the equilibriumsites. In the following we discuss the investigation of the "fast diffusion" byneutron scattering.

Elementary neutron interaction theory shows that the spectral intensityscattered from a single proton in a solid consists of lines and bands on theenergy loss and gain side (caused by the particle's vibrational states) and, forthe proton at rest, a delta function centered at energy transfer = 0 (whichcorresponds to the Mössbauer line for gamma ray scattering). If the proton isdiffusing, the line at = 0 is broadened and proportional to a sum ofLorenzians (quasielastic spectrum) [1,2], namely:

Sinc(Ô,eo)= ,(g) J£fl*Q) ^Ql("2> + Phononlines (1)

where TiQ = (k0—

kJ and = E0—

Ex are the momentum and the energytransfer during scattering, respectively. Sinc ( ), ) is the incoherent scatteringlaw1. „ and µ are the eigenvalues and weight factors of the system of rateequations which describes the probability of occupancy of the interstitialsublattices µ=\,.. , ( =6 for tetrahedral sites in a bcc host lattice). The

1 "Incoherent" since, in the case of hydrogen, coherency is practically destroyed by the strongspin dependence of the (n, p) interaction. For deuterium, incoherent and coherent scattering are

of the same order of magnitude.318

Investigation of Metal-Hydrogen Systems by Means of Neutron Scattering 143

rate equations are usually derived under the following assumptions("Chudley-Elliott model" [3]): (i) The time which the H atom spendsbetween the interstitial sites %¡ (jump time) is small compared to the mean resttime at a site, (ii) The jump probability to all neighbouring sites is the same,and subsequent jumps are statistically independent. The introduction of theDebye-Waller factor exp

—

Q2 (u2} implies that vibrations and diffusivemotions are independent. As a matter of fact, the Lorentzian in Eq. (1)dominates at small . At large side lines appear due to the vibrationalmodes.

For small Q, Eq. (1) is always dominated by a single Lorentzian with a

half width = Q2 D, (2)

independent of the lattice type and the details of the diffusive jump.Therefore, for Q values below, say, 0.3 Â"1 a measurement of vs. Q2 yieldsthe hydrogen selfdiffusion constant D. At larger Q, the ß-dependence of thewidth is related to the geometry of the diffusive step.

The determination of D is an important application of quasielasticscattering. To judge its applicability, we compare it with alternative methods :

(i) Obviously, the scattering is a "bulk" method. Typical sample sizes are mm

or cm, and powders can be studied as well. Diffusion along microcracks or

grain boundaries does not enter if they are sufficiently extended in space(compared to 1/ß). (ii) The driving force in a density gradient experiment (ase.g. in permeation [4] or in Gorsky-effect measurements [5]) is related to thegradient of the chemical potential grad c ( µ/ôc), and one obtains

D* = (öß/öc)cM(c,T) (3)where ( µ/ôc) is the derivative of the chemical potential with respect tohydrogen concentration, and M is the mobility of the dissolved hydrogen.Consequently, D* shows critical slowing down at the top of the ( ')coexistence curve for NbHx [5]. On the other hand, incoherent scattering isrelated to the motion of an individual proton. The mutual interaction does notact as the driving force, and the value for a dilute system, kTjc, centers.Therefore one defines a "reduced" diffusion constant [6]

fl' = fl'PcUW4 (4)which depends only on the mobility. Multiplying D* by (kT/c)/(ößßc)c, a

value D' is obtained which agrees quite well with D, the "neutron" or singleparticle value [6]. It should be pointed out that D is also obtained by other"single particle" methods, for instance NMR spectroscopy. Obviously, D or

D' has no critical slowing down.The scattering method works at not too small concentrations (a few 0.1

atomic % for in Nb, and 5 to 10 atomic % in other cases) because ofparsitic host scattering. D-values > 10~8cm2/s can be measured if highresolution spectrometers are available. The backscattering spectrometer IN

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144 T. Springer

Il I III I I I I I l_l_I_L_l_I_I_I_I_l_l50 75 100 125 50 75 100 125

Time of flight [15.5 µ5/ 3 ] ——

Fig. 1. Typical time-of-flight spectra for neutron scattering on in a NbH0 02 single crystal.Solid line: quasielastic part (centered at zero energy transfer), calculated by means of the

multiple jump model (see text). Peak at the left: phonon line [8]

10 at the Institut Laue-Langevin has a resolution half width of0.2 µ whichallows such an accuracy. Dissolved deuterium produces incoherent as well as

coherent scattering. Therefore, interference effects between pairs of deu-terium atoms and also with the host lattice come into play. At sufficientlylow concentrations, these may be small. In general, the interpretation of thespectra is complex [7]. It has been treated by Monte Carlo calculations. At

high concentrations one might be able to study the diffusion of interstitialvacancies instead of deuterium atoms.

Figure 1 shows typical measured spectra, and Fig. 2 presents the width vs. Q2 for quasielastic lines of in Nb and V [8]. A comparison of thevanadium results with Gorsky effect experiments reveals a deviation from a

simple Arrhenius curve which may be analogous to the change of theactivation energy already observed for hydrogen in niobium [5,9] and intantalum [10],

From a technological point of view, diffusion constants in "hydrogenstorage" alloys, as Ti2NiHx [11] are of special interest. For Ti2Ni powder thediffusion constant was roughly estimated from the measured dischargingtime td and the grain size S (50

—

100 µ), which leads to D = S2/6 td= 10~6cm2/s for

—

2 at 155°C. Quasielastic scattering however, yielded1.5 · 10~8 cm2/s. This discrepancy shows that "evaporation" may occur froman internal interface of this phase into a low concentration phase nearer to thesurface (where D is high). As well, diffusion could take place alongmicrocracks or along grain boundaries. Such grain boundaries were actuallyobserved by electron microscopy, with subgrains of about 10 µ size. Thisvalue leads to a D consistent with the bulk selfdiffusion constant and the

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û2rj"2]—*Fig. 2. Width of the quasielastic spectra for small scattering vectors Q. The slope of vs. Q2

yields the selfdiffusion constant D [8] (1 meV = 10"3eV)

discharging time. Similar studies were carried out on FeTiH01_10 [12]. Webelieve that quasielastic scattering is a reliable method to determine bulkselfdiffusion constants, in particular if the sample is not a good crystal, or, as

well, for "inconvenient" conditions, as high temperatures or pressures.

