Theory of Rayleigh–Brillouin optical activity light scattering applicable to chiral liquids

Abstract

It has long been understood that dilute samples of chiral molecules such as rarefied gases should exhibit Rayleigh optical activity. We extend the existing theory by accounting for molecular dynamics and correlations, thus obtaining a more general theory of Rayleigh–Brillouin optical activity applicable to dense samples such as neat liquids.

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1 Introduction

It was predicted a little over fifty years ago by Barron and collaborators that chiral molecules should exhibit Rayleigh optical activity (RayOA): differential Rayleigh scattering with respect to left- and right-handed circular polarisations of light.^1–3 In the theoretical descriptions of RayOA published to date, each molecule is effectively held static in position and orientation and scattered intensities due to different molecules are added incoherently.^1–11 These descriptions provide expressions for the total intensity of the analysed signal and are best suited to dilute samples such as rarefied gases in which correlations between molecules are unimportant.^3,12–15

In this paper, we extend the existing theory of RayOA by accounting for the translational and rotational dynamics of the molecules and adding scattered fields due to different molecules coherently, thus obtaining a more general theory that describes Rayleigh–Brillouin optical activity (RayBOA). Our theory provides expressions for not only the total intensity of the analysed signal but also the underlying frequency spectrum. It is applicable to dilute samples such as rarefied gases as well as dense samples such as neat liquids in which correlations between molecules are important.

Raman optical activity (ROA)^{1–3,16–18} is the inelastic sister of RayOA. The theory of ROA does not need to be extended like the theory of RayOA, however, as one can add Raman scattered intensities due to different molecules incoherently at essentially all sample densities.^3,19

(Linear) RayOA is not to be confused with hyper Rayleigh optical activity (HRS OA)^20–24 and its extensions,^25,26 which are distinct, nonlinear optical phenomena.

For large biological scatterers, the terminology “circular intensity differential scattering” (CIDS) is often used^27–31 for what is essentially RayOA.

2 Theory of RayBOA

Let us consider weak, monochromatic, off-resonant, planar light incident upon a non-conducting fluid of small, diamagnetic, chiral molecules, as illustrated in Fig. 1. Fluctuations of the optical properties within the scattering volume give rise to Rayleigh–Brillouin scattering away from the forward direction,^32–54 a fraction of which is analysed at a detector in the far field. In what follows, we derive expressions for dimensionless circular spectral differentials and dimensionless circular intensity differentials which serve as convenient measures of the Rayleigh–Brillouin optical activity exhibited by the sample.

Fig. 1 Our scattering geometry, illustrated for neat (1R,5R)-α-pinene and an SCP configuration.

Our derivation borrows heavily from theoretical descriptions of Rayleigh–Brillouin scattering by Landau and Lifshitz¹² and Berne and Pecora¹³ and is essentially an amalgamation of these with Barron's mechanism of Rayleigh optical activity.^1–3 See also ref. 55–58. We work in an inertial frame of reference with time t and position vector r = xx̂ + yŷ + zẑ, where x, y and z are right-handed Cartesian coordinates and x̂, ŷ and ẑ are the associated unit vectors. The Einstein summation convention is to be understood with respect to unprimed Greek indices α, β, … ∈ {x,y,z} and primed Greek indices α′,β′, … ∈ {X⁽ⁿ⁾,Y⁽ⁿ⁾,Z⁽ⁿ⁾}, where X⁽ⁿ⁾, Y⁽ⁿ⁾ and Z⁽ⁿ⁾ are molecule-fixed Cartesian coordinates for the nth molecule. Complex quantities are decorated with tildes and unit vectors are decorated with carets. We use SI units throughout.

2.1 The sample

Within the scattering volume V, we model the sample as a collection of vibronically polarisable molecules that can translate with position vectors R⁽ⁿ⁾ = R⁽ⁿ⁾(t) and rotate with Euler angles ϑ⁽ⁿ⁾ = ϑ⁽ⁿ⁾(t), φ⁽ⁿ⁾ = φ⁽ⁿ⁾(t) and χ⁽ⁿ⁾ = χ⁽ⁿ⁾(t) (n ∈ {1, …}).⁵⁹ In the interests of generality, we say nothing about the explicit forms of the positions and orientations of the molecules, except that they are such that the sample is optically homogeneous and isotropic on average.

