Neutron depolarization theory in superconductors

Journal of Magnetism and Magnetic Materials 104-107 (1992) 527-528 North-Holland

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Neutron depolarization theory in superconductors M . T . R e k v e l d t a n d F.F. van Zijl Interfacultair Reactor lnstituut, Delft University of Technology, Delft, Netherlands Neutron depolarization appears to give essential information about the magnetic fluxlines in superconductors. For a fluxline structure of parallel fluxlines in the z direction and random structure of the circular cross-sections in the xy plane, neutron depolarization formulae are derived which relate the shortening of the polarization vector in the xy plane, Df = P / P o = e x p [ - kBto/d] with the mean magnetic induction B and the average diameter d of the fluxlines. Inhomogeneous distribution of fluxlines in direction and density leads also to depolarization which is described as Di,o.p = exp[c~-B2~t,,( 1 - 3"o,p)/Yp]. Here the indices o and p refer to the depolarization perpendicular and parallel to the z axis, 6 is the correlation length of the inhomogeneity, 3'o.p is the mean square direction cosine of the fluxline magnetizations (3'~, +23'o = 1) and k and c are known constants. From the total depolarization D = DfD i as a function of the magnetic induction in principle the linear and quadratic term can be separated if 6 is assumed to be independent of B.

1. Introduction All n e u t r o n scattering techniques have in c o m m o n that they study inhomogeneities. Because of the magnetic m o m e n t of the n e u t r o n the inhomogeneity may be m a g n e t i c also. N e u t r o n depolarization m e a s u r e s only the m a g n e t i c interaction of the n e u t r o n with the sample. T h e r e f o r e only i n f o r m a t i o n of the magnetic i n h o m o g e n e i t i e s is obtained. N e u t r o n depolarization is based on the m e a s u r e m e n t of the polarization c h a n g e after transmission of the b e a m t h r o u g h the sample. T h e m e a n r o t a t i o n of the polarization vector delivers directly the m e a n m a g n e t i z a t i o n a n d the s h o r t e n i n g of the vector, called the depolarization, is a m e a s u r e of the size of the m a g n e t i c inhomogeneity. T h e t e c h n i q u e is used mainly to study m a g n e t i c i n h o m o g e n e i t i e s in f e r r o m a g n e t i c d o m a i n structures [1]. In 1969 it has b e e n applied for the first time to study the inhomogen e o u s fluxlines in s u p e r c o n d u c t o r s [2,3]. Recently, probably b e c a u s e of the new interest in superconductivity, a revival can be observed of applying n e u t r o n depolarization in studying the fluxline structure a n d flux creep in s u p e r c o n d u c t o r s [4-9]. A l t h o u g h most of the a u t h o r s recognize the possibilities of n e u t r o n depolarization in s u p e r c o n d u c t o r s , quantitative results are not r e p o r t e d , for a part because the e x p e r i m e n t a l conditions were not optimal to distinguish the depolarization of the m a g n e t i c fluxlines from the depolarization of the d e m a g n e t i z a t i o n fields a r o u n d the sample. However the m a i n r e a s o n is that no satisfying theory is p r e s e n t to i n t e r p r e t the d e p o l a r i z a t i o n results in terms of well defined quantities of the superconductor. T h e p r e s e n t p a p e r aims to fill this gap. 2. Theory First the depolarization f o r m u l a e will be derived for a polarized b e a m t r a n s m i t t i n g a s u p e r c o n d u c t o r in the

x direction. T h e s u p e r c o n d u c t o r consists of a slab of thickness t o in the x direction with parallel fluxlines in the z direction. In the xy p l a n e no o r d e r i n g is assumed. T h e effect of ordering will be considered later on. F o r simplicity reasons the fluxlines are assumed to bc cylinders with d i a m e t e r d in which a magnetic induction B c is present. Because the fluxline should contain only o n e fluxquant, B c and d are c o n n e c t e d by

Bcd2rr/4 = h /2e.

(1)

T h e depolarization can be found by evaluating the average rotation matrix which describes the polarization c h a n g e after transmission t h r o u g h the sample [1]. This theoretical average contains the terms (cos 9 ) and (sin 9 ) by which the polarization c o m p o n e n t s in the xy plane change. H e r e 9 is the average rotation angle of the polarization in the sample. T h e depolarization occurs by the variance A 9 in 9. So, in the xy plane,

D~x = ( c o s ( 9 + A g ) ) = (cos 9 ) ( 1

-

- (h9)(sin

((A9)2))

9),

(2)

Ox~ =
(3)

Using this, the depolarization of the polarization vector in the xy p l a n e can be f o u n d from the s u b d e t e r m i n a n t of the depolarization matrix in the xy plane, 2 O t = D 2 ---D ~ x2 +Ox~, = 1 - ((A9,)2) _ ((A92)2).

