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Submitted on 1 Jan 1978
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ON NEUTRON SCATTERING FROM AN
IMPERFECT FLUX-LINE LATTICE
E. Brandt
To cite this version:
ON NEUTRON SCATTERING FROM AN IMPERFECT FLUX-LINE LATTICE
E- H . Brandt
Ames Laboratory-USDOE and Dept. of Physios, Iowa State University, Ames, Iowa 50011, U.S.A.
Abstract.- The diffraction of neutrons from the imperfect vortex lattice of a type-II superconductor is investigated. It is found that flux pinning leads not only to a broadening of the Bragg reflec-tions but also to diffuse scattering into the first Brillouin zone, which at high fields exceeds Bragg scattering.
In recent years there has been much effort to interpret the volume pinning force in type-II super-conductors in terms of elementary pinning interac-tions between flux lines (FLs) and crystal lattice defects/1-2/. Up to now the statistical summation theories required for this interpretation are still unsatisfactory. It is therefore desirable to get mo-re insight into the natumo-re of these interactions. This may be achieved by neutron diffraction, which probes the magnetic field inside the superconductor. Flux pinning changes the diffraction pattern of the ideal flux line lattice (FLL) in two ways : the Bragg reflections are broadened/3/ and a finite scattering intensity occurs for scattering wavevec-tors in the first Brillouin zone (BZ)/4/. A relati-onship between the scattered intensity and the pin-ning interactions may be established in 5 steps.
1.- The differential scattering cross section for neutrons in Born approximation is related to the Fourier transform jl(k.) of the magnetic field H(r) inside the specimen by (in SI units)
da/dfi = (yoyn/4lJ>o)2|H(k)|2 (1)
where y = 1.91, 4> is the flux quantum, and lc the scattering vector.
2. For a given FL displacement field with Fourier transform JKk), the Ginzburg-Landau (GL) theory yields for small FLL strains, for small FL tilting angles, and for lc well inside the first BZ
H(k) = (<l)o/uo)(l+k2/k^)~1li:k x |s_(k) x z | (2)
where z^ is the unit vector along the applied field, k£ % (l-b)A2, b = B/Bc2, and \ = K? is the field
penetration depth. Equation(2) applies to k < 0.7k,, /
where k„ = (2b) /C is the BZ radius. An extension a
of (2) to larger k, which is required for a rigorous calculation of the profiles of Bragg reflections, has not been obtained yet for 0<b<l. Inserting (2) in (1) we get for scattering into the first BZ
da/dfi = 0.23(l+k2/k2)-1(|k5(k)|2 + k2|s0c)|2) (3)
3. As a next step we express the displacement field j[(k.) in terms of the pinning force field P(k) = $(k)j[(k) where j|(k.) is the elastic matrix of the FLL. The GL result for k < 0.7 k ("continuum
appro-B ximation") and arbitrary b and K is /5/
where n = B/<j) , c6 6 is the shear modulus, and c n ( k )
£ c11(l+k2/k2)"1(l+k2/k^)"1and c„„(k) % c„„(l+k2/k£)" are the k-dependent compressional and tilting modu-li of the FLL, k2 = k2+k2+k2 and k2 = 2(l-b)/£2.
x y z Y
The Labusch parameter a emerges from statistics of pinning forces/6/ and for consistency must be small aL/ c1i « k | and OL/CX i«k?£k^. Insertion in (3) of
pinning forces requires the contribution to j[(k_) of the intrinsic FLL defects to be eliminated somehow. Fortunately, the elementary defects exhibit mere shear strains (point defects, edge dislocations) or tilt strains (screw dislocations), which all satis-fy div £ = ks_ = 0 and thus do not contribute to (3) if k = 0 ; this can be achieved by choosing the in-cident neutron beam parallel to the applied field. In this scattering geometry we get simply
da(kx,k ,0)/dfl = 0.23n2(l+k2/k|)2(cnk2+aL)~2
|k?(kx,ky0)|2 (5)
On leave from Max-Planck-Institut fur Metallfor-schung,Institut fur Physik, Stuttgart, Germany
JOURNAL DE PHYSIQUE
Colloque
C6,
supplément au n°
8,
Tome
39,
août
1978,
page
C6-644
Résumé.- Nous examinons la diffraction des neutrons thermiques par le réseau des vortex d'un supra-conducteur de deuxième espèce. Nous montrons que l'ancrage des vortex non seulement introduit un élargissement des pics de Bragg mais aussi une composante diffuse dans la première zone de Brillouin qui, à champ élevé, est plus intense que la composante de Bragg.
