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Crystal field states in CeCu4Al
Solid State Communications 149 (2009) 2240–2243
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Crystal field states in CeCu4 Al T. Toliński a,∗ , A. Hoser b , S. Rols c , A. Kowalczyk a , A. Szlaferek a a
Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznań, Poland
b
Helmholtz-Zentrum, Glienicker Straße 100, D-14109 Berlin, Germany
c
Institut Laue Langevin, 6 rue Jules Horowitz B.P. 156 F-38042, Grenoble Cedex 9, France
article
info
Article history: Received 6 August 2009 Received in revised form 2 September 2009 Accepted 4 September 2009 by M. Grynberg Available online 9 September 2009 PACS: 65.40.Ba 71.20.Lp 71.27.+a 75.30.Mb 75.30.Cr 78.70.Nx
abstract Heavy fermion CeCu4 Al compound has been studied by inelastic neutron diffraction (INS), heat capacity and magnetic susceptibility measurements. A single Crystal Electric Field (CEF) peak has been detected in the INS spectra, which may be explained by a quasi-quartet state suggested by the analysis of the Schottky anomaly contributing to the magnetic part of the specific heat. The Kondo interactions have been included in the analysis of the magnetic part employing the simplified resonance level model. The resulting Kondo temperature of about 5–10 K is somewhat larger than in the previous studies. The magnetic susceptibility confirms the CEF level scheme of the type 0–93 K and provides the values of the CEF parameters. © 2009 Elsevier Ltd. All rights reserved.
Keywords: D. Heat capacity D. Heavy fermions D. Kondo effects E. Neutron scattering
1. Introduction In Ce-based compounds the position of the 4f states with respect to the Fermi level determines the appearance of a variety of fascinating phenomena, like mixed-valence (MV) behavior, formation of the heavy fermion (HF) state, Kondo effect or critical quantum point. CeCu4 M compounds can reveal most of these effects depending on the element M. We have studied the magnetic, transport, thermodynamic and electronic properties of numerous RT4 M compounds (R: rare earth, T : transition metal, M: B, Al, Cu, Si, Ga), crystallizing mostly in the hexagonal CaCu5 -type structure [1–3]. This structure is also accommodated by a few of the CeCu4 M compounds. The magnetic, transport and structural investigations have already been carried out on the CeCu4 Al compound, which is a heavy fermion system [4–8] with a large effective mass and paramagnetic behavior down to the lowest temperatures. However, the type of the ground state and the energy level split by the
∗
Corresponding author. Tel.: +48 61 8695 282; fax: +48 61 8684 524. E-mail address: [email protected] (T. Toliński).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.09.002
Crystal Electric Field (CEF) are not widely evidenced and finally established. We obtain a set of data based on the inelastic neutron scattering (INS), heat capacity and magnetic susceptibility measurements. Heat capacity provides the energy level scheme from the analysis of the Schottky anomaly. Employing INS is very useful owing to the fact that the cross section is proportional to the dynamic magnetic susceptibility. The CEF excitations should develop the inelastic peaks in the INS spectrum. To study the magnetic response for Ce-based compounds one has to estimate the phonon contribution. It can be done either by angular dependence of INS or by employing a nonmagnetic analogue. We have used the YCu4 Al reference compound in order to distinguish the Ce contribution. In Ref. [9] first thorough INS measurements for CeCu4 M (M = Al, Ga) were performed. A single excitation was observed at 5.5 meV. However, a peak at 17meV was found for CeCu5 and two excitations (5.6 meV and 7.3 meV) were predicted from specific heat studies [9]. We carry out a comparison of the INS, specific heat and magnetic susceptibility data providing the values of the CEF parameters. YCu4 Al is used as the nonmagnetic reference compound.
T. Toliński et al. / Solid State Communications 149 (2009) 2240–2243
a
2241
b
Fig. 1. Neutron time-of-flight spectra for CeCu4 Al and the reference compound YCu4 Al at the temperature of 4 K (a) and 200 K (b) for wavelength of 1.8 Å.
