Halbritter, J.

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J;f rIl - Ex ~l-. 8e///:1 0-6·~( ,~, f YI)3/70-6 IExterner BerichtJ. HalbritterGeselhcl13H für K~~rnfonf~hung m. b.KERNFORSCHUNGS ZENTRUM KARLSRUHEInstitut für Experimentelle KernphysikFORTRAN-Program for the computation of theSurface Impedance of Superconductors"Gesellschaft für Kernforschung, m.b.H., KarlsruheJuni 1970Als Manuskript vervielfältigtFür diesen Bericht behalten wir uns alle Rechte vorGESE LLSCHAFT FüR KE RN FORSCHUNG M. B. H.KARLSRUHE3/70-6Externer BerichtKERNFORSCHUNGSZENTRUM KARLSRUHEJ. HalbritterInstitut für Experimentelle KernphysikFORTRAN-Program for the computation of theSurface Impedance of SuperconductorsGesellschaft für Kernforschung, m.b.H., KarlsruheJuni 1970Abstract[TE ( •• )] : temperature,with m the mass, e the charge and N the densityof (transport) electrons;critical temperature;ohserved ~(O)/kTc value;London penetration depth at zero temperatureand infinite mean free path t:[FO] : frequency;[TC][FAK][DLON]The material parameters ror a superconductor in the weakcoupling limit areThe program (see tables) computes the surface 1mpedanceZ for the actual parameters of measuremen~~:the symbols 1n the brackets [ ••• ] are used in theFORTRAN-program (see tables). The number [ITJ of tempera-ture values is limited to < 20 by the program.-I. SUMMARYThe FORTRAN-program presented in this report computesthe surface impedance Z of superconductors in theMeissner state for frequencies lower than the energygap. The formulae used are exact within the frame workof the BCS-theory for weak-coupling superconductors.Strong coupling effects can be introduced by usingmeasured ~/kTc values. In the program the temperaturedependence of the gap parameter ~(T) 1s approx1matedby (~(T)/~(O»2 = cos ! (T/Tc )2.With the described input data the program computes thesurface impedance Z = R + iX for specular "S" and diffuse"D" reflection of the charge carriers at the surface. Inthe output the surface resistance R [RS]or [RDl is prin-ted in Ohm. whereas the surface reactanc~ X = w~o~ isgiven through the penetration depth ~ = f Re{H(z.t)/H(O.t)}dz[XSJ or [XD}. 0(2)- 2 -dimension of a cooper pair (coherence length)at T = 0 and t = ~.with the Fermi velocity vF;mean free path of the electrons in the normalstate.[XKOHJ[FREI]In the program the unit for temperature is degree Kelvinand the unit for length Angstrom.The accuracy of computation can be increased by loweringthe factor [REST]. The lower limit of this factor in ourprogram (see tables) is given by about 5.10-7 becausethe number of abcissas [IQ] is limited by 103 • For-4 %REST ~ 10 the absolut accuracy of Z will be about 1 •the relative accuracy (for normal materials like leador niobium) will be about 1 %0. For a further increase ofaccuracy one should use adapted abcissas.