(4) Two c o n t r i b u t i o n s to the guished; first that of the ond that of the variance passed by a n e u t r o n . F o r

0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

variance A 9 can be distinindividual fluxlines a n d secof the n u m b e r of fluxlines small r o t a t i o n angles 9 ' in

528

M.T. Rekveldt, F.F. can Zijl / Neutron depolarization theory in superconductors

one fluxline with circular cross-section in the xy plane, ( ( A ¢ I ) 2) = N~((A~o') 2) follows immediately from (~v') = w d c B c / 4 ,

fluxlinc distribution the total depolarization can be described by In D

- kBt o

d

(q0 '2 ) = 3(dCSc) 2, ( ( A q N ) 2 ) = O . 0 5 N , ( d c B c ) 2,

(5)

with c a c o n s t a n t of the n e u t r o n b e a m given by c = 4.67* 1014 a (m 2T 1), A the wavelength of the neutron b e a m a n d N~ the average n u m b e r of fluxlines passed by a n e u t r o n over a distance t. If N is the density of fluxlines p e r m 2, t h e n the average n u m b e r of fluxlines, t r a n s m i t t e d by a n e u t r o n over a distance t is given by the n u m b e r c o n t a i n e d in a surface 2 d t a r o u n d the n e u t r o n path, is

Because of the assumption that the positions of the fluxlines are not correlated, we apply random statistics on this n u m b e r , which m e a n s that this n u m b e r is described by a Poisson distribution with ( ( A N t ) 2 ) = N t and (7)

( A ¢ ~ ) = U,q~ '2 = 3 X , ( dcB,.) 2.

Even in a sequence of badly o r i e n t e d o r d e r e d lattices of fluxlincs, as it is the case in a polycrystalline sample with the total volume of fluxlines relatively small, this variational assumption is still reasonable. F r o m comparison of eqs. (5) and (7) we may conclude that in a first approximation the depolarization of the individual fluxlines can be neglected. Using this and eq. (1), the depolarization in eq. (4) can also be written as

(8)

D t = 1 - kBt/d

with k = 2 c 2 h / 2 e and B = N h / 2 e . Eq. (8) describes the depolarization in the xy p l a n e for a small thickness t = t o / N ' . T h e s u b d e t e r m i n a n t of the depolarization over the sample thickness t o can easily be found by factorizing, D ~ = D f f " = (1 - k B t o / f

,d ) iv' = exp[ - k B t o / d

].

(9)

According to cq. (9) the depolarization in the fluxlines is described by an e x p o n e n t similar as in magnetic domain structures, except that the f o r m e r contains a linear term in the m e a n magnetic induction, while the depolarization by m a g n e t i c inhomogeneities, e.g. inhom o g e n e o u s fluxline distribution, contains a term quadratically in the deviation of the local m a g n e t i c induction from the m e a n induction B, also found in magnetic d o m a i n structures [1]. So, in an arbitrary

(10)

Here # is the size of the i n h o m o g e n e i t y a n d B2(1 Tll)/Tll is the square of the deviation of the local induction from the m e a n m a g n e t i c induction in the sample. If we assume t h a t # does not vary with varying B it is possible to e n t a n g l e the linear and quadratic term of eq. (10) from the B d e p e n d e n c e of the depolariation and d e t e r m i n e the fluxline d i a m e t e r d and the i n h o m o g e n e i t y size 8 from the same experiment. M o r e o v e r from the depolarization II and ± to the m e a n m a g n e t i c induction, the local direction cosines of the fluxlines can be d e t e r m i n e d from

(6)

N t = 2diN.

c2B 281° . YI~

y,i = 1

2 In DII ln(D~D,) "

(11)

3. Summary and conclusions N e u t r o n depolarization f o r m u l a e have b e e n derived for s u p e r c o n d u c t o r s containing m a g n e t i c fluxlines. A b o v e the i n h o m o g e n e i t y of the fluxline itself an inhom o g e n e o u s fluxline distribution may be present. Assuming that the latter does not vary with the m e a n magnetic induction, the two c o n t r i b u t i o n s to the neutron depolarization can be d e t e r m i n e d from the B d e p e n d e n c e of the depolarization. M o r e o v e r the local direction cosines of the fluxlines can be d e t e r m i n e d from the same experiment.

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