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1 277 Progress of Theoretical Physics, Vol. 20, No. 3, September 1958 The Magnetic Properties of the Praseodymium and the Neodymium Metals Tsuyoshi MURAO Department of Physics, Tohoku University, Sendai (ReceivEd May 17, 1958) Along the same line on which we have pteviously discussed the cubic cerium metal, the magnetic -properties of the succeeding rare earth metals, praseodymium and neodymium, are investigated. Because these metals have the structure of h. c. p., their crystalline field potentials cannot be uniquely determined so as to fit the experimental data. The analysis is, however,.qualitatively satisfactory.. ntroduction The magnetic properties of the rare earth metals have been discussed on the bases of Zener's mechanism for indirect exchange interaction 1 > and of the crystalline field potential 2 l for 4{ electrons. The characteristic features of these metals are thought to be -due to the simultaneous effects of them and have actually been understood to some extent from this point of view. n the present paper we investigate the effect of the crystalline field on the magnetic properties of hexagonal rare earth metals. For this purpose we choose the praseodymium -and the neodymium metals, because the effect of.the crystalline field predominates the exchange field in these metals. The understanding of the crystalline field in these metals will be useful for further researches on complicated phenomena like ferromagnetic-antiferromagnetic transitions observed in other rare earth metals. Pr, unlike Ce and Nd, has no antiferromagnetic phase.s), 4 l Pr shows a Shottky type anomaly in the atomic heat, 3 > whose maximum lies at a comparatively lower temperature than Ce and Nd. The magnetic susceptibility of Pr at low temperatures seems to tend to a finite value at 0 K. 4 > Nd has two anomalous peaks at 7. 5 K and at 19 K 3 l besides the same kind of anomaly in specific heat as Pr. Both peaks may be regarded -a.s the cooperative phase transitions and the former of them can be considered to correspond to the Neel point. 4l The latter peak is suggested to be due to the interactions among the multipoles of the 4{ electrons."> The anisotropic magnetic susceptibility of the Nd -single crystal has recendy been reported. 6 l The electrical resistivity curves of Pr and Nd are concave downward over a wide temperature range and Nd shows another anomaly in the lower temperature range. 7 > The phenomena mentioned above in the praseodymium and the neodymium metals can be explained semi-quantitatively by the view of the Stark splitting of the 4{ electrons -due to the crystalline field. We shall show in 2 the calculation of the crystalline field potential, the results of which will be applied to Pr and Nd in 3 and 4, respectively;

2 278 T. Murao n the present paper, we confine ourselves to analyzing the specific heat and the magnetic susceptibilities above the cooperative transition points. n 5 discussion of our results will be made. 2. Calculation of the crystalline field While both metals, Pr and Nd, have the h. c. p. structure in the temperature range to which our interest will be confined, they have cubic closest packing structure at higher temperatures. 8 ' Calculation of the crystalline field is performed on the model similar to that for the cerium metal which has already been reported by us 2 l : the valence electrons. are treated as free electrons and smeared out uniformly over the whole of the metal, so that they have no contribution to the crystalline field discussed below. Therefore, the crystalline field can be obtained from the hexagonal closest packing of the ionized atoms. as where The crystalline field potential for 4[ electrons in the h. c. p. structure is expressed. (1) (2) (l-m)! -ve 2,. Ctn,=2 -(l ---~-)l ~ -R P1 (coso,.) cosmcp,,.. +m. n n n eq. (2), r", (}" and cp" are the spherical coordinates of the A-th electron in the atom whose nucleus is taken as the origin, and Rn, (}n and Cfn the spherical coordinates of the nucleus of the n-th atom, and v the valency of the ionized atom. On account of the symmetry of the crystal structure the sine term does not appear in eq. (2). n the lattice sum for L> 4, the direct sum method is applied over the range Rn < 4a, where a denotes the lattice constant along the a-axis. For l = 2, the convergency of the series of the direct sum is rather slow. We therefore adopted the Fourier method, 9 } whose convergency for l = 2 is very rapid, and carried out for h 71 < 4 a in the reciprocal space. The calculation is performed for cla=1.579 and and the results are shown. in Table 1 and Fig. 1. To verify we calculated c 40 by the both methods for cla=l.613. n Table 1 cell, n, indicates the contribution from ( n-1) a < R n < na for direct summation and from nl a< hn < ( n + 1) a for the Fourier method, except for n = 1 where h:,. < 21 a. As is seen in Table 1, the larger l is, the more rapid the convergency is. c 20 needs some comment. As can be seen in Fig. 1, the value of c 20 is approximately zero when cla=v 8/3= This is due to the symmetry of the lattice. When we carry out the lattice sum for c 20, the contributions cancel one another noticeably within the 1st cell and the total contribution from the 1st cell is reduced by a factorof , which is just the order of magnitude to be cancelled by the contributions:

3 The Magnetic Properties of the Praseodymium and the Neodymium Metals 279' Table 1. The calculated values of Ctm (-ve2fal+l) should be multiplied t? each value in the Table. c~; 5 =c66*x (2X10395/12!). c/a Method cell c:w Direct sum Total Fourier Total Direct 3 sum Total Fourier Total from other cells. ( cf. Table 1). The convergency of the lattice sum is nevertheless very rapid because the cancellations of the lattice sum in the 2nd and 3rd cells are less than those in the 1st cell and they show the characteristic convergency of the Fourier method. n the calculation of the energy levels of 4{-orbitals, the contribution from the term V2 is equally important as the other terms, and c 20 1s so small in magnitude and so sensitive to the value of cja that it cannot have any rigorous meaning in view of our model used for the calculation of the crystalline field. n what follows, therefore, we modify D 20 somewhat arbitrarily. (cf. eq. (3) and (4)). Eq. (1) can be conveniently expressed in terms of equiv!j.lent operaters as far as our interest is confined to the manifold of ] 2 =const., 10 J i.e., < o c6o c66 cja <zo Fig. 1. The calculated values of Cim ( -ve~/a 1 ~ 1 ) should be multiplied to each value of the ordinate- 0

4 280 T~ Murao where (3) 12()=3]. 2 -](]+1), ]40=35}. 4-30}(]+ 1)] }. 2-6](]+ 1) +3]2 (]+ 1) 2, Jno=231} J(J+ 1)}z }z ]2(}+ 1) 2 } }(}+ 1)}z Jz2-5PCJ+ 1) 3 +40]2 (}+ 1) 2-60}(}+1), Jrm=H (J.. +i]y)6+ (J,.-i]y)a], (4) n eq. ( 4), a 20, due to his notation. for example, corresponds to the a in Stevens' notations/ 0 l and r is also 3. Praseodymium The most stable ionic state of Pr is Pr 3 + whose ground state is 3 ~. The next excited state is 3 H 5 which is estimated theoretically to be higher by 2000 cm- 1 than the ground state. 11 l Therefore, we discard the possibility of Pr 3 + being in the 3 H 5 state. The splitting of the energy level, 3H 4, by the crystalline :field (3) can be obtained in the same way as in the case of Ce 2 l : n the present case, howevel:', the difficulty arises from the fact that the relative magnitude of 1, where l takes 2, 4 and 6, cannot be determined uniquely. There is no available calculation for the 4{ wave functions in the rare earth atoms. We therefore employ for the radial part of the 4{ wave function the Slater function containing the effective nuclear charge Z* as a parameter which will be determined below. n Fig. 2, the energy levels are shown as a function of Z* for the observed values of a ( = A) and c /a ( = ). 12 l the following eigenfunctions. The ) 's in the :figure indicate!al)=nal ( ±4)+ CAll =F2)) JA2)=NA2(J ±4) + CA2J =F2)) JBl)=B.Bl (J +3) + C.Bll-3)) B2)=NB2(1 +3)+ Cn2J-3)) C)= ± 1) D)=O) (singlet), (singlet), (singlet). (5) n eq. (5), the state +4 ), for example, represents the normalized wave function of ].= + 4. N.'lf is the normalization constant and is equal to (1 + C 81 l 2 ) -l/ 2, where s stands for A or B and i for 1 or 2. As can be seen from eq. ( 4), only the states

