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Critical current density and induced magnetic field in ceramic superconductors
Physics Letters A 365 (2007) 341–344 www.elsevier.com/locate/pla
Critical current density and induced magnetic field in ceramic superconductors N.A. Bogolyubov Institute of Inorganic Chemistry Russian Academy of Sciences, Siberian Branch Lavrent’eva, 3, Novosibirsk 630090, Russia Received 1 September 2006; received in revised form 12 January 2007; accepted 16 January 2007 Available online 30 January 2007 Communicated by J. Flouquet
Abstract It is shown that the transport critical current density in HTSC ceramics and induced magnetic field are homogeneous functions of coordinates. The dependence of the critical current density in HTSC ceramics on the magnetic field is reported. © 2007 Elsevier B.V. All rights reserved. PACS: 74.25.-q; 74.25.Sv Keywords: HTSC ceramics; Critical current density; Magnetic field
1. Introduction The weak links between randomly oriented superconducting grains of the high-temperature superconducting ceramics form the random three-dimensional Josephson net. The dependence of the critical current density jc on the magnetic field H is one of the central questions in the theory of critical state in such materials. If external magnetic field is absent only magnetic field induced by transport current is present in a sample. Distribution and value of transport critical current density in system are determined by the magnetic field and distribution of this field in a sample depends on cross-section shape of sample. Bean [1,2], Kin and Anderson [3,4] models which were developed for hard superconductors and some other models [5,6] for long time have been used in order to take into account the effect of magnetic field on the critical current density in ceramic superconductors [7–10]. This picture is useful but it can hardly be said to account completely for the remarkable properties of HTSC ceramics. We have studied temperature dependency of critical current in HTSC ceramics having various cross-section shapes. It was determined that the critical current in HTSC ceramics was hoE-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.01.024
mogeneous function of cross-sectional dimensions. It is very impotent fact. It allows getting the distribution law of critical current and induced magnetic field in the sample. In this Letter the validity of previous conclusions is verified. We analyze results of the comparative experiment where various samples have been studied. Using our experimental results we have found that jc is power function of magnetic field in any HTSC ceramics having randomly oriented superconducting grains. 2. Experimental The purpose of the present work is to investigate various ceramics samples which consist of randomly oriented grains. Such samples were prepared from high-purity oxides or carbonates. Mixing components were powdered, pressed and calcined in air. Then pellets were powdered, pressed and calcined in air again. No texturing or special oxidation processes were used. Some details of synthesis are described in Ref. [11]. The samples under study had the form of a ring. Rings were machined with the lathe. Special lathe tools were used. Sample 1 (Y-123) had rectangular cross-section shape. Sample 2 had triangular cross-section shape and only one bismuth phase (Bi-2223). Sample 3 had round cross-section shape and contained two superconducting phases (Bi-2212 and Bi-2223). Using two-phase
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sample permitted the study of critical current in two Josephson net in the same sample to be carried out. Samples 1, 2, 3 ware characterized by the superconducting transition temperature (Tc ) 91 K, 108.2 K, and 105 K correspondingly. The critical current in superconducting rings and Tc was measured with the help of contactless transformer method [12–15]. The sample, the primary and secondary coils were placed in the ferrite pot core. When an alternating current (14 Hz) flowed in the primary coil a current of the opposite direction was induced in superconducting sample. Its magnitude must be such that the magnetic flux concentrated in the central kernel of a pot and flowing through ring and coils orifices remained constant and equal to zero. As a result, the magnetic induction in the various parts of the pot was equal to zero. I.e., the sample was placed in a jacket where magnetic field was equal to zero. At that very instant when the amplitude of a current in the ring attained or exceeded the critical current value, the signal was induced