Critical current anisotropy in YBCO superconducting samples

Physica C 309 Ž1998. 284–294

Critical current anisotropy in YBCO superconducting samples L.M. Fisher a b

a,)

, A.V. Kalinov a , S.E. Savel’ev a , I.F. Voloshin a , V.A. Yampol’skii

b

All-Russian Electrical Engineering Institute, 12 Krasnokazarmennaya Str., 111250 Moscow, Russian Federation Institute for Radiophysics and Electronics, Ukraine Acad. Sci., 12 Acad. Proskura Str., 310085 KharkoÕ, Ukraine Received 28 April 1998; revised 16 October 1998; accepted 16 October 1998

Abstract The critical current densities Jcab and Jcc in YBCO superconductors are shown to be simultaneously determined by means of measurements of a rectangular sample response to an external low-frequency ac magnetic field. A method is based on the correlation of the critical current density and ac magnetic susceptibility and takes into account for both shielding currents Jcab and Jcc. The magnetic field dependencies of Jcab and Jcc have been measured for melt-textured YBCO samples. The anisotropy parameter JcabrJcc is found to increase with the dc magnetic field H. q 1998 Published by Elsevier Science B.V. All rights reserved. Keywords: High-Tc superconductors; Flux pinning and creep

1. Introduction In recent years, the electrodynamics of hard superconductors has drawn attention of many research groups Žsee Refs. w1–20x and references therein.. The electromagnetic properties of such materials are described by the critical state model w21,22x in a rather wide interval of amplitudes and frequencies Žsee, for example, Ref. w17x. where pinning of the magnetic flux inside a superconductor plays a dominant role. The main characteristics of hard superconductors are the critical current density Jc and its magnetic field dependence. It is well known that high-Tc superconductors are strongly anisotropic media and the value of Jc in the ab plane Ž Jca b . is much higher than Jc along the c-axis Ž Jcc .. The study of the magnetic field dependence of both current components is necessary to understand better the nature of the critical current capability in these materials. Usually the parameter Jc is measured by a direct four-probe method. However, good high-Tc samples exhibit very high values of the critical current density and, therefore, the four-probe method turns out to be difficult to realize owing to problems with current contacts. Besides, this method allows, as a rule, to determine only Jca b current component. Thus, a development of some contactless methods is of a great importance. A variety of contactless methods to measure the critical current density were suggested by many authors beginning from Bean’s pioneer papers where the relationships of the critical current density to the width of the magnetization curve w21x and to the third harmonics of the ac response of a sample w23x were established. A

)

Corresponding author. Tel.: q7-095-361-9226; Fax: q7-095-361-9226

0921-4534r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 6 0 7 - 8

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simple contactless method based on measurements of the low-frequency surface impedance was proposed in Ref. w16x. A principal idea of the method is connected to the one-to-one relation between the surface impedance and the critical current density. This method was effectively used to determine Jc and its dependence on the value of the dc field, its orientation, temperature and so on in different samples. The results obtained were in good agreement with the direct measurements of Jc by the four-probe method w24x. At first glance, this method as well as the four-probe one permits us to measure the Jca b component only. Indeed, the main contribution to the surface impedance of thin plates is defined by the shielding current flowing along the sample surface which coincides usually with the ab plane. Therefore, the measurements of the critical current density Jcc represent a very complex problem. To solve this problem, it is important to realize that the surface impedance of a superconducting plate of finite thickness contains information about both current components Jcab and Jcc. According to Maxwell’s equations, the shielding current is closed and flows not only in the ab plane parallel to the sample surface but along the c-axis as well. The relative contribution of the last component to the surface impedance of an isotropic superconductor is negligible if the sample thickness d is much smaller than its width a. However, for anisotropic materials, this contribution may be ignored only in the case when a more stringent condition is fulfilled: ard 4 Jca brJcc ' g ,

Ž 1.

where g is the parameter of anisotropy. If this inequality is not met, the surface impedance contains information about both components of the current. This means that it is possible, in principle, to extract this information. Equations to determine both critical currents Jca b and Jcc from surface impedance measurements are presented in this paper. The surface impedance of YBCO melt-textured samples has been measured and the dependencies of Jca b and Jcc on the external dc magnetic field H have been got. We have found out that the parameter g increases rapidly with H for H ) 5 kOe.

