HIGH- TEMPERATURE SUSCEPTIBII ITY AND. . . 307

ture of 595'K. This result is in reasonable agree-ment with experiment, considering that the smallantiferromagnetic intrasublattice exchange inter-

action (found'o to be —3.3 K from fitting the seriesexpansion to the susceptibility data) has of neces-sity been neglected in the present computation.

'P. J. Wojtowicz, Phys. Rev. 155, 492 (1967).2G. S. Rushbrooke and P. J. Wood, Mol. Phys.

1, 257 (1958).3The notation used here for the lattice parameters dif-

fers from that of Ref. 1 but is in keeping with the morewidely accepted notation introduced in Ref. 2.

S. Freeman and P. J. Wojtowicz, Phys. Rev. 177,882 (1969).

~In the original version of Eq. (2) proposed in Ref. 1the constant of proportionality was denoted by p. We usethe symbol K here to avoid confusion with the critical in-

dex y of Eq. (1).6G. S. Rushbrooke and P. J. Wood, Mol. Phys. 6, 409

(1963).S. Freeman and P. J. Wojtowicz, J. Appl. Phys. 39,

622 (1968).K. Obayashi and S. Iida, J. Phys. Soc. Japan 25,

1187 (1968).K. Miyatani and K. Yoshikawa, J. Appl, Phys. 41,

1272 (1970).' P. J. Wojtowicz, Phys. Letters 11, 18 (1964).

P H Y SIC AL REVIEW B VOLUME 6, NUMBER 1 1 JULY 1972

Effect of J Mixing on the g Values of the I ~ Ground State of Pu3+ Ion and Am + Ion inCubic Symmetry Sites*

D. J. Lam and S. -K. ChantAxgonne National L aboratoxy, Axgonne, Illinois 60439

(Received 1 October 1971)

The g values of the I'7 ground state and the energy separations between the I'& ground stateand the I'8 excited state of Pu ' and Am

' ions in a cubic crystal field have been calculated ac-counting fully for intermediate-coupling and J-mixing effects. The results are compared withthe recent experimental values of these ions in alkaline-earth fluorides and thorium dioxides.

I. INTRODUCTION

Edelstein et al. ' reported the g values of the I',ground state of the Pu~' ion in cubic sites of CaF~,SrF~, and BaF3 measured by the electron-paramag-netic-resonance (EPR) technique to be l. 297+ 0.001, 1.250+0. 002, and 1.187 +0.004, respec-tively. They pointed out the importance of account-ing for the effects of intermediate coupling andcrystal-field mixing of higher J states into theground state in interpreting these results. Theyshowed in detail the change in the nature of theground state due to the intermediate-coupling ef-fect, but only briefly discussed the J-mixing prob-lem. Abraham et al. reported the g values of Pu 'and Am" ions in ThO~ and their results were inter-preted in the same manner as those of Edelsteinet al.

The present authors have studied the effects ofintermediate coupling and crystal-field mixing ofJ states for the 5f5 configuration in octahedral sym-metry sites and found that to ensure convergence ofthe low-lying crystal field energy levels of the 5f'configuration, the truncated basis set should includeten J manifolds of the free ion in the crystal fieldcalculations. s It is expected that the same problemswould be encountered in the cubic symmetry sites

of interest to the interpretation of the EPR.In this communication, we present the results of

the investigations of the effects of strong crystalfield interaction on the low-lying energy levels ofPu~' and Am ' ions in cubic symmetry sites. Boththe calculated g values and the energy separations&E between the I', ground state and the I'8 excited .

state of these ions are shown and compared withthe experimental results obtained by EPR'~ andelectron-nuclear-double-resonance (ENDOR) ex-periments of Kolbe and Edelstein. ' The purposeis to demonstrate, within the framework of theelectrostatic model, quantitatively the effect of Jmixing on the g values and &E. It is emphasizedthat no meaningful conclusion for crystal-fieldparameters can be deduced by comparing only gvalues. However, a meaningful conclusion can beobtained by comparing both the g values and &Esimultaneously with their respective experimentalvalues.

