4
VOLUME 83, NUMBER 10 PHYSICAL REVIEW LETTERS 6SEPTEMBER 1999 Asymmetry of Critical Exponents in YBa 2 Cu 3 O 72d M. Charalambous,* O. Riou, P. Gandit, B. Billon, P. Lejay, and J. Chaussy Centre de Recherches sur les Très Basses Températures, CNRS (associé à l’université Joseph Fourier), BP166, 38042 Grenoble Cedex 09, France W. N. Hardy, D. A. Bonn, and Ruixing Liang Department of Physics, University of British Columbia, Vancouver, British Columbia, Canada V6T1Z1 (Received 9 February 1999) The critical behavior of YBa 2 Cu 3 O 72d close to the zero field superconducting critical point (T c 92 K) was investigated using a high precision, high resolution microcalorimeter and detwinned single crystals with mg mass. A detailed analysis of the specific heat reveals a pronounced asymmetry of the critical exponent a above and below T c : a 1 0.5 6 0.2, a 2 20.2 6 0.3, which does not heal for different choices of the critical temperature T c or the phonon background C p , and persists in the overdoped regime. This abnormal behavior cannot be explained within the framework of regular second order phase transitions. PACS numbers: 74.25.Bt, 64.60.Fr Despite intensive efforts there is still a very poor under- standing of the physics of the high-T c superconductors. One way to investigate this question is a detailed analysis of the normal-superconducting transition. Such a transi- tion is normally expected to fall in the universality class of the 3D XY model ( 4 He transition) [1], since the order parameter has both amplitude and phase fluctuations al- though the charged nature of the pairs can lead to more complicated behaviors [2,3]. For high-T c one expects that an analysis of the transition, and in particular of the critical exponents, can give a clue on the underlying mi- croscopic mechanism. For example, preformed pairs [4] would favor phase fluctuations, leading to a 3D XY transi- tion, whereas an amplitude-driven transition, e.g., because the energy scale for phase fluctuations is much higher than T c [5], would have a mean-field behavior. The critical ex- ponents that depend on the number of components of the order parameter [6] would also allow one to decide on more complicated order parameters [7]. Since in a superconductor there is no external pa- rameter conjugate to the order parameter, one can only measure the exponents a, n, and z . There have been several attempts to measure such exponents. Precise mea- surements of the magnetic penetration depth below T c in YBa 2 Cu 3 O 72d (YBCO) single crystals [8] revealed a large critical region below T c (10 K) and n 0.66 6 ... , in agreement with the 3D XY exponent. If the hy- perscaling relation is obeyed below T c this imposes a small value of a close to zero. The situation is less clear above T c , where such measurements cannot be performed. Above T c measurements of microwave conductivity in YBCO single crystals [9] yield a combination of n and z and were interpreted in terms of 2D Gaussian fluctua- tions, whereas frequency dependent microwave conduc- tivity measurements in YBCO films [10] were interpreted in terms of critical fluctuations. Specific heat measure- ments have the enormous advantage of measuring a both above and below T c in the same experiment and on the same sample. However, large error bars in the exponents arise from the large phonon contribution and the wide transitions of samples. First, specific heat data in YBCO single crystals were interpreted in terms of Gaussian fluc- tuations [11]. More recently, C p data were interpreted within the 3D XY framework [12 – 15], accommodating also data in large magnetic fields [16]. However, those exponents are not obtained over more than one decade in reduced temperature and a clear discussion of the error bars for the exponent at zero field is nonexistent. We present here a detailed study of the C p transition of detwinned YBCO single crystals. The size of singularity in our crystals is 1.3 – 3 times larger than the ones reported in literature. The samples have exceptionally sharp transi- tions and large disorder lengths. Our measurements show definite incompatibilities with the 3D XY predictions and provide evidence that the critical exponents are asymmet- ric around T c . This fact is incompatible with a regular second order phase transition. Our very sensitive ac microcalorimeter [17] allows us to measure the specific heat with a relative resolution dCC of 10 24 . Our absolute precision is 10%. Our sample holder is a polymer membrane freely suspended from a copper matrix; thermometers and heaters are thin films de- posited and lithographically patterned within a central area of 0.6 mm of diameter. Their thermal response times are 0.1 ms. The crystal responds to a temperature change within 5 ms. Much longer times characterize the ther- mal coupling of the crystal to the copper matrix (0.5 s) resulting to quasiadiabatic conditions. At 3 Hz the tem- perature oscillation front propagates coherently through- out the sample. The total specific heat of the central area of the membrane is 1.5mJK at 100 K, which is about 3 times less than our smallest YBCO sample. The sample holder contribution is reproducible within 30%, and is sub- tracted from the data. The excellent sensitivity of our 2042 0031-9007 99 83(10) 2042(4)$15.00 © 1999 The American Physical Society

