J. 31. D~xow et al. : Contributions to “Crystal Fields” from Kinetic Energy 155

phys. stat. sol. (b) 111, 155 (1982)

Subject classification: 13

Department of Physics, University of Warwickl)

Contributions to cc Crystal Fields” from Electron Kinetic Energy Terms BY J. M. DIXON, R. CHATTERJEE~), and J. A. MCINNES

The magnitude of contributions to the Bt crystal field coefficient, from second-order kinetic energy terms, is estimated and compared with the value from other sources. Its relative value is small and nev‘er gr”&ter than approximately 10%. If the parameters chosen in the paper for the 3s oxygen state are tibed, the contribations to B$ B& and B! are 1 t o 2 cm-l.

Die Hohe der Beitrage zum Kristallfeldkoeffizienten Bg aus den Termen zweiter Ordnung der kinetischen Energie werden berechnet und mit dem Wert aus anderen Quellen verglichen. Es wird gefunden, daI3 sein relativer Wert klein und nie groBer als etwa 10% ist. Wenp die in der Arbeit gewiihlten Parameter fur den 3s-Sauerstoffzustand benutzt werden, sind die aeitriige zu B;, 4 und Bt etwa 1 bis 2 cm-1.

1. Introduction

Recently Stevens [l] suggested that electron kinetic energy terms could provide contributions, in second-order perturbation theory, to spin Hamiltonian parameters, which are indistinguishable from crystal-field-like terms in first order. Following this idea we have tried to estimate how large these terms are likely to be for the magnetic insulator Gd3+ in La(C,H,SO,), . 9 H,O. The mechanism is given below,

I n (1) la,) denotes a one-electron 4f state of Gd3+ with orbital angular momentum component m. Ie ) represents a 3s state of a nearby oxygen ion a t position R - in the ethylsulphates there are nine in number, the point group symmetry being C3h. (r is a spin component and Po denotes a projector [2] for a specific degenerate manifold. This describes the excitation of an electron from a 4f orbital m up into an excited 3s orbital of an oxygen neighbour and then back to the central ion to another 4f orbital m’ - notice it is not necessary that m = m’.

2. Theory

Stevens used orthonormal Wannier-like states but in this case we shall use an ortho- normal linear combination of all the seven 4f orbitals and all nine of the 3s neigh- bouring orbitals. As the overlap between the 4f and,the 3s orbits is expected to be small a convenient set of orthonormal functions can b? generated using the method of Lowdin [3]. If the original atomic orbitals are denoted by Iqp) and the new cam-

’) Warwick, Coventry CV4 7AL, Great Britain. ‘) Visitor from the Physics Department, The University of Calgary, Calgary, Alberta, Canada

T2N 1x4.

156 J. M. DIXON, R. CHATTERJEE, and J. A. McImEs

R+R’ J (4)

where we have used a self-evident notation and have dropped terms between sites which are quadratic in the overlap. The largest terms in (4) will be

(5 a) -1/”’

(1 + 8 ) 4 f ” f 4 f ” (1 + 8 G L (CpZI o2 I& 9

The 3s atomic wave functions we have assumed to have a Slater form and given by

where r’ = Ir - RI and have the same form about each site [4]. The 4f wave func- tions were assumed to be R(r) Yr(19,q) about the Gd3+ site, where

4

i = l R ( r ) = r3 C Ci e-Zir; (7)

the parameters C , and 2, are from Freeman and Watson [5] and are given in Table 1. Zi has dimensions a r l .

Tab le 1

i Zi c4

1 12.554 1923.8151 2 7.046 329.66724 3 4.697 43.274827 4 2.678 1.5047469

3.1 M a t r i x elements of V e between sites

Equation (5a) reduces to evaluating integrals given below,

where we have used @ = 0: since the overlaps are approximately

(8)

unity. Using

Contributions to "Crystal Fields" from Electron Kinetic Energy Terms 157

Rotenberg e t al.'s equation (1.31) [6] the integral in (8) can be reduced to

d I d21

[--6 - 6B __ - B

dCL2

where

The integral in (9b) can then be easily evaluated [7] using

where the modified Bessel functions are given by [8]

and

The angular integrals in (9b) using (10) to (12) are trivial and the remaining radial integrals can be evaluated using (11) and (12) with standard techniques [8]. To obtain an order of magnitude for B we used Slater's rules [4] for the two configura- tions of the oxygen ion Is2 2s2 2p5 3s1 and ls2 2s2 2p4 3s2. The former configuration gives B = 0.0166 at. units whereas the latter gives B = 0.2000 at. units. Because of the complicated nature of the radial integrals for equation (5a), even though they can be evaluated analytically, we have evaluated them by computer for a range of values of B.

