Dispersion of light

Dispersion
The phenomenon of splitting up visible light into its component colors is called
dispersion. The following are a few examples where one might have seen the
phenomena -
● White light through a prism
● Due to oil on road
● Formation of rainbow in the sky
● Dispersion in a diamond

Refractive index of a medium (n) = c/v

Where,
c = speed of light in vacuum and v = speed of light
in a medium

Snell’s law
It describes the relationship between the angle of
incidence and angle of refraction.

Refraction by a prism
A ray OP, incident on a refracting surface AB, gets refracted along PQ. The angle of
incidence and the angle of refraction are i1 and r1 respectively. The ray PQ is incident on
the surface AC. Here the light goes from an optically denser medium to an optically
rarer medium and the ray is reflected along QR. The angle of refraction is i2. If the
prism were not present, the incident ray would have passed undeviated along OPML.
Because of the prism, the final ray goes along QR. The angle LMR= δ is called the angle
of deviation. The angle of deviation is given by,

δ = i1 + i2 − A

Dispersion by a prism
The refractive index of a material depends slightly on the wavelength of light. The
relation between the two may be approximately described by the equation -

n = n0 + A/λ2

A is a small positive constant known as Cauchy’s constant.

The refractive index decreases as the wavelength increases. For visible light, it is
maximum for the violet end and minimum for the red end.

Because of the difference in refractive indices, light of different colours bend through
different angles on refraction. If white light passes through a glass prism, the violet rays
deviate the most and the red rays deviate the least. Thus, white light is separated into its
various component colours.

Diffraction of light
When light is obstructed by an object, the rays bend round the corner. This
phenomenon is known as diffraction. A diffraction grating is made by making many
parallel scratches on the surface of a flat piece of some transparent material.
There are two types of diffraction grating - reflection grating and transmission grating.
Diffraction in grating
A parallel bundle of the rays will fall on the grating. Rays and wavefront form an
orthogonal set so the wavefront will be perpendicular to the rays and parallel to the
grating. According to Huygens’ Principle every point on a wavefront acts as a new
source, and each transparent slit becomes a new source so the cylindrical wavefront
spreads out from each.
Due to the interference phenomenon, when a peak falls on a valley the waves cancel
and no light exists at that point, this is called destructive interference.

(2n+1) (λ/2) = d sin(Θ)

When peaks fall on peaks and valleys fall on valleys the waves add up to form bright
spots, this is called constructive interference.

n λ = d sin(Θ)

Where n is the order of diffraction (an integer), λ is the wavelength of incident light, d is
the separation between the slits and Θ the angle measured about the normal at the
center of the plane of the slits.
The spacing between the bright spots is dependent on the wavelength of the incident
light. If white light undergoes diffraction, the diffraction pattern would be like this -

This is because different wavelengths have different locations for their bright spots
upon diffraction. Therefore, diffraction is an alternative way of observing the spectrum
of light.

Spectral Lines
Bohr’s model
Assumptions -
1. Electrons move around the nucleus in circular orbits.
2. Energy of an electron is fixed and can only move from one fixed energy
state to another.
3. The jump occurs when radiation of v = (E2-E1) / h is absorbed.
4. Angular momentum is quantized. mevr = nh/2π.
Principal quantum number (n) - 1,2,3,4,...
Energy at the nth state = En = -RH(1/n2), where RH is the Rydberg constant = 2.18×10–18 J
or 13.6 eV.

Hydrogen Spectrum
Say a beam of white light (which consists of photons of all visible wavelengths) shines
through a gas of atomic hydrogen. A photon of wavelength 656 nanometers has just the
right energy to raise an electron in a hydrogen atom from the second to the third orbit.
Given a stream of photons, only photons with this particular wavelength will be
absorbed. When they are absorbed, a number of the photons of this wavelength and
energy will be missing from the general stream of white light.
Similarly other photons will have the right energies to raise electrons from the second to
the fourth orbit, or from the first to the fifth orbit, and so on. Only photons with these
exact energies can be absorbed. All of the other photons will stream past the atoms
untouched. Thus, hydrogen atoms absorb light at only certain wavelengths and
produce dark lines at those wavelengths in the spectrum we see.

