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Neutron scattering from tunneling protons in metals
Physica 141B (1986) 305-311 North-Holland, Amsterdam
NEUTRON SCATTERING FROM TUNNELING PROTONS IN METALS J. KONDO Electrotechnical Laboratory, Tsukuba Research Center, Ibaraki, Japan Received 26 November
1985
We consider a two-level system in metals, in which a proton is tunneling between two neighbouring sites. Cross section of inelastic neutron scattering from this system is calculated. The interaction with the metallic electrons reduces the level separation and causes a level broadening. The ratio of the former to the latter turns out to increase as a negative power of the temperature as the temperature decreases. Thus, the proton may make a coherent motion at very low temperatures. ‘Ihe cross-over temperature where this ratio goes through unity is shown to be of the order of the level separation at T = 0. This result explains a recent neutron experiment on the proton in niobium with a small amount of oxygen.
1. Introduction In previous papers, we considered a two-level system in metals [l, 21, where a particle can jump between two sites with a transfer matrix A. The particle was assumed to interact with the metallic electrons. An important consequence was that the transfer matrix of the particle is renormalized by the overlap integral between two electronic wave functions, each wave function being what is realized when the particle is fixed in one of the sites. As an application of the result, we calculated [3] probability of the particle being found in a neighbouring site, when it was initially found at the initial site. The probability was found to be proportional to the time, and the proportionality constant W, the hopping rate, was closely related to the diffusion constant. Theoretical consequence was that the diffusion constant is proportional to a power of the temperature, and this power law was actually found in diffusion of the positive muon in copper [4]. Another typical two-level system is a proton trapped by an oxygen in Nb metals. It has been proposed that the proton jumps between two neighbouring tetrahedral sites near an oxygen, and it was shown that there is no trend for the proton to precipitate even at very low temperatures [5]. This implies that the trapped state is stable, and the proton remains in that state at low temperatures. When the proton jumps between
two sites with transfer integral A, symmetric and antisymmetric states are the stationary states with level separation 24. Recently, this level separation as well as level broadening were measured by inelastic neutron scattering [6-81. In these experiments, a small level separation (-2 K) was clearly observed, and the line broadens very rapidly as the temperature is raised from 1 K to 4 K (under a magnetic field enough to destroy superconductivity of Nb [S]). Such a rapid change of the level broadening must come from the effect of metallic electrons. In fact, the authors show that the Korringa-type relaxation explains the level broadening [7]. The purpose of this paper is to calculate inelastic line shape of neutron scattering from a two-level system in metals and analyse these experimental results. In the next section, we present a qualitative consideration which involves the most important aspect of this paper. In the following section, we present a detailed calculation of the cross section.
2. Qualitative cousideration We first summarize results of previous investigation [l, 31. The transfer integral A is renormalized as
03784363/86/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
@=>A)
>
J. Kondo
306
I Neutron scattering from tunneling protons in metals
where D is an energy of the order of the band width of metallic electrons, K is a dimensionless coupling constant (see (38)). Suppose the particle was at one of the two sites at t = 0. The probability P of the particle to be found in the other site at t would be P = sin* A& ,
(2)
if there were no effect of the interaction except renormalization of A. Actually the metallic electron causes a dissipation or a damping of the particle motion. Thus, coherent increase of P is disturbed after a short time r,, namely P ceases to increase quadratically at TV. After that, P again increases quadratically until a time --TV is reached. A net result is a linear increase of P = Wt (fig. l), where W = A&,
.
(3)
A detailed calculation
At this point, the particle begins to w4ff. make a coherent oscillation between the two sites. The temperature at which W- Aeff occurs is obtained from
where we have neglected numerical factors in (4). This gives us the cross-over temperature T,,: , A \ K/Cl-K) kT,, = A(;) .