in the secondary coil. It had the shape of a sharp peak. The primary current amplitude I1 was recorded at the instant of the signal appearance. The critical current in the sample was calculated by the formula Ic = n1 I1 , where n1 is the number of turns in the primary coil. More details are given in Refs. [14,15]. The critical current in the sample having initial sizes was measured at a number of fixed temperatures. Then the sample was lathed and the critical current for the new cross section was measured at the same temperatures. The same technique was used for critical temperature determination. The only sinusoidal signal was induced in the secondary coil when the temperature was higher than critical temperature. If descending temperature became slightly less than Tc then distorted waveform showed up in the form of a coat hanger (at constant I1 ). The appearance (or disappearance at increasing temperature) this waveform showed that critical temperature was reached. The measuring current in the sample was about 0.1 mA, and in all cases its density was less than 10 mA/cm2 . The width and height of sample 1 were varied in proportion, and all six sections were similar. The critical current in this sample was measured in temperature range from 73.3 K to Tc . Sample 2 was studied in temperature range from 7.3 K to Tc at nine cross sections. Initially the base X of triangular cross sections was varied at constant height Y . Then height was varied at constant base. As a result, cross section had shapes of acuteangled (twice), equilateral, acute-angled, right-angled, obtuse, acute-angled (twice), and equilateral triangles. The sample 3 was studied in the range from 62 K to Tc . Its diameter was changed from 2.7 to 0.8 mm. 3. Results and discussions We have established earlier [14,15] that transport critical current in ceramic HTSC is homogeneous function of the sample sizes, i.e. for any given k [16]: Ic (kX, kY, T ) = k p Ic (X, Y, T ).
(1)
Our results [14,15] have showed that index p is independent on cross-section shape, temperature, type of the Josephson net
Fig. 1. Critical current in sample 1 (Y-123) as function of length of the base X at various temperatures: 86.3 K (curve 1), 83.15 K (curve 2), 79.95 K (curve 3), 77.33 K (curve 4), 73.3 K (curve 5). Dashed line corresponds to IG = Ic (X, T )/Ic (X0 , T ). X0 = 1.555 mm.
of the sample. For the most part Bi-based samples were studied in previous works. Present investigation of YBCO sample 1 confirms Eq. (1) in full. Any homogeneous function may be represents as [16] Y p , Ic (X, Y ) = X F (2) X and vice versa. Fig. 1 shows critical current dependence on length of base X. All isotherms are power function of X. The width and height of the sample were varied in proportion and so the ratio Y/X = const. The deviations of the value of exponent p determined for different isotherms of this sample from the average value are of random nature; their values coincide within confidence intervals. I.e., the exponent is independent of the temperature. The average value of the exponent is equal to 1.39 ± 0.03. Here and below, the confidence intervals are determined by using fractile of the Student distribution with a confidence probability of 0.95. By introducing relative critical currents IT = Ic (X, Y, T )/Ic (X, Y, T0 ), IG = Ic (X, Y, T )/Ic (X0 , Y0 , T )
and (3)
we have shown earlier that the critical current may be presented as Ic (X, Y, T ) = f (T )G(X, Y ).
(4)
T0 , X0 , Y0 in Eq. (3) are any experimentally studied values of T , X, Y . Function f (T ) depends on temperature and specific properties of the sample only. Function G(X, Y ) depends only on the sample cross-sectional dimensions. The same examination of the new experimental results shows that the critical current in any sample may be presented as Eq. (4) in this case. Let us introduce relative current IG by dividing the values of critical currents for each isotherm by the current determined at the same temperature for X = X0 (and Y = Y0 ). The values of IG calculated for different isotherms are quite close. Their deviations from mean values for each X are of random nature. So function IG is independent on temperature. The scale of
N.A. Bogolyubov / Physics Letters A 365 (2007) 341–344
Fig. 2. Critical current in sample 2 (Bi-2223) as function of temperatures for nine cross-sections. Cross-section shapes: (1) acute-angled triangle (curves 1, 2, 4, 7, 8), (Q) equilateral triangle (curves 3, 9), (1) right-angled triangle (curve 5), (a) obtuse triangle (curve 6). Dashed line corresponds to IT = Ic (X, Y, T )/Ic (X, Y, T0 ). T0 = 63.05 K.
