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Neutron scattering study of deuterium short-range order in VDx
Journal of
ALL~
AHD COM~DUHD$ ELSEVIER
Journal of Alloys and C o m p o u n d s 231 (1995) 121-125
Neutron scattering study of deuterium short-range order in V D x U . K n e l l a, H . W i p f a, G . L a u t e n s c h l ~ i g e r b, R . H o c k b, H . W e i t z e l b, R . R . A r o n s c, E . R e s s o u c h e c alnstitut f iir Festk,~rperphysik, Technische Hochschule Darmstadt, Hochschulstrafle 6, D-64289 Darmstadt, Germany bFachbereich Materialwissenschaften, Technische Hochschule Darmstadt, Petersenstrafle 20, D-64287 Darmstadt, Germany CDRFMC, MDN, Centre d'Etude Nucl~aire de Grenoble, 17 Avenue des Martyrs,, F-38041 Grenoble, France
Abstract The short-range order ot the D interstitials in two (a' phase) VD x powder samples (x = 0.75 and 0.82) was investigated at room temperature by diffuse neutron scattering. The study demonstrates a (within experimental accuracy complete) blocking of interstitial sites in the two nearest shells around a given D atom, a reduced occupation probability for the sites in the third shell and an increased occupation probability of the sites in the fourth, fifth and sixth shells. These results prove directly the existence of a strong short-range repulsion between the D atoms. Keywords:
Diffuse scattering; Neutron diffraction;Short-range order; Metal-hydrogen systems
1. Introduction The most direct information on the short-range order of the hydrogen interstitials is derived from the diffuse scattering intensity as determined by neutron diffraction [1,2]. Two such studies were indeed recently carried out on the deuterided a (or a ' ) phase systems NbD x [3,4] and V D x [5] in which the D atoms occupy tetrahedral sites :in a b.c.c. Nb or V lattice (the symbols H and D distinguish between the two hydrogen isotopes whereas the word hydrogen applies to both of them). The studies were performed on the deuterated systems since D has more favorable scattering cross sections [6]. They reported a strong shortrange repulsion between the hydrogen interstitials, as already previously suggested from band structure calculations [7,8], from the structure of hydride phases [9] and from the hydrogen solubility data [10-16]. In this paper, we report the results of an extension of our previous study [5] on the VD x system, now carried out for the two concentrations x = 0.75 and 0.82.
2. Samples and experimental results The experiments were performed at room temperature on two VD x powder samples with x = 0.75 and 0.82 (mean particle size, about 30 ~m). The samples were enclosed in an AI container (x = 0.75) or in a V 0925-8388/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0925-8388(95)001785-2
container (x = 0.82) with slightly different diameters of about 1 cm. Multiple scattering events represented about 13% (x = 0 . 7 5 ) o r about 10% (x = 0 . 8 2 ) o f the t o t a l scattering intensity [17]. We performed background reference scans, carried out with a hydrogenfree V sample in the AI container (x = 0.75) and with the empty V c o n t a i n e r (x=0.82). Our neutron data were taken with the powder diffractometer DN5 at the Siloe reactor of the Centre d'Etude Nucl6aire de Grenoble. The wavelength of the incident neutrons was 1.349 ~,. For more details, see [5]. Fig. 1 presents the powder diffractogram of the VD0.82sample, obtained as the difference between the scattering intensities from the sample plus V container and empty container. The diffraetogram shows Bragg peaks, specified on the bar below, and an additional diffuse scattering intensity that is most directly observable between the Bragg peaks. The diffuse scattering intensity reflects the shortrange order of the D atoms which will be described by correlated probabilities cj for the occupation of the interstitial sites in a shell j around a given D atom. More specifically, cj is the probability that an interstitial site in shell j is occupied, provided that the site in the center of the shell is also occupied. In the presence of correlation effects, the cj differ from the simple occupation probability c = x / 6 of a tetrahedral site (there are six tetrahedral sites per V atom). We
122
U. Knell et al. / Journal of Alloys and Compounds 231 (1995) 121-125
w h e r e o-i.v = 5 . 0 8 b [6] is the incoherent scattering cross-section of V, m v is the mass of a V atom, kBT is the t h e r m a l energy, e x p [ - 2 W v ( Q ) ] is the D e b y e Waller factor of the V, k i is the w a v e v e c t o r of the incoming neutrons, and kf_~ or ke.+l are the wavevectors of the scattered n e u t r o n after the annihilation or creation respectively of a p h o n o n with energy hto. T h e later two w a v e v e c t o r s are given by
1600
VD0s2 •~ 1350 • .;'. . , , S-..~.~..... "i 850
6oo VDo.82 I
• ~.
