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Notes on Ultrasound Physics and Instrumentation

May 16, 2025October 25, 2025By logancollins No Comments

PDF version: Notes on Ultrasound Physics and Instrumentation – by Logan Thrasher Collins

Fundamentals of ultrasound waves

Sound waves such as those in ultrasound are longitudinal waves where particles oscillate backwards and forwards along the wave’s direction of propagation. This forms regions of higher (compression) and lower (rarefaction) density. Sound waves occur through the exchange of kinetic energy from molecular movement with the potential energy from elastic compression and stretching of bonds.

The speed of sound c depends on the medium. For example, in air c = 330 m/s while in water c = 1480 m/s. The frequency of a sound wave depends on the source producing it. Frequency is measured in Hertz (Hz) where 1 Hz = 1 cycle/second. Sound waves with frequencies greater than or equal to 20 kHz are referred to as ultrasound. The wavelength of a sound wave is λ = c/f. It is measured in mm (or other units of length). A sound wave’s phase describes what position the wave starts at within a cycle of oscillation and is measured in degrees or radians. Phase shifts are typically measured relative to a phase of 0° (or 0 rad). A sound wave’s amplitude corresponds to its “loudness” and describes the height of the wave’s peaks (or depth of its troughs). Cosine-based or complex exponential equations can describe sound waves in a similar way as they describe electromagnetic waves. These equations include amplitude as a coefficient, frequency as a factor multiplied by time, and phase shift as a term added to time.

Ultrasound pressure, power, and intensity

A sound wave’s excess pressure is the difference between its peak amplitude and the normal ambient pressure of the medium. When a medium is compressed, the excess pressure is positive and when it is rarified, the excess pressure is negative. In practice, the ambient pressure is often quite low and can be ignored, in which case pressure is used rather than excess pressure. Excess pressure and pressure are measured in Pascals (Pa), which are equivalent to N/m².

When an ultrasound wave passes through a medium, it deposits energy (measured in Joules) into said medium. The rate at which a source produces this energy is the power, which is measured in Watts or J/s. An ultrasound wave’s power is unevenly distributed across the beam and often is more concentrated near the beam’s center. Intensity is a measure of the power flowing through a unit area perpendicular (or normal) to the wave’s direction of propagation and is measured in W/m² or W/cm². Intensity is proportional to the wave’s pressure squared (I ∝ p²).

Ultrasound and its medium

As mentioned, the medium (material) an ultrasound wave passes through determines the sound’s speed of propagation. Specifically, the material’s density and stiffness determine the speed of sound propagation. Density ρ is often measured in kg/m³. For instance, bone’s density ρ_bone = 1850 kg/m³ and water’s density ρ_water = 1000 kg/m³ (or 1 g/cm³). Stiffness k describes how much a material resists deformation by force F. It equals the amount of pressure (stress) needed to change the thickness of a material by a given fraction and is measured in Pa. The ratio of change in thickness ΔL over original thickness L₀ is called tensile strain ε. The increase (or decrease) in length of a material due to applied force is denoted ΔL = L – L₀. Stress σ or fractional change in thickness is given by change in thickness divided by original thickness of the sample. Stress over strain is the elastic modulus E (also called Young’s modulus). Here, A is the cross-sectional area of a sample perpendicular to the applied force. Note that these formulas apply only for tensile stress (which induces tensile strain), the type of stress where a material is compressed or elongated.

The relationship between a sound wave’s speed and the properties of density and thickness can be modeled by masses on springs where ρ = m = density (corresponding to masses in the model) and k = stiffness (corresponding to the spring stiffness in the model). The relationship between c, ρ, and k is given by the following equation.

As a result of these properties, the speed of sound varies in different tissues. A table of the approximate speeds of sound across selected tissues is provided.

Acoustic impedance z measures the response of a medium’s particles in terms of their velocity v to a sound wave of a given pressure p. Acoustic impedance can also be expressed in terms of density and stiffness or in terms of density and the sound’s speed. The latter form of definition is called the characteristic acoustic impedance of the tissue. It has units of kg m^–2 s^–1, also called rayl.

Ultrasound reflection and refraction

When an ultrasound wave traveling through a medium of acoustic impedance z₁ encounters a new medium of different acoustic impedance z₂, some of the wave is transmitted and some is reflected back. The equations in the following discussion will initially assume that the wave approaches the interface at a 90° angle (perpendicular) until stated otherwise.

To maintain continuity across the interface, the following equations must hold. Here, p_i and v_i are initial pressure and velocity, p_r and v_r are reflected pressure and velocity, and p_t and v_t are transmitted pressure and velocity.

Additionally, the intensity transmitted I_t across the interface equals the incident intensity I_i minus the reflected intensity I_r.

By using the equation for the definition of acoustic impedance z = p/v, the following two equations are true. Algebraically manipulating these equations leads to the third equation below. R_p is called the amplitude (or pressure) reflection coefficient of the interface. It is important because it decides the amplitude of the echoes produced at various interfaces within the tissue.

Note that if the acoustic impedance of the first medium is greater than the acoustic impedance of the second medium (i.e. z₁ > z₂), then R_p is negative and the reflected wave is inverted (across the x axis). For most interfaces between soft tissues, R_p is quite small and so most of the wave is transmitted to produce further echoes deeper within the tissue. This is useful for ultrasound imaging. Since the interface between air and soft tissue has a much larger R_p, the ultrasound source must be placed in direct contact with the patient’s skin to avoid air blocking transmission. This also means that gas-containing tissues like lungs and gut effectively block imaging beyond the region with the gas.

Another way to describe reflection is in terms of the intensity reflection coefficient R_I. Since I ∝ p², this means that R_I = R_p².

Since the incident intensity is I_i = I_t + I_r and the transmitted intensity is I_t = I_i – I_r, one can also define an intensity transmission coefficient T_I as the first equation below. Since the transmitted pressure is p_t = p_i + p_r, one can also define a pressure transmission coefficient T_p as the second equation below.

It is useful to realize that, because energy at interfaces must be conserved, the following equation holds true.