3. Investigation of the Single Diffusive StepAs stated before, the dependence of the quasielastic spectrum on the

scattering vector Q at larger Q provides atomistic information on thediffusive step itself. So far, the systems : in fee palladium, and in the becmetals : niobium, tantalum and vanadium, have been studied most exten-

sively. The quasielastic scattering on in PdH seems to follow closely theChudley-Elliott model2 (Ch-E model), with jumps between adjacent octa-hedral sites [13] (for ßPdH see [14]).

For the bec hydrides in Nb, Ta, and V a number of experiments existsand it has been recognized rather early that the quasielastic width increaseswith Q more slowly than anticipated from the simple Ch-E model,suggesting the existence of some kind of correlated or long distance jumps[16]. This idea has been worked out quantitatively in detail for in singlecrystals of Nb, Ta, and V [8]. Experiments on single crystals at the D7 andIN4time-of-flight spectrometers in Grenoble with 0.5 < Q < 2.5 Â"1 at 290,430 and 580 were carried out at various orientations and values of thescattering vector Q. The one-phonon scattering superimposed on the

2 More recently, it has been suggested that also jumps to larger-distance neighbours come intoplay [15].

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146 T. Springer

2.0

0.5

©

nunmscattering plane (110)model :(1)

oo0 = 581

1.6

1.2

=431

dfo

= 293 i

^ 0.8

OA

t.% ° < * °

NbHoûz , 581 scattering plane (110)

° x* CP

0[ -1.0arj-

2.0 3.0];

Fig. 3. (a) Apparent jump rate / from a fit by means of the Chudley-Elliott model with nearestneighbour jumps, vs. scattering vector Q. At 293 the model holds (1 / is the same for all Q). Itfails at large 7*(b) Same fit procedure for double jump model (see text) : the same is obtained for

all Q, which means that the model is consistent with the experiment [8]

quasielastic line was substracted, using a Debye spectrum with a multi-phonon expansion.

As a first step of interpretation, the Ch-E model (with jumps betweennearest neighbour tetrahedral sites) was applied with the mean rest time andthe integrated quasielastic intensity I(Q) as fitting parameter. Figure 3 (a)shows that, for room temperature, the same is obtained for all values of Qproving the consistency of the simple Ch-E model. However, at elevatedtemperatures the apparent values of decrease with increasing Q whichmeans that the model fails. Agreement was established with a double jumpmodel, which means : the hydrogen can pass immediately from a tetrahedralsite to a secoHiZ-nearest neighbour site (with a probability 1/ 2) withoutspending a significant time at the nearest-neighbour site (see Fig. 3 b). Thefraction of such jumps found from this model was rather high. This naturallyleads to a generalization of the model [8,17] : One assumes that the hydrogenalternates between an excited mobile "state" where it spends, in the average, atime „, carrying out numerous jumps between nearest neighbour tetrahedralsites (rate l/rj, and a "trapped" state during an average time ,. The numberof jumps in the excited state is obviously / . In general terms, the physics

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NbH002:581 Kmodel (4)

° nno30° 70

Fig. 4. Apparent jump rate from fits of different models with single crystal experiments:(2) model with (a/2) [100] jumps. (3) double jumps of the (110) type. (4) [110] jump-sequence

model with 1/t, = 3.3, 1/ , = 1.12, 1/ , = 0.641012s_1 [17]

behind this model we interprete as follows : A hydrogen is trapped at a site(called "1") due to the strong relaxation of the surrounding host lattice whichlowers the ground state. After a while, thermal excitation throws the over

the barrier to an adjacent site "2". For the first instance, the lattice is stillunrelaxed, so that the barrier is lower (E0 instead of E0 + ).Consequently, the particle proceeds quickly to another site "3", or movesback to "1". Therefore, at high T, it follows a sequence of jumps over theunrelaxed barriers until relaxation occurs, at some distant site. On the otherhand, at low T, the hydrogen normally rests sufficiently long at "2" to allowrelaxation, and the barrier is high.

Figure 4 shows the description of experiments for in Nb by this model :The model parameters tu ß, , were found to be independent of theorientation (as shown in the Figure) and also from the value of Q. This means

that the model consistently describes the experimental results. For com-

parison, Fig. 4 includes also the "double jump" model explained at thebeginning (with 2/ = 1.2) and a model which allows 2nd neighbour jumpswith (a/2) [100] jump vectors. The latter obviously fails to describe theexperimental results. Experiments were also carried out on in Ta wheresimilar conclusions can be drawn as before, namely : the validity of the simpleCh-E model at room temperature and the appearance of double jumps atelevated temperature [8]3.

In addition to the quasielastic width, the experiments yield also thequasielastic intensity I(Q). For in Nb and Ta, I(Q) follows a harmonic andisotropie Debye-Waller factor: exp (

—

Q2 <w2» where <w2> is of order of

3 The generalized jump-sequence model has not been worked out in this case.

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148 T. Springer

10.2

10 0

9 8

9.6

9.4J.i^ 9.6'

9.4

92

90

a-VH007 model (1)

10

=408

2.0 3.002[) 2]-

4.0 5.0 6.0

Fig. 5. Intensity of the quasielastic line In / vs. Q2, for in V (using the double jump model). Atlower temperatures, HQ) follows a harmonic Debye-Waller factor [8]

0.01—0.02 Â2, in reasonable agreement with values calculated from vib-ration spectra (see Chapter 5). Deviations from a "normal" Debye-Wallerfactor reported earlier by ourselves [18] and others [19] are mostly due toimproper extrapolation procedures for the integration of the wings of thequasielastic line4. For in V at elevated temperature, the quasielasticintensity I{Q) does not follow a Debye-Waller factor (Fig. 5). The curve InI(Q) vs. Q2 starts with an initial slope consistent with the estimated value of<w2>. At larger Q(> 2 Â~ '), however, the slope increases considerably. Thisis presumably caused by the finite "flight time" of the H from site to site. For770 one estimates tf/(rf + ) = 30 % with rf > a ]/T/4 vth. This means thatthe flight period ( for in V is not negligible (a = lattice parameter). Theflight period is responsible for a rather broad spectrum whose contributionfalls far beyond the quasielastic spectrum at larger Q5.