"~
~ ,
, , ~, ,_. ~
J 20
kf,~l
, ~ , ~ , ~ 40 6o 8o scattering angle 20 (degree) I 110
I 200
,
100
I I I 211 220 310
Fig. 1. Powder diffractogram of the VD0s2 sample. The figure shows the difference between the scattering intensities from sample plus V container and empty container in a plot vs. scattering angle. The bar below the scattering data indicates the position of the Bragg peaks of the sample,
m e n t i o n that s h o r t - r a n g e o r d e r is frequently characterized b y differently defined p a r a m e t e r s 5 [1] or aj [2], w h e r e the relation b e t w e e n cj, ei and aj is given by cj - c = ~/c = aj(1 - c ) . T h e aim of o u r study is the d e t e r m i n a t i o n of the cj f r o m a theoretical description of the diffuse scattering intensity. F o r such a description, the system V D x is ideally suited since the c o h e r e n t scattering cross-seetion of D is a b o u t 300 times larger than the c o h e r e n t cross-section of V [6], so that we can c o m p l e t e l y neglect the c o h e r e n t scattering f r o m the V atoms. In this case, the total (coherent and i n c o h e r e n t ) diffuse scattering intensity of the V D samples consists of (i) the c o h e r e n t elastic scattering f r o m the D short-range order, (ii) the c o h e r e n t elastic scattering f r o m the static displacements of the D atoms, (iii) the incohere n t elastic s c a t t e r i n g f r o m the D a t o m s , (iv) the c o h e r e n t and i n c o h e r e n t inelastic scattering f r o m the D a t o m s and (v) the i n c o h e r e n t elastic and inelastic scattering f r o m the V a t o m s [5]. T h e contributions ( i ) - ( i v ) have b e e n quantitatively discussed in o u r previous p a p e r [5], so that we do not r e p e a t the equations here. T h e final contribution (v) is the only one that results f r o m a scattering f r o m the V atoms. T h e scattering cross-section (dtr/dg2)v p e r V a t o m due to this contribution can be written as [1,2]
=
k~
+
2
to 1/2
(2)
w h e r e m N is the mass of a neutron. T h e quantities Q ' Q,, and are Ik i - k f _ 11 and Iki - kf.+ 11 respectively and Zv(to ) is the p h o n o n density of states of the V host lattice, n o r m a l i z e d to unity. In our n u m e r i c a l calculation, we used for Zv(to ) and for the D e b y e - W a l l e r f a c t o r e x p [ - 2 W v ( Q ) ] e x p e r i m e n t a l values for hydrogen-free V [18,19] since the vibrational p r o p e r t i e s of the V lattice do not substantially change in the prese n c e of the hydrogen. It is i m p o r t a n t to point out that the short-range o r d e r contribution (i) is the m o s t relevant of the contributions above. It d e p e n d s sensitively on the cj, which allows their reliable d e t e r m i n a t i o n , and it exceeds all the other scattering contributions f r o m the D atoms, and it does so (as we shall see) even for the s u m of these contributions. This allows the use of simplifying a p p r o x i m a t i o n s for the contributions (ii) ad (iv), as discussed in [5], without impairing the reliability of the data analysis. Fig. 2 presents our results for the diffuse scattering intensity f r o m the D a t o m s of the investigated V D x samples. T h e figure does not show the B r a g g peaks,
. 600
(a)
400
/
~ 20o ~ ~ 0 ~. ~ 600 ~f~ .~ 400
. . . ~_~o_ ~ ° ~ ~ ~ '
\'D~,7~ _~o - - t~ ~
......................................................... ..... displacements + incoherent elastic • inelastic , ~ , ~ , ~ , ~ , ' ' ' ' (b) o o VD0.82 ~it~ - fit
20O =
v
~
exp[-2Wv(Q)]+
dto 2 m v t°
o~
0
(k~._ 1 Q,2 e x p ( _ 2 W v ( a , ) ] X \
+
ki
exp(hto/k BT)-x
kf.+ 1 a,,2 exp[_2Wv(Q,,)] \ ] k----~ ~ - - ~ )J
(1)
:'---'-"displacements + incoherent elastic + inelastic I ~ J ~ I 20 40I 60I 80 100 scattering angle 20 (degree)
Fig. 2. Diffuse scattering intensity of the D atoms in (a) VDo.v5and (b) VDo.s2 plotted vs. the scattering angle: - fits to the data, performed under consideration of the scattering contributions (i)(iv); - - - , sum of the intensity contributions (ii), (iii) and (iv). For more details, see the text.
U. Knell et al. / Journal of Alloys and Compounds 231 (1995) 121-125
and their close neighborhood, since the Bragg peaks do not characterize the diffuse scattering behavior. Further, besides carrying out a very minor absorption correction [17], we subtracted from the original data a small contribution calcu]tated for multiple scattering [17] and, more importantly, the contribution (v) from the incoherent scattering of the V. In the case of the VD0.75 sample, contribution (v) was obtained from the reference scan on a hydrogen-free V sample. For the VD0.s2 sample, the reference scan was performed with the empty V container, so that it did not yield contribution (v). Therefore this contribution was calculated from Eqs. (1) and (2). We mention finally that we did not account for the very minor effects of absorption and multiple scattering in the data analysis of our previous study on the VD0.75 sample [5], so that there are slight differences (well within the quoted errors) between the previous and the present results for this sample.
The data in Fig. 2 show, for both samples, the diffuse scattering intensity from the D atoms, specified by
the
contributions
(i)-(iv).
