What Type of Materials Tend to Produce a Continuous Spectrum

Continuous Spectrum

Positive Maps on C*-Algebras

F. Cipriani , in Encyclopedia of Mathematical Physics, 2006

Normal and Singular States

When observables with continuous spectrum have to be considered and one chooses the algebra $\mathcal{B}\left(h\right)$ of all bounded operators, the above formula, although still meaningful, does not describe all states on $\mathcal{B}\left(h\right)$ but only the important subclass of the normal ones. To this class, which can be considered on any von Neumann algebra $\mathcal{M}$ , belong states ϕ which are σ-weakly continuous functionals. Equivalently, these are the states such that for all increasing net $a_{\alpha }\in {\mathcal{M}}_{+}$ with least upper bound $a_{\alpha }\in {\mathcal{M}}_{+},\varphi \left(a\right)$ is least upper bound of the net ϕ(a _α ).

In general, each state ϕ on a von Neumann algebra $\mathcal{M}$ splits as a sum of a maximal normal piece and a singular one. Singular traces appear in noncommutative geometry as very useful tools to get back local objects from spectral ones via the familiar principle that local properties of functions depend on the asymptotics of their Fourier coefficients.

This is best illustrated on a compact, Riemannian n-manifold M by the formula

${\displaystyle {\int }_{M}f\text{dm}=c_n\cdot {\tau }_{\omega }\left(M_f{\left|D\right|}^{-n}\right)}$

which expresses the Riemannian integral of a nice function f in terms of the Dirac operator D acting on the Hilbert space of square-integrable spinors, the multiplication operator M _f by f, and the singular Dixmier tracial state τ _ω on $\mathcal{B}\left(H\right)$ . Here the compactness of M implies the compactness of the operator M _f |D|⁻ⁿ and τ _ω is a limiting procedure depending only on the asymptotic behavior of the eigenvalues of M _f |D|⁻ⁿ . Similar formulas are valid on self-similar fractals as well as on quasiconformal manifolds. Local index formulas represent cyclic cocycles in Connes' spectral geometry (see Noncommutative Geometry and the Standard Model; Noncommutative Geometry from Strings; Path-Integrals in Noncommutative Geometry).

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Supernovae

David Branch , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.A The Continuous Spectrum

The shape of a star's thermal continuous spectrum is determined primarily by the temperature at the photosphere. The absolute brightness depends also on the radius of the photosphere. In a supernova, the temperature and radius of the photosphere change with time. When a star explodes, its matter is heated and thrown into rapid expansion. The luminosity abruptly increases in response to the high temperature, but most of this energy is radiated as an X-ray and ultraviolet "flash" rather than as optical light. As the supernova expands, it cools. For about 3  weeks the optical light curve rises as the radius of the photosphere increases, and an increasing fraction of the radiation from the cooling photosphere goes into the optical band. Maximum optical brightness occurs when the temperature is near 10,000   K. At this time the radius of the photosphere is 10¹⁵  cm, or 70   AU, almost twice the radius of Pluto's orbit. The matter density at the photosphere is low, near 10⁻¹⁶  g/cm³.

After maximum light, further cooling causes the light curve to decline. Owing to the expansion and consequent geometrical dilution, the photosphere recedes with respect to the matter, i.e., matter flows through the photosphere, and deeper and deeper layers of the ejecta are gradually exposed. Eventually, at a time that depends mainly on the amount of matter ejected but that ordinarily is a matter of months, the ejected matter becomes optically thin, the continuous spectrum fades away, and the "photospheric phase" comes to an end. The supernova then enters the transparent "nebular" phase.

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Hyperspectral Imaging

José Manuel Amigo , in Data Handling in Science and Technology, 2019

1.2.4 Hyperspectral images

Hyperspectral images are the images in which one continuous spectrum is measured for each pixel [9]. Normally, the spectral resolution is given in nanometers or wave numbers (Fig. 4).

Figure 4. Representation of the image of a cookie measured with a hyperspectral camera in the wavelength range of 940–1600   nm (near infrared, NIR) with a spectral resolution of 4   nm. The spectra obtained in two pixels are shown and the false color image (single channel image) obtained at 1475   nm. The single channel image selected highlighted three areas of the cookie where water was intentionally added. This water is invisible in the VIS region (nothing can be appreciated in the real color picture). Nevertheless, water is one of the main elements that can be spotted in NIR.

Hyperspectral images can be obtained from many different electromagnetic measurements. The most popular are visible (VIS), NIR, middle infrared (MIR), and Raman spectroscopy. Nevertheless, there are many other types of HSI that are gaining popularity like confocal laser microscopy scanners that are able to measure the complete emission spectrum at certain excitation wavelength for each pixel, Terahertz spectroscopy, X-ray spectroscopy, 3D ultrasound imaging, or even magnetic resonance.

Hyperspectral images are the only type of images where we can talk about spectral resolution (also known as radiometric resolution in remote sensing field). The spectral resolution is defined as the interval or separation (gap) between different wavelengths measured in a specific wavelength range. Obviously, the more bands (or spectral channels) acquired in a smaller wavelength range, the higher the spectral resolution will be.

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Light Sources

Graeme G. Lister , John F. Waymouth , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

V.B Radiation Mechanism

The radiation mechanism in incandescent lamps is the continuous spectrum resulting from deflection and unquanitized radiation of valence electrons (traveling freely through the conduction band of the solid) by collisions with nuclei, each other, and lattice phonons. The optical depth is great enough that the internal radiation flux reaches the blackbody level throughout the UV-visible-IR spectrum. A large index of refraction causes reflection losses in the escape of this radiation from the surface, yielding an effective emittance of ∼0.45 in the visible, decreasing to approximately 0.10 to 0.15 at 10  μm wavelength in the IR. Thus, tungsten emits a larger fraction of total radiation into the visible (giving approximately 30–40% higher luminous efficacy) than a blackbody at the same temperature, with a comparable increase in this visible fraction (and luminous efficacy) with increasing temperature (see Fig. 7). For efficient incandescent lamps, the temperature of the radiator must be as high as possible, yet consistent with constraints on service life. Visible spectral power distribution is blackbody-like, with a color temperature of 50 to 100   K higher than true: the CRI is in the high 90   s.

FIGURE 7. Plots of luminous efficacy versus temperature for tungsten and blackbody incandescent radiators. The higher luminous efficacy for tungsten is the result of its higher radiant emittance in the visible in comparison to the IR range (see Fig. 3).

