Annalen der Physik. 7. Folge, Band 38, Heft 3,1981, S. 179-191 J. A. Barth, Leipzig bepolarized Light Scattering from Paramagnetic Liquids in a Magnetic Field By N. P. MALOMUSH,S. D. LATUSHKIN, and K.-P. G R ~ D E R Sektion Physik der Staatlichen Universitiit Odessa (UdSSR) Abstract. In order to identify spectral features and molecular movement types the magnetic field's influence on rotational and diffuse shear modes of paramagnetic liquids is analysed. Consideration of the corresponding changes in depolarized (VH) light scattering spectra illustrates a systematic method of studying both paramagnetic and diamagnetic liquids. Depolarisierte Lichtstreuung durch paramagnetische Fliissigkeiten im magnetischen Feld Inhaltsubersicht. Zwecks Identifizierung spektraler Bemnderheiten und Typen molekularer Bewegungen wird der EinfluS eines iiuBeren magnetischen Feldes auf Rotations- und Diffusionsschermoden in paramagnetischen Flussigkeiten analysiert. Die Untersuchung der entsprechenden Veriinderungen im Spektrum der depolarisierten (VH) Lichtstreuung illustriert eine systematische Untersuchungsmethode sowohl von paramagnetischen als auch diamagnetischen Fliissigkeiten. 1. Introduction With the development of high resolution laser light scattering techniques fine structure has been discovered in depolarized Rayleigh scattering spectra of some organic liquids. It has been supposed [2, 31 that the VH scattering spectrum consists of two components of different width and comparable intensities. Recent investigations [13 have shown a doublet structure in the sharp central component. There exist various interpretations of the given spectra peculiarities. The appearance of the doublet is explained both by the scattering from transverse phonons and by the scattering from fluctuation internal angular momentum [4], or by scattering from orientational fluctuations [5]. The emergence of the narrow Lorentzian component is due.t o the relaxation of reorientations (relaxation time is about the Debye time of molecular dipole moment relaxation). The broad Lorentzian component comes from stress tensor relaxation [5]. Existing experimental investigations of the temperature and angular dependencies of V H scattering spectra do not allow to settle the question of the simple correspondence of spectral features and molecular motion types. The situation calls for the elaboration of a number of effective controllable influences on every type of molecular motion in liquids. It is also necessary to make a detailed analysis of angular and polarization dependencies of the differential light scattering cross section. N. P. IV~ALOMUSE, S.D. LATUSHKIN, and K.-P. G R ~ ~ N D E R 180 One of the possible test is t o probe the system by a magnetic field. The modes most sensitive t o the magnetic field are rotational and shear modes of paramagnetic liquids and solutions. The latter are especially important because of the strong coupling of collective modes of diamagnetic solvent and paramagnetic admixture. The coupling makes it possible t o change the dynamics of the molecular motion in diamagnetic liquids. Thus we may exert a controllable influence on those spectral features which are ascribed t o different types of molecular motions both in paramagnetic and diamagnetic liquids. The present paper is devoted t o the investigation of depolarized light scattering from paramagnetic liquids in a magnetic field. Section 2 gives the description of polarization and angular dependencies of scattering tensor and the differential light scattering cross section. The results of the research hold good both for paramagnetic and diamagnetic liquids. Section 3 deals with a short account of hydrodynamics of paramagnetic liquids with a symmetric stress tensor in a magnetic field. In the framework of the traditional approach [6] the stress tensor of paramagnetic liquids contains a n antisymmetric part, contrary to the total angular momentum conservation law. Section 4 gives the analysis of the magnetic field influence on rotational and shear modes of paramagnetic liquids. Light scattering from rotational degrees of freedom is considered in connection with the phenomenon of optical fluctuation activity. The set of tests suggested makes it possible to explicitly single out contributions of innumerated molecular motions into scattering cross section. The main results are presented a t the end of the paper. 