With
the
help
of the
expressions for the scattering cross-sections of these
four contributions, as given in [5], we performed fits to the scattering intensities iin Fig. 2. Unknown quantities were the correlated occupation probabilities cj, together with a normalization factor that was identical for all four contributions. We considered short-range order effects for the D atoms up to the fifteenth shell of tetrahedral sites. This means that the cj could differ from c for j <~ 15, whereas they were identical with c for j > 15. A further simplification was that all cj in the range 8 ~ < j ~< 15 were assumed to be identical. We applied this simplification since the quality of the fits required (especially for the data at small scattering angles) a consideration of more than seven shells whereas, at the same time, essentially only the average of the cj of the outer shells was of relevance and could, accordingly, be determined. We considered finally 1000 shells in our summation for the scattering cross section from the static displacements of the D atoms (contribution (ii); see [5]). This turned out to be sufficient since parallel calculations with a smaller number of shells (usually 100) yielded almost the same result, The results of the fits for the diffuse scattering intensities in Fig. 2 are indicated by full curves. The broken curves show solely the sum of the intensities from the scattering contributions (ii)-(iv). It can be seen that these contributions represent only a minor part of the scattering intensity of the D atoms. This shows that the scattering contribution (i) from the short-range order of the D atoms is in fact the dominant contribution te, the diffuse scattering intensity of the D atoms, Our results for the correlated occupation probabilities cj are presented by the full data points in Fig.
123
shell numbcrj 1 2 5 to 15 I I ' ' I .... I .... t'" ~- 0.2 VD075c ~0.125 .~ ~ ~ ~ ...........................................* , , - n 6 - r o ~ ~ 0.1 ~.
(a)
~ = ~ 0.2 m "~
,
, I .t , , ~ I I ' ' I .... VD0.82 c=0.137 ................................
~. . . . . . . . . .
. ~ . t . . . . I' "
i f - IF - i f - o - - ~ - i f i - - - 0 - o o o t
~ 0.1 Co) 06
,
~~ .~ , . , . 1 2 3 distance Rj [/~]
, 4
Fig. 3. Correlated occupation probabilities cj for (a) VD0.TS and (b) VD0~: vs. the distance Rj between a site of shell j and the site in the center of the shells: the top of the figure shows the number j of the respective shell; . . . . , the occupation probabilities c = 0.125 and 0.137 for VD0m and VD0.s2 respectively: O, 1 ~
3, plotted vs. the distance Rj between a site of shell j and the site in the center of the shells (the top of the figure shows the shell number j). The broken lines indicate the occupation probabilities c of the two respective samples. For j 1> 16, the open circles show finally the fixed values cj = c which were not a variable parameter in the fits.
3. Diseussion The results in Fig. 3 are essentially identical for both investigated samples. They show that a D atom blocks (within experimental accuracy completely) the interstitial sites in the first and in the second shell. This blocking of the sites in the two nearest shells demonstrates clearly the existence of a strong short-range repulsion between the D atoms, as suggested in the previous studies [3,5,7-16]. For the sites of the third shell, we find a c 3 value that is definitely smaller than c, corresponding to a partial blocking of the sites of this shell. This behavior demonstrates that the repulsion between the D atoms extends up to the third shell, or up to distances of 2 ,h.. The results in Fig. 3 show further increased cj for the sites of the fourth, fifth and sixth shell where, in particular, c 5 is clearly above the average value c. Finally, we find very slightly reduced cj for the sites in the more distant shells. Let us now consider small m o m e n t u m transfers hQ.
U. Knell et al. / Journal o f Alloys and Compounds 231 (1995) 121-125
124
In the limit Q---> E, where e tends to zero but remains much larger than the reciprocal dimension L of the bulk material, the scattering cross-section (do-/dY2)SRO per D atom, resulting from the short-range order contribution (i), can be written as (Eqs. (1.12), (9.4), (9.7) and (17.4) of [1], using the nomenclature of [4]; see also [12]) lim O~,
= ~
SRO
(3) 4~r
where o-c,D is the coherent cross-section of the D atoms. The quantity fbulk is the thermodynamic factor for the bulk modes of the hydrogen [4,12] and the angular brackets on the right-hand side of Eq. (3) indicate an orientational average, which is required for a powder sample because fbulk depends on the direction of Q. Since our subsequent discussion remains semiquantitative anyway, we present only the simple expression
c(Ol~/Oc) 2[(1-2o-)/(1 - - o.)]cpoB AV E fbulkka T + kB T
(4)
