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Effects of crystal anisotropy on magnetization and magnetic order in superconducting RBa2Cu3O7−x
Physica 148B (1987)285-288 North-Holland, Amsterdam
EFFECTS OF CRYSTAL ANISOTROPY ON MAGNETIZATION AND MAGNETIC O R D E R IN S U P E R C O N D U C T I N G RBa2Cu307_ x R.N. SHELTON, R.W. M c C A L L U M a, M.A. D A M E N T O a, K.A. G S C H N E I D N E R Jr. a, H.C. KU b'*, H.D. YANG b, J.W. LYNN c, W.-H. LI e and Q. LI c Department of Physics, University of California, Davis, CA 95616, USA aAmes Laboratory and Department of Materials Science & Engineering, Iowa State University, Ames, 1A 50011, USA bArnes Laboratory and Department of Physics, Iowa State University, Ames, 1A 50011, USA ~Department of Physics, University of Maryland, College Park, MD 20742 and National Bureau of Standards, Gaithersburg, MD 20899, USA Received 1 August 1987
Two distinct experiments are used to study the influence of crystal anisotropy on superconductivity in YBa2Cu307 and magnetic order in the superconducting state of ErBa2Cu307. Magnetization data on a single crystal of YBa2Cu30 7 reveal pronounced anisotropy. Analysis of the magnetization versus applied field hysteresis loops implies strong anisotropy of the critical current density. For ErBa2Cu3OT, neutron diffraction measurements indicate that the Er moments order two dimensionally at low temperatures (~0.5 K), with chains of spins coupled ferromagnetically, while adjacent chains align antiparallel.
1. Introduction
The intense pace of research activity on superconducting oxides following the initial discovery of high transition temperatures (To) in the K2NiF4-type compound La2_xBaxCuO 4 [1] has recently been focused on the compound YBa2Cu30 7 and its isostructural rare earth analogues [2-7]. After the initial attainment of superconductivity at 95 K [2], there have been reports published in the scientific literature of evidence for Tc's as high as 240 K [8, 9]. Among all of these materials there are some common structural features; namely, the presence of C u O layers and a significant degree of anisotropy due to an elongated c-axis in the orthorhombic unit cell. Early band structure work on compounds with the K2NiF4-type structure [10-12] emphasized this reduced dimensionality and the impact of these crystallographic features on the superconducting properties. In this work we illustrate how these anisot* Permanent address: Department of Physics, National Tsing Hua University, Hsinchu, Taiwan, 30043, R.O.C. 0378-4363/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
ropic crystallographic features affect both the superconducting state and the nature of magnetic order in this class of compounds. For a single crystal of YBa2Cu307 with a superconducting transition temperature (Tc) of 88 K, we present magnetization hysteresis in an applied field for two orientations of the crystal with respect to the field. For a polycrystalline specimen of ErB a 2 f u 3 0 7 ( T c = 95 K), neutron diffraction data show that the Er moments are also sensitive to the crystal anisotropy and exhibit a two-dimensional ordering.
2. Experimental details
Single crystals of YBazCu307 were grown by a technique described in detail in a previous paper [13]. Using single crystal X-ray diffraction data, we determined that the c-axis of the orthorhombic unit cell was perpendicular to the largest face of the rectangular parallelpiped with a~proximate dimensions 0.50 by 0.24 by 0.08 m m . Magnetic properties were measured in a commercial SQUID magnetometer [14]. The neutron experi-
286
R.N. Shelton et al. / Effects o f crystal anisotropy on RBa2Cu~O 7
anisotropy ratio of 15 which is similar to that reported by Dinger and coworkers [16]. Bean [17] applied a critical state model to a cylindrically shaped sample of radius R in order to relate the critical current density (Jc) to the magnetization (M) by the formula: Jc = 30M/R. The units in this expression are A / c m 2 for Jc, e m u / c m 3 for M and cm for R. Since our crystal is not cylindrical, we approximate R as the geometric mean of one-half of the length of each side of the rectangular face perpendicular to the applied field. We estimate Jc for each of the two crystal orientations from the difference in the magnetization at an applied field of 40kOe. For H parallel to the c-axis so that the currents are induced in the ab planes, we calculate J~ = 2.1 x 105 A / c m 2. In contrast, for H perpendicular to the c-axis so that the currents are induced perpendicular to the ab planes, we compute Jc~ = 4.6 X 104 A / c m 2. The ratio jc/j c l l z = 4.6 is indicative of the anisotropy of the superconducting properties. The relationship between the hysteresis loops of fig. 1 and the critical current density extends the importance of crystal orientation to the
merits were conducted at the research reactor at The National Bureau of Standards using a pyrolytic graphite monochromator and filter at a wavelength of 0.23509nm. Details of sample preparation of the polycrystalline ErBa2Cu30 7 are given elsewhere [15].