(5)
As the temperature is lowered further, the particle will oscillate many cycles following (2), before it is perturbed. In this situation, the particle motion may be called coherent and the hopping rate W will be of the order of Aeff. Thus, W/A,,, vs. T will look like fig. 2. Now, we expect that Aeff also levels off below To. At T = To, the value of Aeff is
[3] shows K
= kT,, ,
W A -A eff = rKkT For the positive muon in copper, the values of these parameters were A = 0.065 K, K = 0.3. We take D = 6 eV as a typical band width of copper. Then, W/Aeff = 0.0023 at T = 1 K, which is much smaller than 1. This means that the particle state is perturbed many times before the particle makes a coherent motion (a motion back and forth between the two sites). In this case the particle motion may be called hopping. As the temperature goes down, W/Aeff increases as (4) indicates. Finally, one will have sinZ&fft
I
it
I /
f‘
-
OK7 of a particle found in a neighbouring
LL
l -------: F
=ff
0
c
Fig. 1. Probability
where use is made of (5). Thus, we expect that Aeff will remain to be the order of kT,, below To. Aeff vs. Twill look like fig. 3. Aeff begins to deviate from (1) below T = A lk and levels off below T = T,,. We thus arrive at an important conclusion: the cross-over temperature is the same order of magnitude as the effective level separation at T = 0 (devided by k). kT,, given by (5) is nothing but energy scale of our problem. Such an energy scale was introduced by Yamada et al. [9]. We now discuss spectroscopic aspect of our problem. Here, two quantities are relevant: the level separation Aeff and the level broadening r.
site.
TO
Fig. 2. W/A,,, vs. T.
t
J. Kondo
I Neutron scattering from tunneling protons in metals Table I
A In- eff
0
307
T+
In+
K
A (K)
T o svpsr(K)
To (K)
0.2 0.3 0.4
15 40 110
2.44 2.30 2.22
2.0 2.0 2.0
A+A kT 0 super= A( D s)XI(l-K) ,
0
(9)
Fig. 3. In A,,,/ TOvs. In TIT,.
We may expect -1
7,
-r.
(7)
where A, is the superconducting gap. With A, = 16 K for Nb, we find Tosuper as shown in table I. They are just right magnitude to explain the 15% increase.
Then, from (3) we have 3. Calculation
w -=A eff
Aeff
of the cross section of neutron
scattering
I”
Then, at T 9 To, where WfAeff & 1, we have A,,,/I’ e 1. At T 6 T,,, (3) is no longer valid, but we may expect A,,,/r % 1. Thus, T,, is also the cross-over temperature for the spectroscopic problem. We now analyse the experimental results on the proton two-level system in NbO, [7,8]. We are interested in protons in normal Nb [8]. We first determine T,, for this system. As I mentioned, kT,, is the level separation at T = 0. This can be obtained from the peak position at T = 1 K for the proton concentration 0.0001. We find T,, = 2 K. Since To is the cross-over temperature as well, it is quite natural that the line broadens and almost vanishes at T = 4.2 K. Such a rapid change of the line shape as the temperature is changed from 1 K to 4.2 K can be explained only in terms of the effect of the metallic electrons. From To = 2 K, we can deduce the value of A based on (5). We may take D = 4 eV as a typical band width. For three values of K, we find A, whose values are shown in table I. Now, we compare the level separation in normal Nb with that in superconducting Nb [7]. The latter at T = 1 K is larger than the former at T = 1 K by about 15%. Here, we assume that the effect of the superconducting gap on the renormalization of A is expressed as
In this section we calculate the cross section of inelastic neutron scattering from a two-level system in metals and confirm (7) or (8). b 1 and b, will be the particle operators for the site 1 and 2, respectively. a, will be the electron operator with wave number k and spin s. Then the unperturbed Hamiltonian will be Ho =
2 b
s,aLa,
-
$ (bib, + bib,)
- p(b;bl + bib,).
(10)
The first term is the kinetic energy of metallic electrons, the second is the transfer energy of the particle and the third is its chemical potential. p is considered to be negative and large. The potential energy between the particle and the electrons will be expressed by H’
=
c
va:,,ak~[ei(k-“).R1b:bl
+ ei(k-k’)‘hb;b2]
.
kk’s
(11) R, and R, are locations of the site 1 and 2, respectively. V is the Fourier transform of the potential, whose k- and k’-dependencesAre neglected. The condition under which particle’s de-, ? grees of freedom can be presented by b,, b,
308
J. Kondo
I Neutron
scattering from
operators has been discussed in ref. 2. We assume that the condition is satisfied in our problem. The transfer energy is diagonalized by
tunneling protons
The matrix element of V, between states is expressed by V,(K)(
Ci =
-$ (b, + b2),
(12)
with c_ representing the ground state (E = -A/ 2). Then, we have .
in metals
mle-iK’RJn)
these two
(17)
,
where
(18)
K=k-ko.
VN(~) is the Fourier transform of V,(r) . Then the differential scattering cross section is
where the second summation should be over (+ = + and - . E, includes the chemical potential:
&
~27TlVN(K)12;
I(ml
e-iK’Rln)12
x S(E, - E, - w) (a=+).