for fbulk that holds in the case of complete elastic isotropy. In this equation, /.t is the chemical potential of the hydrogen, P0 is the number of tetrahedral interstitial sites per unit volume, B -- 157 × 10 9 N m 2 [20] is the bulk modulus of the sample, o" ~--0.36 [20] is its Poisson's ratio and AV= 2 . 6 A 3 [21] is the volume increase of the sample per D atom. We point out that the above restriction Q >>1/L implies that the concentration fluctuations of the hydrogen, which determine the scattering cross-section, have a wavelength much smaller than L (bulk modes [4,12]). The first term on the right-hand side of Eq. (4) is the (ordinary) thermodynamic factor ftherm-'~-'C(O[~/ Oc)/kBT [4], which is positive for stability reasons and can be derived from hydrogen solubility measurements [10,11]. However, the quality of the available experimental data [10,11] does not allow a reliable determination of ftherm for the present D concentrations. What only can be concluded from the data is that ftherm is very large, in fact certainly larger than 10. The second (and positive) term on the right-hand side of Eq. (4), with a numerical value of about 4.2, reflects the influence of elastic coherency stresses [1,4,12]. These stresses arise in the presence of concentration fluctuations of the hydrogen, raise the elastic energy of the sample and suppress therefore the amplitude of the fluctuations. According to the numerical estimates above, both terms on the right-hand side of Eq. (4) together yield a value for fb~k which is certainly larger than 10, so that (1/fbu,k) in Eq. (3) is expected to be smaller than 0.1. The limit Q--->e lim (do./d£2)SRO of Eq. (3) can be compared with the general expression for (do./d/2)SRO,
which depends on the correlated occupation probabilities cj and is, for example given in Eq. (1) of our previous paper [5] (see also [1,2]). From the derivation of this general expression, it is clear that it describes the effects of hydrogen fluctuations well within the bulk material, so that we can replace the limit Q---> E by the limit Q---->0. For this limit, the comparison between Eq. (3) and the general expression for (do./ dO)sRo shows that the cj must fulfill the thermodynamic relation 0<1-c+
~ Nj(cj - c) j=l
=
(f~k)
<0.1
(5)
where Nj is the number of tetrahedral sites in shell j and the summation extends over all shells j. The number 0 to the very left reflects the fact that fbutk is positive and the number 0.1 to the right expresses the above estimate for (1/fbul k). From the present correlated occupation probabilities cj, as given in Fig. 3, we +0.9 and 0.09(_0.8) +0.9 forVD0.75 obtain (1/fbu,k) ~ 0.008(_0.8) and VO0.82 respectively. These results agree with the theoretical expectation in Eq. (5). It is further worth mentioning that complete blocking of the first two shells and the reduced correlated occupation probability c 3 in Fig. 3 yield, for both samples, a negative value of about -0.2 for the expression 1 - c + E3=l N~(cj - c) which includes the first three terms of the sum in Eq. (5). The remaining terms Ej~4 Nj (cj - c) of the sum must therefore be at least as large as about 0.2. This means that the short-range repulsion, which leads to the blocking effects in the first three shells, necessarily requires cj for outer shells that are larger than the average occupation probability c, as experimentally observed for the fourth to sixth shells. We conclude with a comment on the slightly reduced cj for the sites in the outer shells j i> 7. These slightly reduced cj may indicate the influence of coherency stresses which can also be considered to reflect a repulsive contribution to the total long-range elastic interaction between the hydrogen interstitials [1,4,12]. However, such an interpretation is certainly speculative since, for an elastically isotropic material, the repulsive long-range contribution (from the coherency stresses) is exactly compensated by an attractive long-range contribution which influences the concentration dependence of the chemical potential tt[1,12]. This means that, again for an elastically isotropic material, we do not expect a net long-range repulsion as required for the slightly reduced cj of the outer shells. On the contrary, the present sample material is definitely not elastically isotropic, so that the different orientational averages for the chemical potential and the scattering cross-section (of a powder sample) may indeed lead to a situation, in which the
U. Knell et al. / Journal of Alloys and Compounds 231 (1995) 121-125
resulting scattering cross section must be described by a n effective net long-range repulsion between the hydrogen atoms.
Acknowledgment
[6] V.F.Sears, Neutron News, 3 (1992) 27. [7] A.C. Switendick, Z. Phys. Chem., N.F., 117 (1979) 89. [8] A. Mokrani and C. Demangeat, Z. Phys. Chem., N.F., 163 [9] [10] [11] [12]
The present work was supported by the Bundesrninisterium ftir Forschung und Technologie.
[13] [14]
References
[15] [16]
[1] M.A. Krivoglaz, Theory of X-ray and Thermal-Neutron Scattering by Real Crystals, Plenum, New York, 1969. [2] B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading, MA, 1969. [3] R. Hempelmann, D. Richter, D.A. Faux and D.K. Ross, Z. Phys. Chem. N.F., 159 (1988) 175. [4] H. Wipf, J. V61kl and G. Alefeld, Z. Phys. B, 76 (1989) 353. Eq. (5) of this reference is misprinted; it should read E 1 = P2/B and Ebulk = p2[Cll. [5] U. Knell, H. Wipf, G. Laulensehl~iger, R. Hock, H. Weitzel and E. Ressouche, J. Phys.: Condens. Matter, 6 (1994) 1461.
125
[17] [18] [19] [20] [21]
(1989) 541. D.G. Westlake, J. Less-Common Met., 91 (1983) 1. E. Veleckis and R.K. Edwards, J. Phys. Chem., 73 (1969) 683. W. Rummel, Siemens Forsch.-Entwickl.-Ber., 10 (1981) 371. H. Wagner, in G. Alefeld and J.V61k (eds.) Hydrogen in Metals I, in Top. Appl. Phys., 28 (1978) 5. G. Boureau, J. Phys. Chem. Solids, 45 (1984) 973. M. Sotzek, H. Siebler, H. Wipf and U. Leuth~iusser, J. LessCommon Met., 104 (1984) 21. A.I. Shirley and C.K. Hall, Phys. Rev. B, 33 (1986) 8084, 8099. P. Meuffels and W.A. Oates, J. Less-Common Met., 130 (1987) 403. I.A. Blech and B.L. Averbach, Phys. Rev., 137 (1965) Al113. P.H. Dederichs, H. Schober and D.J. Sellmyer (eds.), LandoltBOrnstein, New Series, Vol. 13a, Springer, Berlin, 19 , p. 163. M. Kamal, S.S. Malik and D. Rorer, Phys. Rev. B, 18 (1978) 1609. A. Magerl, B. Berre and G. Alefeld, Phys. Status Solidi A, 36 (1976) 161. H. Peisl, in G. Alefeld and J. V61kl (eds.), Hydrogen in Metals I, in Top. Appl. Phys., 28 (1978) 53.