(card 1)(card 3)(card 2)(card 4)17500.DIFFUSE REFLEXION1750.2.2\30.SPECUlAR-4.17'2.05+':::.'result in:- 3 -fuR f = 0.3000E 10 HERTZ THE BCS-THEORY RESUlTS2As an example the following input-data:==============================~=========;=:===::~AleRJAlPARAMETERS:TC = 7.20 K GAP(T=O)/KTC = 2.05l(~OOC~PEN. CEPTH(T=O,l/l=O) = 280.0 ANGSTROMCCHERENCE lENGTH(T=O,l/L=O)f = 1750.0 ANGSTROM~eAN fREE PATH = 17500.0 ANGSTROM.3000E lOO.4170E 010.2000E 01G.IOOOE-02SURfACE IMPEOANCE IN T~E SUPERCONOUCTING STATE000000000000000000000000000000000000000000000000000000000000000000000000000000034 56189W"g~"~~"""~~nn~~~"n"~~n»M.~"R»WO"~«~~"~U.MU~~~~U~~~~~~M~MD~WR"nnNg~n""W1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I 1 I 1 I 1 1 I 1 1 I I I 1 I I I 1 I 1 1 1 I 1 1 1 I 1 1 I I I ! 1 1 1 1 1 1 1 1 I 1 1 1 I 1 I 1 1 I2222222222222222222222222222222222222222222222222222222222222222222222222222223333333 H3 3 3 3333333333333333333333333333333333333333333333333333 3 333333333333334 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 47.2- 4 -The program (see tables) can be used also for a compu-tation of the penetration for w = 0 (or nw « ~) if oneomits the subroutine NONAN in the program-deck.- 5 -II. INTRODUCTIONIn plane geometry the surface impedance Z = R + iX isgiven by the penetration of the magnetic field H(z,t)into the wall:Z = iw~o J dz H(z,t)/H(O,t) = iw~o A(O,t)/B(O,t) (3)owith z the distance from the surface and A the vectorpotential: ~o H = vxA. The fields H(z,t) or A(z,t) arethe solutions of the following equations and boundaryconditions 1,3-5).Maxwell's equation (a 2/at 2 can be neglected forfrequencies lower than 1013 Hz):material equation:oj(z,t) = - J dt' J dz'Q(z-z', t+t') A(z',t');boundary conditions for the fields:aA(z,t)/az Iz=o = B(O,t), A(~Jt) = 0;boundary conditions for the charge carriers at thesurface:specular or diffuse reflection.According to 3) with the Fourier transformed integralkernel Q(k,w) the surface impedance can then be written:[AK)1 eo dkspecular reflection: Zs = ilJ 000 .... J 211 k +lJoQ(k,oo)-110(4)'ITdiffuse reflection: Zo = ilJ ooo Z dkin{1+lJ oQ(k,oo)/k2;K J dqtn(1+Q(q.oo)~IQ(O,oo) IK 2q2»o- 6 -For Q evaluation of (4) the k-integration is trans-formed in a q-integration by: q = k ~F 6(0)16(T)[OX( ••• )J, i.e. the momenta k will be measured in unitsof a coherence length. With the abreviations21C co dqZs = i\.looo 15 L(T, R,. (0) - f """';:'2-:2"1"'""""'-------'IT 0 K q +Q(q,oo)/IQ(O,oo)1Before we can start with the integration dq in subrou-tine HAUPT and compute the integrand in subroutineINTQ ( ••• ) the integralkernel Q(k,oo) should be evaluated.According to 1) we split Q in an in 00 analytical term QAand a non-analytical term QO:(4) can be written:(8)(6)(9)fivF 1+ e.e.