- Email: [email protected]
The magnetochiral effect and related optical phenomena
Chemical Physics 245 Ž1999. 165–173 www.elsevier.nlrlocaterchemphys
The magnetochiral effect and related optical phenomena Georges H. Wagniere `
)
Institute of Physical Chemistry, UniÕersity of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland Received 30 October 1998
Abstract The magnetochiral effect has only recently been experimentally detected and measured. The interest for the effect lies less in any foreseeable practical application, than rather in the symmetry principles that govern it. In this article we first show how the magnetochiral effect is related to other optical and magneto-optical phenomena, such as natural optical activity, the Faraday, Cotton–Mouton effects and quadratic magnetic field-enhanced optical activity. The general principles of the measurements of magnetochiral birefringence and magnetochiral dichroism are then briefly described. The basic theory of the non-linear-optical, inverse magnetochiral birefringence is presented and possibilities of its detection are explored. The emphasis of this paper lies in the discussion of selection rules and in symmetry considerations. q 1999 Elsevier Science B.V. All rights reserved.
1. Nature of the magnetochiral effect As the name implies, the magnetochiral effect occurs in chiral media in the presence of a magnetic field. The effect is related both to natural optical activity which manifests itself only in chiral media, and to magnetic optical activity which is observed in media of any symmetry but requires a static magnetic field parallel to the direction of propagation of the incident light. Natural optical activity has been known since the early 19th century, its prominent pioneers being Arago, Biot and Pasteur w1x. Magnetic optical activity was discovered in 1846 by Faraday w2,3x. Both effects are circular differential, corresponding to a different response for left ŽL. and right ŽR. circularly )
Corresponding author. Fax: q41-1-635-6813; E-mail: [email protected]
polarized light. Outside resonances one measures the difference of refractive index n LŽ l . y n RŽ l . Žrespectively, as natural optical rotatory dispersion, ORD; or as magnetic optical rotatory dispersion, MORD., inside absorption bands the difference of the absorption coefficient ´ LŽ l . y ´ RŽ l . Žrespectively, as natural circular dichroism, CD; or as magnetic circular dichroism, MCD.. The magnetochiral effect, in contrast, is not circular differential. In a chiral medium the index of refraction and the absorption coefficient for arbitrarily polarized light undergo a shift in value when a static magnetic field is applied collinear with the direction of propagation of the incident radiation. The sign of the shift changes if the relative direction of field and beam is inverted. For a given relative direction, the sign of the effect changes on going from one chiral medium to its enantiomer Žmirror image.. This situation is summarized in Table 1. The
0301-0104r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 0 2 3 - 3
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
166
Table 1 Relative magnetochiral shifts of the refractive index n and the absorption coefficient ´ for parallel and antiparallel configurations of the static magnetic field B0 Magnetic field configuration
B0 ≠≠ B0 ≠x
Enantiomer d
l
nq D n ´ qD´ ny D n ´ y D´
ny D n ´ yD´ nq D n ´ q D´
absolute magnitude of the magnetochiral signals in molecular media is predicted to be 10y3 to 10y2 times smaller than that for magnetic circular birefringence or dichroism. For a diamagnetic medium in a magnetic field of 5T, this leads to an estimate for D nrn of the order of 10y8 to 10y6 . The magnetochiral effect has only recently been detected in molecular media. The measurement of magnetochiral birefringence described in Section 4 confirms the above estimate for D nrn. In Section 5 we will see that for highly chiral compounds, exhibiting at the same time strong magnetic circular dichroism, magnetochiral dichroism D ´r´ of the order of up to 10y3 may be observed. Although, as of yet, the magnetochiral effect is of no apparent practical use, it will be shown that it obeys symmetry rules which are of quite fundamental significance. The magnetochiral effect was theoretically predicted in the course of the last forty years. A term responsible for magnetochiral birefringence is already to be found in a paper by Groenewege w4x which is mainly concerned with magneto-optical rotation. Baranova and Zel’dovich w5x predicted the existence of a ‘‘new linear magnetorefractive effect’’ of similar origin. Independently, the occurrence of the corresponding effect in absorption w6–10x and emission w11x was deduced. The names ‘magnetochiral birefringence’ and ‘magnetochiral dichroism’ were suggested by Barron and Vrbancich w10x and have been adopted.
2. Theory of magnetochiral birefringence As is well known, the interaction between a molecular system and the radiation field may be
expanded into a multipole series: Hint s ym P E Ž t . y m P B Ž t . y Q := E Ž t . q . . . , with E Ž t . s Ey exp Ž yi v t . q Eq exp Ž qi v t . , B Ž t . s By exp Ž yi v t . q Bq exp Ž qi v t . . Ž 1. In the long-wavelength approximation for ordinary refraction and absorption, the electric dipole term may in general be considered as dominant. For our purposes this is, however, insufficient and we must include the magnetic dipole and electric quadrupole contributions in our discussion as well. Simple graphs may conveniently visualize the relation of the magnetochiral effect to other optical phenomena w9,12,13x. These figures are directly connected to the quantum mechanical expressions for the radiation-induced polarization p in a molecule. We assume for the time being that the frequency of incident radiation lies outside molecular resonances. Fig. 1a represents ordinary Rayleigh scattering or refraction. An incident photon of frequency v promotes the molecular system from an initial state < a: into a non-stationary superposition of states < k :, whereupon the photon is re-emitted Žscattered. within a time of the order of 2 prv . In general, this phenomenon is mainly due to the electric dipole interaction with the radiation field. Fig. 1b shows the contribution which is responsible for natural optical activity. Here the magnetic dipole interaction with the radiation v Ž M . also comes into play. In non-isotropic media there may be a contribution from the electric quadrupole interaction v Ž Q ., as well. Fig. 1b translates into the expression w9x: ² a < m < k :² k < m P By < a: p Ž1. Ž v ;y v Ž M . . s Ý "Ž vka y v . k
½
q
² a < m P By < k :² k < m < a: "Ž vka q v .
5
.
Ž 2. In the corresponding susceptibility, we obtain products of electric transition moments ² a < m < k : which are parity-odd polar vectors, and of magnetic transition moments ² k < m < a: which are parity-even axial vectors. The optical activity tensor of the form m m is therefore odd with respect to space inversion, and the effect occurs in non-centrosymmetric media only; in fluids, the molecules must be chiral.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
167
is even with respect to parity, and that the effect occurs in all matter. Magnetochiral birefringence arises through the combination of a magnetic dipole interaction with the static magnetic field and a magnetic dipole, or electric quadrupole interaction with the radiation. This is shown in Fig. 1d leading to the expression p Ž2. Ž v ;y v , Ž M . ,0 Ž M . . sÝ k
Ý l
² a < m < l :² l < m P By < k :² k < m P B0 < a: "2 Ž v l a y v . v k a
q 5 similar terms .
Fig. 1. Graphs for the radiation-induced molecular polarization due to the optical effects discussed in the text. The general form of the tensor products of the corresponding susceptibility are indicated. m stands for a Žparity-odd. electric dipole, m for a Žparity-even. magnetic dipole matrix element. The overall symmetry with respect to parity Žspace inversion. P and with respect to time reversal T w10x is indicated at right for each case.
Ž 4.
