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Multicomponent Cluster States in Dilute Mixed Molecular Crystals, with Application to 1B2u Naphthalene Excitons
dc.contributor.authorHong, Hwei‐kwanen_US
dc.contributor.authorKopelman, Raoulen_US
dc.date.accessioned2010-05-06T23:12:41Z
dc.date.available2010-05-06T23:12:41Z
dc.date.issued1972-11-01en_US
dc.identifier.citationHong, Hwei‐Kwan; Kopelman, Raoul (1972). "Multicomponent Cluster States in Dilute Mixed Molecular Crystals, with Application to 1B2u Naphthalene Excitons." The Journal of Chemical Physics 57(9): 3888-3898. <http://hdl.handle.net/2027.42/71104>en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/71104
dc.description.abstractWe develop the theory of Frenkel excitons for multicomponent cluster states in medium‐dilute mixed molecular crystals. This theory can be applied to both secondary host traps induced by a single impurity and to multicompositional, chemical, or isotopic clusters. A Green's function technique, which is a generalization of the Koster‐Slater method, is developed and utilized. Symmetry properties of such clusters are discussed, with emphasis on interchange equivalent sites in nonsymmorphic crystals. The optical spectra of naphthalene‐h8 in naphthalene‐d8 can now be further analyzed, with the help of some numerical calculations on multicomponent cluster states. Using our recently acquired dispersion relation for the 1B2u naphthalene exciton state we fit satisfactorily both the fine structure and the ``hyperfine'' structure, without any additional parameters, except for the experimentally known trap‐depths of a few isotopic impurities. This corroborates both the importance of the exciton superexchange effect and the validity of the exciton dispersion formula.en_US
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dc.publisherThe American Institute of Physicsen_US
dc.rights© The American Institute of Physicsen_US
dc.titleMulticomponent Cluster States in Dilute Mixed Molecular Crystals, with Application to 1B2u Naphthalene Excitonsen_US
dc.typeArticleen_US
dc.subject.hlbsecondlevelPhysicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumDepartment of Chemistry, The University of Michigan, Ann Arbor, Michigan 48104en_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/71104/2/JCPSA6-57-9-3888-1.pdf
dc.identifier.doi10.1063/1.1678859en_US
dc.identifier.sourceThe Journal of Chemical Physicsen_US
dc.identifier.citedreference(a) J. Frenkel, Phys. Rev. 37, 17, 1276 (1931); (b) A. S. Davydov, Theory of Molecular Excitons (McGraw‐Hill, New York, 1962); (c) A. S. Davydov, Usp. Fiz. Nauk 82, 393 (1964) [Sov. Phys. Usp. 7, 145 (1964)]; (d) A. S. Davydov, Theory of Molecular Excitons (Plenum, New York, 1971).en_US
dc.identifier.citedreferenceFor example, if both guest A and guest B are present in small amounts, the concentration of clusters, consisting of A plus B, would be CACB,CACB, which is smaller than either CACA or CBCB in such limit.en_US
dc.identifier.citedreferenceJ. Hoshen has discussed this problem within the coherent potential approximation. See J. Hoshen, Ph.D. thesis, Tel‐Aviv University, 1971.en_US
dc.identifier.citedreferenceSince the impurity states inside the host band (or virtual states) are generally broad, no such states have been identified experimentally in dilute mixed crystals [however, see J. C. Laufer and R. Kopelman, J. Chem. Phys. 57, 3202 (1972)]. Theoretical treatments, however, are quite abundant in the literature. See, for example, B. S. Sommer and J. Jortner, J. Chem. Phys. 50, 187, 822 (1969); Y. A. Izyumov, Advan. Phys. 14, 569 (1965). A. Shibatani and Y. Toyozawa, J. Phys. Soc. Japan 25, 335 (1968). In order to detect virtual states at low impurity concentration, an impurity with much larger oscillator strength than the host is called for. In other words, chemically mixed crystals rather than isotopically mixed crystals are more amenable to such an investigation. Mixed crystals with large concentration of very shallow traps have been studied by H. K. Hong and G. W. Robinson [J. Chem. Phys. 52, 825 (1970); 54, 1369 (1971)]; however, it seems no longer proper to refer to the guest states as virtual states.en_US
dc.identifier.citedreference(a) D. M. Hanson, J. Chem. Phys. 52, 3409 (1970). (b) H. K. Hong and R. Kopelman, Phys. Rev. Letters 25, 1030 (1970). (c) H. K. Hong and R. Kopelman, J. Chem. Phys. 55, 724 (1971).en_US
dc.identifier.citedreference(a) D. S. McClure and E. F. Zalewski, Molecular Luminescence, edited by E. C. Lim (Benjamin, New York, 1969). (b) H. K. Hong (unpublished).en_US
dc.identifier.citedreferenceH. K. Hong and R. Kopelman, J. Chem. Phys. 55, 5380 (1971).en_US