2. Polarization Properties of the Differential Light Scattering Cross Section In phenomenological theory the scattered light intensity is defined by a correlation of dielectric tensor fluctuations : Here J f ( d ~k,, el, e2) = < d & ~ ( tr),~EL(O, 0))mk =z e2i; e:ke&lm (2.2) where A m , & are the frequency and wave vector shifts respectively, el, e2,are polarization vectors of incident and scattered waves. From now on, when speaking about scattering cross section and scattering tensor, we mean the functions M ( A o , &), Tiklmrespectively. The requirements of the absence of light absorption lead t o dielectric tensor hermiticity. This means that besides a symmetric part the fluctuating dielectric tensor may also contain an antisymmetric part : BEik +i = d&& dE&. (2.3) For paramagnetic liquids with small molecules (of NO, 0, type) the fluctuations of the symmetric part of the dielectric tensor are modelled by the irreducable parts of the symmetric stress tensor. The antisymmetric part of dielectric tensor fluctuations is not connected with stresses in liquids. Depolarized Light Scattering from Paramagnetic Liquids 181 One of the reasons for the appearance of the antisymmetric part is the absence of the balance between the internal and vortex rotations in liquids. Let us consider the properties of the scattering tensor and cross section tensor which follow from the symmetry of the system. Here we shall deal with the most simple and experimentally most easily realizable case of the longitudinal magnetic field only. The field will be called longitudinal if its induction is directed along the vector k. - k, where k, and k, are the wave vectors of the incoming and outgoing waves. In the longitudinal field the system is invariant under the rotations around the direction specified by n. Thus the scattering tensor and the tensor of optoelastical constants form the representation of the rotational group mentioned above. The tensors should also satisfy the requirements of the permutation symmetry. Accordingly, the scattering tensor is the sum offour tensors with different symmetry with regard to the transposition between the pairs of indices : -k -k (2.5) where T” is symmetric under the transposition of indices inside the first pair i, k and antisymmetric in the second pair I , m. Generally, the scattering tensor is not invariant under the transposition of pairs. This follows from the non-commutativity of operators describing dielectric tensor fluctuat ions. The structure of the fourth rank tensor is defined by the combination of vector n components, of the isotropic symmetric tensor dik and the totally antisymmetric third rank isotropic tensor 6?ikl. We shall accept the “transverse-longitudinal” classification of tensor combinations worked out in the theory of turbulence [7, 81. Let us consider the transverse-transverse contributions to scattering tensor. In the coordinate system M, N, L (see Fig. 1) the following components are different from zero : Tiklna = T%lm T%a T%m T%m TMHHM ,THHNN, TMNMN, THHHNetcSince M, N are arbitrary and due to the rotational invariance follows: (2.6) THN~WN = TNHNM, T H H H N= - TNNNH and analogous components T HHHH,TMHNN (2.7) and T H N H N ,TMNNH satisfy the identity T ~ W H H=HTHMNN -k TMNMN f THNNH (2.8) generalizing the Millionchikov identity in the theory of isotropic turbulence [a]. Here another identity is pointed out : TMHMN -k TMHNH-I- T M N M M -k T N M M H =0(2.9) Let us turn from the special M ,N, L coordinate system to the laboratory system. The relevant tensor transformation is performed in a standard way with the help of the following relations : (2.10) (2.11) -N.P. MaLoam~S. ~, 190 >..- a. LATUSHKIN, and K.