IC/90/58

NATIONAL CENTRE FORTHEORETICAL PHYSICS

NONLOCAL GINZBURG-LANDAU EQUATIONSI. PURE CASE

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Hong-Hua Xu

Xiao-Xing Wu

and

Chien-Hua Tsai

1990M1RAMARE-TRIESTE

IC/90/58

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

NONLOCAL GINZBURG-LANDAU EQUATIONSI. PURE CASE*

Hong-HuaXu"

International Centre for Theoretical Physics, Trieste, Italy,

Xiao-Xing Wu

Institute of Condensed Matter Physics and Department of Physics,Jaio-Tong University, Shanghai 200030, People's Republic of China

and

Chien-Hua Tsai

Institute of Condensed Matter Physics and Department of Physics,Jiao-Tong University, Shanghai 200030, People's Republic of China

andCenter of Theoretical Physics,

Chinese Centre of Advanced Science and Technology (World Laboratory),Beijing 100080, People's Republic of China.

MIRAMARE - TRIESTE

April 1990

* To be submitted for publication.** Permanent address: Institute of Condensed Matter Physics and Department of Physics, Jiao-Tong

University, Shanghai 200030, People's Republic of China.

T

ABSTRACT

Starting from the BCS Hamiltonian, we succeeded to establish, in the pure case, nonlo-cal Ginburg-Landau equations which reduce to their conventional form in the local limit and leadnaturally to a reversal of the magnetic field in the neighbourhood of a vortex in type U/l supercon-ductors. The form factor resulted from first principle calculation is in nice agreement with neutronscattering data.

1. INTRODUCTION

Following general arguments in the theory of second order phase transition, Ginzburgand Landau * proposed, for a description of superconductors, their well-known equations, whichwere lately derived microscopically by Gor'kov 2 from the BCS theory3. Werthamer 4 andTewordt5 clarified the premises underlying Gor'koc's derivation, and succeeded to extend G-L equations to a wider temperature range of validity. The necessity for a generalization to thenonlocal case becomes evident with the discovery 6 of phase transitions of different nature at Hc\in type D/l and K/2 superconductors, implying a magnetic field reversal in the neighbourhood of avortex and an attraction between vortices at distances when the G-L parameter is within a certainrange near 2 */2.

There had been a number of theoretical efforts 7>8 attempted at an attack of this problem.But none of them provided a satisfactory first principle solution in the sense that the nonlocal effectrelated, particularly, to spatial variation of the order parameter is correctly and fully included.

Measurements 9 on high Te materials revealed extreme small coherence lengths, imply-

ing rapid variations of the order parameter. Fully nonlocal G-L equations are, therefore, highly

desirable by the discovery of superconducting ceramics.

Using the closed time path Green's function (CTPGF) technique 10 , we formulate realtime Gor'kov equations and an associated electric current expression in terms of the retardedGreen's functions at finite temperatures. Nonlocal G-L equations with gradient terms of the orderparameter included to all orders are then derived using the fluctuation-dissipation theorem. Theyreduce to conventional G-L equations in the local limit. The kernels characterizing full nonlocaleffects in our equations have an asymptotic behaviour k~2 for k -+ oo leading naturally to a fieldreversal and an attraction between vortices in type U/l superconductors. A first principle calcula-tion gives a form factor in good agreement with neutron scattering data 1] without any adjustableparameters.

For simplicity, we set in this paper h = c ~ 1. However, dimension will be restored

whenever confusion may arise.

2. PRELIMINARY

We initiate from the BCS Hamiltonian in its familiar form

H*J^fd3xti(x) \-^[V+ieA(x)}2-//L.(z)-A f d( 2 1 )

For the moment, we find (2.1) the best starting point. Experimental evidence agrees unanimously at

the pairing of carriers in either conventional or high Tc superconductors12. As an effective pairing

Hamiltonian, (2.1) does not, in fact, necessarily result from the phonon mechanism. Superconduc-tivity arising from any mechanism is describable in tenns of (2.1), provided the pairing interactioncan be approximated by its last tenn on the right-hand side. We emphasize, therefore, (2.1) is themost general microscopic Hamiltonian at the moment, and we choose it to start with.

