TUNNELLING MOTION AND INTERNAL FRICTION OF HYDROGEN IN METALS

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TUNNELLING MOTION AND INTERNAL FRICTION OF HYDROGEN IN METALS

I. Svare

To cite this version:

I. Svare. TUNNELLING MOTION AND INTERNAL FRICTION OF HYDROGEN IN METALS.

Journal de Physique Colloques, 1987, 48 (C8), pp.C8-281-C2-286. �10.1051/jphyscol:1987840�. �jpa-

00227144�

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TUNNELLING MOTION AND INTERNAL FRICTION OF HYDROGEN IN METALS

I. SVARE

P h y s i c s D e p a r t m e n t , U n i v e r s i t y of T r o n d f i e i m , N - T - H - . N - 7034 T r o n d h e i m , N o m a y

Abstract: Tunnelling motion of hydrogen in metals is possible in

--

excited states and through two-phonon transitions. Asymmetry between the H sites blocks the tunnelling and permits only slow classical hopping. We discuss in a simple model the various tunnelling rates, and we get quantitative agreement between many types of experimental data for H, D and T in Pd, V, Nb and Ta.

PACS no. 6240, 6320 M, 6630 J.

1

.

Introduction.

Internal friction (IF) from hydrogen in metals has a maximum when the hoppin? rate T-1 of the H impurities movinu between inter- stitial positions equals the frequency of the externally applied mechanical or ultrasonic strain that perturbs neighbouring sites diff- erently. The H motion can also be studied with diffusion techniques, and inelastic neutron scatterinu (INS) can give hydrogen vibration frequencies and tunnel splittings. All these results must be simply related to the potential between the sites. But the very different experimental hopping rates may appear bewildering. For example, in niobium at 100 K, diffus'on measurements point to a hopping rate for H of order r-l s 1

o1

O s-* [ 1 1

,

while IF due to H near 0 and N impurities gives T-' t lo5 s-l [ 2 1 . Other t y ~ e s of IF such as from H in dislocations has % 10 s-1 around 140 K [ 3 1 , and from H bound to 0

impurities has r- 2 lo8 s-I around 2.5 K 1 4 1

.

It has been problematic to find the connection between these very different rates of motion, and there is no widely accepted theory that will explain all the aspects of the diffusion, hopping and tunnelling of hydrogen in metals. Plany modern theories of diffusion use the small polaron concept [ 5 1 where self-trapping of the H severely limits the tunnelling rate, but with limited success.

We think that all these cases can be treated uniformly with a simple model of hydrogen tunnelling, with transition rate calculations similar to, but of higher phonon order than those used for tunnelling systems in glasses. We neglect the self-trapping, but stress that asymmetry in the energies of the neighbouring wells, caused by lattice strain near defects and impurities, prohibits tunnelling transitions.

The various cases of hydrogen motion in metals are shown schematically in Fig.1, and the fastest rate for each H dominates. Qnly classical hopping over the potential barrier is possible for H moving to and from im~urities, or along with dislocations. However, far from defects, or between two equivalent sites near an impurity, the H can tunnel in several ways that will be brieflv reviewed here, and pub-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987840

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JOURNAL DE PHYSIQUE

Impurity

Dislocation Fig.1. Schematic drawing of H motions in the metal potential. Large asymmetry A between wells permits only classical hopping towards impurities, case a), and in moving dislocations where the final state is drawn dashed, case b). Rapid tunnelling is possible when A

"

^0;

in a regular lattice, case c), and between equivalent sites near an impurity, case d)

.

lished in more detail elsewhere. We find that a qualitatively con- sistent model with a minimum of adjustable parameters can be con- structed to cover a11 cases [6,7].

2. Tunnelling processes for diffusion.

We have recently discussed [ 6 1 the diffusion of H between octa- hedral sites in 8-PdHx as tunnelling by interference in the excited vibration states En having tunnel splitting AETn, which leads to the transition rate between neiahbouring sites

This is possible only if AETn is larger than the asymmetry A between the two wells [ 8 1 . In cases with large A, only classical hoppinq over the barrier is possible with approximate rate

T

-

1 13

class * 10 ex?(-Ea/kT)

.

Hence in cold-worked O-PdHx where the many dislocations give large A between neighbouring sites, the internal friction peak corresponding to the rate (2) is prominent [ 9 1 , while the IF peak corresponding to tunnelling in the second excited state with (1) dominate in well- annealed samples. Assuming a sinusoidal barrier potential of height Vo (' Ea+E1/2) = 4000 K, we could calculate the En and A E T ~ from simple formulas and fit a great amount of experimental data with this single adjustable parameter VO [ 6 1 .