Fig. 3. Temperature dependence of critical current in sample 3 (Bi-2212 + Bi2223) for four cross sections. Diameters of cross sections: 2.30 mm (curve 1), 2.05 mm (curve 2), 1.55 mm (curve 3), 0.88 mm (curve 4). Dashed line corresponds to IT = Ic (X, Y, T )/Ic (X, Y, T0 ). T0 = 77.33 K.
figure does not allow us to demonstrate this, and only dashed line drawn through mean values is shown on Fig. 1. Figs. 2 and 3 show temperature dependences of critical currents in samples 2 and 3. As the temperature decreases, a critical current in sample 2 increases smoothly. This sample had various crosssections triangle shapes but all Ic (T ) curves were similar to each other. The turn up of sample 3 Ic (T ) curves is associated with a superconducting transition of the Bi-2212 phase at temperatures about 80 K and with the advent of a now Josephson net formed by two phases Bi-2223 and Bi-2212. Functions IT (T ) of the samples are similar to their Ic (T ) curves and tend to zero as the temperature increases and tends to Tc . I.e., this functions describe specific properties of the sample material. Fig. 4 shows critical current dependences on the cross-section areas of the samples. All dependences are power functions of area: Ic ∼ S p/2 . Therefore Ic = AS p/2 f (T ).
(5)
343
Fig. 4. Critical current in samples 1, 2, 3 as function of cross-section area. Rectangular sample 1 (Y-123): 2, E, and 1 correspond to isotherms 86.3 K, 83.15 K, 77.33 K. corresponds to triangular sample 2 (Bi-2223). Isotherms: 46.66 K (line 1), 63.05 K (line 2), 88.2 K (line 3), 92.9 K (line 4), 98.21 K (line 5), 101 K (line 6). ! corresponds to round sample 3 (Bi-2212 + Bi-2223). Isotherms: 62.5 K (line A), 72.73 K (line B), 82.55 K (line C), 87.55 K (line D), 92.45 K (line E), 97.1 K (line F). Dashed line IG corresponds to IG = Ic (S, T )/Ic (S0 , T ); S0 = 2.72 mm2 ; 1, P and ! correspond to samples 1, 2 and 3 accordingly.
I.e., Ic is homogeneous function of cross-section dimensions and G(X, Y ) = AS p/2 . The deviations of the values of exponent p determined for different isotherms of sample 2 from the average value are of random nature and determined values coincide within confidence intervals, i.e. the exponent is independent of temperature. For sample 3 having two superconducting phases we have the same results. So we may say that the exponent is independent on the type of Josephson net. For samples 1, 2, and 3 average value of exponent p is equal to 1.39 ± 0.03, 1.36 ± 0.02, and 1.34 ± 0.03 correspondingly. These values coincide within confidence intervals i.e. p is independent on cross-section shape. Fig. 4 illustrates these facts. We find the average of averages: p = 1.36 ± 0.03. Dashed line IG = Ic (S, T )/Ic (S0 , T ) on Fig. 4 demonstrates that IG is independent of the temperature and any specific properties of the samples. This function is geometric description of the critical state. It is quite possible that factor A in Eq. (5) depends on the shape of the sample cross section. Introducing relative current for sample i we eliminate specific properties of this sample i.e. function fi (T ), and multiplier A is eliminated at the same time. If our results may be presented in form of the currents IT and IG then critical current has the form of Eq. (4) or Eq. (5). We arrive at the conclusion that transport critical current in ceramic HTSC inclusive randomly oriented superconducting grains in zero magnetic field is homogeneous function of cross-section shape dimensions having universal exponent irrespective of temperature range, type of superconducting material, Josephson net existing in the sample, method which is used for varying of a sample cross section. Taking into account this result let consider similar closed domains M and M1 which represent cross section of a sample. For every point x, y in domain M there is a point x1 , y1 in domain M1 and x1 = kx, y1 = ky. For so important types of cross sections as rectangle, circle, rhombus and others the zero of coordinate system one must locates in the center of these
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geometrical figures. Using Eq. (1) we write p jc (x1 , y1 ) dx1 dy1 = k jc (x, y) dx dy. M1
M
Here jc is a density of transport critical current. If we carry out the change of variables in left-hand side and take into account that Eq. (1) is valid in any cases then we get: jc (kx, ky) = k p−2 jc (x, y).