An atom can absorb energy in various ways (absorption of photons; due to collisions
upon heating), which raises it to a higher energy level - excitation. An atom remains
excited for only a very brief time ~10-8s after which it drops back spontaneously to its
ground state, with the simultaneous emission of light - giving rise to the emission
spectrum. The jump can be directly to the ground state (n=1), or to another state in
between and then to the ground state.

Figure: top - absorption spectrum, bottom - emission spectrum.

Each type of atom has its own unique pattern of electron energy levels, and no two sets
of orbits are exactly alike. Therefore, each type of atom shows its own unique set of
spectral lines, produced by electrons moving between its unique set of energy levels.

Different Series in Hydrogen Spectrum

The emission spectrum of atomic hydrogen has been divided into a number of spectral
series, with wavelengths given by the Rydberg formula. The spectral series are
important in astronomical spectroscopy for detecting the presence of hydrogen. The
energy differences between levels in the Bohr model, and hence the wavelengths of
emitted or absorbed photons, is given by the Rydberg formula.

Where,
Z is the atomic number,
n’ is the principal quantum number of the lower energy level (nl in the figure),
n is the principal quantum number of the upper energy level (nh in the figure),
R is the Rydberg constant (1.09677×107 m−1 for hydrogen)
 Series Discovered by Transition to Region of the EM spectrum

Lyman Theodore Lyman (1906) n’ = 1 Ultraviolet

Johann Balmer
Balmer n’ = 2 Visible region
(1885)

Friedrich Paschen Infrared

Paschen n’ = 3
(1908)

Frederick Sumner Brackett

Brackett n’ = 4 Infrared
(1922)

Types of spectra / Classification of spectra

A German physicist Gustav Kirchoff (1824-1857) investigated the properties of a spectra
in the laboratory and discovered that there are three kinds which are produced under
different physical conditions.

● Continuous spectra
A hot opaque solid, liquid or gas which is under high pressure will emit a
continuous spectrum. The number of energy levels available is large, thus
leading to an availability of continuous energy difference range.
 ● Emission spectra
A hot gas under low pressure (i.e. much less than atmospheric) will emit a series
of bright lines on a dark background. Such a spectrum is called a bright line or
emission spectrum.
As the gas is heated, its atoms gain kinetic energy and collide with their
neighbours causing their electrons to be raised to excited states. As the electrons
drop down, photons will be emitted with many different energies and
wavelengths corresponding to the particular electron energy level scheme for the
gas.

● Absorption spectra
When light from a source that has a continuous spectrum is shone through a gas
at a lower temperature and pressure, the continuous spectrum will be observed
to have a series of dark lines superimposed on it. This kind of spectrum is known
as a dark line or absorption spectrum.
If it is the case that the energy of some of these photons is exactly equal to the
difference between the ground state and an excited state of an atom in the
unknown gas, then that photon will be removed from the incident light. The
excited electron will quickly return to the ground state emitting a photon
however, the emitted photon need not be emitting along the same direction as
the absorbed photon but is usually emitted in a different direction.

Broadening of Spectral Lines

One might expect that with a highly sensitive optical instrument, the emission spectrum
might have bright lines that are infinitesimally small in their thickness. But the observed
lines show a wavelength spread around the line center. This gives a width to the spectral
lines. Below is an intensity vs frequency plot where vo = (Ea-Eb)/h corresponding to a
transition from a to b.

v1 and v2 are where the intensity is the half of the

maximum intensity (i.e. Io/2).
v2 - v1 is known as the Full Width at Half Maximum
(FWHM).

