Theoretical Study of the Mean and Resultant Change of Interaction Frequency for Oxygen-Oxygen Ion Dipoles in CaF2 Crystal

ABSTRACT The purpose of this paper is to analyze and study the angular variation, in the presence of the external magnetic field, of the mean and resultant displacements of the dipole-dipole interaction frequency, O-O ion dipoles in CaF2 crystal. The dipolar interaction frequencies of six sites of equivalent ions are described by their geometrical parts of their tensors and analyzed for four different cases. The conclusion is that the theoretical method provides a satisfactory approach in cases for which there are adequate data.


INTRODUCTION
Many studies involving experimental techniques have been done in an attempt to understand the behavior of oxygen centers in CaF 2 crystals. Calcium fluoride is a suitable material for a variety laser optical applications in the ultraviolet spectral region. It can be used for generating high-order harmonics and white light from ultra short laser pulses (Megerle et al., 2009). On the other hand, one of the big unsolved problems in the application of CaF 2 as an optical material is the oxygen contamination in the bulk and at the surface. Oxygen is readily incorporated into the bulk during crystal growth. For this reason, we say that one of the main obstacles to exploiting CaF 2 is the presence of oxygen centers, having an optical absorption in the UV region (Sils et al., 2007). Therefore, a comprehensive effort is undertaken for calculating geometrical structures and optical properties of oxygen centers and related defects in CaF 2 .
In growing process, substitution oxide ion impurities are compensated by anion vacancies V A and form O -2 ‫ـ‬V A dipoles. The properties of this dipole have been studied experimentally. However, accurate theoretical modeling of the atomic scale mechanisms in the O -2 ‫ـ‬V A dipole is still lacking (Mysovsky et al., 2005). As a result of the photo dissociation when the temperature exceeds 200 º K, the concentration of the O -2 ‫ـ‬V A dipoles decreases and other defect species, such as isolated substitutional Oions and complexes of O -2 with neutral and charged anion vacancies are observed. The Oimpurities formed during photo dissociation, were studied experimentally by means of EPR spectroscopy, which showed no optical absorption associated with Oions (Radzhbov et al.,1986). From a theoretical point of view, modeling of anion vacancies and oxygen impurities face several problems, because accurate modeling of the magnetic properties of these defects requires describing a large number of atoms near them quantum mechanically. In addition, vacancies and oxygen impurities aggregate into complex defects which induce atomic displacements in several tens of atoms near the defect site. For example, to clarify the electron structure and atomic geometry of oxygen-vacancy dipoles, needed to be taken into account where the supercell containing 96 atoms (Shi et al., 2007).
The purpose of this paper is to make an accurate calculation of oxygen-oxygen ions defect in the bulk CaF 2 and to predict their spectroscopic properties. Detailed quantitative calculations have been performed to analyze the angular dependence of the mean and resultant change of interaction frequency for oxygen-oxygen dipole center in the presence of an external magnetic field.

THEORETICAL PROCEDURE
In an ordinary paramagnetic crystal, the nature of the mutual interactions between paramagnetic ions and their effects on the paramagnetic resonance frequency are considered in this work.
The most obvious interaction between these ions is dipole-dipole interaction, which arises from the influence of the magnetic field of one ion on the dipole moments of neighbouring ions. Furthermore if an external magnetic field acts on the crystal, the local field at each ion must be added in a vector form to it. If the local field is small compared with the external magnetic field, only the component of the former parallel to the latter is important and this component varies from site to site, giving a displacement to the resonance interaction frequency of each ion (Abragam et al., 1970). If the direction cosines of are (l,m,n), equation (2) can be expanded: Which is a symmetric tensor type interaction.
Using the foregoing approach, an expression describing the dipolar interaction frequency has been found by expanding the vector relationships for , and and performing the indicated scalar products. The resulting expansion can be expressed in a compact notation as (Graybeal,1988)
In the presence of an external magnetic field applied along an arbitrary axis, the resonance frequency of each ion is displaced due to the effect of the magnetic dipole-dipole interaction. In the high field approximation, the displacement in frequency can be represented in one dimension for a pair of like spins, i.e., one frequency component, by the following modified formula (Graybeal, 1988 ;Kasir, 1994).

………..…….(6)
where each spin is quantized along the magnetic field M is the projection of the spin along the principle c-axis of the crystal, the magnetogyric ratio and ( , , ) are the angles between the magnetic field vector and the selected coordination system of axes respectively.

CALCULATION AND RESULTS
In this work, the model postulated for the location of the oxygen center in CaF 2 is shown in Fig. (1), presumed that Oions replace two adjacent fluorine ions (other possibilities may occur), and these types of defects may occur either naturally or produced by photo dissociation of other complex oxygen impurities.
Taking into account the nearest and next nearest neighbours, one may distinguish six different sites according to the coordinate system (xyz) assumed for the O --Ocenter in Fig. (1). Each site represents a state of a group of equivalent ions. Calculations have been done to determine the position vector for each individual ion taking into account that the distance between fluorine-fluorine ions parallel to in CaF 2 is 0.2725nm. The normalized maximum dipolar interaction frequencies were calculated using the following parameter values: Fluorine magnetogyric ratio=40.07 MHz/Tesla, g=2.01, S=1/2 and =9.2732x10 -24 Joules/Tesla (Kasir, 1994).
The direction cosines of for each site, ( Fig. 2), were calculated Table (1) and then combined with the , to get the geometric parts of tensors for each site which result from the main interaction axes. The values of the normalized interaction frequency and the geometric parts of tensors obtained are shown in (Table 2), for the six different sites. x 1 2 S1 S2 S3 S4 S5 x , z y -y -z S6 Fig. 2: The six different sites according to the coordinate system assumed in Fig. 1.

Origin of xyz axis
When applying a magnetic field along definite directions, a detailed geometrical calculations have been done to relate the angles ( , , ) done by the magnetic field and the coordinate system, for the four different combinations of axes, with the angle α designated between and the principal axis under consideration.
Using the physical model discussed earlier and the known alignment of the crystal with respect to , it turns out that we expect four types of angular variation of the mean interaction frequency and four types of angular variation of the resultant displacement of interaction frequency as the magnetic field is rotated.  Theoretical calculations were employed to evaluate the tabulated parameters to study the angular variation of mean interaction frequency obtained by the effects of the six sites and the angular variation of the resultant displacement of interaction frequency using the complete version of equation (6), when a constant applied magnetic field B=3340 Gauss (resonance frequency for fluorine ion at B=3340 Gauss is equal to 13.383 MHz), is rotated by an angle ranged from 0º to 180º, for the following cases: Case I: Rotation axis of //[100] direction. The angular variation of mean interaction frequency and the angular variation of the resultant displacement of interaction frequency, for magnetic field orientations in the [100] plane of the crystal were analyzed for two cases correspond to two different combinations of coordination systems with the direction of . These cases are illustrated in Fig. 3, 4, 5 and 6.

Rotation axis//[110]: case 4 CONCLUSION
Theoretical and experimental studies of angular and magnetic fields effects of dipoledipole interaction frequencies, provide a means of extending the frequency range of available and convenient resonance centers in particular solid state laser active medium, such as oxygen centers in CaF 2 crystals. These investigations marked the importance of intense studies of magnetic and geometric properties of the laser medium, i.e., considering the response of the paramagnetic material at the atomic level to the effects of strength and orientation of the applied external magnetic field. Tuning range of dipole-dipole interaction frequencies can be extended by various combinations of magnetic field strengths and geometrical orientations of active medium crystal.