General Introduction BY A. D. BUCKINGHAM Dept . of Theoretical Chemistry University of Cambridge Received 1 1 iIz Fehrrrury 1970 Most magneto optical effects have their root in the Zeeman splitting of energy levels. The symmetry properties of the magnetic interaction are contrasted with those of an electric interaction. In molecular spectroscopy measurement of the Zeeman splittings yields the magnetic moments of molecules in both ground and excited states. When lines are broad the new technique of magnetic circular dichroism is particularly useful for assigning spectra and determining magnetic moments. The Cotton-Mouton effect in absorption bands is briefly considered. It is appropriate that an International Symposium on magneto optical effects should have been arranged by our Society for Michael Faraday was indeed very active in this field.In 1846 he announced the discovery of the effect that now bears his name,l and which forms a central part of the business of this Symposium. The other great name associated with magneto optics is Zeeman who discovered his effect just half a century after Faraday found his. The Netherlands Physical Society celebrated the centenary of Zeeman’s birth with a Conference in Amsterdam in 1965,2 and in some respects that meeting was the forerunner of the present Symposium. In this introductory talk I shall not attempt to give a complete survey of our subject but shall deal with the background in an attempt to set the stage for the proceedings to follow. Early work in magneto optics is described in three well-known books by Zeeman,3 Wood and S c h u t ~ .~ The basic actions of a magnetic field on matter are to align the electronic and nuclear angular momenta and to induce orbital motion. These interactions affect the macroscopic equations of electromagnetism and also make the important phenomena of electron and nuclear magnetic resonance possible. These basic properties of magnetic interactions are expressed in the hamiltonian describing a molecule in a uniform magnetic induction Bo : where Po is the hamiltonian of the free molecule m is the electronic magnetic moment operator, pus = eA/2rne is the Bohr magneton L and S are the total electronic orbital and spin angular momentum operators in units of ti m(N) = y(N)tiI(N) is the magnetic moment of nucleus N and x ( ~ ) and dN)(d) the diamagnetism and diamagnetic nuclear shield-ing operators : m = -pus(L+2S) (2) p o is the permeability of a vacuum and in S.I.it has the value 47z x lo-’ J s2 C-2 m-I. 8 GENERAL INTRODUCTION The significance of the diamagnetic term was appreciated long before the advent of quantum mechanics through considerations of the Larmor precession of an electron with an angular velocity eBo/2rn about the axis of the magnetic field (the z-axis). This leads to an induced magnetic moment - (e2B0/4rn,)~($ +z) whose direction is opposite to that of the field; the bars indicate an average over the electron cloud and the sum extends over all the electrons. The hamiltonian (1) is sufficient for most purposes. However when considering the effects of spin-orbit coupling it may be necessary to include small additional term^."^ The magnetic moment operator (2) commutes with the parity operator 9 (which changes the sign of all spatial coordinates) and has even parity.Since Y2 = 1 the eigenvalues of 9 are + 1 and - 1 corresponding to even and odd parity respectively. 9 commutes with So (and the total angular momentum operator J) so the eigen-functions $,JM of Z0 (where J and M determine the magnitude and z-component of the total angular momentum and n defines the other constants of the motion of the molecule) are either of even or odd parity and associated with magnetic moments ($,JM I m I $nJM). In this respect the magnetic moment operator differs funda-mentally from the electric dipole moment p = CeirI whose parity is odd.A molecule can only possess a permanent electric dipole moment (and hence show a first-order Stark effect) if $n is degenerate; thus the H-atom spectrum exhibits large Stark splittings because the excited stationary states are of mixed parity (e.g. (l/J2) [t,bZs+- t,kzp,]). A magnetic field splits the (2J+ 1)-fold degeneracy associated with each value of the total angular momentum J and this fact is of great importance. If rn is 1 Bohr magneton (0.9274 x J T-I = 0.9274 x erg gauss-l) then in a magnetic flux density of 1T (i.e. lo4 gauss) the Zeeman energy J = 0.5 cm-l i.e. of an order of magnitude that is suitable both for high-resolution optical spectroscopy and for paramagnetic resonance. The fundamental difference