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DETECTION OF IMPURITY TUNNELING IN SOLIDS VIA COHERENT PHONON COUPLING
AND DIRECT NEUTRON SCATTERING
R. Casella
To cite this version:
R. Casella. DETECTION OF IMPURITY TUNNELING IN SOLIDS VIA COHERENT PHONON
COUPLING AND DIRECT NEUTRON SCATTERING. Journal de Physique Colloques, 1981, 42
(C6), pp.C6-923-C6-925. �10.1051/jphyscol:19816275�. �jpa-00221563�
JOURNAL DE PHYSIQUE
CoZloque C6, suppldment au n o 12, Tome 42, d6cembre 1981 page c6-923
DETECTION OF IMPURITY TUNNELING I N SOLIDS V I A COHERENT PHONON COUPLING AND DIRECT NEUTRON SCATTERING
R.C. Casella
National Measurement Laboratory, National Bureau of Standards, Washington, D.C. 20234, U.S.A.
Abstract.- A theoretical treatment is given of the observation of molecular tunneling in solids by the coherent interaction of the tunnel-split excita- tions with acoustic phonons and by direct neutron inelastic-scattering.
Results are applied to the case of rotation tunneling of CN dumbbells in KBr and KC1, and to the motion of H atoms in two-well traps associated with oxygen impurities in niobium. Comparison is made with experiment.
Coherent forward resonant scattering of phonons from the tunnel-split ground and libronically excited configurations of impurity molecules in solids results in mixed modes, with non-zero component amplitudes for both the phonon and the coherently excited impurity complexes. When observed via inelastic neutron scattering, one sees a perturbed acoustic branch in the neighborhood of the wave vector
-+
k at which the component modes are degenerate [I], [2]. The mixed mode10
> is given by relation (I), where IEX, -+ R > represents the impurity excitation at siteit.
10 > = Iphonon,
%
>+
i1
IEX,3
>< EX, itl~l~honon, -+ k >-b
(1) -+ R
where the sum on R of the T-matrix elements extends only over lattice sites occupied by the dumbbell impurities. The mixed modes have been described in terms of vector cubic harmonics and analyzed group theoretically in terms of a model [ 2 ]
Fig. 1. Energy level schematic for the unperturbed impurity complex: the A lowest level and the neutron-active lg low-lying Eg and Tzg states vs. the coupling strength g of the hindering potential for the model in which the impurity dumbbells are aligned along
< 111 > directions. The first excited
libronic level is of type E
.
Thetunnel split excitation witfin the libronic ground state is of type T2g.
B is the rotation constant; L is the angular-momentum quantum number when g = 0.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816275
C6-924 JOURNAL DE PHYSIQUE
which accomodates the neutron scattering data [ 3 ] , [ 4 ] as well as the infrared and Raman scattering data [ 5 ] for KC~:CN- and KB~:CN-. The more easily observed phonon admixture with the first libronically excited tunnel state is determined to be of E character. (See Fig. 1.) If the tunnel-split second excited state (T
g 2g
character) were to lie close to the E excitation in these systems it ought couple g
with equal strength [2]. The inability to observe the T interaction via neutron 2g
scattering indicates that it lies above the potential barrier and is subject to lifetime broadening due to multi-phonon emission. This destroys the coherence required for its observation via the neutron probe. The observation of a broad T
2g state in the Raman spectra considerably above the E [ 5 ] is in accord with this
g
picture. Moreover, the model leads to the conclusion that the tunnel-split ground state in these systems ought be observable by neutron scattering (via its coupling to the acoustic phonon) only in the T configuration (~ig. 1). This prediction
2g
has been confirmed experimentally in KBr : C N 131.
At low temperatures, dilute (< 0(1%) concentration) hydrogen atoms in metals such as niobium and tantalum, also doped with 0(1%) concentration oxygen atoms, become trapped at the latter sites in what are believed to be two-well centers.
Tunneling 06 the hydrogen between the two potential minima near the oxygen provides another interesting system in which molecular (atomic) tunneling in solids can be studied under quite different conditions. Because of the very large low-energy n p scattering length it is possible to study the excitations directly via neutron scattering. This has been done both for transitions among the tunnel-split components of the vibronic ground state [6] and also for transitions from them to the first and second vibronically excited states 171. Here the picture is complica- ted by local strains which can produce relative displacements in the energy of the component well minima. This greatly reduces the ratio of inelastic to elastic neutron scattering within the ground doublet according to a model developed in 161.