This broadening can be caused by the emission source

or can arise due to instrumental limitations, we are
interested in the former.
Effects causing the broadening of spectral lines
1. Natural broadening due to uncertainty
2. Collisional / pressure broadening
3. Doppler broadening
4. Rotational broadening
5. Zeeman effect

These effects can be grouped for materials as homogeneous broadening - where all atoms
experience the same effect and inhomogeneous broadening - where individual atoms
experience different effects.

Natural Broadening

From Heisenberg’s Uncertainty Principle we have τ△E = h / 2π.

Since τ (lifetime of the state of the electron) is small and finite, there is an uncertainty in
energy corresponding to a state. This inturn results in the existence of a natural
linewidth. This is a type of homogeneous broadening.

Collision / Pressure broadening

For a gas at a given pressure, radiating atoms interact with the neighboring atoms via
collision and this affects the emission line width strongly. Because of the interaction, the
energy levels get perturbed and the shift of these energy levels occur. Due to this the
line profile gets broadened.
FWHM = 1/πτo where τo is the mean flight time between two successive collisions.
This is a type of homogeneous broadening.

Doppler broadening (thermal broadening)

In the atmosphere of a star, the atoms have random velocities due to their thermal
energy. At any instant some of the atoms travel towards us and others away when they
emit photons. The effect of this is to produce a Doppler shift in the absorption lines of
the spectrum. This Doppler broadening is a lesser effect than collisional broadening.
Doppler broadening is inhomogeneous broadening.

Rotational Broadening

The lines in the spectrum of a rotating star are broadened because light from the
receding limb is redshifted and light from the approaching limb is blueshifted. The
effect is dependent on the speed of rotation, the radius of the star and the inclination of
the axis of the star wrt to the observer.

Figure: left - thermal/doppler broadening; right - rotational broadening

Zeeman Effect

The Zeeman effect is the effect of splitting of a spectral line into several components in
the presence of a static magnetic field. It is named after the Dutch physicist Pieter
Zeeman, who discovered it in 1896 and received a Nobel prize for this discovery. Since
the distance between the Zeeman sublevels is a function of magnetic field strength, this
effect can be used to measure magnetic field strength, e.g. that of the Sun and other stars
or in laboratory plasmas. When the spectral lines are absorption lines, the effect is called
inverse Zeeman effect.

Figure: left - splitting of spectral lines due to Zeeman effect; right - Zeeman effect on one of the
sun spots.

George Ellery Hale was the first to notice the Zeeman effect in the solar spectra,
indicating the existence of strong magnetic fields in sunspots. Such fields can be quite
high, on the order of 0.1 tesla or higher. Today, the Zeeman effect is used to produce
magnetograms showing the variation of magnetic field on the sun.

Appendix
Spectrum from astronomical objects
Different celestial objects produce different types of spectra. The spectrum of an object is
one means of identifying what type of object it is.

Stellar spectrum
The overall shape of intensity plot of a star approximates a black body curve with a
peak wavelength. The characteristic absorption line features including strong lines for
Hα, Hβ, Hγ and Hδ - the Balmer Series can be seen. Using such a graph, the effective
temperature can be determined.

Emission Nebulae
An emission nebula is a nebula formed of ionized gases. Emission spectra can be
observed in emission nebulae such as M42, the Great Nebula in Orion and the Eta
Carinae nebula. The distinctive colour (pinkish-red colour) is due to the spectral line
emission when an ionised electron recombines with a proton to form neutral hydrogen.
Galaxy Spectra
As galaxies vary in structure and relative composition of star type and gas their spectra
will vary.

The two galaxies show prominent Hα emission lines suggesting active star formation in
them.

Quasar Spectra
Quasars exhibit very bright emission features relative to a low intensity continuum in
their spectra. Indeed it was only through careful analysis of the spectra of quasars that
astronomers realised they were not just faint stars.
Note: the Lymann alpha line (n=2 to n=1) has been redshifted a lot - expected 1216
Ångstroms; measured 4000 Ångstroms - as it is from a far away quasar and is receding
away from us.