-P. GB~~KDEB Fig. 1. Coordinate system of the experimentalset-up. L axis points along the external field. dl, N axes are chosen perpendicular to the field Here z,y, z = {xi> denote the parameters of the laboratory coordinate system; X I , y’,z’ axi correspond to M , N , L axes; ni = -is the directing cosine of the L axis in the laa21 boratory system. The transformation leads t o T%m = TMMNN Aik Aim +1 (TMNMN + TMNNM) (Ail Akm + A i m A t.2) + 71 (TMMMN- TM N M M+ TMMNM T N M M M ) x n h+ + + (2.12) - (Ail A k [ nim Aimnk, A h n i l )9 1 T%n = - -2- ( T N M M M- T M M N M (Aiin,,+A,nim--din;nkl-Akmnil), ) 1 T%m = N TMNNM ( A) @ , 2 ( T M N M1 T%m == - Aklnim + (TMNMN - TMNNM ( A)i 4 X . m - A d i m ) . Aimnk.2 (2.13) - A k m n a ) (2.14) 3 (2.15) There is only one longitudinal contribution describing the scattering of the symnletricsymmetric type : Tzlm = TLLLLn%% nlnm* (2.16) Now we look a t the construction of non-zero transverse-longitudinal contributions of the type: TMMLL, TMLML> TMLNL. (2.17) &polarized Light Scattering from Paramagnetic Liquids 183 Taking into consideration that reflections in vector n plane are forbidden, we get the following representation of tensors Tan,TEa,T w and Tm : 1 T%m + =T(TLLMi¶l T M M L L ) (na.nkAlm 1 2(TL&MM + nlnmAik) - nlnmAik) - TMMLL) (%ik f + -1 4(TLMLM + T L ~ M+LTMLLM+ TMLML) x (AGnkm fA k p i m + + Akmnil) 1 + 4(TLMLM+ TMLLN+ TLMNL+ TMLNL) + x (n&nkm T%m = + Atcmn~d Aimnknl- 1 + 4( T L M L M -AimnPi nknl%m f I + (TLMLN TMLNL - TMLLN- T L M N L ~ - na.nmnkl - nknmnil) + TMNLLna.nkndnm* T M L M L - T M L L M - T L M M L ) (Ailnknm (2.20) - Akln%.nm 1 +A~mnind+ 4 (TLMLN -I- TLMML - TMLNL - TMLNL) x ( n i n l n k m - nknlnim + nn.nmnkl - nknmnil). (2.21) Summing up the results we see that the scattering tensor ofthe symmetric-symmetric type in a magnetic field is defined by eight independent contributions, and the tensors TEa, Tadand Taaby four and three independent contributions.. I n the isotropic case (in the absence of the field) the TEE tensor is described [9] by five independent functions, Taaby one function, TEaand Tadare equal to zero. Some coefficient functions a s T A N A M , for instance, differ from zero only in the presence of the field. Now let us discuss the field dependence of the functions. Rotate the N and L axes around the M axis a t 180°, and simultaneously reverse the direction of the external field [6] N 4 - N , L 4 -L, H --f -H. The system is invariant under this transformation. The coefficient functions transform either as (2.22) N. P. MALOMUSH,S. D. LATUSKKIN, and K.-P. GXUNDE~ 184 Thus the coefficient functions of a n even number of indices N and L are even in the field, and the functions having an odd number of indices N and L are odd in the field. The induced changes in T M M N N , T M N M N , T M x L L , T M L M L , T A L L 5 and the like T M L N L , T L L M N type functions type functions are quadratic in the field. The TMM", are linear in the field and vanish with it. Note the phenomenological character of the analysis which we conclude by showing it in the explicit form of the scattering cross section of ss and aa type. By choosing the polarization vectors el and e2 in the form : - (k, - Bk,) sin% 1-Vl-p 2 e - 1 e2= 1! where -BZ [(k,- &) sin + [klkz]ekl wz2 + [klk2]eka cos Yz-,2 (2.23) (2.24) j3 = cos e, describing (by appropriate selection of angles yl,yz,ql, y2, 0 ) every possible experimental geometry, we get for the scattering cross sections : Depolarized Light Scattering from Paramagnetic Liquids 185 Siniilar relations may be written for mixed type cross sections. To determine the nineteen coefficient functions T it is possible t o point out a corresponding number of independent experiments but we shall not do it for the lack of space. We wish t o proceed t o the models of dielectric tensor fluctuation correlationfunctions. I n a phenomenological approach is described by linear relations : (2.27) k k = n%rn@ltn n g m A l m , A h = e l d (s - la),, where alm is the fluctuating symmetric stress tensor in liquids, S is the density of spin, I is the average density of inertia moments, and 17:ilm, are tensors of optoelastical constants. The latter depend on the external field and, like the scattering tensor, form a representation of the rotational group. I n the absence of the field tensor ll:Elmvanishes, and the tensor I&, is determined by only two independent contributions. This is not true for the scattering tensor since it depends both on the field and on the wave number k. To determine the order of magnitude of different contributions t o nPb, let us write it as follows: + + aC ~ M M M N + I 6klnim (dil12km + ~ M M N M- f 6kmniZ)* -~MNMM -k ~NMMMI (2.28) I n the absence of the field the system is isotropic and therefore in (2.28) only the first two terms differ from zero. The coefficients of the n M M M N type are odd in the field. In the weak field the appearance of the dipole magnetic moment results in additional physical effects. According t o