Asymmetry of Critical Exponents in

  • Upload
    ruixing

  • View
    219

  • Download
    1

Embed Size (px)

Citation preview

Page 1: Asymmetry of Critical Exponents in

VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER1999

dry

ar

2042

Asymmetry of Critical Exponents in YBa2Cu3O72d

M. Charalambous,* O. Riou, P. Gandit,† B. Billon, P. Lejay, and J. ChaussyCentre de Recherches sur les Très Basses Températures, CNRS (associé à l’université Joseph Fourier), BP166,

38042 Grenoble Cedex 09, France

W. N. Hardy, D. A. Bonn, and Ruixing LiangDepartment of Physics, University of British Columbia, Vancouver, British Columbia, Canada V6T1Z1

(Received 9 February 1999)

The critical behavior of YBa2Cu3O72d close to the zero field superconducting critical point(Tc � 92 K) was investigated using a high precision, high resolution microcalorimeter and detwinnesingle crystals withmg mass. A detailed analysis of the specific heat reveals a pronounced asymmetof the critical exponenta above and belowTc: a1 � 0.5 6 0.2, a2 � 20.2 6 0.3, which does notheal for different choices of the critical temperatureTc or the phonon backgroundCp , and persists inthe overdoped regime. This abnormal behavior cannot be explained within the framework of regulsecond order phase transitions.

PACS numbers: 74.25.Bt, 64.60.Fr

ents

deO

uc-

see inrror

itytedsi-

how

r

to

lea

de-reaarenger-

-gh-rea

pleb-ur

Despite intensive efforts there is still a very poor undestanding of the physics of the high-Tc superconductors.One way to investigate this question is a detailed analyof the normal-superconducting transition. Such a trantion is normally expected to fall in the universality clasof the 3DXY model (4He transition) [1], since the orderparameter has both amplitude and phase fluctuationsthough the charged nature of the pairs can lead to mcomplicated behaviors [2,3]. For high-Tc one expectsthat an analysis of the transition, and in particular of thcritical exponents, can give a clue on the underlying mcroscopic mechanism. For example, preformed pairswould favor phase fluctuations, leading to a 3DXY transi-tion, whereas an amplitude-driven transition, e.g., becauthe energy scale for phase fluctuations is much higher thTc [5], would have a mean-field behavior. The critical exponents that depend on the number of components oforder parameter [6] would also allow one to decide omore complicated order parameters [7].

Since in a superconductor there is no external prameter conjugate to the order parameter, one can omeasure the exponentsa, n, and z. There have beenseveral attempts to measure such exponents. Precise msurements of the magnetic penetration depth belowTc

in YBa2Cu3O72d (YBCO) single crystals [8] revealed alarge critical region belowTc (�10 K) and n � 0.66 6

. . . , in agreement with the 3DXY exponent. If the hy-perscaling relation is obeyed belowTc this imposes asmall value ofa close to zero. The situation is less cleaaboveTc, where such measurements cannot be performAbove Tc measurements of microwave conductivity iYBCO single crystals [9] yield a combination ofn andz and were interpreted in terms of 2D Gaussian fluctutions, whereas frequency dependent microwave condtivity measurements in YBCO films [10] were interpretein terms of critical fluctuations. Specific heat measurments have the enormous advantage of measuringa both

0031-9007�99�83(10)�2042(4)$15.00

r-

sissi-s

al-ore

ei-

[4]

sean-then

a-nly

ea-

red.n

a-uc-de-

above and belowTc in the same experiment and on thsame sample. However, large error bars in the exponearise from the large phonon contribution and the witransitions of samples. First, specific heat data in YBCsingle crystals were interpreted in terms of Gaussian fltuations [11]. More recently,Cp data were interpretedwithin the 3D XY framework [12–15], accommodatingalso data in large magnetic fields [16]. However, thoexponents are not obtained over more than one decadreduced temperature and a clear discussion of the ebars for the exponent at zero field is nonexistent.

