INFRARED ABSORPTION AND ANHARMONICITY OF THE U-CENTER LOCAL MODE THEORY AND DISCUSSION H. Bilz, D. Strauch, B. Fritz To cite this version: H. Bilz, D. Strauch, B. Fritz. INFRARED ABSORPTION AND ANHARMONICITY OF THE U- CENTER LOCAL MODE THEORY AND DISCUSSION. Journal de Physique Colloques, 1966, 27 (C2), pp.C2-3-C2-18. 10.1051/jphyscol:1966201. jpa-00213061 HAL Id: jpa-00213061 https://hal.archives-ouvertes.fr/jpa-00213061 Submitted on 1 Jan 1966 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. INFRARED ABSORPTION AND ANHARMONICITY OF THE U-CENTER LOCAL MODE THEORY AND DISCUSSION by H. BILZ,D. STRAUCH Institut fur Theoretische Physik der Universitat Frankfurt and B. FRITZ Physikalisches Institut der Technischen Hochschule Stuttgart RbumB. - On discute, dans le cadre d'une theorie gkntrale de;l'absorption infrarouge des modes localises utilisant les fonctions thermodynamiques de Green, les resultats expkimentaux d'un travail precedent de Fritz et al. [I]. On montre qu'il importe de tenir compte de la polarisa- bilitb de l'ion H- pour calculer la frequence des modes locaux et rendre correctement compte des bandes locales. Le couplage anharmonique entre le mode local et les modes de reseau trks peu perturb& explique de faqon satisfaisante l'effet de la tempkrature et de la frequence sur l'absorption dans la region de la bande principale et des bandes laterales. On discute les analogies et les differences de com- portement des ions H- et Dm.I1 apparait que le mode d'absorption local montre d'ktroites ana- logies avec l'absorption infrarouge des oscillateurs de dispersion dans les cristaux ioniques par- faits ; cela signifie, en particulier, que la fonction d'amortissement et l'energie propre anharmo- nique dependent de la frequence d'une faqon analogue dans les deux cas. Abstract. - The experimental results of a previous paper by Fritz et al. [I] are discussed within the framework of a general theory of infrared absorption of localized modes using thermody- namic Green functions. It is shown that taking into consideration the polarizability of the H- ion is important for the calculation of the local mode frequency and a correct treatment of the side bands. The temperature and frequency dependence of the absorption in the main band and in the side band region are satisfactorily explained by the anharmonic coupling of the local mode to the nearly unperturbed lattice modes. The similarities and differences in the behaviour of H- and D- centers are discussed. It turns out that the local mode absorption shows close analogy to the infrared absorption of dispersion-oscillators in perfect ionic crystals ; this means especially that the damping function and the anharmonic self-energy have a similar frequency dependence in both cases. 1. Local mode frequency and oscillato~ strength. are obtained which exceed the observed values by In a recent publication [I] experimental results 40-60 percent (40 % NaCI : H-, 46 % KC1 : H-, have been reported on the infrared absorption bands 50 % NaI : H-, 64 % KI : H-). of U-centers. The aim of this communication is a To remove this discrepancy, changes of force cons- theoretical interpretation of these results. tants have to be taken into account. The U-center is a' negative hydrogen ion which In the following discussion, a simplified shell model is substituted for an anion in an alkali halide. This of the perturbed lattice will be used with springs defined ion differs in mass, polarizability and interaction as follows (Fig. 1) : with its neighbours from those of the host lattice. a) core-shell springs which are denoted by g-, gf The very small mass m, of the hydrogen ion gives and g, for anions, cations and the hydrogen ion, rise to a localized mode whose frequency o, is gene- respectively ; rally more than two times larger than the maximum b) shell-shell springs to nearest and next nearest lattice frequency. If one considers the mass defect neighbours denoted by f,,f, and the corresponding only (isotopic model [43], [2]) frequency values for o, ones involving the hydrogen ion f ;, f ;. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1966201 C2-4 H. BILZ, B. FRITZ AND D. STRAUCH We have examined the effects of a variable polariza- bility in alkali halides with different anion radii (a, cc r&,,). This is suggested by the behaviour of the spatial extension of the ground state wave func- tion; latter increases somewhat by going from LiF (1 .73 a.u.) to KC1 (2.28 a.u.) and RbBr (2.50 a.u.) 151. A simple shell model calculation which makes use of eqns. (1) and (2) shows, that the modification of the shell-shell constant f is larger in KI than it is in KC1. The difference is, however, smaller than sug- gested by the calculation of ref. [43]. We obtain 2 f'/gH x 1 from eqn. (2). Thus an increase in the size of the impurity ion brings a large contribution to the frequency change. From this discussion we conclude, that the change of the nearest neighbour force constant f is small at least as far as chlorides and bromides are concerned, and that its influence on the dielectric susceptibility FIG. 1. - Shell model of the U-center in a diatomic can be neglected in a first approximation. This differs cubic lattice. The force constants are explained in the from the treatments of the U-center by Timusk and text. Klein [6] and by Xinh 171. In these papers the polari- zability of the hydrogen ion is neglected and conse- The shell-shell springs f which represent the repul- quently a strong change off has to be assumed in sive overlap forces between the electron clouds can order to be consistent with the observed local mode in principle be calculated for the perfect crystal from frequency. As will be discussed in a later section, a Born-Mayer potential. The shell-core springs g this change leads to serious consequences for the are related to the crystal polarizabilities aiof the ions by structure of the calculated side bands. The shell model allows to calculate the oscillator strength of the local mode transition. The effective charge e* is related to the spring constants and the where Ye means the charge of the shell [3]. polarizability by the formula : The local mode frequency o, is in a rough approxi- mation given by Since both f' and gH determine the local mode fre- quency it is important to know their relative magnitude. Here 2, is the total ionic charge which is nearly - 1. This problem has been investigated by Fieschi, It was shown above that gH/2f' z 1 ; furthermore Nardelli and Terzi [2]. In their calculation a constant Y z 1, f' FS f. This way we obtain for NaCI, KC1 a, polarizability of = 1.9 A3 is assumed ; the latter and KBr values of e$ z 0.5 e. This should be compa- was calculated for LiH by Calder et al. [4]. The shell red with the results of paper [I], where effective charges charge is Y z - 1, which also can be justified from are obtained by fitting both, the infrared and ultra- the calculations of the ground state wave function violet U-center absorption in identical samples to a by Gourari and Adrian [5]. It was shown in ref. [2] Smakula equation of the form : that the nearest-neighbour spring constants f are practically unchanged in NaCl and KCl. That means that the lowering of the observed local mode frequency o, compared with the theoretical one is mainly caused by the small value of gH as compared with g-. For Here K(o) = Zn(I,/l)/d and f,,,,, is the oscillator the case of KI they find a decrease of the value off strength in the infrared and ultraviolet region, respecti- by about 50 percent. vely. The term in brackets is the local field correction INFRARED ABSORPTION AND ANHARMONICITY C2-5 in a medium with refractive index n (taken at the absorption and induce relatively sharp absorption bandpeak frequency). peaks. Latter have been observed by Sievers [16] Calibrations of the oscillator strength f,, in the and Weber 1171. ultraviolet give values of approximately 0.8 in KC1 and ELM describes the response of the local mode which KBr [8]. In these crystals the ratio forf,,/f,, is E 0.7 [I]. is coupled to all the other existing modes. This part This can be related to effective charges e: by setting shows close analogy to zLattice,that means that the eE2 = e2.hR. The result of e: E 0.75 e seems to be local mode has the properties of a dispersion oscillator not inconsistent with the estimate given above. like the Reststrahlen oscillator in an alkali halide. This conclusio~l together with the fact that the E,, is a crossing term which arises from the coupling oscillator strength f,, turns out to be nearly indepen- of the local mode to the Reststrahlen oscillator and dent of temperature [l] confirms the strongly ionic the other band modes. For illustration, eqn. (6) is character of the defect. Our explanation differs from represented in figure 2 (upper half) by diagrams which that given by Hardy [9] who considers strong electron- describe the different contributions to the absorption phonon interaction in the IS, ground state which in terms of photon-phonon (dipole moment) and leads to an essentially temperature dependent oscil- phonon-phonon (anharmonic) interactions.