But ultrasound waves often do not approach interfaces at a 90° angle, so the equations in this section must be modified via trigonometry to account for other angles. First, note that the incident angle θ_i will equal the backscattered angle θ_r (this is true in the 90° case as well).

When a sound wave in a less dense tissue (slower sound wave) crosses into a tissue of greater density (faster sound wave), the transmitted wave bends away from the normal and thus θ_t > θ_i.

Likewise, when a sound wave in a tissue of greater density (faster sound wave) crosses into a tissue of less density (slower sound wave), the transmitted wave bends towards the normal and thus θ_t < θ_i.

Ultrasound scattering

When ultrasound encounters an object which is small compared to the wavelength, it scatters in all directions, though slightly more energy is typically backscattered towards the transducer than away from it. Scattering specifics depend on the shape, size, and acoustic properties (z, k, ρ) of the object.

When many similar small objects are close together (e.g. red blood cells), constructive interference can occur, which is useful. In the case of the blood, this is the basis for Doppler ultrasound, which measures blood flow. By comparison, when many small objects are far apart, complex interference patterns occur. This leads to a phenomenon in images known as “speckle”, which is usually (but not always) considered a form of undesirable noise.

Absorption and relaxation of ultrasound

As ultrasound moves through tissue, it loses energy due to absorption, resulting in heat. There are two ways this occurs: relaxation absorption and classical absorption. The effects of relaxation absorption are typically much more dominant.

Relaxation absorption depends on the elastic properties of tissue, occurring when the tissue returns to its original state after rarefaction or compression by ultrasound. This is quantified by the relaxation time τ = 1/f_r, which is how long the tissue takes to return to its original state after the ultrasound’s effect.

Relaxation is characterized by a relaxation absorption coefficient β_r which is given by the first two equations below. Here, f_r is the frequency of relaxation, f is the frequency of ultrasound, and B₀ is a material-specific constant. In practice, tissues contain a range of values of τ and f_r, so the third equation below is a more general formula where the overall relaxation absorption coefficient is proportional to the sum of the various contributions. Higher values of β_r mean more energy is absorbed into the tissue.

It is useful to note that in tissues, the relationship between the relaxation absorption coefficient β_r and the frequency f is approximately linear.

Classical absorption is less important in tissues since, at clinical frequencies, the relaxation absorption is strongly dominant as mentioned. That said, overall absorption does consist of a combination of relaxation and classical absorption (though the latter may be approximated away sometimes). Classical absorption occurs because of friction between particles as they are displaced by ultrasound, causing loss of energy to heat. This loss is characterized by the classical absorption coefficient β_class ∝ f².

Attenuation coefficients

When an ultrasound beam propagates through tissue, the sum of the absorption and scattering is described as attenuation, which causes an exponential decrease in the pressure and intensity of the ultrasound as a function of the propagation distance x through tissue.

The following equations describe the loss of ultrasound intensity and pressure as the wave moves through tissue. Here, µ is the intensity attenuation coefficient and α is the pressure attenuation coefficient. The value of µ is equal to twice the value of α (so, µ = 2α). Both have units of cm^–1, though the value of µ is often given in units of decibels (dB) per cm, where the conversion factor is µ(dB cm^–1) = 4.343µ(cm^–1). It is useful to note that each 3 dB decrease corresponds to a decrease in intensity by a factor of 2.

Approximate frequency dependences of µ are given in the table below. As an example, in soft tissue, the value of µ = 1 dB cm^–1 for 1 MHz ultrasound and µ = 2 dB cm^–1 for 2 MHz ultrasound. Note that the value of µ for fat is calculated differently than the others via the equation µ(f) = 0.7f^1.5 dB.

Ultrasound transducers

Ultrasound imaging is performed using ultrasound transducers. A gated frequency generator first produces short periodic voltage pulses, which are then amplified and fed into the transducer via a transmit-receive switch. Because the transducer transmits high power pulses and receives low intensity signals from the reflected ultrasound waves, the transmit and receive circuits must be isolated from each other. Amplified voltage is converted by a shaped piezoelectric material (typically lead zirconate titanate, which is abbreviated PZT) in the transducer into a mechanical pressure wave which is transmitted into the tissue.

After reflecting and scattering from boundaries within the tissue, pressure wave signals return to the transducer and are converted back into voltages by the piezoelectric material. The voltages must pass through low-noise preamplifier before digitization. Further amplification and signal processing facilitates display of images on a computer.

When oscillating voltage is applied to one end of shaped PZT material, the thickness of the PZT element oscillates at the same frequency as the voltage. By placing this element in contact with skin, mechanical pressure waves are transmitted into tissue. The element has a resonant frequency f₀ which is determined by its thickness T and the speed of the ultrasound wave in the PZT material c_PZT. The value of c_PZT is ~4000 m/s.

For most ultrasound devices, the transducer element’s thickness must be designed to equal one half the wavelength of ultrasound in the PZT material λ_PZT, so T = λ_PZT/2. This facilitates use of the resonant frequency.

Because of the much higher acoustic impedance of PZT material compared to skin (~18 times higher), a large amount of the energy would be reflected if the PZT was placed directly onto the skin’s surface. This would mean the mechanical wave traveling into the tissue would lose most of its energy.

To prevent this energy loss from happening, transducers possess a matching layer with a z_matchingvalue between z_skin and z_PZT as given by the equation below. The thickness of the matching layer is typically made to be 1/4 of the ultrasound’s wavelength in its material T = λ_matching/4. All this improves the transmission and reception efficiency. Sometimes multiple matching layers are used to further improve efficiency.

At the back of a PZT element, there is a damping layer, typically made of some backing material and epoxy. This damping layer prevents the PZT from continuing to oscillate (at a decaying rate) after each voltage pulse. This continued oscillation would blur the boundaries between the short pulses, which can decrease axial resolution (as will be described soon).

Although transducers have a central frequency f₀, they typically cover a range of frequencies (e.g. a 3 MHz transducer might cover a range of 1-5 MHz). Higher mechanical damping leads to broader transducer bandwidth. Transducer bandwidth is described as the frequency range over which the sensitivity is greater than half the maximum sensitivity level. The relationship between bandwidth and f₀ is often quantified by the quality factor Q, which is the ratio of f₀ to the bandwidth. Low values of Q mean larger bandwidths. Note that 2f₀ is the second harmonic frequency.

