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Analytic calculation of the nonzero fast wave reflection coefficient from the tunneling equation with absorption C. S. Ng and D. G. Swanson Physics Department, Auburn University, Auburn, Alabama 36849 (Received 17 September 1993; accepted 16 November 1993) An analytic series expression of the nonzero fast wave reflection coefficient has been found from the tunneling equation which models the wave propagation in weakly inhomogeneous media near a back-to-back resonance-cutoff pair. Explicit calculations are done for both second ion cyclotron and second electron cyclotron harmonic cases. Numerical values of.this expression are compared to those calculated by solving an integral equation iteratively. They, agree well with each other whenever both the series and the iteration converge. This result is based on a new method of evaluating integrals derived from the integral equation without having to solve the equation itself. This new method may have more applications and deserves further research and development. 1.INTRODUCTION Waves propagating in weakly inhomogeneous media through a back-to-back resonance-cutoff region, in general, experience effects of transmission, reflection, and mode conversion with or without absorption. These phenomena converslon ~ ~ ~ ~ ~ ~ ~ ~~~icl can be modeled by different kinds of high order ordinary these is a fourth order equation of the form (e.g., Refs. 14) f lD+) 2Z(fnr +f) (I) + yf =° with A and y being real constants. We call this the tunneling equation without absorption. This equation can model many different physical situations. We will explicitly consider two of them: (i) second ion cyclotron harmonic with y>-1; (ii) second electron cyclotron harmonic with r <-1. We can generalize (1) to include localized absorption:3 .5 4' !l)A2Z(4/44j) +yr=htz)(' ++), (2) where h (z) is the absorption function which generally must fall off at least as fast as z- I as Iz } oo0.Analytic questions athe scattering parameters i.e Analtic expressions for the scattering parameters, i.e., transmission, reflection, and mode conversion coefficients, for Eq. (1) have long been known. Solutions for Eq. (2), and thus scattering parameters, can be found by solving an integral equation iteratively, making use of the numerical solutions of Eq. (1). These scattering parameters are expressed in terms of integrals involving h (z), solutions of Eqs. (1) and (2) along the z-axis. It has, been pointed out that odd order derivatives should be added to Eqs. (1) and (2), with a much more complicated expression in the right hand side of Eq. (2), in order to conserve energy. 3 However, from previous numerical results the scattering parameters found by both sets of equations are close, especially for the reflection coefficient R2 ? So we will use the simpler equations for our analysis. We believe that the method we develop here should also work for the energy conserving equations, since the two sets of equations are very similar. Phys. Plasmas 1 (4), April 1994 Recently, by extending solutions of Eqs. (I) and (2) into the complex z-plane, 6some of these integrals were shown to be zero identically. Thus, fast wave transmission official entsa al To each othe promote sides and the fast wave reflection coefficient (which is idenzero fro theou side which encounters resonance typically zero) from the side which encounters the resonance before the cutoff, were shown to be independent of absorption. However, analytic expressions for other scattering parameters which do change with absorption have been unknown so far. Here, we go further along this direction and develop a new method to calculate some of those nonzero integrals analytically. We note that the above method is not the only approach used to solve the mode conversion problem. Other methods have been developed, including, e.g., direct numerical integration, 7 finite element, 8 finite difference,9 phase space methods and order reduction.' 