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PDF version: Notes on x-ray physics – Logan Thrasher Collins

Thomson scattering and Compton scattering

Electrons are the main type of particle that can scatter x-rays. Elastic or Thomson scattering occurs when a non-relativistic electron is accelerated by the electrical component of an incoming electromagnetic field from an x-ray. The accelerated electron then reradiates light at the same frequency. Since the frequency of the input light and output light are the same, this is an elastic process.
The intensity of the re-emitted radiation at an observer’s location depends on the angle Χ between the incident light and the observer. Because of the sinusoidal wave character of light, the scattered intensity at the observer’s location is given by the proportionality equation below.

Light that encounters the electron is scattered if it is incident on the region defined by the electron’s classical radius. This region is called the Thomson scattering length r₀. For a free electron, r₀ = 2.82×10^-5 Å.

Compton scattering occurs when an electron scatters a photon and the scattered photon has a lower energy than the incident photon (an inelastic process). For Compton scattering, a fraction of the incident photon’s energy is transferred to the electron.

The amount of energy lost via Compton scattering where the incident photon has energy E₀ = hc/λ₀ and the scattered photon has energy E₁ = hc/λ₁ is described by the following equation. Here, ψ represents the angle between the paths of the incident photon and the scattered photon.

Scattering from atoms

X-rays are scattered throughout the volumes of atomic electron clouds. For x-rays that scattered in the same direction as the incident x-rays, the strength of scattering is proportional to the atom’s Z-number. In the case of an ionic atom, this value is adjusted to equal the atom’s number of electrons. Note that this assumes free electron movement within the cloud.
By contrast, x-rays that are scattered at some angle 2θ relative to the incident x-rays exhibit lower scattering magnitudes. Each of the x-rays scattered at angle 2θ will possess different magnitudes and phases depending on where they were scattered from within the atomic cloud. As a result, the scattering amplitude for the x-rays at angle 2θ will be a vector sum of these waves with distinct magnitudes and phases.

A wavevector k is a vector with magnitude 2π/λ that points in the direction of a wave’s propagation. The difference between the wavevector of the incident wave k₀ and the wavevector of the scattered wave k₁ is equal to a scattering vector Q (that is, Q = k₀ – k₁). The magnitude of Q is given by the following equation.

The atomic scattering factor f describes the total scattering amplitude for an atom as a function of sin(θ)/λ. By assuming that the atom is spherically symmetric, f will depend only on the magnitude of Q and not on its orientation relative to the atom. Values for f can be found in the International Tables for Crystallography or computed using nine known coefficients a_1,2,3,4, b_1,2,3,4, and c (which can also be looked up) and the following expression. The coefficients vary depending on the atom and ionic state. The units of f are the scattering amplitude that would be produced by a single electron.

If the incident x-ray has an energy that is much less than that of an atom’s bound electrons, the response of the electrons will be damped due to their association with the atom. (This no longer assumes free electron movement within the cloud). As a result, f will be decreased by some value f_a. The value f_a increases when the incoming x-ray’s energy is close to the energy level of the electron and decreases when the incoming x-ray’s energy is far above the energy levels of the electrons.
When the incident x-ray’s energy is close to an electron’s energy level (called an absorption edge), the x-ray is partially absorbed. With this process of partial absorption, some of the radiation is still directly scattered and another part of the radiation is re-emitted after a delay. This re-emitted radiation interferes with the directly scattered radiation. To mathematically describe the effect of the re-emitted radiation’s phase shift and interference, f is adjusted by a second term f_b (which is an imaginary value). Far from absorption edges, f_b has a much weaker effect (it decays by E^-2). The total atomic scattering factor is then given by the following complex-valued equation.

Refraction, reflection, and absorption

A material’s index of refraction can be expressed as a complex quantity n_c = n_Re + in_Im. The real part represents the rate at which the wave propagates through the material and the imaginary part describes the degree of attenuation that the wave experiences as it passes through the material.
The reason that a material can possess a complex refractive index involves the complex plane wave equation. The wavenumber k = 2π/λ₀ is the spatial frequency in wavelengths per unit distance and it is a constant within the complex plane wave equation (λ₀ is the wave’s vacuum wavelength). The complex wavenumber k_c = kn_c is the wavenumber multiplied by the complex refractive index. As such, the complex refractive index can be related to the complex wavenumber via k_c = 2πn_c/λ₀ where λ₀ is the vacuum wavelength of the wave. After inserting 2π(n_Re + in_Im)/λ₀ into the complex plane wave equation, a decaying exponential can be simplified out as a coefficient for the rest of the equation. The decaying exponential represents the attenuation of the wave in the material. Once this simplification is performed, the equation’s complex wavenumber is converted to a real-valued wavenumber.