The product of two magnetic dipole and of only one electric dipole matrix elements m mm shows that the corresponding susceptibility tensor is odd w.r. to parity. The effect occurs only in non-centrosymmetric media, in particular, in isotropic media that are chiral. Furthermore, it may be deduced w6,13x that an odd number of magnetic dipole andror electric quadrupole interactions leads to a circular differential effect; an even number thereof does not. The magnetochiral effect is consequently not circular differential. To further illustrate this point, we show in Fig. 2 the graphs for two other magneto-optical effects: Fig. 2a represents the well-known Cotton– Mouton–Voigt effect w14x which is quadratic in the static field, occurs in all media and is not circular
Magnetic optical activity is represented by Fig. 1c. One incident vertex symbolizes the interaction with the static magnetic field B0 , denoted by 0Ž M ., the other the electric dipole interaction with the radiation of frequency v . The corresponding mathematical expression for the induced polarization becomes p Ž2. Ž v ;y v ,0 Ž M . . sÝ k
Ý l
² a < m < l :² l < m P Ey < k :² k < m P B0 < a: "2 Ž v l a y v . v k a
q 5 similar terms .
Ž 3.
The products of molecular vectors of the kind mm m Žpolar vector = polar vector= axial vector. let us conclude that the tensor for magnetic optical activity
Fig. 2. Graphs for the Cotton–Mouton–Voigt effect Ža. and for the quadratic magnetic field-induced contribution to natural optical activity Žb.. The former effect occurs in all matter, the latter one only in chiral media.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
168
differential. In Fig. 2b we have replaced the electric dipole interaction with the radiation by a magnetic dipole interaction. We obtain a circular differential contribution to natural optical activity which is quadratic in the static magnetic field w15,16x and which, somewhat misleadingly, also has been named the ‘quadratic Faraday effect’. The recent experimental observation of this effect is reported in Refs. w17,18x. 3. Theory of magnetochiral dichroism To study absorption or emission, the quantity of interest is the transition probability per unit time: 2p w Ž a ™ b, v . s 2 <² b < Hint Ž v . < a:< 2d Ž v b a y v . . " Ž 5. If we introduce the molecule–radiation interaction hamiltonian Ž1. into Ž5., we will obtain a series of terms, the first one being the electric dipole–electric dipole term Že–e. responsible for ‘ordinary’ electric dipole absorption or emission. The electric dipole– magnetic dipole Že–m., and electric dipole–electric quadrupole Že–q. cross-terms lead, in a chiral medium, to circular dichroism. If, in addition to the interaction with the radiation, we assume the states < a:, < b :, to be perturbed by a static magnetic field B0 parallel to the direction of propagation of the incidentremitted radiation, then to first order in B0 , the Že–e. terms will contain contributions to magnetic circular dichroism ŽMCD., and the Že–m. and Že–q. terms will lead to expressions describing magnetochiral dichroism w6,11x. For a transition between two non-degenerate molecular states and after isotropic averaging, the contribution to magnetochiral dichroism reads: Dw Ž a ™ b; Ž e y m . , B0 . 2 ² < < : ² < < : s 3 Re Ý vy1 nb Ž n m a P a m b 3" n/b =² b < m < n: q ² b < m < a: P ² a < m < n: = ² n < m < b : . q
² < < : ² < < : Ý vy1 na Ž b m n P a m b n/a
=² n < m < a: q ² b < m < a: P ² n < m < b : = ² a < m < n: . = g Ž v b a , v . Ž EyP Bq= B0 . .
Ž 6.
Here, as in Ž4., we encounter products of an electric dipole transition moment and of two magnetic dipole transition moments, of the type m mm. An expression similar to Ž6. is obtained for the corresponding electric dipole–electric quadrupole terms Že–q. w6x. As is known from the theory of magnetic optical activity w3,19x, there is a variety of contributions to MCD and MORD, depending on if a molecule possesses magnetically degenerate electronic states. In diamagnetic molecules of low symmetry one encounters only so-called B-terms, reflecting the mixing of non-degenerate molecular states under the influence of the static magnetic field. In molecules of higher symmetry, exhibiting threefold or higher rotation axes, bisignate A-terms are to be expected and in addition, if the ground state is magnetically degenerate, also temperature-dependent C-terms. The A- and C-terms derive from the splitting of the degenerate levels by the applied magnetic field. An analogous situation prevails in the magnetochiral effect. Expression Ž6. describes the magnetochiral B-terms. The theory of magnetochiral A- and C-terms is discussed in Refs. w9,10x. The interested reader is reminded that expression Ž5., the ‘Fermi Golden Rule’, is not the only way to calculate the transition probability. One may equivalently consider the radiation-induced molecular polarization p, as in Section 2, but including damping w9,10x. The corresponding molecular susceptibility is then complex. The imaginary part of this complex expression represents the absorptionremission cross-section.
4. Measurement of magnetochiral birefringence The magnetochiral effect occurs in all chiral media and is not averaged out in fluids and gases. The relative shift of the refractive index D nrn to be expected in a diamagnetic molecular system Ž B-term. outside resonances in a magnetic field of 5 T is of the order 10y8 to 10y6 , which is very small. We here briefly describe the recent detection and measurement of magnetochiral birefringence in liquids by interferometry w20,21x. The magnetochiral birefringence will tend to be overshadowed by the larger effects of natural and magnetic optical activity. It is
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
therefore necessary to fully exploit the different symmetry properties of these various phenomena. As illustrated in Ref. w20x, an incident beam of linearly polarized light of 633 nm produced by a He–Ne laser is divided into two sub-beams. The probe consists of two compartments, one containing the d-, the other the l-enantiomer of the substance to be investigated. The sub-beams travel in opposite directions through both compartments in which there is a static axial magnetic field of 5 T. The experiment is devised in such a way that natural and magnetic optical rotation are compensated to zero, while the magnetochiral signal is enhanced. One sub-beam gets phase-retarded by the magnetochiral effect, the other one phase-accelerated. The two sub-beams are brought to interference, and the magnitude and sign of the phase shift are then measured. To enhance the weak signals, high concentrations of the material samples are required. Therefore, the first measurements w20x have been performed with pure liquids. For the highly optically active compound 3-Žtrifluoroacetyl.-camphor one measures under the stated conditions D n s q6 = 10y8 referred to the l-enantiomer in a static field configuration parallel to the direction of propagation of the light beam. For the compound l-carvone one measures in the same conditions q2.5 = 10y8 . The device described in Ref. w20x should also be applicable to the measurement of the magnetochiral birefringence in chiral cubic or uniaxial crystals. In the latter case, the optic axis of the crystal should be collinear with B0 and with the light beam direction. It is to be noted that this kind of measurement requires the availability of both enantiomorphous crystal forms. The magnetochiral effect may be considered as a special case of magneto-spatial dispersion effects that occur in certain crystal classes of reduced symmetry for particular directions of light propagation w22–27x. However, as we are mainly interested in the optical properties of isotropic chiral molecular fluids and of uniaxial chiral crystals along the optic axis, we will here not go into the details of these additional aspects in anisotropic media. 5. Measurement of magnetochiral dichroism The recent first measurement of magnetochiral dichroism in a molecular probe has been performed
169
by Rikken and Raupach in emission w28x. These authors have investigated solutions of the chiral trisŽ3-trifluoroacetyl-camphor. –Eu3q complex: The sample was excited with unpolarized light of 350 nm, and the luminescence collected by optical fibers in the directions parallel and antiparallel to the applied alternating magnetic field. The axial magnetic field, oscillating with a frequency of 0.9 Hz, had a peak strength of 0.9 T. Specific emission wavelengths were selected by a grating monochromator. The signal detection was performed phase-sensitively. The registered signals showed opposite signs for the d- and l-enantiomers and were identified as being essentially due to the 5 D 0 ™7 F1 and 5 D 0 ™7 F2 transitions of the ion. As may be inferred from the theory w5–11x, the order of magnitude of the magnetochiral anisotropy factor Žper unit of magnetic field strength. g Ž MChD. s
I≠ ≠ y I≠ x I≠ ≠ q I≠ x
,
Ž 7a .