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Radio Astronomy, Planetary

Samuel Gulkis , Imke de Pater , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.A Thermal (Blackbody) Radiation

Any object in thermodynamic equilibrium with its suroundings (having a temperature above absolute zero) emits a continuous spectrum of electromagnetic radiation at all wavelengths, including the radio region. This emission is referred to as thermal emission. The concept of a "blackbody" radiator is frequently used as an idealized standard which can be compared with the absorption and emission properties of real materials. A blackbody radiator is defined as an object that absorbs all electromagnetic radiation that falls on it at all frequencies over all angles of incidence. No radiation is reflected from such an object. According to thermodynamic arguments embodied in Kirchhoff's law, a good absorber is also a good emitter. The blackbody radiator emits the maximum amount of thermal radiation possible for an object at a given temperature. The radiative properties of a blackbody radiator have been well studied and verified by experiments.

A blackbody radiator is an idealized concept rather than a description of an actual radiator. Only a few surfaces, such as carbon black, carborundum, platinum black, and gold black, approach a blackbody in their ability to absorb incident radiant energy over a broad wavelength range. Many materials are spectrally selective in their ability to absorb and emit radiation, and hence they resemble blackbody radiators over some wavelength ranges and not over others. Over large ranges of the radio and infrared spectrum, planets behave as imperfect blackbodies. Later, we will see how the deviations from the blackbody spectrum contain information about physical and chemical properties of these distant objects.

An important property of a blackbody radiator is that its total radiant energy is a function only of its temperature; that is, the temperature of a blackbody radiator uniquely determines the amount of energy that is radiated into any frequency band. Planetary radio astronomers make use of this property by expressing the amount of radio energy received from a planet in terms of the temperature of a blackbody of equivalent angular size. This concept is developed more fully in the following paragraphs.

The German physicist Max Planck first formulated the theory that describes the wavelength dependence of the radiation emitted from a blackbody radiator in 1901. Planck's theory was revolutionary in its time, requiring assumptions about the quantized nature of radiation. Planck's radiation law states that the brightness of a blackbody radiator at temperature T and frequency ν is expressed by

(1) $\text{B}=\left(2\text{h}{\nu }^3/{\text{c}}^2\right){\left({\text{e}}^{\text{h}\nu /\text{k}\text{T}}-1\right)}^{-1},$

where B is the brightness in watts per square meter per hertz per radian; h is Planck's constant (6.63   ×   10⁻³⁴  J sec); ν is the frequency in hertz; λ   = c/ν is the wavelength in meters; c is the velocity of light (3   ×   10⁻⁸  m/sec); k is Boltzmann's constant (1.38   ×   10⁻²³  J/K); and T is the temperature in kelvins.

Equation (1) describes how much power a blackbody radiates per unit area of surface, per unit frequency, into a unit solid angle. The curves in Fig. 1 show the brightness for three blackbody objects at temperatures of 6000, 600, and 60   K. The radiation curve for the undisturbed Sun is closely represented by the 6000   K curve over a wide frequency range. The other two curves are representative of the range of thermal temperatures encountered on the planets. It should be noted in Fig. 1 that the brightness curve that represents the Sun peaks in the optical wavelength range, while representative curves for the planets peak in the infrared. This means that most of the energy received by the planets from the Sun is in the visible wavelength range, while that emitted by the planets is radiated in the infrared. Radio emissions are expected to play only a small role in the overall energy balance of the planets because the vast majority of the power that enters and leaves the planets is contained within the visible and infrared region of the spectrum.

FIGURE 1. Blackbody radiation curves at 6000, 600, and 60   K. The 6000   K curve is representative of the solar spectrum.

A useful approximation to the Planck radiation law can be obtained in the low-frequency limit where h ν is small compared with kt(hν   ⪡ kT). This condition is generally met over the full range of planetary temperatures and at radio wavelengths. It leads to the Rayleigh–Jeans approximation of the Planck law, given by

(2) $\text{B}=2{\nu }^2\text{k}\text{T}/{\text{c}}^2=2\text{k}\text{T}/{\lambda }^2.$

The Rayleigh–Jeans approximation shows a linear relationship between physical temperature and the Planck brightness B. The brightness is also seen to decrease as the inverse square of the wavelength, approaching infinity as the wavelength gets shorter and shorter. The Planck brightness, on the other, hand reaches a maximum value at some wavelength and decreases at longer and shorter wavelengths. The Rayleigh–Jeans approximation matches the Planck law at wavelengths considerably longer than the wavelength of peak brightness. However, for shorter wavelengths, the approximation gets progressively worse. At a temperature of 100   K and a wavelength of 1   mm, the error is ∼8%.

Planetary radio astronomers estimate the radio power emitted by the planets by measuring with a radio telescope the power flux density received at the Earth. Figure 2 illustrates the geometry involved in the measurement of power from an ideal blackbody radiator. The spectral power (per unit frequency) emitted by an elemental surface element of the blackbody of area dA into a solid angle dΩ is given by B cos(θ)dΩdA, where θ is the angle between the normal to the surface and the direction of the solid angle dΩ. The total power (per unit frequency interval) radiated by a blackbody radiator is obtained by integrating the brightness over the surface area and over the solid angle into which each surface element radiates. The total spectral power density produced by a spherical blackbody radiator of radius r at a distance d from the blackbody is given by

FIGURE 2. Relationship between the brightness and power radiated by a blackbody spherical radiator of radius r and the flux density at a distance d from the blackbody.

(3a) $\text{S}=\frac{1}{4\pi {\text{d}}^2}\int \int \text{B}cos(\theta )\text{d}\text{s}\text{d}\Omega$

(3b) $=2\pi \text{k}\text{T}{(\text{r}/\text{d})}^2/{\lambda }^2.$

The double integral represents integration over the surface area of the emitting body and over the hemisphere into which each surface element radiates. The quantity S is called "flux density." Flux density has units of power per unit area per unit frequency. A common unit of flux density is the flux unit (f.u.) or jansky (Jy), which has the value 10⁻²⁶  W m⁻² Hz⁻¹.

