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Interaction of electromagnetic waves in hard superconductors
PHYSICA ELSEVIER
Physica C 251 (1995) 50-60
Interaction of electromagnetic waves in hard superconductors F. P&ez-Rodriguez a,* I.V. Baltaga b, K.V. II'enko b N.M. Makarov b V.A. Yampol'skii b, L.M. Fisher c, A.V. Kalinov c, I.F. Voloshin c a Instituto de Fisica, UniversidadAut6noma de Puebla, Apdo. Post. J-48, Puebla, Pue 72570, Mexico b Institute for Radiophysics and Electronics, Ukrainian NationalAcademy of Sciences, 12 Acad. Proskura St., 310085 Kharkov, Ukraine c All-Russian Electrical Engineering Institute, 12 Kransnokazarmennaya St., 111250 Moscow, Russian Federation Received 23 March 1995
Abstract
Nonlinear interaction of low-frequency radio-waves in hard superconductors is studied both theoretically and experimentally. It is shown that this interaction gives rise to a new nonlinear phenomenon, namely, jumps in the temporal dependence of the electric field at the surface of the sample. Heights and positions of the jumps are determined by the amplitudes, frequencies, and phase difference of the interacting waves. Necessary conditions for the existence of the jumps are discussed. The calculations are carried out in the framework of a critical-state model with an arbitrary magnetic field dependence of the critical current density. Experimental results are found in good agreement with theoretical ones.
1. Introduction
The nonlinearity in the material equation, which relates the current density to the electromagnetic field, is responsible for the violation of the superposition principle and leads to the interaction of electromagnetic waves in conducting media. As such, the characteristic properties of the nonlinear mechanism manifest themselves as features of such interaction. In view of this, the study of nonlinear radio-wave interaction in hard superconductors is of great interest. In recent years the electrodynamics of hard superconductors has drawn attention of many research groups (see Refs. [1-19] and references therein). The electromagnetic properties of such materials are de-
* Corresponding author.
scribed by the critical-state model [20,21] in a rather wide interval of wave amplitudes and frequencies (see, for example, Ref. [16]), where the pinning of the magnetic flux inside a superconductor plays a governing role. In this model, the Maxwell equation specifying the distribution of magnetic induction B has the following form: 4rr curl B = - - j c ( B ) c
E IEI
.
(1)
Here E is the electric field, jc(B) is the critical current density, c is the speed of light in vacuum. The critical-state equation is essentially nonlinear. The nonlinearity in the right-hand side of Eq. (11) is stipulated by the dependence jc(B), as well as by the factor E / I E I. The nonlinearity associated with the latter is of principal interest, since it does not occur in any other nonlinear media. Because of this nonlinearity, unusual effects are observed in hard supercon-
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 3 8 5 - 1
F. P~rez-Rodrlguezet al./ PhysicaC 251 (1995)50-60 ductors (see, for example, Refs. [6] and [14]). As is shown below, this nonlinearity gives rise to a peculiar phenomenon in the radio-wave interaction, namely, jumps in the electric field at the sample surface versus time. A brief report on this effect was published in Ref. [22]. High-Tc ceramics being in the critical state represent a convenient object to observe nonlinear electrodynamic effects. Therefore, we shall carry out theoretical calculations for high-Tc ceramics. As was shown in Ref. [3], these specific nonlinear Josephson media can be described by an equation similar to Eq. (11). It contains in the right-hand side an additional factor/z, representing the effective magnetic permeability (0 < / x < 1). The parameter /x allows for intragranular currents preventing magnetic flux penetrating into the grains. The results for traditional hard superconductors correspond to /x = 1. Section 2 is devoted to the statement (Subsection 2.1) and theoretical study of the problem. In Subsection 2.2 we discuss the origin of jumps in the temporal dependence of the electric field E(t) at the surface of a superconducting plate, caused by the radio-wave interaction. Subsection 2.3 contains formulas for the function E(t) with an arbitrary dependence j¢(B). Results for two usual geometries (plate and cylinder) are presented. Besides, some necessary conditions for the existence of E(t) jumps are formulated therein. In Section 3 a comparison of the theoretical curves for E(t) with data, obtained in experiments with yttrium ceramic sample, is done. The paper contains an Appendix.
51
geometry. The vector B contains z component and the vector E has y component only:
B(x, t ) =
{0, 0,
B(x,
t)};
E( x, t) = {0, E( x, t), 0}.
(3)
In this situation B(x, t) is an even function of x, whereas E(x, t) is an odd one. Our goal is to calculate the electric field at the boundary x = d, which we define as E(t),
E( t) = E( d, t) = - E ( - d ,
t).
(4)
It is related to the derivative of the magnetic flux per unit plate width with respect to time by the equation 1 dqb
E(t)=
2c d t '
d
~ = f _ d B ( X , t) dx.
(5)
In this geometry, the critical-state equation for ceramics can be written as follows: 0B
- O---x=
4 xr/z
c Jc(B) sign E.
(6)
It should be noted that this equation is valid only in those regions of the sample where the electric field is nonzero. In other regions, where E = 0, the distribution of the magnetic induction B(x, t) turns out to be frozen. It keeps the same shape since the last moment of prehistory when E ~ 0. Boundary conditions for Eq. (6) are
B( d, t ) = B ( - d ,
t ) = IxH(t).
(7)
2.2. Origin of jumps in the E(t) dependence 2.
Theory
Let us follow the evolution of the distribution
B(x, t) inside the plate. We start from the moment t = t o when the magnetic field H(t) at the sample
2.1. Problem statement Let us consider a superconducting plane-parallel plate of thickness 2 d placed in an external AC magnetic field directed along the plate surfaces (along the z-axis) and having the form
H(t) = H 1 cos(toll ) + H 2 COS(WEt+X).
(2)
The x-axis is normal to the plate boundaries with its origin x = 0 at the middle of the sample. The magnetic induction B(x, t) and tlectric field E(x, t) depend on a single spatial coordinate x in this
surface reaches its absolute maximum value H
F. Pgrez-Rodrlguez et al. / Physica C 251 (1995) 50-60
52
B
a) ~ TriO)
/Tmax
B
b)
pH(2) .CJm~ ......................................
5
4
-d
~
o
i
d
c) --(2)
5
5
6
6
-d 7 8
7 8
x
three regions. The electric field is nonzero within the regions ~(t) < [ x [ < d. Here the sign of the derivative OB/Ox is opposite to the one at the initial moment t = t 0. In the region I x I < ~(t), the electric field remains zero and the distribution B(x, t) retains the form it had at the moment t - - t o. The sequence of plots demonstrating the change of the distribution B(x, t) with time is shown by broken lines 1-3 in Fig. l(a). At t - - t l , the external field H(t) reaches its next extreme (minimum) value -/4- m(1) ltl" At that point of time the electric field becomes zero everywhere inside the sample and the distribution B(x, t1) takes the form shown by line 3 in Fig. l(a). We emphasize that the relationship between HmOa)x and H(~)n is such that the distribution B(x, t 1) has two symmetrically located sharp peaks. The second stage of the evolution of the distribution B(x, t) begins at the moment t = t 1 when B(x, t) has the form shown by line 3 in Fig. l(b) (this line coincides with line 3 in Fig. l(a)). This stage continues up to the point of time t = t 2 at which the external field H(t) reaches its new maximum value H~)x . We assume that this maximum value is less than the initial one, H~2)x < Hm(~x. Plots 3 - 5 in Fig. l(b) demonstrate the variation of the B(x, t) distribution within time interval t 1 _< t < t 2. As a result of the second stage two sharp peaks in the B(x, t) distribution are transformed into two zigzags (compare line 3 with 5 in Fig. l(b)). The following (third) interval t 2 ___ t < t 3 of monotonical variation of H(t) is very important in our analysis. During this time the form of the distribution B(x, t) changes from broken line 5 up to 8 in Fig. 1(c). It is important that the new minimum value H ( t 3) = Hm(3~ is assumed to be less than H ~ ). Due to this fact, there exists a time moment t = tjump, when the B(x, /jump) f u n c t i o n coincides with B(x, t 1) (see plots 7 in Fig. 1(c) and 3 in Fig. l(b)). At this moment the positions of the planes I xl = ~(tjump), dividing the regions where OB/Ot = 0 and OB/Ot ~ O, undergo a sudden change. The reason is that reference plot 5 in Fig. 1(c) has a zigzag form. The jump of ~(t) leads to a jump of the derivative of
Fig. 1. The spatial distribution of the m a g n e t i c induction B(x,t) inside the plate for different time intervals o f the m o n o t o n i c a i variation of H ( t ) : (a) t o < t < / 1 ; (b) t I _< t2; (c) t 2 ~ t _< t 3.