E,=A,-p=$uA-p
(14)
m
=
The interaction follows:
H’ is written
= &[ei(k-k’Pl
-cc
xe
(15)
v,,
1 (nlei”‘R(t)e-iK’Rln)
IvN(K)12
in terms of c’s as
_ ei(k-k’PRz] .
(16)
-‘“’ dt >
(19)
where w is the energy gain of the neutron, and the argument t indicates the Heisenberg representation. When the initial state of our system is thermally distributed, the average ( IZI In) must be replaced by the thermal average ( ): cc
c:c++
We have neglected the term involving cIc_, because it is a constant potential for the electrons. Let H = H,, + H’, and let us denote the eigenstate of H by n: Hln)
= EnIn) .
We now consider a neutron, and denote the neutron-particle interaction by V,(r - R),where r and R are the coordinates of the neutron and the particle, respectively. Let the incident neutron state and the initial state of our system be
&
a Ivd’d2
eik’rlm) .
(eisc’R(t)
e-iut
dt.
(20)
In our second-quantization should be represented by e
kR
=
e
i~‘Rlb;bl
+ eiK’Rzb;b2
formalism,
eirceR
.
With (12), we find (ei”‘R(t) e-iK’R) = cos’ y + sin2 7
state be
e-iu.R)
-cc
eiko’rln) ,
and those of the scattered
I
((CIC,
+
(A(M)
ClC_)‘)
,
(21)
where A = ctc,
+ c;c_
(22)
309
J. Kondo I Neutron scattering from tunneling protons in metals
With the use of (22) we have
and
-R,.
a=R,
(23)
The first term of (21) is independent oft and gives us an elastic cross section. Thus, the inelastic cross section is e-‘“’ dt .
K(u) = 2 K,(u) 7
(30)
K,(U) =“- c ( zk’_&)c&)c~c_,) U’ The Fourier
coefficient
.
(31)
of K,(U) is defined by
(24)
K,(u) = p-’ 7 &(iq)
It is convenient to consider a retarded body Green’s function defined by
two-
Before calculating kc, we must calculate the one-particle Green’s function defined by
KR(t) = -ie(t)(
(25)
(A(t
eeiqU .
(32)
-m
A(
- AA(t))
,
G,(u) = - (~c,(~)c:) -
where 0(t) = 1, t>O, and 0(t) = 0, t < 0. If one finds its Fourier transform
= p-’ z e’,(iq)
KR(co)= [ KR(t) eiw’ dt
emioiU,
(33)
I
,
--CD
the right-hand
u>o UC0
wi = (2i + 1)@3 -l .
side of (24) is expressed
as
(34)
e‘, is expressed with the use of the self-energy as
m
I
e-‘“’
(A(t
dt
= i
KR’w~p~ ,l(-o) .
-cc
G,(iq)
(26)
Now, KR(u) is obtained as analytic continuation of the thermal (Matsubara) Green’s function. We define K(u) by
=
1 io, - E, - &,(iq)
(35)
*
The lowest-order contribution to JZO is represented by fig. 4 and is expressed by (36)
K(u) = - (TA(u)A) =
- (A(u 1 - (AA(u))
u>o, UCO,
(27)
where A(u) = euHA eeuH. We expand K(U) as a Fourier series: K(U) = p-’ 7 k(iv,) e-‘“” ,
(28)
where v, = 2alp-‘. The sum over 1 is extended over all integers. Then KR(u) is obtained by an analytic continuation:
pair excitation ener‘kC’ - ek is the electron-hole gy. fk is the Fermi distribution function. In the 2nth order of V-, we calculate the diagram shown in fig. 5, where n electron-hole pairs are initially produced and then absorbed. There are n! such diagrams, all of which give us the most divergent terms of the self-energy. In a previous paper [lo], we have shown how to calculate such diagrams. Here, we present the result for the analytically continued G’, : Kl2
1 E - EV + iri2
KR(u) = @iv,+
0 + is).
(29)
’
(37)
J. Kondo
310
I Neutron scattering from tunneling protons in metals
-u
j
oi
Fig. 4. Lowest-order self-energy correction. The thick solid line represents the particle, the thin solid line represents the electron and the dotted line represents the interaction.
Fig. 6. Unperturbed
two-body Green’s function.