We take the light to satisfy Maxwell's equations in the form

together with the constitutive relation
where D̃ = D̃(r,t), Ẽ = Ẽ(r,t), B̃ = B̃(r,t), P̃ = P̃(r,t) and M̃ = M̃(r,t) are the complex displacement, electric, magnetic, polarisation and magnetisation fields and ω is the angular frequency of the incident light.^3,6,60

Working in the domain of linear optics to first order in multipolar expansions whilst neglecting local field corrections, we take

with
where [small mu, Greek, tilde]⁽ⁿ⁾_α = [small mu, Greek, tilde]⁽ⁿ⁾_α(t), [capital Theta, Greek, tilde]⁽ⁿ⁾_αβ = [capital Theta, Greek, tilde]⁽ⁿ⁾_αβ(t) and are the complex electric dipole, electric quadrupole and magnetic dipole moments of the nth molecule and α⁽ⁿ⁾_αβ = α⁽ⁿ⁾_βα = α⁽ⁿ⁾_αβ(t), and A⁽ⁿ⁾_α,βγ = A⁽ⁿ⁾_α,γβ = A⁽ⁿ⁾_α,βγ(t) are the vibronic† electric dipole–electric dipole, electric dipole–magnetic dipole and electric dipole–electric quadrupole polarisability tensors of the nth molecule.^3,6 The laboratory-fixed components of the polarisability tensors are related to the molecule-fixed components via relations like
for example, where is the direction cosine tensor for the nth molecule.^3,59

2.2 Incident light

We take the incident light (superscript i) to satisfy Maxwell′s equations in the formtogether with the constitutive relationwhere D̃ⁱ = D̃ⁱ(r,t), Ẽⁱ = Ẽⁱ(r,t), B̃ⁱ = B̃ⁱ(r,t) are the complex displacement, electric and magnetic fields of the incident light, ε is the average permittivity of the sample and γ is the average optical activity parameter of the sample.^3,6,60 The solutions of (2) and (3) are circularly polarised plane electromagnetic waves and superpositions thereof.³

Let us assume that the incident light propagates only a short distance through the sample, as will typically be the case in an experiment using a small cuvette, for example. Accordingly, we neglect the optical rotation of the incident light (γ → 0) and take

with
where E⁽⁰⁾, ẽⁱ, κⁱ and nⁱ are the electric-field amplitude, complex polarisation vector, wavevector and propagation vector of the incident light and is the average refractive index of the sample with γ = 0.

2.3 Scattered light

We take the scattered light (superscript s) to be the difference between the light and the incident light defined above, as

D̃^s = D̃ − D̃ⁱ, Ẽ^s = Ẽ − Ẽⁱ B̃^s = B̃ − B̃ⁱ, (4)

where D̃^s = D̃^s(r,t), Ẽ^s = Ẽ^s(r,t), B̃^s = B̃^s(r,t) are the complex displacement, electric and magnetic fields of the scattered light.^12,13

Using (3) and (4), we obtain

D̃^s ≈ εẼ^s − γ∇ × Ẽ^s + δD̃ (5)

with

δD̃ ≈ D̃ − (εẼ − γ∇ × Ẽ),

where δD̃ = δD̃(r,t) embodies fluctuations with respect to the average optical properties of the sample, being the difference between the displacement field D̃ and the form it would have if the sample had homogeneous and isotropic constitutive relations (εẼ − γ∇ × Ẽ).

Using (1), (2), (4) and (5) as well as the vector identity

∇ × (∇ × D̃^s) = −∇²D̃^s + ∇(∇·D̃^s),

we find that D̃^s must satisfy the wave equationwhich shows that the fluctuations embodied by the difference δD̃ drive waves in the displacement D̃^s of the scattered light.

