The role of Fermi-Dirac statistics in hydrogen internal friction in amorphous alloys

Volume 134, number 6

PHYSICS LETTERS A

16 January 1989

THE ROLE OF FERMI-DIRAC STATISTICS IN HYDROGEN INTERNAL FRICTION IN AMORPHOUS ALLOYS W. ULFERT and H. KRONMULLER Inslitut für Physik, Max-P/a nck-Institut für Metallforschung, Heisenhergstrasse 1. 7000 Stuttgart 80, ERG Received 26 October 1988; accepted for publication 17 November 1988 Communicated by D. Bloch

Internal friction experiments in hydrogen charged amorphous PdNiP-alloys are analysed on the basis ofFermi—Dirac statistics for the occupation probability of hydrogen interstitial sites. A quantitative understanding of the relaxation strength requires the assumption of a repulsive interaction of nearest-neighbor hydrogen atoms and of an anisotropy ofthe elastic dipole tensor of h~./ trA—O.3.

The study of structural relaxation phenomena by means of internal friction or magnetic relaxation studies has become a powerful tool for the charac-

Fig. 1 shows a series of IF peaks and of the so-called modulus defects as measured in an inverted torsion pendulum for (Pd40Ni35P-,5) _~H with x varying

terization of amorphous metallic alloys [1—3]. In general these experiments have to be performed above room temperature and therefore require a complicated analysis because reversible as well as irreversible processes are superimposed [1—6]. The irreversible processes in general are suggested to be related to the annealing of so-called excess free volumes [7,8] which have been conserved during the rapid quenching process. The situation is somewhat more favorable if we consider the diffusion properties of hydrogen isotopes dissolved in amorphous alloys. In this case the internal friction (IF) or magnetic after-effects can be observed below room temperature where by appropriate preannealing we are free from irreversible changes of the amorphous matrix. As in the case of ordered or disordered binary alloys it is expected that also in multicomponent amorphous alloys hydrogen atoms give rise to Snoektype IF effects. In fact internal friction peaks due to hydrogen have been observed in many amorphous transition metal—metalloid as well as in transition metal—transition metal alloys [1,9—111.The IF peaks in these materials reveal some very surprising properties related to the high mobility of hydrogen and to the wide spectrum of interstitial sites available for hydrogen atoms,

from 0.0025 to 0.09. The most characteristic features of the internal friction peaks and of the modulus defects are the following ones: (1) The halfwidth of the internal friction peaks and of the modulus defects of hydrogen are much broader than Debye peaks which are expected for atomic relaxation processes governed by a single relaxation time. (2) The peak height and the total modulus defect depend linearly on hydrogen concentration. (3) The peak temperature and the modulus defect shift to lower temperature with increasing hydrogen concentration CH. (4) The shape of the internal friction peaks remains nearly invariant for varying hydrogen concentration. The existence of broad relaxation peaks in amorphous alloys is well known and usually attributed to the presence of a distribution of relaxation times. In the case of thermally activated processes with Arrhenius equations for the relaxation times TR=zOexp(ER/kT),

(1)

this means that there exists a distribution function of activation enthalpies ER, due to variations of site energies E0, and saddle point energies E~.In prin-

0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

385

Volume 134, number 6

PHYSICS LETTERS A

16 January 1989

a

TmLKJ

Pd~

0Ni35 P25

160 150

__ 20

60

140

t30

120

110

\\

100

140

160

220

1

T[K1

___________________________

75

80

85

90

95

100

105

1/k7’ [1/eV]

b 2

-~

~ ‘\

-

4

3

-

‘.

at

/0

Fig. 2. Angular frequency w, at the peak maximum temperature T~being correlated to the most probable relaxation times t by the expression wf= I, as a function of 1 /kTm. The fitted straight lines show the validity of eq. (I). The activation parameters of 4 r0 and ER, are given in table 1.

H

~

-~

~.

50

S~,,

~

-6

D

-8 20

or the distribution function thalpies, given by

N.~,9 60

100

140

160

220

P(E

T[K]

R

Fig. 1. Hydrogen concentration dependence of the background reduced internal friction peaks and the corresponding modulus defect curves of Pd40N135P25. The internal friction peaks below 50 K with the break-down of the damping at 50 K maybe a Pcculiarity ofPd, discussed as the “well-known 50K anomaly”,

ciple also the pre-exponential factor ‘r0 should be described by a distribution function. However, within the framework of a harmonic oscillator model, T0cc 1 / holds, i.e., a variation of ER by a factor of two changes m only by a factor of \/~whereas the vanation in t may amount to several orders of magnitude. Therefore T~in eq. (1) may be considered as a constant for varying ER values. Under these assumptions the spectrum of internal friction may be written as a superposition of Debye-type relaxations

