Infrared Absorption and Anharmonicity of the U-Center Local Mode Theory and Discussion H

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 analogies 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 anharmonique 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 polarizability 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 function; 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 suggested 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 polarizability 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 approximation given by

Since both f' and gH determine the local mode frequency 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. It turns lator strength. out, that interference terms between the different contributions to eqn. (6) drop out by introducing 2. Theory of infrared absorption. - The U-center complex renormalized (linear) dipole moments (MI) local mode gives rise to an infrared absorption band (Wehner [IS]). An instructive example is the derivation which shows the following main features (see [I]) : of eqn. (8) for the side band absorption. a) a strongly temperature-dependent resonance peak whose half-width increases approximately as T~ ; E ------( crystal Zz b) side bands due to the additional absorption or emission of a lattice phonon. To explain these features, we derive in the following an expression for the dielectric constant 8 as a function of frequency and temperature in the region of the local mode frequency. In the unperturbed lattice the dielectric constant is described by

Here E~(o)is the response of the (( Reststrahlenx oscillator which is anharmonically coupled to all the other lattice modes. E~~(w)means the contribution of the non-linear dipole moments. This process is well known from infrared absorption in the h~mopolarcrystals Ge, Si, Diamond [lo]. In the presence of defects the dielectric constant contains additional contributions :

~(m)= ~~attice(~)+ EBM(~) f ELM(^) + ECR(~). (6) eLatticeis the contribution described in eqn. (5) slightly modified by the perturbation of the lattice modes. EBM is an impurity activated one-phonon absorption in addition to ; the theory of this absorption, FIG. 2. - Representation of eqn. (6) by diagrams which is absent in the pure crystal, has been treated (upper half); of the renormalized dipole moment MIL in the harmonic approximation by several authors [ll- (7 b), anharmonic self energy 17~(7 e), and renormalized Green function gl (lower half). 141. The resonant modes [13-151 which may appear in 0 vertex for photon-phonon coupling (M,) the impure crystals are characteristic for this type of vertex for phonon-phonon coupling (V,). C2-6 H. BILZ, B. FRITZ AND D. STRAUCH

In the region of the local mode frequency E,, is We follow in the presentation of this theory the the most important contribution and will be discussed treatment given by Wehner [18] in a modified form in the following. Possible influences of other parts which is appropriate for the perturbed crystal. of eqn. (6) will be discussed in a later section. E,, Apart from the violation of selection rules due to may be treated in close analogy with the treatment the disturbance of lattice periodicity the different of dispersion oscillators in ideal lattices [18], 1191. terms in the dielectric constant are of the same form The theory of the infrared absorption in diatomic as for the unperturbed crystal and lead to dispersion cubic crystals with point defects has been developped formulas of the same structure as known from the by Klemens [20], Maradudin [12], Visscher [21], perfect crystals. Hanamura and Inui [22], Dawber and Elliott [14] Taking into account all important two- and three- Szigeti [23], and several other authors. phonon processes, we derive the following dispersion formula (Appendix A) :

- Here MIL(w) is the renormalized frequency-dependent dipole moment of the local mode :

- 6 2 J) 3-j, - j g J - 24 Mi,j k V-i - j - k L).,j k) . (7d) i,i i,i,k nL(w) is a complex damping function the real part of which describes a frequency-dependent shift A,(w) of the local mode frequency :

- 96 2 I V4(L, i,j, k) 12.{ Reg:(i, j, k) + iImg:(i, j, k)) i,j,k

2(m, * mi) = +.- [(~j+2)i(ni+3I.(~(~~~~~)~-~~+ iz[B(wi 3. mi - 0) - B(c()~+ mi + o)] ] (7f) INFRARED ABSORPTION AND ANHARMONICITY C2-7

2(oi + coi + o,) + in[6(wi + oj + wk - a) - 6(oi + oj + ok+ a)] (75) (ai + oj fco;)2 - o2 ]

V, and M, are expansion coefficients of the crystal (c < lop4). Therefore we may consider the imaginary potential V and the dipole moment M in terms of part of the dielectric constant only. normal coordinates (see Appendix A). The different terms of eqn. (7) are represented by 3. Structure sf the damping function. - A good diagrams in figure 2. The renormalized one-particle survey of the different types of phonon interactions Green functionis obtained by solving a Dyson equation. contributing to the damping function l7, of eqn. (7e) An essential simplification of the formulas arises is achieved by means of the diagrams introduced from the fact that the refractive index of the crystal above (Fig. 3). It seems to be useful to classify them is practically unchanged for concentrations of interest according to the order of the anharmonic potential

order of diagram contribution to argvmenl of d fu. temperature dependence @'%-,) damping ~(LJ)in < ;h-equency denorn?lnafor of A, s urnmalion process difference process a half widIh(D7 fq~wj)~- LJ 1 +ni+n. n.- ni 0 J J I b li2 side band (~2q12-d2 l+nL+ni=j l+ni n,-n,--,n; (2nd harmonic 1 C [!I forbidden 1 - a y 7,: u~r 2n'+l; 2nL+l= 1 I1 - b V, 0QL 1 2 (I+n,)( ltnj )n, a holf width (q+ 9 ? 0,) - w ' (l+n,)(it~)(kn,)-n,njnt -(l+n,) ninj b 0' =(l+q)(l+n,) (1+nj)nk 111 V; c side(not observed)band 'f2q- -t%i2- w' -l+n I( nK 3rd harmonic d (no, observed) j2qf q j2- d2 1 $0 half width(sca1t)