In the ethylsulphates the oxygen ions form triangles. Three of the ions are in the same plane as the Gd3+ site and their spherical polar co-ordinates are (0.252 nm, n/2, p = (2% + 1) 4 3 ) [9]. Six of the ions form two separate triangles above and below the latter plane and their co-ordinates are, respectively, (0.237 nm, 6, 'p = = 2nn/3) and (0.237 nm, 3t - 6, 'p = 2nn/3) with 6 approximately 40". In the coni- puter programme we calculated the sum of the radial integrals for each value of R for every value of B. We also calculated overlap integrals using the computer for the same values of R and B.

2.2 Radial integrals on site

I n equations (5b), (5c), the angular integrals are trivial [6] and the radial integrals, although straightforward for the (&I o2 I & ) matrix element, are laborious for (pzl v2 I&). We simply give the results below,

(&I v2 = -B2/10 at. units , ('p4mfJ 0 2 I&) = -43.828 at. units .

Overlaps in (5b), (5c) can be obtained from the last term in (9a) from the computer by putting the first two terms in this equation equal to zero and dividing by B2.

158 J. N. DISON, R. CHATTERJEE, and J. A. MCINNES

2.3 Effective operators

Equation (1) can now be rewritten as

where y is the sum of the products of the radial integrals and overlaps in equation (5). If we define (h2/2mag)2 y2/(E3, - E 4 f ) to be E, when y is evaluated for the largest R = R, and E, for the smallest R = R, we can then rewrite (14) in the form

(0 .1667~~ - 0.25358,) C or’(Z,) + + (0 .0205~~ - 0.03488,) C Of’(Z,) + + (0 .0063~~ + 0.0131s,) x O!’(Z,) +

i

i

i

+ (0 .0096~~ + 0 .0014~~) x (O?)g(Z,) + 0?)g(Zt)) . (15) i

To obtain (15) we have used the positions of the ions in ethylsulphates given by Abragam and Bleaney [9] and the following relationships due to Stevens [lo]:

&+a,*- = C A,,#,* [Om--m*~+]i (n’) , n’, i

(n’) I J a&+a,.+ = f C A,,.,. [om-,. (1 + 2Sz)li ,

n’, i

where

The OF) are one-electron orbital operators defined by Smith and Thornley [ll], the coefficients N,, are listed by Stevens [lo], and the sum over i is over the magnetic electrons of the Gd3+ ion. The operators OF) are related to the Racah defined tensor C i or T:. For example, 0; [11] is related to Racah’s Cq or Ti [12] by

3. Results

Below we tabulate values x (Table 2 ) , the sum of radial integrals and the 3s-4f overlap, as a function of R,, R,, and B (Table 3).

We obtained the absolute Gd energies of the 4f and 2p states from the calculations of Hermann and Skillman [13] and an energy of 3s1 relative to 2p from Moore [14]. Hence we could estimate ESS - E4f which we took to be 1.22 Ryd. Thus we expect the very largest value of ()z2/2maa)2/(Es, - E4f) to be approximately 1 Ryd. As an example, we consider the case of B = 0.2 at. units. We find for (5a) and R = R,,

Contributions to “Crystal Fields” from Electron Kinetic Energy Terms 159

Table 2 Values of z in units of 10-3 a;’

B (l/u?) for R, for R,

0.11 0.0098 0.0106 0.20 0.1 149 0.1280 0.30 0.4966 0.5740 0.38 0.9997 1.2014

5 = 1.149 x 10-4 at. units and for R = R,, x = 1.280 x at. units. For (5b) the radial overlap for R = Rl is -1.828 x at. units whereas for R = R, i t is -1.825 x 10-4 at. units. I n the case of (5b) the kinetic energy matrix element can be found from equation (13). Similarly the radial overlaps for (5c) are the same as for (5b). These values include the radial part of each overlap in (5b) and (5c) and we have set (1 + s)afm-4f.l m (1 + 8)3;%3s~g M 1. Thus the on-site 4 f 4 f matrix ele- ments of v2 are by far the largest for the case B = 0.2 at. units. We find for R = R,, y = 28.036 x and for R = R,, y = 27.994 x loe3, where we have dropped the very small contributions from x and the 3s-3s on-site terms. Therefore

E~ = (2.8036)2 x Ryd and E, = (2.7994)2 x Ryd .

