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Unformatted text preview: PHYSICAL REVIEW Letters to the Editor UBLI CA TI ON of brief reports of important discoveries in physics may be secured by addressing them to this department. The closing date for this department is five weeks prior to the date of issue. No proof will be sent to the authors. The Board of Editors does not hold itself responsible for the opinions expressed by the corre spondents. Communications should not exceed 600 words in length
and should be submitted in duplicate. Bound Electron Pairs in a Degenerate
Fermi Gas* LEON N. COOPER Physics Department, University of Illinois, Urbana, Illinois
(Received September 21, 1956) T has been proposed that a metal would display
superconducting properties at low temperatures if
the one—electron energy spectrum had a volume—inde
pendent energy gap of order A2161}, between the
ground state and the ﬁrst excited state}2 We should
like to point out how, primarily as a result of the
exclusion principle, such a situation could arise. Consider a pair of electrons which interact above a
quiescent Fermi sphere with an interaction of the kind
that might be expected due to the phonon and the
screened Coulomb ﬁelds. If there is a net attraction
between the electrons, it turns out that they can form
a bound state, though their total energy is larger than
zero. The properties of a noninteracting system of such
bound pairs are very suggestive of those which could
produce a superconducting state. To what extent the
actual manybody system can be represented by such
noninteracting pairs will be discussed in a forthcoming
paper. Because of the similarity of the superconducting
transition in a wide variety of complicated and differing
metals, it is plausible to assume that the details of metal
structure do not affect the qualitative features of the
superconducting state. Thus, we neglect band and
crystal structure and replace the periodic ion potential
by a box of volume V. The electrons in this box are
free except for further interactions between them which
may arise due to Coulomb repulsions or to the lattice
vibrations. In the presence of interaction between the electrons,
we, can imagine that under suitable circumstances there
will exist a wave number go below which the free states
are unaffected by the interaction due to the large energy
denominators required for excitation. They provide a
ﬂoor (so to speak) for the possible transitions of elec
trons with Wave number k,> qo. One can then consider
the eigenstates of a pair of electrons with 161, k2> (10. For a complete set of states of the two—electron system
we take planewave product functions, go(k1,k2; r1,r2) VOLUME 104. NUMBER 4 NOVEMBER 15. 1956 = (1/ V) exp[i(k1' r1+k2r2)] which satisfy periodic
boundary conditions in a box of volume V, and where
r1 and r2 are the coordinates of electron one and elec
tron two. (One can use antisymmetric functions and
obtain essentially the same results, but alternatively
we can choose the electrons of opposite spin.) Deﬁning
relative and center—ofmass coordinates, R=%(r1+r2),
r= (1‘2— r1),K=(k1+k2) and k: % (k2— k1), and letting
5K+ek= (h2/m) (iK2+k2), the Schrodinger equation
can be written (5KI ek—E)ak+zkr akz(k[H1l k’)
><B(K—K’)/5(0)=0 (1) ‘I’(R,r)= (1/\/V)e"K'Rx(r,K),
X(r7K)=Zk (“k/VVFW": 1
(lellk')=(—V— f We haveiassumed translational invariance in the metal.
The summation over k’ is limited by the exclusion
principle to values of In and [22 larger than go, and by
the delta function, which guarantees the conservation
of the total momentum of the pair in a single scattering.
The K dependence enters through the latter restriction. Bardeen and Pines3 and Frohlich4 have derived
approximate formulas for the matrix element (k I H1 [ k’) ;
it is thought that the matrix elements for which the
two electrons are confined to a thin energy shell near
the Fermi surface, elzegzep, are the principal ones
involved in producing the superconducting state?—4
With this in mind we shall approximate the expressions
for (MB 1 k’) derived by the above authors by (lellk')="lFl if kogk,k’$km
=0 otherwise, where (2) and 0 phonona (3) where F is a constant and (h2/m) (kmzwk02)22hw20.2
ev. Although it is not necessary to limit oneself so
strongly, the degree of uncertainty about the precise
form of (lellk’) makes it worthwhile to explore the
consequences of reasonable but simple expressions.
