Volume 26A, number 12 PHYSICS L E T T E R S 6 May 1968

G R A V I T A T I O N - I N D U C E D H A L L E F F E C T I N T Y P E I S U P E R C O N D U C T O R S *

G. PAPINI Department of Physics, University of Saskatchewan Regina Campus,

Regina, Saskatchewan, Canada

Received 5 April 1968

A gravitationally-induced time-independent Schiff-Barnhill field can give rise to a Hall effect in type I superconductors. The case of a rotating superconducting slab is considered in particular.

The re is r ea son to be l ieve that, as repeatedly stated: [1-3], the usua l s ta t ic exper iment with c ro s sed e lec t r ic and magnet ic f ields cannot give r i s e to a Hall effect in type I superconduc tors . Here i t i s impor tan t to r e m e m b e r that the usual l i nea r equations of superconduct iv i ty imply that no e lec t ros ta t i c f ield can exis t ins ide a supe rcon - ductor . This i s unl ike the case of a microwave e lec t r i c f ield which pene t r a t e s a superconductor and can give r i s e , at l eas t in p r inc ip le [4,5], to a f requency-dependent Hall effect. The purpose of this paper i s to show that the occu r r ence of a Sch i l l -Barnh t l l f ield [6,7] in superconduc tors of the f i r s t kind subjec t to a s ta t ionary gravi ta t ional f ield makes the ex is tence of a s ta t ic Hall c u r r e n t theore t ica l ly poss ib le . It i s hardly n e c e s s a r y to emphas ize at this point that, in the f r a m e of a l i nea r theory, no means exist other than g r av i t a - t ion to probe the ins ide of a superconductor with s ta t ic e lec t r ic f ields and that the Hall effect, if detected, would provide the appropr ia te tool to c a r ry out such an inves t igat ion. Beside having an i n t e r e s t in i t se l f - the effect has not so far been observed - a poss ib le exper iment could also y ie ld i n t e r e s t i ng data about the number of cu r r en t c a r r i e r s per unit vo lume or the band s t ruc tu re of superconduc tors , and tes t some of the exis t ing theor ies of superconduct iv i ty [5].

The t e r m in the Hamil tonian r~spons ib le for the Schi f f -Barnhi l l f ield is - ~ m c boo where boo r e p r e s e n t s the grav i ta t iona l f ield as in ref . 7. We now cons ider the combined effect on the s u p e r - conductor of hoo and of an externa l magnet ic field. Although our ca lcula t ions apply to any s ta t ionary gravi ta t ional f ield of pe rmanen t or ' f ic t i t ious ' na ture , we r e f e r more specif ical ly in what follows to the much l a r g e r centr i fugal

* Supported by the National Research Council of Canada.

force genera ted by rapid ro ta t ion of the s u p e r - conductor about an axis . The express ion for hoo i s then boo = o~2r2/c 2 where o~ i s the veloci ty of ro ta t ion ~tbout the axis . Also we a s s u m e our superconduct ing spec imen to have the shape of a s lab so that the gravi ta t ional field in one d i rec t ion only is re levant . If the axis of ro ta t ion of the s lab is chosen as z -ax i s and the s lab i t se l f l ies in the (x, z ) -p lane , then the Schi f f -Barnhi l l f ield genera ted by ro ta t ion is in the x -d i rec t ion . We choose the vec tor potent ial aHo(- Q) r ep re sen t ing the externa l f ield in momen tum space in the y- d i rec t ion and Q along x so that the usual t r a n s - v e r s e gauge condit ion is sat isf ied. The notat ions and the method to evaluate the Hall c u r r e n t a r e those of refs . 9 and 4.

The effect of boo on the superconductor can be s tudied by adding to the f ree Hamil tonian of the s y s t e m [4] the pe r tu rba t ion t e r m

~1 = -½ mc2 ~ c~-Q ,~ck~h oo(-Q ') k,e

whose choice s t ems na tura l ly f rom the Hami l to - n ian descr ip t ion of non re l a t iv i s t i c par t i c les in weak s ta t ionary gravi ta t ional f ield [7,8]. The ex- t e r n a l magnet ic f ield can be in t roduced as in eq. (9.5) of ref. 4. For s impl ic i ty we neglect band s t r u c t u r e effects ent i re ly . The r e i s no d iamag- ne t ic contr ibut ion to the Hall cu r r en t for the geomet ry chosen. This is shown below and may be a l so es tabl i shed independently f rom the gene- r a l r e l a t iv i s t i c formula t ion of e l ec t romagne t i sm [10] as applied to our case. In the der iva t ion of the equations of motion for pa i r s of quas i -pa r t i c l e opera to r s we have kept only t e r m s which con t r i - bute to the c u r r e n t l inea r ly in the product aH.(-Q)/*nn(-Q') . A se l f - cons i s t en t solution of