For x-rays, a material’s complex refractive index for wavelength λ is related to the atomic scattering factors of atoms in the material using the following equation. N_i represents the number of atoms of type j per unit volume and f_j(0) is the atomic scattering factor in the forward direction (angle of zero) for atoms of type j. Recall that r₀ is the Thomson scattering length.

The refractive index is a function of the wavelength. For most optical situations, as the absorption maximum of a material is approached from lower frequencies, the refractive index increases. But when the radiation’s frequency is high enough that it passes the absorption maximum, the refractive index decreases to a value of less than one.
The refractive index is defined by n = c/v, where v is the wave’s phase velocity. Phase velocity is the rate at which a wave’s phase propagates (i.e. how rapidly one of the wave’s peaks moves through space). Rearranging the equation, v = c/n is obtained. When the refractive index is less than one, the phase velocity is greater than the speed of light. However, this does not violate relativity because the group velocity (not the phase velocity) carries the wave’s energy and information. For comparison, group velocity is the rate at which a change in amplitude of an oscillation propagates.
Anomalous dispersion occurs when the radiation’s frequency is high enough that the refractive index of a material is less than one. As a result, x-rays entering a material from vacuum are refracted away from the normal of the refracting surface. This is in contrast to the typical case where the radiation would be refracted toward the normal of the refracting surface. In addition, the refracted wave is phase shifted by π radians.
The complex refractive index is often expressed using the equation below. Here, δ is called the refractive index decrement and β is called the absorption index. Note that n_Re = 1 – δ and n_Im = β (as a comparison to the previously used notation). Recall that n_Im = β describes the degree of a wave’s attenuation as it moves through a material.

The refractive index decrement can be approximately computed using the average density of electrons ρ, the Thomson scattering length r₀, and the wavenumber k = 2π/λ₀. Note that this approximation is better for x-rays that are far from an absorption edge.

With most materials, the resulting real part of the index of refraction is only slightly less than one when dealing with x-rays. For example, a typical electron density of one electron per cubic Angstrom yields a δ value of about 5×10^-6.
Snell’s law applies to the index of refraction for x-rays and is given as follows.

Because the index of refraction for x-rays is slightly less than one, total external reflection can occur when x-rays are incident on a surface at angles less than the critical angle θ_critical. This stands in contrast with the total internal reflection that commonly occurs with visible light.

The critical angle can be approximated with a high level of accuracy using the following equation (derived from the Taylor expansion of the cosine function). With typical values of δ on the order of 10^-5, θ_critical is often equal to just a few milliradians (or a few tenths of a degree). These small angles relative to the surface are called grazing angles.

Because grazing incident angles facilitate x-ray reflection, special curved mirrors can be used to focus x-rays. The curvature of these mirrors must be small enough that the steepest incident angle is less than θ_critical. It should be noted that, even when undergoing total external reflection, x-rays do penetrate the reflecting material to a depth of a few nanometers via an evanescent wave.

The absorption index β is related to the value f_b using the following equation where r₀ is the Thomson scattering length. Recall that f_b represents the effects of scattering from absorption and remission of x-rays with energies that are close to the absorption edges of a material.

Using the process explained earlier for computing the decaying exponential exp(-2πn_Imx/λ₀) that represents the attenuation of a wave’s amplitude as it travels through a material, the decay of a wave’s intensity as it travels through a material can also be found. Recall that λ₀ is the wavelength in a vacuum. Because intensity is proportional to the square of the amplitude, the equation below describes the exponential decay of a wave’s intensity in a material. (This decaying exponential function is multiplied by the equation of the wave). Here, μ is called the absorption coefficient and is defined as the reciprocal of the thickness of a material required to decrease a wave’s intensity by a factor of 1/e. The absorption coefficient is a rough indication of a material’s electron density and electron binding energy.

The correspondences between the atomic configurations associated with an x-ray absorption edge and the commonly used name for said absorption edge are given in the following table. The subscripts used with the configurations represent the total angular momenta.

