Flux penetration of a hard superconducting tube

Volume 28A, n u m b e r 1

FLUX

PENETRATION

P H Y S I C S L E T T E RS

OF

A HARD

21 October 1968

SUPERCONDUCTING

TUBE

C. R. W I S C H M E Y E R

Rice University, Houston, Texas, USA Received 3 September 1968

Observed flux jumps c o r r e l a t e with sharply i n c r e a s e d Joulean dissipation when a flux front p e n e t r a t e s the bore of a N b - Z r tube. Varying the wall thickness exhibits a m a x i m u m of dissipation.

M a n y o b s e r v a t i o n s of t h e p e n e t r a t i o n of c y l i n drical and tubular hard superconductors by an axial field H have been reported. Sufficiently high r a t e s of c h a n g e of f i e l d a r e o b s e r v e d to p r o d u c e a s u c c e s s i o n of flux j u m p s . In N b - Z r t u b e s , v e r y h i g h v a l u e s of dH/dt a r e o b s e r v e d to p r o d u c e a f i r s t flux j u m p b e f o r e t h e flux f r o n r e a c h e s t h e b o r e [1,2]. M o d e r a t e v a l u e s , w h e r e v i s c o u s e f f e c t s m a y b e n e g l e c t e d , t r i g g e r t h e f i r s t flux j u m p w h e n t h e f i e l d p e n e t r a t i o n r e a c h e s t h e b o r e [3,4]. The latter situation (which we encountered inc i d e n t a l l y i n t r a c i n g t h e m a g n e t i z a t i o n of t u b e s of d i a m e t e r s 6 t o 19 m m , w a l l t h i c k n e s s e s 0.2 to 1.5 m m , a t dH/dt ~ 1 to 25 O e / s e c ) i s e l u c i d a t e d b y c a l c u l a t i o n of t h e J o u l e a n p o w e r d e n s i t y , w h i c h e x h i b i t s a w o r s t c a s e i n t e r m s of w a l l t h i c k n e s s . We assume the Anderson-Kim critical-state chara c t e r i z a t i o n [5], i g n o r i n g d e p a r t u r e t h e r e f r o m b e l o w Hcl. F u r t h e r , w e a s s u m e h o m o g e n e i t y of t h e sample. It i s e a s y to s k e t c h t h r e e c h a r a c t e r i s t i c s t a g e s of t h e p e n e t r a t i o n p r o c e s s . S e e fig. l a . (1) r e p r e s e n t s t h e r a d i a l d i s t r i b u t i o n of i n d u c t i o n , o r t h e f l u x f r o n t , c o n t a i n e d w i t h i n t h e w a l l t h i c k n e s s w; (2) d e p i c t s t h e flux f r o n t a s it r e a c h e s t h e i n n e r w a l l s u r f a c e , w i t h t h e f i e l d H ' s t i l l z e r o in t h e b o r e ; a n d (3) s h o w s t h e s u b s e q u e n t r i s e of H ' . T h e v e r t i c a l l y s h a d e d a r e a r e p r e s e n t s flux

" Fig. l a . Typical radial dependence of introduction for applied fields in the neighborhood of the penetration value. The r a t i o of wall thickness to outside radius is 0.25.

14

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FRACTIONAL 0

0.25

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RADIAL 0.50

i

'

0.90 I

I.OO

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I/

/

O"

~oo

~eo~G 1

I

0.75

'

//

200

DEPTH

",.

/ \ \\

77~sG

I

Fig. l b . Computed Joulean power d e n s i t i e s associated with flux f r o n t s contained within the tube wall (lower curves) and corresponding power densities i m m e d i a t e ly upon breaking through the wall of a hollow cylinder. The sar~ple is Nb 0 75Zr 0 25, with otc = 1.2 × 10 ~ G A / c m z and Bo = 1".2 kG h't 4.2°K. Outer radius is 4 ram, and dH/dt = 1 G / s e c . Z e r o depth connotes the outside surface.

