New Concepts

215 Pages·1977·2.915 MB·English

by  W. E. Wallace, S. G. Sankar, V. U. S. Rao, D. K. Hoffman, R. Ruedenberg, J. G. Verkade, K. D. Warren, K. Schubert, C. Linarès, A. Louat, M. Blanchard (auth.)

STRUCTURE
AND BONDING
Volume 33
Editors:
J. D. Dunitz, Ziirich • P. Hemmerich, Konstanz
J. A. Ibers, Evanston. C. K. Jorgensen, Gen~ve • J. B. Neilands,
Berkeley • D. Reinen, Marburg-R.J.P. Williams, Oxford
With 85 Figures and 47 Tables
galreV-regnirpS
Berlin Heidelberg New York 1977
ISBN 3-540-08269-7 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-08269-7 Springer-Verlag NewYork Heidelberg Berlin
Library of Congress Catalog Card Number 67-11280
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically
those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and
storage in data banks. Under § 54 of the German Copyright Law where copies are made for other then for private use, a fee is
payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
O by Springer-Verlag Berhn Heidelberg 1977
Printed in Germany
The use of general descriptive names, trade marks, etc. in this publication, even if the former are not especially identified, is not to
be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used
freely by anyone.
Typesetting R. & J.Blank, Miinchen. Printing and bookbinding: Briihlsche Universit~itsdruckerei, ne3lezG
Contents
Crystal Field Effects in Rare-Earth Intermetallic Compounds
W.E.Wallace, .S G. Sankar, V. U. .S Rao
Molecular Orbital Bonding Concepts in Polyatomic Molecules: A Novel
Pictorial Approach
D. K. Hoffman, R. Ruedenberg, J. G. Verkade 75
Ligand Field Theory of f-Orbital Sandwich Complexes
K. D. Warren 97
The Two-Correlations Model, a Valence Model for Metallic Phases
K. Schubert 931
Rare-Earth--Oxygen Bonding in the LnMO4 Xenotime Structure:
Spectroscopic Investigation and Comparative Study of Ligand Field Models
C. Linarbs, .A Louat, M. Blanchard 971
STRUCTURE AND BONDING is issued at irregular intervals, according to the material received.
With the acceptance for publication of a manuscript, copyright of all countries is vested exclusive-
ly in the publisher. Only papers not previously published elsewhere should be submitted. Like-
wise, the author guarantees against subsequent publication elsewhere. The text should be as clear
and concise as possible, the manuscript written on one side of the paper only. Illustrations should
be limited to those actually necessary.
Manuscripts will be accepted by the editors:
Professor Dr. Jack D. Dunitz Laboratorium fur Organische Chemic der Eid-
gen6ssischen Hochschule
CH-8006 Ziirich, Universit~tsstrafie 6/8
Professor Dr. Peter Hemmerich Universit~it Konstanz, Fachbereich Biologic
D-7750 Konstanz, Postfach 733
Professor James A. Ibers Department of Chemistry, Northwestern University
Evanston, Illinois 60201/U.S.A.
Professor Dr. .C Klixbiill nesnegr~CJ 51, Route de Frontenex,
CH-1207 Gen~ve
Professor Joe B. Neilands University of California, Biochemistry Department
Berkeley, California 94720/U.S.A.
Professor Dr. Dirk Reinen Fachbereich Chemic der Universitiit Marburg
D-3550 Marburg, Gutenbergstra~e 18
Professor Robert Joseph P. Williams Wadham College, Inorganic Chemistry Laboratory
Oxford OX1 3QR/Great Britain
SPRINGER-VERLAG SPRINGER-VERLAG
NEW YORK INC.