The multi-step model discussed so far does not imply any time-memory ofthe diffusing hydrogen. "Memory" means that the probability of a jump

4 For in Pd there is strong evidence for an anomalously large <»2> [13].5 In the harmonic approximation, the Debye-Waller factor can be also obtained from the one-

phonon inelastic scattering intensity of the band modes, /b. One gets/b(e)/e2<y>oce-e:<"!> +

...

(5)Actually it was observed that ¡JQ2 follows a "normal" Debye-Waller factor [8]. This isconsistent with the explanation given before : The "free-flight" spectrum, "lost" in the integratedquasielastic line, is not recovered in the one-phonon part.

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leading to a certain site depends on the previous site(s) where the particlecomes from (e.g. a sequence following a certain lattice direction may be more

frequent than another). Models with memory were formulated theoreticallyfor H in bcc metals [20], based on more general concepts [21]. Such memoryeffects may play a role for hydrogen diffusion, in particular at elevatedtemperatures, but they have not yet been applied for interpretation.

As a summary it can be stated (i) that H diffusion at elevated temperaturesis dominated by double or multiple jump sequences occuring over tetrahedralsites, and (ii) for H in V, that the time for the diffusive jump t, is not negligible[21a]. We assume that this statement holds also for other systems if thetemperature is sufficiently high. (Finally, rf may dominate diffusion com-

pletely, leading to a limiting value of D = s2/6r; at high temperatures.)

4. Diffusion, with TrapsIt is known experimentally that impurities, as nitrogen or oxygen in bcc

metals act as trapping centers for the dissolved H atoms [22]. In particular,this effect influences the diffusion process [23]. To study this in an atomisticscale, quasielastic scattering experiments on H diffusion in niobium withdilute nitrogen impurities were carried out at different concentrations,temperatures, and as a function of the scattering vector [24].

In a simplified theoretical picture, the behaviour of the quasielasticscattering of such a system can be approximately understood in terms of thefollowing two-step model : The hydrogen alternates between free diffusion fora certain time interval (rf in the average), with a diffusion constant D0 as givenfor the non-doped crystal, and an immobile or trapped "state" in the vicinityof the impurity atom during an average time 0. Obviously, this separationholds only for a low impurity concentration such that the trapping regionsare separated. As a chemical reaction scheme for a trapping atom we canwrite :

+ (free) <± NH (trapped).For small Q, the scattering process "averages" over a large volume in space(

—

1 /Ö3), and, therefore over a long diffusive path, thus including the trappedand the free state as well. The resulting quasielastic spectrum has a widthg2 űeff where De(f is the selfdiffusion constant in the impurity-doped metal.On the other hand, at large Q, one observes a small volume region i.e. more or

less the single diffusive steps. Consequently, the spectrum at large Q shouldconsist of two parts : (i) a narrow line due to the H atoms being trapped ;its width and intensity are related to 1/ 0 and to the fraction of atomsbeing trapped, respectively; and (ii) a broad line whose width corresponds tofree diffusion ( l/ where is the mean rest-time in the undisturbedlattice). This two-state model was worked out quantitatively [24] (see also[25,26]). The resulting scattering law has two eigenvalues corresponding to

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150 T. Springer

Fig. 6. The weights R¡, R2 and widths Ax, A2 of the superimposed quasielastic lines for the two-

state model, for hydrogen diffusion in a metal with trapping centers (schematic; logarithmicscale). (Q) corresponds to the quasielastic line for hydrogen diffusion in the pure metal

the widths of two superimposed Lorentzians. Their ^-dependence isschematically shown in Fig. 6, together with the intensities of the two spectralcomponents.

The measurements were carried out on NbH^Hj, with = 0.4 and 0.7, andy = 0.4 and 0.3 atomic %, respectively, using the IN 10 spectrometer at theGrenoble reactor. Its resolution is 0.2 µ , and the width of the energy"window" covered is 7 µ . Figure 7 shows the intensity / integrated

this "window" in units of the total scattering intensity I0. As expected, forsmall Q the quasielastic intensity falls entirely into the energy window, i.e.I/I0 = 1. For large Q, I0/i decreases with temperature since the fraction oftrapped atoms is lowered with increasing temperature. Figure 8 shows thecharacteristic times ( and 0 vs. temperature, as obtained from a fit of thespectra with the theoretical formalism explained before. Obviously, 0 doesnot depend on the trap concentration as expected, since 0 is an inherentproperty of the trap. On the other hand, 1/ { should be proportional to thetrap concentration and to D0. The proportionality with respect to : was

confirmed. The proportionality to D0 should yield the same activation energyfor if1 and for D0, which is not the case (93 instead of 76meV). Thediscrepancy can be resolved by the assumption that a impurity is saturatedafter it has trapped one atom [26a], Introducing the T-dependentconcentration of non-saturated traps, xef{

—

y I(T)¡I0, one obtains an

apparent activation energy for the true trapping rate tf"e'ff = zf1 (xeff/x) of

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1.5

1.00.8

'» 0.6

OA

0.2

0.13.0 3.5 AO 4.5 5.0 5.5 6.0

1000/r- Fig. 7. The quasielastic intensity /t for hydrogen diffusing in Nb with dilute traps, integratedover an energy window of width 7/ieV, in units of the total quasielastic intensity (measured atvery low temperature), I0. At large Q, the intensity ratio is related to the decrease of trapping

probability with increasing temperature [24]

1000/ [ 1]--Fig. 8. The characteristic times 0 (trapping-time) and (time of free diffusion) for the two-state

model for in Nb, at two concentrations cN [24]

79meV, in good agreement with the known value for D0. The resultsdemonstrate that the simple concept of a two-step process is able to describethe diffusive behaviour of a system with dilute impurities. However, theassumption that the diffusion constant during the "free" diffusion periodZ>0, is equal to its value in undoped Nb may fail at not too low concentrationswhere the fluctuations of the lattice parameter are no more negligible.