Introducing the density matrix

(2.2)

and defining the CTPGF

tf (y) >,Gp(x,y)=i[ (2.3)

with < .. . > implying a statistical average, e.g.,

- i < rPiMz)^T+(y) > = -iTr{pTp^{x)tf{y)} . (2.4)

We have the following equations of motion for Gp after Gor'kov's factorization 2 in the light ofCooper pairing

kGp(x,y)=6p-(x-y)-i\o2GT(x,x)<j2Gp(x,y) (2.5)

where 6* (z — y) is the four-dimensional delta function defined on the closed time path, &% thePauli matrix and __„

. / & 2 U 2 , 0K=\ _ (2.6)

whereas G£ is the transpose of Gp. In terms of the retarded Green's function Gr and the correlationfunction Ge Eq.(2.5) can be cast into the single time form 10

kGr(x,y) =£>l(x-y)-i\o2GT

c{x,x)a2GT{x,y) . (2.7)

The diagonal elements of Gc( x, x) are

l \ i[ M)tf() ^()^() >] (2.8a)

fGf (i,x) = -ji[< 4>t(x)i>i(x) >-< ^(x)^(x) >] . (2.86)

In relatively weak magnetic field, the Pauli paramagnetism can be neglected to restore spin sym-metry. Eq.(2.8) then implies

G\\x,x)=G?(x,x) (2.9)

and the terms containing Gf can be included in the chemical potential, Moreover, the off-diagonalelements of Ge define the order parameter A (x):

A(x) = -i\Gf(x,x) = j\[< iMz) iM*) > ~ < i M * W z ) >] (210a)

A*(x) = -i\G?(x,x) = U[< ^(x)^(x) >-< 0|(x)0f(x) >] . (2.106)2

4

We obtain, therefore, the following real time form of Gor'kov's equations

- \ 1 * «L. It A ( T*\ \ *— +

_T 2 J Gr(x,y) = 6*(x — y) . (2.11)

The equivalent integral equation is

- - CM - / 0, A(z) \ = -

Gor(x, y) being the normal state retarded Green's function satisfying

-y). (2.13)

The relation Ga(x,y) = G%(y,x) enables us to write (2.12) alternatively in terms of the advancedGreen's function Ga{x, y)

y ^ : ( 2 )A ( 2 ^ 5 < « ( ^ I y ) . (2.14)

Meanwhile the current is given by

T(x) = - (r^-EVT - V7] + — T(x)\ [< tf(x)Wy) > + < V>t(zWi/) >l*'v •(2.15)

It is not difficult to show that the following relation holds for the Heisenberg operators ̂ and ^

[ (^)]^ ) ] ^.iu (2.16)

with [A, B]+ =AB + BA. We then have

< Vf(x)Vr(V> > = i U + «p ( - ^ J - ) | [£"(?.*) " GiJ(y,x)] (2.17)

and similarly

[ ^ f ] (2.18)

So that (2.15) leads to the real time expression of the electric current

[(2'n)-Ad?k \-T+ eA(x)] f(ko)ImG?{k,x) (2.19)J L •>

Ifm

where under the integral sign, /(k0) is the Fermi function and GA k, x) is the Fourier componentof Gr(x,x)

GAx.y) = j{2>x)-A<?kGr(k,x) exp [-it • (it-T) + iko(U - ty)]^ . (2.20)

T

Since the normal state functions <?„- contributes nothing to the current, we have

7 ( ) = — [ {2T<V* d? k\-t + eA(x)] f(ko)Im8G?(k,x) (2.21)771 J L J

where

SGf (x,x) = f <?zl<?z2G%(x,zt)A*(z1)d}l(z1,z2)Mz2)Gl2(z2,x) . (2.22)

We obtain, from the definition of the order parameter (2.10a) and using the fluctuation-dissipationtheorem, the following equation

A(z) = -iJ(2ir)-Ad*kGlHk,x) = -ij'(27r)-4d4kth ^ I [5j2(Jt,x) - G?(k,x)\(2.23)

where

Gi2(x,a:) = - j^zG^r{x,z)Mx)Gf{zix) (2.24)

and

Gl2(x,x) = ~ J <?zGla'(x,z)A(z)G%(z,x) (2.25)

Eqs.(2.11), (2.21) and (2.23) form the basis of our nonlocal theory of type II superconductivity.