Fig. 7. A correction to the two-body Green’s function. Fig. 5. Most divergent nth order self-energy correction.
where K=2V2p2
where Hi and H,, when continued to the complex plane, have a cut on the real axis and are analytic elsewhere. We note that (42) can be written as
sin* k,]R, - R,j
(
l-
k;lR,_R212
>
9
tiU(iq) = - &
2
$ 1
T=rKkT,
f(~)~l(z)~2(~
-
iq)
dz
,
i
(39)
(43) where
f(z) = is the state density of the electrons. Aeff is defined in (1). (37) has been obtained under the restriction 1E + p I< kT and A < kT. We now turn to kc. The lowest term is shown in fig. 6 and is expressed by
p
Kr’(iq)
= /3-l
1
= p-l
C H,(io,)H,(iw,) i-j=,
and the path of the integral is a small circle around z = iq. With the properties of Hi and Hz, the path can be deformed as shown in fig. 8. Thus, we have
1
C i_j=r iq - E, iwi - E-,
.
(41)
When each line is decorated by self-energy diagrams separately, the free one-particle Green’s functions in (41) will be replaced by the perturbed ones such as (37). Furthermore, we have diagrams like fig. 7, where each of n electronhole pairs are connected to two particle lines. There are n! such diagrams, all of which give us the most divergent contribution. They give us a renormalization factor of KU as we mention later. It can be shown that & in general has a form tiS(iq)
&y
,
(42)
--HI(& -is)]
+ H,(E + iq)
x [H2(e + ia) - H2(c - is)]} d.s , (44) ,where use has been made of f(.s + iv,) = f(8). Now, from (29) and (30), we have KR(co) - KR(-W)
= c [&(w L7 - K,(-w
So, we set iu,d
&OJ +
+ ia) + is)] .
is in (44). Here, we pre-
.I. Kondo I Neutron scattering from tunneling protons in metals
311
4. Conclusion Cross section of inelastic neutron scattering from a two-level system in metals is calculated. The interaction with metallic electrons reduces the level separation and causes a level broadening. The former goes as TK, whereas the latter as T. Since K < l/2 [ll], the ratio of the former to the latter increases as T goes down. When the ratio goes through unity, the proton motion may change from a hopping motion to a coherent one. The expression of the cross-over temperature is derived. It turns out that the cross-over temperature is of the order of the level separation at T = 0. This temperature is estimated to be -2 K for the proton trapped in the neighbourhood of impurity oxygen in niobium metal. This explains the rapid broadening of the neutron absorption line as the temperature raised ,from 1 K to 4 K.
Fig. 8. Integration path in (43).
sent the result of summation of the diagrams. For example, we have H,(E + iS)Z&(E - w - is) 1
E - E’, + irl2
E -
0
-
E_,
-
Xl2
’
(45) If one looks at (37), one finds the effect of diagrams of fig. 7 is to multiply a factor (DIkT)K’2. We note that we must replace the particle lines in fig. 7 by the perturbed G;,. We have done this only for the right most parts of the lines. Decoration of other parts gives us next divergent contributions. Using (45) and similar expressions in (44), we carry out the integration over E, setting f(e) to f(- p). We finally find i
KR(W)- KR(-W) = A&
f(-p)(
%)
K’2
epw - 1
Acknowledgement
The author would like to express his thanks to Professor H. Wipf for the preprint of the neutron experiment. Discussion with him motivated the present work.
References [l] [2] [3] [4]
1’
[6]
(46)
[7]
If one divides (46) by f(- CL), one has the cross section per tunneling proton. Thus we find that Aeff is the level separation and r the line broadening for the spectroscopic measurement. From (39), (4) and (I), we find that (8) holds. This confirms our argument of the previous section.
[8]
r x G
1 (o -
1
A,,,)* + r* + (W + A,,,)’ + r*
[5]
[9] [lo] [ll]
J. Kondo, Physica 84B (1976) 40, 207. J. Kondo, Physica 124B (1984) 25. J. Kondo, Physica 125B (1984) 279; 126B (1984) 377. R. Kadono, J. Imazato, K. Nishiyama, K. Nagamine, K. Yamazaki, D. Richter and J.-M. Welter, Phys. Lett. 109A (1985) 61. A. Magerl, J.J. Rush, J.M. Rowe, D. Richter and H. Wipf, Phys. Rev. B27 (1983) 927. H. Wipf, A. Magerl, S.M. Shapiro, S.K. Satija and W. Thomlinson, Phys. Rev. Lett. 46 (1981) 947. A. Magerl, H. Wipf, K. Neumaier, A.J. Dianoux and IS. Anderson, preprint. H. Wipf, D. Steinbinder, K. Neumaier, P. Gutsmiedl, A. Margel and A.J. Dianoux, private communication. K. Yamada, A. Sakurai and M. Takeshige, Prog. Theor. Phys. 70 (1983) 73. J. Kondo, Physica 123B (1984) 175. K. Yamada and K. Yosida, Prog. Theor. Phys. 68 (1982) 1504.