3. Results and discussion
The magnetization loops taken at 10 K for the rectangular single crystal of YBa2Cu307 are shown in fig. 1. We present data for two orientations of the crystal; namely, with the applied field parallel to the c-axis and perpendicular to the c-axis. Both curves are highly symmetric. Referring to fig. 1, we identify the lower critical fields, H ~ and Hcl, as the point at which the virgin magnetization curve deviates from a linear dependence on the applied field, provided corrections for the demagnetization factors are applied. Directly from the graph we obtain H~I = 3 kOe and H ~ = 0.3 kOe. After correcting for the demagnetization factors these values become H~I = 6 k O e and H c ~ - - 0 . 4 k O e , yielding an
6.0
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-4.0
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i -20.
1
-10. Applied
0. Field
i 10.
I 20.
I 30.
I 40.
I 50.
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Fig. 1. Magnetization hysteresis loops at 10 K for a single crystal of YBa2CU3OT. The orientation of the applied field (H) with respect to the crystallographic c-axis is indicated for each curve.
R.N. Shelton et al. / Effects of crystal anisotropy on RBa2Cu307
quantity Jc which is crucial to most applications of these superconducting oxides. With respect to applications, we note that the magnitudes of our critical current densities are slightly lower than those reported in an earlier study on single crystals with volumes comparable to that of our specimen [16]. This difference could be due to a change in oxygen concentration or the overall ordering of oxygen vacancies in the crystals. Both of these parameters will affect flux pinning and Jc. A second manifestation of the influence of crystal anisotropy is evident in low temperature neutron diffraction experiments on ErBa2Cu307 . This polycrystalline sample has a superconducting onset of 95K and remains superconducting to the lowest temperatures measured. In order to identify the contribution from the magnetic scattering, we use a subtraction technique [18] whereby a direct subtraction of the high temperature data from the data below the ordering temperature isolates the magnetic response. This type of data is presented in the top portion of fig. 2. Starting at low scattering angle and increasing 20, there is a rapid increase in the scattering, reaching a maximum at 20 = 17.8 °, followed by a long tail extending toward larger angles. The position of the peak coincides with a {1/2, 0, 0} type reflection, so we determine immediately that the scattering is antiferromagnetic in nature. The pronounced asymmetry of this peak is a classic profile of a two-dimensionally ordered system [19], which is undoubtedly driven by the anisotropic nature of the orthorhombic lattice. Specifically, the Er sublattice is simple orthorhombic with an E r - E r separation distance along the c-axis about three times longer that the separation of Er ions along the a- and b-axes. Assuming the low magnetic ordering temperature (T~ ~0.5 K) is caused by dipolar interactions, these interactions will be reduced greatly along the c-axis direction, thus giving rise to the observed two-dimensional order. The solid line through the data in the top of fig. 1 is a fit using a model which assumes that the scattering originates from a Bragg line. The fit is excellent;
287
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(b) Fig. 2. a) Magnetic intensity observed at low temperatures. The solid curve is a least-squares fit of the data to a twodimensional model (see text), b) Temperature dependence of the peak intensity. The solid curve is a guide to the eye. The inset shows the magnetic configuration of spins.