(6)
The density of a transport critical current in ceramic superconducting materials at zero external magnetic fields is homogeneous function of coordinates having exponent p − 2. Let us consider two arbitrary and similar domains d and d1 on the sample cross-section surface. L and L1 are bounding curves of these domains. Using integrated form of Maxwell’s equation we can write for induced magnetic field in domain d1 : Hx (x1 , y1 ) dx1 + Hy (x1 , y1 ) dy1 = jc (x1 , y1 ) dx1 dy1 . L1
d1
(7) The change of variables x1 = kx, y1 = ky in Eq. (7) and using Eq. (6) give Hx (kx, ky) − k p−1 Hx (x, y) dx L
− Hy (kx, ky) − k p−1 Hy (x, y) dy = 0.
On account of arbitrariness of d and L expressions in square brackets are equal to zero. Then Hx (kx, ky) = k p−1 Hx (x, y), Hy (kx, ky) = k H (kx, ky) = k
p−1
p−1
(8)
Thus in ceramic HTSC sample the value of magnetic field induced by transport critical current and its components are homogeneous functions of coordinates with exponent p − 1. The value of critical current flowing through any domain d may by written as Ic = d jc dx dy and Ic = L H dl. In these expressions procedure of integration is not deal with temperature variable. And so linear multiplier f (T ) contains in expressions for H and jc . The transport critical current density in HTSC ceramic is homogeneous function of coordinates and so it satisfies Euler differential equation [16]: x
∂jc ∂jc +y = (p − 2)jc . ∂x ∂y
jc (H ) = QH
.
The transport critical current density and value of induced magnetic field (and its components) in any HTSC ceramic samples are homogeneous functions of coordinates having exponents p − 1 and p − 2 correspondingly. There is the power connection between the critical current density and magnetic field. The value of transport critical current depends on cross section area of the sample and independent on cross section shape (possibly accurate within constant factor A).
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13] [14]
On the other hand critical current density in HTSC ceramic depends on magnetic field: jc = jc (H ), and at the same time H is homogeneous function of coordinates. As a result we have: p−2 p−1
4. Conclusion
References
Hy (x, y),
H (x, y).
So, critical current density in HTSC ceramic is power function of magnetic field value jc ∼ H −1.78 . It is quite possible that constant of integration Q depends on the shape of a sample cross section and its overall dimensions. This constant contains function f (T ) as f (T )1/p−1 . Using for homogeneous functions jc (x, y) and H (x, y) forms similar to Eq. (2) and polar coordinates we have: jc ∼ r p−2 , H ∼ r p−1 . I.e., if r tends to center of a sample cross-section then jc diverges and H goes to zero (when we consider ceramic HTSC sample as continuum). We have studied polycrystalline (ceramic) cuprate samples. The weak links between randomly oriented superconducting grains of this materials form the random 3d Josephson net. If any texturing or aligning processes are used during a sample preparation then d decreases. Our results cannot be used for such systems and must be modified. In particular, value of exponent p must be changed. On the other hand it seems that our results are valid for the case of granular, conventional superconductor in an insulating matrix.