NEUTRON SCATTERING FROM TUNNELING PROTONS IN METALS J. KONDO Electrotechnical Laboratory, Tsukuba Research Center, Ibaraki, Japan Received 26 November
1985
We consider a two-level system in metals, in which a proton is tunneling between two neighbouring sites. Cross section of inelastic neutron scattering from this system is calculated. The interaction with the metallic electrons reduces the level separation and causes a level broadening. The ratio of the former to the latter turns out to increase as a negative power of the temperature as the temperature decreases. Thus, the proton may make a coherent motion at very low temperatures. ‘Ihe cross-over temperature where this ratio goes through unity is shown to be of the order of the level separation at T = 0. This result explains a recent neutron experiment on the proton in niobium with a small amount of oxygen.
1. Introduction In previous papers, we considered a two-level system in metals [l, 21, where a particle can jump between two sites with a transfer matrix A. The particle was assumed to interact with the metallic electrons. An important consequence was that the transfer matrix of the particle is renormalized by the overlap integral between two electronic wave functions, each wave function being what is realized when the particle is fixed in one of the sites. As an application of the result, we calculated [3] probability of the particle being found in a neighbouring site, when it was initially found at the initial site. The probability was found to be proportional to the time, and the proportionality constant W, the hopping rate, was closely related to the diffusion constant. Theoretical consequence was that the diffusion constant is proportional to a power of the temperature, and this power law was actually found in diffusion of the positive muon in copper [4]. Another typical two-level system is a proton trapped by an oxygen in Nb metals. It has been proposed that the proton jumps between two neighbouring tetrahedral sites near an oxygen, and it was shown that there is no trend for the proton to precipitate even at very low temperatures [5]. This implies that the trapped state is stable, and the proton remains in that state at low temperatures. When the proton jumps between
two sites with transfer integral A, symmetric and antisymmetric states are the stationary states with level separation 24. Recently, this level separation as well as level broadening were measured by inelastic neutron scattering [6-81. In these experiments, a small level separation (-2 K) was clearly observed, and the line broadens very rapidly as the temperature is raised from 1 K to 4 K (under a magnetic field enough to destroy superconductivity of Nb [S]). Such a rapid change of the level broadening must come from the effect of metallic electrons. In fact, the authors show that the Korringa-type relaxation explains the level broadening [7]. The purpose of this paper is to calculate inelastic line shape of neutron scattering from a two-level system in metals and analyse these experimental results. In the next section, we present a qualitative consideration which involves the most important aspect of this paper. In the following section, we present a detailed calculation of the cross section.
2. Qualitative cousideration We first summarize results of previous investigation [l, 31. The transfer integral A is renormalized as
03784363/86/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
@=>A)
>
J. Kondo
306
I Neutron scattering from tunneling protons in metals
where D is an energy of the order of the band width of metallic electrons, K is a dimensionless coupling constant (see (38)). Suppose the particle was at one of the two sites at t = 0. The probability P of the particle to be found in the other site at t would be P = sin* A& ,
(2)
if there were no effect of the interaction except renormalization of A. Actually the metallic electron causes a dissipation or a damping of the particle motion. Thus, coherent increase of P is disturbed after a short time r,, namely P ceases to increase quadratically at TV. After that, P again increases quadratically until a time --TV is reached. A net result is a linear increase of P = Wt (fig. l), where W = A&,
.
(3)
A detailed calculation
At this point, the particle begins to w4ff. make a coherent oscillation between the two sites. The temperature at which W- Aeff occurs is obtained from
where we have neglected numerical factors in (4). This gives us the cross-over temperature T,,: , A \ K/Cl-K) kT,, = A(;) .
(5)
As the temperature is lowered further, the particle will oscillate many cycles following (2), before it is perturbed. In this situation, the particle motion may be called coherent and the hopping rate W will be of the order of Aeff. Thus, W/A,,, vs. T will look like fig. 2. Now, we expect that Aeff also levels off below To. At T = To, the value of Aeff is
[3] shows K
= kT,, ,
W A -A eff = rKkT For the positive muon in copper, the values of these parameters were A = 0.065 K, K = 0.3. We take D = 6 eV as a typical band width of copper. Then, W/Aeff = 0.0023 at T = 1 K, which is much smaller than 1. This means that the particle state is perturbed many times before the particle makes a coherent motion (a motion back and forth between the two sites). In this case the particle motion may be called hopping. As the temperature goes down, W/Aeff increases as (4) indicates. Finally, one will have sinZ&fft
I
it
I /
f‘
-
OK7 of a particle found in a neighbouring
LL
l -------: F
=ff
0
c
Fig. 1. Probability
where use is made of (5). Thus, we expect that Aeff will remain to be the order of kT,, below To. Aeff vs. Twill look like fig. 3. Aeff begins to deviate from (1) below T = A lk and levels off below T = T,,. We thus arrive at an important conclusion: the cross-over temperature is the same order of magnitude as the effective level separation at T = 0 (devided by k). kT,, given by (5) is nothing but energy scale of our problem. Such an energy scale was introduced by Yamada et al. [9]. We now discuss spectroscopic aspect of our problem. Here, two quantities are relevant: the level separation Aeff and the level broadening r.