therefore, we conclude that strong correlations exist in the ab plane, while the correlation length along the c-direction is small. Assuming a collinear spin structure, our data
288
R.N. She#on et al. / Effects o f crystal anisotropy on RBa2Cu307
a r e c o n s i s t e n t with t h e spin c o n f i g u r a t i o n s h o w n in t h e inset o f t h e b o t t o m o f fig. 2. T h e spins f o r m chains a l o n g t h e b - a x i s with p a r a l l e l spins, w h i l e a d j a c e n t chains a l o n g t h e a - a x i s a r e ant i p a r a l l e l . F i x i n g t h e s c a t t e r i n g angle at 20 = 17.8 ° , t h e l o w e r p o r t i o n of fig. 2 shows the d e v e l o p m e n t o f m a g n e t i c s c a t t e r i n g with t e m p e r a t u r e . N o t e t h a t the i n t e n s i t y c o n t i n u e s to i n c r e a s e r a p i d l y at 0 . 3 K which is t h e l o w e s t t e m p e r a t u r e a t t a i n e d . By c o m p a r i n g this m a g netic i n t e n s i t y to the {001) n u c l e a r i n t e n s i t y , we d e t e r m i n e that t h e o r d e r e d m a g n e t i c m o m e n t (/x z ) is a l r e a d y 2.9/z B. A s t h e t e m p e r a t u r e is l o w e r e d f u r t h e r , w e e x p e c t this i n t e n s i t y to cont i n u e to i n c r e a s e until t h e full g r o u n d state m o m e n t is o b t a i n e d . In all o f o u r d a t a , we h a v e no e v i d e n c e t h a t t h e C u ions o r d e r . In c o n c l u s i o n , we h a v e p r e s e n t e d two sets of e x p e r i m e n t a l d a t a t h a t d e m o n s t r a t e t h e influence of crystal anisotropy on the collective p h e n o m e n a of s u p e r c o n d u c t i v i t y a n d m a g n e i c o r d e r in t h e high T o x i d e s . N e u t r o n d i f f r a c t i o n r e v e a l s a classical t w o - d i m e n s i o n a l o r d e r i n g o f t h e E r ions in E r B a z C u 3 0 7. Single crystals of Y B a z C u 3 0 7 s h o w a m a g n e t i z a t i o n in t h e s u p e r c o n d u c t i n g state t h a t is highly a n i s o t r o p i c . T h i s result e x t e n d s to t h e q u a n t i t y Jc w h i c h is crucial to m o s t a p p l i c a t i o n s o f t h e s e s u p e r c o n d u c t i n g oxides.
Acknowledgements A m e s L a b o r a t o r y is o p e r a t e d for t h e U S D e p a r t m e n t of E n e r g y by I o w a S t a t e U n i v e r s i t y u n d e r c o n t r a c t no. W - 7 4 0 5 - E N G - 8 2 . This w o r k was s u p p o r t e d b y T h e Office of B a s i c E n e r g y Sciences. R e s e a r c h at M a r y l a n d was s u p p o r t e d b y t h e N S F , D M R 83-199936 a n d D M R 8620269.
References [1] J.G. Bednorz and K.A. Mueller, Z. Phys. B 64 (1986) 189. [2] M.K. Wu, J.R. Ashburn, C.J. Torng, P.H. Hor, R.L. Meng, L. Gao, Z.J. Huang, Y.Q. Wang and C.W. Chu, Phys. Rev. Lett. 58 (1987) 908. [3] R.J. Cava, B. Batlogg, R.B. van Dover, D.W. Murphy, S. Sunshine, T. Siegrist, J.P. Remeika, E.A. Rietman, S. Zahurak and G.P. Espinosa, Phys. Rev. Lett. 58 (1987) 1676. [4] A.R. Moodenbaugh, M. Suenaga, T. Asano, R.N. Shelton, H.C. Ku, R.W. McCallum and P. Klavins, Phys. Rev. Lett. 58 (1987) 1885. [5] P.H. Hor, R.L. Meng, Y.Q. Wang, L. Gao, Z.J. Huang, J. Bechtold, K. Forster and C.W. Chu, Phys. Rev. Lett. 58 (1987) 1891. [6] J.Z. Sun, D.J. Webb, M. Naito, K. Char, M.R. Hahn, J.W.P. Hsu, A.D. Kent, D.B. Mitzi, B. Oh, M.R. Beasley, T.H. Geballe, R.H. Hammond and A. Kapitulnik, Phys. Rev. Lett. 58 (1987) 1574. [7] D.W. Murphy, S. Sunshine, R.B. van Dover, R.J. Cava, B. Batlogg, S.M. Zahurak and L.F. Schneemeyer, Phys. Rev. Lett. 58 (1987) 1888. [8] J.T. Chen, L.E. Wenger, C.J. McEwan and E.M. Logothetis, Phys. Rev. Lett. 58 (1987) 1972. [9] S.R. Ovshinsky, R.T. Young, D.D. Allred, G. DeMaggio and G.A. Van der Leeden, Phys. Rev. Lett. 58 (1987) 2579. [10] L.F. Mattheiss, Phys. Rev. Lett. 58 (1987) 1028. [111 J, Yu, A.J. Freeman and J.