4. The p i n n i n g f o r c e f i e l d E ( k ) i s r e l a t e d t o t h e elementary pinning f o r c e f i e l d s Ei(k) of a l a r g e number, N , of weak pinning c e n t e r s a t p o s i t i o n s a .
-1
d i s t r i b u t e d a t random by
kP(k)12 = =
E
I s i ( k ) 1 2I_
_
(6)- -
The sum over t h e c r o s s terms P.P. v a n i s h e s because -1-J
of d e s t r u c t i v e i n t e r f e r e n c e of phase f a c t o r s exp { i k ( a . - a . ) } belonging t o pinning c e n t e r s of t h e
-1 -J
same type a t d i f f e r e n t p o s i t i o n s . This a p p l i e s t o a s t i f f FLL. E l a s t i c i t y of t h e FLL l e a d s t o corre-
-
l a t i o n of t h e P . ( k ) , s i n c e a p p l i c a t i o n of one f o r c e-1
-
changes a l l FL p o s i t i o n s and thus changes a l l o t h e r f o r c e s . This c o r r e l a t i o n i s p a r t l y accounted f o r by i n t r o d u c t i o n of
aL
i n (4) ;a
L removes t h e d i v e r - gence of (5) a tk
= 0 . Besides t h i s , t h e e l a s t i c response d e p r e s s e s t h e f o r c e s below t h e v a l u e s ob- t a i n e d f o r a s t i f f and p e r f e c t FLL. This e f f e c t i s n e g l e c t e d below.5 . The pinning f o r c e s may be d e r i v e d from t h e per- t u r b a t i o n
F = d 3 { - a ~ 2 ~ 4 + y ~ / i ~ - ) ~ 2 x ~ 2 (7) of t h e GL f r e e energy by a method developed i n / 7 / .
The s p a t i a l f u n c t i o n
a,
B,
y, andx a r e small devia-
t i o n s from t h e mean ; f o r s m a l l p i n n i n g c e n t e r s w i t h diameter << they may be r e p l a c e d by a d e l t af u n c t i o n 6 ( ~ - a . ) w i t h weight ao,
Bo,
yo o rx0.
We-1
d i s c u s s t h r e e s p e c i a l c a s e s :
Short range f o r c e s . - Pinning c e n t e r s w i t h co- r e i n t e r a c t i o n a t b < 0.3, o r magnetic i n t e r a c t i o n a t b < 1 / 2 ~ ~ , e x e r t a f o r c e P . o n l y on t h e FL w i t h -1
-
p o s i t i o n R. c l o s e s t t o a . . This g i v e s L(&)=ll;exp -1 -1 (-ikRi) and d ~ / d R = 0.23n2 ( c l lk2+%)-'k2fr
C2;
(8) i I n (8) t h e n o n l o c a l i t y parameterskh
and ky do n o t appear.Long range f o r c e s . - Magnetic i n t e r a c t i o n a t b>>1 / 2 r 2 y i e l d s d o l d S 1 = 0 . 2 3 ~ ~ ~ ( c l ~ k ~ + % ) - ~ ( ~ + k ~ / l c $ ) (1+k2/k2.)-* h B4/u; (9) This i s 2, k - 4 f o r
a
/cll<<k2<<$ and c o n s t . ( k ) L f o r k2>>lc$=ki( 1-b) /b.
