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THE THEORETICAL INTERPRETATION OF THE shf INTERACTION OF Cr+ CENTRES IN ALKALI
HALIDES
R. Bauer, K. Differt
To cite this version:
R. Bauer, K. Differt. THE THEORETICAL INTERPRETATION OF THE shf INTERACTION OF Cr+ CENTRES IN ALKALI HALIDES. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-117- C9-120. �10.1051/jphyscol:1973920�. �jpa-00215395�
JOURNAL DE PHYSIQUE CoIloque C9, supplkniet7t au ttfl 11-12, Tome 34, Nouernbre-Dkcernbre 1973, page C9-117
THE THEORETICAL INTERPRETATION OF THE shf INTERACTION
OF Cr+ CENTRES IN ALKALI HALIDES
R . BAUER and K. D I F F E R T
Institut fiir Physik am Max-Planck-Institut fur Metallforschung, Stuttgart, und Institut fiir theoretische und angewandte Physik
der Universitiit Stuttgart, Germany
RbsumB. - Les resultats experimentaux de I'interaction superhyperfine (shf) des centres para- magnetiques dans les halogenures alcalins ne peuvent pas Stre expliques par les modeles de cova- lence et recouvrement utilises jusqu'ici. Quand on tient cornpte de la deformation des ions Cr+
par des etats excites causee par le potentiel cubique cristallin, I'interaction shf anisotropique avec le premier voisin peut Stre expliquec sans utilisation des parametres ajustes. La contribution des
&tats excites etendus dans I'espace a la densite de spin donne I'ordrc de grandeur correcte de I'inter- action isotropique avec les voisins plus eloignes. I 1 sera montre que se~lle une perturbation du type potentiel des ions ponctuels est capable de donner les resultats corrects. Cornme il a ete constate recemment, la partie angulaire du potentiel cubique cristallin est une contribution impor- tante negligee dans I'approxirnation des ions libres. Les conceptions theoriques utilisees pour les complexes dcs ions des metaux de transition ne peuvent pas Ctre appliquees aux defauts substitu- tionnels de ces ions dans les sels sans modifications et extensions importantes.
Abstract. - The experimental results of the superhyperfine (shf) interaction in alkali halides cannot be understood on the basis of the models of covalency and overlap worked out so far. Taking into account the deforn~ation of tlie C r , ions by excited states admixed by the cubic lattice poten- tial the anisotropic slif interaction with nearest neighbours is explained without adapted para- meters. The contribution of the spatially extended excited states to the spin density gives the correct order of magnitude for the isotropic interaction with the more distant neighbours. It will be shown that only a perturbation of the kind of the point ion potential gives the correct results. As stated recently, the angular dependent part of tlie cubic potential produced by the point ions proves to be one of the leading terms neglected in the free-ion approximation. The theoretical concepts known for the transition-metal ion conlplexes cannot be applied without substantial modifications and extensions to substitutional defects of these ions in salts.
I . Introduction. - The shf interaction of substitu- tional Cr' 3d5 in NaF, NaCI, and NaBr with neigh- bours up t o the ninth order have been measured with the ENDOR technique by Ziegler [I], [2]. Among the quantum theoretical concepts applied t o this defect so far, the overlap model o r free-ion approxima- tion [3], [4] has the advantage of being a parameter- free abitiitio calculation. This approximation uses the free-ion ground state orbitals Y J ~ , , orthonorn~nlized in the crystal according t o
as an approximation of Lhe generalized symmetric Wannier functions [5] of the solid, whicfi depend on the free-ion overlap S,,,(A, B) between orbit, '1s l on different sites A a n d 5. This npproxlrnation gives the correct order of magnitude for the isotropic slif inter- action of the first shell of neighbours, while that of the more distant neighbours turns out to be an order of magnitude t o o small.
The difference between the total value of the aniso-
tropic interaction with nearest neighbours a n d the classical dipolar value,
Ah:'' = Cte
{ N,Z <
Cr:,,I
CI;,,, > 2 -comes out positive, the a-overlap being always larger than the rr-overlap, whereas the experimental values are negative for all salts [I]. The constant Cte in (2) contains the g factors a n d magnetic moments of both the free electron atid the CI--nucleus and the radial matrix element
<
3p1
I/r3 1 3p>
of the anion. Nu and N , are normalization constants differing n o t much from unity.Several models of covalency [7], [8]. [9], [lo] have been proposed t o overcome this deficiency of the theoretical description. but, as Ziegler [ I ] has shown, none of them can explain the experimental data. even
\vlien two additional covalency parameters are fitted.
In the present note, we consider excited states of the C r t ion in NaCI admixed by tlie cubic lattice potential, and their contribution to the nearest-neighbour aniso- tropic shf interaction Ah, givcn by eq. (2). Furthermore, we discuss charge-tr-ansfcr excitcd states.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973920
R. BAUER A N D K. DIFFERT
Overlap integrals betli,een the nearest-neighboi~r chlo/~ine orbitals and the outer orbital of tile ground state of Cr', Cu +, Ag', CuO, Ago
and of excited states of Cu + and Ag' in various clilorides, with nearest-neighbour distance ro Ion
r.,,/.\, strc.
Config.
orbital - CI- 3p, C I 3pn CI- 3s CI- 2s CI- Is
Crf : NaCl 2.80, NaCl
3d5 6s
3d
-
0.077 5
- 0.046 5 0.058 2 0.007 5 0.000 89
2. The contribution of free-ion excited states to the ground state in the lattice. - The contribution of the cubic part of the lattice potential as an extension of the free-ion approximation has recently been dis- cussed for ionic crystals with the NaCl st]-ucture [4]
and has been applied to the silver halides [lo] to explain the electron density distribution and some features of the interatomic interaction. Hartree-Fock exciled-state orbitals have been computed and are at present available for copper and silver. The details of these calculations will be published elsewhere [I I]. As to the nearest neighbour overlap, Cu' in CuCl with a nearest neighbour distance r , = 2.34
A
is analogous to C r f in NaCl with r , = 2.80A,
both having rota- tional symmetric ground states with the configura- tions 3d" IS and 3d5 'S, respectively. In both cases the overlap integrals of the 3d orbitals with all nearest- neighbour chlorine orbitals are nearly the same, as it is shown in table I. Due to the higher nuclear charge, tlie d orbitals of Cuf are more contracted than those of Cr'. This effect is compensated by tlie smaller interatomic distance in CuCI. In table I. the overlap for AgCl with r , = 2.75A
is included, too, in order to show that very large values of the overlap integrals are characteristic for the excited states under conside- ration. In table 11, the similarity of optical spectraOptical energy levels of copper and chromium, in eV. After [ I 21
First ionization potential I,,t
Energy above ground state
-
C r - 3d5 6S Cu'.3d10 I S
0'0 no cubic configura- Cr 3d-s' 2G -
4.77 tion Cr- 3d44dl 6s - 5.13
Cu- 3d94dl IS 11.35
Cuo : CuC1 - Ago : AgCI
-- 2.34, ZnS 2.75, NaCl 3d104sl 2S 4d"J 5sl 2s
4s 5s
- - 0.272 4 0.182 1
- - 0.270 6 0.210 9 0.044 9
0.004 5
data of ~ r ' and Cuf [I21 is shown. There is quite a similar behaviour of Cr' in NaCl and Cu' in CuCI.
In tlie following calculations the Cuf orbitals in CuCl and the energy values and the symmetry of Cr' in NaCl are used.
Figure 1 sliows the radial part of the excited 4s and 4d orbitals of Cu' together with the radial
FIG. 1. - Radial part of orbitals for CI- and for excited states of CLI'. The ions are separated by the interatomic distance in
CuCI, = 2.34 .\.
functions of the nearest neighbour anion in CuCI, the parity being indicated for the 3s and 3p functions. The 3d orbitals are completely concentrated within the range of tlie Madelung potential shown in the upper part of figure 2, and are hence raised towards the vacuum level by tlie total amount of the Madelung energy. The excited-state 4d orbital, having its maxi- mum at tlie nearest nelghbour nucleus, is only slightly shifted by If,,(,.) but strongly affected by the cubic potential V,(r). This results in a splitting into the twofold e,, term and the threefold tlg term. The corresponding splitting of tlie 3d-level is much smaller.
The first order perturbations of the energies asso- ciated with tlie diagonal elements
<
A:I I
A:>
of the perturbation matrix are represented for C r f i n figure 3 together wltli tlie valence and conduction band of NaCI and tlie clial-ge-transfer excited states CrO 3d5 4s'.
T H E THEORETICAL INTERPRETATION OF THE shf INTERACTION O F Cr CENTRES C9-1 19
FIG. 2. - Radial functions for the spherical (Vo) and the cubic
( V 4 ) term in the potential at a cation vacancy in the NaCl
structure. The dashed curves give the point ion approximation.
The solid curves give the likely form including ion size effects for an alkali halide.
Errotlrtn : On the lower part of this figure the formula
has to be read V.41.) -
J
V ( Y ) Y;(R) dR.The point ion approximation of the potentiiils.
given by the dashed lines in figure 2, is taken from [I 31.
The solid lines show the likely form of the po~entials including ion size effects similar to those discussed by Wood [I41 and others [15], [16]. The positons of the Cro 4s' levels relative to the C r f 4d' level. as well as the position of the Cr' 3d5 ground state, are nearly independent o f the detailed form of the lattice poten- tial, whereas the distance of all the excited states from the C r f 3d5 ground state depends strongly on it.
The perturbation of the 3d orbitals through second order depends on the transition matrix elements of the excited states as well as on the diagonal elements.
Hence, one gets a quick result of the perturbation calculation giving the correct order of magnitude by inserting the energy levels perturbed to first order into the expression for the first order perturbation of the orbitals.
As to the interaction with nearest neighbours in the NaCl structure, the e, orbital with nonvanishing overlap corresponds to the d, orbital, and the t,, orbi~als to d,, the two symmetry types having niatl-ix elements of opposite signs in the cubic potential, as it is indicated in figure 4 where in the shaded region V,(t.) has the same sign. The selection rules of the cubic potential and the angular part of the matrix elements are given in [lo]. Starting with the matrix elements
i
Energy Levels ofT i J
t2g :4 -
p,C r + : Na Cl
d
FIG. 4. - Angular part of d orbitals for Crl- located in the-17 ~ r + 3 d ? 6~ cubic poteritial in NaCI, and the overlap with the p-orbitals
V, ( r l L$ Irl -r7
ol' a nearest ncighbour anion. In thc shaded regions the cubic config. free cation in a lattice
m tion site part of thc potential has thc same sign.
FIG. 3. - Energy levels of frcc C r ' and of C r l : NaCl perturbed f i r t r ~ r ~ r ~ ~ : On thc lo\ver part of this ligi~re please read r:;. I,,, :
by the latticc potential to first order. dn ++ ~n instead of r 1 2 . fz,, : d n ++ l>n.
C9- 1 20 R. BAUER A N D K . D I F F E R T
belonging to the lowest excited state,
<
3dI
V4I
4d>,
and taking into account all higher excited states of appropriate symmetry, including the continuum, by a factor of two, one obtains an effective transition element which leads together with the energy levels in figure 3 to the effective admixture coefficients
and to the perturbed orbitals
Inserting the 3d(') orbitals instead of the 3d orbitals into (2) results in
With the small 8-overlap and the large n-overlap of the excited orbitals (comp. Table I), and due to the opposite signs of ua and a,, the a-contribution is slightly reduced, whereas the n-contribution is strongly increased rendering ~ 6 1 ~ ' ~ ' negative. From the quoted values one obtains ~ 6 : ~ ' ~ ' = - 0.18 M H z for Cr' : NaCI, the experimental value being Ab, = - 0.12 M H z . This overestimation of the negative n-contribution may be due to an overestimation of the cubic poten- tial V4(r), the point ion approximation of which is reduced by ion size effects. To some extend, it may be caused by using Cu' orbitals instead of Cr' orbitals.
For higher order neighbours, a more detailed investigation shows that the excited state orbitals admixed to the ground state by the cubic potential give, without fitting any parameter, the correct order of magnitude for all shf interactions, in particular for the isotropic interaction of the second shell cation on
the (1 10)-site and the fourth shell cation on the (200)-site.
3. Covalency models considering charge transfer states. -The configuration interaction with the charge- transfer excited states corresponding to CrO 3d5 4s' and a pa-hole on the nearest neighbour (( ligands )), being well known in the theory of transition-metal ion complexes 161, [8], has been used by Simanek and Miiller [I71 to explain the hf interaction of transition metal ions in salts. The two states with the spin of the 4s electron parallel and antiparallel to the 3d spins are shown in figure 3. These states lie well above the lower ones of the 4d orbitals of the cubic excited states of Cr', and their separation in the crystal is very small compared with their distance to the ground state, as Vo(t.) raises the two states by a different amount, according to their different radial extension.
As the admixture coefficients are hence nearly equal for the two spin orientations, these states are not likely to contribute to the shf interaction. In addition, Ziegler [I] has pointed out that these states, fitted to explain Ab,, would give even the wrong sign for the isotropic interaction of the higher order neighbours.
In general, the charge transfer state takes part in the formation of the ground state of a coniplex, as the binding energy of a con~plex is increased with the reduction of ionicity of the central ion and the ligands associated with the charge transfer. In a lattice where the (( ligands )) are in cubic sites, on the other hand, the state wit11 the maximal ionicity is the most stable one. It follows that the theory of the transition-metal ion complexes is not altogether applicable to the crystalline state.
Acknowledgements. - The authors are indebted to Prof. A. Seeger.
The work has in part be sponsored by the Deutsche Forschungsgemeinschaft.
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