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Semiclassical Analysis of the Extrema in the Velocity Dependence of Total Elastic‐Scattering Cross Sections: Relation to the Bound States
dc.contributor.authorBernstein, Richard B.en_US
dc.date.accessioned2010-05-06T22:53:20Z
dc.date.available2010-05-06T22:53:20Z
dc.date.issued1963-06-01en_US
dc.identifier.citationBernstein, Richard B. (1963). "Semiclassical Analysis of the Extrema in the Velocity Dependence of Total Elastic‐Scattering Cross Sections: Relation to the Bound States." The Journal of Chemical Physics 38(11): 2599-2609. <http://hdl.handle.net/2027.42/70899>en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/70899
dc.description.abstractThe phenomenon of extrema in the velocity dependence of the total elastic cross section Q(v) for atom—atom scattering in the thermal‐energy region is shown to be a quite general one, whenever the interaction potential consists of both attractive and repulsive parts and the resulting well has a ``capacity'' for one or more discrete levels. The phase shift vs angular‐momentum dependence exhibits a maximum; since this maximum is a function of the de Broglie wavelength, the cross section exhibits an undulatory velocity dependence. A semiclassical analysis of the extrema velocities (and undulation amplitudes) is presented. Suitable plots are suggested from which one may deduce certain information on the interatomic potential and the diatom bound states. The following rule is proposed: the observation of m maxima in the elastic atom—atom impact spectrum implies the existence of at least m discrete vibrational levels of zero angular momentum for the diatom.en_US
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dc.publisherThe American Institute of Physicsen_US
dc.rights© The American Institute of Physicsen_US
dc.titleSemiclassical Analysis of the Extrema in the Velocity Dependence of Total Elastic‐Scattering Cross Sections: Relation to the Bound Statesen_US
dc.typeArticleen_US
dc.subject.hlbsecondlevelPhysicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumChemistry Department, University of Michigan, Ann Arboren_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/70899/2/JCPSA6-38-11-2599-1.pdf
dc.identifier.doi10.1063/1.1733558en_US
dc.identifier.sourceThe Journal of Chemical Physicsen_US
dc.identifier.citedreferenceR. B. Bernstein, J. Chem. Phys. 37, 1880 (1962).en_US
dc.identifier.citedreferenceSee, for example, (a) R. B. Bernstein, J. Chem. Phys. 36, 1403 (1962); also the basic paper: (b) K. W. Ford and J. A. Wheeler, Ann. Phys. (N.Y.) 7, 259, 287 (1959).en_US
dc.identifier.citedreferenceIn the present paper attention is restricted to collisions between unlike atoms, in which one is in the 1S01S0 state (i.e., an atom belonging to Group II or VIII) while the other may be either 1S01S0 (Group II) or 2S½2S12 (Group I), yielding the single molecular state 1∑+1∑+ or 2∑+,2∑+, respectively. Extension is straightforward to the case of two 2S½2S12 atoms, yielding both 1∑+1∑+ and 3∑+3∑+ molecular states (the singlet state with a relatively deep “binding” well, the triplet with only a shallow “van der Waals” well). However, the analysis becomes cumbersome in the general case for scattering of state‐unselected beams. See the Appendix for an enumeration of the possible molecular electronic states. Where the colliding atoms are identical, the following modifications are required: (a) for spinless atoms, only doubly weighted even‐ or odd‐order phases are to be used in the summation for Q according as the atoms are bosons or fermions, respectively. This has the effect of halving the necessary number of phases (at any given collision energy) but at the same time making for poorer “statistics” in the semiclassical treatment. (b) for atoms with spin, proper weighting (according to the multiplicity of the molecular state) of the above‐mentioned even‐ or odd‐type cross‐section sums is required. Normally, different potentials are used for each molecular state, so that Pauli exclusion is automatically taken into account [see reference 17(a)].en_US
dc.identifier.citedreferenceR. B. Bernstein, (a) J. Chem. Phys. 33, 795 (1960); (b) 34, 361 (1961); (c) 38, 515 (1963); re Appendix I, an important related paper by E. M. Baroody, Phys. Fluids 5, 925 (1962), had been overlooked.en_US
dc.identifier.citedreferenceH. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc. (London) A144, 188 (1934).en_US
dc.identifier.citedreferenceThe question of the absolute accuracy of the over‐all MM approximation treatment is discussed in another paper: R. B. Bernstein and K. H. Kramer, J. Chem. Phys. 38, 2507 (1963).en_US
dc.identifier.citedreferenceJ. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954).en_US
dc.identifier.citedreferenceE. W. Rothe, P. K. Rol, and R. B. Bernstein, Phys. Rev. (to be published).en_US