We present here a detailed study of theCp transition ofdetwinned YBCO single crystals. The size of singularin our crystals is 1.3–3 times larger than the ones reporin literature. The samples have exceptionally sharp trantions and large disorder lengths. Our measurements sdefinite incompatibilities with the 3DXY predictions andprovide evidence that the critical exponents areasymmet-ric aroundTc. This fact is incompatible with a regulasecond order phase transition.

Our very sensitive ac microcalorimeter [17] allows usmeasure the specific heat with a relative resolutiondC�Cof 1024. Our absolute precision is 10%. Our sampholder is a polymer membrane freely suspended fromcopper matrix; thermometers and heaters are thin filmsposited and lithographically patterned within a central aof 0.6 mm of diameter. Their thermal response times�0.1 ms. The crystal responds to a temperature chawithin 5 ms. Much longer times characterize the themal coupling of the crystal to the copper matrix (�0.5 s)resulting to quasiadiabatic conditions. At 3 Hz the temperature oscillation front propagates coherently throuout the sample. The total specific heat of the central aof the membrane is�1.5mJ�K at 100 K, which is about3 times less than our smallest YBCO sample. The samholder contribution is reproducible within 30%, and is sutracted from the data. The excellent sensitivity of o

© 1999 The American Physical Society

Page 2: Asymmetry of Critical Exponents in

VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER 1999

metallic thermometer (100 mV�K) allows one to measuretemperature oscillations as small as 5 mK, with a reso-lution of 0.5 mK, at an average temperature of 90 K,corresponding to a specific heat resolution of 0.5 nJ�K.The dc temperature is incremented by steps as small as5 mK; at each point the temperature is stabilized with driftssmaller than 1 mK�h. The calorimeter is optimized atthe temperature range 40–160 K; recently a similar tech-nique was extended in the temperature range 1–10 K [18].Since the electronic specific heat of YBCO is very sensi-tive to oxygen doping [19], we used macroscopic samples(10 35 mg), the smallest dimension of which is largerthan 30 mm, in the purpose to achieve good doping ho-mogeneity during the annealing process. Specific carewas taken in order to define precisely the oxygen content.We measured several twinned and detwinned single crys-tals with our best samples being detwinned. We presentdata on samples named YBCO3, UBC1, UBC2, and UBC2overdoped. The YBCO3 sample was grown in a goldcrucible resulting in a slight contamination of Cu chainswith Au [20]. Samples UBC1 and UBC2 were grown onY -stabilized zirconia crucibles resulting in a very low im-purity level. The central result of this paper is shown inFigs. 1 and 2. Despite the fact that the singular part ofthe specific heat resembles a l shape similar to the oneof the 4He transition, the derivative shows that this is nottrue. From Fig. 1, we observe an enormous difference be-tween dC1�dt and dC2�dt (the 6 sign corresponds todata above/below Tc), with a measured ratio dC1�dt

dC2�dt of�10 close to Tc for our best sample, whereas the theo-retical ratio within the 3D XY model is 1.03, indicat-ing dramatic difference between our data and the 3D XYpredictions.

The 3D XY model would also predict [21,22] for thesingular part of the specific heat C6

sing � B6 1A6

a6 jtj2a6

with B1 � B2, a1 � a2 � 20.013 6 0.003, and

FIG. 1. Derivative of the total specific heat of four YBCOsingle crystals. All samples are detwinned and optimallydoped, except for sample UBC2over, which is overdoped. Theasymmetry of the derivative peak reaches a ratio of 10 forsample UBC2, whereas the prediction of 3D XY is a ratioof 1.03.