(9)
[15] [16]
C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. C.P. Bean, Rev. Mod. Phys. 36 (1964) 31. P.W. Anderson, Y.B. Kim, Rev. Mod. Phys. 36 (1964) 39. Y.B. Kim, C.F. Hempstead, A.R. Strnad, Rev. Mod. Phys. 36 (1964) 43. M. Xu, D. Shi, R.F. Fox, Phys. Rev. B 42 (1990) 10773. F. Stuki, J. Rhyner, G. Blatter, Physica C 181 (1991) 385. H. Dearch, G. Blatter, Phys. Rev. B 38 (1988) 11391. A.A. Zhukov, V.V. Moshchalkov, D.A. Komarkov, V.P. Shabatin, A.A. Bosh, S.N. Gordeev, D.V. Shelomov, Jpn. J. Appl. Phys. 29 (1990) L760. X.N. Xu, A.M. Sun, S.A. Aruna, X. Jin, X.X. Yao, Y. Feng, J. Supercond. 10 (1997) 151. A.B. Riise, T.H. Johansen, H. Bratsberg, J. Supercond. 11 (1998) 353. V.S. Kravtchenko, M.A. Zhuravleva, Y.M. Uskov, O.G. Potapova, N.A. Bogolyubov, P.P. Bezverkhy, L.L. Makarshin, Superlattices Microstructures (Suppl. A) 21 (1997) 87. E.A. Harris, J.E. L Bishop, R.L. Havill, P.J. Ward, J. Phys. C 221 (1988) L673. G.P. Meisner, C.A. Taylor, Physica C 169 (1990) 303. N.A. Bogoliubov, Fiz. Nzk. Temp. 23 (1997) 808, Low Temp. Phys. 23 (1997) 606. N.A. Bogoliubov, Fiz. Nzk. Temp. 25 (1999) 1243, Low Temp. Phys. 25 (1999) 929. J. Aczel, Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966.
Critical current density and induced magnetic field in ceramic superconductors N.A. Bogolyubov Institute of Inorganic Chemistry Russian Academy of Sciences, Siberian Branch Lavrent’eva, 3, Novosibirsk 630090, Russia Received 1 September 2006; received in revised form 12 January 2007; accepted 16 January 2007 Available online 30 January 2007 Communicated by J. Flouquet
Abstract It is shown that the transport critical current density in HTSC ceramics and induced magnetic field are homogeneous functions of coordinates. The dependence of the critical current density in HTSC ceramics on the magnetic field is reported. © 2007 Elsevier B.V. All rights reserved. PACS: 74.25.-q; 74.25.Sv Keywords: HTSC ceramics; Critical current density; Magnetic field
1. Introduction The weak links between randomly oriented superconducting grains of the high-temperature superconducting ceramics form the random three-dimensional Josephson net. The dependence of the critical current density jc on the magnetic field H is one of the central questions in the theory of critical state in such materials. If external magnetic field is absent only magnetic field induced by transport current is present in a sample. Distribution and value of transport critical current density in system are determined by the magnetic field and distribution of this field in a sample depends on cross-section shape of sample. Bean [1,2], Kin and Anderson [3,4] models which were developed for hard superconductors and some other models [5,6] for long time have been used in order to take into account the effect of magnetic field on the critical current density in ceramic superconductors [7–10]. This picture is useful but it can hardly be said to account completely for the remarkable properties of HTSC ceramics. We have studied temperature dependency of critical current in HTSC ceramics having various cross-section shapes. It was determined that the critical current in HTSC ceramics was hoE-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.01.024
mogeneous function of cross-sectional dimensions. It is very impotent fact. It allows getting the distribution law of critical current and induced magnetic field in the sample. In this Letter the validity of previous conclusions is verified. We analyze results of the comparative experiment where various samples have been studied. Using our experimental results we have found that jc is power function of magnetic field in any HTSC ceramics having randomly oriented superconducting grains. 2. Experimental The purpose of the present work is to investigate various ceramics samples which consist of randomly oriented grains. Such samples were prepared from high-purity oxides or carbonates. Mixing components were powdered, pressed and calcined in air. Then pellets were powdered, pressed and calcined in air again. No texturing or special oxidation processes were used. Some details of synthesis are described in Ref. [11]. The samples under study had the form of a ring. Rings were machined with the lathe. Special lathe tools were used. Sample 1 (Y-123) had rectangular cross-section shape. Sample 2 had triangular cross-section shape and only one bismuth phase (Bi-2223). Sample 3 had round cross-section shape and contained two superconducting phases (Bi-2212 and Bi-2223). Using two-phase