should be roughly equal to that of the product of the dissymmetry factors for natural circularly polarized emission ŽCPE. w29x g Ž CPE . s
I L y IR I L q IR
,
Ž 7b .
and of magnetic circular dichroism ŽMCD; per unit field strength. w30x: g Ž MCD. s
2Ž ´ L y ´ R .
´L q´R
.
Ž 7c .
I and ´ denote the emission intensity and absorption coefficient, respectively. Rikken and Raupach have estimated g ŽMChD. to be of the order of 5 = 10y3 for the 5 D 0 ™7 F1 , and 4 = 10y4 for the 5 D 0 ™7 F2 transition in the investigated complex, which is remarkably high. This is related to the fact that we here have A-terms. The measured signals in the range 590–620 nm agree with this order of magnitude and obey the expected symmetry rules for the magnetochiral effect. A first measurement of magnetochiral dichroism in absorption is reported by the same authors w31x for the optically active uniaxial crystal a-NiSO4 P 6H 2 O. The applied magnetic field, collinear with the incident radiation and with the optic axis, was alternated at a frequency of 0.9 Hz, as for the emission experi-
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
170
ment. The magnetochiral dichroism, registered over the range 900–1500 nm, is of an order of magnitude comparable to that recorded in the previous example in emission. Highly chiral complexes or crystals containing ions of transition metals or of rare-earth elements appear indeed to be well-suited to study magnetochiral dichroism.
6. Magnetochiral effect in atomic and nuclear physics Under the influence of a parity-non-conserving interaction, such as that caused by the parity-violating weak neutral currents, an atomic system is predicted to become optically active and also to show magnetochiral dichroism. The theoretical model described in Ref. w32x is based on the transitions between the spin–orbit coupled states 2 P3r2 and 2 P1r2 of the hydrogen atom. In the unperturbed atom, these transitions are magnetic dipolerelectric quadrupoleallowed. The parity-violating interaction couples the 2 S1r2 states to the 2 P1r2 states, thereby inducing in the 2 P3r2 ™2 P1r2 transitions weak electric dipole components. If, in addition, we assume the atom to be in a static magnetic field, the degeneracies of the states are lifted, and the transitions considered occur between energy-split pure eigenstates of Jz . It is then established that for any transition between particular states for which D m j s "1, there is a difference in the intensity of light emitted parallel and antiparallel to the axis of quantization, as summarized in Table 2. This is an expression of magnetochiral dichroism. The relative intensity shift D I changes sign if the
Table 2 Predicted w32x relative magnetochiral emission intensity shift in a parity broken atom for forward q k and backward y k propagation along the axis of quantization D mj
Propagation
Polarization
Relative intensity
y1
qk yk qk yk
L R R L
IqD I IyD I IyD I IqD I
q1
According to the Zeeman selection rules, the radiation is either left ŽL. or right ŽR. circularly polarized.
axis of magnetic quantization is inverted and, being proportional to the parity-violating interaction, also would change sign if the chirality were reversed. Furthermore, we notice that if the magnetic field is absent, and the transitions become degenerate, then the emitted light in both directions, qk and yk, is of same total intensity but elliptically polarized. The atomic model considered then exhibits circular dichroism. From this we conclude that the magnetochiral effect and natural optical activity are of same origin. The lifting of magnetic degeneracy distinguishes the former from the latter. Although this simple model may appear rather academic, it nevertheless illustrates basic symmetry principles. These principles should be essentially valid in any parity-non-conserved system described by a hamiltonian which commutes with the total angular momentum containing both spin and orbital contributions. Interestingly, also for atomic nuclei one finds w33x: ‘‘Parity violation results in circular polarization of g emission from unpolarized nuclei or fore–aft asymmetry in g emission from polarized nuclei’’. A polarized nucleus is in a definite magnetic substate. The superposition of multipoles of different parity in the emission of radiation due to parity-violation leads to the predicted and measured w33,34x fore–aft asymmetry in the emission intensity. This is then, from our point of view, a manifestation of magnetochirality.
7. Magnetochiral effect and molecular evolution The parity-non-conserving influences in the universe essentially all derive from the primary parityviolating weak interactions. It is not our aim here to speculate on the complex mechanisms by which these basic forces have led to the general asymmetry of the universe, much less on the origins of the homochirality of life. We are rather interested in parity-non-conserving influences of secondary origin and in the way they further induce chirality. These are: Ž1. Already present chiral matter, which by direct physical and chemical interaction leads to, or catalyzes, the formation of additional chiral products. This is the basis of most asymmetric syntheses.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
171
dichroism this should exert a similar enantioselective influence as 2 w8x. All three effects mentioned here are local and may therefore, with respect to the sign of the induced chirality, go either way. Our immediate concern is the relative importance of 3, as compared to 1 and 2. We are aware of the smallness of the magnetochiral effect. On the other hand, the necessary conditions may be met almost anywhere in the universe. Wherever there is light and there are static magnetic fields not exactly perpendicular to the direction of lightpropagation, a chiral influence will be exerted w8,36,37x. The magnetochiral effect may thus play a role in enantioselective processes in the universe.
8. Inverse magnetochiral birefringence: a non-linear optical effect
Fig. 3. Graphs representing the nonlinear optical inverse Faraday effect and inverse magnetochiral birefringence, in comparision with the related linear optical phenomena.