If a planet radiates like a blackbody and subtends a solid angle of Ω steradians at the observer's distance, then the flux density produced by the planet is given by (using the Rayleigh–Jeans approximation)

(4a) $\text{S}=2\text{k}\text{T}\mathrm{\Omega }/{\lambda }^2$

(4b) $=\text{B}\mathrm{\Omega }$

The convention generally adopted for calculating Ω for a planet is to use the polar (PSD) and equatorial (ESD) semidiameter values in the expression

(5) $\mathrm{\Omega }=\pi \times \text{PSD}\times \text{ESD}.$

The American Ephemeris and Nautical Almanac (AENA) provides values for PSD and ESD. A web site (http://ssd.jpl.nasa.gov) operated by the JPL Solar System Dynamics Group also provides these data. Equations (4) and (5) can be combined to yield the expression

(6a) $\begin{array}{cc}\hfill \text{S}=5.1\times {10}^{-34}\text{T}{\theta }_{\text{E}}{\theta }_{\text{P}}/{\lambda }^2& \hfill {\text{Wm}}^{-2}{\text{Hz}}^{-1}\hfill \end{array}$

(6b) $=5.1\times {10}^{-8}\text{T}{\theta }_{\text{E}}{\theta }_{\text{P}}/{\lambda }^2\text{Jy}$

where θ_E and θ_P are the apparent equatorial and polar diameters of the planets in seconds of arc and λ is in meters.

Even though the planets do not radiate like a blackbody, planetary radio astronomers express the observed brightness in terms of the temperature of an equivalent blackbody that would produce the same brightness. This temperature is called the brightness temperature T _B, defined as follows [from Eq. (4)]:

(7) ${\text{T}}_{\text{B}}=\text{B}{\lambda }^2/2\text{K}=\left(\text{S}/\mathrm{\Omega }\right){\lambda }^2/2\text{k}.$

The brightness temperature for a planet can be calculated once the flux density S and solid angle Ω are known. The brightness temperature approximates the physical temperature the more the planet behaves like a blackbody radiator.

For problems that deal with the energy budget of the planets, it is necessary to know the total amount of power radiated over all frequencies. Once again the concept of the blackbody is useful. The Planck radiation law can be integrated over all frequencies and solid angles to obtain the relationship known as the Stefan–Boltzmann law, given by

(8) $\text{R}=\sigma {\text{T}}^4,$

where R is the rate of emission, expressed in units of watts per square meter in the mks system. The constant σ has a numerical value of 5.67   ×   10⁻⁸  W m⁻² K⁻⁴. Using the blackbody concept, we can now estimate the energy balance of planets that absorb visible light and radiate energy into the infrared.

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Hyperspectral Imaging

Nicola Falco , ... Jon Atli Benediktsson , in Data Handling in Science and Technology, 2019

Abstract

Advances in sensor technology and spectroscopy have provided with high spectral resolution imagery known as hyperspectral images. Their peculiar characteristics of providing a continuous spectrum of the material investigated make them extremely valuable for a variety of applications, ranging from land-cover classification to plant biophysical parameter estimation and monitoring. Concerning the task of image classification, while the information provided by these images is noteworthy, their processing poses important challenges in terms of information extraction and classification performance. This chapter provides a general discussion on hyperspectral supervised classification strategies and then focuses on classification ensembles and feature extraction based on edge-preserving filtering and operators from the mathematical morphology framework for spectral-spatial techniques. For both discussions, the necessary steps of the processing pipeline are addressed and examples of experimental results carried on hyperspectral images are provided and discussed.

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Advances in Computers: Improving the Web

Dalibor Mitrović , ... Christian Breiteneder , in Advances in Computers, 2010

5.4.2 Tonality

Tonality is the property of sound that distinguishes noise-like from tonal sounds [11] . Noise-like sounds have a continuous spectrum while tonal sounds typically have line spectra. For example, white noise has a flat spectrum and consequently a minimum of tonality while a pure sine wave results in high tonality. Tonality is related to the pitch strength that describes the strength of the perceived pitch of a sound (see Section   2.4). Sounds with distinct (sinusoidal) components tend to produce larger pitch strength than sounds with continuous spectra.

We distinguish between two classes of features that (partially) measure tonality: flatness measures and bandwidth measures. In the following, we first describe bandwidth measures (bandwidth, spectral dispersion, and spectral rolloff point) and then we focus on flatness measures (spectral crest, spectral flatness, subband spectral flux, and entropy).

Bandwidth. Bandwidth is usually defined as the magnitude-weighted average of the differences between the spectral components and the spectral centroid [69]. The bandwidth is the second-order statistic of the spectrum. Tonal sounds usually have a low bandwidth (single peak in the spectrum) while noise-like sounds have high bandwidth. However, this is not the case for more complex sounds. For example in music we find broadband signals with tonal characteristics. The same applies to complex tones with a large number of harmonics that may have a broadband line spectrum. Consequently bandwidth may not be a sufficient indicator for tonality for particular tasks. Additional features (e.g., harmonicity features, see Section   5.4.6 and flatness features, see below) may be necessary to distinguish between tonal and noise-like signals.

Bandwidth may be defined in the logarithmized spectrum or the power spectrum [49, 102, 106]. Additionally, it may be computed within one or more subbands of the spectrum [103, 107].

In the MPEG-7 standard the measure for bandwidth is called spectral spread [65, 104]. Similarly to the bandwidth measures above, the MPEG-7 audio spectrum spread (ASS) is the RMS deviation from the spectrum centroid (MPEG-7 ASC descriptor, see Section   5.4.1). Measures for bandwidth are often combined with that of spectral centroid in literature since they represent complementary information [49, 103, 107].

Spectral dispersion. The spectral dispersion is a measure for the spread of the spectrum around its spectral center [94]. See Section   5.4.1 for a description of spectral center. In contrast to bandwidth, the computation of spectral dispersion takes the spectral center into account instead of the spectral centroid.

Spectral rolloff point. The spectral rolloff point is the N% percentile of the power spectral distribution, where N is usually 85% or 95% [86]. The rolloff point is the frequency below which N% of the magnitude distribution is concentrated. It increases with the bandwidth of a signal. Spectral rolloff is extensively used in music information retrieval [85, 108] and speech/music segmentation [86].

Spectral flatness. Spectral flatness estimates to which degree the frequencies in a spectrum are uniformly distributed (noise-like) [109]. The spectral flatness is the ratio of the geometric and the arithmetic mean of a subband in the power spectrum [103]. The same definition is used by the MPEG-7 standard for the audio spectrum flatness descriptor [65]. Spectral flatness may be further computed in decibel scale as in Refs. [110, 111]. Noise-like sounds have a higher flatness value (flat spectrum) while tonal sounds have lower flatness values. Spectral flatness is often used (together with spectral crest factor) for audio fingerprinting [111, 112].