With these matrix elements, the eigenvalue equation becomes
‘7" N(K,e)de
1=—IF f W, (4)
so E—E—gK where N (K ,e) is the density of twoelectron states of
total momentum K, and of energy e= (hZ/m)k2. To a
very good approximation N (K ,e)zN (K ,eo). The result
ing spectrum has one eigenvalue smaller than God£13,
while the rest lie in the continuum. The lowest eigen
value is Eo= 60+ fig—A, where A is the binding energy
of the pair A: (em— eo)/ (61’5— 1), (5) 1189 1190 where B=N(K,e) IF The binding energy, A, is inde—
pendent of the volume of the box, but is strongly
dependent on the parameter 6. . Following a method of Bardeen,5 by which the
coupling constant for the electron—electron interaction,
which is due to phonon exchange, is related to the high
temperature resistivity which is due to phonon absorp—
tion, one gets [~32an 10—6, where p is the high—tem
perature resistivity in esu and n is the number of
valence electrons per unit volume. The binding energy
displays a sharp change of behavior in the region 621
and it is just this region which separates, in almost
every case, the superconducting from the nonsuper—
conducting metals.5 (Also it is just in this region where
the attractive interaction between electrons, due to
the phonon ﬁeld, becomes about equal to the screened
Coulomb repulsive interaction.) The ground—state wave function, eik"N(K,e(k)) d. Xo(r,K)=(C0nSt)fm represents a true bound state which for large values of r
decreases at least as rapidly as const/rz. The average
extension of the pair, [(rzﬁvjli’, is of the order of 10”4 cm
for Aszc. The existence of such a bound state with
nonexponential dependence for large 7 is due to the
exclusion of the states [a <ko from the unperturbed
spectrum, and the concomitant degeneracy of the lowest
energy states of the unperturbed system. One would
get no such state if the potential between the electrons
were always repulsive. All of the excited states xn>o(r,K)
are very nearly plane waves. The pair described by xo(r) may be thought to have
some Bose properties (to the extent that the binding
energy of the pair is larger than the energy of interaction
between pairs).6 However, since N (K ,e) is strongly de
pendent on the total momentum of the pair, K, the
binding energy A is a very sensitive function of K,
being a maximum where K =0 and going very rapidly
to zero where szm— 160. Thus the elementary excita—
tions of the pair might correspond to the splitting of
the pair rather than to increasing the kinetic energy of
the pair. In either case the density of excited states (dN/dE)
would be greatly reduced from the free—particle density
and the elementary excitations would be removed from
the ground state by what amounted to a small
energy gap. , If the manybody system could be considered (at
least to a lowest approximation) a collection of pairs
of this kind above a Fermi sea, we would have (whether
or not the pairs had signiﬁcant Bose properties) a model
similar to that proposed by Bardeen which would
display many of the equilibrium properties of the
superconducting state. The author wishes to express his appreciation to LETTERS TO THE EDITOR Professor John Bardeen for his helpful instruction in
many illuminating discussions. * This Work was supported in part by the Ofﬁce of Ordnance
Research, U. S. Army. 1 J. Bardeen, Phys. Rev. 97, 1724 (1955). 2 See also, for further references and a general review, J.
Bardeen, Theory of Superconductivity H andbnch der Physik
(Springer—Verlag, Berlin, to be published), Vol. 15, p. 274. 3 J. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955). 4 H. Frtihlich, Proc. Roy. Soc. (London) A215, 291 (1952). 5 John Bardeen, Phys. Rev. 80, 567 (1950). °It has also been suggested that superconducting properties
would result if electrons could combine in even groupings so that
the resulting aggregates would obey Bose statistics. V. L. Ginz
burg, Uspekhi Fiz. Nauk 48, 25 (1952); M. R. Schafroth, Phys.
Rev. 100, 463 (1955). Magnetic Resonance in Manganese
Fluoride B. BLEANEY* Clarendon Laboratory, Oxford, England
(Received September 24, 1956) UCLEAR magnetic resonance of the ﬂuorine
nuclei in Man has recently been observed by
Shulrnan and Jaccarino,1 who found a much greater
“paramagnetic shift” of the resonance ﬁeld than would
be expected from simple magnetic dipole ﬁelds of the
manganese ions. Electronic paramagnetic resonance of
Mn2+ ions present as impurities in the isomorphous
crystal Zan has previously been observed by Tinkharn,2
who made detailed measurements of the ﬂuorine hyper—
ﬁne structure which results from overlap of the mag
netic electrons onto the ﬂuorine ions. The purpose of
this note is to point out that the shift of the nuclear
resonance can be estimated from Tinkham’s data, and
that good agreement with the measured value is found.
With an obvious extension of Tinkharn’s nomen
clature, the Hamiltonian for the system can be
written as 38=~gNBNHI+ZNI.ANSN, (1) Where I is the spin operator for a ﬂuorine nucleus, A” is
the hyperﬁne structure constant for interaction between
this nucleus and the Nth manganese ion whose spin
operator is SN. Owing to the rapid change of spin
orientation .for the manganese ions, we must take a
weighted mean of the different values of the projection
M of S” on the direction of the applied ﬁeld. This
mean is M=ZM M exp(—WM/kT)/ZM exp(—WM/kT), which cannot be evaluated from ﬁrst principles in a
substance such as Man where strong internal ﬁelds
are acting. However, we may relate it to the measured
susceptibility, since, per mole, x=NgﬁM/H. Hence we have, for the quantum of energy required to ...
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 Conductivity, Superconductivity

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