Structure of Neutron-Scattering Peak in both s -wave and s -wave states ++ ± of an Iron pnictide Superconductor Seiichiro Onari1, Hiroshi Kontani2, and Masatoshi Sato2 1 Department of Applied Physics, Nagoya University and JST, TRIP, Furo-cho, Nagoya 464-8602, Japan. 2 Department of Physics, Nagoya University and JST, TRIP, Furo-cho, Nagoya 464-8602, Japan. (Dated: January 6, 2010) Westudytheneutron scattering spectrum in iron pnictidesbased on therandom-phaseapproxi- mationinthefive-orbitalmodel,forfully-gappeds-wavestateswithsign reversal(s±)andwithout 0 1 sign reversal (s++). In the s++-wave state, we find that a prominent hump structure appears just above the spectral gap, by taking account of the quasiparticle damping γ due to strong electron- 0 electron correlation: As the superconductivity develops, the reduction in γ gives rise to the large 2 overshoot in the spectrum above the gap. The obtained hump structure looks similar to the reso- n nance peak in the s±-wave state, although the height and weight of the peak in the latter state is a muchlarger. Inthepresent study,experimentally observed broad spectral peak in iron pnictidesis J naturally reproduced byassuming thes++-wave state. 6 PACSnumbers: 74.20.-z,74.20.Rp,78.70.Nx ] n o Since the discovery of superconductivity in iron pnic- tivity ρcr for vanishing T is about 20 µΩcm. However, c imp c - tideswithhightransitiontemperature(Tc)nexttohigh- experimental ρcimrp reaches ∼ 750 µΩcm, which corre- r p Tc cuprates[1],thestructureofthesuperconducting(SC) sponds to the minimum metallic conductivity 4e2/h per u gap has been studied very intensively. The SC gap in layer [26]. Since this result supports a conventional s- s manyironpnictidesisfully-gappedandband-dependent, wave state without sign reversal (s -wave state), we . ++ t as shown by the penetration depth measurement [2] and haveto resolvethe discrepancybetweenneutronscatter- a m theangle-resolvedphotoemissionspectroscopy(ARPES) ing measurements and the impurity effects. [3, 4], except for P-doped Ba122 [5]. The fully-gapped In this letter, we study the dynamical spin suscepti- - d state is also supported by the rapid suppression in 1/T1 bility χs(ω,Q) based on the five-orbital model [9] for n (∝Tn; n∼4−6) below Tc [6–8]. boths++ ands± wavestates,anddiscussbywhichpair- o Inironpnictides,thenestingoftheFermisurface(FS) ing state the experimental results are reproducible. In c [ between hole- and electron-pockets is expected to in- the normal state, χs(ω,Q) is strongly suppressed by the duce the antiferromagnetic (AF) fluctuations in doped quasiparticledamping γ due to strong correlation. How- 2 metal compounds. Since fully-gapped sign-reversing s- ever, this suppression diminishes in the SC state since γ v 2 wavestate (s±-wavestate)is a naturalcandidate [9,10], isreducedastheSCgapopens. Forthisreason,apromi- 7 it is urgent to clarify the sign reversalin the SC gap via nent hump structure unrelated to the resonance mecha- 4 phase-sensitiveexperiments. Oneofthepromisingmeth- nismappearsinχs(ω,Q)justabove2∆inthe s++-wave 3 ods is the neutron scattering measurement: Existence of state. In the s±-wave state, very high and sharp reso- . 