Spectral crest factor. The spectral crest factor is a measure for the "peakiness" of a spectrum and is inversely proportional to the flatness. It is used to distinguish noise-like and tone-like sounds due to their characteristic spectral shapes. Spectral crest factor is the ratio of the maximum spectrum power and the mean spectrum power of a subband. In Ref. [111], the spectral crest factor is additionally logarithmized. For noise-like sounds the spectral crest is lower than for tonal sounds. A traditional application of spectral crest factor is fingerprinting [103, 111, 112].

Subband spectral flux (SSF). The SSF has been introduced by Cai et al. [60] for the recognition of environmental sounds. The feature is a measure for the portion of prominent partials ("peakiness") in different subbands. SSF is computed from the logarithmized short-time Fourier spectrum. For each subband the SSF is the accumulation of the differences between adjacent frequencies in that subband. SSF is low for flat subbands and high for subbands that contain distinct frequencies. Consequently, SSF is inversely proportional to spectral flatness.

Entropy. Another measure that correlates with the flatness of a spectrum is entropy. Usually, Shannon and Renyi entropy are computed in several subbands [103]. The entropy represents the uniformity of the spectrum. A multiresolution entropy feature is proposed by Misra et al. [113, 114]. The authors split the spectrum into overlapping Mel-scaled subbands and compute the Shannon entropy for each subband. For a flat distribution in the spectrum the entropy is low while a spectrum with sharp peaks (e.g., formants in speech) has high entropy. The feature captures the "peakiness" of a subband and may be used for speech/silence detection and automatic speech recognition.

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INTERACTION OF ELECTRONS WITH PHOTONS

V.B. BERESTETSKII , ... L.P. PITAEVSKII , in Quantum Electrodynamics (Second Edition), 1982

§ 90 Synchrotron radiation

According to the classical theory (Fields, §74), an ultra-relativistic electron moving in a constant magnetic field H emits a quasi-continuous spectrum with a maximum at the frequency

(90.1) $\omega \sim {\omega }_o{\left(\epsilon /m\right)}^3,$

where

(90.2) $\begin{array}{l}{\omega }_0=\upsilon \left|e\right|\text{H}/\left|\text{p}\right|\\ \approx \left|e\right|\text{H}/\epsilon \end{array}$

is the frequency of revolution of an electron having energy ɛ in a circular orbit (in a plane perpendicular to the field). ^† We shall assume that the longitudinal velocity of the electron (parallel to H) is zero, as can always be achieved by a suitable choice of the frame of reference.

Quantum effects in synchrotron radiation originate in two ways: from the quantization of the motion of the electron, and from the quantum recoil when a photon is emitted. The latter is determined by the ratio ħω/ɛ, and this must be small if the classical theory is applicable. It is therefore convenient to use the parameter

(90.3) $x=\frac{\text{H}}{{\text{H}}_o}\frac{\left|\text{P}\right|}{m}\approx \frac{\text{H}\epsilon }{{\text{H}}_{o}m}\approx \frac{\hslash {\omega }_o}{\epsilon }{\left(\frac{\epsilon }{m}\right)}^3,$

where H ₀ = m ²/|e|ħ (= m ² c ³/|e|ħ) = 4.4 × 10¹³ G. In the classical case, χ ˜ ħω/ɛ ≪ 1. In the opposite limit (χ ≫ 1), the energy of the emitted photon ħω ˜ ɛ, and (as we shall see below) the significant region of the spectrum extends to frequencies at which the electron energy after the emission is

(90.4) $\epsilon \text{'}\sim \text{m}{\text{H}}_o/\text{H}.$

If the electron remains ultra-relativistic, the field must satisfy the condition

(90.5) $\text{H}/{\text{H}}_0\ll 1.$

The quantization of the electron motion itself is expressed by the ratio ħω₀/ɛ ħω₀ is the interval between adjacent energy levels for motion in a magnetic field.

Since

$\hslash {\omega }_o/\epsilon =\left(\text{H}/{\text{H}}_o\right){\left(m/\epsilon \right)}^2,$

it follows from (90.5) that ħω₀ ≪ ɛ, i.e. the motion of the electron is quasi-classical for all values of χ. That is, the non-commutativity between the operators of dynamical variables of the electron (quantities of order ħω₀/ɛ) may be neglected, while the non-commutativity of these operators with those of the photon field (quantities of order ħω/ɛ) is not neglected. ^†

The quasi-classical wave functions of stationary states of an electron in an external field can be put in the symbolic form

(90.6) $\psi =\frac{2}{\surd \left(2\widehat{H}\right)}u\left(\widehat{p}\right)e^{-\left(i/\hslash \right)\widehat{H}t}\varphi \left(r\right),$

where ϕ(r) ˜ exp(iS/ħ) are the quasi-classical wave functions of a spinless particle (S(r) being its classical action); u(p) is the operator bispinor

$u\left(\widehat{p}\right)=\left(\frac{\begin{array}{l}\surd \left(\widehat{H}+m\right)w\\ 1\end{array}}{\surd \left(\widehat{H}+m\right)}\left(\sigma \cdot \widehat{p}\right)w\right),$

obtained from the bispinor plane-wave amplitude u(p) (23.9) on replacing p and ɛ by the operators ^‡

$\widehat{p}=\widehat{p}-eA=-i\hslash \nabla -eA,\widehat{H}=\surd \left({\widehat{p}}^2+m^2\right),$

where P is the generalized momentum of the particle in a field with vector potential A(r). The order of the operator factors in ψ is immaterial, since their non-commutativity is neglected, and the spin state of the electron is determined by the three-dimensional spinor w.

In order to calculate the probability of photon emission in the quasi-classical case, it is more convenient to start not from the final formula (44.3) of perturbation theory but from a formula in which the integration with respect to time has not yet been carried out. For the total (over all time) differential probability we have ^§

(90.7) $dw={\displaystyle \sum _f|a_{fi}{|}^2\frac{d^{3}k}{{\left(2\pi \right)}^3},a_{fi}={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{V}_{fi}\left(t\right)dt}}$

(cf. QM, (41.2)); the summation is over final states of the electron.