dc.identifier.citedreferenceOne source of the inaccuracy may be in the use of the parabolic approximation for η(β)η(β) near ηm;ηm; the range β2−β1β2−β1 was found to be slightly greater than that for which the assumption is valid. Also, there is some uncertainty in the factor G(K)G(K) whose evaluation requires numerical differentiation of θ(K),θ(K), to yield θ0′.θ0′.en_US
dc.identifier.citedreferenceIt is interesting to note that, in addition to definite observations of such extrema in the case of atom‐atom collisions11,12, there is the possibility that small undulations appearing on graphs of Q(υ)Q(υ) for certain charge‐changing ion‐atom (and ion‐molecule) collisions13 may originate from similar considerations.en_US
dc.identifier.citedreferenceH. U. Hostettler and R. B. Bernstein, Phys. Rev. Letters 5, 318 (1960).en_US
dc.identifier.citedreference(a) E. W. Rothe, P. K. Rol, S. M. Trujillo, and R. H. Neynaber, Phys. Rev. 128, 659 (1962); (b) P. K. Rol and E. W. Rothe, Phys. Rev. Letters 9, 494 (1962).en_US
dc.identifier.citedreferenceSee, for example, E. A. Mason and J. T. Vanderslice, J. Chem. Phys. 29, 361 (1958) and references cited therein.en_US
dc.identifier.citedreference(a) See, for example, H. Harrison and R. B. Bernstein, J. Chem. Phys. 38, 2135 (1963). The analogous nuclear problem has been studied by (b) R. S. Caswell, National Bureau of Standards Technical Note No. 159 (1962), and (c) A. E. S. Green, Phys. Rev. 99, 772, 1410 (1955).en_US
dc.identifier.citedreferenceHowever, note that N. Bernardes and H. Primakoff [J. Chem. Phys. 30, 691 (1959)] have suggested the possibility of observing the Raman spectra of certain van der Waals molecules (rare‐gas dimers) in the liquid state.en_US
dc.identifier.citedreferenceBefore proceeding, it should be pointed out, in all fairness, that the number of bound states is completely determined from the potential, so that no new information can come from the extrema‐counting procedure. For example, for the LJ (12,6) potential the number (n0n0) of vibrational states is a function only of the single parameter B.14a For approximation purposes the following simplified formula can be used: n0−½  =  0.27 B½;n0−12=0.27B12; this equation predicts n0n0 within ±1.±1. More exact formulas, for various potentials of interest, are given in reference 14 (a).en_US
dc.identifier.citedreference(a) P. Swan, Proc. Roy. Soc. (London) A228, 10 (1955). Pauli‐excluded states are not involved here; (b) N. Levinson, Kl. Danske Videnskab. Selskab. Mat.‐Fys. Medd. 25, No. 9 (1949).en_US
dc.identifier.citedreferenceComputations were carried out by the “exact” numerical integration (Runge‐Kutta‐Gill) procedure of reference 4(a), slightly modified to deal with the very small values of A.en_US
dc.identifier.citedreferenceSee R. A. Buckingham and J. W. Fox, Proc. Roy. Soc. (London) A267, 102 (1962) for similar results for a square‐well potential bounded by the long‐range r−6r−6 attraction.en_US
dc.identifier.citedreferenceFrom a practical point of view, of course, the point is a moot one, since experimental limitations preclude measurements down to the very low velocity range under discussion.en_US
dc.identifier.citedreferenceFor an LJ (12,6) well of high capacity (e.g., B>2000B>2000), it is possible to develop quantitative relations such as the following: ΔE0→1≡hν≅10.7ϵ∕B12;E1=12μν12≅0.0581ϵB, so that ΔE0→1/E1 ≅ 184 ,ΔE0→1∕E1≅184B−12, where ν1ν1 is the velocity of the N  =  lN=l extremum and ν is the classical fundamental frequency of the diatom (calculated from the curvature of the well at the minimum). Thus for B  =  2000,B=2000, the ratio ΔE0→1/E1ΔE0→1∕E1 is about 0.2%, i.e., the collision energy at the N  =  1N=1 maximum is about 500 times the ν  =  0→1ν=0→1 excitation energy for the diatom. Also, by combining (the very approximate) Eq. (23c) with the relation given in footnote 16, one may estimate directly from the undulation amplitude the number of bound states: n0 ≈ 0.73/U.n0≈0.73∕U.en_US
dc.identifier.citedreferenceH. Pauly [Z. Naturforsch 15a, 277 (1960)] examined the K�N2K�N2 system (high B?) and did not resolve extrema. However, unexplained (and quite possibly unrelated) undulations appear in some of the Q(T)Q(T) curves for CsCl scattering, by H. Schumacher, R. B. Bernstein, and E. W. Rothe, J. Chem. Phys. 33, 584 (1960).en_US
dc.identifier.citedreference(a) R. B. Bernstein, Bull. Am. Phys. Soc. 7, 217 (1962); (b) R. B. Bernstein, A. Dalgarno, H. S. W. Massey, and I. C. Percival, Proc. Roy. Soc. (London) (to be published).en_US
dc.identifier.citedreferenceG. Herzberg, Molecular Spectra and Molecular Structure. I. Diatomic Molecules (D. Van Nostrand, Inc., New York, 1950), 2nd Ed.en_US
dc.owningcollnamePhysics, Department of


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