site.
TO
Fig. 2. W/A,,, vs. T.
t
J. Kondo
I Neutron scattering from tunneling protons in metals Table I
A In- eff
0
307
T+
In+
K
A (K)
T o svpsr(K)
To (K)
0.2 0.3 0.4
15 40 110
2.44 2.30 2.22
2.0 2.0 2.0
A+A kT 0 super= A( D s)XI(l-K) ,
0
(9)
Fig. 3. In A,,,/ TOvs. In TIT,.
We may expect -1
7,
-r.
(7)
where A, is the superconducting gap. With A, = 16 K for Nb, we find Tosuper as shown in table I. They are just right magnitude to explain the 15% increase.
Then, from (3) we have 3. Calculation
w -=A eff
Aeff
of the cross section of neutron
scattering
I”
Then, at T 9 To, where WfAeff & 1, we have A,,,/I’ e 1. At T 6 T,,, (3) is no longer valid, but we may expect A,,,/r % 1. Thus, T,, is also the cross-over temperature for the spectroscopic problem. We now analyse the experimental results on the proton two-level system in NbO, [7,8]. We are interested in protons in normal Nb [8]. We first determine T,, for this system. As I mentioned, kT,, is the level separation at T = 0. This can be obtained from the peak position at T = 1 K for the proton concentration 0.0001. We find T,, = 2 K. Since To is the cross-over temperature as well, it is quite natural that the line broadens and almost vanishes at T = 4.2 K. Such a rapid change of the line shape as the temperature is changed from 1 K to 4.2 K can be explained only in terms of the effect of the metallic electrons. From To = 2 K, we can deduce the value of A based on (5). We may take D = 4 eV as a typical band width. For three values of K, we find A, whose values are shown in table I. Now, we compare the level separation in normal Nb with that in superconducting Nb [7]. The latter at T = 1 K is larger than the former at T = 1 K by about 15%. Here, we assume that the effect of the superconducting gap on the renormalization of A is expressed as
In this section we calculate the cross section of inelastic neutron scattering from a two-level system in metals and confirm (7) or (8). b 1 and b, will be the particle operators for the site 1 and 2, respectively. a, will be the electron operator with wave number k and spin s. Then the unperturbed Hamiltonian will be Ho =
2 b
s,aLa,
-
$ (bib, + bib,)
- p(b;bl + bib,).
(10)
The first term is the kinetic energy of metallic electrons, the second is the transfer energy of the particle and the third is its chemical potential. p is considered to be negative and large. The potential energy between the particle and the electrons will be expressed by H’
=
c
va:,,ak~[ei(k-“).R1b:bl
+ ei(k-k’)‘hb;b2]
.
kk’s
(11) R, and R, are locations of the site 1 and 2, respectively. V is the Fourier transform of the potential, whose k- and k’-dependencesAre neglected. The condition under which particle’s de-, ? grees of freedom can be presented by b,, b,
308
J. Kondo
I Neutron
scattering from
operators has been discussed in ref. 2. We assume that the condition is satisfied in our problem. The transfer energy is diagonalized by
tunneling protons
The matrix element of V, between states is expressed by V,(K)(
Ci =
-$ (b, + b2),
(12)
with c_ representing the ground state (E = -A/ 2). Then, we have .
in metals
mle-iK’RJn)
these two
(17)
,
where
(18)
K=k-ko.
VN(~) is the Fourier transform of V,(r) . Then the differential scattering cross section is
where the second summation should be over (+ = + and - . E, includes the chemical potential:
&
~27TlVN(K)12;
I(ml
e-iK’Rln)12
x S(E, - E, - w) (a=+).
E,=A,-p=$uA-p
(14)
m
=
The interaction follows:
H’ is written
= &[ei(k-k’Pl
-cc
xe
(15)
v,,
1 (nlei”‘R(t)e-iK’Rln)
IvN(K)12
in terms of c’s as
_ ei(k-k’PRz] .