2. Theoretical calculation Let us consider a superconducting sample having a form of a parallelepiped of thickness d, width a, and length l. The origin of the coordinate system is situated at the middle of the sample, the x-, y- and z-axes are directed along the sample thickness, width, and length, respectively. We assume that the next inequalities are fulfilled: d - a < l.

Ž 2.

Let this sample be placed in an external magnetic field directed along the sample length Žalong the z-axis. and having the form: H Ž t . s H q h cos Ž v t . .

Ž 3.

ŽSince we will be interested later in the surface impedance of the sample region near z s 0, we put l s ` in our theoretical analysis.. The magnetic field penetration along the x- as well as along y-directions is considered. ™ ™ This means that the magnetic induction ™B Ž x, y,t . and the electric field EŽ x, y,t . depend™on two transverse spatial coordinates x and y. The vector B contains the z-component only and the vector E has the x- and ycomponents: ™

B Ž x , y,t . s  0,0, B Ž x , y,t . 4 ;

™

E Ž x , y,t . s  E x Ž x , y,t . , E y Ž x , y,t , . ,0 4 .

In this situation B Ž x, y,t . is an even function of x and y, whereas E x, y Ž x, y,t . are odd ones.

Ž 4.

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Fig. 1. Sketch of the quarter of a sample cross-section and the distribution of the shielding critical current at the time-moment t s 0.

Our goal is to determine the connection between the surface impedance of the parallelepiped-like sample and the components of the critical current density Jca b and Jcc. However, to solve this problem, we should first have a picture of the shielding currents flow inside the plate. 2.1. About a form of current lines inside a parallelepiped-like sample The general solution of this problem is quite difficult. For simplicity, let us consider the case that the external magnetic field H increases from zero to Hmax . There are only Meissner’s currents in the sample at H - Hc1 Žwhere Hc1 is the lower critical magnetic field.. The current density has a maximum in the middle of each sample edge in the xy-planes Žsee Fig. 1.. The current lines are approximately parallel to the corresponding edge and the noticeable curvature appears at a distance of about of the London penetration depth l from each corner of the plane. The critical state with the current density equaled the critical value Jc begins to form in the region with the maximum Meissner current at ™ H ) Hc1 . Corresponding picture of the current lines is™shown in Fig. 1. It is evident that owing to the condition < J < s Jc and the equation of the current continuity div J s 0, the current is parallel to the corresponding edge in the region where the critical state exists. The current between these critical region is closed by the Meissner current. In accordance with the analysis given above, the spatial size of the Meissner region is of order of l. As it was shown in Ref. w25x, the character of the current and magnetic field distributions does not change in the presence of the ac magnetic field. Below the scale of order l with respect to the sizes of the sample will be neglected. This means that one can consider that the sample is divided into regions where shielding currents flow parallel to edges. In the frame of the usual critical state model, the current experiences jumps of its directions near of the sample corners. Such a current distribution in a parallelepiped-like sample is well known and used rather frequently Žsee, for example, Ref. w26x and references therein.. 2.2. Surface impedance Our goal is to calculate the first harmonics Ev of the electromagnetic motion force Žemf. E Ž t . in the closed path around the sample which is related via Faraday’s low to the derivative of the magnetic flux F : E Ž t. sy

v Ev s

2p

1 EF c Et 2p r v

H0

1 sy

dr2

ar2

EB

H d xHyar2d y E t , c ydr2

E Ž t . exp Ž i v t . d t.

Ž 5. Ž 6.

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It is convenient to divide Ev by the imaginary part ImŽ Evn . for the same sample in the normal state: EvrIm Ž Evn . ' Z s R y iX ,

X n ' Im Ž Evn . s adh vrc.

Ž 7.

The complex quantity Z represents the dimensionless surface impedance of the sample. To find it one should obtain the distribution B Ž x, y,t .. It may be calculated if one applies Bean’s critical state equation. Since B Ž x, y,t . is even function of both coordinates, this equation is worth considering in the region 0 - x - dr2, 0 - y - ar2 only Žsee Fig. 1.. The form of the equation is different for different parts of the sample:

EB Ex

s"

4p c

EB

Jcy Ž B . ,

Ey

s0

Ž 8.

in region 1 where the current of the critical density Jcy flows along the y-axis and:

EB Ey

s"

4p c

EB

Jcx Ž B . ,

Ex

s0

Ž 9.

in region 2 where the current Jcx exists. Regions 1 and 2 are labelled in Fig. 1. The boundary conditions for these equations can be written as follows: B Ž "dr2, y,t . s B Ž x ," ar2,t . s H q h cos Ž v t . .