Beam geometry and resolution

FUS transducers produce a very complicated wave pattern close to the face of the transducer (the near-field or Fresnel zone). This complicated pattern is not usually useful since it has many parts where the intensity is zero. Beyond the near-field zone, the wave pattern is much simpler and decays exponentially with distance (the far-field or Fraunhofer zone). The boundary between the two zones is called the near-field boundary (NFB) and occurs at a distance Z_NFB away from the face of the transducer. The following equation (where r is the radius of the transducer) can be used to calculate the Z_NFB value.

After the NFB, the FUS beam diverges (spreads out laterally) with an angle of deviation θ which is given by the equation below.

For the far-field zone, the lateral shape of the beam approximates a Gaussian function. The full width at half maximum (FWHM) defines the lateral resolution of the beam. It is given by the equation below, where σ is the standard deviation of the Gaussian. This value is unique to each FUS beam at the specific desired depth. It can be calculated by the following equation.

Single element transducers also produce ultrasound side lobes where the first zero of the side lobe at angle φ is the same equation as the FUS beam’s divergence angle equation. In ultrasound imaging, the side lobes can cause artifacts if they are backscattered from tissue outside of the imaged region.

Axial resolution is the closest distance two boundaries can be relative to each other (in a direction parallel to the FUS beam’s propagation) while still allowing them to be resolved as two distinct features. It is given by the equation below where p_d is the pulse duration and c represents the speed of the ultrasound in the tissue.

The reason that axial resolution works this way is because the echoes of beams returning from two different boundaries are distinguishable so long as these boundaries are spaced widely enough that they do not overlap in time.

Some typical values of axial resolution are 1.5 mm at a frequency (1/c) of 1 MHz or 0.3 mm at a frequency of 5 MHz. But it should be noted that attenuation of the FUS increases at higher frequencies, so there is an important tradeoff between penetration depth and axial resolution. (Very high frequencies such as 40 MHz can be used for imaging the skin at high resolution).

Single flat ultrasound transducers possess relatively poor lateral resolution. Concave curved transducers can achieve better resolutions. (It should be noted that the transducer equations above may be somewhat altered in the case of curved transducers rather than flat transducers). To make a curved transducer, one can add a curved plastic lens in front of the piezoelectric element or the piezoelectric element itself can be made in a concave curved shape.

The shape of a transducer’s curvature can be described by an “f-number”, which is a value equal to the focal distance divided by the aperture dimension where the aperture dimension is determined by the size of the transducer element.

Lateral resolution for a bowl-shaped curved (focused) transducer is calculated using the following equation below where λ is the ultrasound wavelength, F is the focal distance, and D is the diameter of the transducer. The focal distance F is where the lateral beamwidth is narrowest and is approximated as the radius of curvature (ROC) of the lens or PZT element. This approximation is valid except in the case of very high curvature.

When deciding on the focusing power of a transducer, there is a compromise between high spatial resolution and depth over which good spatial resolution is achievable. For a strongly focused transducer, locations further away from the beam’s focal plane diverge more sharply than for a weakly focused transducer. This can be quantified by the on-axis depth-of-focus (DOF) which equals the axial distance over which the beam’s intensity is at least 50% of its maximum value.

Transducer arrays

Contemporary FUS systems typically use arrays consisting of many small piezoelectric elements (rather than single-element transducers). These arrays allow 2D imaging via electronic steering of the beam through tissue while the transducer is held at a fixed position. Sophisticated electronics produce a dynamically changing focus during pulse transmission and signal reception, which maintains high resolution throughout the image. Linear and phased array transducers represent the two main types of arrays.

A linear array consists of many (often 128-512) rectangular piezoelectric elements where the space between elements is called kerf and the distance between their centers is referred to as pitch. Each element is mechanically and electrically isolated from its neighbors by filling the kerf regions with acoustically isolating material. The elements are not focused. Pitch is designed to range from λ/2 to 3λ/2 (where λ is ultrasound wavelength in tissue). Linear arrays are usually about 1 cm wide and 10-15 cm long.

Linear arrays work by using separate voltage pulses to excite a small number of elements at slightly different times where the outer elements are excited first and the inner elements are excited after a short delay. This creates an effectively curved wavefront with a focal point at a certain distance from the array. After all backscattered echoes have been received, another beam consisting of a distinct subset of elements performs the same steps. This is repeated in sequence until all of the groups of elements have completed the procedure. If even numbers of elements were used for each group, the entire process can be repeated again with odd numbers of elements to cover the focal points between those acquired before.

Linear array focusing occurs only in one dimension. By contrast, the elevation plane (the direction perpendicular to the image plane) cannot be focused unless a curved lens is included to produce focus in this dimension. Linear arrays are most often used for applications involving large fields of view and relatively low penetration depth.

Phased arrays are typically around 1-3 cm in length and 1 cm in width. They are used in applications like cardiac imaging where there is only a small region of the body through which the ultrasound can enter without running into bone or air.

As with linear arrays, phased arrays apply voltage pulses at slightly varying times to excite elements and produce an effective wavefront with a certain focal length. But phased arrays must employ beam steering to reconstruct a full 2D image. This occurs by changing the pattern of excitation to sweep the effective wavefront beam across a range of directions to cover the image plane.

Phased arrays also employ a process called dynamic focusing to optimize lateral resolution over the full depth of imaged tissue. This involves dynamically changing the number of elements used to produce a wavefront with varying focal lengths. At deeper regions in the tissue, the number of elements needed to position the focus (where optimal lateral resolution is achieved) is higher than at shallower regions in the tissue. Dynamic focusing allows high lateral resolution across the full depth of the scan. However, dynamic focusing is relatively slow since multiple scans are needed to build up a single line of the image. It should be noted that the length of each element determines the “slice thickness” for the image’s elevation dimension.