5 The Magnetic Properties of the Praseodymium and the Neodymium Metals 281 that are different from each other by 6 in the value of ]. can combine together, and the upper (or lower) signs of the former double signs in the right-hand sides of eq. (5) correspond to the upper (or lower) signs of the latter ones. The possible ground state is either \D) or \B1) depending on Z*. n either case, that the ground state of Pr:l+ is a singlet explains the fact that the praseodymium metal has no antiferromagnetic phase. Furthermore, the levels crowd relatively near the ground state and this fact is also consistent with the fact that the Shottky type anomaly of the atomic heat appears in a comparatively lower temperature range than in the cases of the cerium and the neodymium metals. With these theoretical backgrounds, our calculations are carried out for the atomic heat and the magnetic susceptibility. The fair agreements with the observations are. obtained for Z*= 10.5 with D 20 multiplied by 2. The multiplication factor is necessary, for without such a modification for D 20, the results are not good regardless of the choice of Z*. n Table 2, we show the energy eigenvalues corresponding to the eigenfunctions in eq. (5) together with coefficients C,, in the same equation. The eigenvalue of the ground state is taken as zero. ~ 1S ~ ~ i:;l Fig. 2. The energy levels of the 4f electrons of Pr3+ (3H 4) in the crystalline field as a function of Z*. Table 2. The energy eigenvalues and the coefficients of the eigenfunctions of the 4f electrons in the Pr metal. (Energy eigenvalue)/ (10-5 ve2fa) c.i A1) A2) B1) B2> r !c> D> By using the energy eigenvalues, E,, given in Table 2 and the multiplicity, n,, shown in eq. (5), we can calculate the atomic heat due to the localized 4felectrons from the following equation : L1C=Nk:x2 { (1/Z 11 ) (1',n,Ele-Eix - (1/Zo)2(.E,n,E,e-E x)2}, (6)

6 282 T. Murao n above equations E/s express the numerical values given in Table. 2 (i.e: the value measured in the unit of (10-5 veja)) and x=10-5 (ve 2 ja)jkt. The magnetic susceptibilities are calculated in the same way as in the case of Ce 2 > and expressed as where X = N(gp.R) 2 _1_(p +-1-G ) u kt Zo x R ' X = N(gp.R) 2 _1_(p +-1-G).l kt Zo.l x.l ' F 1 = (17240 e-sl.:j.e e ) + 2e , (7) (8) -111!" -400.e -84,3" -132" G.l= { 331 e -e +408 e -e Co ~ E..._ 1 u... '<:! 1.0 ~. ~ s T K Fig. 3. The part of the atomic heat due to the 4f electtons of Pr metal. The calculated curve is shown together with the experimental ones. T K Fig. 4. The magnetic susceptibilities calculated for the Pr metal are shown together with the experimental curve.

