3.1 Introduction

The heart of the book is this chapter.

In this chapter, the theoretical material developed in Chapter 2 is used to calculate explicit expressions in relation to the intensity of light scattered into any direction from a collection of molecules.
The basic equations for all of the optical activity phenomena under discussion are contained in these expressions.

Light scattering studies have always included polarization phenomena.

Tyndall's early investigations with aerosols showed that linear polarization was an important feature of light scattered at right angles.

Secondary waves are scattered in all directions when a wave encounters an obstacle.

Foreign matter can be found in droplets of water or dust in the atmosphere and suspended in liquids within a medium.
Light scattering can occur in transparent materials that are completely free of contaminants.
The first observations of these frequencies in scattered light were made by Krishnan and Raman in 1928.

A perfectly transparent medium does not scatter light.

Consider a plane wave propagating in a medium in which identical numbers of molecule of one type are found in equivalent volume elements.
Waves from different parts of a single volume element have the same phase if the volume element's dimensions are smaller than the wavelength of the incident light.

Through interference with the unscattered component of the incident wave, forward scattering can survive.

No medium can be perfectly homogeneity.

In the limiting case of a rarified gas, the molecule execute thermal motions with a mean free path length much greater than the wavelength of the light.
The total scattered intensity is the sum of the individual scattered intensities, and the phase difference between the waves is positive, so on average destructive interference occurs half the time and constructive interference the rest of the time.
Lord Rayleigh is to blame for this conclusion.

In a liquid, the total scattering power per molecule can be an order of magnitude smaller than in a gas.

Cabannes extended the fluctuation theory to1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556 For theories of light scattering in dense media, we refer to works such as Bhagavantam.

The general fluctuation theory of light scattering will not be incorporated into our treatment, but we will assume that the total scattered intensity is the sum of the scattered intensities from each molecule.

This model provides valid results for most of the optical activity phenomena discussed in the book.

The light quotients of isotropic and anisotropic scattering contributions are used in the expressions for the polarization properties of Rayleigh and Raman.

The isotropic and anisotropic contributions to Rayleigh scattering are dependent on sample density, so the results of Rayleigh scattering only apply to ideal gases.
The results are always incoherent for both isotropic and anisotropic vibrational Raman scattering.

There is complete constructive interference from all of the scatterers and that part of the scattered light in the forward direction is fully coherent.

The model used to derive the results for optical rotation is correct at all sample densities.

The radiation fields generated by the electric and magnetic multipole moments in a molecule are what we consider to be the origin of scattered light.

The last three terms are required for the calculation of electric field-induced birefringence and related phenomena.

In 1979b, Baranova and Zel'dovich gave a penetrating discussion of the validity of this 'local multipole' ap proximation to spatial dispersion.

Circular and linear dichroism are used to describe the different absorption coefficients for the corresponding light.

Lord Rayleigh said that light scattering is the cause of the light's refraction.

Modern treatments can be found in the books of van de Hulst,Newton and Jenkins and White.
A phase change is equivalent to an altered wave velocity when the individual molecule scatters a small part of the incident light and the forward parts of the resulting spherical waves combine and interfere with the primary wave.
Waves scattered into the forward direction from any point in the medium interfere more with the transmitted wave than the non forward scattered light.
It's natural to formulate a theory of'refringent polarization effects' from Lord Rayleigh's scattering model without introducing an index of refraction.
Kauzmann was the first to present a scattering theory of optical rotation, but it was limited to small angles of rotation.
A light beam of arbitrary azimuth, ellipticity and degree of polarization incident may be oriented and absorbing.
For the infinitesimal changes in azimuth, ellipticity, degree of polarization and intensity of the light beam, expressions are derived.

The circular and linear birefringence and dichroism description is what the conventional theories of refringent polarization effects start from.

The transition to a molecular theory is made by linking the Refractive index to the bulk electric polarization and magnetization of the medium, which are related in turn to an appropriate sum of the electric and magnetic multipole moments in individual molecule by the light wave.
The use of an index of refraction can obscure some of the fundamental processes responsible.
The general case of circular and linear birefringence and dichroism can be interdependent with the infinitesimal scattering theory.
This general stituation can be accommodated by the Refractive index theories.
The properties of an optical element are represented by a real four-by-four matrix, the elements of which are functions of refractive index components, which is what the Muller calculus describes.
The Jones two-vector is similar to the Jones two-by-two matrices.
The only way to describe a pure-polarized beam is with the Jones vector, and only the Mueller calculus can incorporate changes in the degree of polarization.
For further discussion of the methods, we refer to the books.

