Radiation Mechanism

Nature of Radiation

Radiation, or electromagnetic radiation (EMR), is a form of energy that travels through space. It is characterized by its wavelength or frequency and can be described as a wave or a particle. The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from radio waves to gamma rays.

A spectrum is a representation of the distribution of electromagnetic radiation as a function of wavelength or frequency. The spectrum can be continuous, showing a range of wavelengths, or discrete, showing specific wavelengths corresponding to particular energies. Devices used to measure the spectrum of EMR are called spectrometers or spectrographs. The visible spectrum is the part of the electromagnetic spectrum that can be seen by the human eye. The wavelength of visible light ranges from 390 to 700 nm.

The electromagnetic spectrum is divided into several regions based on wavelength or frequency:

• Radio Waves: Wavelengths longer than 1 mm, with frequencies below 300 GHz.
• Microwaves: Wavelengths between 1 mm and 1 m, with frequencies between 300 GHz and 300 MHz.
• Infrared (IR): Wavelengths between 700 nm and 1 mm, with frequencies between 300 THz and 300 GHz.
• Visible Light: Wavelengths between 390 nm and 700 nm, with frequencies between 430 THz and 790 THz.
• Ultraviolet (UV): Wavelengths between 10 nm and 390 nm, with frequencies between 790 THz and 30 PHz.
• X-rays: Wavelengths between 0.01 nm and 10 nm, with frequencies between 30 PHz and 30 EHz.
• Gamma Rays: Wavelengths shorter than 0.01 nm, with frequencies above 30 EHz.

The quantum of electromagnetic radiation is a photon, whose energy is given by $E = h \nu$ where $\nu$ is the frequency of the photon.

Continuous Spectrum

A continuous spectrum is one that contains all wavelengths of light within a certain range, without any gaps. There are several processes that can produce a continuous spectrum:

• Thermal Radiation: Objects at a finite temperature emit thermal radiation, which produces a continuous spectrum. At room temperature, most of the emission is in the infrared (IR) spectrum, though above around 525${}^{\circ}$C enough of it becomes visible for the matter to visibly glow. This visible glow is called incandescence. Blackbody radiation is a concept used to analyze thermal radiation in idealized systems. A perfect blackbody emits a continuous spectrum of radiation at all wavelengths, depending on its temperature. The intensity of the radiation at each wavelength is described by Planck’s law.
• Synchrotron Radiation: Charged particles moving at relativistic speeds in a magnetic field emit a continuous spectrum of radiation, known as synchrotron radiation. When non-relativistic charged particles are accelerated in a magnetic field, they emit radiation in a continuous spectrum, which is also known as cyclotron radiation.
• Bremsstrahlung: When charged particles, such as electrons, are accelerated or decelerated, they emit radiation in a continuous spectrum. This is particularly significant in high-energy astrophysical processes. Bremsstrahlung emitted from plasma is sometimes referred to as free–free radiation.
• Compton Scattering: High-energy photons can scatter off charged particles, resulting in a continuous spectrum of lower-energy photons.
• Rayleigh Scattering: Scattering is an absorption followed by an instantaneous emission at the same wavelength but in a new direction. The scattering of light by particles much smaller than the wavelength of light can produce a continuous spectrum, although it is more commonly associated with the scattering of specific wavelengths.
• Mie Scattering: This is the scattering of light by particles comparable in size to the wavelength of light, which can also produce a continuous spectrum.
• Cerenkov Radiation: Charged particles moving faster than the speed of light in a medium emit a continuous spectrum of radiation, known as Cerenkov radiation.
• Fluorescence: When certain materials absorb light at one wavelength and re-emit it at another, they can produce a continuous spectrum of light. This is often seen in fluorescent materials.
• Phosphorescence: Similar to fluorescence, but the re-emission of light occurs over a longer time scale, resulting in a continuous spectrum of light emitted after the excitation source is removed.
• Recombination Radiation: When free electrons recombine with ions, they can emit a continuous spectrum of radiation as they lose energy. This is also called free-bound radiation.
• Ionization: When atoms or molecules are ionized, they can emit a continuous spectrum of radiation as the electrons transition between energy levels. This is also called bound-free radiation.

Line Spectrum

A discrete or line spectrum is one that contains only specific wavelengths of light, corresponding to the energies of transitions between quantized energy levels in atoms or molecules. The line spectrum can be produced by:

• Atomic Transitions: When electrons in atoms transition between energy levels, they emit or absorb photons at specific wavelengths, resulting in a line spectrum. The wavelengths of these lines are characteristic of the element and can be used to identify the element in a sample. These are also called bound-bound transitions, as they involve transitions between bound states of electrons in atoms.
• Molecular Transitions: Similar to atomic transitions, molecules can also have quantized energy levels, leading to emission or absorption lines in the spectrum. These lines are often associated with vibrational and rotational transitions in molecules.
• Emission Lines: When atoms or molecules are excited, they can emit photons at specific wavelengths, producing an emission line spectrum. This occurs in hot gases or plasmas, where the emitted light is characteristic of the elements present.
• Absorption Lines: When light from a continuous source passes through a cooler gas, certain wavelengths are absorbed by the atoms or molecules in the gas, resulting in an absorption line spectrum. The absorption lines correspond to the same wavelengths as the emission lines of the elements in the gas.