-1/2

The contribution to B& i.e. the coefficient of C CO,(i) will be approximately 7.4 cm-l. i

Table 3 Overlaps in at. units

0.11 -0.1699 - 0.1656 0.20 - 1.8277 - 1.8249 0.30 - 7.2886 - 7.5257 0.38 - 14.0656 - 14.9952

4. Discussion and Conclusions

If we compare the value above with Bi which is normally used [15], i.e. Bg NN 200 cm-l, we see that the kinetic energy contribution is very small. If we decrease B SO that the 3s orbits become more delocalised then x becomes very much smaller and can be safely neglected. The overlaps however will be larger but the (&I v2 term will be of the same order of magnitude. Thus the contribution in (5b) will increase. However, this only occurs for very small values of B. For values of B in Table 3 one can see that the overlap actually decreases in magnitude as B gets smaller. When we consider (5 c) the (pE/ v2 I&) decreases in magnitude with decreasing B from (13). The overlap will also decrease provided B is in the range in Table 3.

For smaller values of B we expect the largest contribution to the overlap to be approximately constant but with smaller linear and quadratic parts in B. When this is combined with the 3s-3s kinetic energy in (5c), i.e. proportional to B2, this term will become very small. I n the integral (9 b) one can easily see that the largest contri- bution, for small B, to the overlap will come from terms in p3. However, this contains Ir - R12 which will not contain a harmonic with 1 = 3 (choosing a polar axis along R)

160 J. 31. DIXON et al. : Contributions to “Crystal Fields” from Kinetic Energy

so the p3-component of I will vanish. Hence for decreasing B, we expect even smaller values of y than those for B = 0.2 and hence a smaller Bg.

On the other hand, if we increase B, we expect, for the range near that in Table 3, that the magnitude of the overlap will increase thus increasing (5b) and (5c). Such values of B are very unrealistic to describe the 3s state since the B-value for the 2p state, which is very well localised and much more so than the 3s state, is B = 1 . 2 8 3 ~ ~ ~ ~ and the value of B for the 3s must be very much smaller than this. I n fact, by the time B = 1uc1 the overlap has already changed sign and for B very large the overlap will be insignificant. We estimate that, even if B is larger than the values we have taken to estimate the coefficient B:, (5b) and (5c) will increase by no more than a factor of three and thus BE will only be within 10% of the value of Bi from other sources. Thus we conclude that for the specific case of Gd3+ in the ethyl- sulphates the contribution to B; from kinetic energy terms, in second order will be small.

References [I] K. W. H. STEVENS, Physics Rep. Z4, 1 (1976). [2] C. A. BATES, J. N. DIXON, J. R. FLETCHER, and K. W. H. STEVENS, J. Phys. C 1, 859 (1968). [3] P. 0. LOWDIN, Adv. Phys. 5 (Suppl.), 46 (1958). [4] J. S. GRIFFITH, The Theory of Transition Metal Ions, Cambridge University Press, 1964. [5] A. J. FREEMAN and R. E. WATSON, Phys. Rev. 127, 2058 (1962). [6] X. ROTENBERU, R. BIYINS, N. METROPOLIS, and J. I(. WOOTEN, JR., The 3j and 6 j Symbols,

[7] P. M. MORSE and H. FESHBACH, Methods of Theoretical Physics, McGraw-Hill Publ. Co.,

[8] I. S. GRADSHTEYN and I. M. RYZEIP, Tables of Integrals, Series and Products, Academic

[9] A. ABRAGAM and B. BLEANEY, Electron Paramagnetic Resonance of Transition Ions, Claren-

The Technology Press, MIT, Cambridge (Mass.) 1959.

1953.

Press, London 1965 (p. 310).

don Press, Oxford 1970 (p. 278). [lo] K. W. H. STEVENS, Phys. Letters A 47, 401 (1974). [ll] D. SNITH and J. H. M. THORNLEY, Proc. Phys. SOC. 88, 799 (1966). [12] H. A. BUCHNASTER, R. CHATTERJEE, and Y. H. SHIN@, phys. stat. sol. (a) 13, 9 (1972). [13] F. HERMANN and S. SKILLMAN, Atomic Structure Calculations, Prentice Hall Inc., Englewood

[I41 C. E. XooRE, Atomic Energy Levels, Val. 1, Nat. Bur. Standards, Circular 467, 1949. [15] B. G. WYBOCRNE, Phys. Rev. 118, 317 (1966).

Cliffs (X.J.) 1963.

(Received October 5, 1981)