between the perturbations due to external electric and magnetic fields can be illustrated by considering a linear molecule in a lZ vibronic state e.g.OCS. As is well known its microwave spectrum shows only second-order Stark splittings but it exhibits first- and second-order Zeeman effects.IO The magnetic moment in this case is generated by the rotational angular momentum and is proportional to J ; however the constant of proportionality the “ rotational g factor ” x pB is normally - of its value for atoms and molecules with electronic angular momentum. The second-order Stark effect-which depends on the electric dipole moment induced by the external field-arises because parity is not conserved in the presence of the external field (2 = Z 0 - p E does not commute with 9 if the external field E is unaffected by the inversion).Considerations of time-reversal symmetry and spin-operators 11-1 require that molecules with half-integral J must be at least doubly degenerate in the absence of an external magnetic field (Kramers’ theorem) ; furthermore the application of a magnetic field to a system of half-integral J causes a first-order Zeeman splitting of the pairs of states having angular momentum components +M. Where spectral lines are sufficiently sharp direct observation of the Zeeman splittings or shifts provides the most satisfactory way of extracting information about the magnetic moments-and hence about the electronic structure-of molecules in both ground and excited states. In astrophysics the roles are reversed and Zeeman splittings of the spectra of atoms are used to determine magnetic fields on stars-for example flux densities up to 3700 gauss have been detected on the sun during intense sunspot activity.14 Many of the papers presented to the Zeeman Centenary Conference describe Zeeman splittings as do those by Hochstrasser and Lin and 1 A .D . BUCKINGHAM 9 Marzzacco and McClure to be given at this Symposium. Even when the lines are broad large magnetic moments can sometimes be detected as in the work of Malley, Feher and Mauzerall l 5 on porphyrins in solution at lo5 gauss at room temperature and at 77 K. However the differential approach through magnetic circular dichroism is generally superior in these conditions. The modern techniques of atomic and molecular beam resonance spectroscopy optical pumping level crossing experi-ments,18 and various quantum mechanical interference effects l 9 are also relevant and of importance in very accurate measurements of small Zeeman splittings and hyperfine interactions.Paramagnetic resonance transitions have been detected in phosphorescent molecules by sweeping the magnetic field and observing changes in the intensity of the phosphorescence 20-22 ; information is obtained about the mode of entry into and decay from the meta-stable state. This work is represented at this meeting by Sharnoff's paper. As in optical pumping this type of experiment can also yield lifetimes of excited molecules in triplet states. A magnetic field can influence triplet-state lifetimes through the triplet-triplet annihilation probability in solids 2 3 and solutions 2 4 ; the effect is not very large and is most marked in single crystals.Magnetophotoconductive effects have been observed in copper phthalo-cyanine and attributed to the influence of the magnetic field on the charge-carrier generation rate.25 When the spectral lines are broad or when there are difficulties in assigning spectra, the methods of magnetic optical activity are particularly useful. Eight of the thirteen papers to be presented to this meeting are concerned with this particular " magneto optical effect "-the Faraday effect in absorption bands. The method of magnetic circular dichroism is particularly useful as it is absent in transparent regions the solvent does not contribute significantly and it is a direct measure of the difference in absorption of right- and left-circularly-polarized light.The technique is particularly important in assigning the spectra of transition metal complexes and in determining their magnetic moments in excited states ; it is also helpful in organic and biochemistry. There is a very useful recent review of the subject by Schatz and McCaffery.26 Magnetic optical activity is determined by the linear effect of a magnetic induction Bo on the frequency dependent electric polarizability a I aaS = + a$iBoy + + C ~ $ ~ ~ B ~ B ~ + . . . . (4) i.e. by the tensor a('). If the molecular tumbling motion is random g m a y be replaced by its isotropic part a(')eafir where a(l) = I gCXa/jy+fiy ( l ) and cap = 1 