3. DERIVATION OF NONLOCAL ORDER PARAMETER EQUATION

Under the semiclassical approximation, the solution to Eq.(2.13) is 13

(3.1), o

0, exp \-ie ffdT -X(s)]

where Gor( x, y) is the free electron Green's function with the following Fourier transformation

(3.2)

In the same approximation, Eq.(2.11) is solved to give

ie I ds • A (s)

ds • A (s) (3.3)

where GT(xt y) is the advanced Green's function for superconducting State in the absence of a

magnetic field. It satisfies

which can be approximately solved to yieldI3

r> ft- >r\ - \iA P 2 a » u ; t n + r ' f * « + £ ( ^ ' —A(s) \GAk,x)-[ko~E (kiX) + tko0 J ^ J

with E2(k, x) = £*(_*:) + |A(z)|2 . Making use of (3.1) and (3.3), Eq.(2.24) can be expressed,with D% = V* - 2ieA ( i ) , as

[ ] ( z , x ) (3.6)

which leads, after Fourier transformation, to

T-iDx,k0)A(x)]G?(k,x). (3.7)

In obtaining (3.6), the following relation 13

exp —lie / d~T -A is) A(*)=exp[(z -3Vz?jA(i) (3.8)h

has been utilized. A similar expression

Gl2(k,x) = -Gll(k,x)G%(T + iDs,k0)A(x) (3.9)

can be derived from (2.25). The substitution of (3.7) and (3.9) into (2.23) yields

Dx) + iVjfc • Dx - 1] [G£(k)A(x)] - G\Hk,x) [exp(tV* • Dx)

(3.10)

The first term inside the curly brackets under the integral sign on the right-hand side of Eq.(3.10)is easily evaluated with the result >, N( 0) A (x) S( T, A),

fUlo 1

S(T,A)= / du[2E{u,x)rxth~^E{u,x) (3.11)Jo 2

where uo is a characteristic frequency depending upon the specified mechansim of superconduc-tivity, e.g., the Debye frequency in the case of phonon mechanism. The calculations of the secondand third terms can be simplified by introducing the operators

L\ (a) = exp(-iaV* • Dx) + iV* • Dx - 1

= exp(«*V* • Dx) - iVk • Dx - 1 . (3.12)

The said terms are then

-G^ik^yUia) [C?2(fc)A(i)]} . (3.13)

It is easy to see

I(x) = /* da Tda1-^ I(a',x) (3.14)

where

*<?k thj0ko [G?{k,x) exp(-iaV* ^

l : DxDxA(x) . (3.15)

Rewriting

(3.16)fd^y /"

and neglecting terms of orders higher than A(x), (3.15) can be reduced to

~-I(<*,x) = - j t X /*d3yJ(2TT)-*dsk exp

^ l l 2 ^ } : DvDvA(y) .

(3.17)

We consider first the integral

f doc rda'tWGilV^ : * T (3.18)Jo Jo

which yields after simple manipulation

G^(^ + t l P o ) -GjJ(p) - V^G^(p) T . (3.19)

Similarly

" da rda'(VVG22V-aTr : fcT = GL2(F-r,p0) -G2(P) - V?G^(P) T . (3.20)JO

The last terms in (3.19) and (3.20) contribute nothing to I(x)t since they are odd functions of jfc .This means that the factors before k k in (3.18) and (3.20) are proportional to the unit tensor andJ( x) can be expressed, thus

= \N(0) f d3y Ci(T,T)D2vA(y) (3.21)

8

with —

.Ci(x ,y ) — I (2TT)~ d C\{k ,A)e " (3.22)

where

«0+)2 -

x [po -e (p + k ) + t'O*]"1 (3.23)

As a conventional approximation 14, we neglect terms of 0{k/kp) in (3.23). The latter is thenreduced to

/de

/

+ (£ + u) 2]}~ (u2 + u£) . (3.24)

Completing the integration over u and e, we obtain finally from (3.24)

whereEn=*E(wn,x), ujn = (2n+l)nKBT. (3.26)

Combining (3.21), (3.11) and (3.10), we obtain a nonlocal equation for the order parameter

(3.27)

4. DERIVATION OF THE NONLOCAL CURRENT EXPRESSION

In order to derive, in the same sprit, a nonlocal current expression, we transform (2.21),with the aid of (3.1) and (3.3), into

(4.1)

which gives rise, after the Fourier transformation, to

BQf (k,x) = G%(7-iVx + eA(x),ko) \A*{X)G%(? - iV* - eA(x), ko)A(x)]

G?(t+eA(x)tk0). (4.2)

The substitution of (4.2) and (2.21) yields after simple manipulation

mA-(x)C?^(Ii -iDx,ko)A(x)]G?(k,x) . (4.3)

Consider now the integral

The zero-th order contribution of the above expression vanishes because the integrand is an oddfunction of k whereas the higher order contributions do not relate to a gauge invariant current andare therefore neglected. We then have

T(^) = - ~ J(2Ti)-4dtkTf(ko)ImG^(k-iVx,k0){Am(x) [exp(-iVk-Ds) -I]