A1�A2 � 1.03. Let us thus further analyze our data toextract the critical exponents. The phonon backgroundis very well described by Cback � bt 2 ct2 1 d whichgives for the total specific heat (after subtraction of thesample holder) Ctot � B 1

Aa jtj2a 1 bt 2 ct2 1 d,

where the superscripts are dropped for clarity. Becauseof our low noise level, we can analyze the derivative ofdCtot�dt which eliminates d 1 B. This leaves five fitparameters: a, A, b, c, and Tc, which reduce to fourbecause fits are not sensitive to the value of c, the slightbackground curvature. We have a good control over pa-rameters b and Tc which allows us to make a reasonablefit of the data despite the large number of parameters.Tc is restricted in a very small temperature windowfor our samples because the transitions are remarkablysharp (�70 mK). We are able to restrict b, the slopeof the phonon background within 610% of its nominalvalue due to our high absolute precision. Subtractingb 2 2ct from the derivative dCtot�dt, we obtain directlya power law. We use then the straight line slope in alog

dCsing

dt 2 log jtj plot to get a as shown on Fig. 3.A good power law is observed over more than two

decades of reduced temperature both above and below Tc,in the temperature range 4 3 1024 , t , 0.1. We ob-serve two very different slopes resulting in very differentvalues of the exponent: a is big and positive above Tc

(a1 � 0.5), whereas it is small and negative below Tc

(a2 � 20.3). This leads to the stunning result of a di-vergence above Tc and a cusp below Tc. Close to Tc thetransition gets rounded before reaching our experimentalresolution of t � 1025. The transition width dTc is givenby the temperature difference between the two inflection

FIG. 2. Singular part of the specific heat for samples UBC2(dTc � 70 mK) and YBCO3 (dTc � 100 mK). The back-ground subtracted is a two-degree polynomial fit of the datain the range (50–70 K) and (100–120 K). The peak height(11 mJ�gK) is, respectively, 1.8, 1.5, 1.5, 2.8, 2.6, 1.3, and 1.3larger than the one reported in [11–16,19]. No comparison isavailable for the transition width and disorder length in thesesamples. Left inset: C�T data for sample UBC2. Right inset:derivative of the specific heat; gray area represents the roundingof the transition.

2043

Page 3: Asymmetry of Critical Exponents in

VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER 1999

FIG. 3. Derivative of the singular part of the specific heat asa function of reduced temperature. In such a plot a straight lineslope equals a 1 1. Above Tc a power law is observed witha slope of 1.5 (a1 � 0.5), whereas below Tc a power law isobserved with a slope of 0.7 (a2 � 20.3). Inset: simulationof the effect of disorder in the derivative within the 3D XYmodel. Slopes above and below Tc are identical in this scaleand reach the 3D XY value outside the rounded region.

points of C�T � (extrema of the dC�dt curve). The round-ing occurs when the characteristic coherence length whichdiverges close to Tc (j � t2n) reaches a typical disorderlength. Our best samples have transitions widths betweendT � 70 2 100 mK corresponding to a disorder lengthof 500 1500 Å, 50–180 times larger than the coherencelength. This is a very large length for a solid sample,among the largest reported for high-Tc materials, but stillvery small compared to the disorder length in 4He, that isalmost exempt of impurities.

Given the astonishing result of asymmetric exponents,it is important to check how robust is the result against achange of the slope b of the background or a change in Tc.We show on Fig. 4 a1�b, Tc� and a2�b, Tc� as a functionof Tc. The acceptable Tc window is shown gray on Fig. 4,and corresponds to dTc as defined above. The asymmetryof the exponent is pronounced within all the acceptablerange of Tc’s. The exponent becomes symmetric only fora value of Tc unphysically far from the peak temperature(arrow on Fig. 2, right inset). Similarly, the asymmetryis not healed by changing the background slope b. FromFig. 4 one can extract the exponents and their error bars(given from the variation within the gray areas). Thisgives a1 � 0.5 6 0.2 (with a2 � 20.5). To studya2 we use the sample YBCO3 (dTc � 100 mK) eventhough it has a wider transition, since it has the sharpestderivative peak dC2

dt . YBCO3 data are fitted both witha2 � 20.3 or a2 � 20.013, with a slightly better fitfor the 20.3 value. This leads to a2 � 20.2 6 0.3.Such a value is compatible with various models witha number of components of the order parameter N $ 2(accommodating the XY model, the SO(5) model, or theN � 4 d-wave model). For this sample also data aboveTc cannot be fitted with the 3D XY exponent and an

2044

FIG. 4. Exponent a as a function of different choices ofTc for samples UBC2 and YBCO3. Nominal background issubtracted. The gray area corresponds to the rounded region ofthe transition. The point where the two curves meet is denotedby an arrow on Fig. 2, right inset.

excellent fit is again obtained for the Gaussian exponent(a1 � 0.5).