2 2 a &nd(bo-d,) (qr9 ( q)- !V = (1 + rg)(i+nk =(l+nJ In, b , FIG. 3. - Contributions to the anharmonic self-energy DL(w).All diagrams contribute to the shift Ado).Replacing one or two of the anharmonic vertices (0)of the diagrams by the dipole vertex (0)gives the corresponding contributions of the nonlinear dipole moment (V; + Mn- 1 Vn or M:- These diagrams have the same temperature dependence but a changed frequency dependence. C2-8 H. BILZ, B. FRITZ AND D. STRAUCH and the number of local mode phonons invol- by Mn-, and the denominator of the real part by a ved. squared frequency which is of the order of the elec- The temperature dependence of the different terms tronic transition energies (z10' 0, ... lo3 aL). In and the frequency region in which they are important this case the vertices of the diagrams are not coupled are readily evaluated with the help of the Green's to the local mode but immediately to the external functions g, (eqn. (7f), (7g)). Their properties are electric radiation field (see also Fig. 2). tabulated in figure 3. The table contains all processes up to second order in pertubation theory with the 4. Side bands. - We begin with a discussion of addition of a special process arising from (V3)4. This the side band absorption. This part appears like a last term may be important for the half-width of the shifted one-phonon absorption and the structure of main band (Elliott et al. [24]). There is another term these processes seems to be less complicated than of the same order listed in brackets in the table which that of the main band processes. cannot be evaluated at present apart from its tempe- At low temperatures (T < 200K) the half width of rature dependence which is the same as that of the the main U-band, 2T(oL), is much smaller than foregoing term. lo-' o,. We may therefore expand the denominator Figure 3 also gives the contributions of the non- in eqn. (7a) for frequencies which differ from a, by linear dipole moments. We only have to replace Vn more than one percent. Then we find from eqns. (7u), (7dj, and (7b) after some algebra :

with &;(o) = 02 + 2 wL A,(w) .

In eqn. (8) we have neglected M3 and V,. Apart from the two-phonon contribution of IIIb the only and only the summation band remains (x).Further- important diagram in figure 3 is given by Ib. As it + stands, eqn. (8) is completely analogous to that given more we neglect the frequency dependence of the by Szigeti for perfect crystals [25] apart from the eigenvectors and introduce (Appendix A, eqn. (A5)) contribution of the shift function A,(@) to the local mode frequency w,. To investigate the relative importance of cubic anharmonicity (V,) and second order dipole moment (M,) we discuss a density approximation of ELM [26]. That means we consider the case T = 0 OK in conse- where now MI, M2, and V3 are assumed to be frequen- quence of which the occupation numbers niequal zero cy independent parameters. Neglecting for the moment A, in zi we find : INFRARED ABSORPTION AND ANHARMONICITY C 2 - 9

Here p(oi) is the one phonon density of states at the Tableau I lattice frequency oi = o - a,.

Frequency w1=w-q (10~s-')

. - 0 50 100 150 200 Wovenumber ic6'~ FIG. 4. - High-frequency side band of the local mode in KC1 : H - a) experimental. - 6) theoretical : anharmonicity. - c) theoretical : non-linear dipole moment (see text also).