2. Experimental details Induction melting of high purity elements in argon atmosphere was employed to prepare the CeCu4 Al and YCu4 Al compounds. The crystal structure was established by the powder X-ray diffraction technique, using the Co–Kα radiation. The full-pattern Rietveld refinement corroborated the CaCu5 -type structure, space group P6/mmm [8]. Heat capacity measurements were carried out on the PPMS commercial device (Quantum Design) in the temperature range 1.9–300 K, by the relaxation method using the two-τ model. As can be found in papers on similar studies [9–13] the observation of the inelastic lines developed by CEF requires high energies (up to 50 meV) and preferably time-of-flight method. Therefore, the experiments were carried out on IN4 time-of-flight instrument at the Institut Laue Langevin (ILL) in Grenoble using incident neutrons wavelength of 1.1 Å and 1.8 Å. For the INS experiment the samples of the mass of about 6 g were wrapped in aluminum foil. 3. Results The inelastic neutron scattering is a unique technique, which can provide very reliable information on the crystal field excitations. This knowledge is necessary to shed light on the f-electrons physics in the CeCu4 M compounds. To estimate the magnetic contribution for Ce-based compounds we have measured INS for the YCu4 M reference compounds assuming that the phonon contribution is similar both for the Ce- and Y-based samples. In Fig. 1 the INS spectra both for CeCu4 Al and YCu4 Al are shown for the wavelength of 1.8 Å and at temperature of 4 K and 200 K. It is clearly visible that the inelastic peaks developed in YCu4 Al are also present for CeCu4 Al. These phonon excitations are shifted on the energy scale, which results from the softening of the phonons energy due to the lower atomic mass of yttrium compared to cerium. For CeCu4 Al additional feature is indicated with the arrow at about 8 meV, which corresponds to 93 K. The CEF peak observed by Gignoux et al. [9] was placed at 5.5 meV (64 K); however, the analysis of the magnetic contribution to the specific heat carried out by Bauer et al. [6,14] provided the levels scheme 0–65–85 K. Hence, our observation is close to the higher energy excitation of this scheme. To verify our observations derived from the inelastic neutron diffraction we have carried out specific heat measurements for CeCu4 Al and YCu4 Al as the nonmagnetic reference compound. Fig. 2 displays the Cp vs. T dependence for both compounds and the
Fig. 2. Specific heat for CeCu4 Al and the reference compound YCu4 Al. Inset: Cp of YCu4 Al fitted with Eq. (1) and ΘD = 280 K, γ = 9.13 mJK−2 mol−1 . The dashed line is a curve recalculated after mass correction for CeCu4 Al.
inset shows the approximation of the experimental data for YCu4 Al by the Debye formula: Cp (T ) = γ T + 9NR
T
ΘD /T
3 Z
ΘD
x4 ex dx
(ex − 1)2
0
,
(1)
where apart from the phonon contribution represented by the second term the electronic part is additionally included. N = 6 is the number of atoms in the formula unit and x = h¯ ω/kB T . From the fit we have got the Debye temperature ΘD = 280 K and the electronic specific heat coefficient γ = 9.13 mJK−2 mol−1 . The direct subtraction of the specific heat of the reference compound is not optimal due to the difference in the mass of Ce and Y, therefore we have carried out a standard mass correction [15,16] according to the formula: 3 ΘCeCu 4 Al 3 ΘYCu 4 Al
3/2
=
mY
3/2
3/2
3/2
3/2
3/2
+ 4mCu + mAl
(2)
mCe + 4mCu + mAl
Θ (CeCu Al)
yielding the value of the ratio ΘD (YCu 4Al) = 0.92. D 4 Next, before subtraction from the Cp vs. T dependence of CeCu4 Al, the theoretical curve of the inset in Fig. 2 was recalculated with the new Debye temperature ΘD = 258 K, resulting from Eq. (2). In this way derived magnetic contribution to the specific heat of the Ce-based compound is presented in Fig. 3 as the temperature dependence of Cm /T . One can see a peak at about
2242
T. Toliński et al. / Solid State Communications 149 (2009) 2240–2243
Fig. 3. Magnetic contribution to the specific heat of CeCu4 Al analyzed with a sum of Eqs. (3) and (4). ∆i and TK are the energy distance in respect to the ground state and the Kondo temperature, respectively. Two different fits are presented (see text for details).