53
F. P~rez-Rodriguezet al./ Physica C 251 (1995)50-60
the magnetic flux with respect to time. It occurs precisely at the moment t = t j u m p when the field H(t) is passing again through the value Hm¢~),.This means that, in the present situation, a jump of the electric field E(t) should be observed at the sample surface (see Eq. (5)). 2.3. Formulas for the electric field E(t)
In order to solve the set of Eqs. (2), (5), (6), and (7) we write the magnetic-field dependence of the critical current density in the most general form j¢(B) = j o / q J ( B / B * ),
~O(0) = 1.
(8)
The function ~b(B/B *) is assumed to be even and monotonically increasing with I B]. The parameter J0 represents the critical current density at B = 0, and B* is the characteristic scale of the j¢(B) variation. Let us introduce the dimensionless quantities b I = ~H1/B* ,
(9)
0_<~<1,
~=x/d,
T= tOlt;
(10)
and dimensionless functions bs(r) = IxH( t ) / B * =
b 1 COS(T) q- b 2 cos( w2~'/to I + X),
b ( s¢, r ) = B ( x , t ) / B * , (11)
F('r) = (4wjo/IztOlgZ)E( t).
Then Eqs. (5) and (6) together with the boundary conditions (7) can be rewritten as a d
F(~') = 0b - -
O~
1
b-~~fo b(#, ,) d#, ot
= + - ~b(b) '
(1, r)=
b~(r).
(12)
Here the signs " + " or . . . . . are given by sign F(~-). After a simple but lengthy mathematical treatment of Eqs. (12), we obtain the following result for the dimensionless electric field at the surface of the superconducting plate: F('r) = ~
~[b('r)-b~('r)].
b(r) = b(~(z), z),
~ ( r ) =Yc(t)/d.
(13)
(14)
At reasonably high amplitudes of radio-waves (at sufficiently large HCm°a)~),the magnetic flux penetrates into the whole sample volume. In this case there exist time intervals when the function b(¢, r) has no break. We assume for these situations that ~(~') = 0 and b ( r ) is the dimensionless magnetic induction at the middle of the plate (b(r) = b(0, ~')). According to the analysis done in Subsection 2.2, quantities ¢(~-) and b(z) can experience sudden changes (see Fig. l(c)). It follows from Eq. (13) that the jump height AF of the electric field F ( r ) is determined by the height Ab of the jump in the b(~') dependence, A F = dbs ~(-~bzS)Ab. dr b1
b 2 = izH2/B*,
a = 4~rl~jod/cB*;
The quantity b(z) represents the value of the dimensionless magnetic induction b(¢, ~') at the point ¢ = ~(~') where the function b(¢, ~') has a break (see Fig. 1),
(15)
A way to find the function b ( r ) is clear from the analysis carried out in Subsection 2.2. It is reasonable to determine this function separately in each interval of monotonical variation of bs(r). Therefore, one should, first of all, establish these intervals. As is seen in Fig. 1, within a given interval of the monotonicity of bs(r) the quantities ~(r) and b ( r ) are defined by the intersection of two lines. One of them describes the spatial distribution of the magnetic induction b(~, z) within the region ~(r) _< < 1 where the electric field is nonzero. The equation for this distribution can easily be obtained by solving Eq. (12): bs(~) d b'~b(b') = ± a ( 1 (~,T)
¢),
~(~') < ¢ < 1. (16)
The second line represents the plot of the frozen distribution b(~, ~') inside the spatial region ~ < ~(r). In this region the induction b(~, ~-) does not depend on time and is equal to the magnetic induction b(~, Zcxt) at the initial time moment ~'ext of the considered interval of the monotonicity of bs(~') (in this time moment the external field bs reached an extreme value). In the general case, an equation for
F. PJrez-Rodrlguezet aL/ Physica C 251 (1995)50-60
54
the second part of the distribution of b( ~, r ) cannot be written because its shape is determined by entire prehistory of the action of an AC magnetic field on the superconductor. However, the equation for b(~, re~t) can be derived either analytically, or numerically for concrete dependences bs(r) and 0 ( b ) . The procedure to determine the function b ( r ) for a situation allowing for an analytical solution of the problem is shown in the Appendix. Such a procedure was applied in our numerical calculations. Let us use the results of the Appendix for b ( r ) to write the dimensionless electric field F(r) at the surface of the superconducting plate for the case of w2/oJl = 3, X = 0 with b I < 3 b 2 and b I + b 2 < Ce. Then the electric field F ( r ) is a periodic function with period 2w and has the property: F ( r - T r = - F ( r ) . If jc(B) =J0 = const (i.e. at ~b(b) = 1), all integrals in Eqs. (A.14), (A.15), (A.17) are elementary and the formulae for F(r) in the intervals of monotonical variation of bs(r) have the simple forms F(r)
-
dbs dr
r 0=0_
F(7)=
(0) bmax
1,
_ bs(r ) 2b 2 '
db s b~i)n - bs(r ) W 2-b-~ '
T 1 ~ T~___ T2 ~ " i f - - T I ;
0
i
,
got
-4
Fig. 2. Analytically calculated dependences of the dimensionless electric field F(oJt) and magnetic induction bs(OJt) at the surface of the superconducting plate for jc(B) = const, ~o2/~o I = 3, X = 0, b 1= 3, b2 = 5. The jump positions are indicated by arrows.
F(r) '
,
oriented along the sample axis. Hence, jumps of the electric field E(t) at the cylinder surface must exist as well. The corresponding expression for F(~') has the form
rjump_
F ( r ) = d b s b~m:a)x- b s ( r ) dr 2b 2
"k
dbs ~O(bs) = -d-Tr b 2
T2 -~ T_~ Tjump.
X,bs(r)db 1 +
oe
(17) --(2) . Here the quantities bs(r) , b~a)x, Uminl"(1)= _O~;ax. rl ' rj,mp are given by Eqs. (A.1), (A.6), and (A.16). The plot of F(tot) is presented in Fig. 2 for radio-wave a m p l i t u d e s b I = 3, b 2 = 5. The jump positions are indicated by arrows. In the same figure the magnetic induction bs(tot) versus time is displayed. It is seen that the behavior of bs(tot) and consequently of F(tot) are in accordance with the situation studied qualitatively in Subsection 2.2. In particular, the jumps of F ( r ) occur at moments when the field b~(r) passes through its previous extreme value. A similar evolution of the spatial distribution of the magnetic induction takes place in a superconducting cylinder if the AC magnetic field (2) is
×sign ~
.
(18)
Here we use notations of Eqs. (10) and (11) where half of the plate thickness d is replaced by the cylinder radius R, and the coordinate x plays the role of the radial coordinate. The physical meaning of the field b ( r ) is the same as in the case of a plate. Besides, the calculations and formulae for the induction b ( r ) coincide in both geometries. We do not present the expression for the height A F of the jump of the electric field because of its cumbersome form. Note that the jump of the field (18), in contrast to Eq. (15), is determined not only by the value of Ab, but also by the field b ( r ) .
F. P~rez-Rodriguezet al./ PhysicaC 251 (1995)50-60 We have demonstrated above the existence of ~dependent jumps in F(T) (i.e. in E(t)) under a number of assumptions as to the behavior of the induction bs(r) (i.e., of H(t)). However, it is clear that these suppositions are not mandatory. Two events must happen for the appearance of E(t) jumps. During one of the intervals of the monotonical variation of bs(r), the zigzag in the reference plot of b( ~, Text) should be formed. Then, within one of the following time intervals of monotonicity, the point (~(T), b(T)) should pass through this zigzag. In order for these events to take place, the following two necessary conditions must be satisfied. Above all, the magnetic induction bs(r) must have, at least, two maxima (and two minima) with distinct heights (depths). In fact, the situation where all maxima and all minima of b~(r) have respectively equal values is similar to the one-wave problem. In such a case the zigzag in b(~, rext) plot never forms and the jumps of E(t) are absent (see, for example, Refs. [23] and [5]). Let the maxima of bs(r) have distinct heights, whereas all minima are of the same value. Then the zigzag is formed, but the point (~(T), b(T)) never passes through it. The second requirement is less obvious. It should be taken into account when the AC magnetic flux penetrates into the whole sample volume. In this situation, the spatial distributions of the magnetic induction at the beginning and at the end of each bs(r) monotonicity interval could be smooth. Then inside the whole region 0 < sc < 1 the induction b( ~, rext) is described by Eq. (16) with sign " p l u s " ( " m i n u s " ) if bs(%xt) is a maximum (minimum). In this case zigzag peculiarities in the reference plot of b(~, %xt) are absent and consequently there are no F(T) jumps. Clearly, a necessary condition should exist for neighboring maximum bma x and minimum bmi, of function b~(T) in order that the spatial distribution of magnetic induction b( ~, rext) is not smooth. Namely, at least one such a pair of extrema should satisfy the following inequality:
fbm~x dbO(b) < 2 a .