0 theresonancepeakatanestingwavevectorQisastrong nance peak appears at ωres <2∆. We demonstrate that 1 evidence for AF fluctuation mediated superconductors thebroadspectralpeakobservedinironpnictidesisnat- 9 with sign reversal [11–13]. The resonance condition is urally reproduced based on the s++-wave state, rather 0 : ωres < 2∆, where ωres is the resonance energy and ∆ is than the s±-wave state. v magnitude of the SC gap at T =0. The resonance peak Now, we study the 10×10 Nambu BCS Hamiltonian Xi has been observedin many AF fluctuation mediated un- Hˆk composedofthe five-orbitaltight-bindingmodeland r conventionalsuperconductors,like high-Tc cuprates [14– the band-diagonal SC gap introduced in ref. [25]. The a 16], CeCoIn [17], and UPd Al [18]. FSs are shown in Fig. 1 (a). Then, the 10×10 Green 5 2 3 function is given by Neutron scattering measurements for iron pnictides have been performed [19–22] after the theoretical pre- Gˆ(iω ,k) Fˆ(iω ,k) −1 dictions [23, 24]. Although clear peak structure was ob- Gˆ(iωn,k) ≡ Fˆ†(iωn,k) −Gˆ(−inω ,k) served in FeSe Te [20] and BaFe Co As [21], (cid:18) n n (cid:19) 0.4 0.6 1.85 0.15 2 its weight is much smaller than that in high-Tc cuprates = (iωnˆ1−Σˆk(iωn)−Hˆk)−1, (1) and CeCoIn , and the resonance condition ω < 2∆ is 5 res where ωn = πT(2n+1) is the fermion Matsubara fre- not surely confirmed, as we will discuss later. quency, Gˆ (Fˆ) is the 5×5 normal (anomalous) Green Nonmagnetic impurity effect also offers us useful function, andΣˆ is the self-energyin the d-orbitalbasis. k phase-sensitiveinformation. Theoretically,s -wavestate For a while, we assume that the SC gap for the α-th FS ± shouldbeveryfragileagainstimpuritiesduetotheinter- is band-independent; |∆ | = ∆. Hereafter, the unit of α band scattering [25]; the predicted criticalresidual resis- energy is eV, unless otherwise noted. 2 Here,wehavetoobtainthespinsusceptibilityasfunc- (a) 8(b) S++ wave Approx. normal tion of real frequency. For this purpose, it is rather easy Approx. SC to use the Matsubara frequency method and the numer- V] ∆=0.4 Exact normal ical analytic continuation (pade approximation). In the FS1 FS3 /e2B 6 U=1.2 Exact SC µ present study, however, we perform the analytical con- [ tinuation before numericalcalculationin order to obtain FS2 sχ 4 m more reliable results. The irreducible spin susceptibility I 2 in the singlet SC state is given by [13] FS4 0 χˆ0l1Rl2,l3l4(ω,q) = N1 d2x (2) 0 1ω [eV] 2 k Z X x tanh GR (x ,k )ρG (x,k) FIG. 1: (Color online) (a) FSs in iron pnictides. (b) ω- + thanhx2+TρGl1(l3x ,+k )+GAl4l(2x,k) danepdenthdeenncoermofaIlmsχtast(eω,,wQh)efroertthheeusn+i+t-owfaveenesrtgayteis(∆eV=. T0.h4e) 2T l1l3 + + l4l2 “exact result” is obtained by eq. (2), and the “approximate x + tanh FR (x ,k )ρF† (x,k) result” is obtained by eq. (6). 2T l1l4 + + l3l2 x + tanh +ρF (x ,k )F†A(x,k) , 2T l1l4 + + l3l2 Fermi velocity of the α-th FS, we obtain ρ ≈ (2γ[meV]) i µΩcm [25]. Since ρ(T) − ρ(0) ∼ (5T[meV]) µΩcm in where x =x+ω, k =k+q, l =1∼5 representsthe + + i BaFe Co As below 100 K [27], γ(0) due to inelas- d-orbital, and A (R) represents the advanced (retarded) 1.84 0.16 2 tic scattering is estimated as 2.5T which is comparable Greenfunction. ρG(x,k)≡(GA(x,k)−GR(x,k))/(2πi) ll′ ll′ ll′ tothatinover-dopedcuprates. Ifweassumetherelation and ρF(†)(x,k) ≡ (F(†)A(x,k)− F(†)R(x,k))/(2πi) are ll′ ll′ ll′ γ(ǫ)∝(πT+ǫ)innearlyAFFermiliquid[28],weobtain one particle spectral functions. Since ρG,F(x,k) = 0 for ll′ γ(ǫ)∼2.5(T +ǫ/π). |x| < ∆, Imχˆ0llR,l′l′(ω,q) = 0 for |ω| < 2∆. That is, the Now, we calculate Imχs(ω,Q) in both normal and particle-hole excitation gap is 2∆. s -wave SC states, concentrating on the frequency ++ Then, the spin susceptibility χs(ω,q) is given by the ω ∼2∆. To estimate the renormalizationof Imχs(ω,Q) multiorbital random-phase-approximation (RPA) with due to the self-energy,we have to know the value of γ(ǫ) the intraorbital Coulomb U, the interorbital Coulomb with |ǫ|∼∆ in both normal and SC states. Considering U′, the Hund coupling J, and the pair-hopping J′ [9]: that γ(ǫ) = 2.5(T +ǫ/π) ∼ 2∆ at T = 2.2 meV and c ǫ = ∆ ∼ 5 meV in BaFe Co As , in the present 1.85 0.15 2 χs(ω,q)= χˆ0R(ω,q) , (3) study, we simply put γ(ǫ) in the normal state at Tc as "1−Sˆ0χˆ0R(ω,q)# Xi,j ii,jj γ(ǫ)=γ (4) 0 where vertex of spin channel Sˆ0 =U, U′, J and J′ l1l2,l3l4 with γ0 & ∆. In the present model, αSt = 0.84 (0.79) for l = l = l = l , l = l 6=l = l , l = l 6= l = l 1 2 3 4 1 3 2 4 1 2 3 4 for U = 1.3 (1.2) when γ = 0.1 and T = 0.002; the 0 and l = l 6= l = l , respectively. Hereafter, we put 1 4 2 3 T-dependence of α is small when γ is fixed. J = J′ = 0.15, U′ = U −2J, U = 1 ∼ 1.3, and fix the St 0 In the SC state at T ≪ T , γ(ǫ) = 0 for |ǫ| < 3∆ (= c electronnumberas6.1(10%electron-dopedcase). Inthe particle-hole excitation gap 2∆ plus one-particle gap ∆) presentmodel,χs(0,q)takesthemaximumvaluewhenq [12], and its functional form is approximately the same is the nesting vector Q=(π,π/16). Due to the nesting, as that in the normal state for |ǫ|&3∆. Then, we put χs(0,Q)/χ0(0,Q) ≈ 1/(1−α ) is enhanced; α (. 1) St St is the maximum eigenvalue of Sˆ0χˆ0R(0,Q) that is called γ(ǫ)=a(ǫ)γ (5) s the Stoner factor. In strongly correlated systems, χs(ω,q) is renormal- where(i)a(ǫ)≪1for|ǫ|<3∆,(ii)a(ǫ)=1for|ǫ|>4∆, ized by the self-energy correction. In nearly AF met- and(iii)linearextrapolationfor3∆<|ǫ|<4∆. Wehave als,forexample,thetemperaturedependence ofthe self- confirmedthatthe obtainedresultsareinsensitivetothe energy induces the Curie-Weiss behavior of χs(0,Q). At boundaryof|ǫ| (4∆inthe presentcase)between(ii)and themoment,thereisnoexperimentalinformationonthe (iii). Although γ at T ≪ T should be smaller than γ s c 0 k-, ǫ-, and band-dependences of the self-energy. There- atT =T ,wesimplyputγ =γ hereafter,whichcauses c s 0 fore, we phenomenologically introduce a band-diagonal underestimation of the peak height of Imχs. self-energy as ΣˆR(ǫ) = iγ(ǫ)ˆ1. First, we estimate the Figure 1 shows Imχs(ω,Q) obtained by eqs. (2) and k value of γ(ǫ) in the normal state. Since the conduc- (3) for U = 1.2, γ = 0.4 and T = 0.01. In the s - 0 ++ tivity is given by σ = e2 N (0)v2/2γ(0), where wave SC state, we put ∆ = γ and a(3∆) = 0.05. In α α α 0 N (0) and v are the density of states (DOS) and the calculating eq. (2), we use 256 × 256 k-meshes, and α α P 3 1000 x-meshes. Although values of ∆ and γ in Fig. 1 16 (a)S++ wave (b) ∆=0.05 U=1.3 8 γ=0.05 araretioveγry/l∆arg∼e t1oiosbctoaninsisetneonutgwhitnhumexepriecrailmaecnctusr.acIyn, tthhee sχ 12 γ0=0.003 6 0 normal0state, Imχs(ω,Q) is suppressed by large quasi- Im 8 4 4 2 particle damping γ0 ∼ ∆. In the SC state, the gap in Imχs(ω,Q) is 2∆. Since the particle or hole with en- 0 0 8(c) 8(d) ergy |ǫ| < 3∆ is free from inelastic scattering in the SC state, the lifetime of particle-hole excitation with energy s 6 γ0=0.075 6 γ0=0.1 χ |ǫ|<4∆becomeslong. Forthisreason,Imχs(ω,q)shows m 4 4 I a large hump structure for 2∆ . ω . 