X-ray fluorescence and Auger emission

Materials fluoresce after bombardment with x-rays or high-energy electrons. If electrons are used, the emitted light consists of Bremsstrahlung radiation (which comes from the deacceleration of the electrons) and fluorescence lines. The Bremsstrahlung radiation includes a broad spectrum of wavelengths and has low intensity while the fluorescence lines are sharp peaks and exhibit high intensity. If x-rays are used to bombard a material, there is no Bremsstrahlung radiation, but fluorescence lines occur.
Different materials exhibit different characteristic fluorescence lines. These x-ray fluorescence lines are caused by outer-shell electrons relaxing to fill the holes left after the ejection of photoelectrons. However, not all electronic transitions are allowed, only those which follow the selection rules for electric dipoles. These selection rules are given below. J is the total angular momentum and can be computed from the sum of the Azimuthal quantum number L (which determines the type of atomic orbital) and the spin quantum number S (which determines the direction of an electron’s spin).

The nomenclature for x-ray fluorescence lines is based on the shell to which an electron relaxes. If an excited electron relaxes to the 1s shell state, then the fluorescence line is part of the K series. For an excited electron that relaxes to the 2s or 2p state, the fluorescence line is part of the L series. The M series includes relaxations to 3s, 3p, and 3d. The N series includes relaxations to 5s, 5p, 5d, and 5f. As such, the Azimuthal quantum number determines if the fluorescence line falls into the K, L, M, or N series (there are some series beyond these as well which follow the same pattern). The transition within each series that exhibits the smallest energy difference is labeled with α (i.e. Kα), the transition with the next smallest energy difference is labeled with β, and so on. It should be noted that the fluorescence lines are further split by the effects of electron spin and angular momentum and so are labeled with suffixes of 1, 2, etc.
Auger emission is the process where a photoelectron is ejected, an outer shell electron relaxes to fill the hole, and the released energy causes ejection another electron instead of emitting a photon. The energies of emitted Auger electrons are independent of the energies of the incident photons.
The excess energy released by the relaxation of the outer shell electron is equal to |E_core – E_outer|. In order for the last electron ejection to occur, the electron must have a binding energy that is less than the excess released energy from the relaxation. The kinetic energy of the ejected Auger electron is |E_core – E_outer – E_binding|. Note that E_binding is the binding energy of the Auger electron in the ionized atom (which is different from the binding energy in the neutral form of the atom).
Auger emission and x-ray fluorescence are competitive with each other. Fluorescence is stronger for heavier atoms (higher Z-number) since they exhibit larger energy differences between adjacent shells as well as binding electrons more tightly. For the same reasons, Auger emission is stronger from atoms with lower Z-numbers.

Reference: Willmott, P. (2011). An Introduction to Synchrotron Radiation: Techniques and Applications. Wiley.

Cover image courtesy of: Asia Times

PDF version: Notes on wave optics – Logan Thrasher Collins

The wave equation

Because light exhibits wave-particle duality, wave-based descriptions of light are often appropriate in optical physics, allowing the establishment of an electromagnetic theory of light.
As electric fields can be generated by time-varying magnetic fields and magnetic fields can be generated time-varying electric fields, electromagnetic waves are perpendicular oscillating waves of electric and magnetic fields that propagate through space. For lossless media, the E and B field waves are in phase.
By manipulating Maxwell’s equations of electromagnetism, two relatively concise vector expressions that describe the propagation of electric and magnetic fields in free space are found. Recall that the constants ε₀ and μ₀ are the permittivity and permeability of free space respectively.

Since an electromagnetic wave consists of perpendicular electric and magnetic waves that are in phase, light can be described using the wave equation (which is equivalent to the expressions above). Note that the speed of light c = (ε₀μ₀)^-1/2. Electromagnetic waves represent solutions to the wave equation.

Either the electric or the magnetic field can used to represent the electromagnetic wave since they propagate with the same phase and direction. With the exception of the wave equation above, the electric field E will instead be used to represent both waves. Note that either the electric or magnetic field can be employed to compute amplitudes.

Solutions to the wave equation

Plane waves represent an important class of solutions to the wave equation. The parameter k is the wavevector (which points in the direction of the wave’s propagation) with a magnitude equal to the wavenumber 2π/λ. In a 1-dimensional system, the dot product k•r is replaced by kx. The parameter ω is the angular frequency 2πf and φ is a phase shift.