There is a procedure between the basic scattering theory used in this book and the refractive index theory outlined above.

Lord Rayleigh's scattering model can be used to calculate the Refractive indices for linearly and circularly polarized light.

The wavelength of the light affects the thickness of the lamina.

There is a Geometry for forward scattering.

Waves from outside the base of a narrow cone with apex at f tend to interfere with the function of f, so only molecules within the base of a narrow cone with apex at f will contribute effectively to forward scattering.

Since only those molecules close to the axis of the cone will contribute coherently, we can calculate the total scattered electric vector at f.

2 is relative to the transmitted wave.

The phase shift is important in generating the characteristic of linear birefringence.

The incident wave's Stokes parameters can be found in terms of the scattering tensor components by substituting (3.4.4) into (2.3.6).

We will not apply the equations in detail to every one of the phenomena they embrace, but will use them to get the changes in intensity and polarization for the basic optical activity phenomena.

Chapter 4 in which symmetry classifications are developed is where the criteria for deciding whether or not a particular component of a property can contribute to a certain intensity change is elaborated in detail.

In the case of a fluid composed of achiral molecules, this would be possible.

The initial intensity is 0.

The electric field causes anisotropy in the fluid because of a partial orientation of the electric dipole moments.

The result is valid only at transparent frequencies.

Buckingham discussed the application of the equation at absorbing frequencies.

The simultaneous presence of linear dichroism can lead to additional complexity.

The development of an expression for the azimuth change at absorbing frequencies due to Kerr linear dichroism proceeds in an analogous fashion.

For further discussion of this complicated situation, we refer to Kuball and Singer.

The Cotton-Mouton effect can be achieved with the development of similar expressions.

Extending the development of linear birefringence in the previous section would allow for a static electric field.

The theoretical background for the determination of quadrupole moments in fluids is provided here.
This example shows the power and generality of the refringent scattering formalism and provides a glimpse of one of the great achievements of the field.
A similar treatment has been given by two other people.

The electric quadrupole moment will be origin dependent if the quadrupolar molecule also has a permanent electric dipole moment.

This situation requires a bulk observable, the electric field, to depend on an arbitrary origin.

The current treatment is similar to Buckingham and Longuet-Higgins who used the same approach.

The arrangement by which the electric field is generated in the experiment needs to be considered.

The sample is usually contained in a long tube with two wires running parallel to the axis of the tube.
An inhomogeneous electric field is created between the wires when a potential difference is set up.
The light beam goes along the tube between the wires.

The apparent electric quadrupole moment has its origin at the point.

The only contribution that contributes to the natural optical rotation and circular dichroism of isotropic samples is the electric dipole-electric quadrupole contribution.

The results of this section give the complete po larization changes only for nonmagnetic samples which are isotropic in the plane.

They are valid for light propagating along the axis of uniaxial crystals.
Additional terms can contribute to propagation directions in anisotropic media.

The initial and final ellipticities are the ones that count.

The incident light is used and linearly polarized.

Cotton discovered that unpolarized light becomes partially circularly polarized in an absorbing chiral medium in his early experiments.

Measurement of the degree of circular polarization of transmitted light could be useful in situations where it is not possible to prepare the polarization state of the incident light.
Magnetic fields and light scattering by dust particles are possible sources of circular polarization.

Section 1.9.3 requires the magnetic field to be applied along the direction of propagation of the light beam.

A fluid becomes a uniaxial medium.

The magnetic field must be brought into these expressions.

The effect on a fluid is considered first.

This shows that only a field along the propagation direction can generate zero contributions after spatial averaging.

We won't write them out explicitly until Chapter 6.

The first excited state of the hydrogen atom is the only atomic system showing a permanent electric dipole moment.

A uniform static magnetic field can cause anisotropy in a crystal.
It is now necessary to use a quantum-statistical average in place of the classical Boltzmann average since it is the relative populations of quantum states with nonzero spin that determine the magnetic anisotropy.

If the incident light beam is unpolarized, we assume it remains over the sample path length, and only the conventional absorption terms and the magnetochiral terms survive.

The Refractive index is a measure of the strength of the beam of light propagating parallel and antiparallel to the static magnetic field.

Chapter 1 stated that dichroism and magnetochiral birefringence need to be samples of the same type.

Like linear birefringence, gyrotropic birefringence can only be found in oriented media, and the associated polarization changes are subject to all the complications indicated in Section 3.4.4.

Jones derived the two-by-two matrix for determining the effect of a nondepolarizing medium on a light beam incident in certain directions.

The Jones matrix has a number of optical effects, such as absorption, circular dichroism, linear birefringence, and linear dichroism with respect to a pair of axes.