or -1 if aPy is an even or odd permutation of xyz and is zero otherwise.Then the permittivity can be written E = id 0 where i~' = E, = - E,, when Bo is in the z-direction. (:iet ; :) Since every molecule regardless of its symmetry has an a so too does it have an a(1), since Bo is an axial vector ; however distortion of a in an isotropic medium by a polar vector of any strength (e.g. an electric field) could not yield optical rotation about the field (unless the molecules are different in right-handed (x y z) and left-handed ( - x -y - z ) frames as in an optically active medium). and for a molecule in a non-degenerate state described by the ket I n ) may be 4;) = Z 2 h - ' [ f ( w ~ j J - i g ( ~ ~ j n ) I C ~ j n Re {<. I Pa I j ) < j I PS I .>I-The complex polarizability a is defined by the equation pa = 28 J iw Im I<.I I j ) < j I PS I .>I] ( 5 ) where A g have dimensions of o r 2 is the angular frequency of the electri 10 G EN ER A L I N TROD U CT I ON field E wjn = (EJ-En)/h and g and fare absorption and dispersion line shape func-tions of the form illustrated in fig. 1. FIG. 1.-Typical absorption (a) and dispersion (6) line shapes near an absorption frequency a j r l . Right (E+) and left (E-) circularly-polarized light of angular frequency co travelling in the uniaxial z-direction can be represented by the equation where i j k are unit vectors along the x y z axes ; E,t = n* - ik+ is the complex refractive index so right-circularly polarized light travels with a velocity c/n+ in the z-direction with an intensity Z = I. exp (- 2cok+z/c). The actual electric field is the real part of eqn (6) i.e., Re (E*) = exp (-ok,tz/c)[i cos w(t-n*z/c)+j sin o(t-n*z/c)] (7) and the beam associated with E k is comprised of photons whose component of angular momentum in the direction of propagation is T 1.On absorption it induces transitions with AM = + 1 which for a transition to an electronically degenerate E* = E(O) exp [ico(t- n^+z/c)] (i& ij) (6) -pI: M = - l (n+) (n-) M - 0 FIG. 2.-The origin of (a) magnetic circular dichroism and (6) magnetic optical rotation. The A-terms are shown for a non-degenerate ground state and a triply-degenerate excited state A . D . RUCKINCHAM 1 1 excited state from a non-degenerate ground state? are displaced to lower/higher frequency by interaction with Bo (see fig. 2).The difference between n+ and n- leads to optical rotation through an angle 6 = (toz/2c)(iz -n__) (8) about the direction of propagation. The sign of this expression is opposite to that commonly adopted but it agrees with the natural choice that a positive rotation of the electric vector and the direction of propagation form a right-hand screw ; it also fits naturally into the established definitions of the Cotton-Mouton and Kerr constants (see eqn (14)). This sign convention is a point we might discuss at this Symposium. Similarly the difference between k+ and k- leads to an ellipticity 8 = (012/2C)(k.# -k) (9) (10) and the two equations (8) and (9) can be combined in the coniplex form A a) = 4 - io = (cloz/2c)(ti+ -fi-). The importance of studying 8 or # in the vicinity of the absorption frequency ujn is that the contribution of the excited state l j ) can be isolated.Thus the awkward summation in eqn (5) is obviated. If the magnetic moment differs in states 1 p t ) and I j ) the Zeeman shift of the absorption frequencies from cojn leads to opposite displacements of the resonance frequencies for E+ and E- as illustrated in fig. 2. The result is the line shape in magnetic circular dichroism or in magnetic optical rotation that is characteristic of electronic degeneracy in the ground or excited state (called A-terms by Serber 29). Even when neither state possesses a magnetic moment, circular dichroism or rotation may arise from small differences in the intensities of the interaction with E+ and E- ; this is due to the effects of the perturbation -m Bo on the eigenfunctions I n) and I j ) in eqn (5) and is illustrated in fig.3. The familiar line shapes are associated with Serber’s B-terms which are presumably responsible for the magnetic circular dichroism detected in organic carbonyl compounds and reported in the paper by Barth Bunnenberg Djerassi Elder and Records. 0 4 FIG. 3.