ALL~
AHD COM~DUHD$ ELSEVIER
Journal of Alloys and C o m p o u n d s 231 (1995) 121-125
Neutron scattering study of deuterium short-range order in V D x U . K n e l l a, H . W i p f a, G . L a u t e n s c h l ~ i g e r b, R . H o c k b, H . W e i t z e l b, R . R . A r o n s c, E . R e s s o u c h e c alnstitut f iir Festk,~rperphysik, Technische Hochschule Darmstadt, Hochschulstrafle 6, D-64289 Darmstadt, Germany bFachbereich Materialwissenschaften, Technische Hochschule Darmstadt, Petersenstrafle 20, D-64287 Darmstadt, Germany CDRFMC, MDN, Centre d'Etude Nucl~aire de Grenoble, 17 Avenue des Martyrs,, F-38041 Grenoble, France
Abstract The short-range order ot the D interstitials in two (a' phase) VD x powder samples (x = 0.75 and 0.82) was investigated at room temperature by diffuse neutron scattering. The study demonstrates a (within experimental accuracy complete) blocking of interstitial sites in the two nearest shells around a given D atom, a reduced occupation probability for the sites in the third shell and an increased occupation probability of the sites in the fourth, fifth and sixth shells. These results prove directly the existence of a strong short-range repulsion between the D atoms. Keywords:
Diffuse scattering; Neutron diffraction;Short-range order; Metal-hydrogen systems
1. Introduction The most direct information on the short-range order of the hydrogen interstitials is derived from the diffuse scattering intensity as determined by neutron diffraction [1,2]. Two such studies were indeed recently carried out on the deuterided a (or a ' ) phase systems NbD x [3,4] and V D x [5] in which the D atoms occupy tetrahedral sites :in a b.c.c. Nb or V lattice (the symbols H and D distinguish between the two hydrogen isotopes whereas the word hydrogen applies to both of them). The studies were performed on the deuterated systems since D has more favorable scattering cross sections [6]. They reported a strong shortrange repulsion between the hydrogen interstitials, as already previously suggested from band structure calculations [7,8], from the structure of hydride phases [9] and from the hydrogen solubility data [10-16]. In this paper, we report the results of an extension of our previous study [5] on the VD x system, now carried out for the two concentrations x = 0.75 and 0.82.
2. Samples and experimental results The experiments were performed at room temperature on two VD x powder samples with x = 0.75 and 0.82 (mean particle size, about 30 ~m). The samples were enclosed in an AI container (x = 0.75) or in a V 0925-8388/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0925-8388(95)001785-2
container (x = 0.82) with slightly different diameters of about 1 cm. Multiple scattering events represented about 13% (x = 0 . 7 5 ) o r about 10% (x = 0 . 8 2 ) o f the t o t a l scattering intensity [17]. We performed background reference scans, carried out with a hydrogenfree V sample in the AI container (x = 0.75) and with the empty V c o n t a i n e r (x=0.82). Our neutron data were taken with the powder diffractometer DN5 at the Siloe reactor of the Centre d'Etude Nucl6aire de Grenoble. The wavelength of the incident neutrons was 1.349 ~,. For more details, see [5]. Fig. 1 presents the powder diffractogram of the VD0.82sample, obtained as the difference between the scattering intensities from the sample plus V container and empty container. The diffraetogram shows Bragg peaks, specified on the bar below, and an additional diffuse scattering intensity that is most directly observable between the Bragg peaks. The diffuse scattering intensity reflects the shortrange order of the D atoms which will be described by correlated probabilities cj for the occupation of the interstitial sites in a shell j around a given D atom. More specifically, cj is the probability that an interstitial site in shell j is occupied, provided that the site in the center of the shell is also occupied. In the presence of correlation effects, the cj differ from the simple occupation probability c = x / 6 of a tetrahedral site (there are six tetrahedral sites per V atom). We
122
U. Knell et al. / Journal of Alloys and Compounds 231 (1995) 121-125
w h e r e o-i.v = 5 . 0 8 b [6] is the incoherent scattering cross-section of V, m v is the mass of a V atom, kBT is the t h e r m a l energy, e x p [ - 2 W v ( Q ) ] is the D e b y e Waller factor of the V, k i is the w a v e v e c t o r of the incoming neutrons, and kf_~ or ke.+l are the wavevectors of the scattered n e u t r o n after the annihilation or creation respectively of a p h o n o n with energy hto. T h e later two w a v e v e c t o r s are given by
1600
VD0s2 •~ 1350 • .;'. . , , S-..~.~..... "i 850
6oo VDo.82 I
• ~.