Figure 4 shows a comparison of the two limiting cases of eqn. (1 1) with new experimental results on KC1 : H- (curve a). Curve b describes the purely anharmonic case (M2 = 0), while curve c gives the effect of the second order dipole moment only (V, = 0) The density of states is taken from Karo and Hardy =+ localized mode [27, (Fig. lla)] and has been multiplied by ~0;~or ----t band mode m;', respectively. A very small correction factor has II u center been omitted. This way we obtain oeL,(co) which is o neares f neighbours proportional to the absorption constant K(o). second nearest neighbours For long acoustic waves the approximation (10) b is no longer justified but the frequency dependence of Cubic anharmonic coupling parameters between the coupling parameter M2 and V3 has to be taken the U-center and its nearest and second nearest neighbours. The corresponding coupling parameters for into account. As shown in Appendix B. M2 and V3 the second order dipole moment are obtained by sup- are expected to be proportional to mi. This gives the pressing one of the arguments % in the brackets. extrapolated dotted curves shown in figure 4 for both cases. Figure 4 shows that the absorption due to anharmo- [31]. If we assume an unchanged force constant f nicity fits the experimental data in KC1 : H- considera- (see Sect. 1) we have to use the linear combinations bly better than the absorption caused by second order of the eigenvectors of the perfect lattice according to dipole moments analogous to the multi-phonon the representations r:2(~g) and T:(A,,) of the point absorption of perfect alkali halides which also is group 0,(Timmesfeld [30], Sennett [32j). Our calcula- governed by anharmonicity [19], [28]. The same result tions show, that T:2 plays a dominant role for the has been obtained by Xinh [7] for KI : H- (see below). side band absorption. The contributions of the cou- In order to study the influence of the frequency- pling parameters a; and a; are plotted in figure 5. dependent eigenvectors ~(v1 i) (App. A) and the Although the main contribution seems to come from coupling parameters to nearest and next-nearest coupling to nearest neighbours (a;), the share of neighbours [29], [30] (Tab. 1) we have made prelimi- next-nearest neighbours (a;) might also play an nary calculations for KC1 [26] using the eigenvectors important role. Mixing these two contributions results for 512 points in the Brillouin zone given by Kucher in a curve which contains nearly all parts of the phonon C2-10 H. BILZ, B. FRITZ AND D. STRAUCH

We have tried to take care of these complications in the following way : Ad a) Regarding the fact that in the long-wave-length limit the ration of am- plitudes of the ions in a diatomic cubic lattice is equal to unity for acoustic modes and equal to the reciprocal mass ratio for optical modes, we get with the help of the normalization condition

approximately the following relation (eqn. (B3)) : j12(cation I optic) r- - X2(cation1 acoustic)

manion --- C = 2.04 for KBr] . (12) mcation We therefore modify our density approximation by n~ultiplyingp(oi) in the optic frequency region with this factor (App. B). Since normally the ratio r is greater for shorter wavelength, eqn. (12) gives the lower limit of the effect. frequency ur=~-wL(IO'~S-? Ad b) and c) It turns out that the best fit of the FIG. 5. - Side band absorption in KCl. Dashed experimental curve in KBr, in this simple modified lines : experimental curves by Fritz et al. [I].Full lines, density approach, may be obtained if the origin of upper half: theoretical curve using one coupling parameter (a',) to nearest neighbours ; lower half : theoretical curve using one coupling parameter (a;) to second nearest neighbours. Dotted line : theoretical extrapslation for long wavelengths. spectrum. This might be the reason for the quite satisfactory results obtained with the simple density approximation (Fig. 4). This leads us to an investigation of similar density approximations in crystals with unequal anion and cation masses such as KBr, KI, and NaBr. In these cases, acoustic and optic frequencies are separated by a gap. This leads to the following complications : a) Considering the important coupling to nearest neighbours, an increasing mass ratio of anions to cations brings about enhanced displacements of the cations in the optical region, thereby producing a preferred coupling to the optical modes. b) The frequency dependence of the local mode energy, arising from its anharmonic coupling to the other phonons, leads to a shift of the side band fre- FIG. 6. - a) Density of lattice modes in KBr. quencies, especially of the gap position. Solid line : calculated by Cowley [33]. Dashed line : cal- c) With the exception of KBr which has been culated by Karo and Hardy [27]. b) Side band absorption of the U-center in KBr. Solid line : experimental measured with neutron spectrometry, the calculated curve by Fritz et al. [I] ; Dashed curve : simple density density spectra are inaccurate in the optic region approximation (see text). Dotted line : perturbed density including the gap position. approximation by Timusk and Klein [6]. INFRARED ABSORPTION AND ANHARMONICITY C2-ll

the density spectrum is displaced to higher frequencies Frequency ii- w, (l0"i') by about 6 cm-' so that the forbidden frequency region coincides with the gap in the side bands. The agreement between the two curves reached in this fashion is quite satisfactory (Fig. 6, upper half). In the lower part of figure 6 the density functions by Karo and Hardy [27] and by Cowley [33] are given for comparison ; only the latter function shows the gap and some additional features which are reflected in the course of the side band spectra. The displacement of the theoretical curve which 0 AV~.. 50 100 150 200 I was used to fit the gap region towards higher energies Wovenumber (cm'') I '-'' may be justified from a more rigorous treatment of v-v, the two-phonon Green function g,(w) which in eqn. (7 f) is given in the harmonic approximation only.