(19)
brain
This inequality is easily obtained from Eq. (16). Note that the aforesaid conditions impose certain limitations on the amplitudes, frequencies and initial phase difference of the interacting waves.
55
3. Comparison with experiment
For the experimental study of the wave interaction, we used a sample of yttrium ceramics prepared by means of the standard technology from initial oxides YzO3, BaCO 3 and CuO. The cylindrical sample had the radius R = 0.26 cm and the length about 3 cm. The magnetic-field dependence of the critical current density jc(B) and the effective magnetic permeability /x were measured by the method described in Refs. [7] and [15]. It was established that for our sample
Jc(B) =
J0
(20)
1+ [B/B* 19/5'
with j0=285"2 A / c m 2, B * = 3 . 6 5 Oe and /x= 0.67. The penetration field Hp = bpB */tz was about 28 Oe. The AC magnetic field (2) was created by a solenoid wound by a copper wire. To measure a signal proportional to E(t) the monolayer pick-up coil would directly on the sample by a copper wire of 30 I~m in diameter was used. The signal was preliminary amplified by PAR-124 A working in a broadband mode. The measurement process was computer controlled, allowing for rapid data acquisition and calculations. The abrupt jumps of the electric field E(t) were observed at different parameters of the AC field (2).
10 0 -10
H
I
10 0
...... '
ol I
-10 Fig. 3. Dimensionlesselectricfield F(0)1t) and externalmagnetic field H(t) (measured in Oe) at 0)1/21r = 90 Hz, 0)2/2-n = 270 Hz, H1= 70e, H2 = 7.92 Oe, X = -0.12.
56
F. P~rez-Rodrlguez et al. / Physica C 251 (1995) 50-60
F i
.10I~[
2 N~¢'
2
.
"',
-3
I
/H;
4
i
iota
. . . . . . . . . . . . !....
~I/
\
.
-10
r"x
i,
/
\
0~,t
Fig. 4. Dimensionless electric field F( ~o1t) and external magnetic field H(D (measured in Oe) at coI/2"~ = 90 Hz, o)2/2~r = 270 Hz, H i = 8 0 e , H 2 = 10.38 Oe, X = 3.95.
Some of the experimental plots of the external magnetic field H(t) and the dimensionless electric field F(tolt) versus time are presented by solid curves in Figs. 3-7. The parameters of the AC field (2) are listed in the figure captions. The dashed curves show the numerically calculated F(tOlt) dependences at O(b) = 1 + I b l 9/5 in conformity with Eq. (20). The positions of the F(~') jumps are indicated by arrows. In accordance with the theoretical prediction, the
F 10
/\.
0
2
O)l l
0
.
5
10
o,,
10 Fig. 6. Dimensionless electric field F(tolt) free of jumps and external magnetic field H(t) (measured in Oe) at to 1/2"n = 160 Hz, to2/2"rr = 320 Hz, H 1 = 7 0 e , H 2 = 6.06 Oe, X = 0.
jumps of F(~-) occur precisely at the time when the field H(t) passes through its previous extreme value (maximum or minimum). Our experimental study confirmed the theoretical deductions related to the necessary conditions for the existence of E(t) jumps. Plots in Figs. 5, 6 are recorded for the same radio-wave frequencies, but for distinct phase differences. Curve H(t) in Fig. 5 shows the external magnetic field (2) at X * 0. So, the function H(t) has different maxima and minima that provide the jump of F(tolt). Fig. 6 demon-
E
6
-5 -10
¢ i i
10
~H
2 ~
4 ~
6
t
-10 Fig. 5. Dimensionless electric field F(tOlt) and external magnetic field H(t) (measured in Oe) at to1/2,rr = 160 Hz, to2/2"rr = 320 Hz, H 1 = 5 . 3 0 e , H 2 = 7.54 Oe, X = - 2.39.
Fig. 7. Jumpless electric field E(tolt) in arbitrary units at ca1 / 2 w = 90 Hz, to2/2"rr = 270 I-Iz, H 1 = 28 Oe, H 2 = 54.20e, g = 0.
F. P~rez-Rodrlguezet al. / Physica C 251 (1995)50-60 strates another case when X = 0 and all minima of the field H ( t ) are equal. This means that the first necessary condition is violated, and we do not observe the jumps. They are absent in Fig. 7 because the second necessary condition (19) is broken down.
Acknowledgements This work was supported by a Grant 3004 E 9306 of the Consejo Nacional de Ciencia y Tecnolog~a (CONACyT, M6xico) and National Programs in high-Tc superconductivity, projects "Collapse" (Ukraine and Russia).
57
Clearly, the number of solutions of Eq. (A.4) depends on the sign of the first term in the square bracket. Our initial concern is when b I < 3b2,
(A.5)
and Eq. (A.4) has four solutions in half-period (A.3) (see, for example, Fig. 2). This case is completely appropriate to the situation studied qualitatively in Subsection 2.2. Extreme points % < 9-x < 9-2 < r3 and corresponding extreme values of b~(9") are described by 9"0= 0,
bs(0 )
~
vh m(0) ax = b 1 + b 2"
l( bl ) 1/2 9"~=arccos7 1 - 3b2
Appendix
( b2 - b l / 3 ) 3/2
Method for calculation of the dimensionless induction b(9")
b s ( r l ) - b mm °.) = -
rz=xr-r
,,
hl/2 u2
,
bs(r2) = btmZa)x= -b~i)n;
Here we calculate the function b(9") with an example when the ratio of frequencies of interacting radio-waves is w 2 / w I = 3 with a phase difference X = 0. In other words, the dimensionless magnetic induction in Eq. (11) is rewritten in the form
Thus, the first half-period turns out to be divided into three intervals of the monotonical variation of bs(z):
bs(9") = b, cos(9") + b 2 cos(39-).
"l"1 <~ * m- 9"2 = ffl" - - T 1 ;
(A.8)
9"2 ~- 'IT - - 9"1 -~< 9" -~< T3 = 'IT.
(A.9)
(A.1)
It should be emphasized that all formulas given below are valid for plane as well as for cylindrical geometries. Note, that here the inductions bs(9-), b(9-) and the electric field F(9-) are periodic functions with period 2~r and have the properties:
b~(9--Tr) = -bs(9");
b(r-
"rr) = -b(9");
F ( 9 " - 7r) = - F ( 9 " ) .
(A.2)
Therefore, we carry out the analysis for the first half of the period 0 < 9"< "rr
(A.3)
'/'3 ~--- 9-g,
T O ~- 0 _~< T ~
bs(T3) = b~i)n= - b(m°a)x.
9-1;
(A.6)
(A.7)
Note, that the designations for extreme points and extreme values of bs(9") correspond to the ones adopted in Subsection 2.2 (see, also, Fig. 1). The behavior of the induction ~,(9") within intervals (A.7)-(A.9) depends on whether or not the AC magnetic flux penetrates into the whole sample volume. First, we consider the case when the magnetic induction at the middle of the sample is always zero. This happens if the absolute maximum b~2x is less than the dimensionless penetrating field bp,
b~)ax = b 1 + b 2 < bp.
(A.10)
only. As was mentioned in Subsection 2.3, we start our analysis by determining the intervals of the monotonical variation of bs(9"). So, we need to investigate the equation for extreme points of induction (A.1):
As follows from the critical-state equation in Eqs. (12), the value of bp is defined by the equality
sin(r)
Now, following the rule formulated in Subsection 2.3, we can obtain the induction b(9") within each
1-2~
+ cos2(9")
= 0.
(A.4)
fobpdb~b(b) = c~.