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152 T. Springer

At low temperature, an anomaly of the specific heat has been found [27].Recent investigations have shown that this is probably due to H trapped on

impurity atoms. It has been suggested that this anomaly is caused by energystates of the hydrogen atom tunneling in a periodic field in the vicinity of thetrapping atom [28]. Neutron scattering experiments are underway todemonstrate if such states exist.

5. Vibrations of Hydrogen Metal SystemsThis chapter treats the influence of hydrogen atoms in a metal on the

vibrational spectrum of the lattice. Due to the small mass, the H (or D) carriesout vibrations at frequencies beyond the phonon energies of the host lattice.At low hydrogen concentration, these vibrations can be considered as

strongly localized. With increasing concentration they become collective, i.e.optical phonons whose frequency may depend on the wave vector q. Thedispersion curves co(q) can be measured by coherent inelastic neutronscattering (see e.g. [2]).

There is a relevant difference between the optic vibrations for in bccmetals as Nb, Ta or V on the one hand, and for fee metals like Pd or certainrare earth-metals on the other hand. Table 1 summarizes the experimentalresults for a number of such hydrides. For the bcc systems mentioned above,the coupling between the H and the host metal atoms is very strong, leading to

frequencies as high as for chemical bond vibrations (1000—

2000 cm"l). Theinteraction with the H neighbours is nearly negligible, such that thecorresponding optical branches are "flat" (no ^-dependence), and there is nodifference between the optical frequencies for nearly stoichiometric hydridesand for dilute systems. Consequently, there is also no disorder broadening fora highly loaded non-stoichiometric hydride. Figure 9 shows typical curves (q) for the optic and, as well, for the acoustic branches of a NbD singlecrystal in its /J-phase, as measured by coherent inelastic scattering on a tripleaxis spectrometer [29] the orthorhombic distortion in the /f-phase is small,and the curves are presented as if it were a cubic crystal).

For hydrogen or deuterium in the fee palladium a phase the mutualdeuterium coupling is strong. Therefore, the optical branches reveal a

pronounced wave vector dependence, as shown in Fig. 10 which compares theoptic and acoustic branches for PdD0 63 [30] and pure Pd. For thissystem, which is non-stoichiometric, the observed effect of the disorder on thewidth and the position of the phonons (in particular for the longitudinaloptic) has been treated in detail by pseudo random lattice calculations [31].Obviously, the optical branches introduced by the D atoms are rather soft,with energies close to the host lattice modes (as compared to D in the bccmetals). This may be one of the reasons for the influence of or D on thesupraconductivity of palladium. This idea is supported by tunnelingexperiments on PdD0 9 [32] which give information on the electron-phonon

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Table 1. Hydrogen and deuterium in niobium, tantalaum, vanadium and palladiumOptic mode frequencies AcoH (approximate energy of band center) and activation energy for self-

diffusion

Sample Lattice type;proposedH-sites

faoH(eV)'

^act(eV)

Latticeparameter[A]

«NbHo.05 bec; tetrah. 0.11 and 0.18 ( ) 0.10 Nb: 3.30

aNbD0 60 bec; tetrah. 0.113/1/2" and 0.158/[/2 (b)j8NbD0.75 orthorh. tetrah. 0.120/1/2" and 0.170/|/2 (c) ~0.3

j?NbH095 orthorh. tetrah. 0.120 and 0.170 (d)

aTaH0 15 (500°C) bec; tetrah.

aTaD0 22 bec; tetrah.

ßiTaHo-, monoclinic

0.12 and 0.17 (e) 0.14

0.119/|/2and 0.167/1/2" (/)0.13 and 0.18 (g)

Ta: 3.30

bec; tetrah.

(tetrah.loctah.

yVHj.5-1.7 fee; tetrah.

t*VH0 04

PVHo.4o

0.12 and 0.17 (a)0.120 and 0.175 (A)0.055

-0.166 (A)

0.05 V: 3.02

aPdH0.,«PdH0(aPdD„,

fee; octah.fee; octah.fee; octah.

0.056 (j)0.066 (/)0.051/1/2 ( )

0.23 Pd: 3.89

1 For the deuterides, the frequencies are given in units of |/2 to make them comparable to thehydrides.

(a)

(h)

(c)uh

(e)

GO(g)

(h)W

(k)

G. Verdan, R. Rubin and W. Kley, Neutron Inelastic Scattering, Vol.1, p. 223. Proc.IAEA, Vienna 1968 ; see also : W. Gissler, G. Alefeld and T. Springer, J. Chem. Phys. Solids31 (1970) 2361.N. Stump, G. Alefeld and D. Tocchetti, Solid State Comm. 19 (1976) 805.

-

H. Conrad, G.Bauer, G. Alefeld, T. Springer and W. Schmatz, . Physik 266 (1974) 239.V. Lottner et al., Proc. Conf. on Lattice Dynamics (ed. M. Balkanski), p. 247 (1977). . A. Chernoplekov, M. G. Zemlyanov, V. A. Somenkov and A. A. Chertkov, SovietPhysics-Solid State 11 (1970) 2343.J. J. Rush, R. C. Livingston, L. A. de Graaf, H. E. Flotow and J. M. Rowe, J. Chem. Phys.59 (1973) 6570.A. Magerl, . Stump, W. D. Teuchert, V. Wagner and G. Alefeld, to be published (1977).R. Yamada, . Watanabe, . Sato, H. Asano and M. Hirabayashi, J. Phys. Soc. Japan 41(1976) 85.J. J. Rush and H. E. Flotow, J. Chem. Phys. 48 (1968) 3795.M. R. Chowdhury and D. K. Ross, Solid State Comm. 13 (1973) 229; see also: J. Bergsmaand J. A. Goedkoop, in: Inelastic Scattering of Neutrons in Solids and Liquids, Proc.IAEA Vienna 1961, p. 501.W. Drexel, A. Murani, D. Tocchetti, W. Kley, I. Sosnowska and D. K. Ross, J. Phys.Chem. Solids 37 (1976) 1135; see also: Proc. Gatlinburg Conf. on Neutron Scattering,Gatlinburg 1976.J. M. Rowe, J. J. Rush, H. G. Smith, M. Mostoller and H. E. Flotow, Phys. Rev. Letters 33(1974) 1297.

