SICE UNIVERSITY A Study of the Meissner Effect in Type I Superconductors hy Yu Liu A THESIS SUBMITTED IN PARTIAL PULMLiMEIT OP THE REQUIREMENTS IOR THE DEGREE OF MASTER OE ARTS Thesis Director^ signature Houston, Texas May, 1966 'Vu L~< u Abstract Shis report includes an analysis of the flux penetration and expulsion from a normal conducting cylinder and then an account of the observed results of a superconducting ellipsoid. The penetration and expulsion of magnetic flux during the normal to superconducting transition vas governed principally by the eddy current shielding. There was an initial rapid penetration and expulsion of flux as the external magnetic field became equal to the critical field. Cable of Contents Abstract I. Introduction 1 II. Theory 3 III. Description of the Experimental Equipment 13 IV. Description and Interpretation of the Observation 16 V. Discussions. 1? Acknowledgments 22 Bibliography 23 I. Introduction This study was "begun to investigate the time development of the Meissner effect. The Meissner effect is possibly the outstanding characteristic of Type I superconductors. If a superconductor is cooled in a magnetic field, the field is expelled from the material as it passes through the transition temperature. If a superconductor is in a magnetic field greater then the critical field, the superconductit* vity is destroyed. If, however, the field is reduced below the critical field, it is expelled. Since an investigation of the expulsion of the field, when the temperature is lowered, is hampered by the problems associated with the rate of cooling, it was thought possible to investigate the Meissner effect when the magnetic field is reduced at a constant temperature. It appears, however, that the eddy currents induced in the normal regions of the material are dominant in determining the rate of the transition. These investigations were carried out by surrounding the ellisoidal samples of tin or indium with a closely wound test coil connected to an oscilloscope. The sample was then placed in the field of a solenoid and the whole inserted in a cryostat. The oscilloscope pattern was then observed as the current was on or off in the solenoid. 2 This report includes an analysis of the flux penetration and expulsion from a normal conducting cylinder under the conditions of the experiment and then an account of the observed results on a superconducting ellipsoid. These latter restilts can he accounted for qualitatively hut have not proved thus far amenable to quanti¬ tative analysis. z, II. Theory. (1) The Eddy Current a in a Conducting Cylinder. Since the penetration of flux into a superconducting ellipsoid end its expulsion are apparently governed principally by the induced . •'•V; »- eddy currents, the expected eddy current "behavior in a normal conduc¬ tor is analysed as follows: The Maxwell1s equations are aH <7* B = - M * Bt (1) -% Bp V*# « J + Bt (3) Fig. 1. —■>j 2R |*~ When a longitudinal magnetic field is applied to a long cylinder, as show in Fig.l, fields are cylindrical symmetry everywhere, that 3 is, H Hg and E = E0. Then the Maxwell’s equations, in cylindrical coordinates, reduce to ZhL £E ~M>o a Bir (3) 4, dp = 0 7T = rE. ' (4) where o~ is the conductivity, and /£, , the magnetic permeability. Prom eq.(4), we have r L dH* t e “ 0- <?r (5) Substituting eq.(5) into eq.(3), then we get 2 ' dt r *r dt (6) Assume the field 4s sinusoidal dependence on time in order to use a Pourier integral. Let r ul {'wt H* ( , 'i) - f\ lr) e *' (?) then a? A x- f — <! s. 0 sr r a-r (8) Let (-luiMoO-) 'Ar = °<r = f (9) then eq.(8) becomes to -AA -h — -f- A - 0 x df f c|f (io) Solving eq. (10), we get the solution that is finite at r = 0 /4(r) = J0( P - J0(^r) (ii) 5 where Ja idr) is the zeroth order Bessel function. Substituting eq.(ll) into eq.(7), we get C H(r.u.t)= At i Uit = jB Jol**')* (13) iv/t At the surface of the cylinder, r = B and H3(P.,w, t} = He( w)e , then .1 ,*»>*■ _ _. * Hat»)e =BLi*R)e hence K - Ho«*> thenfore (13) Turning on a current gives an external field equal to Hp( 1-e ), where p =R, /l>,, Hence consider the case of H=Hoe*’^ for 17 0 and express it as a Fourier integral, then , , . -/i£ / «Mlt H^)-H0t = / c/ to fid (14) Applying the Fourier transformation, then A/ /“ -/3* -<>* , -2 7T Ip-til") (15) Substituting eq.