Ž 10 .

The problem under consideration can be treated in the same manner as in Ref. w16x. The magnitude h of the ac magnetic field in Eq. Ž3. is assumed to be much smaller than the dc field H: h < H.

Ž 11 .

Due to this inequality, one can neglect the dependence of both critical currents on an alternating part of the magnetic induction. According to Refs. w7,27x, the dc component of B has the uniform distribution wherever the ac field penetrates. This means that one can replace the induction B by the external dc field H in arguments of both functions Jcy Ž B . and Jcx Ž B .. To get expressions for the surface impedance one should solve the set of equations Ž5–10. and make simple but rather cumbersome transformations. Some of them are presented in Appendix A. The final results have different form depending on the value of the ac magnetic field magnitude with respect to penetration fields Hpx s 2p Jcy drc and Hpy s 2p Jcx arc. Here, Hpx and Hpy are the magnitude h at which the ac field reaches the middle of a sample along the x- or y-axes, respectively. At low values of h, one has: Rs

Xs

ch

ž

2

3p ad ch 4p ad

ž

a

d

Jcy

q

a Jcy

ch

Jcx

y

2p Jcx Jcy

d q

5ch

Jcx

y

16p Jcx Jcy

/ /

,

h - Hpx , Hpy ,

Ž 12 .

,

h - Hpx , Hpy .

Ž 13 .

At higher values of h, the result depends on the ratio G s HpyrHpx . For G ) 1, one gets: Rs

2 Hpx

ph

ž

1y

ž

1y

1 3G

y

2 Hpx

q

3h

Hpx 3G h

/

,

Hpx - h, Hpy ,

Ž 14 .

whereas: Rs

2 Hpy

ph

G y 3

2 Hpy 3h

q

G Hpy 3h

/

,

Hpy - h, Hpx

Ž 15 .

for G - 1. Expressions for the imaginary part X of the surface impedance in the last formulae are not presented in view of their very cumbersome forms. Naturally, formula Ž15. can be obtained from formula Ž14. by replacement of the magnetic fields Hpx l Hpy and substitution G ™ 1rG and vice versa. Eqs. Ž12. – Ž15. allow

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the transition to the limiting case G 4 1 Ži.e., a ™ `. which corresponds to an infinite width a of a superconducting plate Žsee Ref. w16x.. 2.3. Determination of Jcx and Jcy In contrast to the expressions for the impedance in Ref. w16x, the impedance in Eqs. Ž12. – Ž15. is defined by both components of the critical current density Jcx and Jcy. Therefore, it is possible to extract these values separately from measurement of the surface resistance of a superconducting sample having a parallelepiped-like form. The simplest way to find these parameters consists in measurement of the dependence of the surface resistance on the magnitude h of the ac magnetic field under conditions h - Hpx , Hpy ŽEq. Ž12... Indeed, the linear term in RŽ h. with respect to h is proportional to Ž arJcy q drJcx ., whereas the squared term is inversely proportional to the product Jcy = Jcx . This two conditions are quite enough to find both critical current densities. Another method is connected with measurement of two identical samples having different widths a1 and a2 . The following expressions may be derived for Jcx , Jcy under conditions h - Hpx , Hpy : Jcx s

ch 2 Ž a1 y a2 . y 3p Ž a1 R 1 y a2 R 2 . 6p

2

a1 a 2 Ž R 2 y R 1 .

,

Jcy s

ch 2

a1 y a 2

3p d a1 R 1 y a 2 R 2

,

Ž 16 .

where R 1 and R 2 are the values of the surface resistance for the samples with widths a1 and a 2 , respectively. Finally, the third method can be suggested. It is based on the intrinsic properties of the surface resistance Žsee Eq. Ž14... A simple analysis shows that the function RŽ h. has a maximum Žso-called size-effect. at G s HpyrHpx ) 1: h s h max s Hpx

2 Ž 2 G q 1. 3Gy1

,

R max s

2

Ž 3 G y 1. . 6p G Ž 2 G y 1 . 1

Ž 17 .