For the much more rapid diffusion of H between the tetrahedral sites in bcc V, Mb and Ta we also have to consider rate contributions from two-phonon tunnelling transitions between the symrnetrised ground states that are split by

Adapting the standard perturbation procedure with virtual excitations to the tunnel-split states El, we find in the case of small A [ 7 1

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as it will be for long wavelength phonons in a regular lattice. HC is the perturbation potential of the phonon time dependent crystal fleld, and $L and $R are the localised H wavefunctions.

The Debye phonon spectrum and the neglect of dispersion are crude approximations, but the numerical integrals (4) fit well to the diffusion data [I], corresponding to case c) in Fig.1, for H in V and for H in Nb and Ta below 250 K with reasonable values for H01. Above 250K in Nb and Ta the tunnelling (1) dominates. The fitted parameters

f o r H d i f f u s i o n a r e l i s t e d i n T a b l e 1 , and t h e v i b r a t i o n e n e r g i e s and tunnel splittings are in good agreement with INS data [10,111.

Table 1. H in bcc metals. Fitted potentials Vo and H O I , and calculated energies Eat El and tunnel splittings ET,, in units K

.

v~

Ea A E ~ ~A E ~ H~ ~ I

The tunnelling (4) is also blocked by large asymmetry A > k 8 ~ between the wells. Therefore the internal friction from H near 0 impurities in Bb [21, case a) in Fiq.1, fit the classical hopping (2) reasonably well with the calculated Eat since a binding energy of H to the impurity of up to 1000 K does not perturb VO too much. We interpret the deviation of the IF peaks from (2) below 100 K as

"leakage" by H that are halfway escaped from the impurities and therefore see a smaller A and can tunnel somewhat. The effect of random asymmetry is also clearly seen in the slow diffusion of H and D in VO 5Nb0 alloy [ 12 1, which as classical hopping (2) is very much

sl6wer f2an the rapid tunnelling in the pure metals.

We see from (4) that the two-phonon tunnelling is less important for the heavier isotopes D and T which have smaller AETI and H01 than H has. The observed diffusion of D and T [I] therefore go as ( 1 ) . 3. Tunnelling motion below 10 K in Mb.

Internal friction of H bound to 0 impurities in Nb below 10 K showstwo peaks where the lowest correspond to the motion of a single H between two nearly equivalent sites, case d) in Fig.1. The observed rates are some 108 times faster than the extrapolation of the H diffusion rate with (4). It has been suggested that the low tempera-

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JOURNAL DE PHYSIQUE

-

1 F i a . 2 . T u n n e l l i n g r a t e s T f o r H

3 -

and D bound t o 0 i m p u r i t i e s i n Ub.

: I F peak t e m p e r a t u r e s A a n d A from

-

[ 4 , 1 3 1 , V from [151, from [ 1 6 1 , and INS l i n e b r o a d e n i n g 0 from

[ 1 1 1 , w i t h u n c e r t a i n t y b a r s . S o l i d l i n e s i n d i c a t e T5 b e h a v i o u r , a n d d a s h e d l i n e i n d i c a t e s s u p e r c o n d u c t - i n a Tc

.

1 2 3 4 5 6 7 8 10 15

T IK1

t u r e t r a n s i t i o n s a r e c a u s e d b17 i n t e r a c t i o n w i t h some e x c i t e d s t a t e o f t h e d e l o c a l i s e d H [ 1 3 1 , o r w i t h t h e s u p e r c o n d u c t i n g e l e c t r o n s [ I l l , and i t h a s been d i s c u s s e d a s t r a n s i t i o n s between t w o - l e v e l t u n n e l l i n g s y s t e m s a s i n g l a s s e s [141.

We p l o t t h e I F peak and INS l i n e w i d t h d a t a [ 4 , 1 1 , 1 3 , 1 5 , 1 6 1 i n a n o t h e r way, and show i n . F i g . 2 t h a t t h i s h a s a c l e a r T5-dependence.

T h i s i n d i c a t e s a d i f f e r e n t two-phonon p r o c e s s t h a t i s g o s s i b l e n e a r i m p u r i t i e s where t h e c r y s t a l f i e l d v i b r a t i o n s c a n have d i f f e r e n t m a t r i x e l e m e n t s i n t h e l e f t a n d t h e r i g h t w e l l

I f we had H O O R =

-

H O O f o r a l l phonons, t h e r e s u l t would b e a T 3 two-phonon p r o c e s s [ 1 7 j . But t h e c o n t r i b u t i o n from phonons w i t h s u c h e x a c t asymmetry w i l l be overwhelmed by l a r g e r c o n t r i b u t i o n s from o t h e r phonon p a i r s when kT > AEo

,

and a l a r g e r ~ 5 r a t e r e s u l t s . Taking HOOL

= 0 a s a n example, w e d e r i v e f o r t h e p r o c e s s where one phonon i s ab- s o r b e d and one l a r g e r by AEo i s e m i t t e d