The last two properties were new.

They have been predicted to occur naturally in certain magnetic and nonmagnetic crystals, and in fluids by the simultaneous application of uniform static electric and magnetic fields parallel to each other.
The organometallic complex methylcyclopentadienylmanganese-tricarbonyl, C9H7MnO3 has been observed as a paramagnetic molecule in the neat liquid state.

The Jones birefringence has a kinship with the other two, since all three of them have the same time-odd property.

This time it is caused by static electric and magnetic fields that travel to the light beam.

It has been observed in fluids and compared with the Jones birefringence where it was found to have the same magnitude.

The Cotton-Mouton effect is related to this effect.

There is an electric analogue of the Faraday effect in some crystals.

One source of linear electric optical rotation is easy to understand.

The fluctuations associated with the two equal and opposite spin lattices in antiferromagnetic crystals are thought to be the cause of the electrical-induced magnetization.

There is interference with the unscattered wave that does not occur on account of the different frequencies.

We will extract explicit expressions for specific situations if necessary.

The last three terms in 3.3.4 do not contribute since the scattered waves are not considered a finite cone of collection.

In the absence of external fields, we only consider fluid samples that are isotropic.

The same expressions apply to both scattering and absorbing frequencies.

In the absence of external static fields, random achiral molecule scattering at transparent frequencies is the most common situation.

The specified products of tensor components must be averaged over all orientations of the molecule.

Since the incident wave interfere with the transmitted wave and generate birefringence phenomena, rasy scattering is meaningless.

The standard expressions for the depolarization ratio are generated by Equation 3.5.9.

There is interest in the part of the light that is circular.

If the molecule is isotropically polarizable, the incident beam is completely circularly polarizable.

In the backward direction, the degree of circularity of the scattered light is the same as in the forward direction.

The lack of antisymmetric scattering gives no more information than the depolarization ratio.

The degree of circularity is the same in the backward direction.

As for pure isotropic scattering, if the incident beam is completely circularly polarized, the near-forward Rayleigh and the forward Raman components are also completely circularly polarized.

The band is affected by symmetric and antisymmetric scattering.

Large 'anomalies' in the depo larization ratio of light scattered from atoms in spin-degenerate ground states can be produced when the incident Frequency is in the vicinity of an electronic absorption Frequency.

There is a possibility of antisymmetric resonance Rayleigh and Raman scattering from the molecule in a degenerate state.
Chapters 4 and 8 discuss these questions in detail.

It is necessary to measure the depolarization ratio in 90* scattering and the degree of circularity or reversal coefficients in 0* or 180* scattering in order to separate them if isotropic.

Information about the effective symmetry of the haem group has been provided by complete polarization measurements.

The relative contributions to a particular band could be determined by decomposing the lineshape into the three characteristic parts, since the shapes of the bands generated by isotropic, anisotropic and antisymmetric scattering are different.

The slight difference in the response to right- and left-circularly polarized light can be seen in the scattering of raytracing samples.

In the forward and backward directions, the azimuth of the light is always the same as it was in the scattering plane.

There are scattered intensities in the light.

The degree of circularity of the scattered light wave gives the same information as the circular intensity difference.

Chapter 7 talks about the symmetry requirements for Rayleigh and Raman scattering.

We note that only strontium can support such scattering.

The 2 scatterings are not related to the sign of the optical activity tensor.

The topic of fluctuations in achiral, rather than racemic systems, has been re-examined.

There are certain resonance Raman scattering situations where cross terms are important.

All samples in a static magnetic field parallel to the incident light beam can show optical rotation and dichroism, just like all samples in a static magnetic field parallel to the incident light beam.

Natural optical activity in which the light scattered at any angle shows a circular component and a circular intensity difference at the same time is different.

Chapter 8 talks about the symmetry aspects of optical activity.

The components of the unperturbed and perturbed dynamic polarizabilities specified in each temperature-independent term always have the same transformation properties, so we note that all molecules can support such scattering.

There are scattering situations.

We will not write them down explicitly because of their complexity.

Chapter 8 shows that they simplify considerably when applied to a specific situation.

Section 1.9.4 shows that light can be scattered from all molecules in a static electric field, and that it can be seen from both the incident and scattered directions.

If one of the following is reversed, the intensity difference will change.
There is no optical activity for light in the forward or backward direction.

For electric Raman optical activity, the dynamic molecular property tensors are replaced by transition tensors.

The resulting contributions to the parameters are not given here.

There are a number of differential light scattering effects that are the same order as electric Rayleigh optical activity.