II. NUMERICAL CALCULATIONS

Inasmuch as the method used to obtain the crys-tal-field states using the truncated basis schemehas been described elsewhere, only a brief outlinewill be presented. Using the Russel-Saundersbasis functions, the Coulomb and spin-orbit ma-

308 D. J. LAM AND S. -K. CHAN

TABLE I. Eigenvectors of Pu3' ions below 13 000 cm ~.

Energy (cm ')

3 141

6 009

7 029

7 206

8 501

9 659

10641

12 118

7

2

92

32

2

7

2

132

152

Eigenvectors~

-0. 8391 I H) +o. 1114 I E3) + 0.2984 l G ))—0.1026 l G3) +0.3566 l G4)

0. 9086 l H) —0.2347 I G() —0.2860 I G4)

0. 9364 l H) —0. 1619 I G() —0. 1965 I G4)+0.1153 l H3)+0. 1313 l I3)

—0. 8624 I E) -0.1604 l Dg) —0.2065 I D2)+0.2006 l D3) —0. 1.681. 1 Ei)+0. 3007 l F3) +0. 1048 I E4)

—0. 7455 l F) +0.2768 l H) —0.1079 I Po)—0. 1475 l Dg) —0.2009 l D2)+0. 1962 l D3) —0. 1964 l F)) +0.3385 I E3)+ 0. 1060 l E4) —0.1033 l D2) + 0. 1.123 I D5)

0. 9217 l F) +0. 1830 l Dg) +0.2235 l D2)—0. 2228 I D3)

0.9194 l H) —0. 1075 l G4) +0.1193 l H()+0, 1517 l H3) +0.1366 l Ii)+0.2407 I I3)

0, 8484 l E) —0. 1939 l H) +0.1414 l D )—0. 1319 l D3) + 0. 1934 l E() —0. 3147 l 4E3

+0. 1386 l G3)

0. 8649 I H) +0. 1132 l H() +0. 1464 I H3)+0. 1844 l I()+0.3594 I I3) —0. 1374l K

0.7168 l H) +0.2203 l I()+0.45981 I3)—0.1842 l Ki) —0. 1541 l L) + 0.2181 l M)+0. 1568 l 2K() +0. 1531 l 2K4) —0. 1728 l 2K5)—0. 1101 l L&) —0. 1812 l L3)

'Only those components & 0. 1 are listed. Notation is that of Nielson and Koster (Ref. 7).

trices are set up and diagonalized. The eigenstatesar~ in the form of a linear combination of the Rus-sel-Saunders basis functions, i. e. ,

[zM) =P (vsl.zMl~~) l

vsi.m),

4 -IA4&r &=-800 cm

A greg 375cm '

where v represents seniority and other quantumnumbers introduced by Racah. The values of theelectrostatic and the spin-orbit parameters ob-tained by Conway and Rajnak' for the Pu~' ion inLaCl~ and those obtained by Conway' for the Am 'ion are used in the present calculation. The eigen-vectors of the Pu ' ion below 13000 cm ' and of theAm ' ion below 15000 cm obtained with theseparameters are shown in Tables I and II, respec-tively. To reduce the computational effort, theJ=~z and ~~ manifolds, which lie more than 1PPPPcm ' above the ground state, have been truncatedto contain seven and six major I.-S components,respectively. All other J manifolds retain the fullcomplements of all I.-S components. These free-ion eigenvectors are then used as basis functionsto construct the crystal-field matrix thereby ob-taining, after diagonalization, the crystal fieldstates

-400—lECP

-800—ly

i@00 =l.260 -I.260

-1.2857-l.295

I I I l

2 4 6 8 IO

NuMBER OF J-MULTIPLETS MIXED

-!600—

-2000

FLG. 1. Eigenvalues of I'7 and I'8 crystal field statesof Pu3' ion (5f5 configuration) as a function of the numberof J multiplets included in the truncated basis set for the

crystal field calculations.