Polarization of Radiation

Electromagnetic radiation can be polarized—linearly, circularly, or elliptically. Polarization refers to the orientation of the electric field vector of the electromagnetic wave.

• Linearly Polarized Radiation: The electric field vector oscillates in a single plane, perpendicular to the direction of propagation. This can be represented as $E(t) = E_0 \cos(\omega t + \phi)$, where $E_0$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase.
• Circularly Polarized Radiation: The electric field vector rotates in a circular motion as the wave propagates. This can be represented as $E(t) = E_0 \cos(\omega t + \phi) \hat{x} + E_0 \sin(\omega t + \phi) \hat{y}$, where $\hat{x}$ and $\hat{y}$ are orthogonal unit vectors in the plane of polarization.
• Elliptically Polarized Radiation: The electric field vector traces out an ellipse as the wave propagates. This can be represented as $E(t) = E_{0x} \cos(\omega t + \phi) \hat{x} + E_{0y} \sin(\omega t + \phi) \hat{y}$, where the amplitudes in the $\hat{x}$ and $\hat{y}$ directions are different.

The polarization of light can be measured using polarimeters. The degree of polarization is a measure of how much of the radiation is polarized. For completely linearly polarized radiation, the degree of polarization is 1, while for unpolarized radiation, it is 0. The degree of polarization can also be expressed in terms of the maximum and minimum intensities of the radiation, as follows:

$$P = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}}$$

where $I_{\text{max}}$ and $I_{\text{min}}$ are the maximum and minimum intensities of the radiation when measured through a polarimeter, respectively.

Bohr’s Model of the Atom

Niels Bohr proposed a model of the atom in 1913 to explain the spectral lines of hydrogen. The key postulates of Bohr’s model are:

1. Electrons move in circular orbits around the nucleus, and these orbits are quantized.
2. The angular momentum of an electron in a given orbit is quantized and is an integer multiple of $\hbar$ (reduced Planck’s constant).
3. Electrons can only occupy certain allowed orbits, and when they transition between these orbits, they emit or absorb photons with energies corresponding to the difference in energy between the orbits.
4. The energy of an electron in a given orbit is quantized and is given by the formula for the total energy of the electron in that orbit.

By Bohr’s second postulate,

$$\tag{2.3.1} mvr = n \hbar$$

where $n = 1,2,3, \ldots$ is called the principal quantum number, $m$ is the mass of the electron, $v$ is its velocity, and $r$ is the radius of the orbit.

The total energy of an electron in the $n^\text{th}$ orbit is

$$\tag{2.3.2} E_n = - \frac{m_e e^4 Z^2}{32 \pi^2 \varepsilon_0^2 \hbar^2 n^2} = - E_0 \frac{Z^2}{n^2}$$

where $E_0 = \frac{m_e e^4}{32 \pi^2 \varepsilon_0^2 \hbar^2} \approx 13.6 \, \mathrm{eV}$ is a constant, $Z$ is the atomic number, and $e$ is the elementary charge.

During a transition, an electron moves from one orbit to another, and the energy of the electron changes. The energy difference between two orbits is given by

$$\tag{2.3.3} E = E_n - E_m = -E_0 Z^2 \left( \frac{1}{n^2} - \frac{1}{m^2} \right)$$

where $n$ and $m$ are the principal quantum numbers of the initial and final orbits, respectively.

Thus, the wavelength of the photon is

$$\tag{2.3.4} \frac{1}{\lambda} = \frac{E_0 Z^2}{hc} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) = RZ^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$$

where $R = \frac{E_0}{hc} \approx 1.097 \times 10^7 \, \mathrm{m}^{-1}$ is the Rydberg constant.

Spectral Broadening

Spectral lines are not infinitely narrow, but have a finite width. The width of the line is called the line profile. The line profile is a function of the wavelength and is denoted by $I(\lambda)$. The area under the line profile is equal to the total intensity of the radiation. There are several mechanisms which cause broadening of spectral lines. The most important ones are:

Natural Broadening

The lifetime of excited states results in natural broadening, also known as lifetime broadening. The uncertainty principle relates the lifetime of an excited state with the uncertainty of its energy. A short lifetime will have a large energy uncertainty and a broad emission. This broadening effect results in an unshifted Lorentzian profile.

Doppler Broadening

This is the broadening of spectral lines due to Doppler shift caused by velocity distribution of particles. The resulting profile is called the Doppler profile and follows a Gaussian distribution.