,J~

IF ( w

T)

— —

JI 1

Wt (ER, T) +w2’r~(ER, T) A (E R, T) dER,

(2) where A(ER, T) dER is the relaxation strength of the interstitials with activation enthalpies ER within the interval [ER, Eg+dER]. Once a value of-v 0 has been chosen eq. (2) may be used to determine A(ER, T) 386

A(ER, T) f~°A(ER,T) dER

P(ER)

of activation en(3)

A(E~,T) AR(T)

where AR ( T) denotes the total relaxation strength of the hydrogen atoms. The scaling factor wr0 may be derived by the 1 / Tm shift of the peak temperatures with varying frequency as shown in fig. 2. From a computer fit using the spectrum parameters given in table 1 we obtained the distribution functions P(ER) shown in fig. 3 for different hydrogen concentrations. From the results of fig. 3 it becomes obvious that the shape ofthe distribution function is more or less independent of CH, whereas the most probable activation enthalpy ER decreases with increasing hydrogen concentrations; also the pre-exponential factor t~increases by a factor of 3. The main result of Table 1 Concentration dependence of the most probable activation parameters of ~, ~oand ER, determined from fig. 4R 2 and thebeing concenkT/x the tration and temperature independent quantity fitparameterwhichisneededtonormalizeP(ER). x ‘~R (10’3s) (eV)

ARkT/x (meV)

0.035 0.05

3.0 4.6

0.36 0.31

6.5 6.3

0.09

10.3

0.26

6.2

Volume 134, number 6

PHYSICS LETTERS A

tensor 8

Pd~ ~

~

6 4

o..

2

>

0

90

0_i

02

03

~

0.4

of the hydrogen atom at the interstitial site

0.5

2(u)>,

(6)

2~A~+(23 —2~)2]>

(7)

AR(T)_G6G_—~OGP~u)(A

Fig. 3. Hydrogen concentration dependence of the normalized distribution functions of the relaxation enthalpies P(ER), definedbyeq.(3).

fig. 3 is the fact that with increasing CH the high activation enthalpies obviously are suppressed; or quite generally, the distribution function P(ER) depends on the hydrogen concentration, A model which at least explains qualitatively some of the experimental results has been proposed by Kirchheim and others [12—15].This model is based on the assumption that the thermodynamics of the site occupation probabilities obey Fermi—Dirac statistics. This model is self-suggesting if it is assumed that multiple occupation of a given site is forbidden, The probability that a site i with site energy E’~is occupied then is given by

—

~.

p [19]. The summations on p andp’ are equivalent characterized to an averaging by over a random all elastic distribution dipoles of which the onare entation of their main axes. After performing these averaging procedures and considering the shear modulus G, appropriate for the experimental conditions, we obtain for the total relaxation strength

ER LeVI

= 1 + exp [ (Eb

16 January 1989

where the quantity

~<~

~

(A

corresponds to the average of the second invariant of the ~ tensors of the H atoms at the Fermi level ti. P(~u)in eq. (6) denotes the density of H sites at the Fermi level. In the case of the validity ofFermi—Dirac statistics due to the factor f~, (1 —fr) only H atoms with site energies within a small energy interval, kT, around the Fermi level ~u contribute to the internal friction, i.e., the fraction of H atoms taking part in relaxation processes is given by C1-1 P(ji)kT. (8) Within the framework of this model the actual dis-

(4)

tribution of is thethat activation in fig.be-3 in essential of the enthalpies saddlepointshown energies,

where ~i corresponds to the chemical potential or the Fermi energy of the H atoms. Modifications of eq. (4) due to interstitial—interstitial interactions on the basis of a mean field theory have been considered by Griessen et al. [16—18]. For a test of the hydrogen Fermi—Dirac model we have tried to determine the internal friction of the

cause kT is of the order of 10 meV whereas the experimental halfwidths of P(ER) are 100—120 meV. For a test of the validity of eq. (6) we now have to introduce2Cu)>. numerical and obthe Here values we can for use .PCu) the results quantity tamed by
1u) /kT]

hydrogen atoms quantitatively. The total relaxation of the compliance S0k/ may be written as ~ ><

[f~(l

(~‘ ~ — ~

—f~)2~ (5)

where and E a denotes the dispersion of the site energies 0 the average site energy. For a>> kTthe Fermi

where i9~corresponds to the average atomic volume and 2~to the (lj)-component of the elastic dipole

level is given by [12,13] (step approximation ofj,) JL=Eo±aerf’(!2CH—1l), (10) 387

