Quadrupole-Driven Non-Fermi Liquid and Magnetic-Field Induced Heavy Fermion States in a Non-Kramers Doublet System
T. Onimaru, K. Izawa, K. T. Matsumoto, T. Yoshida, Y. Machida, T. Ikeura, K. Wakiya, K. Umeo, S. Kittaka, K. Araki, T. Sakakibara, T. Takabatake
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Quadrupole-Driven Non-Fermi Liquid and Magnetic-Field Induced Heavy FermionStates in a Non-Kramers Doublet System
T. Onimaru, ∗ K. Izawa, K. T. Matsumoto, T. Yoshida, Y. Machida, T. Ikeura, K. Wakiya, K. Umeo, S. Kittaka, K. Araki, T. Sakakibara, and T. Takabatake
1, 5 Department of Quantum Matter, Graduate School of Advanced Sciences of Matter,Hiroshima University, Higashi-Hiroshima 739-8530, Japan Department of Condensed Matter Physics, Graduate School of Science and Engineering,Tokyo Institute of Technology, Tokyo 152-8551, Japan Cryogenics and Instrumental analysis Division, N-BARD,Hiroshima University, Higashi-Hiroshima 739-8526, Japan Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan Institute for Advanced Materials Research, Hiroshima University, Higashi-Hiroshima 739-8530, Japan (Dated: September 6, 2018)Orbital degrees of freedom in condensed matters could play important roles in forming a variety ofexotic electronic states by interacting with conduction electrons. In 4 f -electron systems, because ofstrong intra-atomic spin-orbit coupling, an orbitally degenerate state inherently carries quadrupolardegrees of freedom. The present work has focussed on a purely quadrupole-active system PrIr Zn showing superconductivity in the presence of an antiferroquadrupole order at T Q = 0.11 K. Weobserved non-Fermi liquid (NFL) behaviors emerging in the electrical resistivity ρ and the 4 f con-tribution to the specific heat, C f , in the paramagnetic state at T > T Q . Moreover, in magneticfields B ≤ ρ ( T ) and C f ( T ) are well scaled with characteristic temperatures T ’s.This is the first observation of the NFL state in the nonmagnetic quadrupole-active system, whoseorigin is intrinsically different from that observed in the vicinity of the conventional quantum criti-cal point. It implies possible formation of a quadrupole Kondo lattice resulting from hybridizationbetween the quadrupoles and the conduction electrons with an energy scale of k B T . At T ≤ ρ ( T ) and C f ( T ) exhibit anomalies as B approaches 5 T. This is the manifestation of a field-inducedcrossover toward a Fermi-liquid ground state in the quadrupole Kondo lattice. PACS numbers:
I. INTRODUCTION
In metals and alloys, localized d and/or f electronsinherently possessing ‘spin’ and ‘orbital’ degrees of free-dom are dominant sources of not only various magneticphenomena but also unconventional superconductivity.Extensive studies on a lot of ‘spin’-active systems havesuccessfully revealed the variety of the magnetic phenom-ena arising from competition of the inter-site Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the on-site magnetic Kondo effect, which was represented as thewell-known ‘Doniach diagram’. On the other hand, thereis less variety of researches on ‘orbital’-driven physics.Although the ‘orbital’ itself must have potential variety,there are few evidences of interplay between the orbitaldegrees of freedom and the conduction electron.In the f -electron systems, strong intra-atomic spin-orbit coupling forces the spin and orbital degrees of free-dom to be described in terms of the total angular momen-tum J . When the magnetic moment due to J interactsstrongly with itinerant conduction electrons, physicallyobservable quantities follow the Fermi-liquid (FL) modelof Landau, which is known as heavy fermion state. When ∗ Electronic address: [email protected] some conditions were fulfilled unexpectedly, the FL statecould become unstable, instead, an anomalous metallicstate would emerge, so-called non-Fermi liquid (NFL)state.[1–4] In the vicinity of the quantum critical point,there emerges unconventional superconductivity.