ANOMALOUS THERMAL EXPANSION DUE TO PARAMAGNETIC IMPURITY IONS

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ANOMALOUS THERMAL EXPANSION DUE TO PARAMAGNETIC IMPURITY IONS

F. Sheard

To cite this version:

F. Sheard. ANOMALOUS THERMAL EXPANSION DUE TO PARAMAGNETIC IMPURITY IONS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-939-C1-940. �10.1051/jphyscol:19711336�.

�jpa-00214370�

JOURNAL

DE PHYSIQUE Colloque C I, supple'ment au no 2-3, Tome 32, Fe'orier-Mars 1971, page C 1 - 939

ANOMALOUS THERMAL EXPANSION DUE TO PARAMAGNETIC IMPURITY IONS

F. W. SHEARD

Physics Department, University of Nottingham, Nottingham NG 7 2 RD, England

R&urne. - Les ions pramagnetiques peuvent influencer la dilatation thermique d'un cristal iso!ant au voisinage du maximum de la chaleur sp6cifique de Schottky. Quand un ion donne un effet Jahn-Teller dynam~que, un tunnelling entre les differents modes de deformations peut se produire. La sensibilitk des dkompositions des niveaux a la valeur de la barrikre de tunnelling donne une contribution exceptionnellement importante. Sur le plan expCrimental une recherche de I'anomalie peut donner des renseignements utiles sur la nature des dkompositions des niveaux de l'ordre de plusieurs cm-1 que les techniques de resonance ne peuvent a present mesurer. On donne une evaluation numkrique pour Cr2.1- dans MgO.

Abstract. - Paramagnetic ions will influence the thermal expansion of an insulating crystal at temperatures close the Schottky specific heat peak. When the ion exhibits a dynamic Jahn-Teller effect, tunneling between different modes

.

distortion can occur. The sensitivity of the level splittings to the height of the tunnelling barrier then leads to an exce R -

tionally .large contribution. Experimental investigation of the anomaly could thus provide useful information on t e nature of level splittings - several cm-

which are at present inaccess~ble to resonance techn~ques. A numerical estimate is given for Crz+ in MgO.

I. Introduction. - Small concentrations of para- magnetic ions with low-lying energy levels will contri- bute to the thermal expansion of a crystal at tempe- ratures close to the Schottky specific heat peak [I].

Although magnetic contributions to thermal expansion have been measured for ordered magnetic materials there has been no systematic work on the effect of paramagnetic impurities. The only reported anomaly appears to be that of Tilford and Swenson [2], who in studying the thermal expansion of solid argon observed a low temperature peak which they attributed to the ground state splitting of unwanted O2 impuri- ties in their samples.

We suggest here that the contribution of parama- gnetic ions to thermal expansion may be a useful method of obtaining information on the nature of energy level splittings - 10 cm-' which are too large for direct observation by electron or acoustic para- magnetic resonance techniques. The effect of para- magnetic ions on thermal conduction has already been used to detect low-lying energy levels via the resonant scattering of phonons. However lack of knowledge of the form of the thermal phonon relaxation time hampers the interpretation of these experiments. A more direct method is by far-infrared spectroscopy though such experiments lack sensitivity below about 5 cm-'. Thermal expansion, whilst lacking the preci- sion of spectroscopic methods of determining level splittings, provides additional information since it depends essentially on the pressure dependence of the splittings.

11. Griineisen parameter for paramagnetic impurity ions. - The standard Griineisen theory 131 may be generalised to include the effect of two-level magnetic impurities (spins) with level separation Eo. The expan- sion coefficient is then given by (1)

where CL and C, are the heat capacities at constant

volume V of the lattice and spins respectively and

xT is the isothermal compressibility. In this theory the lattice anharmonicity is characterised by a single parameter y, = - (811zw,/811zV)~, where o, is the Debye frequency, and y, = - (alnEo/dlnV), is the corresponding parameter for the spins. Provided the magnitude of y, is not much smaller than y, (- 2 for many solids) the effect of paramagnetic impurities will be observable when there is a sufficient concen- tration that C, - CL. For Eo - 5 OK and a Debye temperature - ³⁰⁰

this requires less than 0.1 ato- mic %.

If the energy levels of the ion are describable by the point-charge model of static crystal field theory it is possible to determine the dependence on crystal volume V quite simply. We consider an iron-group ion in a cubic crystal. The splitting of the orbital levels by the crystal field is then of magnitude A - lo4 cm-'. If the orbital angular momentum is not quenched there are first-order splittings due to spin-orbit coupling of magnitude I - 10' cm-I.

When the lowest level is more than four-fold degene- rate it will be further split by spin-orbit coupling in second-order perturbation theory giving small splittings E, - ^??/A - 1 to 10 cm-I [4]. For example, for

~ e (3 d6 'D) in tetrahedral ZnS, the lowest crystal ~ ' field term is 'E. The first-order spin-orbit splittings are quenched for E orbitals and in second order the 'E term splits into five equally spaced levels with mutual separation E,, -- 15 cm-'. This is in agreement with far-infrared measurements [5]. Since on the point-charge model [6] A oc a-', where a is the distance between the paramagnetic ion and its neighbours, and V cc a we have

y, = - S ( d l n ~ ~ / a l n a ) = - 5. This corresponds to a negative thermal expansion anomaly in the region of the Schottky peak [I].

But for E orbitals in an octahedral environment, such as CrZ+ (3 d4 'D) in MgO, it is thought that the orbit-lattice interaction is sufficiently strong that a description in terms of static crystal field theory is inappropriate. The coupling of the electronic states to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711336

C 1 - 940 F. W. SHEARD the vibrations of the neighbouring atoms, the dynamic

Jahn-Teller effect [7], has been calculated in some detail for this case [8]. The potential energy of the ion and its neighbours may be reduced by a tetragonal distortion in either of three equivalent directions.

Owing to the anharmonicity of the lattice, distortions in other directions lower the energy less so that the equivalent tetragonal distortions are separated by energy barriers. Tunnelling between the three different modes of distortion occurs and the resulting lattice disturbance is dynamic. The lowest eigenstates are separated by a tunnelling splitting which depends exponentially on the tunnelling barrier. Owing to this exponential dependence we may anticipate that the splitting may change rapidly with variations in crystal volume.

From the expressions given by Fletcher and Stevens [8] we can estimate the value of y, associated with the tunneling splitting 6. An approximate form for 6 is

Here R, is the average displacement of a neighbouring ion of mass M , arising from the Jahn-Teller effect and B is the anharmonic force constant, so that BR?

corresponds to the anharmonic potential barrier.

Since R, occurs to a high power in the exponential its variation with volume Vis most important. The manni- tude of Ro is determined by the balance between-the orbit-lattice coupling energy ER, and the lattice elastic energy + ^{Mo R:,} and is given by Ro = c/MO mi.

On the point-charge model the dominant term in the coupling constant

ac a - 6 , hence

taking the lattice Griineisen parameter y, - ^2.2

This shows that R, tends to increase as the lattice expands owing to the weakening of elastic restoring forces. Little is known about the variation of the third-order anharmonic force constant. B which depends on the fourth-order anharmonic terms. But if we define yl ^{= -} (alnB/alnV), we may expect it to be positive and - 2 as for y,. Fortunately the variation of Wwith volume is dominated by the variation of R, :

( 1 )

At 250

thermal expansion measurements give 71, - ^1.51.

However for the tetragonal compressions occurring in the Jahn-Teller effect it is more appropriate to take the Griineisen parameter for a longitudinal mode along the (100) axis which has been estimated [9] to be ~ ~ ( 1 0 0 )

2.21.

We have finally :

In order to fit their theory to acoustic resonance data Fletcher and Stevens [8] choose parameters such that W - 7 which gives y, - 30. We note that the large positive value of y, arises primarily from the depen- dence of Ro on o, and the uncertainty inherent in y ; does not qualitatively alter this.

In addition to the tunnelling splitting the energies of the lowest states depend also on the second-order spin-orbit parameter A2/A and for Cr2+ in MgO there are also splittings due to random lattice strains.

However these latter terms are relatively insensitive to crystal volume changes compared with the tunnelling splitting. Thus for the low-lying states of Cr2+ in MgO, theory based on a strong Jahn-Teller effect predicts a large positive y, 30. Although this is necessarily a crude estimate it contrasts strongly with the small negative value y, = - 513 to be expected in the absence of Jahn-Teller effects.

111. Conclusion. - Experimental investigation of the low temperature thermal expansion anomaly can provide useful information on the nature of zero-field splittings - several cm-'. In particular it could be used as a valuable test of existing Jahn-Teller theory for Cr2+ in MgO. The available experimental infor- mation on this ion is meagre. Acoustic paramagnetic resonance work [lo] has been restricted to the Zeeman levels of a low-lying doublet and thermal conductivity work [l I] has detected only one transition between the zero field levels. The fit between the theory and these experiments is necessarily somewhat tenuous. The value of y, determined from the thermal expansion anomaly would point in a more direct way to the existence of a tunnelling splitting and hence provide more firm support for the theory.

We should mention finally that the generalisation of equation (I) to a multi-level impurity is elementary ; y, then becomes an average of the values associated with the different splittings in the same way as y, is actually an average over all lattice modes [3]. These details are omitted here since we have been concerned with general conclusions based on a rough estimate of y,.

References

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Letters, 1969, 22, 1296. [7] STURGE (M. D.), Solid State Physics, eds. F. Seitz, [3] COLLINS (J. G.) and WHITE (G. K.), Progress in Low D. Turnbull and H. Ehrenreich, Academic Press,

Temperature Physics, ed. C . J. Gorter, North New York, 1967, 20, 91.

Holland, Amsterdam, 1964, 4, chap. 9. 181 FLETCHER (J. R.) and STEVENS (K. W. H.), J . Phys. C [4] BALLHAUSEN (C. J.), Introduction to Ligand Field (Solid ^St. Phys.), 1969, 2, 444.

Theory, Mc Graw-Hill, New York, 1962, chap. [9] GANESAN (S.), Phil. Mag., 1962, 7 , 197.

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