[ ] } (4.4)

Let us define

j\a,x) = - ^ j{!>*)-*<?kkfikJImG^Ck -iVXiko)[A\x) [expC-taV* • Dx) -

}] (4.5)

It is then easy to see

/ • *, «^L (4.6)o ^

with

4 K ,Q ! 'X ) = - — /"(27T)-4d^ktfikoMmG^ik - iV%,k0){A\x) expt-iaVjt • D%)da m J \.

2(fc,x) . (4.7)

Retaining only the linear term in A (x) in (4.7), i.e., exp(—taVjt • Dx) ~* exp( — toV* • V s ) , wethen have

G*(k - iVx,ko){A*(x) exp(-iaVt • Dx) [VkG%(k) DxA(x)]] =

Cpk,!*) l(VGl^^_iVf)A*(y)] DvA(y) .

(4.8)

10

Omitting nonlinear terms V£A "(y) • D™A (y) for n > 1 and m > 1, and using the relation

/ V ^ (4.9)

(4.6) is reduced to

T(x)^^eN(0)fd3yf(2irr3dikeik^-^)Imi'C^(t)A)(-i)A*(y)DvA(y) (4.10)

where the tensor C2 (A; , A) can be expressed as

\f{p,x) . (4.11)

Up to O( k/kF), Eq.(4.11) is reduced to

Po-BHpl-EHE^)]-1-

(4.12)

(4.12) implies that Cz (k , A) is also proportional to the unit tensor, and we obtain by means ofthe theorem of residues

C2(r,A) = (2/3vFfc3)-1 fde f tt^uV^ + Ce + tt)*]^^^^^!)]}"1

J J-vFk n

(4.13)which gives rise, after straightforward calculation, to

(4.14,

Substituting (4.14) into (4.10), we obtain the following nonlocal current

,t)\A(y)\2[<p(y)-eA(y)] (4.15)

where 2 ̂ ( y) is the phase of order parameter

A(y) = |A(y ) | e < 2 ^ ) (4.16)

andA)eik t*'^ . (4.17)

11

5. DISCUSSION OF THE KERNEL FUNCTION

The kernels C\ (k , A) and Cz (k , A) are complicated functions of parameterb = \A(X)\/T!KBT. This complication makes formidable the numerical solution of the nonlocalG-L equations. Fortunately, they are insensitive to variations in b for

6 ^ 0 . 5 . (5.1)

To show this, let us consider the normalized ratios

( 0 A )l ' 2 C5-2)

which are shown in Fig.l. The condition (5.1) is equivalent to

£ i i 2 (5 3)

In the region far from the vortex centre (5.3) is reduced to T/Tck, 1.12 A(T")/A(0) which isvalid for T?Z 0 .8T*c. On the other hand, in the neighbourhood of the vortex centre where we aremost interested in, we have, e.g., j A (x) |/A (T) £ 0.6, and (5.3) is valid for T k, 0.6 Tc. We cansay that within the range To T k, 0.6 Tc the kernel functions can be approximately simplified to

or00

2 n + l "T ' (n ) ( 5 5 )

where, after restoring the length dimension

Zo{T) = h)Fj2-nKBT (5.6)

is the coherence length and

i=0

Within the accuracy of the approximation leading to (5.4), it is reasonable to set T](n)/r}( 1)for n > 2 in (5.5), so that

^ p ^ (5.8)where

- 2 . _ i _ — . . I ( 5 9 )

is a kernel function normalized to 1 at k = 0.

12

Introducing the conventional G-L parameter

and the reduced order parameter/(z)=V(s)M>Cn (5.11)

where ̂ Q(T) = nZn(Tc/T) with n = j fiN(0), the nonlocal current (4.15) is expressed, in termsof the simplified kernel function (5.8), as

T(x) = [4*\2(T)rl f cPyCoC? -t)\f(y)\2 [~V<p{y)-A{y)] (5.12)

where tpo is the flux quantum and, in original dimension,

\2(T)=\2L/2lrtTc/T),

Co(^-jT) = [(2irr3d?kCo(k)eik<7J*>) . (5.14)

The Maxwell equationV x V x T ( i ) =4irT(x) (5.15)

becomes accordingly

2 j£(?-t)\f(y)\2A(y) =

( D / y o ( j ) | / ( y ) | ^ ( y ) . (5.16)