1 -1 4 Partial"3 or approximate expressions1 1 4 of the scattering parameters have also been reported. However, we will concentrate on the above integral equation method of solving Eq. (2) here. We emphasize that the analytical expression for the relcinofiintgvnheiseatntesnetato reflection coefficient given here is exact in the sense that no further approximation is used after we accept Eqs. (1) and () hc r nagnrlmteaia omta a (2), which are in a general mathematical form that may represent many other physical situations. In all other semianalytic methods, additional approximations are made before evaluating the reflection coefficient. We have been successful in calculating the other fast wave reflection coefficient R2 , which is nonzero, for the above two cases (i) and (ii). It seems that this method, with further development, may also be useful in calculating other scattering parameters or integrals of similar nature for other problems. In the next section, we will review the integral equation method and thus the integral expressions of the scattering parameters. In Sec. III, we will discuss the basic ideas of this method and show explicitly how to calculate R 2 analytically for the two cases. We will compare results from this method to those calculated by the integral equation method in Sec. IV. 1070-664X/94/1(4)/615/7/$6.00 y 1994 American Institute of Physics - 915 Downloaded 09 Aug 2001 to 128.255.34.168. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp i (k 3 k g(k) =-ikc+- ,+)p+ z 11.INTEGRAL EQUATION Equation (I) can be solved exactly by using the method of Laplace: 4 (i-a)ln(k+lI) + (i+a)ln(k-1) fj(z)= Cfr [expzg(k)] dk, where a = (I+ y)/2A2. With a suitable choice of constants ci, we can express the asymptotic behavior of fj (note that the definitions in this paper are in general different from those in previous literatures 4 ) as where the rj are contours in the complex k-plane which end at infinity with approach angles of -r/6, 51r/6, or 3ir/2, and ( 0 {1 R 2 0 o f,\ T, 0 c, \f+\ 0 C2 4 f C3 2 0 C34 a+ f C42 I R4 a_- f4 ( - 0-z } (3) j=1,2,3,4, Z- 0° (fzZ > RI I C13 T2 0 C2 3 C3 1 0 R3 I S-I. C4 1 0 C43 (4) for y> -1, and R 1 0 C:4 f_ T2 0 0 C24 f+ C31 0 0 C34 C4 , 0 1 R4 -c0z fz Z- 00 U+ \ a/ 0 T, C13 I R2 C23 0 C32 R3 0 C42 C43 I S (5) for y < -1, with scattering parameter given by T, C13 c14 0 g -t -T2 Rz C23 C24 -S(°) g e2 gg 31 C3 2 R3 C34 _ - g g C41 C4 2 C4 3 R4 0 RI Su=C whereg=exp(-7), q = rTaj, I r-4 -ire-n/2 f '- =>r()-epI=I ii|j+a i,:id~ ~ si=sgn(a), ,4, 4 2 (6) g2 g g 1/21 =I-g 2 , and 1 In 2+z+cr In jzj | zil n 3 IzI _ - Fj(y)h(y)PIk(Y)dY, (8) f 0 -=f 3 -gf,, for y>-1, fo-8g(f 2 +gf 3 ), f,/g, for y < -1. Equation (7) can be solved iteratively using bk=fk as the first trial functions with fk calculated numerically by Eq. (3). This can be done for the three physical solutions, k = 1,2,3, if h(z) fall off at least as fast as z- I as IzI - oo . For the k=4 solution, this 0 fast enough as z can be done only if h(z)a+(z) - 0. Numerically, this method converges in general, but may become divergent when the absorption is very large. After solving Eq. (7), scattering parameters can be found, making use of Eqs. (4)-(6), by f ___4 f10 F.=f'±f,, \I'j=0j7+ 3 1, and _Z /2 (7 = pjt] rep Z23/2+ 2 3.7xp|_ 12 * 2I 2 with II,1 Xp~di|2AZ3/2~-Z/xi2tZ 2 _=-sgn(a) jk= an 2+ 5 =f 4 - - Only I',, f2, f 3 are physically allowed in an unbounded region. Using these we can find an integral equation that solves Eq. (2 ):3'5 '=7fk1z IfI2 Only | k+f4ok+f2,-k+fosk arephsicllr<lowdiaunoude-1, f2 k Sjk=S5k=1 for 4 (r+l 1) > 0, with where 816 -1, Phys. Plasmas, Vol. 1, No. 4, April 1994 C. S. Ng and D. G. Swanson Downloaded 09 Aug 2001 to 128.255.34.168. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp t ~~~ 3~~~ y~~ooe Y- Y oo'7 FIG. 1. Integration contours of 122 for the y> -I case. Ijk A Fj(z)h(z)'Pk(z)dz, FIG. 2. Integration contours of I12 for the y < - I case. (9) where we have already used the fact that Ijk= I.4 - In particular, the integral expression for R 2 is R 2 =g F'22. In the next section, we will show how to calculate this integral I22 analytically, without the need of solving Eq. (3) or Eq. (7) numerically. Since Iz | co on the contour, all terms in this asymptotic series may be kept. Although the integrand is in general a divergent series in z-" for any finite z, the series obtained after integration may become convergent for certain absorption function h(z). First of all, it is easy to expand h(z) in asymptotic series, hz III. INTEGRATION To perform the integration of I22, we must first specify the absorption function h(z). For case (i), the second ion cyclotron harmonic with y -1, ,n=1 Y fo - y+1)>0 where y-z-z0 . This can be done by inverting the two asymptotic series, i 17r(n+ 1/2) h(z) =ii12K[g-I/Z(-a) ]; and for case (ii), the second electron cyclotron harmonic with y <-1, 2 h (z) =A K[g-/F 7 /2 (~- 7/2)], (12) and (-l)'r(n+q) 1 n (10) with the plasma dispersion-function Z(g)=i+i w(g), where w is the error function for complex argument, 15 and F7/ 2 being the relativistic plasma dispersion function ,16,17 and g= (z-zo)/K, zo= -r,2, where K is a real parameter characterizing the strength of absorption. Note that both Z(g) and F7/ 2 (g) are analytic function and have zeros only in the lower half g-plane. Also, solutions Fk and 6 1 Then, f 'k have been shown to be analytic everywhere. following Ref. 6, we can change the path of integration of Ijk, defined in Eq. (9), to the semicircles Co: co and 0 from -'w to 0 for case z=R exp(iO) with R (i), see Fig. 1, and 0 from 1r to 0 for case (ii), see Fig. 2. In Ref. 6, solutions fk and Opk have been extended to the complex z-plane, and it was shown that on these new contours C,, both fl and l&1 ccexp ( r iz) and are thus exponentially small for both cases of - (y + 1) > 0, and that f2 and iP2 c exp ( - iz) and are thus exponentially large. It was then concluded that 111=I12=I21=0, which means that T 1=T 2 =exp(-s7), and R 1=0, independent of absorption. 6 Our approach here to calculate I22 is to expand the integrand F2hT 2 in an asymptotic series on C,. (11) r (q)g' (13) on the path of integration C;. First, we need to shift the argument of the F7,/2 function in Eq. (10) from g-7/2 to A. Let, A,(q) A q(¢_q) = ' ng q The coefficients An(q) can be founding Eq. (13),,by n 1 An(q)=-F-(Y, (-i),r(q+m-1)C'_-mq m=1 7m,: - Phys. Plasmas, Vol. 1, No. 4, April 1994 where Cn'= n/ml (n - m)! is the binomial coefficients. The first few values of An areA 1 = 1, A 2 =0, A 3 = q;Then we can invert Fq by, _n= 2 C~q Fq(~-q) (14) where In C2.+ 1/2.1/2(q)= I n=~1 [-A3 (q)I nD2Cm n) +1/2kL/(") C. S. Ng and D.G. Swanson 817 Downloaded 09 Aug 2001 to 128.255.34.168. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp for -(y D,,(q,m) = Z D.-k(qsmk=O + 1) > 0, where ) Dk(qI ), n= . D.(q,l ) =A.+3(q)1A3(q). Finally we can get the coefficients hn of Eq. (11) by We can also invert the asymptotic series of Z( -g) of Eq. ( 12), using calculations similar to Eqs. ( 14) and ( 15), to get h+. To find the asymptotic series for f2, and 4l2, we first note that only the part of them with leading behavior exp 4 i(y+ a In y) on C, contribute to the integral I22, so 21,riAI 2 2= f (17) (15) An=-A~e~l~n~l(7/2). F2 hq 2dz= fc F,(y)h(y)'I',(y)dy, (16) with C = -ze ira/2 [a-4i(n-I)][a , ',+I,~:=12 afzk = afl- 1+ [a-~i(n-1)1a' l~k-, (:Fnab11 E Y ye+2i(Y+ajny)dy, f (19) n-2 for n>2. m=I To evaluate the y-integrals in Eq. (19), we change the integration contour again. Now, there is only one pole y = 0 in the integrand, and there is also a branch cut. We choose the branch cut to be from y = 0 to 4 ico for the two cases (see Figs. 1 and 2). Then we change the integration contours to go along the semicircles with infinite radius at the other side of the real axis, opposite to C+, and go around the branch cut (see Figs. I and 2) so that the analytic continuation is still valid. The contributions from integrating along the two quarters of the semicircles are zero due to the exp - 2iy factor. Integration along the path going around the branch cut can be found by making use of the Hankel's contour integral, 15 Jo(e-"t) 818 --*-tdt= _2rri) Phys. Plasmas, Vol. 