Volume 28A, number 1

PHYSICS LETTERS

added, a c c o m p a n y i n g an i n c r e m e n t AH, in p a s s i n g f r o m (1) to (2); the diagonally shaded a r e a r e p r e s e n t s that added for an equal i n c r e m e n t in going f r o m (2) to (3), a s flux suddenly c a s c a d e s into the b o r e . (Although not a c t u a l l y p r o p o r t i o n a l to t h e s e a r e a s , flux is a m o n o t o n i c a l l y i n c r e a s i n g function of them). C l e a r l y the v o l u m e to which flux m u s t be supplied (and hence d ~ / d t ) i n c r e a s e s d i s c o n t i n u o u s l y at b r e a k - t h r o u g h . Without c o n c e r n for the m i c r o scopic d e t a i l s [6], i m m e d i a t e l y we i n f e r that the induced e l e c t r i c field at a l l r a d i i i n c r e a s e s d i s c o n t i n u o u s l y a s H p a s s e s through the p e n e t r a t i o n value. No c o r r e s p o n d i n g effect i s r e f l e c t e d in the wall c u r r e n t s . Hence, if we calculate J o u l e a n power d e n s i t y [7] p j a s H p a s s e s through the pene t r a t i o n value, we should expect the c u r v e s b e fore and after b r e a k - t h r o u g h to have no points in common. As an i l l u s t r a t i o n , we select our u n a n n e a l e d Nb0.75Zr0.25 tube ~123, with outside r a d i u s a = = 4 m m , r e g a r d i n g it m o m e n t a r i l y a s a solid cyl i n d e r . In the lower p o r t i o n of fig. lb, the five c u r v e s which tend toward z e r o with i n c r e a s i n g depth a r e plots of the r a d i a l d i s t r i b u t i o n of power d e n s i t i e s for deepest p e n e t r a t i o n s rf/a = 0.25, 0.50, 0.75, 0.90 and 1.00. C o r r e s p o n d i n g v a l u e s of H a r e noted. At H = 9168 G, the field has pene t r a t e d to the axis of the cylinder. If now we r e g a r d our s a m p l e a s a tube, the flux front b r e a k s through when rf = w. T h i s is depicted in fig. l a , for a wall t h i c k n e s s equal to 25% of the outside r a d i u s , in which c a s e the pene t r a t i o n field is c a l c u l a t e d to be 3820 G. As H i s i n c r e a s e d through this value, the d i s t r i b u t i o n of J o u l e a n power d e n s i t i e s a c r o s s the wall section r i s e s abruptly, f r o m the lower to the upper curve m a r k e d 3820 G in fig. lb.

21 October 1968

It i s significant that, for a given outside d i a m e t e r , t h e r e e x i s t s a "worst" t h i c k n e s s , in the s e n s e of g r e a t e s t Joulean d i s s i p a t i o n e n c o u n t e r e d at b r e a k - t h r o u g h . Pj(w), i.e., the power d e n s i t y at the i n n e r wall s u r f a c e a s a function of w, i s shown a s the dashed c u r v e of fig. lb. M a x i m i z i n g r e v e a l s a w o r s t case at w = 31a - B ~/1.27ra c, where otc and B o a r e s p e c i m e n p a r a m e t e r s [5]. S u c c e s s i v e flux j u m p s produced by continued i n c r e a s e in H (with H' > 0) m a y be s i m i l a r l y explained. I n c r e a s i n g H a f t e r a flux jump tends to r e e s t a b l i s h the A n d e r s o n - K i m c r i t i c a l state. P e n e t r a t i o n by the new flux front r e s u l t s in a s i m i l a r though ( s u c c e s s i v e l y ) l e s s s p e c t a c u l a r inc r e a s e in Joulean d i s s i p a t i o n , tending to produce yet a n o t h e r flux jump. F i n a l l y , d e c r e a s e d J o u l e a n d i s s i p a t i o n , due to d e c r e a s e d e l e c t r i c field and wall c u r r e n t s with i n c r e a s i n g H, will no l o n g e r i n i t i a t e f u r t h e r flux jumps. The author gratefully acknowledges his ind e b t e d n e s s to Dr. Y . B . K i m , Bell Telephone L a b o r a t o r i e s , M u r r a y Hill, N.J., w h e r e these e x p e r i m e n t s were done d u r i n g a r e c e n t sabbatical.

References 1. J.M. Corsan, Phys. Letters 12 (1964) 85. 2. F.Lange, Cryogenics 5 (1965) 143. 3. N. Morton, Phys. Letters 19 (1965)457, Cryogenics, to be published. 4. S.L. Wlpf, Phys. Rev. 161 (1967) 404. 5. Y.B.Kim, C.F. Hempstead and A.R. Strnad, Phys. Rev. 131 (1963) 2486. 6. B.D. Josephson, Phys. Letters 16 (1965) 242. 7. C.R. Wisehmeyer, Phys. Rev. 154 (1967) 323.

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