EFFECTS OF J MIXING ON THE g VALUES OF. . . 309

Energy (cm ')

4 218

7 879

8 233

8 438

8 937

10 875

11900

13 286

14 383

'Only those components

Eigenvectors

—0. 8003 I H) +0.1389 I E3)+ 0.3192 l G g)

—0. 1040 I G3) +0.3772 I G4)+0. 1031 l E2) +0. 1017 l E6) +0, 1006 l Ep)

+0. 8899 l 6H) —0.1079 I Fs)—0.2534 I G g)

—0.3061 I G4)

—0. 17361 Gg) —0.2076 I G4)+ 0. 1285 I H3) + 0. 1520 I I3)

—0.3436 I H) +0. 1296 I P2)+ 0. 1584 I D g) + 0.2104 l D2) —0.2147 I D3)+ 0.2062 I Eg) —0.3605 I F3) —0. 1257 I F4)+0. 1221 l D2) +0. 10491 D4) —0. 1311I D5)

—0. 1735 I D() —0.2162 I D2)+0.2201 I D3) —0. 1937 l Fg) +0.3551 l F3)+0. 1372 I F4)

+0.2043 l Dg) + 0.2415 l D2)—0, 2535 l D3) + 0, 1259 l P4)

-0.1106 l G4) +0.1356 I Hg)+ 0. 1699 l H3) +0. 1574 I Ig) + 0.2728 I I3)—0. 1106 I Kg)

—0.22791 H) + 0. l 531 I D2)—0, 1490 l D3) +0.2095 l Ef) 0 3446 I F3)+0.1412 l 'G3)

+0. 1247 I Hi) +0. 1614 I H3)+0.2053 I I() +0.3980 I I3) —0. 17811 Kf)

0 1305 l 4L)+0. 1007 l 2K )

+0, 1978 l Ii) +0.4599 I I3)—0.2208 l Ki) —0. 1944 I L) + 0. 3203 I M)+0, 1920 l 2K)) + 0. 1839 l K4) —0.2154 l K5)—0. 1571l L)) —0.2617 I L3)

ster (Ref. 7).

—0.1073 I E)

0.1069 I E)

0.9223 I H)

0.6676 l E)

—0. 8161 I E)

0. 8970l F)

fi2 0. 8934 l H)

0. 81041 E)

f3 0. 8159 l H)

152 0.5823 I H)

& 0. 1 are listed. Notation is that of Nielson and Ko

TABLE II, Eigenvectors of Am4' ion below 15000 cm '.

(2)

It is seen that the separation of the 1"7 ground stateand the 1'8 excited state changes by a factor of 3as the number of J manifolds included in the basisset increases from two to ten. It is impossible todecide a priori the number of J manifolds to be in-cluded in the basis set without performing a con-vergence test. For reasonable crystal-fieldstrengths, it is necessary3 to include ten J man-

r)=~ (ZMli) ZM)JM

As has been noted previously, ' the calculatedphysical quantities are inaccurate if too small abasis set is used. Figure 1 illustrates the effecton the eigenvalues of the I', and I'8 states of thePu" ion as the number of J manifolds included inthe basis set is increased. The crystal-fieldparameters used are A4(r6) = —800 cm ' and A6(r )= 3'75 cm '. These parameters are related to B4and B6 of Ref. 1 by

84= 8A4(r ) and B~=16A6(re).

ifolds to obtain accurate results for the low-lyingcrystal field states.