D-6900 Heidelberg 1 D-1000 Berlin 33
.P O. Box 105280 Heidelberger Platz 3 175, Fifth Avenue
Telephone (06221) 4 87.1 Telephone (030) 82 20 01 New York, N.Y. 10010
Telex 04-61723 Telex 01-83319 Telephone [phone]
Crystal Field Effects ni Rare-Earth Intermetallic
Compounds
W.E. Wallace, S.G. Sankar dna V.U.S. Rao
Department of Cher~istry, University of Pittsburgh, Pittsburgh, PA. 15260 (USA)
Table of Contents
I. Scope cf the Review ....................................... 2
II. Introduction ............................................ 3
A. The Exchange Ir teraction ................................... 3
B. Tte Crystal Field Ir ter~ction ................................. 3
HI. Calculational Procedures ..................................... 7
A. Methodology for Treating the Interactions ......................... 7
1. The Case of a Single J State ................................ 7
2. Effects of J Mixing ..................................... 9
3. Effect of External or Exchange Fields .......................... 10
B. Interrelationship with Exr eriment .............................. 21
1. Magnetic Susceptibility ................................... 12
2. Heat Capacity and Free Energy .............................. 13
3. Neutron Inelastic Scattering ................................ 13
4. Spin Disorder Resistivity .................................. 14
5. Magnetic Anisotropy .................................... 15
IV. Discussion of Specific Families of Compounds ......................... 17
A. The Rare Earth-Aluminium Compounds ........................... 17
.1 The RA12 Series ....................................... 17
2. The RA13 Series ....................................... 23
B. The Rare Earth-Nickel Compounds ............................. ~7
.1 T~e RNi 2 Series ....................................... ~7
2. The RNi 3 Series ....................................... 31
3. The RNi 5 Series ....................................... 33
C. The Rare Earth-Cobalt Compounds ............................. 36
1. The RCo 2 Series ....................................... 36
2. The RCo s Series ....................................... 38
3. The R2Co17 Series ..................................... 41
D. The Rare Earth-Iron Compounds ............................... 42
E. The Rare Earth-Hydrogen Compounds ........................... 44
F. Samarium Compounds ..................................... 46
1. Influence of J-Mixing .................................... 46
2. Spin Reorientation in SmFe 2 ............................... 47
3. Magnetocrystalline Anisotropy in SmCo 5 ......................... 47
V. Concluding Remarks ....................................... 51
References ................................................ 52
W.E. Wallace, S.G. Sankar and .V U. .S Rao
I. Scope of the Review
The crystalline electric field interaction plays a very significant role in modifying
many properties of metallic rare earth systems - magnetic susceptibility and satura-
tion magnetization, thermodynamic properties such as heat capacity, compressibility
and expansivity, transport properties such as electrical and thermal conductivity and
superconductivity. Several review articles have recently appeared covering certain
specialized aspects of the crystal field interaction (which will be referred to hereafter
as CFI). Cooper and Rhyne (1) have discussed various features of the elemental rare
earths which are influenced by the CFI. Malik et at (2) have summarized observations
on Ce and Sm compounds with emphasis on NMR characteristics. These and other
aspects of Sm compounds have been treated (3) by de ,niW naV Diepen and Buschow.
Liithi (4) has discussed the effects of the CFI with emphasis on elastic constants,
particularly for rare earth pnictides. ecallaW (5) has summarized results obtained
pertaining to the CFI for selected rare earth intermetallics through susceptibility and
thermodynamic (heat capacity) studies. The purpose of the present review is to up-
date the earlier summary, which covered the field only through 1973 and moreover
was in the form of a conference presentation and hence enjoyed only limited circulation.
In Section II we present a brief account of the methods that have been developed
to deal with the CFI and exchange interactions in metallic rare earth systems. Usually
both of these interactions are involved in varying degrees of relative strength. In Sec-
tion III a summary account is given of the calculational procedures involved in deter-
mining the eigenvalues and eigenfunctions and in relating the observable properties of
the system - susceptibility, heat capacity, etc. - to the details of the CFI.
In Section IV we have discussed results for a selection of intermetallic com-
pounds such as RAI2, RAla, RNi2, RNi3, RNis, RCo s and RUe 2 . (R here represents
a rare earth.) There are certain significant omissions in our coverage. Since Gd +3 is in
an S state, the CFI interaction for it is insignificant. Hence little attention is paid to
Gd compounds in this review. Ce and Yb in intermetallic compounds exhibit variable
valency; Ce is found in both the Ce +4 and Ce 3+ states and Yb exists as Yb +2 and
Yb a÷. Because of these complications we have not considered Ce and Yb compounds
except in a few instances. Compounds of Sm represent an interesting special case -
because of J-mixing effects (vide infra) and additionally because of the widespread
interest in SmCo s as an extraordinary permanent magnet material. These compounds
are described in a separate section.
The examples cited in Section IV are frequently drawn from work carried out in
this laboratory. They have been chosen to illuminate the principles involved. The
present review is not indended to be exhaustive, but rather illustrative of principles.
Because of space limitations many important topics are not included - behaviour of
the elemental rare earths (1), amorphous materials (6), liquid rare earth systems (7),
elastic and magnetoelastic properties (4), thermal conductivity (4), etc. The reader
who is interested in these topics is referred to the reviews cited in ref. (1), (4), (6)
and(Z).