Ž2. The influence of circularly polarized light which via circular dichroism in a racemic mixture leads to enantioselective photodestruction or photoenrichment w35x. Ž3. The presence of a static magnetic field parallel to the direction of propagation of incident light of arbitrary polarization. Due to magnetochiral
Optical rectification p Ž2. Ž0;y v , v ., the radiation-induced induction of static electric polarization, occurs only in non-centrosymmetric media. It is, however, forbidden in chiral liquids outside resonances. On the other hand it is possible, to second order in the interaction with the radiation, to induce a static magnetization mŽ2. Ž0;y v , v . in any medium, also a fluid one. The incident light has to be circularly polarized, however. Fig. 3b represents this socalled inverse Faraday effect w38x, the existence of which has been experimentally confirmed w39x. We notice that Fig. 3b is related to that of the ordinary Faraday effect ŽFig. 3a., through an interchange of ingoing and outgoing vertices and by combining with opposite signs the frequencies of the electric dipole interactions with the radiation. We may now
Table 3 Difference in symmetry between the inverse Faraday effect w38,39x and the inverse magnetochiral effect w40–42x
Inverse Faraday effect
Polarization of incident radiation
Enantiomer
Relative sign of induced magnetization
L
d l d l d l d l
q q y y q y q y
R Inverse magnetochiral birefringence
L R
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
172
similarly go from Fig. 3c for ordinary magnetochiral birefringence to Fig. 3d for inverse magnetochiral birefringence w40–42x. While the magnetization induced by the inverse Faraday effect changes its sign for L- and R-circularly polarized light, that due to the inverse magnetochiral birefringence is independent of the state of polarization of the incident radiation. The inverse magnetochiral effect occurs only in chiral media, including liquids, and changes sign for enantiomers ŽTable 3.. For a diamagnetic chiral fluid medium the isotropically averaged magnetization per molecule due to the inverse magnetochiral birefringence is found to be: mŽ2. Ž 0;y v ,q v Ž M . .
v s
3" 2
=
½
Ý Ý Ž m k a P m al = m l k . k
l
References
v Ž vla q vka. v l a Ž v l2a y v 2 .Ž v k2a y v 2 .
y Ž m l k P m k a = m al . =
Ž e"r2 mc .rea0 in cgs units, corresponding to a magnetization of 10y9 to 10y8 G, a magnetic induction of 10y8 to 10y7 G, or 10y12 to 10y11 T w40x. The inverse Faraday effect has been measured with an induction coil technique w39x. The inverse magnetochiral birefringence has not yet been detected experimentally. By using short and powerful laser pulses to increase the induced magnetic induction and its time-derivative, it is conceivable that measurable signals should be obtained. The interest for the inverse magnetochiral birefringence primarily lies in the symmetry principles which govern it. Wherever light of arbitrary polarization interacts with a chiral polarizable medium, it induces a static magnetic field, no matter how small.
Ž vla q vka. 2 vla vk aŽ vla y v . Ž vk a q v .
= Ž Ey= Bq . .
5 Ž 8.
This represents the electric dipole-magnetic dipole contribution. As is the case for magnetochiral birefringence and dichroism, the products of molecular quantities appearing in this expression are of the form m mm, which is odd with respect to space inversion Žsee Sections 2 and 3.. An analogous electric dipole–electric quadrupole contribution may also be derived w41,42x. In a paramagnetic medium there are additional, temperature-dependent terms w40–42x. The magnitude of the magnetization, M s Nm Ž N the number of molecules per unit volume., induced by the inverse Faraday effect is proportional to the intensity of incident radiation. With a light intensity of 10 8 W cmy2 an order of magnitude of 10y6 to 10y5 G has been measured for M w39x. The magnetization due to inverse magnetochiral birefringence is also predicted to be proportional to the light intensity, but should be smaller by the order of
w1x J.P. Mathieu, Handbuch der Physik, vol. 28, Spektroskopie II, Springer, Berlin, 1957, p. 333. w2x J.H. van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, London, 1965, p. 367. w3x A.D. Buckingham, P.J. Stephens, Annu. Rev. Phys. Chem. 17 Ž1966. 399. w4x M.P. Groenewege, Mol. Phys. 5 Ž1962. 541. w5x N.B. Baranova, B.Ya. Zel’dovich, Mol. Phys. 38 Ž1979. 1085. w6x G. Wagniere, ` A. Meier, Chem. Phys. Lett. 93 Ž1982. 78. w7x S. Wozniak, R. Zawodny, Acta Phys. Pol. A 61 Ž1982. 175. ´ w8x G. Wagniere, ` A. Meier, Experientia 39 Ž1983. 1090. w9x G. Wagniere, ` Z. Naturforsch. 39a Ž1984. 254. w10x L.D. Barron, J. Vrbancich, Mol. Phys. 51 Ž1984. 715. w11x G. Wagniere, ` Chem. Phys. Lett. 110 Ž1984. 546. w12x J.F. Ward, Rev. Mod. Phys. 37 Ž1965. 1. w13x G. Wagniere, ` J. Chem. Phys. 77 Ž1982. 2786. w14x A.C. Boccara, J. Ferre, B. Briat, M. Billardon, J.P. Badoz, J. Chem. Phys. 50 Ž1969. 2716, and references cited therein. w15x P.W. Atkins, M.H. Miller, Mol. Phys. 15 Ž1968. 491. w16x R. Zawodny, S. Wozniak, G. Wagniere, ´ ` Mol. Phys. 91 Ž1997. 165. w17x M. Surma, Mol. Phys. 93 Ž1998. 271. w18x M. Surma, Mol. Phys. 90 Ž1997. 993. w19x P.N. Schatz, A.J. Mc Caffery, Quart. Rev. 23 Ž1969. 552. w20x P. Kleindienst, G. Wagniere, ` Chem. Phys. Lett. 288 Ž1998. 89. w21x P. Kleindienst, Doctoral Dissertation, University of Zurich, 1999. w22x V.M. Agranovich, V.L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, Springer, Berlin, 1984. w23x J.J. Hopfield, D.G. Thomas, Phys. Rev. Lett. 4 Ž1960. 357. w24x J.J. Hopfield, D.G. Thomas, Phys. Rev. Lett. 5 Ž1960. 505.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173 w25x D.L. Portigal, E. Burstein, J. Phys. Chem. Solids 32 Ž1971. 603. w26x V.A. Markelov, M.A. Novikov, A.A. Turkin, JETP Lett. 25 Ž1977. 379. w27x R. Zawodny, Adv. Chem. Phys. 85 Ž1993. 307, Part 1. w28x G.L.J.A. Rikken, E. Raupach, Nature 390 Ž1997. 493. w29x H.G. Brittain, F.S. Richardson, J. Am. Chem. Soc. 98 Ž1976. 5858. w30x C. Gorller-Walrand, J. Godemont, J. Chem. Phys. 67 Ž1977. ¨ 3655. w31x G.L.J.A. Rikken, E. Raupach, Phys. Rev. E 58 Ž1998. 5081. w32x G. Wagniere, ` Z. Phys. D 8 Ž1988. 229. w33x E.D. Commins, P.H. Bucksbaum, Weak interactions of leptons and quarks, Sect. 9.5, Cambridge University Press, Cambridge, 1983.
173
w34x E.G. Adelberger, W.C. Haxton, Ann. Rev. Nucl. Part. Sci. 35 Ž1985. 501, and refs. cited therein. w35x O. Buchardt, Angew. Chem. 86 Ž1974. 222, and references cited therein. w36x L.D. Barron, Chem. Soc. Rev. 15 Ž1986. 189. w37x L.D. Barron, J. Am. Chem. Soc. 108 Ž1986. 5539. w38x P.S. Pershan, Phys. Rev. 130 Ž1963. 919. w39x J.P. Van der Ziel, P.S. Pershan, L.D. Malmstrom, Phys. Rev. Lett. 15 Ž1965. 190. w40x G. Wagniere, ` Phys. Rev. A 40 Ž1989. 2437. w41x S. Wozniak, G. Wagniere, ´ ` R. Zawodny, Phys. Lett. A 154 Ž1991. 259. w42x S. Wozniak, M.W. Evans, G. Wagniere, ´ ` Mol. Phys. 75 Ž1992. 99.