Using (90.6), we can write the matrix element V _fi (t) for emission of a photon ω, k in the operator form

$V_{fi}\left(t\right)=-\frac{e\surd \left(4\pi \right)}{\surd \left(2\hslash \omega \right)}{\displaystyle \int \left[{\varphi }_f^{*}{e}^{\left(i/\hslash \right)\widehat{H}t}\frac{u^{+}\left(\widehat{p}\right)}{\surd \left(2\widehat{H}\right)}\right]e^{i\omega t-ik\cdot r}\left(e*\cdot \alpha \right)\frac{u\left(\widehat{p}\right)}{\surd \left(2\widehat{H}\right)}{e}^{-\left(i/\hslash \right)\widehat{H}t}{\varphi }_i{d}^{3}x,}$

where the operators in the square brackets act to the left; the photon field is taken in the three-dimensionally transverse gauge. The factors exp(±iĥt/ħ) convert the Schrödinger operators between them into explicitly time-dependent operators of the Heisenberg representation. We can write V _fi (t) in the form

$V_{fi}\left(t\right)=\frac{e\surd \left(2\pi \right)}{\surd \left(\hslash \omega \right)}〈f\left|Q\left(t\right)\right|i〉e^{i\omega t},$

where

(t) denotes the Heisenberg operator

(90.8) $\widehat{Q}\left(t\right)=\frac{u_j^{+}\left(\widehat{p}\right)}{\surd \left(2\widehat{H}\right)}\left(\alpha \cdot e*\right)e^{-ik\cdot \widehat{r}\left(t\right)}\frac{u_1\left(\widehat{p}\right)}{\surd \left(2\widehat{H}\right)},$

and the matrix element is taken with respect to the functions ϕ _f , ϕ _i .

The summation in (90.7) is taken over all final wave functions ϕ _f , and is effected by means of the equation

${\displaystyle \sum _f{\varphi }_f^{*}\left(\text{r}\text{'}\right){\varphi }_f\left(\text{r}\right)=\delta \left(\text{r}\text{'}-\text{r}\right),}$

which expresses the completeness of the set of functions ϕ _f . The result is

(90.9) $dw=\frac{e^2{d}^{3}k}{\hslash \omega 4{\pi }^2}{\int }^{}d{t}_1{\int }^{}d{t}_2\cdot e^{i\omega }{}^{(t_1-t_2)}〈i|Q^{+}(t_2)Q(t_1)|i〉.$

If the integration is over a sufficiently long time interval, t ₁ and t ₂ can be replaced by new variables

$\tau =t_2-t_1,t=\frac{1}{2}\left(t_1+t_2\right),$

and in the integral over t the integrand may be regarded as the probability of emission per unit time. Multiplying by ħω, we obtain the intensity

(90.10) $dI=\frac{e^2}{4{\pi }^2}{d}^{3}k{\displaystyle \int e^{-i\omega \tau }〈i\left|Q^{+}\left(t+\frac{1}{2}\tau \right)Q\left(t-\frac{1}{2}\tau \right)\right|i〉d\tau .}$

An ultra-relativistic electron radiates into a narrow cone at angles θ ∼ m/ɛ relative to its velocity v. The emission in a given direction n = k/ω therefore occurs over a section of the path in which v turns through an angle ∼ m/ɛ. This section is traversed in a time τ such that $\tau \left|\dot{v}\right|\approx \tau {\omega }_0\sim m/\epsilon \ll 1$ . This region gives the principal contribution to the integral over τ. In the subsequent calculations, we shall therefore expand all quantities in powers of ω₀τ. It may, however, be necessary to retain more than just the leading term in the expansion, because of cancellations which occur since 1 - n ˙ v ∼ θ² ∼ (m/ɛ)².

If the operator

⁺

is reduced to a product of operators which commute (to the necessary degree of accuracy), the taking of the diagonal matrix element 〈i| … |i〉 is equivalent to replacing these operators by the classical (time-dependent) values of the corresponding quantities. This is achieved in the following way.

According to the foregoing discussion, in the expression for

(t) only the non-commutativity of the electron operators with the photon field operator exp(- i k ·

(t)) need be taken into account. We have

(90.11) $\begin{array}{l}\widehat{\text{p}}{e}^{-ik\cdot \widehat{r}}=e^{-ik\cdot \widehat{r}}\left(\widehat{\text{p}}-\hslash k\right),\\ \text{H}\left(\widehat{\text{p}}\right)e^{-ik\cdot \widehat{r}}=e^{-ik\cdot \widehat{r}}\text{H}\left(\widehat{p}-\hslash \text{k}\right).\end{array}\}$

These formulae follow because e ^{−ik ˙}

is the displacement operator in momentum space. Using (90.11), we can take the operator e ^{−ik ˙}

(t) out on the left in (90.8), and write

(t) in the form

(90.12) $\widehat{Q}\left(t\right)=e^{-ik\cdot \widehat{r}\left(t\right)}\widehat{R}\left(t\right),\widehat{R}\left(t\right)=\frac{u_f^{+}\left(\widehat{p}\text{'}\right)}{\surd \left(2\widehat{H}\right)}\left(\alpha \cdot e*\right)\frac{u_i\left(\widehat{p}\right)}{\surd \left(2\widehat{H}\right)},$

where Ĥ′ = Ĥ - ħω,

′ =

- ħ k.

Then

(90.13) ${\widehat{Q}}_2^{+}{\widehat{Q}}_1={\widehat{R}}_2{e}^{ik\cdot {\widehat{r}}_2}{e}^{-ik\cdot {\widehat{r}}_1}{\widehat{R}}_1;$

here and henceforward, the suffixes 1 and 2 denote the values of quantities at the times t ₁ = t − 1/2 τ and t ₂ = t + 1/2 τ. It remains to calculate the product of the two non-commuting operators e ^{ik ˙}

₂ and e ^{−ik ˙}

₁ . This product itself may be regarded as commuting with the remaining factors.