We checked the robustness of this asymmetry againstvarious artifacts, such as disorder effects or a very asym-metric distribution of Tc’s around the optimal doping.Disorder cannot explain the effect. Based on Harris cri-terion it is relevant only for positive values of a andthus should affect the transition only above Tc. Besidesthis criterion, a phenomenological approach consideringthe distribution of Tc ’s within the sample shows that itdoes not produce the observed asymmetry: In the insetof Fig. 3 we show such an effect for a 3D XY transition.If one measures the slope in the rounded region, one mis-takenly finds a smaller (a 1 1) slope and the value of a

appears smaller than the real value. This should not af-fect our data since we systematically exclude the roundedregion from our fits. If it had affected our data, the reala2 should be closer to zero, whereas the real a1 shouldhave a larger positive value. In cleaner samples the ex-ponents would thus have slightly higher values than thereported ones, but those would not become more symmet-ric. A distribution of Tc around optimal doping is alsounable to explain the data since we find asymmetric expo-nents outside the rounded region. In the optimally dopedsamples enhanced rounding is expected on the left side ofTc because there is only one maximum value for Tc. Thiseffect should be reversed in overdoped samples, whereenhanced rounding is expected on the right side of Tc

(one has only minimum value of Tc). We checked for thiseffect by annealing our best crystal UBC2 in adequateconditions in order to push it in the overdoped region. Weobtained x � 0.99 and Tc � 89.2 K; the transition widthis 3 times larger than in optimal doping, even though ourannealing conditions were specially designed to enhancegood doping homogeneity. The derivative of Cp data forthe UBC2 overdoped sample is shown on Fig. 1 togetherwith data for the optimally doped UBC2. The asymmetry

Page 4: Asymmetry of Critical Exponents in

VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER 1999

both in the exponents and in dC1�dtdC2�dt persists and are still

incompatible with the 3D XY prediction. The rounding ofthe transition seems to affect mostly the right side of thetransition (Tc defined as the zero dC�dt point), in agree-ment with the Harris criterion.

Although the exponents are asymmetric, the weightof fluctuations taken separately above and below Tc

is reasonable. Above Tc, C1sing � C0t20.5 gives C0 �

0.218 mJ�gK. From the relation C0 � kB��j304p

2 p�we obtain j0 � 8.2 Å, a very reasonable value for thecoherence length. Below Tc YBCO3 data are fitted withC2

sing � B2 2A2

0.013 jtj0.013 and give A2

0.013 � 97 mJ�gK.Using the relation A2 � �kB�j

3f�R3

f and Rf � 0.8 [23]we find jf � 8.7 Å, again a very reasonable value forthe coherence length below Tc. A possible interpretationof the data is that we observe above Tc the Gaussianfluctuations, whereas the system below Tc is in the criticalregime. Using our specific heat data we can estimate thewidth of the critical regime. By equating the singularpart of the specific heat (using the coefficient C0) withthe mean-field jump (DMF

C � 5 mJ�gK [19]), we find acritical width T 2 Tc � 0.17 K. The critical width willbe 4 times smaller (T 2 Tc � 0.04 K) if a double mean-field jump (strong coupling) is considered [24]. Thusthe critical region above Tc is very close to the roundedregion, and true critical behavior above Tc cannot beobserved, consistent with our observation of Gaussian-likefluctuations. Of course this should also be true below Tc,so it remains a mystery why the critical regime below Tc

would be much larger. The standard Ginzburg criteriongives a critical region twice larger above Tc than below,because thermal fluctuations are less effective below Tc,where the establishment of the order parameter enhancesthe restoring force for fluctuations. A similar result isexpected for the 3D XY model: the coherence lengthbelow Tc is 2.5 times larger than the coherence lengthabove Tc [23,25], indicating a larger energy cost forfluctuations below Tc and a smaller critical region. In4He the critical regime appears symmetric [21].