There are also multidimensional transducer arrays which include extra rows of transducer elements. These multidimensional arrays can focus in the elevation dimension without the need for curved elements or lenses (though they are more complicated devices). Multidimensional arrays with a small number of extra rows (e.g. 3-10) are referred to as 1.5 dimensional arrays. These 1.5D arrays can facilitate some level of focusing in the elevation dimension, though to a limited extent. When multidimensional arrays possess a large number of extra rows (up to the number of elements in each row), they are referred to as 2 dimensional arrays. These can acquire full 3D image data without needing to be moved from their initial position.

Annular arrays represent another class of transducer array. They are useful at very high frequencies (>20 MHz) since linear or phased arrays are quite difficult to create for these frequencies. Annular arrays consist of concentric rings of piezoelectric material alternating with rings acoustically isolating material. Beam forming is accomplished using an analogous strategy to that of phased arrays. The outermost rings are excited first and the innermost rings last, producing an effective focus. Because annular arrays require mechanical motion to sweep the beam through tissue for reconstruction of images, commercially available devices have been developed to precisely control this motion.

When transducer arrays receive signals, they pass through an amplifier to strengthen them before digitization. However, such amplifiers do not provide linear gain for signals with a dynamic range that exceeds 40-50 dB. This is an issue because very strong signals appear from tissue boundaries near the transducer while much weaker signals appear from tissue boundaries deeper in the body. Weak signals can thus be lost when attempting to receive over larger dynamic ranges. A process called time-gain compensation (TGC) is employed to circumvent this issue. TGC increases the amplification factor as a function of time after transmission of an ultrasound pulse. As a result, the weaker backscattered echoes which come later are amplified to a greater degree than the stronger backscattered echoes which come sooner. TGC is controlled by the operator of the instrument, which usually comes with a variety of preset values for clinical imaging protocols.

Parameters for focused ultrasound in practice

There is no universally accepted definition of FUS dose, so various metrics of exposure are used to quantify how much ultrasound is delivered during a therapeutic session. Examples of such metrics (which will be discussed further below) include acoustic pressure or peak negative pressure, mechanical index, frequency, pulse repetition frequency (PRF), and intensity.

Peak negative pressure is often measured in MPa and describes the degree of rarefaction caused by the ultrasound wave in tissue. For low intensity focused ultrasound (LIFU), the greatest mechanical safety risk is from cavitation (bubble formation and collapse). To measure cavitation risk, the mechanical index (MI) is used. MI can be computed using the following equation where P_n is the peak negative pressure, f₀ is the fundamental frequency, and the derating constant of 0.3 adjusts for tissue attenuation (~7% loss per cm per MHz) and has units dBcm^–1MHz^–1. After derating, 0.3P_n has units of MPa. FDA guidelines specify that MI should not exceed 1.9. MI itself is unitless.

An ultrasound wave’s pulse’s duration (PD) is the number of cycles divided by the frequency. For instance, a pulse with 500 cycles of 500 kHz ultrasound would last for 1 ms. The pulse repetition interval (PRI) is the amount of time between the start of one pulse and the start of the next pulse (so it includes both the pulse and the pause after the pulse). Pulse repetition rate (PRR) also known as the pulse repetition frequency (PRF) equals 1/PRI. The pulse duty cycle (PDC) equals PD/PRI and is expressed as a percentage. PD typically ranges from microseconds to seconds, PRI from milliseconds to seconds, and PDC from <1% up to 70%.

One’s choice of a particular PD and PDC comes from two main factors: (i) the duty cycle can have varying neuromodulatory effects (excitatory or inhibitory) depending on its value and (ii) lower PDC values can be leveraged to limit total energy and heat deposition.

A pulse train is a series of pulses, for which the pulse train duration (PTD) equals the total number of pulses times the PRI. Typical PTDs range from less than 1 second to several minutes. The amount of time between the start of one pulse train and the start of the next is called the pulse train repetition interval (PTRI). The amount of time between pulse trains is called the interstimulus interval (ISI). The pulse train duty cycle (PTDC) equals PTD/PTRI.

It should be noted that the PTDC does not have a major influence on neuromodulatory effect, so the ratio is driven by safety such that the ISI is long enough to limit cumulative heating to reasonable levels.

Multiplying PDC by PTDC gives an overall duty cycle equal to (PD/PRI)(PTD/PTRI) which can be further multiplied by the average intensity of the pulses I_avg to obtain average temporal intensity I_{avg_tp}. FDA diagnostic safety guidelines state that average intensity should fall below 720 mW/cm². So, I_{avg_tp}= (PD/PRI)(PTD/PTRI)I_avg generally should not exceed 720 mW/cm².

Total ultrasound application time is the sum of the durations of all pulse trains plus ISIs. It typically ranges from less than 1 minute to over 60 minutes. Longer total ultrasound application time is thought to usually improve efficacy by depositing more energy, though this may not always be true. Energy per unit time might play a more significant role in efficacy, but this is an ongoing area of investigation.

Frequency (or fundamental frequency f₀) is the primary frequency of FUS passing through the tissue. It is typically measured in kilohertz (kHz) or megahertz (MHz). In human neuromodulation applications, frequency typically ranges from 200-700 kHz (or 0.2-0.7 MHz), providing an acceptable tradeoff between amount of energy entering the brain and the size of the focal region. The reason that the upper limit of frequency for human neuromodulation is typically ~700 kHz is because FUS energy attenuation by the skull at 700 kHz is ~75% (though this varies depending on skull morphology) and keeps increasing at higher frequencies.

Recall from the equations in the earlier discussion that lateral resolution involves wavelength λ (and f = c/λ), so frequency influences the focal region’s size. Also discussed earlier, frequency influences the distance from the transducer to the near-field boundary (Z_NFB). Frequency itself is generally not believed to contribute to neuromodulatory effects in a direct fashion, though this is still under investigation. So, the f₀ value is usually selected to create a focal volume of a desired size at a given depth.