In the non-relativistic case, Doppler shift is $f = f_0 \left( 1 + \frac{v}{c} \right)$. If $P_v(v) \, dv$ is the number of particles having velocities between $v$ and $v + dv$, then the corresponding frequency distribution is

$$\begin{aligned} P_f (f) \, df &= P_v (v) \, \frac{dv}{df} \, df \\ &= \frac{c}{f_0} P_v \left[ c \left(\frac{f}{f_0} - 1 \right) \right] \, df \end{aligned}$$

Thermal Doppler broadening is caused by the thermal motion of the molecules. The velocity distribution of the particles is given by the Maxwell-Boltzmann distribution

$$P_v (v) \, dv = \sqrt{\frac{m}{2 \pi k T}} \, \exp\left(-\frac{mv^2}{2kT}\right) \, dv$$

$$\implies P_f (f) \, df = \frac{c}{f_0} \sqrt{\frac{m}{2 \pi k T}} \, \exp\left(-\frac{mc^2}{2kT} \left( \frac{f}{f_0} - 1 \right)^2 \right) \, df$$

This is a Gaussian profile with a standard deviation of $\sigma = \sqrt{\frac{kT}{mc^2}} f_0$ and FWHM

$$\Delta f_\text{FWHM} = \sqrt{\frac{8 \ln 2 kT}{mc^2}} f_0 = 2 \sqrt{2 \ln 2} \, \sigma$$

Pressure Broadening

Impact or collisional pressure broadening is caused by collisions of particles with other particles and follows a Lorentzian profile. Pressure broadening can be classified into the following categories:

• Impact pressure broadening or collisional broadening
• Quasistatic pressure broadening
• Linear Stark broadening
• Quadratic Stark broadening
• Van der Waals broadening
• Resonance broadening

Zeeman Effect

The Zeeman effect is the splitting of spectral lines in the presence of a magnetic field. It occurs due to the interaction of the magnetic field with the magnetic dipole moment of the electron. The effect can be classified into two types:

• Normal Zeeman Effect: This occurs when the magnetic field is weak and the splitting is linear. The spectral lines split into three components: one unshifted component and two shifted components, one on each side of the unshifted line.
• Anomalous Zeeman Effect: This occurs when the magnetic field is strong and the splitting is non-linear. The spectral lines can split into multiple components, depending on the strength of the magnetic field and the quantum numbers of the energy levels involved.

Stark Effect

The Stark effect is the splitting of spectral lines in the presence of an electric field. It occurs due to the interaction of the electric field with the electric dipole moment of the atom or molecule. The effect can be classified into two types:

• Linear Stark Effect: This occurs when the electric field is weak and the splitting is linear. The spectral lines split into two components, one on each side of the unshifted line.
• Quadratic Stark Effect: This occurs when the electric field is strong and the splitting is non-linear. The spectral lines can split into multiple components, depending on the strength of the electric field and the quantum numbers of the energy levels involved.

Cyclotron Radiation

When charged particles like electrons move through a magnetic field, they experience a force that causes them to spiral. This motion causes them to emit radiation, which is important in many astrophysical environments.

An electron in a magnetic field $\mathbf{B}$ feels a Lorentz force:

$$\mathbf{F} = e \left( \mathbf{v} \times \mathbf{B} \right)$$

where $e$ is the electron charge. For a non-relativistic electron, the acceleration is:

$$\mathbf{a} = \frac{\mathbf{F}}{m} = \frac{e}{m} \left( \mathbf{v} \times \mathbf{B} \right)$$

where $m$ is the electron mass. The electron moves in a helical path with radius:

$$\tag{2.6.20} r_c = \frac{mv_\perp}{eB}$$

The pitch angle is $\tan \alpha_c = v_\perp / v_\parallel$, where $v_\perp$ is the velocity perpendicular to the field and $v_\parallel$ is parallel to the field.

The angular frequency of this motion (cyclotron frequency) is:

$$\tag{2.6.21} \omega = \frac{v_\perp}{r_c} = \frac{eB}{m}$$

The electron emits energy (cyclotron radiation) according to the Larmor formula:

$$\tag{2.6.22} P = \frac{2}{3} \frac{e^2}{4\pi \varepsilon_0 c^3} a^2$$

The spectrum is narrow and peaks at:

$$\tag{2.6.23} \nu_\text{max} \approx \frac{\omega_c}{2 \pi} \approx 280 \, \mathrm{GHz} \, \left( \frac{B}{1 \, \mathrm{T}} \right)$$

For reference, the magnetic field at Earth’s surface is $B_\oplus \approx 3.1 \times 10^{-5} \, \mathrm{T}$ and at Jupiter’s surface is $B_J \approx 5 \times 10^{-4} \, \mathrm{T}$.