The magnetochiral effect and related optical phenomena Georges H. Wagniere `
)
Institute of Physical Chemistry, UniÕersity of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland Received 30 October 1998
Abstract The magnetochiral effect has only recently been experimentally detected and measured. The interest for the effect lies less in any foreseeable practical application, than rather in the symmetry principles that govern it. In this article we first show how the magnetochiral effect is related to other optical and magneto-optical phenomena, such as natural optical activity, the Faraday, Cotton–Mouton effects and quadratic magnetic field-enhanced optical activity. The general principles of the measurements of magnetochiral birefringence and magnetochiral dichroism are then briefly described. The basic theory of the non-linear-optical, inverse magnetochiral birefringence is presented and possibilities of its detection are explored. The emphasis of this paper lies in the discussion of selection rules and in symmetry considerations. q 1999 Elsevier Science B.V. All rights reserved.
1. Nature of the magnetochiral effect As the name implies, the magnetochiral effect occurs in chiral media in the presence of a magnetic field. The effect is related both to natural optical activity which manifests itself only in chiral media, and to magnetic optical activity which is observed in media of any symmetry but requires a static magnetic field parallel to the direction of propagation of the incident light. Natural optical activity has been known since the early 19th century, its prominent pioneers being Arago, Biot and Pasteur w1x. Magnetic optical activity was discovered in 1846 by Faraday w2,3x. Both effects are circular differential, corresponding to a different response for left ŽL. and right ŽR. circularly )
Corresponding author. Fax: q41-1-635-6813; E-mail: [email protected]
polarized light. Outside resonances one measures the difference of refractive index n LŽ l . y n RŽ l . Žrespectively, as natural optical rotatory dispersion, ORD; or as magnetic optical rotatory dispersion, MORD., inside absorption bands the difference of the absorption coefficient ´ LŽ l . y ´ RŽ l . Žrespectively, as natural circular dichroism, CD; or as magnetic circular dichroism, MCD.. The magnetochiral effect, in contrast, is not circular differential. In a chiral medium the index of refraction and the absorption coefficient for arbitrarily polarized light undergo a shift in value when a static magnetic field is applied collinear with the direction of propagation of the incident radiation. The sign of the shift changes if the relative direction of field and beam is inverted. For a given relative direction, the sign of the effect changes on going from one chiral medium to its enantiomer Žmirror image.. This situation is summarized in Table 1. The
0301-0104r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 0 2 3 - 3
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
166
Table 1 Relative magnetochiral shifts of the refractive index n and the absorption coefficient ´ for parallel and antiparallel configurations of the static magnetic field B0 Magnetic field configuration
B0 ≠≠ B0 ≠x
Enantiomer d
l
nq D n ´ qD´ ny D n ´ y D´
ny D n ´ yD´ nq D n ´ q D´
absolute magnitude of the magnetochiral signals in molecular media is predicted to be 10y3 to 10y2 times smaller than that for magnetic circular birefringence or dichroism. For a diamagnetic medium in a magnetic field of 5T, this leads to an estimate for D nrn of the order of 10y8 to 10y6 . The magnetochiral effect has only recently been detected in molecular media. The measurement of magnetochiral birefringence described in Section 4 confirms the above estimate for D nrn. In Section 5 we will see that for highly chiral compounds, exhibiting at the same time strong magnetic circular dichroism, magnetochiral dichroism D ´r´ of the order of up to 10y3 may be observed. Although, as of yet, the magnetochiral effect is of no apparent practical use, it will be shown that it obeys symmetry rules which are of quite fundamental significance. The magnetochiral effect was theoretically predicted in the course of the last forty years. A term responsible for magnetochiral birefringence is already to be found in a paper by Groenewege w4x which is mainly concerned with magneto-optical rotation. Baranova and Zel’dovich w5x predicted the existence of a ‘‘new linear magnetorefractive effect’’ of similar origin. Independently, the occurrence of the corresponding effect in absorption w6–10x and emission w11x was deduced. The names ‘magnetochiral birefringence’ and ‘magnetochiral dichroism’ were suggested by Barron and Vrbancich w10x and have been adopted.
2. Theory of magnetochiral birefringence As is well known, the interaction between a molecular system and the radiation field may be
expanded into a multipole series: Hint s ym P E Ž t . y m P B Ž t . y Q := E Ž t . q . . . , with E Ž t . s Ey exp Ž yi v t . q Eq exp Ž qi v t . , B Ž t . s By exp Ž yi v t . q Bq exp Ž qi v t . . Ž 1. In the long-wavelength approximation for ordinary refraction and absorption, the electric dipole term may in general be considered as dominant. For our purposes this is, however, insufficient and we must include the magnetic dipole and electric quadrupole contributions in our discussion as well. Simple graphs may conveniently visualize the relation of the magnetochiral effect to other optical phenomena w9,12,13x. These figures are directly connected to the quantum mechanical expressions for the radiation-induced polarization p in a molecule. We assume for the time being that the frequency of incident radiation lies outside molecular resonances. Fig. 1a represents ordinary Rayleigh scattering or refraction. An incident photon of frequency v promotes the molecular system from an initial state < a: into a non-stationary superposition of states < k :, whereupon the photon is re-emitted Žscattered. within a time of the order of 2 prv . In general, this phenomenon is mainly due to the electric dipole interaction with the radiation field. Fig. 1b shows the contribution which is responsible for natural optical activity. Here the magnetic dipole interaction with the radiation v Ž M . also comes into play. In non-isotropic media there may be a contribution from the electric quadrupole interaction v Ž Q ., as well. Fig. 1b translates into the expression w9x: ² a < m < k :² k < m P By < a: p Ž1. Ž v ;y v Ž M . . s Ý "Ž vka y v . k
½
q
² a < m P By < k :² k < m < a: "Ž vka q v .
5
.
Ž 2. In the corresponding susceptibility, we obtain products of electric transition moments ² a < m < k : which are parity-odd polar vectors, and of magnetic transition moments ² k < m < a: which are parity-even axial vectors. The optical activity tensor of the form m m is therefore odd with respect to space inversion, and the effect occurs in non-centrosymmetric media only; in fluids, the molecules must be chiral.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
167
is even with respect to parity, and that the effect occurs in all matter. Magnetochiral birefringence arises through the combination of a magnetic dipole interaction with the static magnetic field and a magnetic dipole, or electric quadrupole interaction with the radiation. This is shown in Fig. 1d leading to the expression p Ž2. Ž v ;y v , Ž M . ,0 Ž M . . sÝ k
Ý l
² a < m < l :² l < m P By < k :² k < m P B0 < a: "2 Ž v l a y v . v k a
q 5 similar terms .
Fig. 1. Graphs for the radiation-induced molecular polarization due to the optical effects discussed in the text. The general form of the tensor products of the corresponding susceptibility are indicated. m stands for a Žparity-odd. electric dipole, m for a Žparity-even. magnetic dipole matrix element. The overall symmetry with respect to parity Žspace inversion. P and with respect to time reversal T w10x is indicated at right for each case.
Ž 4.