[5]On the other hand, emergence of a different type ofNFL state was predicted theoretically, in case electricquadrupoles of the localized f electrons, which becomeactive only in an orbitally degenerate state, interactwith the conduction electrons, that is the quadrupole(two-channel) Kondo effect.[6] It is quite different fromthe ordinary (single-channel) magnetic Kondo effect interms of scattering process of the conduction electrons;the scattering source is not magnetic dipole but thetime-reversal electric quadrupole. Thereby, the impurityquadrupoles were overcompensated by the conductionelectrons, leading to the residual entropy of (0.5) R ln2and the NFL behavior: the magnetic susceptibility χ and the specific heat divided by temperature, C/T , showln T dependence, and the electrical resistivity follows ρ − ρ ∝ A √ T , where ρ is residual resistivity and A is a coefficient.[6, 7] NFL state and the residual entropywere also expected in a quadrupole Kondo ‘lattice’, inwhich quadrupole moments are periodically placed.[8]Very recently, the peculiar temperature dependence of ρ ( T ) and C ( T ) have been theoretically predicted.[9, 10]Moreover, in the lattice model, new types of electronicordered states have been proposed.[11–13] Nevertheless,there is no experimental evidence on the exotic states,probably because there are rare systems carrying purelyactive quadrupoles.Over the past few decades, experimental efforts havebeen paid to address the issues that could arise fromthe impurity quadrupole Kondo effect.[14] NFL behaviorswere observed in uranium- and praseodymium-based sys-tems with 5 f and 4 f configurations, respectively, suchas U x Y − x Pd [15, 16], UBe [17], U x Th − x Be [18],U x Th − x Ru Si [19], and Pr x La − x Pb [25]. However,there is no firm experimental evidence for the impurityquadrupole Kondo effect, since atomic disorder or un-certainty in the crystalline electric field (CEF) groundstates particularly in the U-based systems prevent theclarification.Recently, a family of the praseodymium-based systemsPr T X ( T : transition metals, X : Al, Zn, and Cd) haveemerged as a prototype to study the quadrupole Kondoeffect, since the CEF ground state is a non-Kramers dou-bly degenerated state, that is labeled as the Γ doublet inthe cubic T d point group, having no magnetic dipoles butelectric quadrupoles.[26–31] In PrIr Zn , an AFQ orderoccurs at T Q = 0.11 K.[27] Although the entropy releaseof R ln2 is expected from an order of the doubly degener-ated CEF ground state, the entropy at T Q estimated fromthe specific heat is only 20% of R ln2. Therefore, thereshould be another mechanism, except the AFQ order,which consumes the rest of the entropy above T Q . Onthe other hand, below T Q , a superconducting transitionsets in at T c = 0.05 K, suggesting a possible interplay be-tween the quadrupole fluctuations and the superconduct-ing Cooper-pair formation.[27, 32–34] The coexistenceof superconductivity with quadrupole order also mani-fests in isostructural compounds PrRh Zn , PrTi Al ,and PrV Al .[35–38] In PrTi Al and PrV Al , thestrong hybridization between the 4 f elections and theconduction electrons was revealed by the Al-NMR and Pr3 d -4 f resonant photoemission measurements.[39, 40] Thelarge Seebeck coefficient divided by temperature S/T ob-served in Pr T Al ( T = Ti, V and Ta) also suggests thestrong hybridization effect, although that for PrIr Zn at B = 0 is two or three orders smaller than those ofPr T Al .[41] PrV Al exhibits NFL behavior above T Q = 0.6 K; the 4 f contributions in both ρ and χ follow √ T for 2 20 K[28], however, in the temperature range,the magnetic degrees of freedom in the first excited statemust interfere in the NFL behavior. Quadrupolar quan-tum criticality induced by application of magnetic fieldhas been proposed.[42]In the present work, we study the transport, thermody-namic and magnetic properties for the non-Kramers dou-blet system PrIr Zn in magnetic fields applied alongthe [100] direction. For B || [100], the AFQ order col-lapses at around 5 T, whose value is much lower thanthe critical fields for the AFQ order in the [110] and[111] field directions.