2
Crystal Field Effects in Rare-Earth Intermetallic Compounds
II. Introduction
In the rare earth metals and intermetallic compounds, the 4f electrons are well
embedded within the core orbitals and therefore do not play a significant role in
chemical bonding. The metallic rare earth systems, with a few exceptions, may be
treated as an assemblage of tripositive ions located in a sea of conduction electrons.
The 4f electrons possess large angular momentum and are therefore responsible for
the interesting and variety of magnetic properties exhibited by these rare earth
materials. With the progressive filling of the 4f electronic level as one passes from
Ce to Lu, the electronic interaction within the atom gives rise to many states with
varying and interesting characteristics.
Two other elements are normally considered along with rare earths because of
their similar chemical properties: La, which is just before Ce in the sequence, and Y,
which comes in the same position as La in the preceding period. These two elements
exa nonmagnetic and therefore they and their intermetallic compounds serve as
"blanks".
A. The Exchange Interaction
In rare earth metals and intermetallic compounds, a strong magnetic coupling, as in-
dicated by their rather high ordering temperatures, is often observed. This is in con-
trast with the behavior of their oxides and salts. The exchange interaction responsible
for this coupling between 4 f orbitals is an indirect one involving polarization of the
conduction electrons. Such an interaction is oscillatory and has long range character.
The theoretical framework on the basis of these ideas was developed by Ruderman
dna Kittel (8), Kasuya (9) dna Yosida (10) and is usually known as the RKKY theory.
This theory explains a variety of magnetic structures exhibited by rare earth inter-
metallics, de Gennes (11), Kasuya (12), Elliott (13), Rocher (14) dna Kittel (15) have
dealt with this interaction mechanism in great detail.
B. The Crystal Field lnteraction
An array of charges in a crystal produces an electric field at any one ion, the -os
called crystalline electric field. The presence of this field causes a Stark splitting of
the free ion energy levels which remits in a substantial modification of magnetic,
electrical and thermal properties of the material. The theory of the crystal field and
its interpretation in terms of group theory are originally due to Bethe (16).
If the origin of the coordinate system is taken at the nucleus of the rare earth
ion, an expression for the electrostatic potential at a point (r, 0, )o~ near the origin
due to the surrounding k ions may be written as
W.E. Wallace, S.G. Sankar and V. U. S. Rao
V(r, ,O ~) ; qk/(Rk -- r) (1)
k
If the rare earth ion has charge ile at ri, 0 i and ~i, its electrostatic energy due to
the perturbing potential, V, is
~CF = ~, Viq = ~, Z hC k~l(/kq (2)
- ri)
i i k
This perturbing potential partially lifts the (2 J + 1) degeneracy of the ground
multiplet of the free rare earth ion. The resulting eigenvalues and eigenfunctions of
the rare earth ions may be evaluated by the straightforward but tedious methods of
perturbation theory (17). This is seldom done because of the convenience of a cal-
culational method introduced by Stevens about 25 years ago (vide infra).
Table .1 Stevens' multiplicative factors associated with equivalent operators for the ground states
of rare earth ions and the calculated Hartree-Fock radial integrals (r n ) in atomic units of length
ao n (20).