}2l1(T).2,+ e.e. }1-m(m+1~)+~2~/(m+i~j2~ir::2+1 l(m+1~)2 + 1"- 7 -~A(k.w) :: a L {m::(2n+1)an>1Qn(k.w) :: ~p(k.w) + ~SCH(k.w)1 l1(T) l1 (T) '"Qp(k. w) :: J d; {th-; - th- (; + w) } •'" 2kBT 2kBT1-wThe analyt1eal term QA 1s def1ned by:(a :: 1TkT/l1(T);~ :: nwll1(T): [oJ ). ~A is eomputed 1nsubrout1ne ISUM. ~A 1s purely real and therefore g1vesfor ~ « 1 the penetrat10n depth. 1.e. ~A deser1bes theeurrent of "Cooper" pa1rs. ~n wh1eh d1sappears for ~ .... 0and T .... 0 eons1sts of a real term ~p and a term ~SCH.deser1b1ng the rf absorption:- 8 -(10)~p and ~SCH will be computed in subroutine NONAN,The integrals (1), (9) and (10) depend on the function Jwhich results from an angle integration:J(u) = 61- { - u2f1(T) (11J(u) will be evaluated in subroutine WINK (",),In the following sections we will discuss in more detailthe various summations or integrations in the subroutines,III. MOMENTUM INTEGRATIONThe q-integration is done by Simpson's rule (formula 25.4,6in 6», The q-range {O,~} is subdivided into the inter-vals [I] = 1, ",,5 <[G(1)], [G(2)]} ,.<.} and the 6th inter-,(12)n2n(21-1)cos ---2b-a-2nb+ari=1Y ::: -- +ival {[O(6)] t/[ä(t)]}, in whieh q is substituted by 1.The abcissas [DX ( )J within these intervals are evafu-ated in the main program, their number i8 given by[ID~I)] (with [ID(I)J :: u and [DELT] :: h in 25.4.6in ). The subdivision 1nto 6 intervals before usingSimpson's rule has been introduced to improve theaccuracy by giving certain q-regions larger weight.For materials w1th, e.g. 6L/~F « 1 small momenta( ) -1/2k ~ 6Lt F are most 1mportant. After evaluation ofQ in subroutines NONAN and ISUM the integrand 1s calcu-lated in subroutine INTQ ( , ) and summed in subrout1neHAUPT. The result - Z/(w~o6L(T,~,w» - is transfered tothe main program, where after some mult1plicat1ons thesurface resistance in Ohm and the penetration depth~(T,t,w) in Angstrom are printed out.- 9 -IV. EVALUATION OF Q(k,w)We start with the smallest fraction of Q, namely theQD in subroutine NONAN to reduce rounding errors, Thelarger QA is computed afterwards in subroutine 1SUM. Anumerical computation of ~D with Simpson's rule i8 notvery precise because of the square root singularity ofthe integrand. More adequate for numerical integrationof square root sigularit1es is the integration byChebyshev Polynomials of the First Kind, i.e. the useof formula 25.4.39 of 6):f(y)dyJ I(y-a) (b-y)' :::ab- 10 -The evaluation of the residue Rn for (9) and (10) istoo dirficult and it is omitted, therefore. A suffi-cient n is chosen according to the requirement thatthere must be enough abcissas (> 5) in the regionwhere f(y) has a sharp maximum.IV.1 ~SCH (k,oo)This term describes the rf absorption, i.e. QSCH includesthe imaginary