[27, 32, 43] Since high quality of thesample is necessary to reveal the inherent behavior aris-ing from the quadrupolar degrees of freedom as described above, we have chosen single-cryslltaine samples with lowresidual resistivity of 0.2 µ Ω cm (residual resistivity ra-tio, RRR > II. EXPERIMENTAL Single-crystalline samples of PrIr Zn used in thepresent work were grown by using the melt-growthmethod described in the previous papers.[34] Magneti-zation was measured using a commercial SQUID mag-netometer (Quantum Design MPMS) between 1.9 and350 K in magnetic fields up to 5 T. Magnetization mea-surements at low temperatures down to 0.045 K wereperformed by a capacitive Faraday method using a high-resolution capacitive force-sensing device installed in a He- He dilution refrigerator.[44] Electrical resistancewas measured by a standard four-probe dc method ina laboratory-built system with a He- He dilution refrig-erator. Thermopower was measured using a laboratory-made probe by applying a temperature difference of 0.04-0.3 K along a bar-shaped sample. Specific heat in mag-netic fields was measured by a relaxation method at tem-peratures between 0.4 K and 300 K. The measurementsat low temperatures down to 0.06 K were done underquasi-adiabatic conditions with a He- He dilution re-frigerator equipped with a superconducting magnet of 12T. III. RESULTS AND DISCUSSIONA. Specific heat Figure 1 (a) shows the temperature dependence of the4 f contribution to the specific heat, C f , of PrIr Zn inmagnetic fields B ≤ 12 T applied along the [100] direc-tion. To estimate the 4 f contribution C f to the totalspecific heat C , we subtracted the nuclear and phononcontributions C nuc and C ph as described below. Thehamiltonian of a nuclear spin of a Pr nucleus, I =5/2,in magnetic field B can be represented as H nuc = A hf J · B − g N µ N B · I , (1)where A hf =0.052 K is a coupling constant of hyper-fineinteraction of a Pr ion [45], and J , g N =1.72, and µ N arethe total angular momentum of 4 f electrons, nuclear g -factor, and nuclear magneton, respectively. The compo-nent of J along the magnetic field direction was regarded C f ( J / K m o l ) B = 01 T235 6 7 810 12 T PrIr Zn B || [100] T Q (a) f S ( J / K m o l ) 12 T B =0 (b) R ln2 R ln2 T (K) 200.03 R ln2 -100 C f ( J / K m o l ) B =3 T3.544.555.5 T Q T * Pr (16c)Zn (96g)Zn T Sch T ( C ) FIG. 1: (color online) (a) 4 f contribution to the specific heat C f of PrIr Zn in magnetic fields up to 12 T applied alongthe [100] direction. The data of C f are vertically offset forclarity. T Q indicates the temperature of AFQ order. The opentriangles are T Sch , where Schottky-type peaks appear as a re-sult of the splitting of the ground state doublet by applyingthe magnetic fields. (b) Entropy S f estimated by integrat-ing C f ( T ) /T in various fields.The arrow indicates character-istic temperature T ( C )0 defined as the temperature where S f reaches R ln2. The inset shows the data of C f for 3 ≤ B ≤ T Q and a crossover temperature T ∗ emerging only in B =4.5 and 5 T, which will be describedin the text. A Pr ion encapsulated into the symmetric Zn-cage, which is formed by four Zn atoms at the 16 c site andtwelve Zn atoms at the 96 g site, is also shown in the inset. as the isothermal magnetization measured at T =0.045 K.Thereby, the nuclear specific heat was estimated by tak-ing the eigenvalues of the hamiltonian into consideration.The nuclear contribution is zero in B =0. Applying mag-netic field, it gradually increases on cooling, e.g. , in B =3 T, C nuc is increased up to about 3.5 J/K mol at 0.1 K.The contribution of the phonon was subtracted by usingthe specific heat of the La analog LaIr Zn . The mainpanel of Fig. 1 (b) shows the entropy S f estimated byintegrating C f /T . The value of S f reaches R ln2 at 2K, supporting that the physical properties at T < T Q = 0.11 K for B = 0 in Fig.1 (a) is the manifestation of the AFQ order.