Neglecting multiple scattering,‡ we take

δD̃ ≈ δD̃ⁱ (7)

withwhere δD̃ⁱ = δD̃ⁱ(r,t) embodies fluctuations coupled directly to the incident light.

Let us assume that the scattered light (like the incident light) propagates only a short distance through the sample. Accordingly, we neglect the optical rotation of the scattered light (γ = 0) and take the solution of (6) with (7) to be

where t′ = t − n|r − r′|/c is the delayed time.^3,60 Let us assume moreover that the fluctuations are slow relative to the angular frequency ω of the incident light. Accordingly, we simplify (9) by takingwhere we have effectively separated the fast and slow time dependencies of δD̃ⁱ by writing δD̃ⁱ(r′,t′) = [δD̃ⁱ(r′,t′)exp(iωt′)]exp(−iωt′) → [δD̃ⁱ(r′,t)exp(iωt)]exp(−iωt′), thus retaining delay for the fast dependencies only.^12,13

2.4 Analysed signal at the detector

Let us focus now on the form of the scattered light at the detector (superscript d). We take the y–z plane to be the scattering plane, without loss of generality. It is convenient to introduce unit vectors x̂^d = x̂, ŷ^d = cos θŷ − sin θẑ and ẑ^d = sin θŷ + cos θẑ aligned with the detector, where θ is the scattering angle.

As the detector lies in the far field, we take

withwhere R is the distance from the centre of the scattering volume V to the detector (R ≫ V^1/3, κR ≫ 1) and κ, κ^d and n^d are the angular wavenumber, wavevector and propagation vector of the scattered light at the detector.^3,60 Using (10) and (11) as well as the vector identity

−κ^d × (κ^d × V) = (κ^d·κ^d)V − κ^d(κ^d·V)

whilst retaining only the leading contributions which fall off as 1/R, we find that the complex electric field Ẽ^di = Ẽ^di(t) of the scattered light at the detector has the formwhere we have taken Ẽ^di ≈ D̃^di/ε, D̃^di = D̃^di(t) being the complex displacement field of the scattered light at the detector. Substituting (8) explicitly into (13) then integrating by parts and neglecting boundary terms as well as forward-scattering contributions, we obtainwith
where [small alpha, Greek, tilde]_αβ = [small alpha, Greek, tilde]_αβ(t), and Ã_α,βγ = Ã_α,βγ(t) are spatial Fourier transforms of polarisability densities, evaluated at the wavevector difference q = κ^d − κⁱ ≠ 0.^12,13 Note that q = |q| = 2ωn sin(θ/2)/c.

We take the analysed signal Ã^di = Ã^di(t) at the detector to be

with
where ã^d is an analysation vector that picks off the desired polarisation component of the electric field Ẽ^di.

2.5 Frequency spectrum and total intensity

The frequency spectrum of the analysed signal Ã^di can be calculated as the temporal Fourier transform of the autocorrelation of Ã^di, aswhere the angular brackets denote a time average, Ω is an angular frequency and τ is a correlation time.¹³ The total intensity I^di of the analysed signal can then be calculated as the integral of the frequency spectrum , as
Note that for Ω < 0, assuming that Ã^di = 0 for Ω < 0.

Substituting (14) and (15) into (16) whilst working to first order in multipolar expansions, we obtain

with
where K is a prefactor that contains the usual ω⁴, E⁽⁰⁾² and 1/R² scalings characteristic of Rayleigh scattering in the far field and [small alpha, Greek, tilde]^di = [small alpha, Greek, tilde]^di(t) and [small chi, Greek, tilde]^di = [small chi, Greek, tilde]^di(t) are convenient shorthands.