26 K, which is triggered off by the crystal field splitting of the ground state level. The gradual occupation of these energy levels develops this Schottky anomaly. The general formula representing the Schottky contribution is given by:
n −1
P CCEF T
∆2i e−∆i /T
R i =0 = 3 n −1 T P
e− ∆ i / T
2
n−1
∆i e−∆i /T i=0 − n−1 P −∆ /T e
i =0
i
(3)
T
=
kB NA TK
πT2
Om n
1−
TK 2π T
ψ0
1 2
+
TK 2π T
,
(4)
where ψ ’ is the derivative of the Digamma function. The solid line in Fig. 3 is a fit carried out by the sum of Eqs. (3) and (4). It provides the estimation of the distance of the two doublets over the ground state ∆1 ≈ ∆2 = 93 K and the Kondo temperature TK = 10 K. The latter is different from our previous estimation [8] based on the electronic specific heat coefficient, magnetic susceptibility, electrical resistivity (TK = 2.5–5 K) and other studies (TK = 2 K [14]). However, inelastic neutron diffraction measurements of Gignoux et al [9] suggested TK ≈ 30 K. Our result for TK derived from Eq. (4) can be explained in two ways. First, the procedure for the extraction of the phonon contribution presented above and any other similar methods are relatively rough approximation and using TK ≈ 5 K does not spoil the fit dramatically. Second, as the authors of Ref. [17] state, their model is not an exact theory for the Kondo problem. As results from Fig. 3 a best agreement with the peak position is obtained when both the excited doublets are close in energy (∼93 K). It may be caused by a structural distortion suspected from the X-ray diffraction studies [8]. Such a quasi-quartet state could also be one of the possible explanations of the only single CEF peak detected in our and literature INS studies. However again,
(5)
Bm n
where and are the Steven’s operators and CEF parameters, respectively [18]. The calculation of the sum of the Curie and Van Vleck susceptibilities was carried out according to the equation:
i=0
where n denotes the number of the energy levels and ∆0 , i. e. the ground state energy is assumed to be zero. For Ce3+ ions in hexagonal surroundings the 4f levels split into three doublets, which implies n = 3. The steep increase of the magnetic specific heat at low temperatures (Fig. 3) stems from the heavy fermion state inherent for the CeCu4 Al compound. Therefore, in the analysis of Cm we follow the previous studies of Bauer [14] and include the Schotte and Schotte formula [17] obtained in frames of the resonance level model reduced for the spin 1/2 to the simple form: CKondo
fixing the energy value of the first excited doublet according to [14] (∆1 = 63 K) and keeping ∆2 ∼93 K does not affect dramatically the quality of the fit (dotted line in Fig. 3). Finally, the temperature dependence of the magnetic susceptibility has been analyzed in frames of the CEF model assuming the hamiltonian for the hexagonal point symmetry: H = B02 O02 + B04 O04 ,
P
,
Fig. 4. Experimental magnetic susceptibility (circles) and calculation with the CEF model (solid and dotted lines).