U v v 1 the equat ions of mot 'on for the ~k+Q 7k opera tors

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Volume 26A, number 12 PHYSICS L E T T E R S 6May 1968

l i nea r in aHo i s given by eq. (43) of ref. 4. When the two-body potential V(k, k ' ) i s a s sumed i n - dependent of the angle between k and k ' , a se l f - cons is ten t solution of the equations of motion for ~ - p a i r s d i f fer ing by Q' can be easi ly found. Of the col lect ive va r i ab l e s , Ak(QI) = 0, Bk(Q') r e - su l t s even in k while

p(Q,) = ½mc2hoo( - Q1) Q,2/(4~e2 ) .

This express ion of p conf i rms the ~xistence of a Sch i l l -Barnh i l l f ield E(Q ' ) = - ~Tnc g Q'hoo (- Q') within the superconductor [7], here r e - d e r i v e d by means of the s e l f - cons i s t en t method of ref. 9. The equations of motion for the T-opera to r s ap- pear ing in the c u r r e n t [4 eq. (27)] can then be solved. These solut ions a r e propor t ional to ky, which accounts for the vanish ing of the d iamag- net ic Hall cu r r en t . We a l so find that the Hall c u r r e n t is t r a n s v e r s e with the only n o n - v a n i s h - ing component r e p r e s e n t e d by Jy. For T = 0 and tiv o Q' << I we obtain

Jv(Q') = 4eShr(0) ~ o E ( Q ') (1) . 3mcQ-~2

where

g o = l im QaHo(-Q) Q-~O

is the magnet ic f ield ins ide the superconductor .

For s impl ic i ty we have a s s u m e d g o un i fo rm throughout the superconductor , which is val id only for thin spec imens . Eq. (1) conf i rms the exis tence of a Hall c u r r e n t even at a t e m p e r a t u r e for which, in the same approximat ion, the f r e - quency dependent Hall c u r r e n t is vanishing [4]. An es t imate of the effect for o~ ~ 10 Hz, g o ~ 10 G, length and th ickness of the slab equal to 3 cm and 10 -5 cm respec t ive ly , y ie lds a Hall potent ial V H ~ 10"13V between the faces of the spec imen. This value is well within expe r imen- tal l imi t s .

1. F. London, Superfluids Vol. 1 (Dover Publications, New York, 1960).

2. J. Bardeen, Handbuch der Physik, band 15 (Springer- Verlag, Berlin, 1956).

3. A. B. Pippard (unpublished) as quoted in D. Shoen- berg, Superconductivity (Cambridge University Press, 1962) p. 49.

4. P.B. Miller, Phys. Rev. 121 (1961) 435. 5. G. Dresselhaus and M. S. Dresselhaus, Phys. Rev.

120 (1960) 1971; H. W. Lewis, Phys. Rev. 92 (1953) 1149.

6. L.I. Schiff and M. V. Barnhi11, Phys. Rev. 151 (1966) 1067.

7. B.S. DeWitt, Phys. Rev. Letters 16 (1966) 1092. 8. G. Papini, Nuovo Cimento 52B (1967) 136. 9. G. Rickayzen, Physical Rev. 115 (1959) 795.

10. J. L. Synge, Relativity: The general theory (North- Holland, Amsterdam, 1960).

* * * * *

A Z I M U T H A L L Y L O C A L I Z E D F L U C T U A T I O N S I N A L E V I T A T E D T O R O I D A L Q U A D R U P O L E *

M. ROBERTS, I. ALEXEFF and W. HALCHIN Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA

ReceiVed 20 February 1968

Long persistence, azimuthally localized, density fluctuations have been observed in both lithium and hy- drogen plasmas injected into an electro-magnetically levitated toroidal quadrupole.

In our e lec t romagne t ica l ly levi ta ted quadru- pole [ I ] , p l a smas have been produced by gun i n - ject ion into the magnet ic f ield zero. Float ing, double Langmui r p robes read ing ion sa tu ra t ion

* Research spohsored by the U. S. Atomic Energy Com- mission under contract with the Union Carbide Cor- poration.

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c u r r e n t were placed in the cen te r of the magnet ic well at different az imuthal posi t ions .

Both the l i th ium p lasma (from a Bostick type gun) and hydrogen p l a sma (from a t i t an ium washer gun) can be cha rac t e r i zed by steep, az imuthal densi ty grad ien ts following in jec t ion (more than one o rder of magnitude drop f rom the point of i n - ject ion, 0 = 0 °, to the point of col l is ion, 8 = 180 °)