(16)
-‘“’ dt >
(19)
where w is the energy gain of the neutron, and the argument t indicates the Heisenberg representation. When the initial state of our system is thermally distributed, the average ( IZI In) must be replaced by the thermal average ( ): cc
c:c++
We have neglected the term involving cIc_, because it is a constant potential for the electrons. Let H = H,, + H’, and let us denote the eigenstate of H by n: Hln)
= EnIn) .
We now consider a neutron, and denote the neutron-particle interaction by V,(r - R),where r and R are the coordinates of the neutron and the particle, respectively. Let the incident neutron state and the initial state of our system be
&
a Ivd’d2
eik’rlm) .
(eisc’R(t)
e-iut
dt.
(20)
In our second-quantization should be represented by e
kR
=
e
i~‘Rlb;bl
+ eiK’Rzb;b2
formalism,
eirceR
.
With (12), we find (ei”‘R(t) e-iK’R) = cos’ y + sin2 7
state be
e-iu.R)
-cc
eiko’rln) ,
and those of the scattered
I
((CIC,
+
(A(M)
ClC_)‘)
,
(21)
where A = ctc,
+ c;c_
(22)
309
J. Kondo I Neutron scattering from tunneling protons in metals
With the use of (22) we have
and
-R,.
a=R,
(23)
The first term of (21) is independent oft and gives us an elastic cross section. Thus, the inelastic cross section is e-‘“’ dt .
K(u) = 2 K,(u) 7
(30)
K,(U) =“- c ( zk’_&)c&)c~c_,) U’ The Fourier
coefficient
.
(31)
of K,(U) is defined by
(24)
K,(u) = p-’ 7 &(iq)
It is convenient to consider a retarded body Green’s function defined by
two-
Before calculating kc, we must calculate the one-particle Green’s function defined by
KR(t) = -ie(t)(
(25)
(A(t
eeiqU .
(32)
-m
A(
- AA(t))
,
G,(u) = - (~c,(~)c:) -
where 0(t) = 1, t>O, and 0(t) = 0, t < 0. If one finds its Fourier transform
= p-’ z e’,(iq)
KR(co)= [ KR(t) eiw’ dt
emioiU,
(33)
I
,
--CD
the right-hand
u>o UC0
wi = (2i + 1)@3 -l .
side of (24) is expressed
as
(34)
e‘, is expressed with the use of the self-energy as
m
I
e-‘“’
(A(t
dt
= i
KR’w~p~ ,l(-o) .
-cc
G,(iq)
(26)
Now, KR(u) is obtained as analytic continuation of the thermal (Matsubara) Green’s function. We define K(u) by
=
1 io, - E, - &,(iq)
(35)
*
The lowest-order contribution to JZO is represented by fig. 4 and is expressed by (36)
K(u) = - (TA(u)A) =
- (A(u 1 - (AA(u))
u>o, UCO,
(27)
where A(u) = euHA eeuH. We expand K(U) as a Fourier series: K(U) = p-’ 7 k(iv,) e-‘“” ,
(28)
where v, = 2alp-‘. The sum over 1 is extended over all integers. Then KR(u) is obtained by an analytic continuation:
pair excitation ener‘kC’ - ek is the electron-hole gy. fk is the Fermi distribution function. In the 2nth order of V-, we calculate the diagram shown in fig. 5, where n electron-hole pairs are initially produced and then absorbed. There are n! such diagrams, all of which give us the most divergent terms of the self-energy. In a previous paper [lo], we have shown how to calculate such diagrams. Here, we present the result for the analytically continued G’, : Kl2
1 E - EV + iri2
KR(u) = @iv,+
0 + is).
(29)
’
(37)
J. Kondo
310
I Neutron scattering from tunneling protons in metals
-u
j
oi
Fig. 4. Lowest-order self-energy correction. The thick solid line represents the particle, the thin solid line represents the electron and the dotted line represents the interaction.
Fig. 6. Unperturbed
two-body Green’s function.