part of Q. For computation integrand of(10) is transformed by substituting u = 1/;; i.e. theinterval (a,b) in (12) is: (0,1). For the integral themaximum of f(y) « 1111+u~-ui becomes more important forlower ~, i.e. the number of abcissas [IS] will increasewith 1/1 ~~For ~ < 1 Qp(k,w) is a slowly varying function with ;the number of abcissas [IP] will therefore be small,Moreover there is no need of a more exact computationof Qp for ~« 1, because Qp is real and small comparedto QA'~A (7) is computed in subroutine ISUM, For ~ = 0 and~T = 0 (00 ~ 2) QA is the only non-vanishing part, Forthe numerical computation the temperature sum in ISUMyields easily a good accuracy. The number of terms inthe sum [MS]must be increased for T + 0 because theterms are c 1/( T/Tc )2. To reduce rounding errors incomputing the sum ISUM starts with the smallest terms.- 11 -REFERENCES1. J. HalbritterExterner Bericht 3/69-2 t (KernforschungszentrumKarlsruhe t 1969)2. T.P. SheahenPhys. Rev. 149 t 368 (1966)........3. G.E.H. Reuter t E.H. SondheimerProc. Roy. Soc. A 195 t 336 (1948)R.B. DinglePhysica XIX t 311 (1953)4. I.M. Khalatnikov, A.A. AbrikosovAdv. in Phys. 8, 45 (1958)...5. J.P. TurneaureThesis (HEPL-report-507 t Stanford t 1967)6. M. Abramowitz, I.E. StegunHandbook of Mathematical Functions (U.S. Govern-ment Prlnting Offlce t Washlngton t C.D. t 1964)- 12 -FORTRAN - PROGRAMC SURFACE IMPEDANCE OF SUPERCONOUCTORS FOR FREQUENCIES SMALLER THAC THE ENERGY GAP.DIMENSION G(6),IO(6),TE(20)REAL*80ELT,DET,DQ(6),DA,OE,DZ,DX(1000),Pl ,001 ,OS(1000)COMPLEX*a CHS,CHO,CICCMPlEX*16 COS(lOOO),CT(lOOO),CS,COCOMMON/HLI/IO,IQ,MS,lS,IPCOMMCN/Hl2/AK,A2,FL,O,GPCOMMON/HL3/PI,DQ,OX,DQ1,DSCOMMON/HL4/CHS,CHO,CICOMMON/HLS/CDS,CT,CS,CDC IT=NUMBER OF TEMPERATUREVALUES.READ (5,1) ITC FO=FREQUENCY (HERTZ),TE(I)=TEMPERATURES (KELVIN).REAC (5,2) FO,(TE(l),I=l,IT>C MATERLAlPARAMETERS(UNITS:OEGREE KElVIN,ANGSTROM)REAO (5,2) TC,FAK,OlON,XKOH,FREIC REST GIVES T~E ACCURACY OF THE COMPUTATION.READ (5,2) REST1 FORMATOS)2 FORMAT(8EIO.4)WRITE(6,2) FO,(TE(I),I=l,JT),RESTt THE MOMENTUMINTEGRAl IS SPlITTEO IN 6 PARTS OVER THE INTERVALLSt (I),G(!+lH.G(l)=1.E-6G(2) = 1. E-2G(3)=0.2G(4)=2.G(5)=S.G(6)=50.C THESE 6 INTEGRALS ARE EVALUATEO BY SIMPSON'S RULE. 1/10(!) GIVFSC THE SPACING OF THE ABCISSAS.10(1)=510(2)=710 (3)= 1510(4)=2010(5)=2010(6}=9IS=(6.E-4/REST)**0.25+1.00 38 N=1,638 IO(N)=!O(N)*ISPI=3.1415926535PH =0.419*10.**(~10)CI=(0.,1.)FlO=XKOH/FREIAL=OLON/XKOHOG=PH*FO/TC/FAKDX(l)=O.DX ( 2)= G( 1 )IQ=2oe 40 N=1, 5ll=ID(N)-lD=10(~)*2IO(N)=llDA=4( N)DZ=e(N+UOElT=(OZ-DA)/D- 13 -DET=DELT*2.DQ(N)=DELT/3.DG 39 K=1,LLDI\=OA+DETOE=OA-DELTIQ=IQ+1DX(IQ)=DElQ=IQ+139 DX( IQ)=DAOE=DZ-DELTIQ=IQ+1DX( IQ)=DElQ=IQ+1400X(IQ)=DZLL=lD(6)-10=10(6)*2lo(6)=lLDA=G(1)Ol= 1./G (6)DElT=(Dl-DA)/DDET=DELT*2.DQ(6)=DELT/3.lQ=IQ+1DX(IQ)=l./DA00 41 K=l,lloA=DA+DETDE=CA-DElTIQ=IQ+1DX( IQ)=1./DE1Q=IQ+141 OX( lQ)=l./DADE=ol-DEl TlQ=IQ+1DX(IQ)=1./DEIQ=IQ+1OX( IQ)=1./DZWRITE(6,100)WRITE(6,10l)WRITE(6,102)WRITE(6,l03)TC,FAK,OLON,XKOH,FREIWRITE(6,l07IFOWRITE(6,108)DU 22 K=l,lTT=TE(K)/TCB=P I12.