-(a) Magnetic circular dichroism and (b) magnetic optical activity when the states are non-degenerate (B-terms) . In the Cotton-Mouton effect there is a phase difference 8 = ~ n z - n x x ) ~ Y l c (1 1) induced in light travelling in the y-direction and initially linearly polarized in th 12 GENERAL INTRODUCTION z- and x-directions (Bo is again in the z-direction). Near absorption bands ^n = n - ik and A c 6 = ( 6 z z - G x x ) ~ y / ~ = 2nyCB& (12) * where C = (Zzz -^n,,)r;o(2n~B~)-~ is the Cotton-Mouton constant.30 The difference in the absorption of E and Ex leads to an optical rotation 4 given by tan 24 = sinh [(L - ~xx)~y/cl/cos [(nz - n,x>~Y/cl (1 3) which reduces to 4 = (k,,-kxx)~Y/2c for small 4.Thus the Cotton-Mouton effect in absorption bands is probably best studied through measurement of optical rotation. The effect is quadratic in the magnetic induction Bo and can therefore be related to a(2) in eqn (4). While magnetic circular dichroism arises from a difference in absorptance of photons having angular momentum m of - 1 and + 1 in the direction of Bo Cotton-Mouton dichroism results from a difference in absorptance of photons polarized parallel (m = 0) and perpen-dicular (m = 1) to Bo as illustrated in fig.4. This interesting new magneto-optical effect is analogous to the electro-optic Kerr effect,31* 32 and has recently been studied by Boccara Ferre Bria Billardon and B a d ~ z . ~ ~ If either the ground or excited state has a magnetic moment its peak intensity is in general proportional to the inverse cube of the line-width compared to an inverse square in magnetic circular dichroism. However because of its proportionality to B,2 the full description of the Cotton-Mouton effect is considerably more complicated than that of the Faraday effect. FIG. 4.-The origin of Cotton-Mouton dichroism and its frequency dependence for a non-degenerate ground state and a triply degenerate excited state as in fig.2. In transparent regions the Cotton-Mouton constant is real and related to the product of the anisotropy in the molecular electric and magnetic polarizabilities and to a small temperature-independent hyperpolarizability (like d2) in eqn (4)) describ-ing the quadratic effect of Bo on a. Measurements have been made on a number of gases using mercury 546.1 nm radiation,34 and recently with a He+Ne laser at 632.8 nm.35 The results lead to accurate values for the anisotropies of molecular magnetic susceptibilities. M. Faraday Phil. Mag. 1846 28[3] 294 ; Phil. Trans. Roy. Soc. 1846 1. Physica 1967 33,l-293. P. Zeeman Researches in Magneto-Optics (Macmillan London 191 3) A . D. BUCKINGHAM 13 R. W. Wood Physical Optics 3rd ed. (Macmillan London 1934).W. Schutz Magnetooptik vol. 16 of Handbuch der Experimentalphysik (Akademische Ver-lagsgesellschaft Leipzig 1936). J. Larmor Aether and Matter (Cambridge University Press 1900) p. 341. A. J. Stone Proc. Roy. SOC. A 1963,271,424. A. Abragam and J. H. Van Vleck Phys. Rev. 1953 92 1448. L. L. Lohr J. Chem. Phys. 1966,45,1362. l o W. H. Flygare W. Huttner R. L. Shoemaker and P. D. Foster J. Chenz. Phys. 1969,50 1714. '' E. P. Wigner Group Theory and its Application to the Quantiim Mechanics of Atomic Spectra, l 2 J. S. Griffith The Theory of Transition-Metal Ions (Cambridge University Press 1961) p. 205. l 3 A. S. Davydov Quantum Mechanics (Pergamon Press Oxford 1965) p.431. l4 H. W. Babcock Physica 1967,33,102. l 5 M. Malley G. Feher and D. Mauzerall J. Mol.Spectr. 1968 26 320. l 6 P. Kusch and V. W. Hughes Handbuch der Physik vol. 37 (Springer-Verlag Berlin 1959) 1 . l 7 A. Kastler J. Phys. Radium. 1950 11 255 ; R. A. Bernheim Optical Pumping (Benjamin, '' T. G. Eck Physica 1967,33 157 '' G. W. Series J. Phys. By 1970 3,84. 2o M. Sharnoff J. Chem. Phys. 1967,46,3263. 21 A. L. Kwiram Chem. Phys. Letters 1967 1,272. 22 J. Schmidt I. A. M. Hesselmann M. S. de Groot and J. H. van der Waals Chem. Phys. Letters, 23 R. C. Johnson R. E. Merrifield P. Avakian and R. B. Flippen Phys. Rev. Letters 1967,19,285. 24 L. R. Faulkner and A. J. Bard J. Amer. Chem. Soc. 1969,91,6495. 2 5 S . E. Harrison J. Chem. Phys. 1969,51,465. 26 P. N. Schatz and A. J. McCaffery Quart. Rev. 1969 23,552. (Errata in 1970 24 329.) 27 M. Born and K. Huang Dynarnical Theory of Crystal Lattices (Oxford University Press 1954), p.189. 28 A. D. Buckingham and P. J. Stephens Ann. Rev. Phys. Chem. 1966,17,399 ; A. D. Buckingham, Adv. Chem. Phys. 1967,12,107. 29 R. Serber Phys. Rev. 1932,41,489. 30 J . W. Beams Revs. Mod. Phys. 1932 4,132. 31 A. D. Buckingham Proc. Roy. SOC. A 1962,267,271. 32 M. P. Bogaard A. D. Buckingham and B. J. Orr Mol. Phys. 1967,13,533. 33 A. C. Boccara J. Ferre B. Briat M. Billardon and J. P. Badoz. J. Chem. Phys. 1969,50,2716. 34 A. 11. Buckingham W. H. Prichard and D. H. Whiffen Trans. Faraday Soc. 1967,63,1057. 35 M. G. Corfield Ph.D. Thesis (University of Bristol 1969). (Academic Press New York 1959) chap. 26. New York 1965). 1967 1,434