7 The Magnetic Properties of the Praseodymium and the Neodymium Metali 283- n these equations; N is the number of atoms, g the Lande g-factor equal to 4/5, P.e the Bohr magneton, and Zo the partition function. X 1 means the susceptibility when. the external magnetic field is parallel to the c-axis and x_l the one when the magnetic: field is perpendicular to the same axis. F 11 and F j_ are the contibutions from the matrix elements between degenerate states, and G 11 and G _L are those from the matrix elements between non-degenerate states. The exchange effect is omitted here. The results are shown in Figs. 3 and 4 together with the experimental data. n the actual calculation v is taken as. 3. n Fig. 3, the experimental curve indicated by (Pr-La) means the excess part of Pr atomic heat over La's.a> The other experimental curve is the one obtained with the Debye temperature 8D= 138 K. 18 > n Fig. 4 X=Xu/3+2X_L/3, which should be compared with the experimental data. X 11 and x_l are very sensitive to the character of the ground state but the.t:e is no observation to be referred to. 4. Neodymium We treat the neodymium metal in the similar way as for the praseodymium metal. The ground state is and the next excited state, 4 ltf2 lies higher by 1800 em > Therefore, only ]=9/2 manifold is considered. The observed values of cja are ) and 1.609, 14 > but the former is taken here with the 'value of J _,.1,,---~~...!:,-A2> /B2>... C> Fig. 5. The energy levels of the 4{ electrons of Nd3+ (419/2) in the crystalline field. a=3.650a. For Nd, there remains- Kramers' doublet on account of the odd number of the 4{ electrons, and all levels are doublets as is shown in Fig. 5. The notations in the figure are similar to the case or Pr. Those eigenstates are represented by the following expressions : A1)=NA1 ( ±9/2)+ CAll ±3/2)) A2)=NAz(l ±9/2)+ CA21 =F3j2)) B1)=NR1 ( ±7 /2)+CB11 =F5/2)) B2)=NB2(1 ±7/2) + CB21 =F5/2)) C)=±l/2) where similar notations have been used as in eq. (5). The calculations have been performed for Z* = 11.0 with D 20 multiplied by a factor (9)

8 :284 T. Murao.2.5. The coefficients in eq. (9) and the energy eigenvalues are listed in Ta.ble 3. The : zero of energy is set on the ground state. Table 3. The energy eigenvalues and the coefficients of the eigenfunctions of the.4[ electtons in the Nd metal JA1) JA2) JB1) B2) (Energy eigenvalue)/ (1o-o ve2/a) c.i JC) 76.7 n this case, the specific heat of the Shottky type can li.lso be cli.lculated by eq. ( 6) with E/s in Table 3 and n/s in eq. (9). The magnetic susceptibilities are expressed l>y eq. (7) with g equal to 8/11 and with F's and G's given by the following: F 1 = e- 137 s..:+3.94 e- 16 7"'+38.7 e- 197 "..,+0.5 e ,. 1 _e-j6.7.: 1 _e-l37.sz 1 _e-p7.!b: Gi = e-16.7.< _ e ~: e-16.7.~: _ e-76.7.: e-76.7.~: _ e e-l37.s.. _ e , (10) _f... ;;;-. -s ~Exp.(0D=141 K) e,.. /', ~ ';- \ y Exp. (Nd-La) '13 J... -u' / '-... "' Calc ,. \,-~...../ T K Fig. 6. The part of the atomic heat due to the 4f electrons of Nd metal. The calculated curve.is shown together with the experimeo.tal ones. Fig. 7. The magnetic susceptibilities calculated for the Nd metal are shown. together with the experimental curves.

9 The Magnetic Properties of the Praseodymium and the Neodymium Metals 285 The results are shown in Fig. 6 and 7, where v=3 is also taken and the notations are similar to those in the case of Pr. account. Again the exchange interaction is not taken into As is seen from Fig. 5, there is one doublet near the ground state and this situation does not change even if some modification of D 20 is made. This explains the experimental fact that the Nd metal shows ijs = Rln 4 below the second cooperative transition point which lies at a comparatively low temperature.a) n Table 2, we show the excess entropy corresponding to the anomalous atomic heat, ilcp. n the last column there are also shown the theoretical values to be compared with. As has been discussed in 1, the cooperative transition points are at 7.5 K and 19 K. Table 4. The entropies related to the excess atomic heats of the Nd metal. T K JS(Nd-La) AS((Jn= 141 K) Rln2=1.38 Rln4=2.75 The magnetic susceptibity has been measured for both single- 6 l and poly-crystals. 4 ),lfi} The observed anisotropy of X is smaller than the result obtained here, but it is qualitatively under-stood that X_1_ is larger than X n and that the sudden change of the slope of the experimental 1/X-T curve is probably related to the effect of the crystalline field. 5. Discussion So far we have used the three parameters, v, Z* and D 20 v determines only the absolute scale of energy as can be seen from Figs. 2 and 5. v=3 seems to be somewhat large compared with the case of the cerium metal where v=2.5 was consistent. 2 l But it would not be unreasonable. The values of Z*, 10.5 and 11.0, should be compared with and 13.3 obtained from Slater's rule for the Pr and the Nd atoms, respectively. Some discrepancy between the Slater function and the wave function calculated numerically by the Hartree method can also be found for the 4{ wave functions of Au and Tl atom. 16 l t is difficult to make clear the meaning of the modification for the D 20 on the basis of the present model of the crystalline field. We can probably say that a small departure from our model for crystalline field may be responsible for this. The results obtained in this paper should be understood to be rather qualitative or semi-quantitative. t cannot be uniquely determined whether the ground state of Pr i&!d) or B1). A better agreement with the observations can be obtained if we choose!d) as the ground state and the realization with proper z* and v relates to the modification as D 20 To explain the susceptibility and the specific heat by taking B1) as the ground state, a: fairly large exchange interaction is necessary. A large exchange effect is not plausible, so that this situation is abandoned in this paper.