"~
~ ,
, , ~, ,_. ~
J 20
kf,~l
, ~ , ~ , ~ 40 6o 8o scattering angle 20 (degree) I 110
I 200
,
100
I I I 211 220 310
Fig. 1. Powder diffractogram of the VD0s2 sample. The figure shows the difference between the scattering intensities from sample plus V container and empty container in a plot vs. scattering angle. The bar below the scattering data indicates the position of the Bragg peaks of the sample,
m e n t i o n that s h o r t - r a n g e o r d e r is frequently characterized b y differently defined p a r a m e t e r s 5 [1] or aj [2], w h e r e the relation b e t w e e n cj, ei and aj is given by cj - c = ~/c = aj(1 - c ) . T h e aim of o u r study is the d e t e r m i n a t i o n of the cj f r o m a theoretical description of the diffuse scattering intensity. F o r such a description, the system V D x is ideally suited since the c o h e r e n t scattering cross-seetion of D is a b o u t 300 times larger than the c o h e r e n t cross-section of V [6], so that we can c o m p l e t e l y neglect the c o h e r e n t scattering f r o m the V atoms. In this case, the total (coherent and i n c o h e r e n t ) diffuse scattering intensity of the V D samples consists of (i) the c o h e r e n t elastic scattering f r o m the D short-range order, (ii) the c o h e r e n t elastic scattering f r o m the static displacements of the D atoms, (iii) the incohere n t elastic s c a t t e r i n g f r o m the D a t o m s , (iv) the c o h e r e n t and i n c o h e r e n t inelastic scattering f r o m the D a t o m s and (v) the i n c o h e r e n t elastic and inelastic scattering f r o m the V a t o m s [5]. T h e contributions ( i ) - ( i v ) have b e e n quantitatively discussed in o u r previous p a p e r [5], so that we do not r e p e a t the equations here. T h e final contribution (v) is the only one that results f r o m a scattering f r o m the V atoms. T h e scattering cross-section (dtr/dg2)v p e r V a t o m due to this contribution can be written as [1,2]
=
k~
+
2
to 1/2
(2)
w h e r e m N is the mass of a neutron. T h e quantities Q ' Q,, and are Ik i - k f _ 11 and Iki - kf.+ 11 respectively and Zv(to ) is the p h o n o n density of states of the V host lattice, n o r m a l i z e d to unity. In our n u m e r i c a l calculation, we used for Zv(to ) and for the D e b y e - W a l l e r f a c t o r e x p [ - 2 W v ( Q ) ] e x p e r i m e n t a l values for hydrogen-free V [18,19] since the vibrational p r o p e r t i e s of the V lattice do not substantially change in the prese n c e of the hydrogen. It is i m p o r t a n t to point out that the short-range o r d e r contribution (i) is the m o s t relevant of the contributions above. It d e p e n d s sensitively on the cj, which allows their reliable d e t e r m i n a t i o n , and it exceeds all the other scattering contributions f r o m the D atoms, and it does so (as we shall see) even for the s u m of these contributions. This allows the use of simplifying a p p r o x i m a t i o n s for the contributions (ii) ad (iv), as discussed in [5], without impairing the reliability of the data analysis. Fig. 2 presents our results for the diffuse scattering intensity f r o m the D a t o m s of the investigated V D x samples. T h e figure does not show the B r a g g peaks,
. 600
(a)
400
/
~ 20o ~ ~ 0 ~. ~ 600 ~f~ .~ 400
. . . ~_~o_ ~ ° ~ ~ ~ '
\'D~,7~ _~o - - t~ ~
......................................................... ..... displacements + incoherent elastic • inelastic , ~ , ~ , ~ , ~ , ' ' ' ' (b) o o VD0.82 ~it~ - fit
20O =
v
~
exp[-2Wv(Q)]+
dto 2 m v t°
o~
0
(k~._ 1 Q,2 e x p ( _ 2 W v ( a , ) ] X \
+
ki
exp(hto/k BT)-x
kf.+ 1 a,,2 exp[_2Wv(Q,,)] \ ] k----~ ~ - - ~ )J
(1)
:'---'-"displacements + incoherent elastic + inelastic I ~ J ~ I 20 40I 60I 80 100 scattering angle 20 (degree)
Fig. 2. Diffuse scattering intensity of the D atoms in (a) VDo.v5and (b) VDo.s2 plotted vs. the scattering angle: - fits to the data, performed under consideration of the scattering contributions (i)(iv); - - - , sum of the intensity contributions (ii), (iii) and (iv). For more details, see the text.
U. Knell et al. / Journal of Alloys and Compounds 231 (1995) 121-125
and their close neighborhood, since the Bragg peaks do not characterize the diffuse scattering behavior. Further, besides carrying out a very minor absorption correction [17], we subtracted from the original data a small contribution calcu]tated for multiple scattering [17] and, more importantly, the contribution (v) from the incoherent scattering of the V. In the case of the VD0.75 sample, contribution (v) was obtained from the reference scan on a hydrogen-free V sample. For the VD0.s2 sample, the reference scan was performed with the empty V container, so that it did not yield contribution (v). Therefore this contribution was calculated from Eqs. (1) and (2). We mention finally that we did not account for the very minor effects of absorption and multiple scattering in the data analysis of our previous study on the VD0.75 sample [5], so that there are slight differences (well within the quoted errors) between the previous and the present results for this sample.
The data in Fig. 2 show, for both samples, the diffuse scattering intensity from the D atoms, specified by
the
contributions
(i)-(iv).