*( l.-T*T)AG=SQRT(SIN(B)Fl=FLO/AGO=OG/AGIF(O-0.5)69,22,2269 GP=AG/T*FAKIS=1./SQRT(O*REST)/5.lP=1./SQRT«2.-0)*REST)/5.00 19 I:: 1, 1QOS(I)=O.19 CDS(I)=Q. ,CALL NONAN \MS=GP/(SQRT(REST)*2.*PIlCAlL ISUM <I- 14 -DQl:::COABS( COS,( U)ENP:::l./OSQRT(OQllAK:::AL*ENP*AGA2.=AK*AKCALL HAUPTCHS=CHS*ENPChO:::CHO*ENPXS=REAL(CHS)*OLONXD=REAL(CHD1*OLONB=(FO/I0.**17)*8.*Pl*PI*OLONRS=AIMAG(CHS)*BRO=AIMAG(CHO)*BWR IT E( 6,109) TE (K) , XS, RS, XO, RD, CHS, CHO22CONTINUE100 FGRMAT(IHl,48H SURFACE IMPEDANCE IN THE SUPERCONDUCTING STATE )101 FORMAT(IHO,48H:::============:::============================:::=====)102 FORMAT(lHO/2X,19HMATERIALPARAMETERS:)103 FORMAT(IHO,6H TC ::: ,F4.2,24H K GAP(T:::O)/KTC ::: ,F4.2/32H LONDIDON PEN. OEPTH(T=O,l/L=O) ::: ,F7.1,9H ANGSTROM/32H COHERENCf LENGTH2(T=0,1/L=0)F ::: ,F7.1,9H ANGSTROM/32H MEAN FREE PATH3= ,F7.1,9H ANGSTROM)107 FORMAT(lHO,9H FOR F = ,EIO.4,29H HERTZ THE BCS-THEORY RESULTS/3H1 Il4X, 9HSPECUL AR-, lOX, 17HO I FFUSE REF L,EX I ON)108 fORMAT(lHO,3X,4HTEM-,6X,4HPEN.,4X,7HSURFACE,5X,4HPEN.,4X,7HSURFACE1/2X,8HPERATURE,3X,5HDEPTH,3X,10HRESISTANCE,2X,5HDEPTH,3X,10HRESIST2ANCE/4X,3H(K),7X,3H(A),6X,5H(OHM),6X,3H(A),6X,5H(OHM))109 FORMAT(lHO,3X,F4.2,3X,F8.1,lX,EIO.3,lX,F8.1,lX,EIO.3,8X,4Ell.4)STOPJ: 1\' f"'- 15 -SUBROUTINE HAUPTDIMENSICN 10(6)REAl*80Q(6),OX(1000),PI, OQ1.0S(1000)CCMPLEX*8 CHS,CHO,CICOMPLEX*16 COS(1000),CS ,CT(lOOO),CIS,CID,CES,CED,CAS,lCAO,COCDMMDN/HlI/ID,IQ,MS,IS,IPCOMMON/Hl2/AK,A2,FL,0,GPCOMMON/Hl3/PI,DQ,OX,OQ1,OSCCMMON/Hl4/CHS,CHO,CICGMMON/HL5/COS,CT,CS,CDM=l1=2CALL INTQ(I,M)Cl S=DX (2 )*CSCIO=DX(2)*CD+Z.*OX(Z)00 40 J=1,6IF(J-6)11,10,1010 M=2CAlL INTQ(l,M)CIS=CIS+CS*OX(2)CID=CIO+CO*OX(Z)11 CES=O.CEO=O.CAS=O.S*CSCAO=O.5*COll=ID(J)DA 39 K=l,LlCALL INTQ(I,M)CES=CES+CSCED=CEOtCOCALl INTQ(I,M)CAS=CAS+CSCAO=CAD+CO39 CONTINUECAll INTO(I,M)CES=CES+CSCED=CED+CDCAlL INTQ(I,H)CIS=CLS+OQ(J)*(4.*CES+(Z.*CAS+CS)}40 CIO=CIO+OQ(J)*(4.*CED+(2.*CAD+CD»A=2./PI*AKCHS=A*CISCHD=2./A/CIORETURNEND- 16 -SUBROUTINE INTQ(l,M)OIMEI\SICN 10(6)REAL*8 DQ(6),OX(1000),Pl, DX2,OA,COMPlEX*8 CHS,CHO,CICOM~lEX*16 CDS(lOOO),CT(lOOO),CS,CO,COGCOMMON/HL1/10,IQ,MS,IS,IPCOMMON/Hl2JAK,A2,Fl,O,GPCOMMON/Hl3/Pl,OQ,OX,OQl,OSCOMMON/HL4/CHS,CHD,CICOMMON/Hl5/CDS,CT,CS,COCDG=COS(l)/DQlOX2=DX(l)*OX(l)OA=OX2*A2GG TO (10,11l,M10 CS=l./(OA+CDGlCD=CDLCG(l.+COG/OA)GO TO 911 CS=1./(A2+CDG/OX2)CO=COlOG(1.