We write

(90.14) $\widehat{L}\left(\tau \right)=e^{-i\omega \tau }{e}^{ik\cdot {\widehat{r}}_2}{e}^{-ik\cdot {\widehat{r}}_1},$

this being the combination of operators which appears in (90.10). The operator e ^iĥτ/ħ is a time-shift operator, and so

$e^{ik\cdot {\widehat{r}}_2}=e^{i\widehat{H}\tau /\hslash }{e}^{ik\cdot {\widehat{r}}_1{e}^{-i\widehat{H}\tau /\hslash }.}$

Substituting this in (90.14) and noting that e ^{ik ˙}

₁ is a displacement operator in momentum space, we find

(90.15) $\widehat{L}\left(\tau \right)=\exp \left\{i\left[\widehat{H}-\hslash \omega \right]\tau /\hslash \right\}\exp \left\{-i\widehat{H}\left({\widehat{p}}_1-\hslash k\right)\tau /\hslash \right\}.$

Differentiating (90.15) with respect to τ and again using the properties of the time-shift operator, we have ^†

(90.16) $\begin{array}{l}d\widehat{L}/d\tau =\left(i/\hslash \right)\exp \left\{i\left(\widehat{H}-\hslash \omega \right)\tau /\hslash \right\}\left[\widehat{H}-\hslash \omega -\widehat{H}\left({\widehat{p}}_1-\hslash k\right)\right]\times \\ \times \exp \left\{-i\widehat{H}\left({\widehat{p}}_1-\hslash k\right)\tau /\hslash \right\}=\left(i/\hslash \right)\left[\widehat{H}-\hslash \omega -\widehat{H}\left({\widehat{p}}_2-\hslash k\right)\right]\widehat{L}\left(\tau \right).\end{array}$

Having thus made use of the non-commutativity of the operators, we can replace all the operators by the corresponding classical quantities (the Hamiltonian Ĥ by the electron energy ɛ). We have identically

$\begin{array}{l}\epsilon \left({\text{p}}_2-\hslash k\right)={\left[{\left({\text{p}}_2-\hslash k\right)}^2+m^2\right]}^{1/2}\\ ={\left[{\left(\epsilon -\hslash \omega \right)}^2+2\hslash \left(\omega \epsilon -\text{k}\cdot {\text{p}}_2\right)\right]}^{1/2}.\end{array}$

The difference

$w\epsilon -\text{k}\cdot {\text{p}}_2=w\epsilon \left(1-\text{n}\cdot {\text{v}}_2\right)$

is small, since from the above analysis 1 - v · n ∼ (m/ɛ)². As far as the first order in this difference,

$\epsilon \left({\text{p}}_2-\hslash k\right)\approx {\epsilon }^{\prime }+\left(\epsilon /{\epsilon }^{\prime }\right)\hslash \left(w-k\cdot v_2\right),$

where ɛ′ = ɛ - ħω. From (90.16), we now find the differential equation for L(τ):

(90.17) $indL/d\tau =(\epsilon /{\epsilon }^{\text{'}})\hslash (\omega -k\cdot v_2)L.$

This equation is to be solved with the obvious initial condition L(0) = 1. Since

${\displaystyle \underset{0}{\overset{\tau }{\int }}{v}_{2}d\tau =r_2-r_1,}$

we have

(90.18) $L\left(\tau \right)=\exp \left\{i\left(\epsilon /\epsilon \text{'}\right)\left(k\cdot r_2-k\cdot r_1-\omega \tau \right)\right\}.$

So far, no use has been made of the specific form of the electron trajectory. Now expressing r ₂ - r ₁ in (90.18) in terms of p ₁ by means of the equation of motion of the electron in the plane perpendicular to the field H (see Fields, §21):

$r_2-r_1=\frac{p_1}{eH}\sin \frac{eH\tau }{\epsilon }+\frac{p_1\times H}{e{H}^2}\left(1-\cos \frac{eH\tau }{\epsilon }\right),$

and expanding in powers of τ gives

(90.19) $k\cdot \left(r_2-r_1\right)-\omega \tau \approx \omega \tau \left\{\left(n\cdot v_1-1\right)+\tau \frac{en\cdot p_1\times H}{2{\epsilon }^2}-{\tau }^2\frac{e^2{H}^2}{6e^2}\right\},$

where in the last term we have put n ˙ v ₁ ˜ 1.

We next transform the remaining factors in (90.13). A direct expansion of the product in R(t), using the matrix α from (21.20), leads to′

(90.20) $\begin{array}{l}R\left(t\right)=w_f^{*}e*\cdot \left(A+iB\times \sigma \right)w_i,\\ A\dot{=}\frac{1}{2}p\left(\frac{1}{\epsilon }+\frac{1}{\epsilon \text{'}}\right)=\frac{\epsilon +\epsilon \text{'}}{2\epsilon \text{'}}v,\\ B=\frac{1}{2}\left(\frac{p}{\epsilon +m}-\frac{p\text{'}}{\epsilon \text{'}+m}\right)\approx \frac{\hslash \omega }{2\epsilon \text{'}}\left(n-v+vm/\epsilon \right),\end{array}\}$

where p′(t) - p(t) - ħ k; terms of higher order in m/ɛ are omitted. Thus we have finally

(90.21) $\begin{array}{l}{e}^{-w\tau }〈i\left|Q_2^{+}{Q}_1\right|i〉=R_2^{*}{R}_{1}L\left(\tau \right),\\ R_2^{*}{R}_1=tr\frac{1}{2}\left(1+{\zeta }_i\cdot \sigma \right)\left(A_2-i{B}_2\times \sigma \right)\cdot e\cdot \frac{1}{2}\left(1+{\zeta }_f\cdot \sigma \right)\left(A_1+i{B}_1\times \sigma \right)\cdot e*.\end{array}\}$

The factors 1/2(1 + ζ ˙ σ) are two-rowed polarization density matrices of the initial and final electron.

Let us consider the radiation intensity summed over the polarizations of the photon and of the final electron, and averaged over the polarizations of the initial electron. These operations give, after a simple calculation, ^†

$\frac{1}{2}{\displaystyle \sum _{polar}{R}_2^{*}{R}_1=\frac{{\epsilon }^2+\epsilon {\text{'}}^2}{2\epsilon {\text{'}}^2}\left(v_1\cdot v_2-1\right)+\frac{1}{2}{\left(\frac{\hslash \omega }{\epsilon \text{'}}\right)}^2{\left(\frac{m}{\epsilon }\right)}^2.}$

With sufficient accuracy we can put

$\begin{array}{l}{v}_1\cdot v_2=v^2-\frac{1}{4}{\tau }^{2}v{\dot{}}^2+\frac{1}{4}{\tau }^{2}v\cdot v¨\\ =1-\frac{m^2}{{\epsilon }^2}-\frac{1}{2}{\omega }_0^2{\tau }^2,\end{array}$

Substitution of these expressions in (90.21) and thence in (90.10) gives

(90.22) $\begin{array}{l}dI=-\frac{e^2}{4{\pi }^2}{\omega }^{2}d\omega d{o}_n\times \\ \times {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left(\frac{m^2}{\epsilon \epsilon \text{'}}+\frac{{\epsilon }^2+\epsilon {\text{'}}^2}{4\epsilon {\text{'}}^2}{\omega }_0^2{\tau }^2\right)\exp \left\{-\frac{i\omega \tau \epsilon }{\epsilon \text{'}}\left(1-n\cdot v+\frac{{\tau }^2}{24}{\omega }_0^2\right)\right\}d\tau .}\end{array}$

This formula shows the frequency and angular distribution of the radiation intensity.