The measurement of the position h max of the maximum and the value R max gives full information on the critical current density Jcx and Jcy as well. Of course, our formulae are not absolutely exact. The curvatures of the current lines near the sample corners are not taken into account. Thus, the possible relative error of the values of Jcx and Jcy obtained with the use of these formulae is about of lrd.

3. Experimental technique The experimental technique is based on the measurement of the surface impedance Z s R y iX by the inductive method w16x. The main idea of this method consists in the following. A plate-like sample with a pickup coil wound around the central part of the sample is placed into the ac magnetic field. A voltage induced in the pickup coil is measured as a function of the external dc magnetic field H. The ac response of the sample gives all information about R and X. The experimental conditions had been chosen in such a way that inequality ŽEq. Ž11.. was fulfilled and we could neglect the effect of the ac field on the critical current density. To measure the dependence of the ac response on the dc magnetic field, melt-textured samples have been used. They were cut out from the homogeneous part of a bulk YBCO ingot. The samples under investigation had the initial size of l s 8 mm, a s 2.55 mm, d s 0.07 mm with the ab plane being parallel to the largest sample surface. The sample thickness d was minimized to insure that the ab plane misalignments were minimal as well as structural inhomogeneities through the thickness d. The experimental RŽ H . curves were measured for the set of the sample widths a s 2.55, 2.05, 1.55, 1.00, and 0.55 mm whereby the other sample sizes were kept constant. A sample width was changed by mechanical grinding with a fine abrasive. All geometrical dimensions in need were determined by an optical microscope with the accuracy of 0.005 mm. Then the sample with the pickup coil was arranged in the central homogeneous field zone of a long solenoid which created ac

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magnetic field hŽ t . s h cosŽ wt . with the frequency vr2p s 133 Hz. The ac magnetic field was parallel to the sample surface and consequently to its ab plane, so that the induced electric currents flowed along the ab plane as well as along the c-direction. The parasitic flux in a gap between the sample and pickup coil which was not connected with the sample was compensated by an additional mutual inductor. The sample with the ac coils was placed into the dc magnetic field H created by a Helmholtz type electromagnet. All our data were obtained in the geometry h 5 H 5 ab plane at the liquid nitrogen temperature T s 77.3 K. The parallelism of H and ab plane required was provided by the following procedure. First of all, the sample was roughly positioned in the horizontal plane, i.e., H proves to be nearly parallel to the ab plane of the sample. Then the fact that the angular dependence of Jc , and consequently R, has a sharp peak at H 5 ab was used Žsee, for example, Ref. w28x.. We inclined the magnet with respect to our sample Žnearby the ab plane. to reach the extremum mentioned and, hence, to provide the condition H 5 ab. To obtain the value of the dimensionless surface impedance, the ac voltage induced in the pickup coil was divided by the voltage induced in the same coil when the sample was in the normal state ŽT ) Tc .. This allowed us to avoid a possible non-ideal bound between the sample and pickup coil. To reduce the demagnetization effects Žthey prove to be fairly small for the plate-like sample., the pickup coil should be wound near the middle of the sample as far as possible from the sample edges perpendicular to the magnetic field. In our experiment, the pickup coil sensitivity zone was about one-eighth of the sample length and it permits us to neglect the demagnetization effects in the further consideration. Our samples were carefully tested to identify the weak link presence. This check was performed by measurement of the dependence of the sample resistance R on the ac magnetic field amplitude h in zero external dc field. The weak link presence must be followed by the deviation from the linear dependence RŽ h. at h < Hp w29x, however, we have not revealed any declination from the linear law, hence our samples did not have measurable weak links.

4. Results and discussion To reconstruct the dependence of both critical current densities Jcc and Jca b on the dc magnetic field, the surface resistance RŽ H . for different samples at various ac field magnitude have been measured. Some results are presented in Figs. 2 and 3. Fig. 2 shows the dependence of the surface resistance on the dc magnetic field H obtained for the set of the samples, differing in their widths a, at the ac magnetic field magnitude h s 200 Oe.