+CDG/OA)*OX29 l=L+lRETURNENDOQ1,OS(1000)- 17 -SUBRCUTINE NONANDIMEI\iSION 10(6)REAL*8 0 ,OX(1000),PI, DQ1,DS(lOOO),DV,DZ,DU,OG,DA,DC,DP,DM,lOW,DWO,OQ(6), OT,02CCMPLEx*a CHS,CHD,CICCMPLEX*16 COS(lOOO),CT(1000),CS,CO,CYCCMMGN/HLI/IO,IQ,MS,lS,IPCOMMON/Hl2/AK,A2,FL,O,GPCOMMON/HL3/PI,OQ,OX,DQ1,05CGMMGN/HL4/CHS,CHO,CICCMMCN/HlS/COS,CT,CS,COOT=0/2 .. *GPIFlCT-O .. 2HO,lO,910 02=01*OToV=Or*(1 .. +02/6 .. *(1.+02120 .. *(1 .. +02/42 .. *(1.+02/72 .. *(1.+O2/110 .. »»)*10EXP (-OT )CT=-OPl' 2.GO TO 89 OT=-OT*2.OV=(1 ..-OEXP(OT»/2.8 8=4*15BS=P 1/B00 48 K=l,IS8=4* ( 1S-K)+2O=B*BSDZ=O.S*(l.+OCOS(O»DU=GP/DZOA=DEXP l -DU)DG=OEXP(-DU+OT)OC=DA/«l.+OG)*(l.+OA»*OVDP=DZ+l.DM=l .. -OZDU=DP*DMOw=DSQRT(OU)OA=OZ*OOWO=OSQRT(OU+DA*(2.+DA»OP=OZ*OSQRTlOZ*OP)DU:::=OA/OUIF(OU-O.l)1,1,61 OU=OU*l2 .. +0A)/2.DM=OU*( 1.-O.S*OU*(1 .. -OU*tl.-l .. 2S*OU*(1.-1.4*OU*(1.-1.S*DU*(1.-11.*lOUI1.*(1 ..-1 .. 5*OU*(1.-13.*OU/9*ll.-1.4*OU»»»»)*OWGe TC 56 OM=DWO-OW5 CY=-CI*DM/2./Ol~FLCALL WINK(CYJOU=DC/tOP/OWOOwU=DWO*OWDM=l.+DZ*(Ol+O)ops= lDM+OWU}*OUDMS=(OM-OWU)*OU00 20 l=l,IQ20 COS(I)=COstI)+DPS*CT(I)OM=OWC+OW .'CY=CI*OM/2./OI+FLCALl WINKJCY)f'- 18 -oe 21 1=1, IQ21 OS( I)=oS(l)+OMS*CT(I)48 CCNTINUECY=-CI*8S*4.DA 22 I=l,IQCOS( I)=(COS( 1)+OS(!) )*CY22 OS ( 1)=0.B=4*IPBS=PI/ß00 47 l<=l,IP8=4*(lP-K)+20=8*B5DZ=1.+O.5*O*(1.+oCOS(O»OU=GP*OZoA=oEXP(-OU)oG=OEXP(-OU-OT)OC=OG/«1.+oG)*(1.+DA»*OVOP=Ol+l.OM=Dl-1.ow=DSQR T( OP*OM)OA=O-O~1OU=DP-ODwO=DSQRT(OA*oUlCY=OWOi'Cl*OW+FLCALL WINK(CY)CY=(1.+DZ*(Dl-O)+CI*OW*OWO)/OSQRT(OP*oU)*OC00 23 1=1,IQ230S(!)=OSCI)+CY*CTCI)47 CONTINUEB=+B5*4.00 24 1=1,IQ24 CDSC!)=CD5C!)+oSCI)*BRETURNENDSLBROUTINE WINK(CLJDIMENSION 10(6)REAL*8 OQ(6),OX(lOOO),PI, D,OQl,OSlI000)COMPLEX*8 CHS,CHO,CICOMPLEX*16 COS(1000),OS ,CT(1000),CD,CC,C2,CLCOMMON/HLI/IO,IQ,MS,IS,IPCOMMON/HL2/AK,A2,FL,O,GPCOMMCN/HL3/PI,DQ,OX,OQl,DSCOMMON/HL4/CHS,CHO,ClCCMMCN/HL5/COS,CT,CS,CC00 20 1= 1, I QO=O)«IJ)CD=O/CLC2=CO*CDU=CDABS(CO)IF(U-0.2)10,10,1110 CT(!)=3./CL*(1./3.-C2*(1./15.-C2*(l./35.-C2*l1./63.-C2*(1./99.-lC2*(I./143.-C2*(1./195.-C2/255.»»»)GO TO 2011 CO=I./CO ,CT( I)=0.15/0*(-2.*CD+(1.+CO*CO)/CI*(CDLOG(CO+CI)-CDLOG{CD-CI»)20 CCNTINUE ~RETURN JENDSUBROUTINE ISUMOIMEt\SICN 10(6)REAL*8 DQ(6),OX(1000),PI, G,COMPLEX*a CHS,CHO,CICOMPlEX*16 COS(1000),CSCOMMCN/HlI/IO,IQ,MS,IS,IPCOMMON/HL2/AK,A2,Fl,O,GPCCMMCN/HL3/PI,OQ,DX,OQ1,OSCGMMCN/HL4/ChS,CHO,CICOMMCN/HLS/CDS,CT,CS,COG=PIIGP00 20 1= 1, I Q20 OS( I .=0.00 30 M=1,MSB= 2*(MS- M) .. 1B=B*GDU=B*B+l.CB=B+O*CIcU=Ce*CB+l.OW=OSQRT(OU)CW=COSQRT( CU)CL=(OW+CW)/2.+FLCP=OW*CWCA=(l.-B*CB+CP)/CPCALL WINK(CL)oe 21 1=1,IQ210S(I)=OS(I)+CA*CT(I)30 CCNTINUE00 22 1=1,IQ22 CCS(I)=DS(!)*G +COS(I)RETURNEND- 19 -OU,DW,OS(lOOO),OQl,CB,CU,CW,CL,CP,CA,CT(1000),CD

Fortran-program for the computation of the surface impedance of superconductors

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Fortran-program for the computation of the surface impedance of superconductors

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