Intensity is defined as power per unit area (and recall the unit of power is Watts or J/s) and is the rate at which energy is transferred by the FUS wave. For ultrasound at any given point in time during the wave cycle, intensity is proportional to the square of the acoustic pressure as described by the equation below where P is the acoustic pressure, ρ is the density of the medium, and c is the ultrasound speed in the medium. Recall that ρc = z, the acoustic impedance. Acoustic intensity is usually measured in watts per square centimeter (W/cm²).

Beyond instantaneous intensity, FUS is often measured by spatial peak pulse average intensity (I_SPPA) and by spatial peak temporal average intensity (I_SPTA). I_SPPA is the average intensity experienced during a single ultrasound pulse. Note that I does not equal P_n²/2z in the case of a ramped pulse. To determine average intensity (I_SPPA) for ramped pulses, the integral of intensity across the pulse is divided by the pulse’s duration PD. Ramped pulses distribute energy more smoothly and help mitigate auditory confounds for LIFU applications.

I_SPTA represents the average intensity of the FUS beam at the point where it is strongest averaged over the pulse duration while accounting for any off periods. It is described by the following equation consisting of the I_SPPA multiplied by the PDC (which is the fraction of PRI that the pulse is turned on).

References:

1. Legon, W. & Strohman, A. Low-intensity focused ultrasound for human neuromodulation. Nat. Rev. Methods Prim. 4, 91 (2024).

2. Smith, N. B. & Webb, A. Introduction to Medical Imaging: Physics, Engineering and Clinical Applications. (Cambridge University Press, 2010).

Notes on Quantum Mechanics

February 10, 2021September 2, 2021By logancollins No Comments

PDF version: Notes on Quantum Mechanics – By Logan Thrasher Collins

The Schrödinger equation and wave functions

Overview of the Schrödinger equation and wave functions

Quantum mechanical systems are described in terms of wave functions Ψ(x,y,z,t). Unlike classical functions of motion, wave functions determine the probability that a given particle may occur in some region. The way that this is achieved involves integration and will be discussed later in these notes.

To find a wave function, one must solve the Schrödinger equation for the system in question. There are time-dependent and time-independent versions of the Schrödinger equation. The time-dependent version is given in 1D and 3D by the first pair of equations below and the time-independent version is given in 1D and 3D by the second pair of equations below. Here, ћ is h/2π (and h is Planck’s constant), V is the particle’s potential energy, E is the particle’s total energy, Ψ is a time dependent wave function, ψ is a time-independent wave function, and m is the mass of the particle. After this point, these notes will focus on 1D cases unless otherwise specified (it will often be relatively straightforward to extrapolate to the 3D case).

For a wave function to make physical sense, it needs to satisfy the constraint that its integral from –∞ to ∞ must equal 1. This reflects the probabilistic nature of quantum mechanics; the probability that a particle may be found anywhere in space must be 1. For this reason, one must usually find a (possibly complex) normalization constant A after finding the wave function solution to the Schrödinger equation. This is accomplished by solving the following integral for A. Here, Ψ* is the complex conjugate of the wave function without the normalization constant and Ψ is the wave function without the normalization constant.

To obtain solutions to the time-dependent Schrödinger equation, one must first solve the time-independent Schrödinger equation to get ψ(x). The general solution for the time-dependent Schrödinger equation is any linear combination of the product of ψ(x) with an exponential term (see below). The coefficients c_n can be real or complex.

Physically, |c_n|² represents the probability that a measurement of the system’s energy would return a value of E_n. As such, an infinite sum of all the |c_n|² values is equal to 1. In addition, note that each Ψ_n(x,t) = ψ_n(x)e^–iEⁿ^t/^ℏ is known as a stationary state. The reason these solutions are called stationary states is because the expectation values of measurable quantities are independent of time when the system is in a stationary state (as a result of the time-dependent term canceling out).

Using wave functions

Once a wave function is known, it can be used to learn about the given quantum mechanical system. Though wave functions specify the state of a quantum mechanical system, this state usually cannot undergo measurement without altering the system, so the wave function must be interpreted probabilistically. The way the probabilistic interpretation is achieved will be explained over the course of this section.

Before going further, it will be useful to understand some methods from probability. First, the expectation value is the average of all the possible outcomes of a measurement as weighted by their likelihood (it is not the most likely outcome as the name might suggest). Next, the standard deviation σ describes the spread of a distribution about an average value. Note that the square of the standard deviation is called the variance.

Equations for the expectation value and standard deviation are given as follows. The first equation computes the expectation value for a discrete variable j. Here, P(j) is the probability of measurement f(j) for a given j. The second equation is a convenient way to compute the standard deviation σ associated with the expectation value for j. The third equation computes the expectation value for a continuous function f(x). Here, ρ(x) is the probability density of x. When ρ(x) is integrated over an interval a to b, it gives the probability that measurement x will be found over that interval. The fourth equation the same as the second equation, but finds the standard deviation σ for the continuous variable x.

In quantum mechanics, operators are employed in place of measurable quantities such as position, momentum, and energy. These operators play a special role in the probabilistic interpretation of wave functions since they help one to compute an expectation value for the corresponding measurable quantity.

To compute the expectation value for a measurable quantity Q in quantum mechanics, the following equation is used. Here, Ψ is the time-dependent wave function, Ψ* is the complex conjugate of the time-dependent wave function, and Q̂ is the operator corresponding to Q.

Any quantum operator which corresponds to a classical dynamical variable can be expressed in terms of the momentum operator –iℏ(∂/∂x). By rewriting a given classical expression in terms of momentum p and then replacing every p within the expression by –iℏ(∂/∂x), the corresponding quantum operator is obtained. Below, a table of common quantum mechanical operators in 1D and 3D is given.

Heisenberg uncertainty principle

The Heisenberg uncertainty principle explains why quantum mechanics requires a probabilistic interpretation. According to the Heisenberg uncertainty principle, the more precisely the position of a particle is determined via some measurement, the less precisely its momentum can be known (and vice versa). The Heisenberg uncertainty principle is quantified by the following equation.