dc.identifier.citedreferenceR. Kopelman, J. Chem. Phys. 47, 2631, 3227 (1967).en_US
dc.identifier.citedreferenceWe have adopted the free molecule approach here because of its simplicity. The merit of such an approach will become clear after Sec. III. In the more rigorous site approach the ϕsϕs would be the site functions and could be different for the same molecular species at different environments.en_US
dc.identifier.citedreferenceThe crystal Hamiltonian takes up different forms as a consequence of the different choices of origins. One could compare Ref. 1(b) and Ref. 1(d) to see such changes. See also p. 42 of Ref. 1(d).en_US
dc.identifier.citedreferenceD. Craig and M. R. Philpott, (a) Proc. Roy. Soc. (London) A290, 583 (1966); (b) A290, 602 (1966); (c) A293, 213 (1966).en_US
dc.identifier.citedreferenceO. A. Dubovskii and Y. V. Konobeev, Fiz. Tverd. Tela 6, 2599 (1964) [Sov. Phys., Solid State 6, 2071 (1965)].en_US
dc.identifier.citedreferenceG. F. Koster and J. C. Slater, Phys. Rev. 95, 1167 (1954); 96, 1208 (1954); G. F. Koster, Phys. Rev. 95, 1436 (1954).en_US
dc.identifier.citedreferenceSee, for example, B. S. Sommer, and J. Jortner, J. Chem. Phys. 50, 187 (1969).en_US
dc.identifier.citedreferenceThere is mounting evidence that this it true. See, for example, D. M. Hanson, R. Kopelman, and G. W. Robinson, J. Chem. Phys. 51, 212 (1969).en_US
dc.identifier.citedreferenceS. D. Colson, J. Chem. Phys. 48, 3324 (1968).en_US
dc.identifier.citedreferenceG. Fischer, J. Chem. Phys. 57, 2646 (1972).en_US
dc.identifier.citedreferenceS. D. Colson, R. Kopelman, and G. W. Robinson, J. Chem. Phys. 47, 27, 5462 (1967); G. W. Robinson, Ann. Rev. Phys. Chem. 21, 429 (1970).en_US
dc.identifier.citedreferenceFor restricted Frenkel case, see Ref. 5(c); for general Frenkel case, see B. S. Sommer and J. Jortner, J. Chem. Phys. 51, 5559 (1969).en_US
dc.identifier.citedreferenceFor time‐reversal symmetry, see L. P. Bouckaert, R. Smoluchowski, and E. Wigner, Phys. Rev. 50, 58 (1936). Notice that Gnα,mβ=N−1∑μ∑k×{Bμα(k)Bμβ∗(k)exp[ik⋅(Rnα−Rmβ)]∕[E−Eμ(k)]}=N−1∑μ∑k̄×(2Re{Bμα(k̄)Bμβ∗(k̄)exp[ik̄⋅(Rnα−Rmβ)]}∕[E−Eμ(k̄)]) where Re is the real part and k̄ only runs through one half of the Brillouin zone. When the crystal is centrosymmetric, the Bμα(k)sBμα(k)s are all real and hence Gnα,mβ=N−1∑μ∑k̄2Bμα(k̄)Bμβ(k̄)cosk̄⋅(Rnα−Rmβ)E−Eμ(k̄)=N−1∑μ∑k̄2Bμβ(k̄)Bμα(k̄)cosk̄⋅(Rmβ−Rnα)E−Eμ(k̄)≡Gmβ,nα. For naphthalene, within the restricted Frankel limit ∣Bμα(k)∣=∣Bμβ(k)∣=1∕2 for all k and μ.en_US
dc.identifier.citedreferenceD. S. McClure, J. Chem. Phys., 24, 1 (1956).en_US
dc.identifier.citedreferenceSee for example, G. Herzberg, Molecular Spectra and Molecular Structure, II. Infrared and Raman Spectra of Polyatomic Molecules (Van Nostrand, Princeton, N.J., 1945), p. 216 on Fermi resonance.en_US
dc.identifier.citedreferenceWe have used the C2C2 interchange convention here. For more discussion on the interchange convention, see Ref. 8.en_US
dc.identifier.citedreferenceA crude estimation of the concentrations of 13C13C mixed pairs and D7D7 perturbed dimers can be made, if we assume that the cluster states are essentially determined by the local configurations. If we take the D7D7 concentration to be 10%, we find that, with 0.65% of C10H8C10H8 the concentrations of 13C13C mixed a and b pairs are 1.66×10−51.66×10−5 whereas the corresponding D7D7 perturbed dimers are 1.52×10−5.1.52×10−5. These estimates are crude because the D7D7 concentrations vary and also because we ignore the effect of nonnearest neighbors on the cluster states.en_US
dc.identifier.citedreferenceAgain, if we use the same crude estimation as in Ref. 24 we find that 29% of the monomers are perturbed by the presence of a simgle D7D7 molecule whereas 66% of them are unperturbed, assuming we have 10% D7.D7.en_US
dc.identifier.citedreferenceSee Ref. 8 and also E. R. Bernstein, S. D. Colson, R. Kopelman, and G. W. Robinson, J. Chem. Phys. 48, 4832 (1968).en_US
dc.identifier.citedreferenceIn terms of absolute concentrations, we find that, with 1.4% C10H8,C10H8, we have 2.78×10−42.78×10−4 perturbed interchange dimers (including r3  =  a,br3=a,b and a+ba+b) which is 35% of the total interchange pairs, whereas the concentration of 13C13C mixed pairs is only 1.72×10−4.1.72×10−4.en_US
dc.identifier.citedreferenceS. D. Colson, D. M. Hanson, R. Kopelman and G. W. Robinson, J. Chem. Phys. 48, 2215 (1968).en_US
dc.identifier.citedreferenceFor a rather complete discussion on the determination of trap depths, see Ref. 15., which also includes an exhaustive tabulation of experimental data by various workers.en_US
dc.owningcollnamePhysics, Department of


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