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sample permitted the study of critical current in two Josephson net in the same sample to be carried out. Samples 1, 2, 3 ware characterized by the superconducting transition temperature (Tc ) 91 K, 108.2 K, and 105 K correspondingly. The critical current in superconducting rings and Tc was measured with the help of contactless transformer method [12–15]. The sample, the primary and secondary coils were placed in the ferrite pot core. When an alternating current (14 Hz) flowed in the primary coil a current of the opposite direction was induced in superconducting sample. Its magnitude must be such that the magnetic flux concentrated in the central kernel of a pot and flowing through ring and coils orifices remained constant and equal to zero. As a result, the magnetic induction in the various parts of the pot was equal to zero. I.e., the sample was placed in a jacket where magnetic field was equal to zero. At that very instant when the amplitude of a current in the ring attained or exceeded the critical current value, the signal was induced in the secondary coil. It had the shape of a sharp peak. The primary current amplitude I1 was recorded at the instant of the signal appearance. The critical current in the sample was calculated by the formula Ic = n1 I1 , where n1 is the number of turns in the primary coil. More details are given in Refs. [14,15]. The critical current in the sample having initial sizes was measured at a number of fixed temperatures. Then the sample was lathed and the critical current for the new cross section was measured at the same temperatures. The same technique was used for critical temperature determination. The only sinusoidal signal was induced in the secondary coil when the temperature was higher than critical temperature. If descending temperature became slightly less than Tc then distorted waveform showed up in the form of a coat hanger (at constant I1 ). The appearance (or disappearance at increasing temperature) this waveform showed that critical temperature was reached. The measuring current in the sample was about 0.1 mA, and in all cases its density was less than 10 mA/cm2 . The width and height of sample 1 were varied in proportion, and all six sections were similar. The critical current in this sample was measured in temperature range from 73.3 K to Tc . Sample 2 was studied in temperature range from 7.3 K to Tc at nine cross sections. Initially the base X of triangular cross sections was varied at constant height Y . Then height was varied at constant base. As a result, cross section had shapes of acuteangled (twice), equilateral, acute-angled, right-angled, obtuse, acute-angled (twice), and equilateral triangles. The sample 3 was studied in the range from 62 K to Tc . Its diameter was changed from 2.7 to 0.8 mm. 3. Results and discussions We have established earlier [14,15] that transport critical current in ceramic HTSC is homogeneous function of the sample sizes, i.e. for any given k [16]: Ic (kX, kY, T ) = k p Ic (X, Y, T ).
(1)
Our results [14,15] have showed that index p is independent on cross-section shape, temperature, type of the Josephson net
Fig. 1. Critical current in sample 1 (Y-123) as function of length of the base X at various temperatures: 86.3 K (curve 1), 83.15 K (curve 2), 79.95 K (curve 3), 77.33 K (curve 4), 73.3 K (curve 5). Dashed line corresponds to IG = Ic (X, T )/Ic (X0 , T ). X0 = 1.555 mm.
of the sample. For the most part Bi-based samples were studied in previous works. Present investigation of YBCO sample 1 confirms Eq. (1) in full. Any homogeneous function may be represents as [16] Y p , Ic (X, Y ) = X F (2) X and vice versa. Fig. 1 shows critical current dependence on length of base X. All isotherms are power function of X. The width and height of the sample were varied in proportion and so the ratio Y/X = const. The deviations of the value of exponent p determined for different isotherms of this sample from the average value are of random nature; their values coincide within confidence intervals. I.e., the exponent is independent of the temperature. The average value of the exponent is equal to 1.39 ± 0.03. Here and below, the confidence intervals are determined by using fractile of the Student distribution with a confidence probability of 0.95. By introducing relative critical currents IT = Ic (X, Y, T )/Ic (X, Y, T0 ), IG = Ic (X, Y, T )/Ic (X0 , Y0 , T )
and (3)
we have shown earlier that the critical current may be presented as Ic (X, Y, T ) = f (T )G(X, Y ).