The product of two magnetic dipole and of only one electric dipole matrix elements m mm shows that the corresponding susceptibility tensor is odd w.r. to parity. The effect occurs only in non-centrosymmetric media, in particular, in isotropic media that are chiral. Furthermore, it may be deduced w6,13x that an odd number of magnetic dipole andror electric quadrupole interactions leads to a circular differential effect; an even number thereof does not. The magnetochiral effect is consequently not circular differential. To further illustrate this point, we show in Fig. 2 the graphs for two other magneto-optical effects: Fig. 2a represents the well-known Cotton– Mouton–Voigt effect w14x which is quadratic in the static field, occurs in all media and is not circular
Magnetic optical activity is represented by Fig. 1c. One incident vertex symbolizes the interaction with the static magnetic field B0 , denoted by 0Ž M ., the other the electric dipole interaction with the radiation of frequency v . The corresponding mathematical expression for the induced polarization becomes p Ž2. Ž v ;y v ,0 Ž M . . sÝ k
Ý l
² a < m < l :² l < m P Ey < k :² k < m P B0 < a: "2 Ž v l a y v . v k a
q 5 similar terms .
Ž 3.
The products of molecular vectors of the kind mm m Žpolar vector = polar vector= axial vector. let us conclude that the tensor for magnetic optical activity
Fig. 2. Graphs for the Cotton–Mouton–Voigt effect Ža. and for the quadratic magnetic field-induced contribution to natural optical activity Žb.. The former effect occurs in all matter, the latter one only in chiral media.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
168
differential. In Fig. 2b we have replaced the electric dipole interaction with the radiation by a magnetic dipole interaction. We obtain a circular differential contribution to natural optical activity which is quadratic in the static magnetic field w15,16x and which, somewhat misleadingly, also has been named the ‘quadratic Faraday effect’. The recent experimental observation of this effect is reported in Refs. w17,18x. 3. Theory of magnetochiral dichroism To study absorption or emission, the quantity of interest is the transition probability per unit time: 2p w Ž a ™ b, v . s 2 <² b < Hint Ž v . < a:< 2d Ž v b a y v . . " Ž 5. If we introduce the molecule–radiation interaction hamiltonian Ž1. into Ž5., we will obtain a series of terms, the first one being the electric dipole–electric dipole term Že–e. responsible for ‘ordinary’ electric dipole absorption or emission. The electric dipole– magnetic dipole Že–m., and electric dipole–electric quadrupole Že–q. cross-terms lead, in a chiral medium, to circular dichroism. If, in addition to the interaction with the radiation, we assume the states < a:, < b :, to be perturbed by a static magnetic field B0 parallel to the direction of propagation of the incidentremitted radiation, then to first order in B0 , the Že–e. terms will contain contributions to magnetic circular dichroism ŽMCD., and the Že–m. and Že–q. terms will lead to expressions describing magnetochiral dichroism w6,11x. For a transition between two non-degenerate molecular states and after isotropic averaging, the contribution to magnetochiral dichroism reads: Dw Ž a ™ b; Ž e y m . , B0 . 2 ² < < : ² < < : s 3 Re Ý vy1 nb Ž n m a P a m b 3" n/b =² b < m < n: q ² b < m < a: P ² a < m < n: = ² n < m < b : . q
² < < : ² < < : Ý vy1 na Ž b m n P a m b n/a
=² n < m < a: q ² b < m < a: P ² n < m < b : = ² a < m < n: . = g Ž v b a , v . Ž EyP Bq= B0 . .
Ž 6.
Here, as in Ž4., we encounter products of an electric dipole transition moment and of two magnetic dipole transition moments, of the type m mm. An expression similar to Ž6. is obtained for the corresponding electric dipole–electric quadrupole terms Že–q. w6x. As is known from the theory of magnetic optical activity w3,19x, there is a variety of contributions to MCD and MORD, depending on if a molecule possesses magnetically degenerate electronic states. In diamagnetic molecules of low symmetry one encounters only so-called B-terms, reflecting the mixing of non-degenerate molecular states under the influence of the static magnetic field. In molecules of higher symmetry, exhibiting threefold or higher rotation axes, bisignate A-terms are to be expected and in addition, if the ground state is magnetically degenerate, also temperature-dependent C-terms. The A- and C-terms derive from the splitting of the degenerate levels by the applied magnetic field. An analogous situation prevails in the magnetochiral effect. Expression Ž6. describes the magnetochiral B-terms. The theory of magnetochiral A- and C-terms is discussed in Refs. w9,10x. The interested reader is reminded that expression Ž5., the ‘Fermi Golden Rule’, is not the only way to calculate the transition probability. One may equivalently consider the radiation-induced molecular polarization p, as in Section 2, but including damping w9,10x. The corresponding molecular susceptibility is then complex. The imaginary part of this complex expression represents the absorptionremission cross-section.
4. Measurement of magnetochiral birefringence The magnetochiral effect occurs in all chiral media and is not averaged out in fluids and gases. The relative shift of the refractive index D nrn to be expected in a diamagnetic molecular system Ž B-term. outside resonances in a magnetic field of 5 T is of the order 10y8 to 10y6 , which is very small. We here briefly describe the recent detection and measurement of magnetochiral birefringence in liquids by interferometry w20,21x. The magnetochiral birefringence will tend to be overshadowed by the larger effects of natural and magnetic optical activity. It is
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
therefore necessary to fully exploit the different symmetry properties of these various phenomena. As illustrated in Ref. w20x, an incident beam of linearly polarized light of 633 nm produced by a He–Ne laser is divided into two sub-beams. The probe consists of two compartments, one containing the d-, the other the l-enantiomer of the substance to be investigated. The sub-beams travel in opposite directions through both compartments in which there is a static axial magnetic field of 5 T. The experiment is devised in such a way that natural and magnetic optical rotation are compensated to zero, while the magnetochiral signal is enhanced. One sub-beam gets phase-retarded by the magnetochiral effect, the other one phase-accelerated. The two sub-beams are brought to interference, and the magnitude and sign of the phase shift are then measured. To enhance the weak signals, high concentrations of the material samples are required. Therefore, the first measurements w20x have been performed with pure liquids. For the highly optically active compound 3-Žtrifluoroacetyl.-camphor one measures under the stated conditions D n s q6 = 10y8 referred to the l-enantiomer in a static field configuration parallel to the direction of propagation of the light beam. For the compound l-carvone one measures in the same conditions q2.5 = 10y8 . The device described in Ref. w20x should also be applicable to the measurement of the magnetochiral birefringence in chiral cubic or uniaxial crystals. In the latter case, the optic axis of the crystal should be collinear with B0 and with the light beam direction. It is to be noted that this kind of measurement requires the availability of both enantiomorphous crystal forms. The magnetochiral effect may be considered as a special case of magneto-spatial dispersion effects that occur in certain crystal classes of reduced symmetry for particular directions of light propagation w22–27x. However, as we are mainly interested in the optical properties of isotropic chiral molecular fluids and of uniaxial chiral crystals along the optic axis, we will here not go into the details of these additional aspects in anisotropic media. 5. Measurement of magnetochiral dichroism The recent first measurement of magnetochiral dichroism in a molecular probe has been performed
169
by Rikken and Raupach in emission w28x. These authors have investigated solutions of the chiral trisŽ3-trifluoroacetyl-camphor. –Eu3q complex: The sample was excited with unpolarized light of 350 nm, and the luminescence collected by optical fibers in the directions parallel and antiparallel to the applied alternating magnetic field. The axial magnetic field, oscillating with a frequency of 0.9 Hz, had a peak strength of 0.9 T. Specific emission wavelengths were selected by a grating monochromator. The signal detection was performed phase-sensitively. The registered signals showed opposite signs for the d- and l-enantiomers and were identified as being essentially due to the 5 D 0 ™7 F1 and 5 D 0 ™7 F2 transitions of the ion. As may be inferred from the theory w5–11x, the order of magnitude of the magnetochiral anisotropy factor Žper unit of magnetic field strength. g Ž MChD. s
I≠ ≠ y I≠ x I≠ ≠ q I≠ x
,
Ž 7a .