Curie’s principle dipole, quadrupole etc. effects and fourth rank tensors of weights 0, 1, 2, etc. we write the third term in (2.28) as follows: (2.29) Thus several first terms of tensors are n%m = nl + 8ik8Zm + n3 (8ilHkm n2(8ildkm f 8imSkl - + 8klHim + + dhHkl 2 + 8ikdZm) &mHiJ), (2.30) (2.31) n%m = pl(dilHkm - dklHim dimH, - d / c m H d The cumbersome relations linking the coefficient functions T with correlation functions of hydrodynamic variables will not be presented here, with the exception of one case given below. It is important to note the following: The VH cross section is determined by the relation Ordinarily the hydrodynamic stress tensor does not contain transverse parts. Therefore the task t o determine the coefficient functions T x N M N with the help of a hydrodynamic model remains to be settled. 13 Ann. Pbyaik. 7. Folge, Bd. 38 N. P. MALOMUSH, S. D. h m - s m , and K.-P. G R ~ N D E R 186 3. Hydrodynamic Model To determine the spectral density correlations responsible for light scattering we will make use of the hydrodynamic model of the paramagnetic liquid with symnietric stress tensor. Some works [4] interprete the appearance of the doublet structure in terms of internal rotational motion with the help of a hydrodynamic model with antisymmetric stress tensor. This approach is, however, not correct since it does not lead t o total angular momentum conservation. This difficulty was previously pointed out [lo, 111. Recently El21 we have obtained hydrodynamic equations for paramagnetic liquids with the syinmetric stress tensor. The basic variables of the “symmetric” fornialisni are linear inomentum density and spin density. On the basis of the corresponding with the niicroscopic model of noninteracting particles with spin 1/2 (Pauli’s model) it has been shown that the densities of linear momentum, spin S and the velocity of transational iiioveinent v of molecular mass centres, are connected by the relation: p = ev 1 + -rot 2 S, (3.1) where p is the mass density. As was shown [12] the symmetric stress tensor in the absence of the field is equal to - 5 Qik div-,21 e (3.2) where 17 is the pressure, 17, E are coefficients of shear and hulk viscosities. In the external field the stress tensor is added by another tensor which is a close analogue of the Maxwell magnetic stress tensor Here H is the inagnetic field vector, 9 is the angular velocity of vortex movement, M = IS is the magnetization density, A is the gyromagnetic ration, x is the susceptibilitg, x is a coefficient of the order of unit (within the phenomenological approach it is impossible t o calculate it exactly). Then we write the eqnation of change for the density of linear monientuin where x i = (x,y} and where we ignore the Senftlebeneffect. And for the magnetization densit 5(3.5) I n (3.5) the first term describes magnetization density precession, the second term describes relaxation, the linear momentum field influence on magnetization is described in term three. The latter is conjugated t o Maxwell stresses. Barnett’s contribution 1 to magnetization may be disregarded. I n order t o obtain natiiral frequencies (dispersion relation) one should introduce Fourier transformations and respective boundary conditions. I n the limit vk% 1 < Depolarized Light Scattering from Paramagnetic Liquids 187 the dispersion relations sought for the transverse components of linear momentum and magnetization are (-icc,)1,2 1 =- + ( ~ i 2 HicElzxzz)kz, )~ (3.6) The relation (3.6) shows the effectively graving imaginary parts of naturalfrequencies. Frequencies themselves are somewhat shifted. I n (3.7) the shift is considerable. The relation (3.7) shows effectively graving imaginary parts of magnetization field natural frequencies. Frequencies themselves are somewhat shifted. We may assume estimates of coefficients in (3.6), (3.7) equal t o ts, v10-2cni2/s, k 105 cm-l, AN lo7, B N 10T (strong fields). Estimates show that in strong fields dissipative and dynamic additions in (3.6) and (3.7) prove t o be small (2 t o 3 per cent). Thus further on we shall write dispersion relations in the form N 1 ( - i ~ )= ~, ~ f iAB, z with ,G = 12B/2eA and where N shows corresponding renormalization of coefficients. U*e are interested in the following correlation spectral densities (obtained with the help of FDT). With the dissipative shear mode approximation we get 1 t <AMA’(k,t ) AILIN(-k, 0 ) ) ~= kRT1 w2 (3-9) 1 ’ +F Yk4 1 (3.10) + ($k2)2 ’ i (cc, + jik2B)2 (3.11) [=’ (w + 1 AB)= +1 1 + (W - /B)’ +- where k,is the Boltzmann constant, T the absolute temperature. The magnetic field leads both t o the increase of width of dissipature and relaxation lines (3.9), (3.10), and t o the appearance of new ones (3.12). Note that an effective increase of viscosity in a magnetic field and a contour shift in (3.11) lead to line broadening, which for typical paramagnetics may account for up t o ten per cent of the line width. In Section 4 we discuss the relevant spectroscopical consequences. 