[27, 32, 33]As shown with the open triangles in the inset of Fig. 1(b), upon applying magnetic field for B ≥ (b) PrIr Zn C f / T ( J / K m o l ) B = T T QuadrupoleKondo LatticeQuadrupoleKondo Lattice T + A T () B = 0 T (ρ) B || [100] (a) ∆ ρ ( T ) / ∆ ρ ( T ) ρ ( ) , T / T ρ ( ) T ( C ) T / 1.00.50 3210 4 3 2 1 T B = 0 8 T 7 6.5 6 5.5 5 C f ( T ) / C f ( T ) ( C ) B = 0 1 T 2 3 4 5 6 FIG. 2: (color online) Scaling plots of (a) the specific heat C f and (b) the electrical resistivity ∆ ρ of PrIr Zn inthe magnetic fields B || [100]. In the temperature region0.8 < T /T ( C )0 , T /T ( ρ )0 < C f and ∆ ρ in magnetic fieldsbetter follow the calculation by using a two-channel Andersonlattice model [10] as shown with the (red) solid curves than the(blue) dotted curve calculated with the impurity quadrupoleKondo model[6]. The inset of (a) shows the temperature de-pendence of C f /T with vertical offsets. The temperaturevariation of ρ/ ( ρ + A √ T ) is shown in the inset of (b). At thevalue of 1, ρ ( T ) follows √ T . The arrows show the character-istic temperature, T ( ρ )0 , where ρ ( T ) deviates from ρ + A √ T relation. shifts to lower temperatures and disappears at B =4.5T. The behavior of the specific heat peak at T Q in themagnetic fields is much different from the previous reportwhere the peak is split into two for 1 ≤ B ≤ a -axis of the dominantcrystal.On the other hand, as shown with the open trianglesin Fig. 1 (a), C f ( T ) in B = 6 T shows a broad peakat 0.6 K, which shifts to 2 K with increasing B up to12 T. The height and width can be explained by takingaccount of the Zeeman splitting of the ground state dou-blet. Because the split singlets lose quadrupolar degreesof freedom, no phase transition occurs in B > C f ( B =0) at around 0.4 K in Fig. 1(a). When C f /T is plot-ted vs ln T in the inset of Fig. 2(a), we find the − ln T dependence between 0.2 and 0.8 K. Since this − ln T de-pendence of C f /T emerges below 2 K, it arises fromthe degrees of freedom of the Γ ground state. Here, wedefine a characteristic temperature, T ( C )0 , as the temper-ature where S f reaches R ln2. This definition followsthe Cox’s definition of the Kondo temperature T K for theimpurity quadrupole Kondo system.[6] As shown in Fig.1(b), a defined T ( C )0 remains at around 0.4 K up to B =3T and increases for B> C f ( T ) /C f ( T ) at various fields,which are well scaled with respect to T /T ( C )0 in the rangeof T /T ( C )0 > B. Electrical resistivity The temperature dependence of the electrical resistiv-ity ρ ( T ) in magnetic fields between 0 and 9 T appliedalong the [100] direction are shown in Fig. 3. It is notedthat all the data are plotted without offset. In B ≤ ρ ( T ) shows upward convex curvature at T Q < T < ρ ( T ) is likely to follow the √ T variation as shown withthe dashed lines. The absolute value of ρ ( T ) for B ≤ ρ ( T ) shows downward curvature.The residual resistivity is increased with increasing themagnetic field.We pay our attention again to the NFL behavior ofthe upward convex curve at T Q < T < ρ ( T ) are replotted in the inset of Fig. 2 (b)as ρ/ ( ρ + A √ T ) vs T , where ρ is the residual resistiv-ity. In the T region where ρ/ ( ρ + A √ T ) stays at 1.0,∆ ρ = ρ ( T ) − ρ follows √ T . The arrows denote the char-acteristic temperature, T ( ρ )0 , where ρ ( T ) starts deviatingfrom the √ T dependence on heating. We represent thedata of ∆ ρ ( T ) / ∆ ρ ( T ) vs T /T ( ρ )0 in Fig. 2 (b), where ρ ( T ) follows the scaling well in the temperature range of0.5 < T /T ( ρ )0 ≤ 3. As shown in the inset of Fig. 2 (b), T ( ρ )0 stays at around 0.35 K up to B =4 T, and increasessignificantly once B exceeds 4 T (see the (red) openeddiamonds in Fig. 5 (a)). The field dependence of T ( ρ )0 coincides with that of T ( C )0 described in the previous sub-section, suggesting the same origin of the both T ( ρ )0 and T ( C )0 .[14] C. Non-Fermi liquid behavior for T > T Q Let us discuss possible mechanisms for the observedNFL behaviors of C f ( T ) and ∆ ρ ( T ) in the magneticfields well scaled with respect to T /T ( C )0 and T /T ( ρ )0 , re- spectively. One is the contribution of a rattling phonon,which gives rise to ∆ ρ ∝√ T .