Fig. 2. Measured dependencies of the dimensionless surface resistance R on the external dc magnetic field H at hs 200 Oe, T s 77 K for the samples of different width a: 1— as 0.255 mm, 2—0.205 mm, 3—0.1 mm, 4—0.055 mm.

290

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Fig. 3. Measured dependencies of the dimensionless surface resistance R on the external dc magnetic field H for the sample of the width as 0.055 mm at different magnitudes h of the ac magnetic field: 1— hs 20 Oe, 2—50 Oe, 3—100 Oe, 4—150 Oe, 5—200 Oe, 6—250 Oe, 7—300 Oe.

It is clearly seen that the dimensionless surface resistance increases with the decrease of a. This fact itself makes us sure that both critical currents Jcc and Jca b give the essential contribution to the surface resistance Žsee Eq. Ž12... Fig. 3 demonstrates the dependence RŽ H . for the sample with a s 0.05 mm at different ac amplitudes h. A set of plots Jca b Ž H . and Jcc Ž H . calculated for each of combination pair of our samples Ždifferent combinations of a1 and a 2 in Eq. Ž16.. have been constructed for the case h - Hpx , Hpy. In spite of the possible inhomogeneities of the samples and instrumental inaccuracy, all ten curves coincide. The relative deviation does not exceed 10%. The averaged results are presented in Figs. 4 and 5. In Fig. 5, the dependence Jcc Ž H . calculated from the plots RŽ H . for different ac field magnitude is also drawn. Let us consider a behavior of the functions Jca b Ž H . and Jcc Ž H .. The first function obeys the low ab Jc A Hy1 r2 . This result agrees well with data of many research groups studied YBCO melt-textured superconductors. Such a dependence is characteristic of a system with planar pinning centers whose role in our case is played by intercrystallite boundaries. In contrast to the critical current density in the ab plane, the function Jcc Ž H . decreases more fast. This fact is in accordance with the statement about the Josephson origin of

Fig. 4. Averaged dependence of the critical current density Jca b on the dc magnetic field.

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Fig. 5. Averaged dependence of the critical current density Jcc on the dc magnetic field reconstructed by means of the surface resistance measurement for the samples of different width Žsolid line. and for different magnitudes of the ac field Ždashed line..

the critical current Jc in high-Tc superconductors and its strong sensitivity to the magnetic field. As a result, the contribution of the shielding currents in the c-direction to the surface impedance increases with H significantly. This feature is brightly exhibited if the plot of the anisotropy parameter g s Jca brJcc as a function of H is drawn. This plot is presented in Fig. 6. According to our results, g augments distinctly at fields H G 5 kOe. Thus, the measurement of the surface impedance of a parallelepiped-like superconductor gives important information on the critical current capability not only for the ab plane but in the c-direction as well. In spite of a crude theoretical description Žwe do not take into account real smoothing off the current lines near the sample corners., our simple approach allows us to reconstruct two independent functions Jca b Ž H . and Jcc Ž H .. It turns out that there is no necessity to take efforts to yield the strong inequality l,a 4 d which was used before. The critical current determination by means of our contactless ac method has come in for a deal of criticism that it is not obvious what current component is measured since the current lines are closed. Namely, this criticism partly stimulated our study. We would like to call attention to the fact that the same problem exists in a widely used method of Jc evaluation via static magnetization curves. Certainly, in the latter case, one can obtain full information about both current components after measuring several samples of different sizes. However, to

Fig. 6. Dependence of the anisotropy parameter G on the dc magnetic field.

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extract this information from experimental data, one should solve a set of two integral equations with respect to two desired functions, whereas in the case of our ac method, we deal with the algebraic system.

5. Conclusions A contactless method to determine the critical current density on the basis of the surface impedance measurement has been widely used in the simplest case ŽEq. Ž1... The surface impedance in this case is defined by a single component of the shielding current. Results obtained by this method were in good agreement with the usual four-probe measurements. The procedure presented in this paper is a natural generalization of the previous method to the situation of two critical current components. Using a widely spread model for the magnetic field distribution inside of a parallepiped-like sample being in the critical state, the one-to-one relation between the surface impedance of such a sample and two components of the critical current density has been established. Moreover, we have made an experimental study of the surface impedance and obtained the field dependence of the currents Jca b and Jcc for a number of melt-textured YBCO samples prepared by different technological groups. The characteristic relation Jca brJcc was close to 10 which is in good agreement with few experimental data presented in literature Žsee, for example, Ref. w30x.. It seems that the field dependence of this relation is not universal and may depend on a sample.