(A.11)
F. P&ez-Rodriguez et al. / Physica C 251 (1995) 50-60
58
interval (A.7)-(A.9). For interval (A.7) we have to choose the minus sign on the right-hand side of Eq. (16): fbb~(') dbkh(b') = - a ( 1 (~:,~-)
~),
~'(r) < ~:< 1. (A.12)
In accordance with Eq. (12) at bs = b~)x, the second equation takes the form
(A.13) Assume ~:= ~(r) and b(~, r) = b ( r ) in Eqs. (A.12) and (A.13) to obtain the function b(r) in an implicit form given by
(A.10) fails, we deal with another, more complex case, in which the magnetic induction b(0, r) at the middle of the sample is always nonzero. Here the first half-period ,tr is separated into several time lapses. During some of them the induction b(0, r) is constant in time. In such cases formulas (A.14), (A.15), and (A.17) defining b(r) work as before. One should choose those of them whose domains of definition cover a given time lapse. Besides, there are other time lapses when the induction b(0, r) varies in time. Then b(r) = b(0, r) and we have to take a fresh look at this function. The method for determining it in this case will be elucidated below. If inequalities (A.5) and I b g(1) in I =
( b2 - bl/3) 3/2 b~/2
< bp < b~m°~ax= b I + b e
(A.19)
f~()dbq,( b ) + fb(')dbq,( b ) = 0, m.x
bs(r )
r 0 = 0 _< r_< %.
(A.14)
Similarly, one can write the formula for b(r) within interval (A.8) such that fbS~')dbO(b) +
fb(')db~O(b)---0, b~(r )
mm
r 1 < r < r 2 = 'rr - r i
(A.15)
The jump in the b(r) dependence appears inside the last interval (A.9). It takes place when r = "/'jump and bs(Tjump)= b(mli)n, "/'jump = "rr --
arccos(l"
-- b l
"I 1/2
3b2 ]
(A.16) "
So, this interval is divided into two subintervals (before and a f t e r ~-jump)"Before the jump, the function 2(z) is defined by
~r)dbO(b) + max
f]'(~)db~(b)= 0 , bs( r )
r 2 = 'n" -- r I _< r <
rj~mp.
(A.17)
After the jump, within the subinterval "/'jump < " r < "r 3 -- 'n',
(A.18)
formula (A.14) for b(r) holds. Thus, we have described situation (A.5), (A.10) when the AC field never penetrates into the whole sample volume. If Eq. (A.5) is valid while Eq.
are satisfied, the jump in the b(r) dependence takes place as before despite the nonvanishing magnetic induction b(0, r). This situation does not differ radically from the one described above, since the jump occurs at the same time moment r = rjumv (A.16). Now we shall study the jumpless case which is realized under condition (A.5) and ( b 2 - b l / 3 ) 3/2
bp < [ b(1.,~[ =
bl/2
(A.20)
Here each interval (A.7)-(A.9) is broken into two time lapses. Thus, there are six time lapses within the half-period (A.3). We denote the time point which separates the first interval (A.7) into the first and second lapses by r[ 0. During the first lapse, r 0 = 0 _< r_< r[ l),
(A.21)
the magnetic induction b(O, r) in the middle of the sample is constant. The equations for determining b(r) are the same as within the whole interval (A.7) in case (A.10), namely, Eqs. (A.13) and (A.12). Therefore, the function b(r) is described by Eq. (A.14). The second lapse r} l) < r < r,
(A.22)
is characterized by the decrease of the induction b(0, r) with time and by the absence of any break in
F. P~rez-Rodrlguez et al. / Physica C 251 (1995) 50-60
the b(~, z) distribution. This distribution is governed by Eq. (A.12). Recall that for such situation b(T) = b(0, ~'). So, assuming ~ = 0 and b(~, T ) = b(0, z ) = b(~') in Eq. (A.12), we obtain the expression for b(T),
fb(j)db~O( b ) = a.
(A.23)
b~( )
Now it is clear that the separating time point T~° is defined by m~ d b q J ( b ) = 2 a , fbb(°)
T O ~- 0 - -
T~ l ) - ~ T 1.
(A.24) At this moment curves (A.13) and (A.12) join at the point ~ = 0. Similarly, one can write the formulas for b(T) within intervals (A.8) and (A.9). The second interval of monotonicity (A.8) is broken into the third and fourth lapses by time point T20). This moment is defined by
fb,(~") b(ral)ndb~b ( b ) = - 2 a ,
T 1 -~< T2(1) _~ T 2 .
(A.25) Since the magnetic induction b(0, T) is constant during the third lapse, Tl _< T < T20),
(A.26)
the function b(~-) is given by Eq. (A.15). The implicit expression for b(T) = b(0, T) inside the fourth lapse is
fb(')db~
(b)b,(,) =
_
a
,
~'z°)_< T<_ T2.
(A.27)
T h e time-point r3°) which separates the third interval (A.9) into the fifth and sixth lapses is defined by fbb(2) max
d b $ ( b ) = 2or,
T2 < ~.~l) < T3"
(A.28)
Formula (A.17) for the induction b(~') works within the fifth lapse z 2 _< r _< ~'30),
T~ J) _< T < T 3 .
At last, let us consider the simplest case when inequality (A.5) is inverted, i.e. 3b 2 < b 1. (A.31) Under this condition, Eq. (A.4) has only two solutions within the half-period (A.3): r 0 = 0 and ~'3 = ~r (see Eq. (A.6)). Consequently, the first half-period represents just one interval of the monotonical variation of bs(z). If the magnetic induction b(0, T) at the middle of the sample is zero, i.e. b(°)max= bl + bz < bp (see Eq. (A.10)), the function b(T) is described by Eq. (A.14). In the opposite situation, when bp < btm°~)x and the magnetic flux penetrates onto the whole sample volume, the half-period (A.3) is divided into two lapses by the time point T~ 1) (see Eq. (A.24)). The induction b(z) is defined by Eq. (A.14) and by Eq. (A.23) within the first (A.21) and the second (r~ l) < T < ~'3 = 'rr) lapse, respectively. The present analysis illustrates two necessary conditions for the existence of the jumps in the b ( r ) dependence formulated in Subsection 2.3. The first of them is violated if Eq. (A.31) holds, because all maxima and all minima of bs(~-) have, respectively, equal values. So, the jumps of b(T) and of E(t) are absent in this case. On the contrary, this condition is satisfied in other cases when Eq. (A.5) holds. However, the jumps take place only under inequalities (A.5) and (A.10). This is because the second necessary condition (19) is broken in case Eqs. (A.5) and (A.20) prevail. Eq. (19), for b(m~i),and -,~ax h(2) -(h(1)-m,.= --b~a)x) of our example, corresponds to
I b~i)n I < bp.
(A.30)
(A.32)
The violation of Eq. (A.32) leads to the absence of zigzag peculiarities in the spatial distribution of the magnetic induction b(~, %xt) at the beginning of each interval of the monotonical variation of bs(T). Therefore, the jumps do not appear. In this Appendix we have studied a reasonably simple example when the ratio of frequencies of interacting radio-waves is (-02//0) 1 = 3 with phase difference X = 0. However, more complex examples, when to2/to I is integer and X ~ 0, can be analyzed analogously.
(A.29)
whereas Eq. (A.23) is valid for the sixth lapse
59
References [1] E.H. Bran&, Phys. Rev. B 34, (1986) 6514. [2] W. Tomasch et al., Phys. Rev. B 37 (1988) 9864.
60
F. Pdrez-Rodriguez et al. / Physica C 251 (1995) 50-60
[3] H. Dersch and G. Blatter, Phys. Rev. B 38 (1988) 11391. [4] K.-H. Miiller and A.J. Pauza, Physica C 161 (1989) 319. [5] L.M. Fisher, N.V. ll'in, N.M. Makarov, I.F. Voloshin and V.A. Yampol'skii, Solid State Commun. 73 (1990) 691. [6] I.V. Baltaga, N.M. Makarov, V.A. Yampol'skii, L.M. Fisher, N.V. ll'in and I.F. Voloshin, Phys. Lett. A 148 (1990) 213. [7] L.M. Fisher, O.I. Lyubimov, N.M. Makarov, I.F. Voloshin, V.A. Yampol'skii and N.V. II'in, Solid State Commun. 76 (1990) 141. [8] M.W. Coffey and J.R. Clem, Phys. Rev. Lett. 67 (1991) 318. [9] M.W. Coffey and J.R. Clem, Phys. Rev. B 45 (1992) 14662. [10] M.W. Coffey and J.R. Clem, Phys. Rev. B 46 (1992) 11757. [11] M.W. Coffey, Phys. Rev. B 46 (1992) 567. [12] E.B. Sonin, A.K. Tagantsev and K.B. Traito, Phys. Rev. B 46 (1992) 5830. [13] T. Hocquet, P. Mathien and Y. Simon, Phys. Rev. B 46 (1992) 1061. [14] L.M. Fisher, N.V. ll'in, I.F. Voioshin, I.V. Baltaga, N.M. Makarov and V.A. Yampol'skii, Physica C 197 (1992) 161.