1, No. 4, April 1994 ia where the contour CH starts at t= ocexp(Oi), come along the positive real axis, turn around the origin counterclockwise, then go back along the positive real axis to tc= oexp(27ri). The y-integral in Eq. ( 19) can be changed to this form by means of a variable transformation t=2yexp(3-ri/2) for y> -1, and t=2yexp(iri/2) for y < - 1. Thus, we can express 122, or R 2 in a series expression, i for n>3, n-I =a n- I ((818) for -(y + 1) > 0, where XYn, m=1 - , k==2,3,4, and aOk= 1. The calculation for a is similar to Eqs. ( 18), with hk =0. Substituting Eqs. (11) and (17) into Eq. (16), we get 27ri)L2 2 2= C I a~tn I~;2 +ZOjJ The coefficients a' and &< can be found by substituting the asymptotic series (17), making use of Eq. (8), into the two differential equations (1) and (2), requiring that all terms vanish and that they satisfy boundary conditions Eq. (4) or (5). The result is zi: 4-4-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ a I= -4 '' FTia Trge =t=i(2b-4/3A2)'j2 R2=0-4- - 2 aAr(4-ict) 00 Y,,' ( =I- 20 ' 1 n=3 nnTNa) ' (20) for 4 (ry 1) >0. This expression is analytic in the sense that all coefficients r; can be calculated exactly by some algebraic recurrence formula, although they are in general rather lengthy to be written out explicitly. This series is useful only if it is convergent, unless we know how to sum it analytically. However, due to the complicated nature of the coefficients, it is beyond the scope of this paper to study the convergence analytically. In the next section, we will present numerical results showing that it does converge, in many cases, to numerically the same values found by solving the integral equation (7) iteratively. IV. COMPARISONS We do our calculations with a deuterium plasma. Table I compares calculations of R2 for case (i), the second ion cyclotron harmonic with y> - 1, for plasma parameters: electron density np, characteristic length of the inhomogeneity L, magnetic field strength B, and plasma temC. S. Ng and D. G. Swanson Downloaded 09 Aug 2001 to 128.255.34.168. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp TABLE 1. Case (i) with ne=2X leX m- 3 , L=3 m, B=3 T, T=1000 eV. oq,~~~~~~~~~~l kil(m 1 2 3 4 5 6 7 8 9 10 11 12 13 K P.(%) P,,(%) q, qa 0.2837 0.5668 0.8488 1.1292 1.4073 1.6828 1.9550 2.2234 2.4875 2.7468 3.0009 3.2492 3.4913 60.495 37.427 16.903 5.6244 1.4016 0.26916 0.041933 5.6902e-3 9.0586e-4 1.6609e-4 3.3683e-5 -4.e-6 diverge 60.498 37.432 16.905 5.6237 1.4014 0.27041 0.043239 6.4453e-3 1.0059e-3 1.6403e-4 2.4967e-5 2.99e-6 2.1e-7 0.9896 0.9891 0.9881 0.9862 0.9826 0.9759 0.9639 0.9450 0.9014 0.8497 0.7986' ~0.77 diverge 0.9893 0.9889 0.9881 0.9863 0.9826 0.9751 0.9599 0.9324 0.8929 0.8505 0.8152 0.794 0.79 1) ~ 0.2 - - a 0.:1- I q - 'I .3 2 1 FIG. 3. Plot of 1-q ., I -q,,, and 1-qf versus K, where q. and q, are the values in Table I, and qf are calculated by Eq. (22). perature T, for different parallel wave vectors k1l. P, is jR212 calculated by solving the integral equation (7) iteratively. Pa is that calculated by the series in Eq. (20). Table II compares these for case (ii), the second electron cyclotron harmonic with y < - 1, where X is the square of the ratio of the plasma frequency to the wave frequency. Previously, it was found numerically that with q R2=e~exp(-qK) with q 1 for case-(i) 7 for case (ii). (2) We also list the q 0 and qa values, calculated by each method,; defined by P-=g4exp( - 2qgK2), and P.