-1 2

T 2ph = ~ P H ~ ~ ~ ( A E ~ ~ ) ( 4 ~ ~ ) - ~ f i ' ~

[fi. ⁺ ( A

-

AE0)/2l ( f ~ w 2

-

AEo)hw d ( b o )

l e x p ( ( h w

-

^AEo)/kT)

-

¹¹¹¹

-

e x p ( - h o / k T ) ] ( 7 ) AEn

The p r o c e s s where two phonons of sum AEo a r e e m i t t e d g i v e s o n l y a s m a l l c o n t r i b u t i o n . The r a t e ( 7 ) d e c r e a s e s s l i g h t l y w i t h A . But o n l y H w i t h s m a l l A

<

AETO a r e s e e n w i t h INS, and t h e c o n t r i b u t i o n t o I F from H w i t h A

>

kT g o e s down w i t h t h e d e p o p u l a t i o n of t h e u p p e r l e v e l . The n u m e r i c a l l y c a l c u l a t e d r a t e ( 7 ) v a r i e s a p p r o x i m a t e l y a s T5 from 2 t o 10 K , and f i t t e d t o t h e H d a t a i n F i g . 2 , w e d e r i v e H 12000 K a s a c r u d e a v e r a g e i n s u p e r c o n d u c t i n g Nb. T h i s m a g n i t u d e o!! H i o . i s

r e a s o n a b l e . I t w i l l g i v e a one-phonon r a t e i n o u r one-dimensional model

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rates shown in Fig.2 are about 100 times slower than the H rates.

Most of this isotope difference can be explained by the factor ( A E ~ ~ ) ~ in (7).

4. Pinning of dislocations by hydrogen.

Dislocations can move relatively easily in pure bcc metals, with rates of order 10 s-1 in Nb at about 40 K. But pinned to hydrogen, the dislocation hopping rate is very much slower, and 10 s-I is then found around 140 K [31. It may seem strange that one single hydrogen that itself is often highly mobile in the metal, can pin a mobile dislocation so that neither can move much. But this locking can now be straightforwardly explained: We assume that the H is strongly bound

(A > 1000 K) in one particular deep site in the dislocation centre.

The dislocation would have to move one lattice vector in a translation to an equivalent configuration, and the H cannot be left behind in a combined motion. But the H tunnelling to its much nearer neighbouring sites is blocked by large A, and the tunnelling rate over the long distance one lattice vector is very small. Hence the combined H plus dislocation motion is possible only by classical hopping of H over the barrier, case b) in Fig.1. The barrier is higher than normal because the H is strongly bound to very distorted lattice in the dislocation.

We find from [31 that the activation energy for the-H-pinned dislocation motion in Nb is about 3700 K, compared to Ea = 2300 K calculated in Table 3. The difference 1400 K is related to the binding energy of the H to the dislocation.

5. Discussion

We have adapted old equations for tunnelling and transition rates in a slightly asymmetric one-dimensional double potential to the many aspects of hydrogen motion in metals. We get in this simple way surprisingly good consTstency between a great variety of experimental data differing in rates by very many orders of magnitude, by using only the adjustable parameters Vo for the static sinusoidal potential, and H O I , HooR and HOOL for the dynamic perturbations of the potentials by the phonons. The motion of hydrogen in metals does not seem to have been fitted to these simple T~ and T~ two-phonon rates before, and it may not have been realised before that the T~ process from H O O R # HOOL can be much faster than the ~3 process from H O O R =

-

HOOL

.

^{Also, the}

importance of asymmetry A for preventing tunnelling transitions has not always been properly understood.

Our approach is related to the calculations for tunnelling systems in glasses [la], but we go beyond the one-phonon rates from

(8). We think that the hydrogen-in-metal system is simpler than glasses with the unknown tunnelling atoms. For H in metals only A and H00 have a distribution of values, while in glasses also AETO is dis- tributed and must be averaged over.

The good fit to experimental data that we have got by neglect- ing the self-trapping effects, is much better than what has been ob- tained with the more complicated small-polaron theories. Therefore we argue that the self-trapping energy does not directly change the tunnelling rate, since it is the same before and after the transition.

The self-trapping only influences the effective potentials VO, which turn out-to be slightly smaller for the heavier isotopes D and T. We

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C8-286 JOURNAL DE PHYSIQUE

note that the small-polaron theories are not used in the calculations for tunnelling systems in glasses, because it is not needed for reasonably good fits to experimental data, and because the self- trapping energy cannot be calculated there. If the self-trapping could be estimated in qlasses and were included as an activation energy for tunnelling, it would probably lead to impossibly small rates around 1 K

.

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