Fit of the data in various temperature ranges suggeststhat “Gaussian” fl uctuations govern the behavior aboveTc all the way from Tc 1 10 K down to Tc 1 0.04 K(rounded region). “Critical” fl uctuations govern the be-havior from Tc 2 0.04 K down to Tc 2 10 K. With theexponents being so different from the two sides, a muchstronger divergence above Tc explains the huge asymme-try of dC

dt . The values of critical exponents may thus beunderstood separately (amplitude fluctuations above Tc,phase fluctuations below Tc), but their asymmetry is notcompatible with a regular second order transition. Com-plications such as a dimensional crossover could changethe apparent value of the exponents but would not explainthe observed asymmetry. To the best of our knowledgeno theoretical explanation exists at present. Some pos-sibilities could be the existence of a weakly first orderphase transition or coincidence of two phase transitions

close to Tc, but clearly more theoretical work is needed.A point remaining to clarify is if the effect will persist inthe underdoped regime and that there is no coupling witha singular phonon background at Tc.

*Deceased.†Corresponding author.Email address: [email protected]

[1] D. S. Fisher, M. P. A. Fischer, and D. A. Huse, Phys. Rev.B 43, 130 (1991).

[2] B. I. Halperin, T. C. Lubensky, and S. K. Ma, Phys. Rev.Lett. 32, 292 (1974).

[3] C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556(1981).

[4] Z. X. Shen et al., Science 273, 325 (1996).[5] V. J. Emery and S. A. Kivelson, Nature (London) 374, 434

(1995).[6] J. Zinn-Justin, Quantum Field Theory and Critical Phe-

nomena (Oxford University Press, New York, 1989).[7] Shou-Cheng Zhang, Science 275, 1089 (1997).[8] S. Kamal, D. A. Bonn, N. Goldenfeld, P. J. Hirschfeld,

R. Liang, and W. N. Hardy, Phys. Rev. Lett. 73, 1845(1994).

[9] S. M. Anlage, J. Mao, J. C. Booth, D. H. Wu, and J. L.Peng, Phys. Rev. B 53, 2792 (1996).

[10] J. C. Booth et al., Phys. Rev. Lett. 77, 4438 (1996).[11] S. E. Inderhees, M. B. Salomon, J. P. Rice, and D. Gins-

berg, Phys. Rev. Lett. 66, 232 (1991).[12] N. Overend, M. A. Howson, and I. Lawrie, Phys. Rev.

Lett. 72, 3238 (1994).[13] N. Overend et al., Phys. Rev. B 54, 9499 (1996).[14] A. Junod, Studies of High Temperature Superconductors,

edited by A. V. Narlikar (Nova Science Publishers, Com-mack, NY, 1996), Vol. 18.

[15] S. Regan, A. J. Lowe, and M. A. Howson, J. Phys.Condens. Matter 3, 9245 (1991).

[16] M. B. Salamon, J. Shi, N. Overend, and M. A. Howson,Phys. Rev. B 47, 5520 (1993).

[17] O. Riou, P. Gandit, M. Charalambous, and J. Chaussy,Rev. Sci. Instrum. 68, 1501 (1997).

[18] F. Fominaya, T. Fournier, P. Gandit, and J. Chaussy, Rev.Sci. Instrum. 68, 4191 (1997).

[19] J. W. Loram, K. A. Mirza, J. R. Cooper, and W. Y. Liang,Phys. Rev. Lett. 71, 1740 (1993).

[20] Substitution of gold in the Cu chains up to 10% wasdetected, with no apparent influence on the height of thesingularity.

[21] J. A. Lipa and T. C. P. Chui, Phys. Rev. Lett. 51, 2291(1983).

[22] A. Singsaas and G. Ahlers, Phys. Rev. B 30, 5103 (1984).[23] T. Schneider and D. Ariosa, Z. Phys. B, Condens. Matter

89, 267 (1992).[24] We note that those numbers set a strict limit to the

application of the Ginzburg criterion in YBCO singlecrystals; they are in accordance with earlier estimates [26]and are much smaller than estimates [1].

[25] From the relations A2j3f � R3

fkB and A1j3 � R3kB

with Rf � 0.8 [23], R � 0.3, and A1�A2 � 1.03, wededuce jf � 2.5j.

[26] C. J. Lobb, Phys. Rev. B 36, 3930 (1987).

2045