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Remarks on the Signs of G Factors in Atomic and Molecular Zeeman Spectroscopy

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Remarks on the Signs of G Factors in Atomic and Molecular Zeeman Spectroscopy MOLECULAR PHYSICS, 2000, VOL. 98, NO. 20, 1597± 1601 Editors’ Note This paper draws attention to the advantages that would be obtained by adopting a new convention for the sign of g factors that would make the g factor for electron spin a negative quantity (g 2), rather than a positive quantity as generally adopted at present. The editorsº ¡ are aware that the proposal made in this paper concerning the conventional sign of the g factor for electron spin will be seen by some readers as controversial. We have nonetheless agreed to publish this paper in the hope that it will stimulate discussion. The editors would welcome comments on this proposal in the form of short papers, which they will then be happy to consider for publication together at a later date. Remarks on the signs of g factors in atomic and molecular Zeeman spectroscopy J. M. BROWN1, R. J. BUENKER2, A. CARRINGTON3, C. DI LAURO4, R. N. DIXON5, R. W. FIELD6, J. T. HOUGEN7, W. HUÈ TTNER8*, K. KUCHITSU9, M. MEHRING10, A. J. MERER11, T. A. MILLER12, M. QUACK13, D. A. RAMSAY14, L. VESETH15 and R. N. ZARE16 1 Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, UK 2 Bergische UniversitaÈ t-Gesamthochschule Wuppertal, Theoretische Chemie, Gauû straû e 20, D-42097 Wuppertal, Germany 3 Department of Chemistry, University of Southampton, Southampton SO9 5NH, UK 4 Universita degli Studi Federico II, Dipartimento di Chimica Farmaceutica e Tossicologica, Via Domenico Montesano 49, I-80131 Napoli, Italy 5 School of Chemistry, University of Bristol, Bristol BS8 1TS, UK 6 Department of Chemistry, MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA 7 Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 8 Abteilung Chemische Physik, UniversitaÈ t Ulm, D-89069 Ulm, Germany 9 Department of Chemistry, Josai University, Keyakidai, Sakado 350-0295, Japan 10 II. Physikalisches Institut, UniversitaÈ t Stuttgart, Pfa enwaldring 57, D-70550 Stuttgart, Germany 11 Department of Chemistry, The University of British Columbia, 2036 Main Mall, Vancouver BC, Canada V6T 1ZI 12 Department of Chemistry, The Ohio State University, Newman and Wolfrom Laboratory, 100 West 18th Avenue, Columbus, OH 43210, USA 13 Laboratorium fuÈ r Physikalische Chemie, ETH ZuÈ rich (Zentrum), CH-8092 ZuÈ rich, Switzerland 14 Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6 15 Department of Physics, University of Oslo, 0316 Oslo 3, Norway 16 Department of Chemistry, Stanford University, Stanford, CA 94305-5080, USA * Author for correspondence. e-mail: [email protected] Molecular Physics ISSN 0026± 8976 print/ISSN 1362± 3028 online # 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals 1598 J. M. Brown et al. (Received 1 March 1999 ; revised version accepted 6 June 2000 ) Various magnetic moments, associated with rotational, vibrational, nuclear spin, electron orbital and electron spin angular momenta, can contribute to the Zeeman e ect in atoms and molecules. They are considered in this paper in the context of the e ective Hamiltonian where relativistic and other corrections as well as the e ects of mixing with other electronic states are absorbed in appropriate g factors. In spherically symmetric systems, the magnetic dipole moment arising from a speci® c angular momentum can be written as the product of three factors: the nuclear or Bohr magneton (which is positive), the g factor (which may be positive or negative), and the corresponding angular momentum (which is a vector). A con- vention is discussed, in which the sign of the g factor is positive when the dipole moment is parallel to its angular momentum and negative when it is antiparallel. This would have the advantage that it could be applied consistently in any situation. Such a choice would require the g factors for the electron orbital and electron spin angular momenta to be negative. This concept can easily be extended to the case of a general molecule where the relation between the dipole and angular momentum vectors has tensorial character. 