[15] L.M. Fisher, V.S. Gorbachev, N.M. Makarov, I.F. Voloshin, V.A. Yampol'skii, N.V. II'in, R.L. Snyder, J.A.T. Taylor, V.W.R. Amarakoon, M.A. Rodriguez, S.T. Misture, D.P. Matheis, A.M.M. Barus and J.G. Fagan, Phys. Rev. B 46 (1992) 10986. [16] L.M. Fisher, N.V. ll'in, I.F. Voloshin, N.M. Makarov, V.A. Yampol'skii, F. Perez Rodriguez and R.L. Snyder, Physica C 206 (1993) 195. [17] J. Wosik, L.M. Xie, R. Chau, A. Samaan and J.C. Wolf, Phys. Rev. B 47 (1993) 8968. [18] M.W. Coffey and J.R. Clem, Phys. Rev. B 48 (1993) 342. [19] P.P. Nguyen, D.E. Oates, G. Dresselhaus and M.S. Dresselhaus, Phys. Rev. B 48 (1993) 6400. [20] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [21] H. London, Phys. Lett. 6 (1963) 162. [22] I.V. Baltaga, K.V. ll'enko, N.M. Makarov, V.A. Yampol'skii, L.M. Fisher, A.V. Kalinov, I.F. Voloshin and M. Pinsky, Solid State Commun. 93 (1995) 697. [23] H.A. Ulmaier, Phys. Status Solidi 17 (1966) 631.
Physica C 251 (1995) 50-60
Interaction of electromagnetic waves in hard superconductors F. P&ez-Rodriguez a,* I.V. Baltaga b, K.V. II'enko b N.M. Makarov b V.A. Yampol'skii b, L.M. Fisher c, A.V. Kalinov c, I.F. Voloshin c a Instituto de Fisica, UniversidadAut6noma de Puebla, Apdo. Post. J-48, Puebla, Pue 72570, Mexico b Institute for Radiophysics and Electronics, Ukrainian NationalAcademy of Sciences, 12 Acad. Proskura St., 310085 Kharkov, Ukraine c All-Russian Electrical Engineering Institute, 12 Kransnokazarmennaya St., 111250 Moscow, Russian Federation Received 23 March 1995
Abstract
Nonlinear interaction of low-frequency radio-waves in hard superconductors is studied both theoretically and experimentally. It is shown that this interaction gives rise to a new nonlinear phenomenon, namely, jumps in the temporal dependence of the electric field at the surface of the sample. Heights and positions of the jumps are determined by the amplitudes, frequencies, and phase difference of the interacting waves. Necessary conditions for the existence of the jumps are discussed. The calculations are carried out in the framework of a critical-state model with an arbitrary magnetic field dependence of the critical current density. Experimental results are found in good agreement with theoretical ones.
1. Introduction
The nonlinearity in the material equation, which relates the current density to the electromagnetic field, is responsible for the violation of the superposition principle and leads to the interaction of electromagnetic waves in conducting media. As such, the characteristic properties of the nonlinear mechanism manifest themselves as features of such interaction. In view of this, the study of nonlinear radio-wave interaction in hard superconductors is of great interest. In recent years the electrodynamics of hard superconductors has drawn attention of many research groups (see Refs. [1-19] and references therein). The electromagnetic properties of such materials are de-
* Corresponding author.
scribed by the critical-state model [20,21] in a rather wide interval of wave amplitudes and frequencies (see, for example, Ref. [16]), where the pinning of the magnetic flux inside a superconductor plays a governing role. In this model, the Maxwell equation specifying the distribution of magnetic induction B has the following form: 4rr curl B = - - j c ( B ) c
E IEI
.
(1)
Here E is the electric field, jc(B) is the critical current density, c is the speed of light in vacuum. The critical-state equation is essentially nonlinear. The nonlinearity in the right-hand side of Eq. (11) is stipulated by the dependence jc(B), as well as by the factor E / I E I. The nonlinearity associated with the latter is of principal interest, since it does not occur in any other nonlinear media. Because of this nonlinearity, unusual effects are observed in hard supercon-
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 3 8 5 - 1
F. P~rez-Rodrlguezet al./ PhysicaC 251 (1995)50-60 ductors (see, for example, Refs. [6] and [14]). As is shown below, this nonlinearity gives rise to a peculiar phenomenon in the radio-wave interaction, namely, jumps in the electric field at the sample surface versus time. A brief report on this effect was published in Ref. [22]. High-Tc ceramics being in the critical state represent a convenient object to observe nonlinear electrodynamic effects. Therefore, we shall carry out theoretical calculations for high-Tc ceramics. As was shown in Ref. [3], these specific nonlinear Josephson media can be described by an equation similar to Eq. (11). It contains in the right-hand side an additional factor/z, representing the effective magnetic permeability (0 < / x < 1). The parameter /x allows for intragranular currents preventing magnetic flux penetrating into the grains. The results for traditional hard superconductors correspond to /x = 1. Section 2 is devoted to the statement (Subsection 2.1) and theoretical study of the problem. In Subsection 2.2 we discuss the origin of jumps in the temporal dependence of the electric field E(t) at the surface of a superconducting plate, caused by the radio-wave interaction. Subsection 2.3 contains formulas for the function E(t) with an arbitrary dependence j¢(B). Results for two usual geometries (plate and cylinder) are presented. Besides, some necessary conditions for the existence of E(t) jumps are formulated therein. In Section 3 a comparison of the theoretical curves for E(t) with data, obtained in experiments with yttrium ceramic sample, is done. The paper contains an Appendix.
51
geometry. The vector B contains z component and the vector E has y component only:
B(x, t ) =
{0, 0,
B(x,
t)};
E( x, t) = {0, E( x, t), 0}.
(3)
In this situation B(x, t) is an even function of x, whereas E(x, t) is an odd one. Our goal is to calculate the electric field at the boundary x = d, which we define as E(t),
E( t) = E( d, t) = - E ( - d ,
t).
(4)
It is related to the derivative of the magnetic flux per unit plate width with respect to time by the equation 1 dqb
E(t)=
2c d t '
d
~ = f _ d B ( X , t) dx.
(5)
In this geometry, the critical-state equation for ceramics can be written as follows: 0B
- O---x=
4 xr/z
c Jc(B) sign E.
(6)
It should be noted that this equation is valid only in those regions of the sample where the electric field is nonzero. In other regions, where E = 0, the distribution of the magnetic induction B(x, t) turns out to be frozen. It keeps the same shape since the last moment of prehistory when E ~ 0. Boundary conditions for Eq. (6) are
B( d, t ) = B ( - d ,
t ) = IxH(t).
(7)
2.2. Origin of jumps in the E(t) dependence 2.
Theory
Let us follow the evolution of the distribution
B(x, t) inside the plate. We start from the moment t = t o when the magnetic field H(t) at the sample
2.1. Problem statement Let us consider a superconducting plane-parallel plate of thickness 2 d placed in an external AC magnetic field directed along the plate surfaces (along the z-axis) and having the form
H(t) = H 1 cos(toll ) + H 2 COS(WEt+X).
(2)
The x-axis is normal to the plate boundaries with its origin x = 0 at the middle of the sample. The magnetic induction B(x, t) and tlectric field E(x, t) depend on a single spatial coordinate x in this
surface reaches its absolute maximum value H
F. Pgrez-Rodrlguez et al. / Physica C 251 (1995) 50-60
52
B
a) ~ TriO)
/Tmax
B
b)
pH(2) .CJm~ ......................................