To find the frequency distribution, we integrate over do _n . If the direction of v is taken as the polar axis, with an angle ϑ between n and v, then

$n\cdot v=\upsilon \cos \vartheta ,d_0=\sin \vartheta d\vartheta d\varphi ,$

and

${\displaystyle \int \exp \left\{\frac{i\omega \tau \epsilon }{\epsilon \text{'}}n\cdot v\right\}d{o}_n=\frac{2\pi \epsilon \text{'}}{i\omega \tau \epsilon v}\left\{\exp \left(\frac{i\omega \tau \epsilon }{\epsilon \text{'}}\right)-exp\left(-\frac{i\omega \tau \epsilon }{\epsilon \text{'}}\right)\right\}}.$

When this is substituted in (90.22), only the first term need be retained, since the second term yields a faster varying exponential (with a factor 1 + v ˜ 2 instead of the small 1 - v ˜ m ²/2ɛ²). Hence

$\frac{dI}{d\omega }=\frac{i{e}^2\omega }{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left(\frac{m^2}{{\epsilon }^2\tau }+\frac{{\epsilon }^2+\epsilon {\text{'}}^2}{2\epsilon \epsilon \text{'}}\tau \right)\exp \left\{-\frac{i\omega \tau \epsilon }{\epsilon \text{'}}\left(1-v+\frac{{\tau }^2}{24}{w}_0^2\right)\right\}}d\tau .$

According to the integral representation of the Airy function ϕ (see QM, §b), the first term reduces to the integral of the Airy function, and the second term to its derivative. The final result is

(90.23) $\frac{dI}{d\omega }=-\frac{e^2{m}^2\omega }{\surd \pi {\epsilon }^2}\left\{{\displaystyle \underset{x}{\overset{\infty }{\int }}\Phi \left(\xi \right)d\xi +\left(\frac{2}{x}+\frac{\hslash \omega }{\epsilon }\chi \surd x\right)}\Phi \text{'}\left(x\right)\right\},$

(90.24) $x={\left(\hslash \omega /\epsilon \text{'}\chi \right)}^{2/3}=\left(m^2/{\epsilon }^2\right){\left(\epsilon \omega /\epsilon \text{'}{\omega }_0\right)}^{2/3}$

(A. I. Nikishov and V. I. Ritus, 1967). The frequency distribution has a maximum when x ˜ 1; for χ ≪ 1 we find (90−1), and for χ ≫ 1 (90.4). In the classical limit, ħω ≪ ɛ′ ˜ ɛ, x ˜ (ω/ω₀)^2/3(m/ɛ)²; the second term in the round brackets is small, and (90.23) becomes the classical formula (Fields, (74.13)).

Figure 15 shows diagrams of the frequency distribution for various values of χ ˙ The quantity

Fig. 15..

$\frac{I}{3I_{cl}/2}\frac{dI}{d\left(\omega /{\omega }_e\right)}$

is plotted against ω/ω_c, where

$\hslash {\omega }_e=\frac{\epsilon \chi }{\frac{2}{3}+\chi },I_{cl}=\frac{2e^2{m}^2{\chi }^2}{3{\hslash }^2}=\frac{2e^4{H}^2{\epsilon }^2}{3m^4}.$

The quantity I _cl is the classical value of the total radiation intensity; cf. Fields, (74.2).

To calculate the total radiation intensity, (90.23) must be integrated with respect to ω from 0 to ɛ. We change to integration with respect to x, noting that

$\hslash \omega =\epsilon \left(1-\frac{1}{1+\chi x^{3/2}}\right),$

and x therefore varies from 0 to ∞. With two integrations by parts in the first term in (90.23), we find

(90.25) $I=-\frac{e^2{m}^2{\chi }^2}{2\surd \pi {\hslash }^2}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{4+5\chi x^{3/2}+4{\chi }^2{x}^3}{{\left(1+\chi x^{3/2}\right)}^4}}\Phi \text{'}\left(x\right)xdx.$

Figure 16 shows a graph of the function I(χ)/I _cl.

Fig. 16..

When χ ≪ 1, the important region in the integral is x ˜ 1. Expanding the integrand in powers of χ and integrating by means of the formula

${\displaystyle \underset{0}{\overset{\infty }{\int }}{x}^v\Phi \text{'}\left(x\right)dx=-\frac{1}{2\surd \pi }{3}^{\left(4v-1\right)/6}\Gamma \left(\frac{1}{3}\nu +1\right)\Gamma \left(\frac{1}{3}\nu +\frac{1}{3}\right),}$

we obtain

(90.26) $I=I_{cl}\left(1-\frac{55\surd 3}{16}\chi +48{\chi }^2-\mathrm{\dots }\right).$

When χ ≫ 1, the important region is that in which χx ^3/2 ˜ 1, i.e., x ≪ 1. In the first approximation, we can therefore replace ϕ′(x) by ϕ′(0) = −3^1/6Γ(2/3)/2 √ π, and the integration then leads to the result

(90.27) $=0.37\frac{e^2{m}^2}{{\hslash }^2}{\left(\frac{H\epsilon }{H_{0}m}\right)}^{2/3}.$

Synchrotron emission causes the occurrence of a polarization of electrons moving in the field (A. A. Sokolov and I. M. Ternov, 1963). To discuss this, we have to find the probability of a radiative transition with spin reversal.

Putting in (90.21) ζ _i = –ζ _f ≡ ζ, |ζ| = 1, we have

$R_2^{*}{R}_1=B_1\cdot B_2-\left(e*\cdot B_1\right)\left(e*B_2\right)-\left(e*\cdot B_1\times \zeta \right)\left(e\cdot B_2\times \zeta \right)-i\left(\zeta \cdot e*\right)\left(e\cdot B_1\times B_2\right).$

Summation over polarizations of the photon gives, after a simple calculation,

(90.28) $\begin{array}{l}{\displaystyle \sum _e{R}_2^{*}{R}_1=\left(B_1\cdot B_2\right)\left(1-{\left(\zeta \cdot n\right)}^2\right)+\left(\zeta \cdot n\right)\left(n\cdot B_1\right)\left(\zeta \cdot B_2\right)+}\\ +\left(\zeta \cdot n\right)\left(n\cdot B_2\right)\left(\zeta \cdot B_1\right)-i\left(\zeta -n\left(n\cdot \zeta \right)\right)\cdot B_1\times B_2.\end{array}$

We shall assume that χ ≪ 1 and seek only the principal term in the expansion of the probability in powers of ħ. Since the expression (90.28) (with B given by (90.20)) already contains ħ², all the remaining quantities ɛ′, including those in the exponent in (90.18), can be replaced by ɛ.