Fig. 7. A correction to the two-body Green’s function. Fig. 5. Most divergent nth order self-energy correction.
where K=2V2p2
where Hi and H,, when continued to the complex plane, have a cut on the real axis and are analytic elsewhere. We note that (42) can be written as
sin* k,]R, - R,j
(
l-
k;lR,_R212
>
9
tiU(iq) = - &
2
$ 1
T=rKkT,
f(~)~l(z)~2(~
-
iq)
dz
,
i
(39)
(43) where
f(z) = is the state density of the electrons. Aeff is defined in (1). (37) has been obtained under the restriction 1E + p I< kT and A < kT. We now turn to kc. The lowest term is shown in fig. 6 and is expressed by
p
Kr’(iq)
= /3-l
1
= p-l
C H,(io,)H,(iw,) i-j=,
and the path of the integral is a small circle around z = iq. With the properties of Hi and Hz, the path can be deformed as shown in fig. 8. Thus, we have
1
C i_j=r iq - E, iwi - E-,
.
(41)
When each line is decorated by self-energy diagrams separately, the free one-particle Green’s functions in (41) will be replaced by the perturbed ones such as (37). Furthermore, we have diagrams like fig. 7, where each of n electronhole pairs are connected to two particle lines. There are n! such diagrams, all of which give us the most divergent contribution. They give us a renormalization factor of KU as we mention later. It can be shown that & in general has a form tiS(iq)
&y
,
(42)
--HI(& -is)]
+ H,(E + iq)
x [H2(e + ia) - H2(c - is)]} d.s , (44) ,where use has been made of f(.s + iv,) = f(8). Now, from (29) and (30), we have KR(co) - KR(-W)
= c [&(w L7 - K,(-w
So, we set iu,d
&OJ +
+ ia) + is)] .
is in (44). Here, we pre-
.I. Kondo I Neutron scattering from tunneling protons in metals
311
4. Conclusion Cross section of inelastic neutron scattering from a two-level system in metals is calculated. The interaction with metallic electrons reduces the level separation and causes a level broadening. The former goes as TK, whereas the latter as T. Since K < l/2 [ll], the ratio of the former to the latter increases as T goes down. When the ratio goes through unity, the proton motion may change from a hopping motion to a coherent one. The expression of the cross-over temperature is derived. It turns out that the cross-over temperature is of the order of the level separation at T = 0. This temperature is estimated to be -2 K for the proton trapped in the neighbourhood of impurity oxygen in niobium metal. This explains the rapid broadening of the neutron absorption line as the temperature raised ,from 1 K to 4 K.
Fig. 8. Integration path in (43).
sent the result of summation of the diagrams. For example, we have H,(E + iS)Z&(E - w - is) 1
E - E’, + irl2
E -
0
-
E_,
-
Xl2
’
(45) If one looks at (37), one finds the effect of diagrams of fig. 7 is to multiply a factor (DIkT)K’2. We note that we must replace the particle lines in fig. 7 by the perturbed G;,. We have done this only for the right most parts of the lines. Decoration of other parts gives us next divergent contributions. Using (45) and similar expressions in (44), we carry out the integration over E, setting f(e) to f(- p). We finally find i
KR(W)- KR(-W) = A&
f(-p)(
%)
K’2
epw - 1
Acknowledgement
The author would like to express his thanks to Professor H. Wipf for the preprint of the neutron experiment. Discussion with him motivated the present work.
References [l] [2] [3] [4]
1’
[6]
(46)
[7]
If one divides (46) by f(- CL), one has the cross section per tunneling proton. Thus we find that Aeff is the level separation and r the line broadening for the spectroscopic measurement. From (39), (4) and (I), we find that (8) holds. This confirms our argument of the previous section.
[8]
r x G
1 (o -
1
A,,,)* + r* + (W + A,,,)’ + r*
[5]
[9] [lo] [ll]
J. Kondo, Physica 84B (1976) 40, 207. J. Kondo, Physica 124B (1984) 25. J. Kondo, Physica 125B (1984) 279; 126B (1984) 377. R. Kadono, J. Imazato, K. Nishiyama, K. Nagamine, K. Yamazaki, D. Richter and J.-M. Welter, Phys. Lett. 109A (1985) 61. A. Magerl, J.J. Rush, J.M. Rowe, D. Richter and H. Wipf, Phys. Rev. B27 (1983) 927. H. Wipf, A. Magerl, S.M. Shapiro, S.K. Satija and W. Thomlinson, Phys. Rev. Lett. 46 (1981) 947. A. Magerl, H. Wipf, K. Neumaier, A.J. Dianoux and IS. Anderson, preprint. H. Wipf, D. Steinbinder, K. Neumaier, P. Gutsmiedl, A. Margel and A.J. Dianoux, private communication. K. Yamada, A. Sakurai and M. Takeshige, Prog. Theor. Phys. 70 (1983) 73. J. Kondo, Physica 123B (1984) 175. K. Yamada and K. Yosida, Prog. Theor. Phys. 68 (1982) 1504.