5
4
-d
~
o
i
d
c) --(2)
5
5
6
6
-d 7 8
7 8
x
three regions. The electric field is nonzero within the regions ~(t) < [ x [ < d. Here the sign of the derivative OB/Ox is opposite to the one at the initial moment t = t 0. In the region I x I < ~(t), the electric field remains zero and the distribution B(x, t) retains the form it had at the moment t - - t o. The sequence of plots demonstrating the change of the distribution B(x, t) with time is shown by broken lines 1-3 in Fig. l(a). At t - - t l , the external field H(t) reaches its next extreme (minimum) value -/4- m(1) ltl" At that point of time the electric field becomes zero everywhere inside the sample and the distribution B(x, t1) takes the form shown by line 3 in Fig. l(a). We emphasize that the relationship between HmOa)x and H(~)n is such that the distribution B(x, t 1) has two symmetrically located sharp peaks. The second stage of the evolution of the distribution B(x, t) begins at the moment t = t 1 when B(x, t) has the form shown by line 3 in Fig. l(b) (this line coincides with line 3 in Fig. l(a)). This stage continues up to the point of time t = t 2 at which the external field H(t) reaches its new maximum value H~)x . We assume that this maximum value is less than the initial one, H~2)x < Hm(~x. Plots 3 - 5 in Fig. l(b) demonstrate the variation of the B(x, t) distribution within time interval t 1 _< t < t 2. As a result of the second stage two sharp peaks in the B(x, t) distribution are transformed into two zigzags (compare line 3 with 5 in Fig. l(b)). The following (third) interval t 2 ___ t < t 3 of monotonical variation of H(t) is very important in our analysis. During this time the form of the distribution B(x, t) changes from broken line 5 up to 8 in Fig. 1(c). It is important that the new minimum value H ( t 3) = Hm(3~ is assumed to be less than H ~ ). Due to this fact, there exists a time moment t = tjump, when the B(x, /jump) f u n c t i o n coincides with B(x, t 1) (see plots 7 in Fig. 1(c) and 3 in Fig. l(b)). At this moment the positions of the planes I xl = ~(tjump), dividing the regions where OB/Ot = 0 and OB/Ot ~ O, undergo a sudden change. The reason is that reference plot 5 in Fig. 1(c) has a zigzag form. The jump of ~(t) leads to a jump of the derivative of
Fig. 1. The spatial distribution of the m a g n e t i c induction B(x,t) inside the plate for different time intervals o f the m o n o t o n i c a i variation of H ( t ) : (a) t o < t < / 1 ; (b) t I _< t2; (c) t 2 ~ t _< t 3.
53
F. P~rez-Rodriguezet al./ Physica C 251 (1995)50-60
the magnetic flux with respect to time. It occurs precisely at the moment t = t j u m p when the field H(t) is passing again through the value Hm¢~),.This means that, in the present situation, a jump of the electric field E(t) should be observed at the sample surface (see Eq. (5)). 2.3. Formulas for the electric field E(t)
In order to solve the set of Eqs. (2), (5), (6), and (7) we write the magnetic-field dependence of the critical current density in the most general form j¢(B) = j o / q J ( B / B * ),
~O(0) = 1.
(8)
The function ~b(B/B *) is assumed to be even and monotonically increasing with I B]. The parameter J0 represents the critical current density at B = 0, and B* is the characteristic scale of the j¢(B) variation. Let us introduce the dimensionless quantities b I = ~H1/B* ,
(9)
0_<~<1,
~=x/d,
T= tOlt;
(10)
and dimensionless functions bs(r) = IxH( t ) / B * =
b 1 COS(T) q- b 2 cos( w2~'/to I + X),
b ( s¢, r ) = B ( x , t ) / B * , (11)
F('r) = (4wjo/IztOlgZ)E( t).
Then Eqs. (5) and (6) together with the boundary conditions (7) can be rewritten as a d
F(~') = 0b - -
O~
1
b-~~fo b(#, ,) d#, ot
= + - ~b(b) '
(1, r)=
b~(r).
(12)
Here the signs " + " or . . . . . are given by sign F(~-). After a simple but lengthy mathematical treatment of Eqs. (12), we obtain the following result for the dimensionless electric field at the surface of the superconducting plate: F('r) = ~
~[b('r)-b~('r)].
b(r) = b(~(z), z),
~ ( r ) =Yc(t)/d.
(13)
(14)
At reasonably high amplitudes of radio-waves (at sufficiently large HCm°a)~),the magnetic flux penetrates into the whole sample volume. In this case there exist time intervals when the function b(¢, r) has no break. We assume for these situations that ~(~') = 0 and b ( r ) is the dimensionless magnetic induction at the middle of the plate (b(r) = b(0, ~')). According to the analysis done in Subsection 2.2, quantities ¢(~-) and b(z) can experience sudden changes (see Fig. l(c)). It follows from Eq. (13) that the jump height AF of the electric field F ( r ) is determined by the height Ab of the jump in the b(~') dependence, A F = dbs ~(-~bzS)Ab. dr b1
b 2 = izH2/B*,
a = 4~rl~jod/cB*;
The quantity b(z) represents the value of the dimensionless magnetic induction b(¢, ~') at the point ¢ = ~(~') where the function b(¢, ~') has a break (see Fig. 1),
(15)
A way to find the function b ( r ) is clear from the analysis carried out in Subsection 2.2. It is reasonable to determine this function separately in each interval of monotonical variation of bs(r). Therefore, one should, first of all, establish these intervals. As is seen in Fig. 1, within a given interval of the monotonicity of bs(r) the quantities ~(r) and b ( r ) are defined by the intersection of two lines. One of them describes the spatial distribution of the magnetic induction b(~, z) within the region ~(r) _< < 1 where the electric field is nonzero. The equation for this distribution can easily be obtained by solving Eq. (12): bs(~) d b'~b(b') = ± a ( 1 (~,T)
¢),
~(~') < ¢ < 1. (16)
The second line represents the plot of the frozen distribution b(~, ~') inside the spatial region ~ < ~(r). In this region the induction b(~, ~-) does not depend on time and is equal to the magnetic induction b(~, Zcxt) at the initial time moment ~'ext of the considered interval of the monotonicity of bs(~') (in this time moment the external field bs reached an extreme value). In the general case, an equation for
F. PJrez-Rodrlguezet aL/ Physica C 251 (1995)50-60
54
the second part of the distribution of b( ~, r ) cannot be written because its shape is determined by entire prehistory of the action of an AC magnetic field on the superconductor. However, the equation for b(~, re~t) can be derived either analytically, or numerically for concrete dependences bs(r) and 0 ( b ) . The procedure to determine the function b ( r ) for a situation allowing for an analytical solution of the problem is shown in the Appendix. Such a procedure was applied in our numerical calculations. Let us use the results of the Appendix for b ( r ) to write the dimensionless electric field F(r) at the surface of the superconducting plate for the case of w2/oJl = 3, X = 0 with b I < 3 b 2 and b I + b 2 < Ce. Then the electric field F ( r ) is a periodic function with period 2w and has the property: F ( r - T r = - F ( r ) . If jc(B) =J0 = const (i.e. at ~b(b) = 1), all integrals in Eqs. (A.14), (A.15), (A.17) are elementary and the formulae for F(r) in the intervals of monotonical variation of bs(r) have the simple forms F(r)
-
dbs dr
r 0=0_
F(7)=
(0) bmax
1,
_ bs(r ) 2b 2 '
db s b~i)n - bs(r ) W 2-b-~ '
T 1 ~ T~___ T2 ~ " i f - - T I ;
0
i
,
got
-4
Fig. 2. Analytically calculated dependences of the dimensionless electric field F(oJt) and magnetic induction bs(OJt) at the surface of the superconducting plate for jc(B) = const, ~o2/~o I = 3, X = 0, b 1= 3, b2 = 5. The jump positions are indicated by arrows.
F(r) '
,
oriented along the sample axis. Hence, jumps of the electric field E(t) at the cylinder surface must exist as well. The corresponding expression for F(~') has the form
rjump_
F ( r ) = d b s b~m:a)x- b s ( r ) dr 2b 2
"k
dbs ~O(bs) = -d-Tr b 2
T2 -~ T_~ Tjump.
X,bs(r)db 1 +
oe
(17) --(2) . Here the quantities bs(r) , b~a)x, Uminl"(1)= _O~;ax. rl ' rj,mp are given by Eqs. (A.1), (A.6), and (A.16). The plot of F(tot) is presented in Fig. 2 for radio-wave a m p l i t u d e s b I = 3, b 2 = 5. The jump positions are indicated by arrows. In the same figure the magnetic induction bs(tot) versus time is displayed. It is seen that the behavior of bs(tot) and consequently of F(tot) are in accordance with the situation studied qualitatively in Subsection 2.2. In particular, the jumps of F ( r ) occur at moments when the field b~(r) passes through its previous extreme value. A similar evolution of the spatial distribution of the magnetic induction takes place in a superconducting cylinder if the AC magnetic field (2) is
×sign ~
.
(18)
Here we use notations of Eqs. (10) and (11) where half of the plate thickness d is replaced by the cylinder radius R, and the coordinate x plays the role of the radial coordinate. The physical meaning of the field b ( r ) is the same as in the case of a plate. Besides, the calculations and formulae for the induction b ( r ) coincide in both geometries. We do not present the expression for the height A F of the jump of the electric field because of its cumbersome form. Note that the jump of the field (18), in contrast to Eq. (15), is determined not only by the value of Ab, but also by the field b ( r ) .