To simplify calculations, Euler’s formula can be used to convert the equation above into complex exponential form. Only the real part describes the wave as the real part corresponds to the cosine term.

Spherical waves are another useful solution to the wave equation (though they are an approximation and truly spherical waves cannot exist). Because of their geometry, the electric field of a spherical wave is only dependent on distance from the origin. As such, the equation for a spherical wave can be written as seen below with origin r₀.

Gaussian beams are a solution to the wave equation that can be used to model light from lasers or light propagating through lenses. If a Gaussian beam propagates in the z direction, then from the perspective of the xy plane, it shows a Gaussian intensity distribution. For a Gaussian beam, the amplitude decays over the direction of propagation according to some function A(z), R(z) represents the radius of curvature of the wavefront, and w(z) is the radius of the wave on the xy plane at distance z from the emitter. Often these functions can be approximated as constants.

Intensity and energy of electromagnetic waves

The Poynting vector S is oriented in the direction of a wave’s propagation (assuming that the wave’s energy flows in the direction of its propagation).

The magnitude of the Poynting vector represents the power per unit area (W/m²) or intensity crossing a surface with a normal parallel to S. Note that this is an approximation since, according to a quantum mechanical description of electromagnetic waves, the energy should be quantized.

Power per unit area (intensity) of plane waves, spherical waves, and Gaussian beams can also be calculated using the equations below. The formula for the Gaussian beam’s power represents the power at a plane perpendicular to the direction of light propagation z.

For electromagnetic waves, instantaneous energy per unit area is difficult to measure, so the average energy per unit area over a period of time Δt is often worked with instead. Since waves are continuous functions, taking their time-average requires an integral.

After using the above integral on the function e^iωt and then taking the real and imaginary parts of the result, the time-averages of the functions cos(ωt) and sin(ωt) are found.

Superposition of waves

Let two waves E₁ and E₂ of the same frequency traveling in the same direction undergo superposition. E₁ and E₂ may or may not possess the same amplitude or phase. The substitution α = –(kx+φ) will be carried out.

If the phases of the waves are different, some special equations are necessary to find the amplitude E₀ and the phase α of the resulting wave.

For the superposition of any number of waves, the equations above can be extended.

Polarization of light

The waves comprising linearly polarized light are all oriented at the same angle which is defined by the direction of the electric field of the light waves. For linearly polarized plane waves with electric fields oriented along the x or y axes that propagate in the z direction, the following equations describe their electric fields.

The superposition of two linearly polarized plane waves that are orthogonal to each other (and out of phase) is the vector sum of each electric field.

The superposition of two linearly polarized plane waves that are orthogonal to each other (and in phase) is computed via the following equation and has a tilt angle θ determined by the ratio of amplitudes of the original waves. This process can also be performed in reverse with a superposed polarized wave undergoing decomposition into two orthogonal waves.

Fig. 2

When two constituent waves possess equal amplitudes and a phase shift of nπ/2, the superposed wave is circularly polarized (as it can be expressed using a sine and a cosine term). Equations for the constituent waves and the superposed wave are given below.

When circularly polarized light propagates, it takes a helical path and so rotates. As such, a full rotation occurs after one wavelength. If a circularly polarized wave rotates clockwise, it is called right-circularly polarized and has a positive sine term. If a circularly polarized wave rotates counterclockwise, it is called left-circularly polarized and has a negative sine term.

If a right-circularly polarized light wave and a left-circularly polarized light wave of equal amplitude are superposed, then they create a linearly polarized light wave with twice the amplitude of the individual waves.

Linearly polarized and circularly polarized light are special cases of elliptically polarized light. For elliptically polarized light, the amplitudes of the superposed waves may differ and the relative phase shift does not need to be nπ/2. As such, the electric field traces an elliptical helix as it propagates along the z direction.

For elliptically polarized light with a positive phase shift φ, it is called right-elliptically polarized if E_0x > E_0y and left-elliptically polarized if E_0x < E_0y.
Most light is unpolarized (or more appropriately, a mixture of randomly polarized waves). To obtain polarized light, polarizing filters are often used.

References

Boudoux, C. (2017). Fundamentals of Biomedical Optics. Blurb, Incorporated.

Degiorgio, V., & Cristiani, I. (2015). Photonics: A Short Course. Springer International Publishing.

Hecht, E. (2017). Optics. Pearson Education, Incorporated.