χ =
N (gJ µB )
ZkB T
2
X
|hn |Ji | mi|2 e
− kEnT B
n,m En =Em
X |hn |Ji | mi|2 − En + 2kB T e kB T , Em − En n,m
(6)
En 6=Em
where i is the x, y and z component of the angular momentum, gJ is the Landé g factor, Z is the partition function, N is the Avogadro’s number and En are the energies of the wave functions above the ground state. Eq. (6) was employed assuming spatial averaging for the polycrystalline sample with a small texture, i .e. 0.45 instead of 1/3 for the hexagonal axis contribution. In the analysis, we assumed that the expected levels scheme is 0–64–93 K. A good fit (Fig. 4, solid line) to the experimental data has been obtained for the CEF parameters B02 = 4.8 K and B04 = −0.12 K. It was also necessary to include the molecular field parameter λ = −10 mol/emu. The above parameters result in the ground state Γ7 level |± 1/2i and the excited Γ9 |± 3/2i and Γ8 level |± 5/2i states. However, a comparable description of the experimental data is obtained assuming that the Γ9 and Γ8 doublets are close each to the other (dotted curve in Fig. 4). In this case B02 = 4.5 K and B04 = −0.22 K are derived. The quasi-quartet state can explain the lack of the second inelastic peak in the INS spectrum in agreement with the specific heat analyzes. The obtained values of the CEF parameters are close to the previous results for the hexagonal compounds PrCu5 (B02 = 5.57 K, B04 = −0.01 K) [19] and PrNi5 (B02 = 5.84 K, B04 = 0.045 K) [20]. It is clearly visible that all the three techniques, i.e. inelastic neutron diffraction, specific heat and magnetic susceptibility provide similar value of the higher excitation state, however the position of the first doublet above the ground state can only be estimated to be in the range 60–95 K. The possibility that no all the inter-level transitions are allowed can partly explain these observations. Further
T. Toliński et al. / Solid State Communications 149 (2009) 2240–2243
studies on single crystal samples can yield better understanding of this effect. 4. Conclusions The inelastic neutron diffraction studies on the heavy fermion CeCu4 Al compound enabled observation of only single crystal electric field excitation at about 8 meV (93 K). The Schottky anomaly extracted from the specific heat by subtraction of the phonon contribution of the nonmagnetic reference compound indicates that the energy level scheme can be of the type doublet – quartet (0–93 K). It can explain the single CEF peak visible in the INS spectra. However, the 0–64–93 K scheme cannot be finally excluded due to the inherent limits of the precision of the phonon contribution determination. The Kondo temperature estimated from the low temperature range of the Cm /T vs. T dependence and the Schotte and Schotte formula takes a value of about 5–10 K. The magnetic susceptibility analyzed in frames of the CEF model confirms the position of the higher energy doublet and supports the possibility that the Γ9 and Γ8 states are close in energy. It implies the CEF parameters B02 = 4.5 K and B04 = −0.22 K. Acknowledgement This work was supported by the funds for science in years 2007–2009 as a research project no. N N202 1213 33 (T. Toliński, A. Kowalczyk).
2243