The reason for the Heisenberg uncertainty principle comes from the wave nature of matter (and not from the observer effect). For a sinusoidal wave, the wave itself is not really located at any particular site, it is instead spread out across the cycles of the sinusoid. For a pulse wave, the wave can be localized to the site of the pulse, but it does not really have a wavelength. There are also intermediate cases where the wavelength is somewhat poorly defined and the location is somewhat well-defined or vice-versa. Since the wavelength of a particle is related to the momentum by the de Broglie formula p = h/λ = 2πℏ/λ, this means that the interplay between the wavelength and the position applies to momentum and position as well. The Heisenberg uncertainty principle quantifies this interplay.

Some simple quantum mechanical systems

Infinite square well

The infinite square well is a system for which a particle’s V(x) = 0 when 0 ≤ x ≤ a and its V(x) = ∞ otherwise. Because the potential energy is infinite outside of the well, the probability of finding the particle there is zero. Inside the well, the time-independent Schrödinger equation is given as follows. This equation is the same as the classical simple harmonic oscillator.

For the infinite square well, certain boundary conditions apply. In order for the wave function to be continuous, the wave function must equal zero once it reaches the walls, so ψ(0) = ψ(a) = 0. The general solution to the infinite square well differential equation is given as the first equation below. The boundary condition ψ(0) = 0 is employed in the second equation below. Since the coefficient B = 0, there are only sine solutions to the equation. Furthermore, if ψ(a) = 0, then Asin(ka) = 0. This means that k = nπ/a (where n = 1, 2, 3…) as given by the third equation below. The fourth equation below shows that this set of values for k leads to a set of possible discrete energy levels for the system

To find the constant A, the wave function ψ = Asin(nπx/a) must undergo normalization. As mentioned earlier, normalization is achieved by setting the normalization integral equal to 1 and solving for the constant A. Note that the time-independent Schrödinger equation can be utilized in the normalization integral since the exponential component of the time-dependent Schrödinger equation would cancel anyways.

Using this information, the wave functions for the infinite square well particle system are obtained. The time-independent and time-dependent wave functions are both displayed below at left and right respectively.

This infinite set of wave functions has some important properties. They possess discrete energies that increase by a factor of n² with each level (and n = 1 is the ground state). The wave functions are also orthonormal. This property is described by the following equation. Here, δ_mn is the Kronecker delta and is defined below.

Another important property of these wave functions is completeness. This means that any function can be expressed as a linear combination of the time-independent wave functions ψ_n. The reason for this remarkable property is that the general solution (see below) is equivalent to a Fourier series.

The first equation below can be employed to compute the n^th coefficient c_n. Here, f(x) = Ψ(x,0) which is an initial wave function. Note that the initial wave function can be any function Ψ(x,0) and the result will generate coefficients for that starting point. This first equation is derived using the orthonormality of the solution set. Note that the formula applies to most quantum mechanical systems since the properties of orthonormality and completeness hold for most quantum mechanical systems (though there are some exceptions). The second equation below computes the c_n coefficients specifically for the infinite square well system.

Quantum harmonic oscillator

For the quantum harmonic oscillator, the potential energy in the Schrödinger equation is given by V(x) = 0.5kx² = 0.5mω²x². This means that the following time-independent Schrödinger equation needs to be solved.

There are two main methods for solving this differential equation. These include a ladder operator approach and a power series approach. Both of these methods are quite complicated and will not be covered here. The solutions for n = 0, 1, 2, 3, 4, 5 are given below. Here, H_n(y) is the n^th Hermite polynomial. The first five Hermite polynomials and the corresponding energies for the system are given in the table. Note that the discrete energy levels for the quantum harmonic oscillator follow the form (n + 0.5)ћω.

As with any quantum mechanical system, the quantum harmonic oscillator is further described by the general time-dependent solution. To identify the coefficients c_n for this general solution, Fourier’s trick is employed (see previous section) where f(x) is once again any initial wave function Ψ(x,0).

Quantum free particle

Though the classical free particle is a simple problem, there are some nuances which arise in the case of the quantum mechanical free particle which greatly complicate the system.

To start, the Schrödinger equation for the quantum free particle is given in the first equation below. Here, k = (2mE)^0.5/ћ. Note that V(x) = 0 since there is no external potential acting on the particle. The second equation below is a general time-independent solution to the system in exponential form. The third equation below is the time-dependent solution to the system where the terms are multiplied by e^–iEt/^ћ. Realize that this general solution can be written as a single term by redefining k as ±(2mE)^0.5/ћ. When k > 0, the solution is a wave propagating to the right. When k < 0, the solution is a wave propagating to the left.

The speed of these propagating waves can be found by dividing the coefficient of t (which is ћk²/2m) by the coefficient of x (which is k). Since this is speed, the direction of the wave does not matter, so one can take the absolute value of k. By contrast, the speed of a classical particle is found by solving E = 0.5mv², which gives a puzzling result that is twice as fast as the quantum particle.

Another challenge associated with the quantum free particle is that its wave function is non-normalizable (as shown below). Because of this, one can conclude that free particles cannot exist in stationary states. Equivalently, free particles never exhibit definite energies.

To resolve these issues with the quantum free particle, it has been found that the wave function of a quantum free particle actually carries a range of energies and speeds known as a wave packet. The solution for this wave packet involves the integral given by the first equation below and a function ϕ(k) given by the second equation below. This second equation allows one to determine ϕ(k) to fit a desired initial wave function Ψ(x,0). It was obtained using a mathematical tool called Plancherel’s theorem.

The above solution to the quantum free particle is now normalizable. Furthermore, the issue with the speed of the quantum free particle having a value twice as large as the speed of the classical free particle is fixed by considering a phenomenon known as group velocity. The waveform of the particle is an oscillating sinusoid (see image). This waveform includes an envelope, which represents the overall shape of the oscillations rather than the individual ripples. The group velocity v_g is the speed of this envelope while the phase velocity v_p is the speed of the ripples. It can be shown using the definitions of phase velocity and group velocity (see below) that the group velocity is twice the phase velocity, resolving the problem with the particle speed. The group velocity of the envelope is thus what actually corresponds to the speed of the particle.

Interlude on bound states and scattering states

To review, the solutions to the Schrödinger equation for the infinite square well and quantum harmonic oscillator were normalizable and labeled by a discrete index n while the solution to the Schrödinger equation for the free particle was not normalizable and was labeled by a continuous variable k.