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1.0 0.5Reduced wave vector

Fig. 9. Optical and acoustic phonon dispersion curves (q) for ß NbD0 75 (in cubic representa-tion) as measured by coherent inelastic neutron scattering for different values and directions of

the phonon wave vector q [29]. Thin line: pure Nb

Reduced wave vectorFig. 10. Optical and acoustic branches for PdD0 63. LO, TO

branches. Dashed: pure Pd [30]longitudinal, transversal

interaction (i.e. on the function a2 F [ ]). They show that the estimatedMcMillan parameter for the optical modes is greater than for the acousticmodes.

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As concerns the acoustic phonons, the dissolved H is responsible forseveral effects. The lattice spacing is increased, which weakens the bindingforces and tends to lower the host lattice phonons. In addition, the hydrogenatoms may contribute electrons to the host lattice, thus shifting the Fermilevel and eventually forming electron bonds. For H in Pd, a lowering of theacoustic branches is observed but, surprisingly, no visible change of the shapeof the branches (Fig. 10). On the other hand, for H in Nb, there is an increaseof the frequencies, and, as well, a drastic change of the curve shape, inparticular the disappearance of the "anomalies" of these curves (see Fig. 9).This behaviour has been discussed in relation to the system Nbj-xMox [33]where the Mo impurity (like the H) adds electrons, and the changes of thebranches were similar to those induced by the H. It has been speculated thatthe disappearance of the "anomalies" is related to the lowering of thetransition temperature for supra-conductivity after hydrogen is dissolved inNb.

In addition to the investigation of the phonon branches of deutendes(measured by coherent inelastic scattering) attention was paid to theincoherent scattering on the dissolved hydrogen atoms themselves in theregion of the host lattice phonons ("band modes"). For phonons at very longwavelengths, the hydrogen atom follows the host lattice atoms in phase andamplitude. The corresponding square amplitudes and therefore the intersityscattered from the H will then be proportional to the host lattice density ofstates. For short wavelength phonons, it has been shown that the vibrationalamplitudes of the H, with respect to the host lattice may undergo drasticchanges. An illustrative case for a bcc Nb crystal is the following : A modewith two Nb corner atoms at the unit cell moving in opposite [001] directionsimplies the motion of a hydrogen at a tetrahedral site perpendicular to thisdirection. The corresponding amplitude will be several times larger thanthe Nb amplitude if the H —Nb bond is rigid. Consequently, for such amode a strong enhancement of the scattering intensity is expected. Such aneffect was actually observed [34] for NbH0 05, as a distinct spectral line near16meV. The q and isotope independence of lines observed for NbD0 85 andNbH0 82 at 18.4meV [35] may have the same physical origin. In general, theunderstanding of these band modes is an open question and needs furtherstudy. The amplitude enhancement influences also the (total) mean squareamplitude <w2> of the hydrogen, as obtained from the Debye-Waller factor.As an approximation one usually writes

(u2) = <W2>band + <M2>localized = fF(Ha)/kT)[Zh(o>)(i/Mha>2)+ & ( - )(1/ 2)] (6)

where Zh ( ) is the host lattice spectrum, assuming the atom follows thehost lattice vibrations. F(x) = (x/2) + x(ex

—

1)"1 is the thermal populationand a>¡ are the localized mode frequencies. Actually, Zh has to be replaced by

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156 T. Springer

-5L-4-

J-H

0.04 0.08 0.12C«0]-

1.20 0.24

Fig. 11. Change of sound velocity (or: slope of frequency vs. wave vector q) of the TXA branchfor in Nb, as a function of q in [110] direction [38]

the spectral density of the dissolved hydrogen atom which is proportional tothe imaginary part of the Greens function GHH ( ) related to the response ofthe atom to a force acting on it [36]. If there are enhanced band modes, the -dependent 1st term in (6) will be strongly increased, in agreement withexperiments (see sect. 3 [8, 34]).

Interesting features were discovered for the low frequency acoustic modescomparing elastic constants from ultrasonic experiments on Nb, Ta and Vhydrides with phonon dispersion measurements. According to the acousticbranches in Fig. 9, deurium loading leads to an increase of the slope at smallq. This is consistent with ultrasonic experiments [37] at 10 MHz which showan increase of the bulk modulus = (cu +2 c12)/3 and of the shear modulusc44 with increasing hydrogen concentration. However, a striking decreasewas obtained for the shear constant c* = (c¡ j

—

ci2)/2. This was also studiedby means of the corresponding phonon branch [110] Tt A for TaD0 22 in theregion of 1012 Hz [38]. Surprisingly, the change of slope due to D loading wasmuch higher than the change of c* from ultrasonic frequencies. For NbH0 15it was possible to observe directly a transition from the value at low to a high-frequency value [38] (Fig. 11).

It would be natural to relate this dispersive transition to a couplingbetween the [110] 1 mode and the diffusive motion of the dissolved atoms : Sitting on a tetrahedral site, the strain field created by the atom hastetragonal symmetry (Fig. 12). A mode of the [110] T[A type obviouslycauses a distortion of this kind. Consequently, this may couple to a diffusivejump which rotates the tetragonal axis of the strain field (dashed arrow inFig. 12). Therefore, a transition could occur between a relaxed low-frequencyshear constant (the atoms follow the deformations caused by the phonon),and a non-relaxed shear constant at high frequency. The transition shouldtake place at a* 1 where is the mean rest time of the atom at atetrahedral site. For the experiment in Fig. 11 this is just fulfilled at q/qm¡¡x =

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o=Nb ·=0• = tetrahedral site

Fig. 12. Displacement of the host atoms around a D atom in Nb, showing virtual forces to beapplied at nearest and next-nearest neighbours (f¡ and f2) [40]

= 0.03 ( calculated from the diffusion constant). Unfortunately, in a certainsense, this argumentation contradicts the experimental facts, namely : (i) the

low-frequency value of c* cannot be explained by orientational relaxationdue to the interstitials, since this requires a T~ 1 dependence of the shift of c*.This was not found [37]. (ii) Snoek effect experiments show that thedeformation tensor is cubic within experimental error [39] (Pt t = P22 = P33,see Chapt. 6). The cubicity was also confirmed by diffuse neutron scatteringexperiments. This implies that, for wavelengths large compared to the latticecell (as for = 0.03), a diffusive jump cannot couple to this mode. A way outof this inconsistency is the assumption that at high frequencies, the long rangedistortion field differs from the distortion studied at zero frequency in theSnoek effect. The problem is still unsolved.