References [1] T. Toliński, A. Kowalczyk, G. Chełkowska, M. Pugaczowa-Michalska, B. Andrzejewski, V. Ivanov, A. Szewczyk, M. Gutowska, Phys. Rev. B 70 (2004) 064413. [2] T. Toliński, A. Kowalczyk, A. Szewczyk, M. Gutowska, J. Phys.: Condens. Matter 18 (2006) 3435. [3] T. Toliński, Modern Phys. Lett. B 21 (2007) 431. [4] A. Eichler, C. Mehls, F.-W. Schaper, M. Schwerin, C. Sutter, F. Voges, E. Bauer, Physica B 206–207 (1995) 258. [5] B. Andraka, J.S. Kim, G.R. Stewart, Z. Fisk, Phys. Rev. B 44 (1991) 4371. [6] E. Bauer, N. Pillmayr, H. Muller, J. Kohlmann, K. Winzer, J. Magn. Magn. Mater. 90-91 (1990) 411. [7] E. Bauer, E. Gratz, N. Pillmayr, Solid State Commun. 62 (1987) 271. [8] A. Kowalczyk, T. Toliński, M. Reiffers, M. Pugaczowa-Michalska, G. Chełkowska, E. Gažo, J. Phys.: Condens. Matter 20 (2008) 255252. [9] D. Gignoux, D. Schmitt, E. Bauer, A.P. Murani, J. Magn. Magn. Mater. 88 (1990) 63. [10] E.A. Goremychkin, R. Osborn, Phys. Rev. B 47 (1993) 14280. [11] E.A. Goremychkin, R. Osborn, I.L. Sashin, J. Appl. Phys. 85 (1999) 6046. [12] K. Hense, E. Gratz, H. Nowotny, A. Hoser, J. Phys.: Condens. Matter 16 (2004) 5751. [13] E. Holland-Moritz, D. Wohlleben, M. Loewenhaupt, Phys. Rev. B 25 (1982) 7482. [14] E. Bauer, J. Magn. Magn. Mater. 108 (1992) 27. [15] J.A. Hoffmann, A. Paskin, K.J. Tauer, R.J. Weiss, J. Phys. Chem. Solids 1 (1956) 45. [16] M. Bouvier, P. Lethuillier, D. Schmitt, Phys. Rev. B 43 (1991) 13137. [17] K.D. Schotte, U. Schotte, Phys. Lett. A 55 (1975) 38. [18] M.T. Hutchings, Solid State Phys. 16 (1964) 227. [19] A. Andreeff, E.A. Goremychkin, H. Griessmann, L.P. Kaun, B. Lippold, W. Matz, O.D. Chistyakov, E.M. Savitskii, P.G. Ivanitskii, Phys. Status Solidi (b) 108 (1981) 261. [20] V.M.T.S. Barthem, D. Gignoux, A. Naït-Saada, D. Schmitt, G. Creuzet, Phys. Rev. B 37 (1988) 1733.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Crystal field states in CeCu4 Al T. Toliński a,∗ , A. Hoser b , S. Rols c , A. Kowalczyk a , A. Szlaferek a a
Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznań, Poland
b
Helmholtz-Zentrum, Glienicker Straße 100, D-14109 Berlin, Germany
c
Institut Laue Langevin, 6 rue Jules Horowitz B.P. 156 F-38042, Grenoble Cedex 9, France
article
info
Article history: Received 6 August 2009 Received in revised form 2 September 2009 Accepted 4 September 2009 by M. Grynberg Available online 9 September 2009 PACS: 65.40.Ba 71.20.Lp 71.27.+a 75.30.Mb 75.30.Cr 78.70.Nx
abstract Heavy fermion CeCu4 Al compound has been studied by inelastic neutron diffraction (INS), heat capacity and magnetic susceptibility measurements. A single Crystal Electric Field (CEF) peak has been detected in the INS spectra, which may be explained by a quasi-quartet state suggested by the analysis of the Schottky anomaly contributing to the magnetic part of the specific heat. The Kondo interactions have been included in the analysis of the magnetic part employing the simplified resonance level model. The resulting Kondo temperature of about 5–10 K is somewhat larger than in the previous studies. The magnetic susceptibility confirms the CEF level scheme of the type 0–93 K and provides the values of the CEF parameters. © 2009 Elsevier Ltd. All rights reserved.