Volume 134. number 6

PHYSICS LETTERS A

with sign + for 2CH> 1 and sign for 2CH< 1. It turns out that eq. (10) describes fairly well the shift of the average activation energy to lower values as observed for increasing C11. For C11=O.09, a=0.l6

16 January 1989 4

—

eVandE~=0,jiandP(u)are—0.l52eVand 1.44 eV From the experimental value of the relaxation strength zlR=GSG’=0.055 at T=l20 K (6) for 1 we obtain from eq. C~1=0.09and 9= 14.7 A (A2(u)>~2=0.l22. In evaluating thts experimental result we may consider the mean values (2,> of the ~ tensor in eq. (7)

r-1

3

E3

2 -~

-‘.

4E2~03eV

1

0

-6

—4

-2

0

2

F 3 LeVJ

which for a-Pd48Ni34P,5 have been found to be <2,> =0.057 [20]. This means that the root-meansquare of the differences A~—2~,(A(,u)> ‘-a, which describes the anisotropy of the ~ tensor is about two times larger than the mean values <2,> themselves, In fact this result is extremely unreliable because the actual anisotropy expected for the components of hydrogen should amount to only 0.3(2,>. In order to get some more information the actual anisotropy of the dipole tensor ~about we performed Monte Carlo calculations of hydrogen in an amorphous structure. The amorphous state was simulated by statistical distortions of the edge lengths of an fcc lattice according to the radial distribution function ofPd 80Si2() [21 ]. The interstitial sites of the H atoms were taken as the original octahedral Sites of the fcc lattice. After putting H atoms into the distorted “octahedral” sites the corner atoms were a!lowed to relax whereas all other atoms were kept fixed. For this model the total energy has been calculated up to the second order in the distortions of the host atoms due to the local disorder and their displacements due to the inserted H atoms. This total elastic energy has been minimized with respect to the displacements of the host atoms and the equilibrium positions of the H atoms. The calculations were performed using potential data determined for Pd from the elastic constants of Pd and for the Pd—H system from inelastic neutron scattering data and from tr ?~of H in Pd [20,22]. The potential data were the following ones: Host—host: U( a0) = U’ (a0) = 0 U”(a0)=0.7 eV/A. a0=2.75 A; host—hydrogen: V( r,,) =0, V (r0) = —0.68 eV/A, U’ (r0) = 1.23 eV/ A r 1 944 A ~.

—

Ftg. 4 shows the resultant site energy distribution of hydrogen which is nearly gaussian with a width of a= 0.15 eV being close to the value reported by 388

Fig. 4. Distribution of site energies PU,,) of hydrogen aloms

0(2-

cupying randomly distorted octahedra.

Kirchheim for Pd50Ni32P,8 [121. The obtained data for the elastic dipole tensor ~ are illustrated in fig. 5. The distribution of tr is centered at 0. 1 7, which is the same as tr ~ of H in Pd. This leads to a mean value for the components 2> 2, l/2 of is(2k> most 0.02=0.057. or aboutThe 0.3<2,>. probable value of (A Accordingly it turns out that the value of (A2>~2=0.l22 calculated on the basis of Fermi— Dirac statistics is more than 6 times too large and consequently for reasonable values of (A2>’ /2 the relaxation strength is about a factor of 40 too small.

~.

One way to reduce this discrepancy consists in a modification of the Fermi—Dirac statistics by the assumption of a repulsive interaction energy between hydrogen interstitials. Repulsive interactions lead to a depopulation of interstitial sites near an occupied one. As a consequence the sharp decrease of the oc-

f \~(A) A

-~

2

P(tr2i)

ii

8 6 4 2 ,

0

02

04

06

/ 15

17

19

A

tr~ Fig. 5. Results of the calculations of the elastic dipole tensor A. Left: normalized distribution function of the root-mean-square of the differences ,~,—2. A. being a measure of the anisotropy of A. Right: normalized distribution function of irA. which is the volume expansion due to one hydrogen atom. Al- ‘/0,,, expressed as fraction of the atomic volume of the host. i3,.