Frequency a,=w-0' f10"s-~

Freouencv o -0, (10"s-li

FIG. 8. - The same as in figure 7 for NaBr : H -.

sities calculated by Karo and Hardy might be improved (Fig. 7 and 8). The size and position of the gap in KI can also be obtained from the one-phonon-density of perturbed crystals measured by Renk [34]. In NaBr again a shift of the gap position has to be taken into account while in KI it seems to be less important. The overall agreement between the measured and calculated curves is nearly as good as for Wovenumber Icm-") KBr. At the first view it seems to be a strange fact that the position of the first and dominant side band maximum always nearly coincides with the critical point at TA(1,0,O) because this point is strictly forbidden by selection rules [35]. As can be seen from the genera1 theory of absorption, the behaviour of the dispersion curves and the discussion of KC1 (Fig. 4), the absorp- Frequency (10"s-') tion is roughly a linear increasing function of frequen- FIG.7. - The same as in figure 6 for K1 : H -. Solid cy for long wavelengths due to the rapidly increasing line in figure 7 a) calculated by Karo and Hardy [27]. coupling parameters and density of states up to Dashed line : modified density (see text). The dotted frequencies w(q) near w,,(l, 0,O) ; for example line in figure 7 6) is calculated by Xinh [7] with a similar wTA(1/2, 1/2, 0). In the neighbourhood of these fre- approximation as used by Timusk and Klein [6]. quencies, the coupling parameters reach their maximum value for the first time, so that for higher fre- Similar calculations have been made for KI (Fig. 7) quencies the absorption constant decreases due to the and NaBr (Fig. 8). In both of these cases only the frequency denominator. Since the acoustic phonon phonon densities by Karo and Hardy [27] are available. branches of the different alkali halides show a very The comparison with the calculations of Cowley et similar behaviour, we would expect that the numerical al. [3], [33], for KBr and NaI indicates how the den- coincidence of the first side band maximum and the C2-12 H. BILZ, B. FRITZ AND D. STRAUCH lowest critical point at w,,(l, 0, 0) is a general feature 5. Linewidth and shift of the main peak. - The of the U-center side bands. width of the main absorption band is determined In order to obtain a deeper insight into the detailed by the lifetime of the local mode ; a temperature structure of the side bands, calculations have to be independent contribution may follow from internal performed similar to that discussed for KC1 in which strain of the crystal due to lattice defects. The experi- the influence of the different coupling parameters is mental results [I] on KC1 : H- and D- clearly show considered. This has been done in two recent papers that the latter effect, if present, results in contributions on KBr :H- by Timusk and Klein 161 and on KI : H- much below 1 cm-', and the temperature dependent by Xinh [7]. Both authors neglect the polarizability line width of H- and D- centers in these crystals of the hydrogen ion and assume a strongly weakened has to be explained by damping of the local mode nearest neighbour force constant f in order to obtain oscillator. the observed value for the local mode frequency The diagrams Ia and IIIa in figure 3 mean a two- (Sect. 1). (Aflf = - 0.43 for KBr and - 0.62 for KI). phonon and three-phonon decay of the localized This force constant change induces a corresponding vibration. At T = 0 OK (no band modes excited) the change of the eigenvectors and leads to resonance corresponding frequency conditions from figure 3 modes at certain frequencies at which the absorption read : constant is enhanced above the original one. In both w, = oi + wj papers anharmonic spring constants a; (Tab. I) to nearest neighbours are taken into account. In addition, and (13) Xinh uses a second parameter y; z 0.1 a; correspon- o, = wi + wj + o,. ding to the value in perfect KI. For all alkali halides (with the exception of LiF and The result of their calculations is shown in figure 6 NaF) the following relation holds and 7, respectively. In both cases the descriptions of the first side band maximum are quite satisfactory uL(D-) < 2 w,,, (band modes) < oL(H-) . (1 4) and comparable with our result for KCI. However, the agreement in the optical phonon region is not Thus two-phonon decay is forbidden for the H- center very good. Klein and Timusk obtain a localized mode because of (13), but allowed for D- centers. We believe in the frequency gap of KBr which has not been that this is the explanation for the low temperature observed experimentally. From the strong spring line width of D- centers in NaCl and KCl, which constant change used by Xinh one would expect a assumes saturation values near 1 cm-' for T-+ 0 OK. similar local mode in KI for the calculated absorption Dotsch et al. [37] have shown that in LiF also H- + OK ; constant, possibly by using an improved shell model. lines retain a larger width of 4 cm-' for T 0 The neglection of the polarizability of the hydrogen in this crystal the small reduced mass brings the H- ion obviously causes the strong weakening of the lines so close to a,,,,, that a two phonon decay occurs. nearest neighbour coupling, which is responsible for The temperature dependence of the damping by the non-observed features in the calculated absorption. two and three phonon interactions is Furthermore the contribution of second nearest neighbours could be important as discussed for KC1 ('). It is interesting to note that in the region of longi- and tudinal acoustic modes the calculated absorption is in all cases remarkably smaller than the observed one. Very recently the shell model has been improved by considering the compressibility of polarizable ions (Schroder [36]). We suppose that an analogous treat- respectively. Above the Debye temperature 0 the ment of the U-center diminuishes the discrepancy former process is practically proportional to T and mentioned above. A weak absorption due to the the latter to T~,but below 0 both processes give a excitation of the local mode and two lattice phonons rather weak temperature dependence of the damping. is seen in the wings of the side bands as well as in the The T2 variation of the half width observed with region of the frequency gap. both isotopes is satisfactorily explained by the scattering of band phonons at the defect, as suggested by (1) We have been informed by Mr. J. Page, Salt Lakecity, Elliott et al. 1241. that a calculation using a correct shell model is in progress. The diagrams required are IIIb and IVa (Fig. 3). INFRARED ABSORPTION AND ANHARMONICITY C2- 13