Ion Ground
state j~r ~j ~j (r 2) (r 4) (r 6 )
-2 2
Ce3+ 2Fs/2 5" 7 32. 5.7 0 1.200 3.455 21.226
- 22. 13 - 22 24"17
Pr 3+ 3H4. 32.52.11 32.5.112 34.5.7.112.13 1.086 2.822 15.726
- 7 - 23.17 - 5.17-19
Nd +3 419/2 32.112 33.113.13 33.7.113.132 1.001 2.401 12.396
13 2-13
Sm3+ 6H5/2 32.5 "7 33-5.7-11 0 0.883 1.897 8.775
- 1 2 - 1 0.758 1.44 5.8
Tb3+ 7F6 32.11 33.5.112 34-7.112.13
- 2 - 23 22
Dy3+ 6Hls/2 32.5.7 33-5.7.11.13 33.7.112.132 0.726 1.322 5.102
- 1 - 1 - 5 0.695 1.22 4.5
H°3+ 815 2.32.52 2.3-5-7.11.13 33.7.112.132
22 2 23
Er3+ 4115/2 32-52-7 32.5.7-11-13 33.7.112.132 0.666 1.126 3.978
1 23 -5
Tm 3+ 3H6 32.11 34.5.112 34.7.112.13 0.640 1.03 3.45
2 - 2 22
yb3+ 2F7/2 32-7 3.5.7-11 33.7.11-13 0.613 0.960 3.104
4
Crystal Field Effects in Rare-Earth Intermetallic Compounds
Because of the fact that the 4f electrons are well shielded from the above p0ten -
tial by the outer electrons, the splitting of the energy levels in rare earth ions is rela-
tively small (with a few exceptions such as Sm and Eu ions) compared to spin-orbit
interaction. Thus it is often sufficient to perform the calculations employing only the
ground multiplet. Under these conditions the quantum mechanics of the interaction
are very much simplified. Stevens introduced what has come to be known as the
method of operator equivalents (18) to deal with the case when only the ground
state multiplet is involved. In this method an operator consisting of standard angular
momentum operators acting on the angular part of the wave functions of the rare
earth ion, which is described in the I LSJM) representation, is determined. Thus we
may write for the crystal field term:
t
n n
~'~CF = ~, ~, B m n O m n (J) ~ (3)
n=O m=--n
where n' can take a maximum value ofn. nB m are the crystal field intensity para-
meters and On m (J) represent polynomials of the angular momentum operators Jz, j2,
J+, J_. The polynomials and the matrix elements associated with several operators for
various values of J have been conveniently tabulated by Hutchings (19). It is to be
noted that:
a) for f electrons n' is 6 and for d electrons it is 4;
b) if there is an inversion center there are no odd n terms; and
c) the site symmetry of the rare earth ions reduces the number of terms which have
to be taken into consideration.
For ions located at points of very high symmetry the coefficients with the same n
may be related to each other.
The nB m coefficients are determined in part by the surrounding ions and in part
by the radial extension and the total angular momentum of the rare earth ion. B m is
evaluated from the expression
nB m = (J lln0II J) (r n ) (1 - )no K m A~ (4)
where (J II nO II J) is a reduced matrix element which is usually designated as aj,/3j and
7J for n = 2, 4 or 6, respectively; (r n) is the expectation value of the n th power of the
4f electron radius. These values are listed in Table 1 for the tripositive rare earth ions
(20). (1 - )no is a factor which takes into account the shielding of the 4fwave func-
tions from the environment by the outer, filled 5s and 5p shells (21) Values of the
constants Kn m occurring in Eq. (4) are given in Table 2. A m is a measure of the lattice
sum and is controlled by the electrostatic potential generated by the surrounding
ions. It is given by the expression
1)e2~,(Zk/R~+l)Ynm(Ok,~Pk )
A m : (- 1) m+l 4rr/(2 n + (5)
k
.E.W ,ecallaW S.G. Sankar dna V.U.S. oaR
nr
Table .2 Values of constants K n used in .qE (4) )a
KO= 1/4(5/u)1/2 K~ = (152 n)l/2 22K = 1/2(15/2 )r* 2/1
KO= 3/16(n)1/2 K 2 = 3/4(5/u)1/2 ~,K = 3/4(5/2 )r* 2/1
K 3 = 3/4(35/u)1/2 K 4 = 3/8(35/2 )r* 2/1
KO= 2/1)n/31(23/1 K~ = 1/8(13.21/2 u)l/2 K g -- 1/32(13 "105hr) 2/1
Kg = 1/16(13-105/n)1/2 K 4 = 3/32 (13" 14/n) 2/1
K~ = 3/16(13.77/u)1/2 K 6 = 1/32(13"21-ll/*r)Â/2
)a Ref. .)701(
where the summation is over all the surrounding ions. Z k is the charge on the k th ion
which is located at (R k, ,kO .)kO9
The eigenvalues and eigenfunctions obtained by diagonalizing the Hamiltonian in
Eq. (3), using crystal field intensity parameters obtained from Eq. (4) and Eq. (5), gives
a representation of the crystal field interaction according to the point charge model. Al-
though the point charge model is a convenient conceptual framework for assessing
the interaction, there are additional effects which may modify the results significantly.
The role played by 5 d and 6s conduction electrons in antishielding has been investi-
gated by Das and Ray (22), while the influence of covalent bonding has been discus-
sed by J4rgensen et .la (23). Further, there is an uncertainty involved in determining
the number of neighbors one has to consider in applying the point charge model to
the metallic state, and in addition the screening effect of the conduction electrons is
difficult to assess. In intermetallic compounds there is often uncertainty as to the
charges associated with the non-rare earth partners. It has therefore become custom-
ary to treat A m (r'*), satisfying the symmetry requirements, as adjustable parameters
to be determined from experiment.