should be roughly equal to that of the product of the dissymmetry factors for natural circularly polarized emission ŽCPE. w29x g Ž CPE . s
I L y IR I L q IR
,
Ž 7b .
and of magnetic circular dichroism ŽMCD; per unit field strength. w30x: g Ž MCD. s
2Ž ´ L y ´ R .
´L q´R
.
Ž 7c .
I and ´ denote the emission intensity and absorption coefficient, respectively. Rikken and Raupach have estimated g ŽMChD. to be of the order of 5 = 10y3 for the 5 D 0 ™7 F1 , and 4 = 10y4 for the 5 D 0 ™7 F2 transition in the investigated complex, which is remarkably high. This is related to the fact that we here have A-terms. The measured signals in the range 590–620 nm agree with this order of magnitude and obey the expected symmetry rules for the magnetochiral effect. A first measurement of magnetochiral dichroism in absorption is reported by the same authors w31x for the optically active uniaxial crystal a-NiSO4 P 6H 2 O. The applied magnetic field, collinear with the incident radiation and with the optic axis, was alternated at a frequency of 0.9 Hz, as for the emission experi-
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
170
ment. The magnetochiral dichroism, registered over the range 900–1500 nm, is of an order of magnitude comparable to that recorded in the previous example in emission. Highly chiral complexes or crystals containing ions of transition metals or of rare-earth elements appear indeed to be well-suited to study magnetochiral dichroism.
6. Magnetochiral effect in atomic and nuclear physics Under the influence of a parity-non-conserving interaction, such as that caused by the parity-violating weak neutral currents, an atomic system is predicted to become optically active and also to show magnetochiral dichroism. The theoretical model described in Ref. w32x is based on the transitions between the spin–orbit coupled states 2 P3r2 and 2 P1r2 of the hydrogen atom. In the unperturbed atom, these transitions are magnetic dipolerelectric quadrupoleallowed. The parity-violating interaction couples the 2 S1r2 states to the 2 P1r2 states, thereby inducing in the 2 P3r2 ™2 P1r2 transitions weak electric dipole components. If, in addition, we assume the atom to be in a static magnetic field, the degeneracies of the states are lifted, and the transitions considered occur between energy-split pure eigenstates of Jz . It is then established that for any transition between particular states for which D m j s "1, there is a difference in the intensity of light emitted parallel and antiparallel to the axis of quantization, as summarized in Table 2. This is an expression of magnetochiral dichroism. The relative intensity shift D I changes sign if the
Table 2 Predicted w32x relative magnetochiral emission intensity shift in a parity broken atom for forward q k and backward y k propagation along the axis of quantization D mj
Propagation
Polarization
Relative intensity
y1
qk yk qk yk
L R R L
IqD I IyD I IyD I IqD I
q1
According to the Zeeman selection rules, the radiation is either left ŽL. or right ŽR. circularly polarized.
axis of magnetic quantization is inverted and, being proportional to the parity-violating interaction, also would change sign if the chirality were reversed. Furthermore, we notice that if the magnetic field is absent, and the transitions become degenerate, then the emitted light in both directions, qk and yk, is of same total intensity but elliptically polarized. The atomic model considered then exhibits circular dichroism. From this we conclude that the magnetochiral effect and natural optical activity are of same origin. The lifting of magnetic degeneracy distinguishes the former from the latter. Although this simple model may appear rather academic, it nevertheless illustrates basic symmetry principles. These principles should be essentially valid in any parity-non-conserved system described by a hamiltonian which commutes with the total angular momentum containing both spin and orbital contributions. Interestingly, also for atomic nuclei one finds w33x: ‘‘Parity violation results in circular polarization of g emission from unpolarized nuclei or fore–aft asymmetry in g emission from polarized nuclei’’. A polarized nucleus is in a definite magnetic substate. The superposition of multipoles of different parity in the emission of radiation due to parity-violation leads to the predicted and measured w33,34x fore–aft asymmetry in the emission intensity. This is then, from our point of view, a manifestation of magnetochirality.
7. Magnetochiral effect and molecular evolution The parity-non-conserving influences in the universe essentially all derive from the primary parityviolating weak interactions. It is not our aim here to speculate on the complex mechanisms by which these basic forces have led to the general asymmetry of the universe, much less on the origins of the homochirality of life. We are rather interested in parity-non-conserving influences of secondary origin and in the way they further induce chirality. These are: Ž1. Already present chiral matter, which by direct physical and chemical interaction leads to, or catalyzes, the formation of additional chiral products. This is the basis of most asymmetric syntheses.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
171
dichroism this should exert a similar enantioselective influence as 2 w8x. All three effects mentioned here are local and may therefore, with respect to the sign of the induced chirality, go either way. Our immediate concern is the relative importance of 3, as compared to 1 and 2. We are aware of the smallness of the magnetochiral effect. On the other hand, the necessary conditions may be met almost anywhere in the universe. Wherever there is light and there are static magnetic fields not exactly perpendicular to the direction of lightpropagation, a chiral influence will be exerted w8,36,37x. The magnetochiral effect may thus play a role in enantioselective processes in the universe.
8. Inverse magnetochiral birefringence: a non-linear optical effect
Fig. 3. Graphs representing the nonlinear optical inverse Faraday effect and inverse magnetochiral birefringence, in comparision with the related linear optical phenomena.
Ž2. The influence of circularly polarized light which via circular dichroism in a racemic mixture leads to enantioselective photodestruction or photoenrichment w35x. Ž3. The presence of a static magnetic field parallel to the direction of propagation of incident light of arbitrary polarization. Due to magnetochiral
Optical rectification p Ž2. Ž0;y v , v ., the radiation-induced induction of static electric polarization, occurs only in non-centrosymmetric media. It is, however, forbidden in chiral liquids outside resonances. On the other hand it is possible, to second order in the interaction with the radiation, to induce a static magnetization mŽ2. Ž0;y v , v . in any medium, also a fluid one. The incident light has to be circularly polarized, however. Fig. 3b represents this socalled inverse Faraday effect w38x, the existence of which has been experimentally confirmed w39x. We notice that Fig. 3b is related to that of the ordinary Faraday effect ŽFig. 3a., through an interchange of ingoing and outgoing vertices and by combining with opposite signs the frequencies of the electric dipole interactions with the radiation. We may now
Table 3 Difference in symmetry between the inverse Faraday effect w38,39x and the inverse magnetochiral effect w40–42x
Inverse Faraday effect
Polarization of incident radiation
Enantiomer
Relative sign of induced magnetization
L
d l d l d l d l
q q y y q y q y
R Inverse magnetochiral birefringence
L R
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173
172
similarly go from Fig. 3c for ordinary magnetochiral birefringence to Fig. 3d for inverse magnetochiral birefringence w40–42x. While the magnetization induced by the inverse Faraday effect changes its sign for L- and R-circularly polarized light, that due to the inverse magnetochiral birefringence is independent of the state of polarization of the incident radiation. The inverse magnetochiral effect occurs only in chiral media, including liquids, and changes sign for enantiomers ŽTable 3.. For a diamagnetic chiral fluid medium the isotropically averaged magnetization per molecule due to the inverse magnetochiral birefringence is found to be: mŽ2. Ž 0;y v ,q v Ž M . .