Assuming a Debye spectrum, one calculates a half- both positive and negative parity. In the limit of low width proportional to Tmwhere m = 2 above Debye frequencies the coupling to the former vanishes while temperature ; below this temperature m increases the coupling to the latter is maintained. In the represen- from 2 towards 7 [38]. The best description for KC1 is tation rs these phonons contain a displacement of the obtained by using an effective Debye-temperature of H--ion itself decreasing with increasing frequency about 120 OK (curve a in Fig. 9). This value is much (Dawber, Elliott [12], Fig. 2). It seems to be not lower than that known from specific heat measure- unreasonable that this effect leads to a very low ments (230 OK) ('). (( effective )) Debye-temperature. A similar behaviour should be observed in other alkali halides. The experimental points of the D- half-width in KC1 (Fig. 9) are fitted by superimposing the two-phonon decay and the scattering contributions. This is done by adjusting the two phonon decay curve (b) to the saturation half-width and adding the curve of the H- half-width (a), but shifting by a factor 0.5 which is the ratio of D- and H- matrix elements. The resulting curve c is a good approximation of the data. The half- width as taken from experiments approximately corresponds to a mean value of the damping function r,(o) which is not known at present but likely is a strongly varying function within the frequency region of the main absorption band. There is a strong correlation between the damping function T,(o) and the shift function AL(w) (eqn. (7e)). They are Kramers-Kronig- transforms of each other. In the local mode frequency region there are other diagrams contributing to the shift than those responsible for the half-width. The most important ones are supposed to be the diagrams Ia, b and IIa, b of figure 3, while the influence of the remaining ones should be negligible. Generally there are positive as well as negative contributions to the shift. One expects negative cubic anharmonic coefficients V, and po-Live quartic coefficients V, if the short-range forces can be deduced from a Born-Mayer-potential [19]. Then fem~erafure the diagrams Ia, IIb lead to a positive and the diagram FIG. 9. -Temperature dependence of the half- IIa to a negative shift. If one adds the summation and width of the local mode in KC1 : H- and KC1 : D-. experimental values by Fritz et al. [I]. The curves difference processes of diagram Ib, the temperature ., dependence of this contribution approximately cancels a, b, care explained in the text. out, leaving a constant term. The reason for this discrepancy may be a preferred In addition to the truly anharmonic shift there is coupling of low frequency (acoustic) phonons. The a contribution from the thermal expansion of the diagrams allow for the coupling of phonons with lattice. Following Cowley [19] we define the quasi- harmonic shift as that caused by thermal expansion (2) In a recent paper Mitra and Singh 1411 have discussed the of the lattice which in turn leads to changes in the half-width and intensities of U-centers in KC1, NaCI, KBr, and CaF2. Some experimental results disagree with those obtained by force constants. The thermal expansion is due to the Fritz et al. [I] especially for the half-width at low temperatures. anharmonicity of all the lattice modes and hence not In their theoretical explanation Mitra and Singh consider in addi- described by the diagrams listed in figure 3. By expe- tion to the scattering process (IIIb of Fig. 3) a constant residual riments with KC1 : RbCl mixed crystals Barth [39] has half-width due to inhomogeneous strain which is not seen by shown that the U-center infrared band position is Fritz et al. (see our discussion above). - Our results are consistent with those obtained by Ivanov et al. [42]. Their paper has sensitive to changes of the average lattice parameter. much in common with our discussion of this section. The shift produced when increasing the lattice constant C2-14 H. BILZ, B. FRITZ AND D. STRAUCH by doping of the KC1 material is negative and given by are proportional to T for high temperatures and retain a constant value for T -, 0 OK. The next important contributions which have been neglected until now, may come from the scattering process which we have where a is the X ray lattice parameter [39]. discussed in connection with the half-width of the This result may help to estimate the effect of a main peak. This process gives a negative contribution similar thermal expansion on the local mode frequency. which behaves as T2for high temperatures ; for low Using the thermal expansion data of pure KC1 the temperatures it dies out. quasi harmonic shift expected from (15) is found to There are experimental hints to a shift of the side be larger than the experimental U-band displacements bands against the local mode frequency : (Fig. 10, curve d). Thus in KC1 the anharmonic pro- a) The low and high frequency side band maxima cesses discussed above are assumed to be responsible have different distances from the main line (Fig. 11, for an additional shift to higher frequencies with Fig. 1 of ref. I). increasing T. The diagram Ia might result in a positive but small shift in KC1 : H- since 0, z 2.4 m,,,,, which means that the energy denominator (Fig. 3) is very large. Furthermore, the contribution of IIa should be of similar magnitude, but has the opposite sign. Therefore the main part of the positive shift compared with the quasi harmonic one might stem from 116. The contribution from this process is larger for D- ions than it is for H- ions, as one can see from eqn. (14). Therefore, A(D-) > A(H-) which is in agreement with experiments (Fig. 10 ; see also figure 4 and table 1 of ref. [I]. The processes discussed