The solutions which are normalizable and labeled by a discrete index are known as bound states. The solutions which are not normalizable and are labeled by a continuous variable are known scattering states.

Bound states and scattering states are related to certain classical mechanical phenomena. Bound states correspond to a classical particle in a potential well where the energy is not large enough for the particle to escape the well. Scattering states correspond to a particle which might be influenced by a potential but has a large enough energy to pass through the potential without getting trapped.

In quantum mechanics, bound states occur when E < V(∞) and E < V(–∞) since the phenomenon of quantum tunneling allows quantum particles to leak through any finite potential barrier. Scattering states occur when E > V(∞) or E > V(–∞). Since most potentials go to zero at infinity or negative infinity, this simplifies to bound states happening when E < 0 and scattering states happening when E > 0.

The infinite square well and the quantum harmonic oscillator represent bound states since V(x) goes to ∞ when x → ±∞. By contrast, the quantum free particle represents a scattering state since V(x) = 0 everywhere. However, there are also potentials which can result in both bound and scattering states. These kinds of potentials will be explored in the following sections.

Delta-function well

Recall that the Dirac delta function δ(x) is an infinitely high and infinitely narrow spike at the origin with an area equal to 1 (the area is obtained by integrating). The spike appears at the point a along the x axis when δ(x – a) is used. One important property of the Dirac delta function is that f(x)δ(x – a) = f(a)δ(x – a). By integrating both sides of the equation of this property, one can obtain the following useful expression. Note that a ± ϵ is used as the bounds since any positive value ϵ will then allow the bounds to encompass the Dirac delta function spike.

The delta-function well is a potential of the form –αδ(x) where α is a positive constant. As a result, the time-independent Schrödinger equation for the delta-function well system is given as follows. This equation has solutions that yield bound states when E < 0 and scattering states when E > 0.

For the bound states where E < 0, the general solutions are given by equations below. The substitution κ is defined by the first equation below, the second equation below is the general solution for x < 0, and the third equation below is the general solution for x > 0. (Since E is assumed to have a negative value, κ is real and positive). Note that V(x) = 0 for x < 0 and x > 0. In the solution for x < 0, the Ae^–κx term explodes as x → –∞, so A must equal zero. In the solution for x > 0, the Fe^κx term explodes as x → ∞, so F must equal zero.

To combine these equations, one must use appropriate boundary conditions at x = 0. For any quantum system, ψ is continuous and dψ/dt is continuous except at points where the potential is infinite. The requirement for ψ to exhibit continuity means that F = B at x = 0. As a result, the solution for the bound states can be concisely stated as follows. In addition, a plot of the delta-function well’s bound state time-independent wave function is given below.

The presence of the delta function influences the energy E. To find the energy, one can integrate the time-independent Schrödinger equation for the delta-function well system. By making the bounds of integration ±ϵ and then taking the limit as ϵ approaches zero, the integral works only on the negative spike of the delta function at x = 0. The result for the energy is at the end of the following set of equations.

As seen above, the delta-function well only exhibits a single bound state energy E. By normalizing the wave function ψ(x) = Be^–κ|x|, the constant B is found (as seen in the first equation below). The second equation below describes the single bound state wave function and reiterates the single bound state energy associated with this wave function.

For the scattering states where E > 0, the general solutions are given by equations below. The substitution k is defined by the first equation below, the second equation below is the general solution for x < 0, and the third equation below is the general solution for x > 0. (Since E is assumed to have a positive value, k is real and positive). Note that V(x) = 0 for x < 0 and x > 0. None of the terms explode this time, so none of the terms can be ruled out as equal to zero.

As a consequence of the requirement for ψ(x) to be continuous at x = 0, the following equation involving the constants A, B, F, and G must hold true. This is the first boundary condition.

There is also a second boundary condition which involves dψ/dx. Recall the following step (see first equation below) from the process of integrating the Schrödinger equation. To implement this step, the derivatives of ψ(x) (see second equation below) are found and then the limits of these derivatives from the left and right directions are taken (see third equation below). Since ψ(0) = A + B as seen in the equation above, the second boundary condition can be given as the final equation below.

By rearranging the final equation above and substituting in a parameter β = mα/ћ²k, the following expression is obtained. This expression is a compact way of writing the second boundary condition.

These two boundary conditions provide two equations, but there are four unknowns in these equations (five unknowns if k is included). Despite this, the physical significance of the unknown constants can be helpful. When e^ikx is multiplied by the factor for time-dependence e^–iEt/ћ, it gives rise to a wave propagating to the right. When e^–ikx is multiplied by the factor for time-dependence e^–iEt/ћ, it gives rise to a wave propagating to the left. As a result, the constants describe the amplitudes of various waves. A is the amplitude of a wave moving to the right on the x < 0 side of the delta-function potential, B is the amplitude of a wave moving to the left on the x < 0 side of the delta-function potential, F is the amplitude of a wave moving to the right on the x > 0 side of the delta-function potential, and G is the amplitude of a wave moving to the left on the x > 0 side of the delta-function potential.

In a typical experiment on this type of system, particles are fired from one side of the delta-function potential, the left or the right. If the particles are coming from the left (moving to the right), the term with G will equal zero. If the particles are coming from the right (moving to the left), the term with A will equal zero. This can be understood intuitively by examining the figure above.

As an example, for the case of particles fired from the left (moving to the right), A is the amplitude of the incident wave, B is the amplitude of the reflected wave, and F is the amplitude of the transmitted wave. The equations of the two boundary conditions are reiterated in the first line below. By solving these equations, the second line of expressions is found. Since the probability of finding a particle at a certain location is |Ψ|², the relative probability R of an incident particle undergoing reflection and the relative probability T of an incident particle undergoing transmission are given by the third line of expressions below.

Also for the example case of particles fired from the left (moving to the right), by substituting back from β = mα/ћ²k and k = (2mE)^0.5/ћ to get the expressions in terms of energy, the following equations are obtained for the reflection and transmission relative probabilities.