With
the
help
of the
expressions for the scattering cross-sections of these
four contributions, as given in [5], we performed fits to the scattering intensities iin Fig. 2. Unknown quantities were the correlated occupation probabilities cj, together with a normalization factor that was identical for all four contributions. We considered short-range order effects for the D atoms up to the fifteenth shell of tetrahedral sites. This means that the cj could differ from c for j <~ 15, whereas they were identical with c for j > 15. A further simplification was that all cj in the range 8 ~ < j ~< 15 were assumed to be identical. We applied this simplification since the quality of the fits required (especially for the data at small scattering angles) a consideration of more than seven shells whereas, at the same time, essentially only the average of the cj of the outer shells was of relevance and could, accordingly, be determined. We considered finally 1000 shells in our summation for the scattering cross section from the static displacements of the D atoms (contribution (ii); see [5]). This turned out to be sufficient since parallel calculations with a smaller number of shells (usually 100) yielded almost the same result, The results of the fits for the diffuse scattering intensities in Fig. 2 are indicated by full curves. The broken curves show solely the sum of the intensities from the scattering contributions (ii)-(iv). It can be seen that these contributions represent only a minor part of the scattering intensity of the D atoms. This shows that the scattering contribution (i) from the short-range order of the D atoms is in fact the dominant contribution te, the diffuse scattering intensity of the D atoms, Our results for the correlated occupation probabilities cj are presented by the full data points in Fig.
123
shell numbcrj 1 2 5 to 15 I I ' ' I .... I .... t'" ~- 0.2 VD075c ~0.125 .~ ~ ~ ~ ...........................................* , , - n 6 - r o ~ ~ 0.1 ~.
(a)
~ = ~ 0.2 m "~
,
, I .t , , ~ I I ' ' I .... VD0.82 c=0.137 ................................
~. . . . . . . . . .
. ~ . t . . . . I' "
i f - IF - i f - o - - ~ - i f i - - - 0 - o o o t
~ 0.1 Co) 06
,
~~ .~ , . , . 1 2 3 distance Rj [/~]
, 4
Fig. 3. Correlated occupation probabilities cj for (a) VD0.TS and (b) VD0~: vs. the distance Rj between a site of shell j and the site in the center of the shells: the top of the figure shows the number j of the respective shell; . . . . , the occupation probabilities c = 0.125 and 0.137 for VD0m and VD0.s2 respectively: O, 1 ~
3, plotted vs. the distance Rj between a site of shell j and the site in the center of the shells (the top of the figure shows the shell number j). The broken lines indicate the occupation probabilities c of the two respective samples. For j 1> 16, the open circles show finally the fixed values cj = c which were not a variable parameter in the fits.
3. Diseussion The results in Fig. 3 are essentially identical for both investigated samples. They show that a D atom blocks (within experimental accuracy completely) the interstitial sites in the first and in the second shell. This blocking of the sites in the two nearest shells demonstrates clearly the existence of a strong short-range repulsion between the D atoms, as suggested in the previous studies [3,5,7-16]. For the sites of the third shell, we find a c 3 value that is definitely smaller than c, corresponding to a partial blocking of the sites of this shell. This behavior demonstrates that the repulsion between the D atoms extends up to the third shell, or up to distances of 2 ,h.. The results in Fig. 3 show further increased cj for the sites of the fourth, fifth and sixth shell where, in particular, c 5 is clearly above the average value c. Finally, we find very slightly reduced cj for the sites in the more distant shells. Let us now consider small m o m e n t u m transfers hQ.
U. Knell et al. / Journal o f Alloys and Compounds 231 (1995) 121-125
124
In the limit Q---> E, where e tends to zero but remains much larger than the reciprocal dimension L of the bulk material, the scattering cross-section (do-/dY2)SRO per D atom, resulting from the short-range order contribution (i), can be written as (Eqs. (1.12), (9.4), (9.7) and (17.4) of [1], using the nomenclature of [4]; see also [12]) lim O~,
= ~
SRO
(3) 4~r
where o-c,D is the coherent cross-section of the D atoms. The quantity fbulk is the thermodynamic factor for the bulk modes of the hydrogen [4,12] and the angular brackets on the right-hand side of Eq. (3) indicate an orientational average, which is required for a powder sample because fbulk depends on the direction of Q. Since our subsequent discussion remains semiquantitative anyway, we present only the simple expression
c(Ol~/Oc) 2[(1-2o-)/(1 - - o.)]cpoB AV E fbulkka T + kB T
(4)