Keywords: D. Heat capacity D. Heavy fermions D. Kondo effects E. Neutron scattering
1. Introduction In Ce-based compounds the position of the 4f states with respect to the Fermi level determines the appearance of a variety of fascinating phenomena, like mixed-valence (MV) behavior, formation of the heavy fermion (HF) state, Kondo effect or critical quantum point. CeCu4 M compounds can reveal most of these effects depending on the element M. We have studied the magnetic, transport, thermodynamic and electronic properties of numerous RT4 M compounds (R: rare earth, T : transition metal, M: B, Al, Cu, Si, Ga), crystallizing mostly in the hexagonal CaCu5 -type structure [1–3]. This structure is also accommodated by a few of the CeCu4 M compounds. The magnetic, transport and structural investigations have already been carried out on the CeCu4 Al compound, which is a heavy fermion system [4–8] with a large effective mass and paramagnetic behavior down to the lowest temperatures. However, the type of the ground state and the energy level split by the
∗
Corresponding author. Tel.: +48 61 8695 282; fax: +48 61 8684 524. E-mail address: [email protected] (T. Toliński).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.09.002
Crystal Electric Field (CEF) are not widely evidenced and finally established. We obtain a set of data based on the inelastic neutron scattering (INS), heat capacity and magnetic susceptibility measurements. Heat capacity provides the energy level scheme from the analysis of the Schottky anomaly. Employing INS is very useful owing to the fact that the cross section is proportional to the dynamic magnetic susceptibility. The CEF excitations should develop the inelastic peaks in the INS spectrum. To study the magnetic response for Ce-based compounds one has to estimate the phonon contribution. It can be done either by angular dependence of INS or by employing a nonmagnetic analogue. We have used the YCu4 Al reference compound in order to distinguish the Ce contribution. In Ref. [9] first thorough INS measurements for CeCu4 M (M = Al, Ga) were performed. A single excitation was observed at 5.5 meV. However, a peak at 17meV was found for CeCu5 and two excitations (5.6 meV and 7.3 meV) were predicted from specific heat studies [9]. We carry out a comparison of the INS, specific heat and magnetic susceptibility data providing the values of the CEF parameters. YCu4 Al is used as the nonmagnetic reference compound.
T. Toliński et al. / Solid State Communications 149 (2009) 2240–2243
a
2241
b
Fig. 1. Neutron time-of-flight spectra for CeCu4 Al and the reference compound YCu4 Al at the temperature of 4 K (a) and 200 K (b) for wavelength of 1.8 Å.
2. Experimental details Induction melting of high purity elements in argon atmosphere was employed to prepare the CeCu4 Al and YCu4 Al compounds. The crystal structure was established by the powder X-ray diffraction technique, using the Co–Kα radiation. The full-pattern Rietveld refinement corroborated the CaCu5 -type structure, space group P6/mmm [8]. Heat capacity measurements were carried out on the PPMS commercial device (Quantum Design) in the temperature range 1.9–300 K, by the relaxation method using the two-τ model. As can be found in papers on similar studies [9–13] the observation of the inelastic lines developed by CEF requires high energies (up to 50 meV) and preferably time-of-flight method. Therefore, the experiments were carried out on IN4 time-of-flight instrument at the Institut Laue Langevin (ILL) in Grenoble using incident neutrons wavelength of 1.1 Å and 1.8 Å. For the INS experiment the samples of the mass of about 6 g were wrapped in aluminum foil. 3. Results The inelastic neutron scattering is a unique technique, which can provide very reliable information on the crystal field excitations. This knowledge is necessary to shed light on the f-electrons physics in the CeCu4 M compounds. To estimate the magnetic contribution for Ce-based compounds we have measured INS for the YCu4 M reference compounds assuming that the phonon contribution is similar both for the Ce- and Y-based samples. In Fig. 1 the INS spectra both for CeCu4 Al and YCu4 Al are shown for the wavelength of 1.8 Å and at temperature of 4 K and 200 K. It is clearly visible that the inelastic peaks developed in YCu4 Al are also present for CeCu4 Al. These phonon excitations are shifted on the energy scale, which results from the softening of the phonons energy due to the lower atomic mass of yttrium compared to cerium. For CeCu4 Al additional feature is indicated with the arrow at about 8 meV, which corresponds to 93 K. The CEF peak observed by Gignoux et al. [9] was placed at 5.5 meV (64 K); however, the analysis of the magnetic contribution to the specific heat carried out by Bauer et al. [6,14] provided the levels scheme 0–65–85 K. Hence, our observation is close to the higher energy excitation of this scheme. To verify our observations derived from the inelastic neutron diffraction we have carried out specific heat measurements for CeCu4 Al and YCu4 Al as the nonmagnetic reference compound. Fig. 2 displays the Cp vs. T dependence for both compounds and the
Fig. 2. Specific heat for CeCu4 Al and the reference compound YCu4 Al. Inset: Cp of YCu4 Al fitted with Eq. (1) and ΘD = 280 K, γ = 9.13 mJK−2 mol−1 . The dashed line is a curve recalculated after mass correction for CeCu4 Al.