0 - H-jeXp, j b M D- 1 c w difference b-o i d - -- quasi harmonic I shift lexlropo/afed) i

FIG. 11. -Damping function and shift function of the local mode in KC1 : H-.

b) The one phonon frequency gap in the side bands is shifted to higher frequencies as discussed above for NaBr and KBr. As mentioned above this is due to a two-phonon Green function and will be discussed in a forthcoming paper. There is in addition a modification of the shape of the side bands due to the frequency dependence of the local mode frequency oL. The absorption constant K(o) and the damping function T,(o) are related to one another (see (7a)) by \ I -15 \ -15 0 100° ZOO0 300°K temperature FIG. 10. - Temperature dependence of the local mode frequency in KC1 : H - and KC1 : D -. Setting the shift A, equal to zero, a zero-order appro- INFRARED ABSORPTION AND ANHARMONICITY C2-15 ximation to TL is easily evaluated from experimental with D- centers is mainly due to the fact that two- data. We have used the Kramers-Kroning relations phonon decay of the local mode is always possible to calculate AL and thus obtained a corrected function for D- centers but normally not for H- centers. The T by again applying formula (16) and inserting ratio of the damping functions is T, : T, z 2 both &: = 02 + 2 oLA,(@). This procedure, .if repeated, experimentally and theoretically at higher temperatures converges and gives the two parts r, and A, which due to the dominant scattering process (sect. 5). we calculated to 3rd order in figure 11. A calculation b) (( Effective )) Debye temperatures for the fitting for the case of LiF (pure material) has been done by of halfwidths to temperature are lower than those Wehner [18] and gives very similar results. obtained from specific heat data, indicating the This might lead to remarkable changes of the slope influence of the frequency dependent coupling mecha- of the absorption curves in regions where AL is a nism. strongly varying function of frequency (e. g. in the IV) The infrared absorption of the local mode of gap region of KBr) via the frequency denominator U-centers in alkali halides shows close analogy to the of E" (eqn. 8). infrared absorption of the perfect alkali halides ((( Reststrahlen )) - oscillator). This is clearly demons- 6. Conclusicsn. - To summarize the discussion of the foregoing sections we would like to stress the trated by the similar behaviour of damping and frequency shift functions in both cases (sect. 5, Fig. 11). following points : It remains to add some remarks on the importance I. LOCALMODE FREQUENCY.- The frequency of of neglected parts of E (eqn. (6)) and on further impro- the local mode is essentially determined by the pola- vements of the theory. rizability of the hydrogen ion [2] (sect. 1). The perturbance of cLatticeby U-centers has not 11. SIDEBANDS. - a) From ref. [I] follows that the been investigated at present. It might be an interesting side bands of some alkali halides can be treated in a problem for itself but doesn't influence the results of rough approximation by neglecting force constant this paper. The investigation of E,,(u) should give changes to nearest neighbours (sect. 1 and 4). interesting insights into perturbed lattice modes with odd parity that do not contribute to the side b) The strongly ionic character of the U-center bands. There are only preliminary results (Sievers, established by the temperature independent oscillator private communication). strength (sect. 1) leads to a dominant role of the The crossing term E,, could be important in cases of anharmonic coupling of the local mode to the other lowlying local modes (D- centers) which can interfere lattice modes compared with the non linear dipole with dispersion oscillators of the perfect crystals. moment. - The ration of intensities of H- and D- There is no evidence at present for such interference centers in equal concentration is experimentally (3) effects. and theoretically [7] nearly two. Further improvements of the theory should lead to c) The possible influence of perturbed phonons a quantitative estimate of the coupling parameters, (resonance effects) cannot be investigated without the relative influence of nearest and second nearest taking into account the influence of second nearest neighbours, the changing of force constants, and neighbours which might be more important (sect. 4). related problems. Investigations along these lines are d) There exists a shift of the side bands which under way. has to be explained by investigating the two-pho- Acknowledgement. - The authors are grateful to non Green function. Dr. Elliott, Dr. Wehner, and Dip1.-Phys. Gross for e) The frequency dependence of the local mode stimulating discussions, and to the Deutsche For- itself leads to a modification of the side band shape, schungsgemeinschaft for financial support. One of the especially in the gap region. authors (B. F.) is indebted to Prof. Pick for his interest 111. MAINBAND. - a) The different behaviour of in this work and wants to thank Dip1.-Phys. Ehret the half-width and shift of H- centers compared for performing numerical calculations.