v s
3" 2
=
½
Ý Ý Ž m k a P m al = m l k . k
l
References
v Ž vla q vka. v l a Ž v l2a y v 2 .Ž v k2a y v 2 .
y Ž m l k P m k a = m al . =
Ž e"r2 mc .rea0 in cgs units, corresponding to a magnetization of 10y9 to 10y8 G, a magnetic induction of 10y8 to 10y7 G, or 10y12 to 10y11 T w40x. The inverse Faraday effect has been measured with an induction coil technique w39x. The inverse magnetochiral birefringence has not yet been detected experimentally. By using short and powerful laser pulses to increase the induced magnetic induction and its time-derivative, it is conceivable that measurable signals should be obtained. The interest for the inverse magnetochiral birefringence primarily lies in the symmetry principles which govern it. Wherever light of arbitrary polarization interacts with a chiral polarizable medium, it induces a static magnetic field, no matter how small.
Ž vla q vka. 2 vla vk aŽ vla y v . Ž vk a q v .
= Ž Ey= Bq . .
5 Ž 8.
This represents the electric dipole-magnetic dipole contribution. As is the case for magnetochiral birefringence and dichroism, the products of molecular quantities appearing in this expression are of the form m mm, which is odd with respect to space inversion Žsee Sections 2 and 3.. An analogous electric dipole–electric quadrupole contribution may also be derived w41,42x. In a paramagnetic medium there are additional, temperature-dependent terms w40–42x. The magnitude of the magnetization, M s Nm Ž N the number of molecules per unit volume., induced by the inverse Faraday effect is proportional to the intensity of incident radiation. With a light intensity of 10 8 W cmy2 an order of magnitude of 10y6 to 10y5 G has been measured for M w39x. The magnetization due to inverse magnetochiral birefringence is also predicted to be proportional to the light intensity, but should be smaller by the order of
w1x J.P. Mathieu, Handbuch der Physik, vol. 28, Spektroskopie II, Springer, Berlin, 1957, p. 333. w2x J.H. van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, London, 1965, p. 367. w3x A.D. Buckingham, P.J. Stephens, Annu. Rev. Phys. Chem. 17 Ž1966. 399. w4x M.P. Groenewege, Mol. Phys. 5 Ž1962. 541. w5x N.B. Baranova, B.Ya. Zel’dovich, Mol. Phys. 38 Ž1979. 1085. w6x G. Wagniere, ` A. Meier, Chem. Phys. Lett. 93 Ž1982. 78. w7x S. Wozniak, R. Zawodny, Acta Phys. Pol. A 61 Ž1982. 175. ´ w8x G. Wagniere, ` A. Meier, Experientia 39 Ž1983. 1090. w9x G. Wagniere, ` Z. Naturforsch. 39a Ž1984. 254. w10x L.D. Barron, J. Vrbancich, Mol. Phys. 51 Ž1984. 715. w11x G. Wagniere, ` Chem. Phys. Lett. 110 Ž1984. 546. w12x J.F. Ward, Rev. Mod. Phys. 37 Ž1965. 1. w13x G. Wagniere, ` J. Chem. Phys. 77 Ž1982. 2786. w14x A.C. Boccara, J. Ferre, B. Briat, M. Billardon, J.P. Badoz, J. Chem. Phys. 50 Ž1969. 2716, and references cited therein. w15x P.W. Atkins, M.H. Miller, Mol. Phys. 15 Ž1968. 491. w16x R. Zawodny, S. Wozniak, G. Wagniere, ´ ` Mol. Phys. 91 Ž1997. 165. w17x M. Surma, Mol. Phys. 93 Ž1998. 271. w18x M. Surma, Mol. Phys. 90 Ž1997. 993. w19x P.N. Schatz, A.J. Mc Caffery, Quart. Rev. 23 Ž1969. 552. w20x P. Kleindienst, G. Wagniere, ` Chem. Phys. Lett. 288 Ž1998. 89. w21x P. Kleindienst, Doctoral Dissertation, University of Zurich, 1999. w22x V.M. Agranovich, V.L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, Springer, Berlin, 1984. w23x J.J. Hopfield, D.G. Thomas, Phys. Rev. Lett. 4 Ž1960. 357. w24x J.J. Hopfield, D.G. Thomas, Phys. Rev. Lett. 5 Ž1960. 505.
G.H. Wagnierer ` Chemical Physics 245 (1999) 165–173 w25x D.L. Portigal, E. Burstein, J. Phys. Chem. Solids 32 Ž1971. 603. w26x V.A. Markelov, M.A. Novikov, A.A. Turkin, JETP Lett. 25 Ž1977. 379. w27x R. Zawodny, Adv. Chem. Phys. 85 Ž1993. 307, Part 1. w28x G.L.J.A. Rikken, E. Raupach, Nature 390 Ž1997. 493. w29x H.G. Brittain, F.S. Richardson, J. Am. Chem. Soc. 98 Ž1976. 5858. w30x C. Gorller-Walrand, J. Godemont, J. Chem. Phys. 67 Ž1977. ¨ 3655. w31x G.L.J.A. Rikken, E. Raupach, Phys. Rev. E 58 Ž1998. 5081. w32x G. Wagniere, ` Z. Phys. D 8 Ž1988. 229. w33x E.D. Commins, P.H. Bucksbaum, Weak interactions of leptons and quarks, Sect. 9.5, Cambridge University Press, Cambridge, 1983.
173
w34x E.G. Adelberger, W.C. Haxton, Ann. Rev. Nucl. Part. Sci. 35 Ž1985. 501, and refs. cited therein. w35x O. Buchardt, Angew. Chem. 86 Ž1974. 222, and references cited therein. w36x L.D. Barron, Chem. Soc. Rev. 15 Ž1986. 189. w37x L.D. Barron, J. Am. Chem. Soc. 108 Ž1986. 5539. w38x P.S. Pershan, Phys. Rev. 130 Ž1963. 919. w39x J.P. Van der Ziel, P.S. Pershan, L.D. Malmstrom, Phys. Rev. Lett. 15 Ž1965. 190. w40x G. Wagniere, ` Phys. Rev. A 40 Ž1989. 2437. w41x S. Wozniak, G. Wagniere, ´ ` R. Zawodny, Phys. Lett. A 154 Ž1991. 259. w42x S. Wozniak, M.W. Evans, G. Wagniere, ´ ` Mol. Phys. 75 Ž1992. 99.