inset shows the approximation of the experimental data for YCu4 Al by the Debye formula: Cp (T ) = γ T + 9NR
T
ΘD /T
3 Z
ΘD
x4 ex dx
(ex − 1)2
0
,
(1)
where apart from the phonon contribution represented by the second term the electronic part is additionally included. N = 6 is the number of atoms in the formula unit and x = h¯ ω/kB T . From the fit we have got the Debye temperature ΘD = 280 K and the electronic specific heat coefficient γ = 9.13 mJK−2 mol−1 . The direct subtraction of the specific heat of the reference compound is not optimal due to the difference in the mass of Ce and Y, therefore we have carried out a standard mass correction [15,16] according to the formula: 3 ΘCeCu 4 Al 3 ΘYCu 4 Al
3/2
=
mY
3/2
3/2
3/2
3/2
3/2
+ 4mCu + mAl
(2)
mCe + 4mCu + mAl
Θ (CeCu Al)
yielding the value of the ratio ΘD (YCu 4Al) = 0.92. D 4 Next, before subtraction from the Cp vs. T dependence of CeCu4 Al, the theoretical curve of the inset in Fig. 2 was recalculated with the new Debye temperature ΘD = 258 K, resulting from Eq. (2). In this way derived magnetic contribution to the specific heat of the Ce-based compound is presented in Fig. 3 as the temperature dependence of Cm /T . One can see a peak at about
2242
T. Toliński et al. / Solid State Communications 149 (2009) 2240–2243
Fig. 3. Magnetic contribution to the specific heat of CeCu4 Al analyzed with a sum of Eqs. (3) and (4). ∆i and TK are the energy distance in respect to the ground state and the Kondo temperature, respectively. Two different fits are presented (see text for details).
26 K, which is triggered off by the crystal field splitting of the ground state level. The gradual occupation of these energy levels develops this Schottky anomaly. The general formula representing the Schottky contribution is given by:
n −1
P CCEF T
∆2i e−∆i /T
R i =0 = 3 n −1 T P
e− ∆ i / T
2
n−1
∆i e−∆i /T i=0 − n−1 P −∆ /T e
i =0
i
(3)
T
=
kB NA TK
πT2
Om n
1−
TK 2π T
ψ0
1 2
+
TK 2π T
,
(4)
where ψ ’ is the derivative of the Digamma function. The solid line in Fig. 3 is a fit carried out by the sum of Eqs. (3) and (4). It provides the estimation of the distance of the two doublets over the ground state ∆1 ≈ ∆2 = 93 K and the Kondo temperature TK = 10 K. The latter is different from our previous estimation [8] based on the electronic specific heat coefficient, magnetic susceptibility, electrical resistivity (TK = 2.5–5 K) and other studies (TK = 2 K [14]). However, inelastic neutron diffraction measurements of Gignoux et al [9] suggested TK ≈ 30 K. Our result for TK derived from Eq. (4) can be explained in two ways. First, the procedure for the extraction of the phonon contribution presented above and any other similar methods are relatively rough approximation and using TK ≈ 5 K does not spoil the fit dramatically. Second, as the authors of Ref. [17] state, their model is not an exact theory for the Kondo problem. As results from Fig. 3 a best agreement with the peak position is obtained when both the excited doublets are close in energy (∼93 K). It may be caused by a structural distortion suspected from the X-ray diffraction studies [8]. Such a quasi-quartet state could also be one of the possible explanations of the only single CEF peak detected in our and literature INS studies. However again,
(5)
Bm n
where and are the Steven’s operators and CEF parameters, respectively [18]. The calculation of the sum of the Curie and Van Vleck susceptibilities was carried out according to the equation:
i=0
where n denotes the number of the energy levels and ∆0 , i. e. the ground state energy is assumed to be zero. For Ce3+ ions in hexagonal surroundings the 4f levels split into three doublets, which implies n = 3. The steep increase of the magnetic specific heat at low temperatures (Fig. 3) stems from the heavy fermion state inherent for the CeCu4 Al compound. Therefore, in the analysis of Cm we follow the previous studies of Bauer [14] and include the Schotte and Schotte formula [17] obtained in frames of the resonance level model reduced for the spin 1/2 to the simple form: CKondo
fixing the energy value of the first excited doublet according to [14] (∆1 = 63 K) and keeping ∆2 ∼93 K does not affect dramatically the quality of the fit (dotted line in Fig. 3). Finally, the temperature dependence of the magnetic susceptibility has been analyzed in frames of the CEF model assuming the hamiltonian for the hexagonal point symmetry: H = B02 O02 + B04 O04 ,
P
,
Fig. 4. Experimental magnetic susceptibility (circles) and calculation with the CEF model (solid and dotted lines).