=keep -2qnK 2 ). They are also plotted in Figs. 3 and 4. We first note that these values calculated by both methods agree with each other very well as long as both methods converge. We have also compared results from these two methods for many other cases over a broad range of parameters and they all agree very well. It has been known previously that the iteration method of solving Eq. (7) becomes divergent for large absorption (i.e., large K) for both cases. It seems that the series method using Eq. (20) for case (i) is still convergent after the iteration method diverges. However, it also has numerical problems for large K, due to the fact that we can only sum a finite number of terms and that the computer has a limited power to handle large numbers. On the other hand, the series method becomes divergent faster than the iteration method for case (ii). Actually, it seems that there is a convergence radius of K < 0.5 for the series. The reason for this difference is that the asymptotic series of the relativistic plasma dispersion function F7 /2 of Eq. (13) diverges faster than that of the plasma dispersion function Z of Eq. (12). Second, we see that the approximation Eq. (21) is good, but not exact. Before the development of this series method, the numerical values found by solving the integral equation iteratively subject to many different kinds of numerical errors, which are difficult to analyze, within the complicated algorithm to calculate fk and 1Obk.Therefore, there was an uncertainty about whether Eq. (21) is really exact, and whether the deviations from the rule in the numerical results are just numerical errors. However, now that the series method is analytic in principle, its results are much more reliable and to the extent that they agree with those calculated by the iteration method, this uncertainty no longer exists. Moreover, the series method provides some analytic explanation of Eq. (21), at least in extreme ~~~. . . . . . . . . . . . -. .I 3 T(eV) 50 100 150 200 250 300 350 400 450 500 550 A2 K P.(%) 93.15 46.58 31.05 23.29 18.63 15.53 13.31 11.64 10.35 9.315 8.468 0.04736 0.09471 0.1421 0.1894 0.2368 0.2841 0.3315 0.3789 0.4262 0.4736 0.5209 0.15192 0.53250 0.99350 1.3983 1.6648 1.7766 1.7568 1.6496 1.4946 1.3215 1.1503 Pa%) 0.15189 0.53254 0.99340 1.3983 1.6646 1.7755 1.7564 1.6492 1.4941 -2 diverge Phys. Plasmas, Vol. 1, No. 4, April 1994 n a 6.9114 6.8615 6.6881 6.4770 6.2201 5.9345 5.6381 5.3328 5.0287 4.7341 4.4494 6.9608 6.8574 6.6927 6.4767 6.2212 5.9382 5.6391 5.3336 5.0295 -4 diverge 7-qu o TABLE I. Case (ii) withX=0.114, n,=4Xl109 m-3 , L=0.15 m, B=3 T. x:7:- qrz,, 2 , :7-qf 5-X 1 ,5 L. ,,~-5 -5-' 0.0 - 0.1 0.2 0.3 0.4 0.5 A; FIG. 4. Plot of 7-q, 7- q., and 7 -qf versus K, where q. and q. are the values in Table II, and qf are calculated by Eq. (23). C. S. Ng and D. G. Swanson Downloaded 09 Aug 2001 to 128.255.34.168. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp cases. Let us consider large /2 , small A1,and small K cases. It is not hard to see that in these cases, we only need to keep the n = 3 terms in the series in Eq. (20). Note that 2 Y3 =2a2A ,,K and yT =14a2A22c, and using approximations r'( + ia) z 4 Ilia, r(3 i-2ia) =2, g~ 1,e=~277, we have R ZtI(-K2) t eXPfK2) R2 2e (l 2 -7K ) = 2 for case (i), exp(-7Kc2) for case (ii), which is just Eq. (21). From numerical calculations, this approximation is very good whenever the condition of large A2, small 7, and small Kis satisfied, although it is also good for some more general cases. Since Eqs. (21) are good approximations already, it is possible to find even better empirical formulas by fitting the results found by the series method. For the ion cases, we found that the q factor can be approximated by the following expression, so that IR21t 2 exp(-qua) qf= 1- qe2 (ao+aiK 2 v), (22) where ao=0.0065(1+0.131 w+0. 103w 2 ), aI =exp( 17.5e-S5 w- 7+0.3w), v=3.085( 1-0.1056w), w= lOO,6iL, and f3i=2yonikBT/B2 is the usual ,6 factor of the plasma. This approximation has been compared with the results of the series method, for the range K2 < 4, so that IR2 12is at least greater than 0.04% of that without absorption, i.e., e, and for k, < k, m,/2, where k, ma is t the k, value when the X-mode is cut off. The error in IR21 rarely exceeds 1%, and this happens only for the cases with w> 2. In all cases, the error is smaller than 1% when IR2 1>i %. Without this correction, i.e., simply using Eq. (21), the error in IR21 may exceed 15% for some cases. The qf factor calculated by Eq. (22) for the cases of Table I is also plotted in Fig. 3. The error between qf and q, seems large in Fig. 3 for some cases. However, we should note that