By performing the same process, but with A = 0 instead of G = 0, corresponding equations can be found for the case of particles fired from the right (moving towards the left).

It is important to note that, since these scattering wave functions are not normalizable, they do not actually represent possible particle states. To solve this problem, one must construct normalizable linear combinations of the stationary states in a manner similar to that performed with the quantum free particle system. In this way, wave packets will occur and the actual particles will be described by the range of energies of the wave packets. Because the actual normalizable system exhibits a range of energies, the probabilities R and T should be thought of as approximate measures of reflection and transmission for particles with energies in the vicinity of E.

Finite square well

The finite square well is a system for which a particle’s V(x) = –V₀ when –a ≤ x ≤ a and its V(x) = 0 otherwise. For this system, the Schrödinger equation is given as follows for the conditions x < –a, –a ≤ x ≤ a, and x > a. Note that the equations for x < –a and x > a are the same since V(x) = 0 in both cases (but the boundary conditions will differ as will be explained soon). As with the Delta-function potential well, the finite square well has both bound states (with E < 0) and scattering states (with E > 0). First, the bound states with E < 0 will be considered. In this case, the Schrödinger equations for the finite square well are as follows.

For the cases of x < –a and x > a where V(x) = 0, the general solutions to the Schrödinger equation are respectively Ae^–κx + Be^κx and Fe^–κx + Ge^κx where A, B, F, and G are arbitrary constants. In the x < –a case, the Ae^–κx term blows up as x → –∞, making this term physically invalid. As a result, the physically admissible solution is ψ(x) = Be^κx. In the x > a case, the Ge^κx term blows up as as x → ∞, making this term physically invalid. As a result, the physically admissible solution is ψ(x) = Fe^–κx. For the case of –a ≤ x ≤ a, the general solution to the Schrödinger equation is ψ(x) = Csin(lx) + Dcos(lx). Note that, because E must be greater than the minimum potential energy V_min = –V₀, the value of l ends up real and positive (even though E is also negative). These solutions are summarized by the following equations.

Since the potential V(x) = –V₀ is an even function (symmetric about the y axis), one can choose to write the solutions to the wave function as either even or odd. This comes from some properties of the time-independent Schrödinger equation. Next, it is again important to constrain these solutions using the boundary conditions which require the continuity of ψ(x) and dψ/dx at ±a.

For the even solutions, the constant C in ψ(x) = Csin(lx) + Dcos(lx) is zero. Because C = 0, the remaining equation is the even function ψ(x) = Dcos(lx) for –a ≤ x ≤ a. So, the continuity of ψ(x) and dψ/dx at +a necessitates the following two equations to hold true. The third equation comes from dividing the second equation by the first equation to solve for κ.

For the odd solutions, the constant D in ψ(x) = Csin(lx) + Dcos(lx) is zero. Because D = 0, the remaining equation is the odd function ψ(x) = Dsin(lx) for –a ≤ x ≤ a. So, the continuity of ψ(x) and dψ/dx at +a necessitates the following two equations to hold true. The third equation comes from dividing the second equation by the first equation to solve for κ.

As κ and l are both functions of E, the κ = ltan(la) and κ = –lcot(la) equations can be solved for E. To do this, it is convenient to use the notation z = la and z₀ = (a/ћ)(2mV₀)^0.5. Simplifying the κ = ltan(la) and κ = –lcot(la) equations using this notation gives the following results. These equations can be solved numerically for z or graphically for z by looking for points of intersection (after obtaining z, E is easily computed).

Let us consider the tan(z) equation. There are two limiting cases of interest. These include a well which is wide and deep and a well which is shallow and narrow. Though not included in these notes, similar calculations can be performed for the –cot(z) equation.

For a wide and deep well, the value of z₀ is large. Intersections between the curves of tan(z_n) and ((z₀/z_n)² – 1)^0.5 occur at nπ/2 for odd n and at nπ for even n. This leads to the following equations which describe values of E_n. From this outcome, it can be seen that infinite V₀ results in the infinite square well case with an infinite number of bound states. However, for any finite square well, there are only a finite number of bound states.

For a shallow and narrow well, the value of z₀ is small. As the value of z₀ decreases, fewer and fewer bound states exist. Once z₀ is smaller than π/2, there is only one bound state (which is an even bound state). Interestingly, no matter how small the well, this one bound state always persists.

The scattering states, which occur when E > 0, will now be considered. In this case, the Schrödinger equations for the finite square well are as follows.

The general solutions to the Schrödinger equation for the finite square well’s scattering states are as follows.

But recall that in a typical scattering experiment, particles are fired from one side of the delta-function potential, the left or the right. Here it will be assumed that the particles are fired from the left side of the well (moving towards the right). Note that similar calculations could be performed for the opposite case. With this assumption, one can realize that the coefficient A represents the incident (from the left) wave’s amplitude, the coefficient B represents the reflected wave’s amplitude, and the coefficient F represents the transmitted (to the right) wave’s amplitude. Finally, the coefficient G = 0 since there is not an incident wave from the right moving towards the left.

There are four boundary conditions, continuity of ψ(x) at ±a and continuity of dψ/dx at ±a. These boundary conditions yield the following equations.

With the above equations, one can eliminate C and D and subsequently solve the system for B and F. This yields the equations below for B and F.

As with the delta-function well, a transmission coefficient T = |F|²/|A|² can be computed across the finite square well. Recall that T represents the probability of the particle undergoing transmission across the well (in this case when moving from the right side to the left side). The probability of the particle undergoing reflection is R = 1 – T.

Since 1/T equals the equation below, whenever the sine squared term is zero, the probability of transmission T = 1.

Recall that a sine (or sine squared) term is zero when the function inside of it equals nπ such that n is any integer.

Remarkably, the above equation is the same as the one which describes the infinite square well’s energies. But realize that, for the finite square well, this only holds in the case of T = 1.

Reference: Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press. https://doi.org/DOI: 10.1017/9781316995433

Cover image source: wikimedia.org

Notes on x-ray physics

August 8, 2019August 8, 2019By logancollins 1 Comment

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.

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