χ =
N (gJ µB )
ZkB T
2
X
|hn |Ji | mi|2 e
− kEnT B
n,m En =Em
X |hn |Ji | mi|2 − En + 2kB T e kB T , Em − En n,m
(6)
En 6=Em
where i is the x, y and z component of the angular momentum, gJ is the Landé g factor, Z is the partition function, N is the Avogadro’s number and En are the energies of the wave functions above the ground state. Eq. (6) was employed assuming spatial averaging for the polycrystalline sample with a small texture, i .e. 0.45 instead of 1/3 for the hexagonal axis contribution. In the analysis, we assumed that the expected levels scheme is 0–64–93 K. A good fit (Fig. 4, solid line) to the experimental data has been obtained for the CEF parameters B02 = 4.8 K and B04 = −0.12 K. It was also necessary to include the molecular field parameter λ = −10 mol/emu. The above parameters result in the ground state Γ7 level |± 1/2i and the excited Γ9 |± 3/2i and Γ8 level |± 5/2i states. However, a comparable description of the experimental data is obtained assuming that the Γ9 and Γ8 doublets are close each to the other (dotted curve in Fig. 4). In this case B02 = 4.5 K and B04 = −0.22 K are derived. The quasi-quartet state can explain the lack of the second inelastic peak in the INS spectrum in agreement with the specific heat analyzes. The obtained values of the CEF parameters are close to the previous results for the hexagonal compounds PrCu5 (B02 = 5.57 K, B04 = −0.01 K) [19] and PrNi5 (B02 = 5.84 K, B04 = 0.045 K) [20]. It is clearly visible that all the three techniques, i.e. inelastic neutron diffraction, specific heat and magnetic susceptibility provide similar value of the higher excitation state, however the position of the first doublet above the ground state can only be estimated to be in the range 60–95 K. The possibility that no all the inter-level transitions are allowed can partly explain these observations. Further
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studies on single crystal samples can yield better understanding of this effect. 4. Conclusions The inelastic neutron diffraction studies on the heavy fermion CeCu4 Al compound enabled observation of only single crystal electric field excitation at about 8 meV (93 K). The Schottky anomaly extracted from the specific heat by subtraction of the phonon contribution of the nonmagnetic reference compound indicates that the energy level scheme can be of the type doublet – quartet (0–93 K). It can explain the single CEF peak visible in the INS spectra. However, the 0–64–93 K scheme cannot be finally excluded due to the inherent limits of the precision of the phonon contribution determination. The Kondo temperature estimated from the low temperature range of the Cm /T vs. T dependence and the Schotte and Schotte formula takes a value of about 5–10 K. The magnetic susceptibility analyzed in frames of the CEF model confirms the position of the higher energy doublet and supports the possibility that the Γ9 and Γ8 states are close in energy. It implies the CEF parameters B02 = 4.5 K and B04 = −0.22 K. Acknowledgement This work was supported by the funds for science in years 2007–2009 as a research project no. N N202 1213 33 (T. Toliński, A. Kowalczyk).
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