(3) In paper, [I] § 3.4 the intensities of the side bands in APPENDIXA RbCl : H - and RbCl : D - crystals were reported to be in a ratio It is convenient to express all important quantities of 1 : 0.29, equal concentrations being assumed. Subsequent mea- suren~entson RbCl and KC1 gave- values for this ratio which are lattice in terms Q. closer to 2.0, in overall agreement with the main band intensities. They are introduced in the following way : The equa- C2-16 H. BILZ, B. FRITZ AND D. STRAUCH tions of motion of the lattice in the harmonic approxi- These expansion coefficients are related to one another mation are : by

Here n denotes the site of the ion considered, m, is its mass, u (z) is the x-component of its amplitude. Eqn. (Al) has the solution

The calculation of the dielectric constant ~(o)can be where i is the phonon quantum number. In an ideal performed by using Green's functions as discussed by lattice, i is given by the wave number-vector q and branch index j and Wehner [18], replacing of the perfect crystal by (i) of the perturbed one.

To get a first approximation of the frequency dependence of the expansion coefficients M,, V,, we neglect (see e. g. Born and Huang [40]). the frequency dependence of the x(vi I i) in (A3) ; then we get The eigenvectors x (: 1 i) are normalized by

The electric dipole moment M and the anharmonic part HA of the crystal Hamiltonian can be expanded in terms of particle displacements

From symmetry considerations there are band modes of positive parity only which may couple to the loca- or of normal coordinates lized mode (which has negative parity) according to M,(L, 1) and V,(L, L, 1). If we treat the U-center as an isotopic defect, we Ai: e-= Jg- neglect changes in force constants ;then the frequencies of the bandmodes of positive parity are equal to 1 A M = z -, z Mn(vl, ..., vn).uv1..... uVn those of the unperturbed lattice. This is true even if we n31 n . vl,...,v, assume the ct spring)) g, between core and (not ="1/= z M,(l, ..., n).A, .....A, deformable) shell of the U-center itself to be changed. nZ 1 1, ...,n The only change in the band modes is that to a lower symmetry of which we may take account by projecting the polarization vectors (of the space symmetry group) on the point symmetry group (0,) with the help of a projection operator P (see e. g. Slater [41] for definitions of P) ; then we have INFRARED ABSORPTION AND ANHARMONICITY

~(vI i) cc sin q.xi

(xia constant vector) For low frequencies we have in good approximation (yj a constant vector) of r (k 1 19) by combining the two proportionalitiea above co (:) cc sin (yj x(v I ii) "JL 031) and then we get (neglecting the frequency dependence Therefore,

"J1 and V3(L, L, I) cc - - in analogy. JG - One has to take in mind that these proportionalities are valid in the low frequency range only. [I] FRITZ(B.), GROSS(U.), BAEUERLE(D.), Phys stat. sol., 1965. 11. 231. [2] FIE~CHI(R.), NARDELLI (G. F.), TERZI(N.), Phys. Rev., 1965, 138, A 203. [3] COWLEY(R. A.), COCHRAN(W.), BROCKHOUSE(B. N.), WOODS(A. D. B.), Phys. Rev., 1960, 119, 980. In the long wave lengtB limit, [4] CALDER(R. S.), COCHRAN(W.), GRIFFITHS(D.), LOWDE(R. D.), J. Phys. Chem. Solids, 1962, x (cation I acoustic) = x (anion I acoustic) , 23, 621. [5] GOURARI(B. S.), ADRIAN(F. J.), Solid State Physics, mc3tion x x (cation I optic) = x (anion ] optic) . 1960, 10, 127 (ed. Seitz & Turnbull). inanion [6] TIMUSK(Th.), KLEIN(M. V.), Phys. Rev., 1966, 141, 664. From eqn. (A2) the following relations hold [7] XINH(N. X.), Sol. State Comm., 1966, 4, 9. [8] TIMUSK(Th.), MARTIENSSEN(W.), 2. Physik, 1963, 176, 305, EROS(S.), KNAPP(R. A.), priv. comm. [9] HARDY(J. R.), Phys. Rev., 1964, 136, A 1745. [lo] LAX(M.), BURSTEIN(E.), Phys. Rev., 1955, 97, 39. [ll] LIFSHITZ(I. M.), NUOVOCim., 1956, 3, SuppI. 716. [12] MARADUDIN(A. A.), Brandeis Lectures 11, 1962, 109. leading to 1131 BROUT(R.), VISSCHER(W. M.), Phys. Rev. Lett., 1962, 9, 546. 2 1 (manion+ mcation). (cation 1 acoustic) = -- , [14] DAWBER(P. G.), ELLIOTT(R. J.), PYOC.Royal Soc., 3 N 1963, A 273, 222. [15] KLEIN(M. V.), Phys. Rev., 1963, 131, 1500. [16] SIEVERS(A. J.), Phys. Rev. Lett., 1964, 13, 310. [17] WEBER(R.), Phys. Letters, 1964, 12, 311. 1 [18] WEHNER(R.), Thesis, FrankfurtlMain, 1965, unpu- x X2 (cation 1 optic) = . 3 N blished; Phys. stat. sol., 1966, 5, 725. [19] COWLEY(R. A.), Adv. Phys., 1963, 12, 421. [20] KLEMENS(P. G.), Phys. Rev., 1961, 122, 443. Combining these equations gives 1211 VISSCHER(W. M.), Phys. Rev., 1964, 134, A 965. [22] HANAMURA(E.), INUI(T.), J. Phys. Soc. Jap., 1963, X2 (cation I optic) ma,,,,,, " --A (B3) 18, 690. X2 (cation ] ac3ust1c) ~?z~~~~~~' 1231 SZIGETI(B.), J. Phys. Chem. Solids, 1963, 24, 225. C2-18 H. BILZ, B. FRITZ AND D. STRAUCH