Core i n t e r a c t i o n a t b > 0.5.- Fora
#
0 , B = y = x = O w e g e t 1 d ~ / d R = 0 . 2 0 ~ ~ > ~ b ~ ( c l l k ~ + ~ ) - ~ k ~ { l ~ ~ 1 - b ) k ~ / b k+
+ k / g 2 ) (10) This i s %kLk-' f o r a L / c l l < < k 2 < < k 4 / 2 k B . I n t h e l a r g e range kg( 1-b) /2b<k<0.7 kg ( 10) reduces t o do/dR ~ 0 . 2 0 ( 8 ~ ~ ) - ~ ~ a ; ~ ~ ~ ( 2 k ~ - l ) - ~ = c o n s t . ( k , b ) (1 1 )I n ( 1 1 ) we have used ~ l l = i ? ~ a H / a B and a = a 1 5 3 ~ 2 p
0 0 c o w i t h l a i l < I . A s i m i l a r r e s u l t i s o b t a i n e d f o r - , Y
#
0 / 4 / . The l a r g e c o n s t a n t s c a t t e r i n g c r o s s s e c t i o n (9) and (10) a t l a r g e i n d u c t i o n s i s e n t i r e l y due t o t h e n o n l o c a l i t y of t h e e l a s t i c response of t h e FLL, i n p a r t i c u l a r t o t h e f a c t o r ( l + k 2I$)-'
i n ell (k).
The f a c t o r s ( l + k 2 / % ) - ' i n t h e "magnetic response"(2) and i n c l l ( k ) c a n c e l i n t h e s c a t t e r i n g c r o s s s e c t i o n . A t l a r g e i n d u c t i o n s t h e i n t e n s i t y of s m a l l a n g l e s c a t t e r i n g exceeds t h a t of t h e Bragg r e f l e c - t i o n s , which v a n i s h e s a s ( ~ - b ) ~ . A t s t i l l h i g h e r i n d u c t i o n s FL t r a p p i n g l e a d s t o a breakdown of t h e above t h e o r y and t o do/& = 0 a t b = 1 . The d e r i v a - t i o n of our r e s u l t s a p p l i e s t o a r b i t r a r i l y weak pin- n i n g and does not r e q u i r e a t h r e s h o l d t o be exceeded a s i n t h e o r i e s of t h e volume pinning f o r c e . Small a n g l e s c a t t e r i n g of c o l d neutrons should thus y i e l d i n s i g h t i n t o t h e p i n n i n g problem beyond t h a t o b t a i - n a b l e from measurements of t h e volume pinning f o r c e .
F i g . 1 : L o n g i t u d i n a l s c a t t e r i n g geometry. Dots :
r e c i p r o c a l f l u x l i n e l a t t i c e p o i n t s , hexagon : f i r s t B r i l l o u i n zone, e l l i p s e s : l i n e s of c o n s t a n t s c a t t e - r i n g i n t e n s i t y f o r k, = 0
,
Ifn:
wavevector of i n c i - dent n e u t r o n s ,&
: s c a t t e r i n g v e c t o r , I b + k l=l&n/Z>B
0
: s c a t t e r i n g a n g l e .ACKNOWLEDGEMENTS.- The a u t h o r wishes t o thank P r o f . J.R. Clem and D r . D.K. C h r i s t e n f o r v a l u a b l e d i s - c u s s i o n s , and t h e Deutsche Forschungsgemeinschaft f o r f i n a n c i a l s u p p o r t .
References
/ 1 / Campbel1,A.M. and E v e t t s , J . , C r i t i c a l
Currents i n Superconductors (Taylor and F r a n c i s , London 1972)
/2/ Kramer,E.J., t o b e p u b l i s h e d i n 3 . Appl. Phys. /3/ Labusch,R., Phys. S t a t . S o l . ( b )
2
(1975) 539 141 Brandt,E.H., t o b e submitted t o Phys. Rev. 151 Brandt,E.H., J . Low Temp. Phys.26
(1977)709 and 735
/6/ Labusch,R., C r y s t a l L a t t i c e D e f e c t s