2.6 Dimensionless circular spectral and intensity differentials

RayOA (and by extension RayBOA) can manifest as an intensity difference with respect to left- and right-handed circular polarisation states in the scattered light (scattered circular polarisation or SCP),¹ the incident light (incident circular polarisation or ICP)² or both simultaneously (dual circular polarisation or DCP_I).⁶⁴

As convenient measures of RayBOA, we identify dimensionless circular spectral differentials of the form

and circular intensity differentials Δ of the formwhere a and c refer to analysed polarisation states of the scattered light, b and d refer to polarisation states of the incident light and , , , , and are coefficients that depend on the specific configuration being considered. Substituting (17) into (18) and (19) and making use of basic symmetry arguments, we obtain the results listed in Table 1 withwhere accounts for isotropic electric dipole-electric dipole scattering, accounts for anisotropic electric dipole-electric dipole scattering, accounts for isotropic electric dipole–magnetic dipole scattering, accounts for anisotropic electric dipole–magnetic dipole scattering and accounts for anisotropic electric dipole-electric quadrupole scattering. Note that and are chirally insensitive whereas , and have equal magnitudes but opposite signs for enantiomorphic samples. It follows that both and Δ have equal magnitudes but opposite signs for enantiomorphous samples, thus serving as signatures of chirality.

Table 1 Coefficients for some important SCP, ICP and DCP_I configurations. The N results are averages of the relevant H and V results, the N standing for “natural” (unpolarised incident light or unanalysed scattered light). Note that the entries in rows one through three match the entries in rows four through six, in accord with the principle of reciprocity^3,61–63

a	b	c	d
R	H	L	H					0
R	V	L	V			0		0	0
R	N	L	N					0
H	R	H	L					0
V	R	V	L			0		0	0
N	R	N	R					0
R	R	L	L

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Let us emphasise here that the circular differentials and Δ are not simply related to each other, in particular that

They provide different insights, as we will see below.

As a quick check on the validity of our results, we note that for the special case of a single molecule held fixed at the origin, we have , , , and , where α², β², αG′, β_G² and β_A² are the usual single-molecule invariants.^3,7 This sees our results for the circular intensity differentials Δ reduce immediately to the (rotationally averaged) results reported previously elsewhere,^2–4,7 as they should.

3 Toy model

For the sake of illustration, let us now evaluate the circular differentials and Δ for a toy model of an enantiopure neat liquid. This model is not meant to provide accurate predictions for real liquids. Rather, we include it to demonstrate the mathematical extraction of spectra, the explicit application of our theory to real liquids being a challenging task that we will return to in future publications.

Considering molecules with quasi-cylindrical symmetry for the sake of simplicity, we take

where α and G′ are isotropic polarisabilities; Δα, ΔG′ and ΔA are polarisability anisotropies and û⁽ⁿ⁾ = û⁽ⁿ⁾(t) is a unit vector dictated by the orientation of the nth molecule.^3,13 Note that Δα = ΔG′ = ΔA = 0 for molecules with spherical rather than quasi-cylindrical symmetry. Substituting (21) into (20), we obtain
with
where α², β², αG′, β² = Δα², β_G² = ΔαΔG′ and β_A²= 2ΔαΔA/3 are the usual single-molecule invariants^3,7 and S = S(Ω) and Θ = Θ(Ω) are dynamic structure factors.^13,65

A simple hydrodynamic model (neglecting intramolecular relaxation) gives

where N is the average number density, χ_T is the isothermal compressibility, T is the temperature, γ is the heat capacity ratio, D_T is the thermal diffusivity, v is the adiabatic speed of sound and Γ is the classical sound attenuation coefficient.^12,13,15,54 A simple rotational diffusion model (neglecting translational effects and orientational correlations between molecules) gives
where Γ_Θ is a tumbling rate.^13,66 The time-domain molecular dynamics that underpin these forms for S and Θ are described explicitly in ref. 12, 13, 15, 54 and 66.