[24] ELLIOTT(R. J.), HAYES(W.), JONES(G. D.), MAC vibrations of the ion ? Can the problem be treated DONALD(H. F.), SENNETT(C. T.), Proc. Royal by a similar theoretical method as in the case of H- Soc., 1965, A 289, 1. impurity ? [25] SZIGETI(B.), PYOC.Royal Soc., 1960, A 258, 377. [26] STRAUCH(D.), Diplomarbeit, FrankfurtIMain, 1965, unpublished. H. BILZ. - The intramolecular vibrations couple [27] KARO(A.), HARDY(J. R.), Phys. Rev., 1963, 129, with the lattice modes especially if the rotation is 2024. hindered, that means at low temperatures. Evidence [28] BILZ (H.), Lecture notes, Scottish University Sum- for this comes out from the investigation by Naraja- mer School, Aberdeen, 1965, to be published. [29] LEIBFIUED(G.), LUDWIG(W.), Solid State Phys., namurti et al. of NO, and CN- (to be published). 1961, 12, 275. (ed. Seitz & Turnbul). The problem seems to be formally related to the U [30] TIMMESFELD(K. H.), Diplomarbeit, FrankfurtIM., center problem and should be treated with a similar 1965, unpublished. Greens's function technique. [31] KUCHER(T. I.), Soviet Physics JETP, 1957, 5, 418. [32] SENNETT(C. T.), J. Phys. Chem., Solids, 1965, 26, 1097. M. HADNI.- Je n'ai pas pu bien entendre votre [33] COWLEY(R. A.), private comm. conclusion sur l'origine des bandes secondaires 1341 RENK(K. F.), Phys. Lett., 1965, 14, 281. d'addition ou de difftrence dans la bande d'absorption [35] LOUDON(R.), PYOC.Phys., Soc. 1964, 84, 379. de KC1 activt avec H- . Quel est le facteur prtpondtrant [36] SCHRODER(U.), Sol. State Comm., 1966, 4, 347. [37] DOTSCH(H.), GEBHARDT(W.), MARTIUS(Ch. V.), l'anharmonicitt mecanique, ou I'effet du moment Sol. State Comm., 1965, 3, 297. electrique ? [38] ZIMAN(J. M.), PYOC.Roy. SOC.,1954, A 266, 436. 1391 BARTH(W.), Diplomarbeit, Stuttgart 1965, unpu- H. BILZ. - A mon avis, il s'agit essentiellement blished. d'anharmonicitt mecanique, mais il existe des thtories 1401 BORN (M.), HUANG (K.), Dynamical Theory of trbs tlabortes concernant l'anharmonicitt dectrique Crystal Lattices, Oxford (1954). 1411 MITRA(S. S.), SINGH(R. S.), Phys. Rev. Lett., 1966, et expliquant plus ou moins bien (plut6t moins bien) 16, 694. les effets secondaires. 1421 IVANOV(M. A.), KRIVOGLAZ(M. A.), MIRLIN(D. N.), RESHINA(I. I.,) Fiz. Tverd. Tela, 1966, 8, 192. BALKANSKI.- Ne pensez-vous pas qu'un calcul [43] JASWAL(S., S.), MONTGOMERY(D., J.), Phys. Rev., direct a partir du premier principe serait mieux plutdt 1964, 135, A 1257. que de consacrer autant d'efforts sur un modble (le (( shell model ))) aussi rudimentaire ? DISCUSSION H. BILZ.- Je crois effectivement que ce serait M. S. LEACH.- What happens in the case when mieux, mais nous avons dtj& pas ma1 de calculs sur the impurity is a molecular ion, for example CN- and le (( shell model D, c'est pourquoi, pour pouvoir compa- NO, ? Is there a coupling with the intramolecular rer des resultats nouveaux, nous sommes liCs au mod6le.