Acknowledgements This work has been done as part of the Russian National Program in Superconductivity, project nos. 95046 and 96046, and was supported by the Russian Foundation for Basic Research, project no. 97-02-16399 as well as INTAS project no. IR-97-1394.

Appendix A. Calculation of the dimensionless surface resistance Here, a set of equations Ž5–10. will be solved and calculations of the spatial distribution of the magnetic induction B Ž x, y,t ., the emf E Ž t ., and the dimensionless surface impedance Z will be done. It is suitable to introduce the following dimensionless variables:

tsvt,

j s 2 xrd, h s 2 yra,

b s Ž B y H . rHpx ,

b 0 s hrHpx ,

G s HpxrHpy .

Ž A.1 .

Then Eqs. Ž5., Ž8. – Ž10. may be rewritten in the form:

Eb Ej

s "1,

Eb

s 0,

Eh

b Ž 1,h ,t . s b 0 cos t

Ž A.2 .

b Ž j ,1,t . s b 0 cos t

Ž A.3 .

for region 1 in Fig. 1,

Eb Eh

Eb

s "G ,

Ej

s 0,

for region 2 in Fig. 1, b s 0 for region 3, and E Žt . sy

v ad c

Hpx

1

1

Eb

H0 d jH0 dh Et .

Ž A.4 .

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Fig. 7. Sketch of the quarter of a sample cross-section and the distribution of the shielding critical current at 0 - t -p r v .

Let us consider the case: b 0 - 1, G

Ž A.5 .

when the ac field does not occupy the whole sample bulk. Since both time-dependent functions E Ž t . and expŽyi v t . differ only in signs within two half-periods Ž0– prv . and Žprv –2prv ., it is enough to find E Žt . for the first half-period Ž0– p .. At the initial time-moment t s 0, one should choose the sign plus in Eqs. ŽA.2. and ŽA.3.. Therefore, the spatial distribution bŽ j ,h ,0. has a form: b s b0 y 1 q j

Ž A.6 .

for region 1, b s b0 y G Ž 1 y h .

Ž A.7 .

for region 2, and bs0

Ž A.8 .

for region 3. The boundaries between regions may be found from the condition of continuity of the magnetic induction. They can be presented as:

j s j 0 s 2 x 0rd s 1 y b 0 , h s h 0 s 2 y 0ra s 1 y bO rG , h s 1 y Ž 1 y j . rG ,

Ž A.9 .

Žsee Fig. 1.. At t ) 0, new regions 3 and 4 arise in the cross-section of the sample Žsee Fig. 7. where the critical currents change the direction of their flowing. Therefore, the distribution B Ž j ,h ,t . becomes: b s b 0 cos t q 1 y j

Ž A.10 .

in region 3, and b s b 0 cos t q G Ž 1 y h .

Ž A.11 .

in region 4, whereas the magnetic induction stays to be frozen within other parts of the sample, i.e., it holds the form of Eqs. ŽA.6., ŽA.7. and ŽA.8.. The boundary between regions 1 and 3 is:

j s j 0 s 1 y b 0 Ž 1 y cos t . r2,

Ž A.12 .

and that between regions 2 and 4 is:

h s h 0 s 1 y b 0 Ž 1 y cos t . r2 G .

Ž A.13 .

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Using Eqs. ŽA.4., ŽA.5., ŽA.6., ŽA.7., ŽA.8., ŽA.9., ŽA.10., ŽA.11., ŽA.12. and ŽA.13., one can easily get the formula for emf: b 02 sin t

2 Ž b 0 y 1 y G . Ž cos t y 1 . y b 0 cos 2t , 4G E Žtyp . , E Ž t . s yE p - t - 2p . E Žt . sy

0-t-p ,

Ž A.14 .

Integrating Eq. Ž6. with this E Žt . and taking into account the normalization of Eq. Ž7., we obtain Eqs. Ž12. and Ž13.. The expressions for the surface resistance ŽEqs. Ž14. and Ž15.. for the cases b 0 , G ) 1 and b 0 ,1 ) G when the full penetration of the ac field into the sample takes place, may be obtained using the same method.

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