(15) into eq.(13)f we get 6 Hz(r•lU, ,t)s: JLf -± dw 37c l» /*■*<" J0t«R) U6) Then the total magnetic flux penetrating through the cylinder is <P = ** jRH (rM,+)rdr Jo 2 <*> *wt H cltu J (cir)rdr {fi 4 iu) Jai*R) t (17) ~ °L. Since eq.(ll) gives f, then /■* ii»t , f 1 ^ 7/3Ti^JmTd J ^ , O© = H (i+iui of Jo(e(fi) (18) From eq.(S), we have i '/ Cf =r - < COyU 0~ C>( = ( - <■ U/yUa O')'1 o * ,2 -4r- = -*<L do< G\ yU O- /toO- 0 Then the path of Integration changes as follows: Fig. 2. w-plane Fig. 3. o( -plane - O© CO 7. hence eq.(18) becomes t ; <p= H.R S^L M = -*<H0R I § — V Ju+i)c* Mt,<r/3 J0l^R) -jit -«Lt = -zitfiRl^TTi) —§ Jj IRJUo'fllj {_ I r ( zJMofp J0llijM,0rjjJ /? of,- " M* p (X* = wxI ~ — T e~^ (19) where pf.-B's are the roots of Jo(c(E)~0. (2) Behavior of Voltage Induced in a Test Coil Viound around the Conductor. (A) i/hen the current to the solenoid is turned on at t= 0, the field at the surface of the conductor is HlR)= ^ Hje" e” l-*° for t > 0 — 0 for t c 0 (20) Applying the result in eq.(19), then the flux inside the sample is 8 -Zf _ . f e 7, tRJ/i,ri} if. e ) T 7T/?Xfe xX * (RlRj/^a^t J0 IRJutr{,) ftP L di^j >■ —ytCfO-t„ J -«„(T *•-* xJ,lRUUrfi) u_ r- g ? - TT/?% u RjT^TjT Jaiftjft(><>'p) /?* 4- *'--' 'TS ae - jrohj /._ _ j* rA_}_ I ) M T' R(jr 0 1 ^P JolfUR'L [qf (21) Then the induced voltage in the test coil is S= w|f _ ,0^,0 2«S!S> + _<L r J$L/ _L . _J L‘5r<4* y/wp jjfifwf) ft* i w ‘ tf-wpl J k( 22) where K is the total number of turns of the test coil. Besides the flux change inside the sample, we should count the flux change in the space between the sample and the test coil. Assume the area of this space be czR2, field outside the sample is always uniform,0 then the flux and the total induced voltage in this space ares (23) Va= C7Tft'fi/rtpe'P* (24) 9 She total Induced voltage in the test coil is equal to the sum of the induced voltage inside the sample and that in the space "between the sample and the test coil. v= v, + \4 -fit = TCR‘uH.f[+c7 pe ( L Rthte<rp ]0(Rlu0<r^ ) J ' , J£_ r- (J. l i. ) p At6°‘ ^ \ f ft1 L A,<r l< 4*-ASP* J (25) (B) Wien the magnet is short circuited, the current falls to zero at a rate governed "by the resistance of the circuit and the inductance of the magnet. Then , OO ,n JtO— £* ’ nft ei-t t-*o ° for t<0 (26) for t>0 (27) The Courier transformation gives hf l«))= f* -('wf . / -pt -ivf . 7 n° **U-+*l- e £ dt +J e e dt J • CO ■> _ Kfj^f e^T J £,-* 6 (, fe - t eo a -/ _ j£( ^ | £ 4*Wtjb (28) 10 Substituting eq.(28) into ea.(13), we get Ho U. u) 2% <o-rt> ( < w>+f3 U0-£/ J (.<*/?) oo 0 (29) As long as the external field is everywhere greater then Hc, the total flux passing through the sample is = 27c / Hzir,u).t)relr <wf j (R e ^ I Tio(r)rclr = Hm •'d 00 //? e </«> (><r j Ajrt** ■ J0l* )rdr , €.-*0 Jo = h-t’ll 112 . (SO) Applying eqs.(17) (18) and (19), we have f cxo e IR ;o(otr;re/r («*•«+1J -JL -**cr 2 e~‘ J,lR4ut<rp) ^ JL \ " e -*l?X 1 HIT&Fp J6l*JMor,i) R 1- d--M0rp i* (31) Similarly R e> d) = He, ( Tli J6 Mt)ralr 4U»f € Jt to(H) du> »£:W ■ /-oo tV-£ Jo L*R) * tu>t = ^ (-^H R) fi £.-90 0 </ + «>> It**) 6t; M.f<r - ^ b*iH0R)hn,')l'--jj- 4/ - J Combining eqe.(Sl) and (S3) together, we get 4= ft - 4k - p-i?*u f -^g ^ # |R/#*<rys J0{RiM*rp ) Shen the induced voltage inside the sample is 13. 0 - fifr = -izftNHc 2pa ' J, IRJ*, o-y) RjjAi rp t J0 lRlMv<rp ) .Again we consider the effect in the space between the sample and the test coil. (35) Vi = A/- - citlfh/pH* e P (36) Then the total Induced voltage in the test coil is V3 v, + K a J,(R!WQ) fC fUM^p JOWMJf) J (37) t 13 III. Description of the Experimental Equipment. (A) Cryogenic Apparatus. The low temperature apparatus (fig.4) was constructed chiefly for the purpose of making measurements at temperatures below the liquid helium teraperature(4.3° K). The sample was mounted on a plastic holder which was connected to a thick brass plate by long German silver tube, of such length that when the holder was lowered into the helium dewar, it rested near the bottom, and the dewar was then bolted securely to the plate.