In order to simplify the equation for order parameter (3.27), we consider first

/•WO 1 TWO 1

S(T,A)-[\N(0))-1 = / (2E)-Hh-/3Ed£- (2E)-Hh-/3B ds

+ r (2E)-lthjfcd£-[\N(0))-1 . (5.17)J—WO

The last two terms on the right-hand side of Eq.(5.17) are evaluated approximately to yield ln( Tc/T),so long as wo /KBT > 1. The first two terms can be combined to give

*> i fr L2 1-1/2

i i • . - . j .. j ^ ^ 2. n + i if rz«+ n* in=0

(5.18)

-<'} -2 g idr

13

We again assume TJ(TI)/T)( 1) & 1 for n =* 2 and (5.17) becomes

S(T,A)-[\N(0)]-'i = ^ ^ - [ ( l + 6 2 ) ~ 1 / 2 -l] + ln(Tc/T) (5.20)

which helps us to obtain from (3.27)

e2CT) jd3yCa(T-T)D2yf(y) + f(x)-2b-2[l-(l + b2rl?2]\ttx)\2f(x)=0 (5.21)

where

£2(T) =a2(T)£2(T), a2(D =7C(3)/12in(rc/r) . (5.22)

In view of (5.1)

2 6 " 2 [ l - ( l + fc2)-1/2] = l - | t 2 + . . .*; 1 (5.23)o

(5.21) is, therefore, further simplified to

(2(T) Jd?yCo& -T)Dlf(y) + /(*) - |/(x)|2/(x) = 0 . (5.24)

We naturally assume that the direction of electric current is perpendicular to that of the gradient ofmodulus of the order parameter. It can then be deduced from (5.12), by taking into account of thecontinuity of current, V Yix) = 0 , and assuming the Coulomb gauge, V • A (x) = 0 , that thephase of order parameter satisfies the Laplace equation

V 2 y3(x)=O. (5.25)

(5.16) and (5.24), together with (5.25) form a closed set of nonlocal Ginzburg-Landau equationsin simplified form. It is easy to see that Eqs.(5.16) and (5.24) reduce to our previous results 15 at

c which reduce further to the conventional Ginzburg-Landau equations in the local limit,

6. PRELIMINARY APPLICATIONS

It is hard to get a rigorous solution to the nonlocal G-L equations (5.16), (5.24) and(5.25). To make an analytical solution of the Maxwell equation (5.16) feasible, we set approxi-mately \f(x) | = 1. Though this seems permissible only in a region away from vortex centre, thesolution of the simplified Maxwell equation

V2A(x) ~\~2(T) Jd3yC0(? -t)A(y) = - ^ - Jd?yC0(? ~T)V<p(y) (6.1)

is quite reasonable even at the centre of a vortex. To show this, we transform (6.1) into

- ^ j &yC0(Z ~1?)V xV<p{y) . (6.2)

14

In the case of a single vortex, the magnetic field H (x) has a cylindric symmetry and is, therefore,

independent of 23. We set

C0(r) = fdx3C0(x) (6.3)

(6.2) is reduced, then, to

^ 2 ( D V 2 f ( r ) - fd2r'Co(r~T')H(T') = -<p0 f d2r'C0(r-r')62(7")?3 (6.4)

where use is made of the condition of a single vortex line lying parallel to the X3 axis,

V x V<p(r) =7r52(r)T3 (6.5)

with~e% the unit vector in the x$ direction. Eq.(6.4) can be easily soved to give

F(r) = tf(r)r3 (6.6)

with H(x) given by

H(r) = ̂ - fdk[\2(T)k2 + Co(k))-xkC0(k)UkT) (6.7)2TT J

where Co{ k) = Co( k) |^«o. and Jo( kr) is the zeroth order Bessel function. We rewrite the kernelfunction (5.9)

^ (6.8)

and the magnetic field (6.7)

^ 1 [°°dy[y2 + CMr}yCo<iy)J0(py) (6.9)[o

where y = k\(T),p = r/X(T) and

. (6.10)

The numerical integration of (6.9) is shown in Fig. 2 which manifests clearly the field reversal of asingle vortex for na(T) £ 1.35. The associated current is

7VW(r)e% (6.11)

Hr) = f^ J~ dk[\2(T)k2 + C0(k)rlk2e0(k)Mkr) (6.12)

where"e% is the unit vector along the azimuthal direction in the (xi , X2) plane. The vector potentialcan be derived from (5.15) and (6.12) with the result

A(r)=A(r)?6 (6.13)/•OO

A(r) = ^- dk[\2(T)k2 + Co(k)]-'iCo(k)Mkr). (6.14)