13 N. P. MALOMUSH, S. D. LATUSHKIX, and K.-P. GRCXDER 188 4. Spectral Lineshape Changes in a Magnetic Field Changes in the character of both transverse-longitudinal diffusive and propagating and relaxation modes of paramagnetic liquids in a magnetic field lead to corresponding alterations in depolarized light scattering spectra. Here we treat liquids like 0,,NO, being almost optically isotropic, which, when modeling dielectric tensor fluctuations, enable US t o consider the latter t o be the consequence of stresses and local internal rotations, only. From (2.27) type relations in the linear field approximation (4.1) d& = e,ofk c2[AjBk AkBi - 2nink BA], and && = c,Aik c,[AiBk - A k B i ] , A i k = eiklAl, (4.2) where vector A is defined by the relation A = S - 152. In (4.1), (4.2) we just' consider traceless combinations since further on our interest is turned to depolarized light scattering only. Unlike the natural optical activity phenomenon, the order of magnitude of a the antisymmetric fluctuation tensor (4.2) is defined by the ratio -where a is the size + + + a of a "Lagrange particle", but not of a inolecule as it is usually the case. The typical size of a "Lagrange particle" (with correlated spin direction) as it was estimated in paper [13] is about (where t Mis the Maxwell stress tensor relaxation time) + a . cm. 10-7 cm numerically. The typical value of -lies within the limits of 10-1 + l/GI a For the example of V H scattering we shall show the controllable depolarized light scattering spectral changes that may be expected in a magnetic field. The total scattering cross section is determined by the following two contributions: with different angular dependencies. It follows from (4.1)- (4.3)and (3.2)that symmetricsymmetric type contributions conditioned by diffusive and propagating inodes differ from zero only in T M L M L , whilst antisymmetric-antisymmetric type contributions are present in both terms. The intensity of light scattered from transverse-longitudinal diffusive modes is described by As a result from the preceding Section there must be observed (in a magnetic field) an increase of the lineshape width by a specific precession frequency (see Fig. 2) Fig. 2. The broadening of M?,(w)-line Depolarized Light Scattering from Paramagnetic Liquids 189 The relative increase of the lineshape width is expected t o be equal t o Pk -N 0,l. V +- The latter holds good for low viscosity liquids i; 0,Ol 0,l cm2/sput in a magnetic field of B 10T [14]. It must be stressed that the increase of the lineshape widths due to transverselongitudinal diffusive modes must be linear in the field. We expect that in strong enough fields these effects may be detected. By introducing the relaxing viscosity N N - 10l2Hz) processes may be achieved a more precise description of high frequency (co The low frequency diffusive regime will be substituted by transverse hypersound proPagitt’1011. On sufficiently viscous paramagnetic liquids with the typical k lo5 cm-l as it is easily t o be seen, that the hypersound wave “dies out’’ on much smaller distances than the sound wave lengh. I n other words, the diffusive line may be traced in light scattering spectra of low-viscosity liquids. Highly viscous paramagnetic liquids (if any) must be treated seperately. Let us discuss the influence of the external magnetic field on the parts of the spectral lines caused by rotational degrees of freedom. To be more precise, we shall trest the influence of the field on antisymmetric-antisymmetric scattering. As has been noted above the latter is nothing else but fluctuation optical activity. The activity is due t o the “packing effect” characteristics of paramagnetic liquids, that is local order of magnetic moments. The