for fbulk that holds in the case of complete elastic isotropy. In this equation, /.t is the chemical potential of the hydrogen, P0 is the number of tetrahedral interstitial sites per unit volume, B -- 157 × 10 9 N m 2 [20] is the bulk modulus of the sample, o" ~--0.36 [20] is its Poisson's ratio and AV= 2 . 6 A 3 [21] is the volume increase of the sample per D atom. We point out that the above restriction Q >>1/L implies that the concentration fluctuations of the hydrogen, which determine the scattering cross-section, have a wavelength much smaller than L (bulk modes [4,12]). The first term on the right-hand side of Eq. (4) is the (ordinary) thermodynamic factor ftherm-'~-'C(O[~/ Oc)/kBT [4], which is positive for stability reasons and can be derived from hydrogen solubility measurements [10,11]. However, the quality of the available experimental data [10,11] does not allow a reliable determination of ftherm for the present D concentrations. What only can be concluded from the data is that ftherm is very large, in fact certainly larger than 10. The second (and positive) term on the right-hand side of Eq. (4), with a numerical value of about 4.2, reflects the influence of elastic coherency stresses [1,4,12]. These stresses arise in the presence of concentration fluctuations of the hydrogen, raise the elastic energy of the sample and suppress therefore the amplitude of the fluctuations. According to the numerical estimates above, both terms on the right-hand side of Eq. (4) together yield a value for fb~k which is certainly larger than 10, so that (1/fbu,k) in Eq. (3) is expected to be smaller than 0.1. The limit Q--->e lim (do./d£2)SRO of Eq. (3) can be compared with the general expression for (do./d/2)SRO,
which depends on the correlated occupation probabilities cj and is, for example given in Eq. (1) of our previous paper [5] (see also [1,2]). From the derivation of this general expression, it is clear that it describes the effects of hydrogen fluctuations well within the bulk material, so that we can replace the limit Q---> E by the limit Q---->0. For this limit, the comparison between Eq. (3) and the general expression for (do./ dO)sRo shows that the cj must fulfill the thermodynamic relation 0<1-c+
~ Nj(cj - c) j=l
=
(f~k)
<0.1
(5)
where Nj is the number of tetrahedral sites in shell j and the summation extends over all shells j. The number 0 to the very left reflects the fact that fbutk is positive and the number 0.1 to the right expresses the above estimate for (1/fbul k). From the present correlated occupation probabilities cj, as given in Fig. 3, we +0.9 and 0.09(_0.8) +0.9 forVD0.75 obtain (1/fbu,k) ~ 0.008(_0.8) and VO0.82 respectively. These results agree with the theoretical expectation in Eq. (5). It is further worth mentioning that complete blocking of the first two shells and the reduced correlated occupation probability c 3 in Fig. 3 yield, for both samples, a negative value of about -0.2 for the expression 1 - c + E3=l N~(cj - c) which includes the first three terms of the sum in Eq. (5). The remaining terms Ej~4 Nj (cj - c) of the sum must therefore be at least as large as about 0.2. This means that the short-range repulsion, which leads to the blocking effects in the first three shells, necessarily requires cj for outer shells that are larger than the average occupation probability c, as experimentally observed for the fourth to sixth shells. We conclude with a comment on the slightly reduced cj for the sites in the outer shells j i> 7. These slightly reduced cj may indicate the influence of coherency stresses which can also be considered to reflect a repulsive contribution to the total long-range elastic interaction between the hydrogen interstitials [1,4,12]. However, such an interpretation is certainly speculative since, for an elastically isotropic material, the repulsive long-range contribution (from the coherency stresses) is exactly compensated by an attractive long-range contribution which influences the concentration dependence of the chemical potential tt[1,12]. This means that, again for an elastically isotropic material, we do not expect a net long-range repulsion as required for the slightly reduced cj of the outer shells. On the contrary, the present sample material is definitely not elastically isotropic, so that the different orientational averages for the chemical potential and the scattering cross-section (of a powder sample) may indeed lead to a situation, in which the
U. Knell et al. / Journal of Alloys and Compounds 231 (1995) 121-125
resulting scattering cross section must be described by a n effective net long-range repulsion between the hydrogen atoms.
Acknowledgment
[6] V.F.Sears, Neutron News, 3 (1992) 27. [7] A.C. Switendick, Z. Phys. Chem., N.F., 117 (1979) 89. [8] A. Mokrani and C. Demangeat, Z. Phys. Chem., N.F., 163 [9] [10] [11] [12]
The present work was supported by the Bundesrninisterium ftir Forschung und Technologie.
[13] [14]
References
[15] [16]
[1] M.A. Krivoglaz, Theory of X-ray and Thermal-Neutron Scattering by Real Crystals, Plenum, New York, 1969. [2] B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading, MA, 1969. [3] R. Hempelmann, D. Richter, D.A. Faux and D.K. Ross, Z. Phys. Chem. N.F., 159 (1988) 175. [4] H. Wipf, J. V61kl and G. Alefeld, Z. Phys. B, 76 (1989) 353. Eq. (5) of this reference is misprinted; it should read E 1 = P2/B and Ebulk = p2[Cll. [5] U. Knell, H. Wipf, G. Laulensehl~iger, R. Hock, H. Weitzel and E. Ressouche, J. Phys.: Condens. Matter, 6 (1994) 1461.
125
[17] [18] [19] [20] [21]
(1989) 541. D.G. Westlake, J. Less-Common Met., 91 (1983) 1. E. Veleckis and R.K. Edwards, J. Phys. Chem., 73 (1969) 683. W. Rummel, Siemens Forsch.-Entwickl.-Ber., 10 (1981) 371. H. Wagner, in G. Alefeld and J.V61k (eds.) Hydrogen in Metals I, in Top. Appl. Phys., 28 (1978) 5. G. Boureau, J. Phys. Chem. Solids, 45 (1984) 973. M. Sotzek, H. Siebler, H. Wipf and U. Leuth~iusser, J. LessCommon Met., 104 (1984) 21. A.I. Shirley and C.K. Hall, Phys. Rev. B, 33 (1986) 8084, 8099. P. Meuffels and W.A. Oates, J. Less-Common Met., 130 (1987) 403. I.A. Blech and B.L. Averbach, Phys. Rev., 137 (1965) Al113. P.H. Dederichs, H. Schober and D.J. Sellmyer (eds.), LandoltBOrnstein, New Series, Vol. 13a, Springer, Berlin, 19 , p. 163. M. Kamal, S.S. Malik and D. Rorer, Phys. Rev. B, 18 (1978) 1609. A. Magerl, B. Berre and G. Alefeld, Phys. Status Solidi A, 36 (1976) 161. H. Peisl, in G. Alefeld and J. V61kl (eds.), Hydrogen in Metals I, in Top. Appl. Phys., 28 (1978) 53.