15

It is obvious that expression (6.7) is regular at the vortex centre. (6.7) and (6.10) lead also naturallyto the quantization of the magnetic flux of a single vortex, i.e.,

f d2rH(r)= £A(r).d~T = <p0 . (6.15)

The form factorF(k)=C0(k)[Xi(T)k2 + Co(k)]-X (6.16)

is measurable with the aid of neutron scattering technique. In Fig.3, our theoretical result is com-pared with the data n obtained at T = 42K on polycrystalline Nto.73Too.27 which has Te =6.9 K and n( Te) = 3.36. We find that the agreement between theoretical and experimental formfactor is satisfactory by taking K - K(TC) = 3.36, or Ka(T) = 3.99 according to (6.10). Themeasured value of \(T) is 780 A which yields, thus, the coherence length £(T") = 235 A, inagreement, too, with the estimated value 190 A n .

Further applications of the present theory and its generalization to impure case are plannedto be given separately.

Acknowledgments

One of the authors (H.H.X.) would like to thank Professor Abdus Salam, the InternationalAtomic Energy Agency and UNESCO for hospitality at the International Centre for TheoreticalPhysics, Trieste.

16

REFERENCES

1. V.I. Ginzburg and Landau, Zh. Eksperim. i Teor. Fiz. <Sov. Phys.) 20,1064 (1950).

2. L.P. Gor'kov, Zh. Eksperim i Tear. Fiz. (Sov. Phys.) 36, 1918 (1959).

3. J. Bardeen, LN. Cooper and J.R. Schrieffer, Phys. Rev. 108,1175 (1957).

4. N.R. Werthamer, Phys. Rev. 132,663 (1963).

5. L. Tewordt, Phys. Rev. 132,595 (1963).

6. J. Auer and H. Ullmair, Phys. Rev. B37,136 (1973).

7. G. Eilenburgerand H. Buttner, Z. Phys. 224,335 <1969);R.M. Cleary, Phys. Rev. Lett 24,940 (1970);K. Ditchtel, Phys. Rev. B4,3016,3022, 3029 (1971);A. Hubert, Phys. Status Solidi (b) 53,147 (1972);S. Grossmann and C. Wissel, Z. Phys. 252,74 (1972);H.C. Leuny, J. Low Temp. 12,215 (1973);L. Kramer, Phys. Rev. B3, 3821 (1973).

8. L. Laplace, F. Mancini and H. Umezawa, Phys. Rep. 10c, 1 (1974).

9. R.J. Cavaet al., Phys. Rev. Lett. 53,16176 (1987).

10. G. Zhou, Z. Su, L. Yu and B. Hao, Phys. Rep. 118,1 (1985).

11. J. Schleten, H. Ullmair and W. Schmatz, Phys. Status Solidi (b) 48, 619 (1971).

12. C.E. Gough et al., Nature 326,335 (1987);T. Witt, Phys. Rev. Lett. 61,1423 (1988);PL. Gammel et al., Phys. Rev. Lett. 50,2592 (19S7).

13. N.R. Werthamer, in Superconductivity, Vol.1, ed. R.D. Parks (Dekker, New Yoik, 1969).

14. A.L. Fetter and J.D. Waiecka Quantum Theory of Many Particle System, Chap.Xm(McGraw-Hill, New York, 1971).

15. Hogn-Hua Xu and Chien-Hua Tsai, Comm, Theor. Phys. (Beijing) in press.

17

FIGURE CAPTIONS

Fig. 1 The normalized ratios Vi( k, b), Eq.(5.2) as functions of k for various b < 0.5. (a) i=l,(b) i=2, and k is scaled by £( T).

Fig.2 Magnetic field reversal in the neighbourhood of a single vortex.

Fig.3 Theoretical form factor (solid curve) for a single vortex somputed with the aid of Eqs.(6.16)and (6.8) with * 0 ( D = 3.99 is compared with neutron scattering data (crosses) at 4.2 iffor Nbo.73Too.n (Tc - 6.9 K, K = 3.36). The dashed line represents a fit to the exper-imental data with the kernel8

C(Jfc) = 11 + ± J - k2 $ + 0.025 Jt4 f4,

18

Parameter : b

0.0

O.4O.O 2.O

19

O.5E-3

0.0 E+0

-O.5E-3

-1.E-310 12

Fig.2

o

LJL

E QOl

QOO1

NbO.73TaQ27 4.2 K

K (Tc) - 3.36

Fig.3

20

Stampato in proprio nella tipografia

del Centro Internazionale di Fisica Teorica