phenomenon of “optical packing activity” is familiar in liquid crystal physics [15]. The ratio of the antisymmetric part of the dielectric tensor fluctuation t o the syma metric one si-, (10-1 f Thus the scattering intensity of aboutisonly A ae A2 a small percentage of total scattering intensity. This problem is treated quite differently by [4]. The first term (4.2) leads t o optical fluctuation activity due t o the absence of the balance between spin density and translational angular momentum (visible rotation). The second term in (4.2) appears only in the external field and describes Faraday’s fluctuation effect. This effect disappears when scattering cross section is calculated in the linear approximation. I n the magnetic field the rotational contribution is changed by magnetic moment precession. According to (3.12) we get N ( 1 w + ,GkzB)2+ ($k2)2 - 1 (o- jikzB)a - + (G??)2 1 (4.7) The precession frequency shift do AB in a field of B 10T may be compared with the half-width of rotational Lorentzian. I n aa scattering cross section there appears a doublet induced by the field. If au scattering intensity may be experimentally registered in V H spectra, the doublet may cause the lowering of the diffusive spectral peak (see Fig. 3). N. P.MALOMUSH, 8.D. LATUSHCINI, 190 and K.-P. GRUTNDER -1 8 XB uFig. 3. MYH line. 1. H = 0. Diffusive and relaxational Lorentzians. 2. H + 0. The relaxatianal Lorentzian shift is comparable with the lowering of the broadening diffusive peak Considering aa scattering we neglected the contribution described by the first term in (4.3) 1 e r e TMNMN sin2 =. ciEBT sin2 (4.8) 2 1 2 - 09 +?2 since it is not influenced by the field although this is the only nonvanishing Contribution to back scattering (0 = n). Thus by combined analysis of field and angular dependencies of depolarized VH scattering cross section it is possible to get sufficiently reliable information about the types of molecular motions, generalizing various features of special lineshapes. It must be stressed that the results of the investigation are obtained for paramagnetic liquids of low viscosity with almost optically isotropic molecules, and the method is applicable for a more general type of research. 6. Conclusions This paper deals with the influence of a magnetic field on depolarized light scattering spectra of paramagnetic liquids. It is considered as a method of systematized testing of molecular motions generating various spectral line features. From symmetry considerationthere have been obtained: the structure of the scattering tensor, the number of independent functions desciibing scattering cross section of various types (88, sa,a,m).These results hold good both for paramagnetic and diamagnenetic liquids. For the example of depolarized VH scattering it has been demonstrated what types of spectral peculiarities may be expected from the influence of a magnetic field on the system. References [l] V. S. STABUNOV, E. V. T r o ~ ~ o and v I. L. FABELINSKY, Zh.Exp. Teor. Fis. Pisma 6,317 (1967). [2] H. C. CBADDOCK, D. A. JACKSON, P. G. POWLES, Mol. Phys. 18, 606 (1968). [3] L. A. ZWBKOV, N. B. ROZFIDESTVENSKAYA, A. S. CEROMOV,Zh. Exp. Teor. Fis. Pismn 11, 173 (1970). [a] N.K. AILAWADI, B. J. BERNE, D. FORSTER, Phys. Rev. A 8, 1172 (1971). [6]T. KEYES,D. KJYELBON, J. Chem. Phys. 84, 1786 (1971). [S] S. R. DE GROOTand P. MAZUR, Non-equilibrium Thermodynamics, North-Holland, Amsterdam 1962. Depolarized Light Scattering from Paramagnetic Liquids 191 [7] s. cEANDRsSEKH.4R, Philos. Trans. A 242, A 866, 667 (1960). [8] A. S. MONLV,9.31. JAGLON, Statistical Hydromechanics, part 11,Nauka, Uoscoxv 1967. 193 B.Y.SELDOVICFI, Zh. Exp. Teor. Fiz. 68, 76 (1972). [lo] I. A. X c L m m , Physica 32, 689 (196G). [ll] P. C. NARTN, P. S. PEVESCHILN, Phys. Rev. A 8, 2401 (1972). N. P. MALOMUSH, Mater. I1 Vsesojuz. Symp. po Akustich. Spectroecop., 4 Izd. [12] S.D.LATUSHKIN, Fan. UZSSR.Tashkent 1978. [13] I. Z.FISHER, Zh. Exp. Teor. Pi. 61,1647 (1971). [14] S. HEW,Xaturforschung We, 111 (1976). [15] P.G. DE GENNES,The Physics of Liquid Crystals, Oxford 1974. Bei der Redaktion eingegangen am 7. &hi 1980. Anschr. d. Verf. : Dr. N. P. NALOXUSH und Dip1.-Phys. S. D. h m s m Sektion Physik der Strtatliohen Universitiit Odessa UdSSR-270012Odessa, ul. Paatera 42 D,pl.-Phys. K.-P. G R ~ D E B D DR-1170Berlin, Pestrtlozzi-Str. 18

1/--страниц