F. P~rez-Rodriguezet al./ PhysicaC 251 (1995)50-60 We have demonstrated above the existence of ~dependent jumps in F(T) (i.e. in E(t)) under a number of assumptions as to the behavior of the induction bs(r) (i.e., of H(t)). However, it is clear that these suppositions are not mandatory. Two events must happen for the appearance of E(t) jumps. During one of the intervals of the monotonical variation of bs(r), the zigzag in the reference plot of b( ~, Text) should be formed. Then, within one of the following time intervals of monotonicity, the point (~(T), b(T)) should pass through this zigzag. In order for these events to take place, the following two necessary conditions must be satisfied. Above all, the magnetic induction bs(r) must have, at least, two maxima (and two minima) with distinct heights (depths). In fact, the situation where all maxima and all minima of b~(r) have respectively equal values is similar to the one-wave problem. In such a case the zigzag in b(~, rext) plot never forms and the jumps of E(t) are absent (see, for example, Refs. [23] and [5]). Let the maxima of bs(r) have distinct heights, whereas all minima are of the same value. Then the zigzag is formed, but the point (~(T), b(T)) never passes through it. The second requirement is less obvious. It should be taken into account when the AC magnetic flux penetrates into the whole sample volume. In this situation, the spatial distributions of the magnetic induction at the beginning and at the end of each bs(r) monotonicity interval could be smooth. Then inside the whole region 0 < sc < 1 the induction b( ~, rext) is described by Eq. (16) with sign " p l u s " ( " m i n u s " ) if bs(%xt) is a maximum (minimum). In this case zigzag peculiarities in the reference plot of b(~, %xt) are absent and consequently there are no F(T) jumps. Clearly, a necessary condition should exist for neighboring maximum bma x and minimum bmi, of function b~(T) in order that the spatial distribution of magnetic induction b( ~, rext) is not smooth. Namely, at least one such a pair of extrema should satisfy the following inequality:
fbm~x dbO(b) < 2 a .
(19)
brain
This inequality is easily obtained from Eq. (16). Note that the aforesaid conditions impose certain limitations on the amplitudes, frequencies and initial phase difference of the interacting waves.
55
3. Comparison with experiment
For the experimental study of the wave interaction, we used a sample of yttrium ceramics prepared by means of the standard technology from initial oxides YzO3, BaCO 3 and CuO. The cylindrical sample had the radius R = 0.26 cm and the length about 3 cm. The magnetic-field dependence of the critical current density jc(B) and the effective magnetic permeability /x were measured by the method described in Refs. [7] and [15]. It was established that for our sample
Jc(B) =
J0
(20)
1+ [B/B* 19/5'
with j0=285"2 A / c m 2, B * = 3 . 6 5 Oe and /x= 0.67. The penetration field Hp = bpB */tz was about 28 Oe. The AC magnetic field (2) was created by a solenoid wound by a copper wire. To measure a signal proportional to E(t) the monolayer pick-up coil would directly on the sample by a copper wire of 30 I~m in diameter was used. The signal was preliminary amplified by PAR-124 A working in a broadband mode. The measurement process was computer controlled, allowing for rapid data acquisition and calculations. The abrupt jumps of the electric field E(t) were observed at different parameters of the AC field (2).
10 0 -10
H
I
10 0
...... '
ol I
-10 Fig. 3. Dimensionlesselectricfield F(0)1t) and externalmagnetic field H(t) (measured in Oe) at 0)1/21r = 90 Hz, 0)2/2-n = 270 Hz, H1= 70e, H2 = 7.92 Oe, X = -0.12.
56
F. P~rez-Rodrlguez et al. / Physica C 251 (1995) 50-60
F i
.10I~[
2 N~¢'
2
.
"',
-3
I
/H;
4
i
iota
. . . . . . . . . . . . !....
~I/
\
.
-10
r"x
i,
/
\
0~,t
Fig. 4. Dimensionless electric field F( ~o1t) and external magnetic field H(D (measured in Oe) at coI/2"~ = 90 Hz, o)2/2~r = 270 Hz, H i = 8 0 e , H 2 = 10.38 Oe, X = 3.95.
Some of the experimental plots of the external magnetic field H(t) and the dimensionless electric field F(tolt) versus time are presented by solid curves in Figs. 3-7. The parameters of the AC field (2) are listed in the figure captions. The dashed curves show the numerically calculated F(tOlt) dependences at O(b) = 1 + I b l 9/5 in conformity with Eq. (20). The positions of the F(~') jumps are indicated by arrows. In accordance with the theoretical prediction, the
F 10
/\.
0
2
O)l l
0
.
5
10
o,,
10 Fig. 6. Dimensionless electric field F(tolt) free of jumps and external magnetic field H(t) (measured in Oe) at to 1/2"n = 160 Hz, to2/2"rr = 320 Hz, H 1 = 7 0 e , H 2 = 6.06 Oe, X = 0.
jumps of F(~-) occur precisely at the time when the field H(t) passes through its previous extreme value (maximum or minimum). Our experimental study confirmed the theoretical deductions related to the necessary conditions for the existence of E(t) jumps. Plots in Figs. 5, 6 are recorded for the same radio-wave frequencies, but for distinct phase differences. Curve H(t) in Fig. 5 shows the external magnetic field (2) at X * 0. So, the function H(t) has different maxima and minima that provide the jump of F(tolt). Fig. 6 demon-
E
6
-5 -10
¢ i i
10
~H
2 ~
4 ~
6
t
-10 Fig. 5. Dimensionless electric field F(tOlt) and external magnetic field H(t) (measured in Oe) at to1/2,rr = 160 Hz, to2/2"rr = 320 Hz, H 1 = 5 . 3 0 e , H 2 = 7.54 Oe, X = - 2.39.
Fig. 7. Jumpless electric field E(tolt) in arbitrary units at ca1 / 2 w = 90 Hz, to2/2"rr = 270 I-Iz, H 1 = 28 Oe, H 2 = 54.20e, g = 0.
F. P~rez-Rodrlguezet al. / Physica C 251 (1995)50-60 strates another case when X = 0 and all minima of the field H ( t ) are equal. This means that the first necessary condition is violated, and we do not observe the jumps. They are absent in Fig. 7 because the second necessary condition (19) is broken down.
Acknowledgements This work was supported by a Grant 3004 E 9306 of the Consejo Nacional de Ciencia y Tecnolog~a (CONACyT, M6xico) and National Programs in high-Tc superconductivity, projects "Collapse" (Ukraine and Russia).
57
Clearly, the number of solutions of Eq. (A.4) depends on the sign of the first term in the square bracket. Our initial concern is when b I < 3b2,
(A.5)
and Eq. (A.4) has four solutions in half-period (A.3) (see, for example, Fig. 2). This case is completely appropriate to the situation studied qualitatively in Subsection 2.2. Extreme points % < 9-x < 9-2 < r3 and corresponding extreme values of b~(9") are described by 9"0= 0,
bs(0 )
~
vh m(0) ax = b 1 + b 2"
l( bl ) 1/2 9"~=arccos7 1 - 3b2
Appendix
( b2 - b l / 3 ) 3/2
Method for calculation of the dimensionless induction b(9")
b s ( r l ) - b mm °.) = -
rz=xr-r
,,
hl/2 u2
,
bs(r2) = btmZa)x= -b~i)n;
Here we calculate the function b(9") with an example when the ratio of frequencies of interacting radio-waves is w 2 / w I = 3 with a phase difference X = 0. In other words, the dimensionless magnetic induction in Eq. (11) is rewritten in the form
Thus, the first half-period turns out to be divided into three intervals of the monotonical variation of bs(z):
bs(9") = b, cos(9") + b 2 cos(39-).
"l"1 <~ * m- 9"2 = ffl" - - T 1 ;
(A.8)
9"2 ~- 'IT - - 9"1 -~< 9" -~< T3 = 'IT.
(A.9)
(A.1)
It should be emphasized that all formulas given below are valid for plane as well as for cylindrical geometries. Note, that here the inductions bs(9-), b(9-) and the electric field F(9-) are periodic functions with period 2~r and have the properties:
b~(9--Tr) = -bs(9");
b(r-
"rr) = -b(9");
F ( 9 " - 7r) = - F ( 9 " ) .
(A.2)
Therefore, we carry out the analysis for the first half of the period 0 < 9"< "rr
(A.3)
'/'3 ~--- 9-g,
T O ~- 0 _~< T ~
bs(T3) = b~i)n= - b(m°a)x.