actual difference is only about 2% in the q factor for the worst cases, and that this happens only for 2> 4. For the electron cases, the approximation is qf=7-A [ I-e-a so that IR2 I1; 2 2 ], (23) exp ( - qyK2), where A=3.602+2.413X, a=- 13.6+17.04/(1 2X)0.0 922. The comparison of this approximation with exact values is limited by the convergence of the series method. The range of jR21 is from 0.15% to 50% of t2, the reflection coefficient without absorption. The range of the X parameter is from 0.114 to 0.457. The range of the temperature T is up 820 to 850 eV. Within these ranges, the error of this approximation in IR21 is less than 0.5%. The qf factor calculated by Eq. (23) for the cases of Table II is also plotted in Fig. 4. We see that the approximation is good for the whole range of data shown on Fig. 4. The large difference between q4and qf for the last data point is due to the error in qg,, because the convergence of the series method is not good for K near 0.5. However, the agreement between qf and q, is still good, so the approximation Eq. (23) may still be good beyond the K <0.5 limit. Phys. Plasmas, Vol. 1, No. 4, April 1994 V. DISCUSSION The fact that these two very distinct methods give essentially the same answers further confirms their own validity respectively, because if either one of these methods is wrong, the chances for their answers to agree with each other accidentally is extremely small. The success of the series method is amazing and shows how powerful complex analysis can be. In the original form of the integral I22 in Eq. (9), we needed to know the values of the solutions F 2 , T 2, and the absorption function h along the real z-axis. The straightforward way to do this is first to find F 2 (z) by a numerical path integration in the complex plane using the method of Laplace like Eq. (3) with a complicated integration and error control scheme. Second, solve the integral equation (7) by doing numerical integrations iteratively along the z-axis, with proper endpoint corrections, to find '+2(Z I),making use of numerical methods to evaluate h(z) for z within different regions on the z-axis. Then find 122 by numerical integration according to the definition Eq. (9). Now, the series method can do the same integration correctly without having to know F2 , 'I'2, and even h on the real z-axis. All it needs is their analytic properties in the complex z-plane and their asymptotic expansions for large complex z values. Also, due to its less complicated algorithm, the series method usually runs much faster on a computer, if we do not care to know the solutions themselves. This powerful idea may have other applications to solve integrals of similar nature, which arise either from the tunneling problem or other physical problems. The main difficulty the series method faces is that it does not converge for all cases. However, since it is essentially an analytical method, there is always a chance that we may find a way to sum the series analytically or to transform it into other convergent forms. Just like we can sum +x+x 2+x 3+ ' ' ' as 1/(1-x), although the series may be divergent itself. Note also that the iteration method also has divergence problems, but the chance to "sum" it analytically is much smaller, since it is basically numerical in nature. We have also tried to apply this method to calculate the conversion coefficient C13, but the resulting series is divergent. However, it is still possible that there is a way to overcome this difficulty. We conclude that the series method has been successful in the calculations of the fast wave reflection coefficient from the tunneling equation and it has advantages in some C. S. Ng and D. G. Swanson Downloaded 09 Aug 2001 to 128.255.34.168. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp aspects. This also shows that the analytic method is powerful, and may have more applications and improvements. ACKNOWLEDGMENT Work supported by U.S. Department of Energy Grant No. DE-FG05-85ER53206-93. 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