4∆ below Tc in 2 2 s++-wave state. 00 0.1 0.2 0.30 0 0.1 0.2 0.3 Unfortunately, we cannot put smaller ∆ and γ in cal- ω ω culating eq. (2) in the five-orbital model, because of the computationtime. Tosolvethisproblem,weperformthe FIG. 2: (Color online) Imχs(ω,Q) for s++-wave (solid line) x-integration in eq. (2) approximately as follows: When and normal (brokenline) states, with γ0 =0∼0.1. γˆ = γˆ1, the retarded (advanced) 10 × 10 Green func- tion is expressedas GˆR(A)(x,k)= Um,α(x+(−)iγ− m,m′ α k Ekα)−1Ukm′,α∗,whereEkα(α=1∼1P0)istheeigenvalueof einqcuraelasteostfhroamt in(bt)h0e.0n5ortmoa(ld)st0a.t1e,fIomrχωs >in 2th∆e. nAorsmγa0l Hˆk and Uˆk is the corresponding 10×10 unitary matrix. state decreases gradually, whereas that in the SC state We promise that Ekα = −Ekα+5 for 1 ≤ α ≤ 5. When depends on γ0 only slightly, since γ(ǫ) ≈ 0 for |ǫ| < 3∆. γ is sufficiently small, then ρG(F)(x,k) ≈ Ul,αδ(x− Therefore, in the case of γ & ∆, Imχs(ω,Q) in the SC ll′ α k 0 Eα)Ul′(+5),α∗, and thus eq. (2) becomes state shows a prominent hump structure, and its peak k k P valueis aboutdoubleofthatinthe normalstate. In(d), 1 f(El)−f(El′ ) experimentalapproximate“sum-rule”atfixedq =Q[21] χˆ0R (ω,q)≈ k k+q is well satisfied. In Fig. 2 (c) and (d), a relatively large l1l2,l3l4 N k l,l′ ω+Ekl −Ekl′+q +iΓll′,kq slopefor|ǫ|<2∆isanartifactoftheapproximationdue XX Ul1,l′Ul3,l′∗Ul4lUl2l∗+Ul1,l′Ul4+5,l′∗Ul3+5,lUl2l∗ ,(6) to large γ0/∆. k+q k+q k k k+q k+q k k h i 8(a) Isotropic S++ wave (b) AnisotropicS++ wave witWhhΓelnl′,kγqi=s aγsfloarrgγe≪as1∆. , however, we have to check 6 γU0 == 10..31 ∆∆mmainx == 00..00375 to what extent eq. (6) is reliable. Considering that the s χ origin of the renormalization of χs is the quasiparticle m4 dampingγ(El)andγ(El′ ),weintroduce thefollowing I k k+q 2 approximation: Γll′,kq =b·max{γ(Ekl),γ(Ekl′+q)} (7) 00 0.1 0ω.2 0.3 0.4 0 0.1 0ω.2 0.3 0.4 SwChersetabte≈fo1r |iEskla|,fi|Ettkli′n+gq|p<ara3m∆e,terre.fleΓctlli′n,kgqt≈he0abinsenthcee FanIGd.n3o:rm(Caollo(rbroonkleinne)linIme)χss(tωat,eQs,)wfoirths+∆+m-waxave=(s0o.l0id7lianned) of quasiparticle damping. In Fig. 1, we show numerical ∆min=0.035. results given by the present approximationwith b=1.3; we replace bγ with γ hereafter since b ≈ 1. Since the Next, we study the effect of band-dependent SC gap 0 0 “exact results” given by eq. (2) is quantitatively repro- observed by ARPES measurements [3, 4]. In Fig. 3 duced, we decide to calculate Imχs(ω,Q) using eqs. (6) (a), we put U = 1.3, ∆ = ∆ = 0.07eV for 1,2,4 max and (7) for more realistic values of ∆ and γ. We veri- FS1,3,4, and ∆ = ∆ = 0.035eV for FS2. In (b), 2 min fiedthatthepresentapproximationworkswellwhenγ is we introduce the anisotropy of the gap function for only comparable to or smaller than ∆. FS3,4 with ratio 2; ∆ = ∆ (1−0.5sin2θ ), where k max k Figure 2 shows Imχs(ω,Q) obtained by eqs. (6) and θ = tan−1(|k |/(|k |−π)) for FS3(4). Here, we k y(x) x(y) (3)forU =1.3andT =0.002. Inthes -waveSCstate, put a(ǫ) in eq. (5) as (i) 0.003/γ for |ǫ| < 3∆ , ++ 0 min we put ∆ = 0.05; although it is a few times larger than (ii) 1 for |ǫ| > 4∆ , and (iii) linear extrapolation for min the gap for Sm1111 with T = 56K, it is enough smaller 3∆ < |ǫ| < 4∆ . In Fig. 3 (a), Imχs(ω,Q) in- c min min than the Fermi energies of electron- and hole-pockets creases rapidly at ω = ∆ + ∆ = 0.105, and it max min [9]. In the numerical calculation, we use 1024×1024 k- shows a peak at ω = 0.14. In (b), the peak is located meshes. Hereafter, we put a(3∆)=0.003/γ . When (a) at ω = 0.125, which is closer to ∆ +∆ = 0.105. 0 max min γ = 0.003, Imχs(ω,Q) in the SC state approximately In Fig. 3 (a) and (b), the width of the hump peak is 0 4 muchsharperthanthatfortheband-independentSCgap fermion Kondo insulator CeNiSn. As shown in Fig. 1 of in Fig. 2, since Imχs(ω,Q) is reduced by damping for Ref. [31], neutron scattering spectrum at q =(0,π,0) in |ω|>4∆ =0.14. WehavealsocalculatedImχs(ω,Q) CeNiSn shows a prominent hump peak structure above min for ∆ =∆ and ∆ =∆ , and verified that the the hybridizationgap below the Kondo temperature T , 3,4 max 1,2 min K obtained result is similar to Fig. 3. which looks very similar to the spectrum observed in Here, we make comparison with experiments. The iron pnictides below T [19–22]. This hump structure c peak height and the weight in Fig. 3 (b) seems to is well reproduced by the dynamical-mean-field theory be consistent with the neutron scattering measurements based on the periodic Anderson model [32]. This fact in iron pnictides [19–22]. In BaFe Co As (T = demonstrates that large hump in Imχs(ω,Q) can ap- 1.85 0.15 2 c 25K), the observed ”resonance energy” is ω = 9.5 pearinstronglycorrelatedsystemswithone-particlegap, res meV [21]. According to ref. [3], ∆ /T ≈ 3.5 and without the necessity of the resonance mechanism. max c ∆min/∆max ≈0.35inmanyironpnictides. (Moresmaller Insummary,we havestudied Imχs(ω,Q)in ironpnic- ∆max,min is reported in ref. [2].) Thus, ∆max+∆min ≈ tides based on the five-orbital model, and revealed that 4.7Tc = 10 meV is comparable to ωres = 9.5 meV a prominent hump structure appears just above 2∆ in in BaFe1.85Co0.15As2. Moreover, finite Imχs(ω,Q) for the s++-wave state, by taking the strongly correlation ω & 0.3ωres in ref. [21] may suggest the existence of effect via γ. This hump structure becomes small as αs SC gap anisotropy. Therefore, the theoretical result in decreasesintheover-dopedregionorq deviatesfromthe Fig. 3 (b) is well consistent with experimental data. We nesting. At present, experimental data can be explained haveverifiedthatthe humpstructureofImχs(ω,q)with in terms of the s -wave state very well. Further ex- ++ q =(π,0) is very small for γ0 ∼0.1. perimental efforts are required to determine the height and width of the ”resonance peak”, and the magnitude 40(a) S+− -wave U=1.2 10(b) S+− -wave U=1.0 relation between ωres and ∆max+∆min. 30 |∆γ|==00..015 8 This study has been supported by Grants-in-Aid for sχ 0 6 Scientific Research from MEXT of Japan, and by JST, m 20 I 4 TRIP. Numerical calculations were performed using the 10 facilities of the supercomputer center, ISSP. 