-H. Xu, Phys. Rev. Lett. 58 (1987) 1035. [12] W. Weber, Phys. Rev. Lett. 58 (1987) 1371. [13] M.A. Damento, K.A. Gschneidner Jr. and R.W. McCallum, Appl. Phys. Lett. 51 (1987) 690. [14] Quantum Design, Inc., San Diego, CA, USA. [15] J.W. Lynn, W.-H. Li, Q. Li, H.C. Ku, H.D. Yang and R.N. Shelton, Phys. Rev. B 36 (1987) 2374. [16] T.R. Dinger, T.K. Worthington, W.J. Gallagher and R.L. Sandstrom, Phys. Rev. Lett. 58 (1987) 2687. [17] C.E Bean, Phys. Rev. Lett. 8 (1962) 250. [18] J.W. Lynn, J.A. Gotaas, R.N. Shelton, H.E. Horng and C.J. Glinka, Phys. Rev. B 31 (1985) 5756. [19] J.K. Kjems, L. Passell, H. Taub, J.G. Dash and A.D. Novaco, Phys. Rev. B 13 (1976) 1446, and references therein.
EFFECTS OF CRYSTAL ANISOTROPY ON MAGNETIZATION AND MAGNETIC O R D E R IN S U P E R C O N D U C T I N G RBa2Cu307_ x R.N. SHELTON, R.W. M c C A L L U M a, M.A. D A M E N T O a, K.A. G S C H N E I D N E R Jr. a, H.C. KU b'*, H.D. YANG b, J.W. LYNN c, W.-H. LI e and Q. LI c Department of Physics, University of California, Davis, CA 95616, USA aAmes Laboratory and Department of Materials Science & Engineering, Iowa State University, Ames, 1A 50011, USA bArnes Laboratory and Department of Physics, Iowa State University, Ames, 1A 50011, USA ~Department of Physics, University of Maryland, College Park, MD 20742 and National Bureau of Standards, Gaithersburg, MD 20899, USA Received 1 August 1987
Two distinct experiments are used to study the influence of crystal anisotropy on superconductivity in YBa2Cu307 and magnetic order in the superconducting state of ErBa2Cu307. Magnetization data on a single crystal of YBa2Cu30 7 reveal pronounced anisotropy. Analysis of the magnetization versus applied field hysteresis loops implies strong anisotropy of the critical current density. For ErBa2Cu3OT, neutron diffraction measurements indicate that the Er moments order two dimensionally at low temperatures (~0.5 K), with chains of spins coupled ferromagnetically, while adjacent chains align antiparallel.
1. Introduction
The intense pace of research activity on superconducting oxides following the initial discovery of high transition temperatures (To) in the K2NiF4-type compound La2_xBaxCuO 4 [1] has recently been focused on the compound YBa2Cu30 7 and its isostructural rare earth analogues [2-7]. After the initial attainment of superconductivity at 95 K [2], there have been reports published in the scientific literature of evidence for Tc's as high as 240 K [8, 9]. Among all of these materials there are some common structural features; namely, the presence of C u O layers and a significant degree of anisotropy due to an elongated c-axis in the orthorhombic unit cell. Early band structure work on compounds with the K2NiF4-type structure [10-12] emphasized this reduced dimensionality and the impact of these crystallographic features on the superconducting properties. In this work we illustrate how these anisot* Permanent address: Department of Physics, National Tsing Hua University, Hsinchu, Taiwan, 30043, R.O.C. 0378-4363/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
ropic crystallographic features affect both the superconducting state and the nature of magnetic order in this class of compounds. For a single crystal of YBa2Cu307 with a superconducting transition temperature (Tc) of 88 K, we present magnetization hysteresis in an applied field for two orientations of the crystal with respect to the field. For a polycrystalline specimen of ErB a 2 f u 3 0 7 ( T c = 95 K), neutron diffraction data show that the Er moments are also sensitive to the crystal anisotropy and exhibit a two-dimensional ordering.