In the absence of a magnetic field, the 1", groundstate contains two degenerate components I', andI', . The effect of an applied magnetic field 1f is tosplit the degeneracy. The manner of splitting de-pends on the magnitude and direction of the field.This Zeeman splitting ~E may be obtained by solv-ing the 2 &2 secular determinant

(8)

where li ) is I I', ), pe is the Bohr magnetron, and

p. = L+g, S, with g, the gyromagnetic ratio. Sincethe crystal-field states are related to the originalRussel-Saunders basis functions 1vSLJM) by thetwo linear transformations in Eqs. (1) and (2), thematrix elements (il p, Ij) may be evaluated as de-scribed in Ref. 3. To compare theory with experi-ment, the conventional approach is to representthe Zeeman splitting by a spin Hamiltonian of theform gp~lN, with S= —,'. In the specific case in

310 D. J. I AM AND S. -K. C BAN

600

Eo 400Co

Ir~

200

(a)

4A4&r &=-l200- I.O- ————A4&r &=- IQQQ

~ earn~ ~~~ A ~r4p 8QQ—A4&r4&=-600

cm-I

cm-I

cm-'

cm I

in cubic symmetry sites as a function of the fourth-order, A, (r'), and the sixth-order, A, (r2), crys-tal field potentials. Four values of A4(r 4) areshown in this figure. The hatched area indicatesthe experimental value of 348+ 25 cm ' deduced byKolbe and Edelstein from ENDOR experiment.Figure 2(b) shows the corresponding g values cal-culated and the experimental g values (hatchedarea) obtained by Edelstein et al. for the Pu" ionin different alkaline-earth fluorides. Figure 3(a)shows the 4Er r for the Am' ion in cubic sym-metry sites. The g value as a function of the crys-tal fieM parameters is shown in Fig. 3(b). Theexperimental g values for the Am ' ion in ThO~were determined by Abraham et al. to be 1.2862+ 0. 0005 and no &E~, ~ value has been publishedfor this system. It is apparent, from the resultsshown in these two figures, that there is no pos-

—I.2

U 700

600

I 4

-I 5—(b)

500—I

400—Co

i 500

200—

0I

IOO

i

200 300 400

A6&r &, cm '500

FIG. 2. (a) Energy separation between the I', groundstate and the I'8 excited state of the Pu ' ion as a functionof cubic crystal field potentials. (b) The g value of theI'7 ground state of the Pu ' ion as a function of cubic crys-tal field potentials. Cross-hatched areas represent ex-perimental values of Hef. 4.

I 00—

-1.0—

(a)

A4&r &=-IOOO

A4&r"& =- 800A4&r &=-6004

cm

cm I

cm-I

which the magnetic field is applied parallel to thefourfold axis of the crystal field, the Zeeman split-ting 4E is given by gp~. Equating this to thesplitting obtained by solving Eq. (3), we have forthe g value

~ gf [(~11 ~22)2 ~ 4 (~12)2)1/2 (4

Here we have used the shorthand notation p,'~

=(&,' I p, If'~7). Similar to the crystal-field split-tings, the g value is sensitive to the effect of Jmixing. In Fig. 1, the numbers shown under theI'7 lines illustrate the variation of the calculatedg values with respect to the size of the truncatedbasis set.

III. RESULTS AND DISCUSSION

Figure 2(a) shows the energy separation betweenthe 1", and I'8 crystal field states of the Pu ' ion

—I.2

D

-I 4

-l.5—

IOO

(b)200 500

A6&r6&, crn '

I

400 500

I'IG. 3. (a) Energy separation between the 1"& groundstate and the I'8 excited state of the Am

' ion as a func-tion of cubic crystal field potentials. (b) The g value ofthe I', ground state of the Am ' ion as a function of cubiccrystal field potentials.