10 286 T. Murao The discrepancies between the present analysis and the experi~ental data can be related to the following facts : (1) The modified hexagonal structure has been reported. 12 ' 17 > (2) The temperature dependency of cja has not yet been observed, which sensitively affects D 20 (3) The conduction electrons may deviate from the free electrons and the polarization of the core electrons, especially those in 5s and 5p orbitals, may arise due to the 4{ electrons. ( 4) There is a lack of our knowledge about the 4 f wave function in the metal. t is possible that there is a marked deviation from the atomic or Slater's function. For example, the larger anisotropy of the susceptibility for the Nd metal which is obtained in the present calculation would be intuitively understood in view of (1) and (3) mentioned above, because these factors have a tendency to mix the levels further than those indicated in Figs. 2 and 5 and diminish the anisotropy. Detailed considerations on these points will be made in the future. Acknowledgements The author should like to thank Professor T. Matsubara of Kyoto University for his -continual discussions and suggestions. He is also indebted _to Professor A. Morita and Professor C. Horie of Tohoku University and to Dr. T. Kasuya and Dr. A. Yanase of Nagoya University for a number of valuable discussions. References 1) T. Kasuya, Prog. Theor. Phys. 16 (1956), 45, 58. 2) T. Murao & T. Matsubara, Prog. Theor. Phys. 18 (1957), ) D.. H. Parkinson, F. E. Simon & F. H. Spedding, Proc. Roy. Soc. A207 _(1951), ) J. M. Lock, Proc. Phys. Soc. B70 (1957), ) A. Yanase & T. Kasuya, the annual conference of Physical Society of Japan held at Osaka in ) D. R. Behrendt, S. Legvold & F. H. Spedding, Phys. Rev; 106 (1957), ) N. R. James, S. Legvold & F. H. Spedding, Phys. Rev. 88 (1952), ) M. F. Trombe & M. Foex, Compt. Rend. 232 (1951), 63. 9) J. Kanamori, T. Moriya, K. Motizuki & T. Nagamiya, J. Phys. Soc. Japan 10 (1955), ) K. W. H. Stevens, Proc. Phys. Soc. A65 (1952), 209. R. J. Elliott & K. W. H. Stevens, Proc. Roy. Soc. A219- (1953), ) R. J. Elliott &. K. W. H. Stevens, Proc. Roy. Soc. A218 (1953), ) W. Klemm & H. Bommer, Z. Aonrg. Allgem. Chern. 241 (1939), ) R. E. Skochdopole, M. Grifel & F. H. Spedding, J. Chern. Phys. 23 (1955), )' W. Klemm & H. Bommer, Z. Anorg. Allgem. Chern. 231 (1937), 131t 15) J. E. Elliott, S. Legvold & F. H. Spedding, Phys. Rev. 94 (1954), ) A. S. Douglas, D. R. Harttee & W. A. Runciman, Proc. Camb. Phil. Soc. 51 (1955}, ) F. H. Ellinger & W. H. Zachariasen, J. Amer. Chern. Soc. 75 (1953), F. H. Spedding, et a!., Prog. Low Temp. Phys. 2 (1957), 368.