The reduction is by a factor
sin228
where 8 is a mixing angle between the left and right single-well states, describing their admixture in tight-binding approximate tunneling eigenstates. (The average is over some assumed distribution of the strain induced relative displacements.) In principle, for zero strain (infinite dilution) the elastic and inelastic differential cross sections for scattering within the ground doublet ought to be comparable. It is important to understand theoretically the neutron induced transitions to the tunnel-split excited states in order to confirm the model directly. If separated considerably from the ground doublet, the excited doublet can be characterized by a mixing angle 4 , generally different from 8. Matrix elements for transitions from either component of the ground doublet to either of the tunnel-split excitations depend upon both 0 and $.Under suitable experimental conditions, however, the differential cross section for excitations from both components of the ground doublet to either component of the excited doublet can be shown to be independent of
e
and of+.
Thus the ratio A of inelastic cross sections,e x c i t e d A = d a f d R ) i n e ~ a s t i c
i s enhanced by a ( c o n c e n t r a t i o n dependent) f a c t o r l / ( s i n 28). 2 The numerator i n
QJ +
t h e e x p r e s s i o n f o r A i s d i m i n i s h e d by t h e t r a n s i t i o n form f a c t o r F ( q l ) where i s t h e momentum t r a n s f e r i n t h e h i g h e r e n e r g y i n e l a s t i c n e u t r o n s c a t t e r i n g e x p e r i m e n t i n v o l v i n g v i b r o n i c e x c i t a t i o n . F o r s m a l l
3 ' , ?(7f1)
% (d'.3)2, whereTf
is t h e d i p o l e t r a n s i t i o n m a t r i x element. F o r s m a l l momentum t r a n s f e r4
i n t h e lower e n e r g y e x p e r i m e n t i n v o l v i n g t r a n s i t i o n s w i t h i n t h e ground d o u b l e t , t h e denominator(6.312) 2 where
i?
d e n o t e s t h e s p a t i a l d i s p l a c e m e n t o f t h e w e l l minima w i t h i n e a c h two-well complex. Thus, i n t h e l i m i t5'
-* 0 , + q + 0 s u c h t h a t r a t i o P of t h e l a r g e r t o t h e s m a l l e r r e m a i n s f i x e d , A remains f i n i t e . I f i n d , i n t h i s l i m i t ,A = ( p 2 / 8 ) (kF/kI) ( H W / V ~ ) / ( s i n 28). 2
Here, ($/kI) i s t h e r a t i o of t h e f i n a l t o t h e i n i t i a l momenta of t h e n e u t r o n i n t h e h i g h e r e n e r g y e x p e r i m e n t , fiw i s t h e v i b r o n i c e x c i t a t i o n e n e r g y , and Vo i s t h e b a r r i e r h e i g h t between t h e p o t e n t i a l minima. G e n e r a l e x p r e s s i o n s h a v e a l s o b e e n d e r i v e d f o r A and v a r i o u s o t h e r c r o s s s e c t i o n r a t i o s when t h e s m a l l
3, 3'
e x p a n s i o n i s n o t j u s t i f i e d [ a ] . Because of t h e d i s p a r a t e c o n d i t i o n s under which t h e h i g h e r[ 0 ( 1 0 0 meV)] and lower [0(0.2 meV] e n e r g y n e u t r o n s c a t t e r i n g e x p e r i m e n t s a r e r u n i n Nb:0(1%) H, 0 ( 1 % ) 0 , many d i f f i c u l t n o r m a l i z a t i o n problems h a v e t o b e d e a l t w i t h i n o r d e r t o d e t e r m i n e A e v e n s e m i q u a n t i t a t i v e l y [ 7 ] . N e v e r t h e l e s s , i t would a p p e a r w o r t h w h i l e .
R e f e r e n c e s
[I]
D. Walton, H. A. Mook, and R. M. Nicklow, Phys. Rev. L e t t .2,
412 (1974).[ 2 ] R. C. C a s e l l a , Phys. Rev. B
0,
5318 (1979).[ 3 ] J. M. Rowe, J. J. Rush, S. M. S h a p i r o , D. G . Hinks, and S. Sussman, Phys. Rev.
B
1,
4863 (1980).[ 4 ] R. M. Nicklow, W. P. Crummett, M. M o s t o l l e r , and R. F. Wood, Phys. Rev. B
2,
3039 (1980).
[5] D. Durand and F. L u t y , Phys. S t a t S o l i d i B
81,
443 (1977).[ 6 ] H. Wipf, A. Magerl, S. M. S h a p i r o , S. K. S a t i j a , and W. Tomlinson, Phys. Rev.
L e t t .
46,
947 (1981).[ 7 ] A. Magerl, J. J. Rush, and J. M. Rowe, p r i v a t e communication ( u n p u b l i s h e d ) . (81 R. C. C a s e l l a , t o b e p u b l i s h e d .