Focussing on a right-angled SCP configuration with vertically polarised incident light for the sake of concreteness, we have

Note that the circular intensity differential Δ here reduces to the previously reported result^2–4,7 when Nχ_Tk_BT = 1 (as in an ideal gas). For a liquid with Nχ_Tk_BT < 1, the isotropic (α²,αG′) contributions to Δ are suppressed relative to the anisotropic (β², β_G², β_A²) contributions, changing the magnitude of Δ and perhaps even the sign; a demonstration of the need for our extended theory. The circular spectral differential offers more information than Δ in that the isotropic and anisotropic contributions can be distinguished as a function of the angular frequency Ω. In the limiting case where Γ_Θ ≫ vq ≫ Γq² > D_Tq², we find in particular that
Using typical values,⁷ this leads us to predict that
although considerable variation is possible, of course.

These features are illustrated in Fig. 2 for ω = 3.54 Prad s⁻¹, n = 1.40, γ = 1.33, Nχ_Tk_BT = 0.100, D_Tq²/2π = 80.0 MHz, vq/2π = 3.00 GHz, Γq²/2π = 160 MHz, Γ_Θ/2π = 10.0 GHz, β² = 0.100α², αG′/c = ±1.00 × 10⁻⁵α² and β_G²/c = β_A²/c = ±1.00 × 10⁻⁴α², where the upper and lower signs correspond to opposite enantiomers. The circular spectral sum (i.e. the spectrum of the S₀ Stokes parameter of the scattered light³) is positive and consists of a narrow Gross (centre) line^53,54 and narrow Brillouin lines^40,45,46,53 superposed with a broad Rayleigh wing.^47–52 The circular spectral difference (i.e. the spectrum of the S₃ Stokes parameter of the scattered light³) has opposite signs for opposite enantiomers, appearing inverted at the Gross and Brillouin lines as we have taken 180αG′ and 28β_G² + 4β_A² to have opposite relative signs. The resulting circular spectral differential also has opposite signs for opposite enantiomers and appears inverted at the Gross and Brillouin lines, where it is suppressed in magnitude in accord with our prediction above.

Fig. 2 Circular spectra predicted for a toy model of an enantiopure neat liquid. The red and blue curves correspond to opposite enantiomers.

4 Conclusions

We have presented a theory of RayBOA applicable to dense samples such as neat liquids. There are many possible avenues for future research.

We have evaluated the circular differentials for a toy model of an enantiopure neat liquid. It remains for us to consider more realistic models for a variety of different samples. Molecular dynamics simulations have recently been applied with success to ROA^19,67,68 and might be developed for RayBOA as well.

We have adopted a microscopic approach, facilitating comparison with the existing theory of RayOA.^1–11 A macroscopic approach is also possible^{12,13,15,37,39} and might yield new insights. Care will need to be taken with the choice of macroscopic constitutive relations,^69,70 which must include electric dipole–electric quadrupole contributions to describe RayBOA correctly even for isotropic samples.

We have focussed on SCP, ICP and DCP_I RayBOA for off-resonant illumination of a fluid by planar light. Interesting new features should emerge for illumination near resonance,^5,64,71,72 anisotropic samples⁵ and illumination by structured light.^8,10,11,73 It should also prove fruitful to consider the influence of static magnetic fields,⁷⁴ static electric fields^31,75 and higher-order multipolar contributions.^9,28

A particularly interesting question is whether Rayleigh–Brillouin optical activity can be used to detect chiral (acoustic) phonons in appropriate samples.^19,76–78

Although our focus in this paper has been on small molecules, similar ideas can be developed for larger scatterers, including large biomolecules.^27–31

We will return to these and related tasks elsewhere.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors gratefully acknowledge support from the Royal Society (URF\R1\191243) and EPSRC (EP/R513349/1). We thank Laurence Barron for useful discussions. RPC is a Royal Society University Research Fellow. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript (AAM) version arising from this submission.

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Footnotes

† By “vibronic”, we mean simply that the polarisabilities account for electromagnetic perturbation of the vibrational and electronic degrees of freedom of the molecules. The initial and final states in the polarisabilities are taken to be the same; the molecules do not undergo real vibronic transitions.

‡ The condition 4πμ₀²ω⁴α²NV^1/3 ≪ 1 should be well satisfied in the visible domain by typical small-molecule liquids for a cuvette of volume V = 1 cm³, say.

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