2. Experimental details
Single crystals of YBazCu307 were grown by a technique described in detail in a previous paper [13]. Using single crystal X-ray diffraction data, we determined that the c-axis of the orthorhombic unit cell was perpendicular to the largest face of the rectangular parallelpiped with a~proximate dimensions 0.50 by 0.24 by 0.08 m m . Magnetic properties were measured in a commercial SQUID magnetometer [14]. The neutron experi-
286
R.N. Shelton et al. / Effects o f crystal anisotropy on RBa2Cu~O 7
anisotropy ratio of 15 which is similar to that reported by Dinger and coworkers [16]. Bean [17] applied a critical state model to a cylindrically shaped sample of radius R in order to relate the critical current density (Jc) to the magnetization (M) by the formula: Jc = 30M/R. The units in this expression are A / c m 2 for Jc, e m u / c m 3 for M and cm for R. Since our crystal is not cylindrical, we approximate R as the geometric mean of one-half of the length of each side of the rectangular face perpendicular to the applied field. We estimate Jc for each of the two crystal orientations from the difference in the magnetization at an applied field of 40kOe. For H parallel to the c-axis so that the currents are induced in the ab planes, we calculate J~ = 2.1 x 105 A / c m 2. In contrast, for H perpendicular to the c-axis so that the currents are induced perpendicular to the ab planes, we compute Jc~ = 4.6 X 104 A / c m 2. The ratio jc/j c l l z = 4.6 is indicative of the anisotropy of the superconducting properties. The relationship between the hysteresis loops of fig. 1 and the critical current density extends the importance of crystal orientation to the
merits were conducted at the research reactor at The National Bureau of Standards using a pyrolytic graphite monochromator and filter at a wavelength of 0.23509nm. Details of sample preparation of the polycrystalline ErBa2Cu30 7 are given elsewhere [15].
3. Results and discussion
The magnetization loops taken at 10 K for the rectangular single crystal of YBa2Cu307 are shown in fig. 1. We present data for two orientations of the crystal; namely, with the applied field parallel to the c-axis and perpendicular to the c-axis. Both curves are highly symmetric. Referring to fig. 1, we identify the lower critical fields, H ~ and Hcl, as the point at which the virgin magnetization curve deviates from a linear dependence on the applied field, provided corrections for the demagnetization factors are applied. Directly from the graph we obtain H~I = 3 kOe and H ~ = 0.3 kOe. After correcting for the demagnetization factors these values become H~I = 6 k O e and H c ~ - - 0 . 4 k O e , yielding an
6.0
4.0
-
/° /
Hlc
{ \
~
Hltc
IT = lo KI 2.0
'~ °~o
A
I$ 0.0
~'o~...
IE
\e~ e
-o.,
,¢,
.5
\°'°\ o L
-2.0
-4.0
¢ -6.0
I -50.
I
I
-40.
-30.
i -20.
1
-10. Applied
0. Field
i 10.
I 20.
I 30.
I 40.
I 50.
(kOe)
Fig. 1. Magnetization hysteresis loops at 10 K for a single crystal of YBa2CU3OT. The orientation of the applied field (H) with respect to the crystallographic c-axis is indicated for each curve.