Volume 134, number u

PHYSICS LETTERS A

CH =

2

5

10

20

16 January 1989

factor of 4. The repulsive interaction energy between nearest-neighbor interstitials of 50 meV may be re-

at.°/0

a -2 e [meVi

(2)a a

2 (2)

_____________________________

0-

0

0.1

02

03

04

E LeVJ Fig. 6. Occupation functionsf(E) for varying hydrogen concentration and repulsive nearest-neighbor interaction energies. P(E) denotes the distribution function of the site energies.

cupation probability at the Fermi level is smeared out and the function f~,(1 —J,) must be replaced by a much broader distribution, A Monte Carlo simulation of the site occupation of an fcc arrangement of 50 x 50 x 50 hydogen sites with gaussian distributed site energies (a= 0. 15) using periodic boundary conditions leads to the occupation probabilityconcentrations functions shown fig. 6 for different hydrogen and invarying repulsive interaction energies As becomes obvious from fig. 6 the broadening of the probability function leads to an increase of the number of H atoms participating in the relaxation processes. Fig. 7 gives the ratio of the concentration of effectively participating hydrogen atoms CHeff(~),and the concentra-

~.

nearest-neighbor alistic according repulsion to the values and 9of meV 35 for meV next-nearfor the est-neighbor repulsion of hydrogen in Pd [23]. From our analysis performed so far we may draw the following conclusions: (1) The observed shift of the mean value of the activation enthalpies corresponds to the shift of the Fermi energy given by eq. (10). (2) A rigorous application of Fermi—Dirac statistics on the IF of hydrogen in amorphous alloys for reasonable values of the elastic dipole tensor ~ leads to relaxation strengths AR which are a factor of 40 too small. (3) The assumption of a repulsive nearest-neighbor interaction between hydrogen interstitials leads to a broadening of the occupation function leading to an increase of the number of relaxing H-interstitials by a factor of4 for repulsive energies of 50 meV. (4) In order to remove the remaining discrepancy of another factor 10 be weahave thatthan real 2> 1/2 of should factortoofassume 3 larger values of (A those determined by our model calculation based on a monoatomic Pd-lattice. This result suggests anisotropies of (A2>~2/tr~.-=0.3. ‘-~

References

tion Cue

1i(O) for zero repulsive energy. In the case of a hydrogen concentration of CH = 0.1 and a repulsive energy of 50 meV the effectively participating concentration of hydrogen atoms increases by a

T = 120 ~c

CH 0 20

at.

0/0

I

C—)

:~

[1] B.S. Berry and W.C. Pritchet, in: Hydrogen in disordered and amorphous solids, eds. G. Bambakidis and R.C. Bowman Jr. (Plenum, New York, 1986) p. 215. [21H.S. Chen and N. Morito, J. Non-Cryst. Solids 72 (1985) 287. [3] H. Kronmüller, Philos. Mag. B 48 (1983)127. 14] N. Morito and T. Egami, Acta Metall. 32 (1984) 603. [5] N. Moser et al., in: Rapidly quenched metals, eds. S. Steeb and H. Warlimont (North-Holland, Amsterdam, 1985) p. 1195. 16] H.-Q. Guo, IEEE Trans. Magn. 20 (1984) 1394. [7] T. Komatsu et al., J. Non-Cryst. Solids 69 (1985) 347. [8] J. Horvath et al., Mater. Sci. Forum 15—18(1987) 523. [9] B.S. Berry and W.C. Pritchet, Scr. Metall. 15 (1981) 637. [l0]K.Agyemanetal.,J.Phys.C5(l981)535.

I

j

I

0

_______________________________ 20 40 60 80 100

1111 U. Stolz et al., Scr. Metall. 20 (1986)1361. 1121 R. Kirchheim et al., Acta Metall. 30 (1982)1059.

c Em eV]

[13] R. Kirchheim et al., Acta Metall. 30 (1982) 1069. 1141 P.M. Richards, Phys. Rev.B27 (1983) 2059. [15] U. Stolz, J. Phys. F 17 (1987)1833. [161 R. Griessen, Phys. Rev. B 27 (1983) 7575.

Fig. 7. The ratio of C’H~ff(e)/CHCff(O)as a function ofthe repulsive interaction energy for varying hydrogen concentrations C11.

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PHYSICS LETTERS A

[17] R. Griessen, in: Hydrogen in disordered and amorphous solids, eds. G. Bambakidis and R.C. Bowman Jr. (Plenum, New York, 1986) p. 153. [18] R. Feenstra, R. Brouwer and R. Griessen, Europhys. Lett. 7 (1988) 425. [191W. Ulferi, Dr. rer. nat. thesis, University of Stuttgart, Stuttgart (1988).

390

16 January 1989

[201 U. Stolz et al., Scr. Metall. 18 (1984) 347. [211 Y. Waseda and W.A. Miller. Phys. Stat. Sol. (a) 49 (1978) K31. [22] J.J. Rush and J.M. Rowe, Z. Phys. B 55 (1984) 283. [23] P.M. Richards, Phys. Rev. B28 (1983) 300.