[46] There is an opticalphonon excitation at around 7 meV (80 K), which is at-tributed to the low-energy vibration of the Zn atom.[47–49] However, as this phonon excitation probably ceasesat T < et al. , which leads to ∆ ρ ( T ) ∝ A √ T and C/T ∝− ln T as mentioned in the introduction.[6]The √ T dependence of ρ ( T ) may be applicable to thepresent data. However, in the present case of PrIr Zn ,the quadrupole moments are not included as impuri-ties but placed periodically, therefore, formation of thequadrupole Kondo lattice is a promising candidate bring-ing about the NFL state.[8] The theoretical analyses onthe temperature variations of ρ and C with the two-channel Anderson lattice model have led the following ρ ( µ Ω c m ) PrIr Zn B || [100] B =01.51.00.50 ρ ( µ Ω c m ) T (K) 498 765 B =0 T Q T Q (a)(b) T FIG. 3: (Color online) Temperature dependence of the elec-trical resistivity of PrIr Zn in magnetic fields of (a) 0 ≤ B ≤ 4T and (b) B ≥ T Q . Thedashed lines are fits to the data using the relation of ∆ ρ ∝√ T . It is noted that the data are plotted without offset. relations; ∆ ρ ( T ) = a a ( T ( ρ )0 /T ) (2) C ( T ) = b − b s TT ( C )0 ! , (3)where a i and b i ( i =1 and 2) are parameters.[10] The cal-culations are shown by the (red) solid curves in Fig. 2 (a)and (b). The curves are in better agreement with our ex-perimental data for wider temperature range than thosecalculated by the impurity quadrupole Kondo model asshown with the (blue) dotted line.In the present case, the NFL behavior does not per-sist down to the low-temperature limit. This is cer-tain because this NFL behavior does not arise fromthe conventional quantum critical point but from thequadrupole Kondo effect as described in the introduc-tion. The quadrupoles are over-screened by the conduc-tion electrons, leading to the unstable electronic statewith the residual entropy of (0.5) R ln2 at zero temper-ature in principle. In real systems, however, the resid-ual entropy should go to zero at T = 0 by a mechanismfollowing the third law of thermodynamics. In fact, inthe magnetic fields B ≤ R ln2to (0.5) R ln2. In zero magnetic field, where the NFL be-haviors of the electrical resistivity and the specific heatappear in the temperature range from 1.5 K to 0.2 K.Taking these experimental results and the analyses onthem, this is the first observation of the NFL state in thenonmagnetic quadrupole-active system, whose origin isintrinsically different from that observed in the vicinityof the conventional quantum critical point. The scalingplots of C ( T ) and ∆ ρ ( T ) using the characteristic tem-peratures T /T ( C )0 and T /T ( ρ )0 , respectively, as shown inFig. 2(a) and (b), imply the possible formation of thequadrupole Kondo lattice resulting from hybridizationbetween the quadrupoles and the conduction electronswith an energy scale of k B T . D. Magnetic-field induced Fermi liquid state Looking close to the data of ρ ( T ) at B = 4 T as shownwith the solid triangles in Fig. 4, we find another kneeat T ∗ = 0.12 K above T Q = 0.08 K. In contrast to thedrop of T Q at B > T ∗ stays at around 0.13 K for B = 4 and 5 T. As shown in the inset of Fig. 1 (b),this T ∗ coincides with the broad peak of C f ( T ) at T ∗ = 0.12 K in B = 4.5 and 5 T. The field dependences of T ∗ observed in the ρ ( T ) and C f ( T ) measurements areplotted with the blue and red closed circles, respectively,in the B − T phase diagram in Fig. 5 (a). The T ∗ line can ρ ( µ Ω c m ) T (K) PrIr Zn B || [100] B =0 T Q T * ρ ( µ Ω c m ) T (K ) FIG. 4: (color online) Temperature dependence of the elec-trical resistivity ρ ( T ) of PrIr Zn in the magnetic field of0 ≤ B ≤ T Q and thecrossover temperature T ∗ , respectively. The inset is the plotof ρ vs T in 4 ≤ B ≤ be a boundary of two states, meaning a phase transitionor a crossover. In the present case, the crossover is likelyto occur at T ∗ , because the peak of C f ( T ) becomes verybroad despite the rather robust T ∗ against the magneticfield. On cooling below T ∗ , ρ ( T ) follows ρ + AT at lowtemperature for 4 ≤ B< A coefficient as a function of B is plotted in Fig. 5(c). It is strongly enhanced at around 4.5 T, where C f /T at 0.07 K is also peaked. Moreover, there is a peak at 5.5T in the Seebeck coefficient divided by temperature, S/T ,at 0.08 K.[50] The coincidence of these peaks at around 5T suggests a peculiar feature of the heavy fermion stateon cooling through T ∗ in a narrow range of B , 3.5