R.N. Shelton et al. / Effects of crystal anisotropy on RBa2Cu307
quantity Jc which is crucial to most applications of these superconducting oxides. With respect to applications, we note that the magnitudes of our critical current densities are slightly lower than those reported in an earlier study on single crystals with volumes comparable to that of our specimen [16]. This difference could be due to a change in oxygen concentration or the overall ordering of oxygen vacancies in the crystals. Both of these parameters will affect flux pinning and Jc. A second manifestation of the influence of crystal anisotropy is evident in low temperature neutron diffraction experiments on ErBa2Cu307 . This polycrystalline sample has a superconducting onset of 95K and remains superconducting to the lowest temperatures measured. In order to identify the contribution from the magnetic scattering, we use a subtraction technique [18] whereby a direct subtraction of the high temperature data from the data below the ordering temperature isolates the magnetic response. This type of data is presented in the top portion of fig. 2. Starting at low scattering angle and increasing 20, there is a rapid increase in the scattering, reaching a maximum at 20 = 17.8 °, followed by a long tail extending toward larger angles. The position of the peak coincides with a {1/2, 0, 0} type reflection, so we determine immediately that the scattering is antiferromagnetic in nature. The pronounced asymmetry of this peak is a classic profile of a two-dimensionally ordered system [19], which is undoubtedly driven by the anisotropic nature of the orthorhombic lattice. Specifically, the Er sublattice is simple orthorhombic with an E r - E r separation distance along the c-axis about three times longer that the separation of Er ions along the a- and b-axes. Assuming the low magnetic ordering temperature (T~ ~0.5 K) is caused by dipolar interactions, these interactions will be reduced greatly along the c-axis direction, thus giving rise to the observed two-dimensional order. The solid line through the data in the top of fig. 1 is a fit using a model which assumes that the scattering originates from a Bragg line. The fit is excellent;
287
i
1500
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Er Ba 2 Cu 3 07 T= 0.:55 K X = 2.3509
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E
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25 I-
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(b) Fig. 2. a) Magnetic intensity observed at low temperatures. The solid curve is a least-squares fit of the data to a twodimensional model (see text), b) Temperature dependence of the peak intensity. The solid curve is a guide to the eye. The inset shows the magnetic configuration of spins.
therefore, we conclude that strong correlations exist in the ab plane, while the correlation length along the c-direction is small. Assuming a collinear spin structure, our data
288
R.N. She#on et al. / Effects o f crystal anisotropy on RBa2Cu307
a r e c o n s i s t e n t with t h e spin c o n f i g u r a t i o n s h o w n in t h e inset o f t h e b o t t o m o f fig. 2. T h e spins f o r m chains a l o n g t h e b - a x i s with p a r a l l e l spins, w h i l e a d j a c e n t chains a l o n g t h e a - a x i s a r e ant i p a r a l l e l . F i x i n g t h e s c a t t e r i n g angle at 20 = 17.8 ° , t h e l o w e r p o r t i o n of fig. 2 shows the d e v e l o p m e n t o f m a g n e t i c s c a t t e r i n g with t e m p e r a t u r e . N o t e t h a t the i n t e n s i t y c o n t i n u e s to i n c r e a s e r a p i d l y at 0 . 3 K which is t h e l o w e s t t e m p e r a t u r e a t t a i n e d . By c o m p a r i n g this m a g netic i n t e n s i t y to the {001) n u c l e a r i n t e n s i t y , we d e t e r m i n e that t h e o r d e r e d m a g n e t i c m o m e n t (/x z ) is a l r e a d y 2.9/z B. A s t h e t e m p e r a t u r e is l o w e r e d f u r t h e r , w e e x p e c t this i n t e n s i t y to cont i n u e to i n c r e a s e until t h e full g r o u n d state m o m e n t is o b t a i n e d . In all o f o u r d a t a , we h a v e no e v i d e n c e t h a t t h e C u ions o r d e r . In c o n c l u s i o n , we h a v e p r e s e n t e d two sets of e x p e r i m e n t a l d a t a t h a t d e m o n s t r a t e t h e influence of crystal anisotropy on the collective p h e n o m e n a of s u p e r c o n d u c t i v i t y a n d m a g n e i c o r d e r in t h e high T o x i d e s . N e u t r o n d i f f r a c t i o n r e v e a l s a classical t w o - d i m e n s i o n a l o r d e r i n g o f t h e E r ions in E r B a z C u 3 0 7. Single crystals of Y B a z C u 3 0 7 s h o w a m a g n e t i z a t i o n in t h e s u p e r c o n d u c t i n g state t h a t is highly a n i s o t r o p i c . T h i s result e x t e n d s to t h e q u a n t i t y Jc w h i c h is crucial to m o s t a p p l i c a t i o n s o f t h e s e s u p e r c o n d u c t i n g oxides.
Acknowledgements A m e s L a b o r a t o r y is o p e r a t e d for t h e U S D e p a r t m e n t of E n e r g y by I o w a S t a t e U n i v e r s i t y u n d e r c o n t r a c t no. W - 7 4 0 5 - E N G - 8 2 . This w o r k was s u p p o r t e d b y T h e Office of B a s i c E n e r g y Sciences. R e s e a r c h at M a r y l a n d was s u p p o r t e d b y t h e N S F , D M R 83-199936 a n d D M R 8620269.
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