(4)
T0 , X0 , Y0 in Eq. (3) are any experimentally studied values of T , X, Y . Function f (T ) depends on temperature and specific properties of the sample only. Function G(X, Y ) depends only on the sample cross-sectional dimensions. The same examination of the new experimental results shows that the critical current in any sample may be presented as Eq. (4) in this case. Let us introduce relative current IG by dividing the values of critical currents for each isotherm by the current determined at the same temperature for X = X0 (and Y = Y0 ). The values of IG calculated for different isotherms are quite close. Their deviations from mean values for each X are of random nature. So function IG is independent on temperature. The scale of
N.A. Bogolyubov / Physics Letters A 365 (2007) 341–344
Fig. 2. Critical current in sample 2 (Bi-2223) as function of temperatures for nine cross-sections. Cross-section shapes: (1) acute-angled triangle (curves 1, 2, 4, 7, 8), (Q) equilateral triangle (curves 3, 9), (1) right-angled triangle (curve 5), (a) obtuse triangle (curve 6). Dashed line corresponds to IT = Ic (X, Y, T )/Ic (X, Y, T0 ). T0 = 63.05 K.
Fig. 3. Temperature dependence of critical current in sample 3 (Bi-2212 + Bi2223) for four cross sections. Diameters of cross sections: 2.30 mm (curve 1), 2.05 mm (curve 2), 1.55 mm (curve 3), 0.88 mm (curve 4). Dashed line corresponds to IT = Ic (X, Y, T )/Ic (X, Y, T0 ). T0 = 77.33 K.
figure does not allow us to demonstrate this, and only dashed line drawn through mean values is shown on Fig. 1. Figs. 2 and 3 show temperature dependences of critical currents in samples 2 and 3. As the temperature decreases, a critical current in sample 2 increases smoothly. This sample had various crosssections triangle shapes but all Ic (T ) curves were similar to each other. The turn up of sample 3 Ic (T ) curves is associated with a superconducting transition of the Bi-2212 phase at temperatures about 80 K and with the advent of a now Josephson net formed by two phases Bi-2223 and Bi-2212. Functions IT (T ) of the samples are similar to their Ic (T ) curves and tend to zero as the temperature increases and tends to Tc . I.e., this functions describe specific properties of the sample material. Fig. 4 shows critical current dependences on the cross-section areas of the samples. All dependences are power functions of area: Ic ∼ S p/2 . Therefore Ic = AS p/2 f (T ).
(5)
343
Fig. 4. Critical current in samples 1, 2, 3 as function of cross-section area. Rectangular sample 1 (Y-123): 2, E, and 1 correspond to isotherms 86.3 K, 83.15 K, 77.33 K. corresponds to triangular sample 2 (Bi-2223). Isotherms: 46.66 K (line 1), 63.05 K (line 2), 88.2 K (line 3), 92.9 K (line 4), 98.21 K (line 5), 101 K (line 6). ! corresponds to round sample 3 (Bi-2212 + Bi-2223). Isotherms: 62.5 K (line A), 72.73 K (line B), 82.55 K (line C), 87.55 K (line D), 92.45 K (line E), 97.1 K (line F). Dashed line IG corresponds to IG = Ic (S, T )/Ic (S0 , T ); S0 = 2.72 mm2 ; 1, P and ! correspond to samples 1, 2 and 3 accordingly.
I.e., Ic is homogeneous function of cross-section dimensions and G(X, Y ) = AS p/2 . The deviations of the values of exponent p determined for different isotherms of sample 2 from the average value are of random nature and determined values coincide within confidence intervals, i.e. the exponent is independent of temperature. For sample 3 having two superconducting phases we have the same results. So we may say that the exponent is independent on the type of Josephson net. For samples 1, 2, and 3 average value of exponent p is equal to 1.39 ± 0.03, 1.36 ± 0.02, and 1.34 ± 0.03 correspondingly. These values coincide within confidence intervals i.e. p is independent on cross-section shape. Fig. 4 illustrates these facts. We find the average of averages: p = 1.36 ± 0.03. Dashed line IG = Ic (S, T )/Ic (S0 , T ) on Fig. 4 demonstrates that IG is independent of the temperature and any specific properties of the samples. This function is geometric description of the critical state. It is quite possible that factor A in Eq. (5) depends on the shape of the sample cross section. Introducing relative current for sample i we eliminate specific properties of this sample i.e. function fi (T ), and multiplier A is eliminated at the same time. If our results may be presented in form of the currents IT and IG then critical current has the form of Eq. (4) or Eq. (5). We arrive at the conclusion that transport critical current in ceramic HTSC inclusive randomly oriented superconducting grains in zero magnetic field is homogeneous function of cross-section shape dimensions having universal exponent irrespective of temperature range, type of superconducting material, Josephson net existing in the sample, method which is used for varying of a sample cross section. Taking into account this result let consider similar closed domains M and M1 which represent cross section of a sample. For every point x, y in domain M there is a point x1 , y1 in domain M1 and x1 = kx, y1 = ky. For so important types of cross sections as rectangle, circle, rhombus and others the zero of coordinate system one must locates in the center of these
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geometrical figures. Using Eq. (1) we write p jc (x1 , y1 ) dx1 dy1 = k jc (x, y) dx dy. M1
M
Here jc is a density of transport critical current. If we carry out the change of variables in left-hand side and take into account that Eq. (1) is valid in any cases then we get: jc (kx, ky) = k p−2 jc (x, y).