1. Introduction implies that the direction of the moment is chosen The g factor, ® rst introduced by Lande [1] to explain such that the torque imposed by the ® eld acts to align anomalous Zeeman e ects in atomic spectra, now has a m in the direction of B. The same sign is used in classical more wide-ranging signi® cance. It is generally utilized to electrodynamics, and therefore is in agreement with the represent the linear relationship between magnetic direction of macroscopic moments induced by any elec- moments and angular momenta. For nuclei, g factors tric current distribution. A common example is the are usually positive but are occasionally negative, e.g. permanent magnetic dipole moment of a small planar 3He, 17O. The underlying convention is that for positive conducting loop, which can be expressed as g factors the magnetic moment of the nucleus is oriented m iAn 2 parallel to its angular momentum but for negative g ˆ ? … † factors the reverse is true. Such is the usage in particle where i is the current in the wire, A is the area enclosed physics and in nuclear magnetic resonance (NMR). In by the loop, and n its normal vector de® ned by the electron spin resonance spectroscopy (ESR) the opposite right hand rule [4]. The? direction of i is de® ned in accor- convention is used by most authors, that is the g factor, dance with the transport of positive charges. e gS, is chosen to be positive even though the electron spin Zeeman e ects in diamagnetic species observed at and magnetic moment vectors are antiparallel, as ® rst high resolution are caused by rotational, vibrational, shown by Dirac [2]. The results of many magnetic ® eld and nuclear magnetic moments (neglecting magnetiz- experiments do not depend on the relative orientations ability contributions). They all have in common that a of the two vectors. For example, the theory of magnetic linear relation holds between the magnetic moment and oscillations in metals (de Haas± van Alphen e ect, used the angular momentum. Thus, a rotational moment mrot e for probing Fermi surfaces) shows that the `sign of gS is can be formulated [5]: indeterminate and it is natural to assume that it is posi- rot rot [ ] mi ·Ngij Jj ; 3 tive’ 3 . In this paper, we wish to discuss the situation ˆ … † where several di erent kinds of moment are present, and where J is the rotational angular momentum in units of where their relative orientations are of great importance. h-, grot is the rotational g tensor, i and j refer to the In particular, we explore the implications of the sign of molecule-® xed axes, and summation is implied over ge (and of ge , the orbital electronic g factor) on terms in S L j x;y;z. The nuclear magneton, · eh-=2m [6], is the molecular Hamiltonian, for high resolution Zeeman ˆ N ˆ p a positive scaling factor (e and m are the charge and studies of molecular species. p mass of the proton, h- is the Planck constant divided by 2º). The other angular momenta considered below are, like J, all understood to be given in units of h-. 2. The eŒective Zeeman Hamiltonian Linear and symmetric-top molecules in degenerate The basic Hamiltonian for the magnetic ® eld inter- vibrational states can exhibit a vibrational angular action is momentum pz about the ® gure axis which gives rise to H m B; 1 a magnetic moment [7] ˆ ¡ · … † where m is the magnetic dipole moment operator and B mvib · gvibp ; 4 the magnetic ¯ ux density. Various contributions to m z ˆ N z … † are considered below. The minus sign in equation (1) where the vibrational g factor, gvib, is a scalar. Signs of g factors 1599 rot vib Experiments have shown [8] that gii and g can be orbital magnetism, are both caused by circulating nega- vib positive or negative depending on the details of the tive charges (a similar remark applies to mz , equation nuclear± electron couplings in di erent molecules. Clas- (4)). sically speaking, the sum of all nuclear currents induced The situation with the electron spin magnetic moment in a rovibrating molecule may or may not be out- is at ® rst sight more di cult since the particle can be weighted by that of the electronic ones in forming the considered point-like, without spatial extension [12]. total m, according to equation (2). Thus, the classical picture of a spinning charged Atomic nuclei also show measurable magnetic dipole sphere does not apply. However, Dirac has shown that moments of either sign when their nuclear spin angular the relativistic version of the SchroÈ dinger equation gives momenta, I, do not vanish: a half-odd spin angular momentum component Sz, and nuc nuc a magnetic moment [2] m ·Ng I; 5 ˆ … † eh- where gnuc, a scalar, means the nuclear g factor. The me S ; 8 S ˆ ¡ m magnetogyric ratio, ® · gnuc=h-, is frequently used as e … † ˆ N an alternative description in NMR spectroscopy because which shows that the spin magnetic moment points anti- ! ®B gives the angular transition frequency between parallel to the spin. This again can be written ˆ neighbouring states directly [9]. me · ge S ; 9 S ˆ B S … † In equations (3), (4) and (5), we note the common where ge 2. Later it was found that quantum elec- S ˆ ¡ principle that the g factors determine the orientations of trodynamic corrections have to be applied [13, 14], and the magnetic dipole moments relative to those of the that small amounts of directional dependence must be angular momenta.
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