6. The Mutual Interaction of Dissolved Deuteriumand the Interaction with the Host Lattice

This chapter describes static effects of the dissolved deuterium6, whichcan be studied by the coherent elastic scattering integrated over the

6 scatters predominantly incoherently. Therefore, this chapter deals with dissolveddeuterium where the coherent contribution is large.

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158 T. Springer

quasielastic line. This scattering is determined by interference effects.Therefore, it yields information on the mutual deuterium interaction, and, as

well, on the interaction of the deuterium with the host lattice atoms. Thecorresponding cross section per deuterium atom can be written as follows [40]

(d<r/dß)q.el. = (l/c-)c2(q) |Jt [b'D + /¿>hQs,,(q)e^<]|2. (7)

For simplicity, the incoherent scattering contribution of the deuterium isomitted. b'D and b'h are the coherent scattering amplitudes for the deuteriumand the host atoms, respectively. They include the corresponding Debye-Waller factors (73' = b exp

—

Q'2<[u2}). The sum goes over the interstitialsublattices (N=6 for tetrahedral sites in Nb), with sublattice postion vectors

µ. The vector q = Q—

Ghkl is the distance of the scattering vector Q from a

reciprocal lattice vector Ghkl. The function c (q) is the Fourier transform ofthe deuterium distribution, s (q) =£sr exp (zqr) is the Fourier transform of thelattice distortion caused by the dissolved D atom, where sr is the displacementof the host atom at a vector distance r for the D atom (see Fig. 12). For a

random distribution and without lattice distortion, Eq. (7) yields thewellknown isotropie Laue scattering da/dß = b'¿. The distortion is re-

sponsible for the term ib'h Qs ... which interferes with the Laue term b'D,leading to an intensity varying smoothly between the Bragg spots. Obviouslythe factorization in Eq. (7) holds for not too high concentrations such that thedisplacements add linearily. Both, the fluctuation c(q)2, and the distortionfactor in Eq. (7) have been extensively investigated by coherent neutronscattering. The distortion scattering was studied with the D7 spectrometer atthe Grenoble Reactor [40] where the integrated quasielastic intensity was

separated from the inelastic part by time-of-flight analysis. The samples were

single crystals of NbD0 026. Figure 13 shows typical results for Q in the (111)plane, after correction for incoherent contributions and background.Theoretical calculations of s (q) in Eq. (7) were based on the concept ofKanzaki forces [41,42]: The effect of the distortion is described by a set ofvirtual forces f"at neighbour atoms which produce the same strain as theinterstitial hydrogen atom. From these f" the elastic dipole tensor can bederived, with the Cartesian components Pi} = rnf). The Fourier transformsSj are related to the Fourier transformed virtual forces j) by the dynamicalmatrix of the host lattice, namely

] = - ^)/ ^) (8)where is known from the measured dispersion curves.

For tetrahedral or octahedral sites in a bec lattice, there are only diagonalcomponents Pu = P22 = A and P33 = B. The calculated curves shown inFig. 13 are based on the known values of bh and bD, and on the trace of thedipole tensor, 2A + B. This quantity is directly related to the (measured)volume expansion per dissolved deuterium. The Debye-Waller factor was

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aiti— Fig. 13. Diffuse coherent scattering on NbD0 026. Solid curve " ": only forces f,. "TC": f, andf2 such that A = B, with D on tetrahedral sites. "OC": D on octahedral sites, with A =B [40]

introduced as a fitting parameter, with wD = 0.035 Â2. Assuming that equalvirtual forces ft occur only at the 4 nearest neighbours (leading to A =/=B), one

is obviously lead to a contradiction with the scattering experiments (seeFig. 13), and as well, with the fact that the Snoek effect is negligible [39],which means A = (see sect. 5). To fulfill this condition of cubicity, forces f2have been applied also to second nearest neighbours (Fig. 12), leading to aratio {f\lr1)l(f2lr2) = 77. This assumption, introduced into Eqs. (7) and (8),yields very good agreement with experiment, also for orientations not shownin the figure. According to these measurements, the occupancy of octahedralsites was ruled out.

The results show that diffuse scattering experiments on such systems yielddetailed information on the forces or the displacements of the host atomssurrounding a dissolved hydrogen or deuterium atom. The calculation ofsuch displacements can be considered as an important quantity to testelectronic models of the hydrogen-metal interaction. The displacements are

also relevant for the atomistic understanding of the diffusive step.The fluctuation part of the coherent scattering c(q)2 can be studied for

small scattering vectors (Q = q < G), i.e. near the center of the Brillouinzone, Here the ^-dependence of the distortion factor in Eq. (7) can beneglected. Such experiments were carried out at the small angle scatteringfacitlity DI 1 at the Grenoble reactor, again on D in Nb. It is known [44] thatthis system reveals a phase separation between a dilute phase ("lattice gas")and an a.' phase ("lattice liquid") which are separated by a coexistence region

7 More recently, the cubic symmetry of the dipole tensor has been confirmed also by X-rayHuang scattering [43].

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160 T. Springer

*20

140S 0

SII [111]lfll = 0.12 iNbDjy,

h rs /

20C 400

ÖII [100]lûl = 0.07Â"

h /

f

600 150 350 550Temperature [ ]—

Fig. 14. Fluctuation scattering [see Eq. (9)] for deuterium in NbDv extrapolated to the depressedspinodal temperature Ts. = 0.31 = critical concentration. Tc = critical temperature [47]

in the (T, c) diagram, with a critical point at Tc = 440 and c = 31 atomic %.At Tc, the derivative of the concentration with respect to the chemicalpotential (dc/ µ) diverges. This critical behaviour is caused by the long-rangeelastic interaction of the dissolved deuterium atoms.