(6)
The density of a transport critical current in ceramic superconducting materials at zero external magnetic fields is homogeneous function of coordinates having exponent p − 2. Let us consider two arbitrary and similar domains d and d1 on the sample cross-section surface. L and L1 are bounding curves of these domains. Using integrated form of Maxwell’s equation we can write for induced magnetic field in domain d1 : Hx (x1 , y1 ) dx1 + Hy (x1 , y1 ) dy1 = jc (x1 , y1 ) dx1 dy1 . L1
d1
(7) The change of variables x1 = kx, y1 = ky in Eq. (7) and using Eq. (6) give Hx (kx, ky) − k p−1 Hx (x, y) dx L
− Hy (kx, ky) − k p−1 Hy (x, y) dy = 0.
On account of arbitrariness of d and L expressions in square brackets are equal to zero. Then Hx (kx, ky) = k p−1 Hx (x, y), Hy (kx, ky) = k H (kx, ky) = k
p−1
p−1
(8)
Thus in ceramic HTSC sample the value of magnetic field induced by transport critical current and its components are homogeneous functions of coordinates with exponent p − 1. The value of critical current flowing through any domain d may by written as Ic = d jc dx dy and Ic = L H dl. In these expressions procedure of integration is not deal with temperature variable. And so linear multiplier f (T ) contains in expressions for H and jc . The transport critical current density in HTSC ceramic is homogeneous function of coordinates and so it satisfies Euler differential equation [16]: x
∂jc ∂jc +y = (p − 2)jc . ∂x ∂y
jc (H ) = QH
.
The transport critical current density and value of induced magnetic field (and its components) in any HTSC ceramic samples are homogeneous functions of coordinates having exponents p − 1 and p − 2 correspondingly. There is the power connection between the critical current density and magnetic field. The value of transport critical current depends on cross section area of the sample and independent on cross section shape (possibly accurate within constant factor A).
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13] [14]
On the other hand critical current density in HTSC ceramic depends on magnetic field: jc = jc (H ), and at the same time H is homogeneous function of coordinates. As a result we have: p−2 p−1
4. Conclusion
References
Hy (x, y),
H (x, y).
So, critical current density in HTSC ceramic is power function of magnetic field value jc ∼ H −1.78 . It is quite possible that constant of integration Q depends on the shape of a sample cross section and its overall dimensions. This constant contains function f (T ) as f (T )1/p−1 . Using for homogeneous functions jc (x, y) and H (x, y) forms similar to Eq. (2) and polar coordinates we have: jc ∼ r p−2 , H ∼ r p−1 . I.e., if r tends to center of a sample cross-section then jc diverges and H goes to zero (when we consider ceramic HTSC sample as continuum). We have studied polycrystalline (ceramic) cuprate samples. The weak links between randomly oriented superconducting grains of this materials form the random 3d Josephson net. If any texturing or aligning processes are used during a sample preparation then d decreases. Our results cannot be used for such systems and must be modified. In particular, value of exponent p must be changed. On the other hand it seems that our results are valid for the case of granular, conventional superconductor in an insulating matrix.
(9)
[15] [16]
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