2 0 0 0.1 0.2 0.3 0 0 0.1 0.2 0.3 ω ω FIG. 4: (Color online) Imχs(ω,Q) for s±-wave (solid line) [1] Y.Kamihara,etal.,J.Am.Chem.Soc.130,3296(2008). and normal (broken line) states, with U =1.2 and 1.0. [2] K.Hashimotoetal.,Phys.Rev.Lett.102,017002(2009). [3] D.V.Evtushinskyetal.,NewJ.Phys.11,055069(2009). We also analyze the resonance peak for the s±-wave [4] K. Nakayamaet al., Europhys.Lett. 85, 67002 (2009). state in Fig. 4. In this case, the spin wave without [5] K. Hashimoto et al., arXiv:0907.4399. damping is observed as the “resonance peak” at ωres < [6] Y.Kobayashietal.,J.Phys.Soc.Jpn.78(2009)073704. 2∆. Figure4showsthenumericalresultsfor(a)U =1.2 [7] H. Mukudaet al., J. Phys.Soc. Jpn. 77, 093704 (2008). and (b) U = 1.0 in the case of ∆ = −∆ = 0.05. [8] G. Fuchset al., arXiv:0908.2101. 1,2 3,4 [9] K. Kuroki et al., Phys. Rev. Lett. 101, 087004 (2008); In (a), a very sharp and high resonance peak appears at K. Kurokiet al., Phys.Rev.B 79, 224511 (2009). ω = 0.85 < 2∆, consistently with previous theoretical res [10] I. I. Mazin et al., Phys.Rev.Lett. 101, 057003 (2008). studies [23, 24]. The case (b) with U = 1.0 corresponds [11] D. K. Morr and D. Pines, Phys. Rev. Lett. 81, 1086 to the ”heavily overdoped” since αSt =0.69 and Tc ∼0. (1998). The obtained resonance peak in Fig. 4 by taking γ(ǫ) [12] A. Abanov and A. V. Chubukov, Phys. Rev. Lett. 83, into account is too large to explain experiments even in 1652 (1999). the caseofα =0.69. InBi-basedhigh-T cuprates,the [13] T.TakimotoandT.Moriya,J.Phys.Soc.Jpn.67, 3570 St c (1998). width of the resonance peak is wide due to the sample [14] S. Iikubo,et al., J. Phys. Soc. Jpn. 74, 275 (2005). inhomogeneity (i.e., nanoscale distribution of T ) [16]. c [15] M. Ito, et al., J. Phys. Soc. Jpn. 71, 265 (2002). However,weightof the peak is 10 times larger than that [16] H. F. Fong, et al., Nature 398, 588 (1999). in BaFe1.85Co0.15As2 [21]. [17] C. Stock,et al., Phys.Rev.Lett. 100, 087001 (2008). In the present study, we have neglected the impurity [18] N. K. Sato, et al., Nature410, 340 (2001). effect since its influence on χs(ω,Q) is expected to be [19] A. D. Christianson, et al., Nature456, 930 (2008). small. Infact, inthe singlebandmodel,the reductionin [20] Y. Qiu, et al., Phys.Rev. Lett.103, 067008 (2009). χ0 due to the impurity self-energy is almost canceled by [21] D. S. Inosov,et al., arXiv:0907.3632. [22] J. Zhao, et al., arXiv:0908.0954. the impurity vertex correction [29]. Moreover, impurity [23] T. A. Maier and D. J. Scalapino, Phys. Rev. B 78, effect tends to enhance χs(ω,Q) in the modified FLEX 020514(R) (2008); T. A. Maier et al., Phys. Rev. B 79, approximation in nearly AF metals [30]. 224510 (2009). Before closing the study, we shortly discuss the heavy [24] M. M. Korshunov and I. Eremin, Phys. Rev. B 78, 5 140509(R) (2008). [29] N. Bulut, Physica C, 353, 270 (2001). [25] S. Onari and H. Kontani, Phys. Rev. Lett. 103 (2009) [30] H. Kontani, Rep.Prog. Phys. 71, 026501 (2007). 177001. [31] H. Kadowaki et al., J. Phys.Soc. Jpn.63, 2074 (1994). [26] M. Sato et al.,arXiv:0907.3007. [32] T. Mutou and D. S. Hirashima, J. Phys. Soc. Jpn. 64, [27] A.S.Sefat et al.,arXiv:0903.0629. 4799 (1997). [28] B. P. Stojkovic and D. Pines, Phys. Rev. B 56, 11931 (1997).