EFFECTS OF J MIXING ON THE g VALUES OF. . . 311

sibility of deducing any meaningful crystal-fieldparameters from the experimental g value alone.It is possible, however, to obtain these parametersusing both the AEr, r, value and the g value ob-tained for the same ion in the same host latticetogether. Taking the &Er r value and the mea-

7. 8sured g value for the Pu ' ion in CaF~, the best fitbetween calculation and experiments can be ob-tained by using A4(x4) = —600 cm ' and A6(r~) = 400cm '. The A4(x4) value is within the + 25% limitof Ref. 1 but the ratio of the sixth- and fourth-order terms, in the notation of B6/B„would be—1.33 instead of —0. 2. It should be noted thatthese parameters can only be considered as ademonstration of the possibility and not as param-eters with physical significance at the present time.The reason is that Kolbe and Edelstein deduced the&Er r value using matrix elements of the Zeemanr7 rsand hyperfine interactions calculated with crystalfield wave functions which takes into account thecrystal field mixing of only two J manifoMs. Until

these matrix elements are recalculated with theJ-mixing effect taken fully into account, it is notpossible to estimate the error of &Er r and the

7 8derived crystal field parameters. It should beemphasized that the present discussion has beendeveloped within the framework of the electrostaticmodel. If the usual assumption of this model, thatthere is no overlap of the ligand ions with the cen-tral ion of interest, is relaxed, then other effectssuch as orbital reduction, ' electrostatic-chargepenetration, and overt configuration interation'should be taken into account, in addition to the ef-fects discussed, to interpret the experimental re-sults.

ACKNOW( LEDGMENT

The authors wish to thank Dr. ¹ Edelstein andDr. F. Y. Fradin for many helpful discussions and

gratefully acknowledge Dr. Edelstein for providingthe ENDOR experimental results before publication.

*Work performed under the auspices of the U. S. AtomicEnergy Commission.

~Present address: Cavendish Laboratory, Cambridge,United Kingdom.

'N. Edelstein, H. F. Mollet, W. C. Easley, and R. J.Mehlhorn, J. Chem. Phys. 51, 3281 (1969).

M. M, Abraham, L. A. Boatner, C. B. Finch, and

R. W. Reynolds, Phys. Rev. B 3, 2864 (1971).3S. -K. Chan and D. J. Lam, in Plutonium 1970, edited

by W. N. Miner (The Metallurgy Society of AIME, New

York, 1970), pp. 219-232.W. Kolbe and N. Edelstein, Phys. Rev. B ~4 2869

{1971).

J. G. Conway and K. Rajnak, J. Chem. Phys. 44,348 (1966).

~J. G. Conway, J. Chem. Phys. 41, 904 (1964).~C. W. Nielson and G. F. Koster, Spectroscopic Co-

efficients for p", d", and f"Configurations (MIT Press,Cambridge, Mass. , 1963).

K. W. H. Stevens, Proc. Roy. Soc. (London) A219,542 (1953).

M. M. Ellis and D. J. Newman, J. Chem. Phys. 49.4037 (1968).

K. Rajnak and B. G. Wybourne, J. Chem. Phys. 41.,565 (1964).

PHYSICAL REVIEW B VOLUME 6, NUMBER 1 1 JULY 1972

Critical Exponents for the Heisenberg Model*

Morgan K. Grover, Leo P. Kadanoff, and Franz J. WegnertDepartment of Physics, B~ocen University, Providence, Rhode Island 02912

(Received 27 March 1872)

A recursion relation obtained by Wilson is used for the numerical calculation of critical in-dices for the d =3 classical Heisenberg model. Some of the results obtained are y=1.36 and

y =1.24.

The renormalization-group (HG) approach to thetheory of critical phenomena has been used byWilson' to calculate critical exponents for the d=3Isingmodel and by Wilson, Fisher, Pfeuty, and

Wegner to obtain perturbation expansions~-' in e= 4 —dof the critical indices for the classical Hei-senberg, X-F, and Ising models. Wegner' hasshown how the RG approach leads tothermodynam-ic scaling and to deviations from the scaling

laws. In this note we report the application ofWilson's recursion relation to the numerical cal-culation of critical indices for the d= 3 classicalHeisenberg model.

Wilson's formulation, in the g =0 approxima-tion, leads to an effective Hamiltonian for spinfluctuations of wave vector jk~ &2-~, p an integer: