OBSERVATION OF MOLECULAR VIBRATION FREQUENCIES OF DIATOMIC HALOGEN CENTERS IN ALKALI HALIDE CRYSTALS BY E. S. R.

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OBSERVATION OF MOLECULAR VIBRATION

FREQUENCIES OF DIATOMIC HALOGEN

CENTERS IN ALKALI HALIDE CRYSTALS BY E. S.

R. W. Dreybrodt

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C 4, supplkment au no 8-9 Tome 28, Aotit-Septembre 1967, page C 4-89

OBSERVATION OF MOLECULAR VIBRATION FREQUENCIES

OF DIATOMIC HALOGEN CENTERS IN ALKALI

HALIDE CRYSTALS

BY E. S. R.

by W. DREYBRODT,

Physikalisches Institut der Universitat Frankfurt am Main, Germany

RBsurn6. - Les spectres E. S. R. des centres halogenes a deux atomes du type X: et X Y - dans les halogknures alcalins montrent des positions de raies dkpendantes de la temperature en cond- quence des variations des constantes de la structure hyperfine. On discute par I'analyse les mkca- nismes qui conduisent 9. la dkpendance observk de la tempkrature. Des vibrations de la molecule et des voisins les plus proches contribuent aux parties des constantes de la structure hyperfine qui dependent de la tempkrature. Ces parties sont separees par l'experience. Les frequences des vibrations du centre de la molecule sont dkterminees. En les comparant avec les frkquences correspon- dantes des molkules libres l'inffuence du rkseau sur les constantes de force est discutee.

Abstract. - The E. S. R. spectra of diatomic halogen-centers of the type X; and X Y - in alkali halides show temperature dependent line positions due to changes of the hyperfine structure constants. From their analysis the mechanisms leading to the observed temperature dependence are discussed. Vibrations of the molecule and of the nearest neighbours contribute temperature dependent parts to the hfs constants. These are separated experimentally and the vibration frequencies of the molecule-centers are determined. By comparing them to those of the corresponding free molecules the influence of the lattice on the force constants is discussed.

Introduction. - I n the interpretation of E. S.

R.-

spectra of paramagnetic centers in crystals one usually assumes the paramagnetic ion and the lattice to be static. Lattice vibrations, however, change the distances

p i between the ion and its ligands thus producing periodic changes of the electronic wavefunction, which, assuming the Born-Oppenheimer approximation, contains the p i as parameters. As the hfs-constants are calculated from matrix elements which contain the electronic wavefunction, they are functions of the parameters pi. Thus the interaction of phonons with the center leads to temperature dependent parts in the hfs-constants. This temperature dependence can be calculated by expanding the hfs-constants into a Taylor series t o second order in the p,. Calculating the expectation values of p, and p: with the harmonic oscillator function < l ) of the lattice and averaging over all lattice vibrations leads to

A11 the thermal average values

show to a good approximation the same temperature dependence for T

<

612. Thus eqn (1) simplifies to

Where C i j are constants and (p

-

'p,) is the mean square deviation of any ligand from the lattice site and is given in the Debye-approximation [ l ] by

Here B is the Debye-temperature of the lattice and m

is the mass if the ligands.

If one considers the case of a diatomic molecule center the vibration of the internuclear separation R of the molecule leads to additional terms in the Taylor- Series. They are

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C 4

-

90 W. DREYBRODT

If one assumes that the vibration functions

<

v

l

of the molecule are eigenfunctions of a Morse potential the matrix elements can be calculated [2] and lead to a temperature dependent contribution

where o is the frequency of the molecular vibration and L contains molecular constants. Thus for a diatomic molecule center the total temperature dependence of the hfs-constants is given by

Experimental Results.

-

We have observed this behavior of the hfs-constants for the V,-center in KC1 [3] and K F [4] and for the mixed centers FCl- and FBr- in KC1 and NaCl. The centers FCl- and FBr- have been investigated by several authors [5

-

71

Both are shown to be [l l l]-oriented molecules occupie- ing a single anion lattice site.

The position of the E. S. R.-lines from all these centers are dependent on temperature. Figure 1 shows the

-

0 c .P v) I

S

lu 2663 2665 2667 2669 [~auss] Magnetic Field

FIG. 1.

-

Temperature dependent line positions of a line from the FCl --spectrum.

position of an E. S. R.-line of the FC1--spectrum at two different temperatures. The line shift AB is caused by

changes of the hfs-constants and is shown to be pro- portional to A(T) by its angular dependence [8]. Figure 2 shows the temperature dependence of the line shift AB = B(T)

-

B (T = 0 OK) for a line of the FBr--spectrum in KCI. The plot contains the extrapo- lated zero-point contributions. The experimental curve (a) can be fitted by eqn (6). Curve (b) shows the shift arising from the molecular vibration (eqn (5)) and curve (c) shows the shift from the vibration of the ligands. The triangles on curve (c) represent the diffe-

FIG. 2.

-

Temperature dependence of the line shift from a line of the FBr --spectrum.

rences between the measured points on curve (a) and the corresponding values on curve (b). An analoguous behavior has been found for all of the other systems cited above. The accuracy of the fits has been tested and from this the error in the determined frequencies is estimated to be about

+

20

%.

Table 1 shows the fitting parameters L and G from eqn (6) and the obtained frequencies of the molecular

vibrations. They are compared to the frequencies of the free molecule-ions F,, Cl,, FCl- and FBr-, which have been estimated after the method of Person

[g] from the measured vibration frequencies of the first excited states [l01 of the corresponding neutral molecules. The vibration frequencies of both the

[ 11 1 ]-oriented FCI- and FBr--enters depend strong- ly on the host crystal and are considerably lower than those of the free molecules. This indicates a strong influence of the lattice on the force-constants. However, in the case of the Xi-centers the frequencies of the free molecules are approximately equal to those of the molecules in the crystal, indicating a small influence of the lattice to the force constants.

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OBSERVATION OF MOLECULAR VIBRATION FREQUENCIES C 4 - 9 1

Frequencies o of the molecular vibrations compared to the frequencies of of the free molecules

. . .

FC1- in KC1

. . .

FC1- in NaCl FBr- in KC1

. . .

FBr- in NaCl

. . .

W f -1

10+13 sec L [Gauss]

1

G [Gauss]

l

F;-center in LiF as a function of the internuclear distance R in the following way : The energy of the F;-center is separated into two parts : The energy of the chemical bond between the two F - ' / ~ ions of the free molecule and the interaction of these two ions with the lattice. The first energy can be calculated either by the method of Person [9] or by an M. 0.-calculation [l l]. The interaction energy with the lattice is calculated from the electrostatic energy, the Born-Mayer repulsion energy and the polarization energy, all as a function of the displacements of the two F-'l2-ions in the [110]- direction and the corresponding relaxation displacements of the nearest neighbours. Expanding them into a Taylor-Series and minimizing with respect to the relaxation displacements leads to the lattice energy of the molecule as a quadratic expression in R.

The total force constant of the center is given by the sum of the force constant K, of the free molecule and the force constant K, of the lattice interaction energy. Since K, is about a factor of 3 smaller than K, the

vibration frequency of the VK-center is essentially determined by the properties of the free molecule. Figure 3a shows schematically these two energies and the total energy of the V,-center.

For F, in K F and Cl, in KC1 one expects the force constant K, to be even smaller, as the larger polarisi- bility of the lattice ions leads to larger negative contributions to K,. Furthermore the Born-Mayer energy in both lattices becomes smaller causing a further decrease of K,. Thus the force constants of both XL-

centers are expected to approach that of the free molecules.

For the case of the mixed FC1--center in KC1 we have calculated the lattice potential in a similar way. As the center is more complicated and the equilibrium positions of the ions are not known even approximately,

3nternuclear Distance R FIG. 3.

-

Schematical representation of the potentials as a function of the internuclear distance. E, is the energy of the free molecule-ion. Et is the lattice-interaction energy. Et is the total energy. a) is for the case of the X?-centers. b) for the case of the XY--centers.

the Taylor-series has to be expanded about different positions of the molecule, and by an iteration process the energy is obtained as a function of R alone. The result shows that the lattice force constant is negative, thus reducing the total force-constant of the molecule. Schematically this is shown in figure 3b. The calcu-

lation however leads to a value of K, which is too small

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C 4 - 9 2 W. DREYBRODT amounts to K, its increase leads to an increase of

I

K, ( and to a decrease of the total force constants. Thus at least the low values of the vibration frequencies of the FC1-- centers can be understood qualita- tively. A similar behavior is expected to hold for the FBr--center in KC1 and NaCl.

Acknowledgements.

-

The author would like to thank Professor Dr. W. Martienssen, Professor Dr.

H. A. Muser, D. Silber and B. Sammel for numerous valuable discussions. The crystals were grown by the Frankfurt Crystal Unit whose financial support from the Deutsche Forschungsgemeinschaft is appreciated.

[l] JAMES (R. W.), The Optical Principles of the Diffrac- tion of X-Rays, London, 1962.

[2] MAKHANEK (A. G.), Opt. and Spec*., 1961, 11, 6.

[3] KANZIG (W.), Phys. Rev., 1955, 99, 1890, CASTNER (T. G.), KANZIG (W.), Phys. Chem. Solids, 1957, 3, 178 ; WOODRUFF (T. O.), KANZIG (W.), Phys. Chem. Solids, 1958, 5, 268.

[4] BAILEY (C. E.), Phys. Rev., 1964, 136, A 1316. [5] WILKINS (W.), GABRIEL (R.), Phys. Rev.; 1963, 132,

1950.

[6] DREYBRODT (W.), SILBER (D.), Phys. stat. sol., 1966, 16, 215.

[7] SCHOEMAKER (D.), Phys. Rev., 1966, 149, 693. [8] DREYBRODT (W.), Phys. stat. sol., 1967, 21,

99.

[9] PERSON (W. B.), J. Chem. Physics, 1963, 38, 109. [l01 HERZBERG (G.), Spectra of Diatomic Molecules,

New York, 1950.

[l l ] DAS (T. P.), JETTE (A. N.), KNOX (R. S.), Phys. Rev., 1964, 134, A 1079.

[l21 DIENES (W.), HATCHER (R.), SMOLUCHOWSKI (R.), Phys. Rev. Lett., 1966, 16, 25.