14 T applied alongthe [100] direction, respectively. The isothermal mag-netization shows metamagnetic behavior at around 5 T,where d M/ d B shows a peak. The peak shifts slightly tohigher fields and becomes broader with increasing tem-peratures, which is likely to be connected to the Schot-tky anomaly due to the splitting of the Γ doublet in themagnetic field for B> M/ d B . Note that the d M/ d B is largely enhanced onlyin the narrow range of B , 3.5
There are some possible mechanisms to form the ex-otic Fermi liquid state at T 14 T applied along the [100] di-rection. The data are vertically offset for clarity. The solidand dotted lines indicate the data for ascending the magneticfields. Fig. 5 (b), C f /T does not diverse in the vicinity of 4.5T on cooling to 0.07 K at all. It seems that the Doniachpicture is not valid for this pure quadrupole system. Al-though it is still an open question why the quantum crit-icality was not detected in PrIr Zn , a peculiar featureof the quadrupole must give rise to the unique electronicphenomena. IV. SUMMARY We report the low-temperature transport, thermo-dynamic and magnetic properties on a cubic systemPrIr Zn in which the non-Kramers doublet groundstate has purely the quadrupolar degrees of freedom.In the moderately wide temperature range at T >T Q ,non-Fermi liquid behaviors were clearly observed in theelectrical resistivity ρ and the specific heat C f . Both ρ and C f in magnetic fields B < T , suggesting the formation of thequadrupole Kondo lattice due to the hybridization be-tween the quadrupoles and the conduction electrons. Itindicates that the NFL behavior observed in the presentsystem is intrinsically different from that observed inthe vicinity of the conventional quantum critical point.Furthermore, ρ and C f exhibit anomalies at T ∗ =0.13K in the vicinity of 5 T, where the coefficient A for ρ = ρ + AT , C f /T , and S/T , have significant enhance-ment as a function of B . The concomitant increase ind M/ d B indicates formation of a magnetic-field inducedFermi liquid ground state to remove the residual entropyin the quadrupole Kondo lattice. These observations im-ply that the Doniach picture relevant to the spin Kondosystems should not be valid for this purely quadrupole-active system, possibly leading us beyond the Doniachpicture. Acknowledgements The authors would like to thank A. Tsuruta, K.Miyake, H. Kusunose, S. Hoshino, J. Otsuki, Y. Ku- ramoto, K. Uenishi, Y. Yamane, I. Ishii, T. Suzuki and K.Iwasa for helpful discussions. We also thank Y. Shibatafor the electron-probe microanalysis performed at N-BARD, Hiroshima University. The magnetization mea-surements with MPMS and specific heat measurementswith PPMS were carried out at N-BARD, HiroshimaUniversity. This work was financially supported byJSPS KAKENHI Grant Numbers 21102516, 23102718,26707017, and 15H05884, 15H05886 (J-Physics), and byThe Mazda Foundation Research Grant, Japan and Hi-roshima University Fujii Research Fund. [1] H. v. L¨ohneysen, A. Rosch, M. Vojta, and P. Wolfle, Rev.Mod. Phys. , 1015 (2007).[2] K. Umeo, H. Kadomatsu, and T. Takabatake, J. Phys.:Cond. Matter , 9743 (1996).[3] J. Custers, P. Gegenwart, H. Wilhelm, K. Neumaier, Y.Tokiwa, O. Trovarelli, C. Geibel, F. Steglich, C. P´epin,and P. Coleman, Nature , 524 (2003).[4] Q. Si, S. Rabello, K. Ingersent, J. L. Smith, Nature ,804 (2001).[5] N. D. Mathur, F. M. Grosche, S. 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