9-1;
(A.6)
(A.7)
Note, that the designations for extreme points and extreme values of bs(9") correspond to the ones adopted in Subsection 2.2 (see, also, Fig. 1). The behavior of the induction ~,(9") within intervals (A.7)-(A.9) depends on whether or not the AC magnetic flux penetrates into the whole sample volume. First, we consider the case when the magnetic induction at the middle of the sample is always zero. This happens if the absolute maximum b~2x is less than the dimensionless penetrating field bp,
b~)ax = b 1 + b 2 < bp.
(A.10)
only. As was mentioned in Subsection 2.3, we start our analysis by determining the intervals of the monotonical variation of bs(9"). So, we need to investigate the equation for extreme points of induction (A.1):
As follows from the critical-state equation in Eqs. (12), the value of bp is defined by the equality
sin(r)
Now, following the rule formulated in Subsection 2.3, we can obtain the induction b(9") within each
1-2~
+ cos2(9")
= 0.
(A.4)
fobpdb~b(b) = c~.
(A.11)
F. P&ez-Rodriguez et al. / Physica C 251 (1995) 50-60
58
interval (A.7)-(A.9). For interval (A.7) we have to choose the minus sign on the right-hand side of Eq. (16): fbb~(') dbkh(b') = - a ( 1 (~:,~-)
~),
~'(r) < ~:< 1. (A.12)
In accordance with Eq. (12) at bs = b~)x, the second equation takes the form
(A.13) Assume ~:= ~(r) and b(~, r) = b ( r ) in Eqs. (A.12) and (A.13) to obtain the function b(r) in an implicit form given by
(A.10) fails, we deal with another, more complex case, in which the magnetic induction b(0, r) at the middle of the sample is always nonzero. Here the first half-period ,tr is separated into several time lapses. During some of them the induction b(0, r) is constant in time. In such cases formulas (A.14), (A.15), and (A.17) defining b(r) work as before. One should choose those of them whose domains of definition cover a given time lapse. Besides, there are other time lapses when the induction b(0, r) varies in time. Then b(r) = b(0, r) and we have to take a fresh look at this function. The method for determining it in this case will be elucidated below. If inequalities (A.5) and I b g(1) in I =
( b2 - bl/3) 3/2 b~/2
< bp < b~m°~ax= b I + b e
(A.19)
f~()dbq,( b ) + fb(')dbq,( b ) = 0, m.x
bs(r )
r 0 = 0 _< r_< %.
(A.14)
Similarly, one can write the formula for b(r) within interval (A.8) such that fbS~')dbO(b) +
fb(')db~O(b)---0, b~(r )
mm
r 1 < r < r 2 = 'rr - r i
(A.15)
The jump in the b(r) dependence appears inside the last interval (A.9). It takes place when r = "/'jump and bs(Tjump)= b(mli)n, "/'jump = "rr --
arccos(l"
-- b l
"I 1/2
3b2 ]
(A.16) "
So, this interval is divided into two subintervals (before and a f t e r ~-jump)"Before the jump, the function 2(z) is defined by
~r)dbO(b) + max
f]'(~)db~(b)= 0 , bs( r )
r 2 = 'n" -- r I _< r <
rj~mp.
(A.17)
After the jump, within the subinterval "/'jump < " r < "r 3 -- 'n',
(A.18)
formula (A.14) for b(r) holds. Thus, we have described situation (A.5), (A.10) when the AC field never penetrates into the whole sample volume. If Eq. (A.5) is valid while Eq.
are satisfied, the jump in the b(r) dependence takes place as before despite the nonvanishing magnetic induction b(0, r). This situation does not differ radically from the one described above, since the jump occurs at the same time moment r = rjumv (A.16). Now we shall study the jumpless case which is realized under condition (A.5) and ( b 2 - b l / 3 ) 3/2
bp < [ b(1.,~[ =
bl/2
(A.20)
Here each interval (A.7)-(A.9) is broken into two time lapses. Thus, there are six time lapses within the half-period (A.3). We denote the time point which separates the first interval (A.7) into the first and second lapses by r[ 0. During the first lapse, r 0 = 0 _< r_< r[ l),
(A.21)
the magnetic induction b(O, r) in the middle of the sample is constant. The equations for determining b(r) are the same as within the whole interval (A.7) in case (A.10), namely, Eqs. (A.13) and (A.12). Therefore, the function b(r) is described by Eq. (A.14). The second lapse r} l) < r < r,
(A.22)
is characterized by the decrease of the induction b(0, r) with time and by the absence of any break in
F. P~rez-Rodrlguez et al. / Physica C 251 (1995) 50-60
the b(~, z) distribution. This distribution is governed by Eq. (A.12). Recall that for such situation b(T) = b(0, ~'). So, assuming ~ = 0 and b(~, T ) = b(0, z ) = b(~') in Eq. (A.12), we obtain the expression for b(T),
fb(j)db~O( b ) = a.
(A.23)
b~( )
Now it is clear that the separating time point T~° is defined by m~ d b q J ( b ) = 2 a , fbb(°)
T O ~- 0 - -
T~ l ) - ~ T 1.
(A.24) At this moment curves (A.13) and (A.12) join at the point ~ = 0. Similarly, one can write the formulas for b(T) within intervals (A.8) and (A.9). The second interval of monotonicity (A.8) is broken into the third and fourth lapses by time point T20). This moment is defined by
fb,(~") b(ral)ndb~b ( b ) = - 2 a ,
T 1 -~< T2(1) _~ T 2 .
(A.25) Since the magnetic induction b(0, T) is constant during the third lapse, Tl _< T < T20),
(A.26)
the function b(~-) is given by Eq. (A.15). The implicit expression for b(T) = b(0, T) inside the fourth lapse is
fb(')db~
(b)b,(,) =
_
a
,
~'z°)_< T<_ T2.
(A.27)
T h e time-point r3°) which separates the third interval (A.9) into the fifth and sixth lapses is defined by fbb(2) max
d b $ ( b ) = 2or,
T2 < ~.~l) < T3"
(A.28)
Formula (A.17) for the induction b(~') works within the fifth lapse z 2 _< r _< ~'30),
T~ J) _< T < T 3 .
At last, let us consider the simplest case when inequality (A.5) is inverted, i.e. 3b 2 < b 1. (A.31) Under this condition, Eq. (A.4) has only two solutions within the half-period (A.3): r 0 = 0 and ~'3 = ~r (see Eq. (A.6)). Consequently, the first half-period represents just one interval of the monotonical variation of bs(z). If the magnetic induction b(0, T) at the middle of the sample is zero, i.e. b(°)max= bl + bz < bp (see Eq. (A.10)), the function b(T) is described by Eq. (A.14). In the opposite situation, when bp < btm°~)x and the magnetic flux penetrates onto the whole sample volume, the half-period (A.3) is divided into two lapses by the time point T~ 1) (see Eq. (A.24)). The induction b(z) is defined by Eq. (A.14) and by Eq. (A.23) within the first (A.21) and the second (r~ l) < T < ~'3 = 'rr) lapse, respectively. The present analysis illustrates two necessary conditions for the existence of the jumps in the b ( r ) dependence formulated in Subsection 2.3. The first of them is violated if Eq. (A.31) holds, because all maxima and all minima of bs(~-) have, respectively, equal values. So, the jumps of b(T) and of E(t) are absent in this case. On the contrary, this condition is satisfied in other cases when Eq. (A.5) holds. However, the jumps take place only under inequalities (A.5) and (A.10). This is because the second necessary condition (19) is broken in case Eqs. (A.5) and (A.20) prevail. Eq. (19), for b(m~i),and -,~ax h(2) -(h(1)-m,.= --b~a)x) of our example, corresponds to
I b~i)n I < bp.
(A.30)
(A.32)
The violation of Eq. (A.32) leads to the absence of zigzag peculiarities in the spatial distribution of the magnetic induction b(~, %xt) at the beginning of each interval of the monotonical variation of bs(T). Therefore, the jumps do not appear. In this Appendix we have studied a reasonably simple example when the ratio of frequencies of interacting radio-waves is (-02//0) 1 = 3 with phase difference X = 0. However, more complex examples, when to2/to I is integer and X ~ 0, can be analyzed analogously.
(A.29)
whereas Eq. (A.23) is valid for the sixth lapse
59
References [1] E.H. Bran&, Phys. Rev. B 34, (1986) 6514. [2] W. Tomasch et al., Phys. Rev. B 37 (1988) 9864.
60
F. Pdrez-Rodriguez et al. / Physica C 251 (1995) 50-60
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