For an ordinary gas, the density fluctuations should diverge and,therefore, critical scattering should appear at Tc. However, the situation isqualitatively different for a "lattice gas" [45]. The concentration fluctuationsof the dissolved D atoms are associated with the local strain caused by thelong range elastic interaction opposing the attractive mutual interaction ofthe D atoms. For a "macroscopic experiment" (e.g. a thermodynamicmeasurement), these strain fields relax via the surface or the grain boundariesof the sample. Neutron scattering experiments, however, probe the short-range fluctuations of the concentration which are unable to relax. This leadsto a lowering of the spinodal temperature. Consequently, at the criticalconcentration and Tc the fluctuations do not diverge. This leads to theconcept of a Q-dependent spinodal temperature in writing

CT / c (q)2 =

r_ > (Q) (9)

where C is a quantity analogous to the Curie constant. For a macroscopicexperiment, \/Q is of the order of the sample thickness, and Ts is the usualspinodal temperature, as shown in phase diagrams. The suppression of Ts andtherefore of the critical scattering at Tc has been experimentally observed[46,47]. Recent results, showing / oc c2 (Q) are presented in Fig. 14. Anessential experimental difficulty was the careful separation of the elasticfluctuations from the rather strong contribution due to phonon modes. Thefigure clearly shows a Curie-Weiss behaviour near Te as in Eq. (9) with an

extrapolated spinodal temperature Ts (Q) far below TQ [The -dependence ofthe distortion factor in Eq. (7) can be neglected]. Further measurementsshowed also an anistropy of Ts (Q) as theoretically expected, related to the

336


elastic anisotropy of the host lattice. Unfortunately, the curves in the figurecannot be extended below the "macroscopic" critical temperature Tc sincephase separation occurs.

Resumé

Comparing the present state of neutron scattering investigations withthe results presented, for instance, at the last Conference of the"Bunsengesellschaft für Physikalische Chemie" on "Hydrogen MetalSystems" (Jülich 1972), we may make the following observations. First of all,quasielastic scattering at small Q has developped into a reliable method todetermine selfdiffusion constants in simple metals, as well as in complicated"technological" alloys. At larger scattering vectors Q, quasielastic experi-ments on single crystals show that the fast hydrogen diffusion is related tojump sequences as soon as the temperature is high. In this field, a progress inthe theoretical understanding was achieved recently, by working out detailedconcepts for the scattering law. Nevertheless, the basic atomistic in-terpretation of the diffusive jump mechanism is still in an early state, being a

complicated non-classical many-body problem [48]. This statement holdsalso with regard to the understanding of the isotope effect of diffusionconstants [5], and in view of the fact that apparent activation energies fordiffusion are smaller than the localized mode frequencies for hydrogen in Nb,Ta, or V (Table 1). Diffusion in the presence of trapping impurities can beinterpreted by a two-state model. However, this model still leaves open thephysics of the trapping process.

During the last years, a great wealth of experimental results was obtaineddealing with localized modes of hydrogen in metals, and with the interactionof the dissolved atoms with the host latticephonons. Also in this case there is alack of interpretation by "first principle" electron models8, in particularfor bcc systems. This holds also for the understanding of the displacements ofthe host atoms produced by the dissolved deuterium atom. On the otherhand, the mutual interaction of dissolved hydrogen, being essentially elasto-mechanic, is much better understood, for instance the behaviour of criticalscattering in the ( ') region ofNbD^. More experiments would be extremelyinteresting and could be related to existing theories [50] for instanceconcerning the size effect of critical phenomena.

AcknowledgementThe author acknowledges stimulating discussions with G. Alefeld, K. Kehr, V. Lottner,

A. Magerl and D. Richter.

8 As concerns calculations of the electronic structure of hydrides we refer to recent papers [49]and the literature quoted therein.

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162 T. Springer

References

1. . Sköld, in: Topics in Appi. Phys. 28, Hydrogen in Metals I, (G. Alefeld, J. Völkl, ed.)Springer Verlag, Berlin 1978, p. 267.

2. T. T. Springer, in: Topics in Appi. Phys. 28, Hydrogen in Metals I (G. Alefeld, J. Völkl, ed.)Springer Verlag, Berlin 1978, p. 75; On neutron spectroscopy in general see: Topics inCurrent Physics 3, Dynamics of Solids and Liquids by Neutron Scattering. Springer Verlag,Berlin 1977.

3. C. T. Chudley and R. Elliott, Proc. Phys. Soc. 77 (1961) 353.4. N. Boes and H. Züchner. J. less comm. metals 49 (1976) 226.5. G. Schaumann, J. Völkl and G. Alefeld, Phys. Stat. Sol. 42 (1970) 401, see also J. Völkl, this

conference.6. J. Völkl, Ber. Bunsenges. Physik. Chem. 76 (1972) 797.7. D. K. Ross and D. L. T. Wilson, Proc. IAEA Symp. Neutron Inelastic Scattering, vol. II,

p. 383 (1978).8. V. Lottner, A. Heim and T. Springer, Z. Physik 32 (1979) 157.9. D. Richter, . Alefeld, . Heidemann and . Wakabayashi, J. Phys. F 7 (1977) 569.

10. J. Völkl, H. G. Bauer, U. Freudenberg, M. Kokinidis, G. Lang, . . Steinhauser and G.Alefeld, Proc. Int. Conf. Friction and Ultrasonic Attenuation in Cryst. Sol. VI, Tokyo(1977).

11. J. Töpler, E. Lebsanft and R. Schätzler, J. Phys. F, Metal Physics 8 (1978) L25.12. E. Lebsanft and J. Töpler, to be published (1978), and this conference.13. K. Sköld and G. Nelin, J. Phys. Chem. Solids 28 (1967) 2369.

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G. Nelin and K. Sköld, J.Phys. Chem. Solids 36 (1975) 1175.

14. I. S. Anderson, D. K. Ross and D. J. Carlile, IAEA Symp. Neutron Inelastic Scattering,Vol. II, p. 421, Vienna 1978.

15. R. Kutner, I. Sosnowska and W. Kley, Phys. Stat. Sol. (B) 89 (1978) 29.16. L. A. de Graaf, J. J. Rush, . H. FlotowandJ. M. Rowe, J. Chem. Phys. 56(1972)4574.-

W. Gissler and H. Rother, Physica 50 (1970) 380, and W. Gissler, Ber. Bunsenges. Physik.Chem. 76 (1972) 770.

17. V. Lottner, J. W. Haus, . Heim and K. Kehr, J. Chem. Phys. Sol. in print (1979).18. N. Wakabayashi, B. Alefeld, K. W. Kehr and T. Springer, Solid State Commun. 15 (1974)

503.19. W. Gissler, B. Jay, R. Rubin and L. A. Vinhas, Phys. Lett. 43 A (1973) 279.

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G. Bauer et al.[40].

20. J. W. Haus and K. W. Kehr, Solid State Comm. 26 (1978) 753.21. E. W. Montroll and G. Weiss, J. Math. Phys. 6 (1965) 167.21a. W. Gissler and N. Stump, Physica 65 (1973) 109.22. H. Y. Chang and C. Wert, Acta Metall. 21 (1973) 1233.

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P. Kofstadt, W. E. Wallace and L.J. Hyvonen, J. Am. Chem. Soc. 81 (1959) 5019.

23. W. Munzing, J. Völkl, H. Wipf and G. Alefeld, Scripta Metall 8 (1974) 1327.24. D. Richter and T. Springer, Phys. Rev. 18 (1978) 126.25. . W. Kehr and D. Richter, Solid State Comm. 20 (1976) 477.26. K. S. Singwi and . Sjölander, Phys. Rev. 119 (1960) 863.26a. G. Pfeiffer and H. Wipf, J. Phys. F 6 (1976) 167.27. G. J. Sellers, A. C. Anderson and H. K. Birnbaum, Phys. Rev. B 7 (1974) 2771.28. G. Alefeld and M. Wipf, pers. Commun. (1978).29. V. Lottner, A. Kollmar, T. Springer, W. Kress, H. Bilz and W. D. Teuchert, Proc. Conf. on

Lattice Dynamics (ed. M. Balkanski), p. 247 (1977), for pure Nb see: Y. Nakagawa and A.D. B. Woods, Phys. Rev. Lett. 11 (1963) 271.

30. J. M. Rowe, J. J. Rush, H. G. Smith, M. Mostoller and H. E. Flotow, Phys. Rev. Lett. 33(1974) 1297, for pure Pd see: A. Miller and B. N. Brockhouse, Can. J. Phys. 49 (1971) 704.

31. A. Rahman, K. Sköld, G. Pelizarri, S. . Sinha and . Flotow, Phys. Rev. 14 (1976) 3630.

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32. A. Eichler, H. Wühl and B. Strizker, Solid State Comm. 17 (1975) 213, see also B. Stritzkerand W. Buckel, . Phys. 257 (1972) 1, and W. Buckel, this conference.

33. . M. Powell, P. Martel and A. D. B. Woods, Phys. Rev. 171 (1968) 727.34. V. Lottner, H. R. Schober and W. J. Fitzgerald, submitted Phys. Rev. Lett. 1979, see also this

conference.35. S. M. Shapiro, Y. Nöda, . O. Brun, J. Miller, H. Birnbaum and T. Kajitani, Phys. Rev.

Lett. 41 (1978) 1051.36. P. H. Dederichs, C. Lehmann, H. R. Schober, . Scholz and R. Zeller, J. Nucl. Materials

69/70 (1978) 176.37. A. Magerl, . Bérre and G. Alefeld, Phys. Stat. Sol. (a) 36 (1976) 161.38. A. Magerl, W. D. Teuchert and R. Scherm, J. Phys. C (Solid State Phys.) 11 (1978) 2175.39. J. Buchholz, J. Völkl and G. Alefeld, Phys. Rev. Lett. 30 (1973) 318.40. GS. Bauer and W. Schmatz, Proc. of'2ème Congrès International sur l'Hydrogène dans

les Métaux', (1977) and G. S. Bauer, E. Seitz, H. Horner and W. Schmatz, Solid StateComm. 17 (1975) 161.

41. H. Kanzaki, J. Phys. Chem. Sol. 2 (1957) 24 and 107.42. J. D. Eshelby, Solid State Phys. 3 (1956) 79 (New York Acad. Press).43. H. Metzger and H. Peisl, J. Phys. F (Metal Phys.) 8 (1978) 391.44. G. Alefeld, Ber. Bunsenges. Physik. Chem. 76 (1972) 746.45. J. Cahn, Acta Metall. 10 (1962) 907.46. H. Conrad, G. Bauer, G. Alefeld, T. Springer and W. Schmatz, . Physik 266 (1974) 239.47. W. Münzing, . Stump and G. Goeltz, J. Appi. Cryst. (1978), in print, and Proc. IAEA

Symp. Neutron Inelastic Scattering; Vol.11 p. 317, Vienna 1978.48. see e.g. A. M. Stoneham, J. Nucl. Materials 69/70 (1978) 109 and references therein, and

D. Emin, M. 1. Baskes and W. D. Wilson, this conference.—

Z. physik. Chem. Neue Folge114 (1979) 231

49. see Hydrogen in Metals, Topics in Applied Physics 28/29 (G. Alefeld, J. Völkl, ed.) Springer-Verlag, Berlin 1978): E. Wicke and H. Brodowsky, Vol. II, p. 73, and A. C. Switendick,Vol. I, p. 101.

50. R. Bausch, H. Horner and H. Wagner, J. Phys. C. (Solid State Phys.) 8 (1975) 2559.

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Investigation of Metal-Hydrogen Systems by Means of Neutron Scattering