With the expansions

$\begin{array}{l}{B}_1=\frac{\omega }{2\epsilon }\left(n-v+\frac{1}{2}\tau \dot{v}+v\frac{m}{\epsilon }\right),\\ B_2=\frac{\omega }{2\epsilon }\left(n-v-\frac{1}{2}\tau \dot{v}+v\frac{m}{\epsilon }\right),\\ r_2-r_1=\tau v+\frac{{\tau }^3}{24}\ddot{v},\end{array}$

and substituting (90.28) in (90.21) and thence in (90.10), we find the differential transition probability per unit time (dw = dI/ħΩ). The integration over d ³ k is carried out by means of the formula

(90.29) ${\displaystyle \int f\left(k_{\mu }\right)e^{-ikx}\frac{d^{3}k}{\omega }=-f\left(i{\partial }_{\mu }\right)\frac{4\pi }{{\left(x_o-i0\right)}^2-x^2},}$

where in this case

$x_o=\tau ,x=r_2-r_1,x^2=x_0^2-x^2={\tau }^2\left(\frac{m^2}{{\epsilon }^2}+\frac{{\tau }^2{\omega }_0^2}{12}\right).$

The result of the calculation is

$w=\frac{\alpha }{\pi }\frac{{\hslash }^2}{m^2}{\left(\frac{\epsilon }{m}\right)}^5{w}_0^3{\displaystyle \oint \frac{dz}{{\left(1+z^2/12\right)}^3}\left[\frac{3}{z^4}+\frac{5}{12z^2}+\left(\frac{1}{z^4}+\frac{5}{12x^2}\right){\left(\zeta \cdot v\right)}^2-\frac{2j}{z^3{\omega }_0}\zeta \cdot \dot{v}\times v\right],}$

where z = τω₀ɛ/m and the contour of integration passes below the real axis and is closed in the lower half-plane. After this integration we finally obtain for the total probability of a radiative transition with spin reversal

(90.30) $w=\frac{5\surd 3\alpha }{16}\frac{{\hslash }^2}{m^2}{\left(\frac{\epsilon }{m}\right)}^5{w}_0^3\left(1{-}_9^2{\zeta }_1^2-\frac{8\surd 3}{15}\frac{e}{\left|e\right|}{\zeta }_{\perp }\right),$

where ${\zeta }_{\left|\right|}=\zeta \cdot v,{\zeta }_{\perp }=\zeta \cdot H/H$ . This formula is valid for both electrons (e < 0) and positrons (e > 0).

The probability (90.30) is independent of the sign of the longitudinal polarization ζ‖ but depends on that of ζ_⊥. The polarization resulting from the emission is therefore transverse. ^† For electrons, the probability of a transition from a state with the spin parallel to the field (ζ_⊥ = 1) to a state with the spin antiparallel to the field is greater than that of the opposite transition. The radiative polarization of the electrons is therefore antiparallel to the field, and the degree of polarization in a stationary state is (when ζ|| = 0)

$\frac{w\left({\zeta }_{\perp }=-1\right)-w\left({\zeta }_{\perp }=1\right)}{w\left({\zeta }_{\perp }=-1\right)+w\left({\zeta }_{\perp }=1\right)}=\frac{8\surd 3}{15}=\mathrm{0.92.}$

Positrons are polarized, to the same degree, parallel to the field.

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Rugged autonomous vehicles

R. Zalman , in Rugged Embedded Systems, 2017

2.2 Vibration and Other Environmental Factors

Another important aspect of the automotive electronics is the stress coming from vibration. In general, the vibration can be divided into harmonic (single frequencies) and random where a continuous spectrum of frequencies are affecting the electronic components. For the harmonic vibrations, the frequency range varies (depending on the placement of the ECUs) from around 20–100  Hz up to around 2   kHz with accelerations in the range of 15–19   × g. For random ones, high levels of vibrations, with values around 10   Grms, ² are present in electronics mounted directly on the engine and on the transmission paths.

One other aspect of mechanical stress is shock. Without considering the requirements for assembly line (e.g., drop tests), during the operation life, the automotive electronics shall withstand mechanical shocks in the range of 50–500   × g.

The effects and failure modes for vibration are mainly affecting the mechanical assemblies of the electronic components on the PCB inside of the ECUs. The connections of the electronic components (e.g., soldering, connectors) are subject to fatigue due to vibrations and the failure modes result, generally, in loss of contact. Nevertheless, vibrations and shocks can also lead to damage inside the integrated electronic components resulting in failures like:

–

Substrate cracks;

–

Cracks and chip peeling for die bonding;

–

Wire deformation, wire breaking, and wire short-circuits for wire bonding.

For automotive electronic components, special design methods and technologies are used in order to mitigate these effects. Similarly, the characterization (and testing) of automotive electronic components is following strict standards in which the environmental parameters which may affect the operational life of a component are tested (e.g., thermal, mechanical, electrical stress). As an example [3], the mechanical stress is tested for shocks with five pulses of 1500   × g acceleration, a harmonic vibration test covers 20   Hz to 2   kHz in each orientation with a 50   × g acceleration peak, a specific drop test on all axes is performed, etc.

One other important environmental factor specific to automotive industry is the fluid exposure. The ECUs exposure to fluids is considered harsh, especially for ECUs mounted outside of the driver compartment, due to the possible exposure mainly to humidity and salt water. Depending on the particular position of each ECU, exposure to fuel, oil, brake fluid, transmission fluid, and exhaust gases is also possible.

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Green's Functions

G. Rickayzen , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

V Green's Functions in the Complex Plane

It was seen in Section II that the Green's functions for Hermitian operators can be written in the form of Eq. (25) . It we allow for the possibility of a continuous spectrum, this can be written

(50) $\text{G}(\text{x},\text{x}\prime ,\text{E})={\int }_{-\infty }^{\infty }\frac{\text{<mi mathvariant="italic" is="true">dy</mi>}}{\text{E}-\text{y}}\text{A}(\text{x},\text{x}\prime ,\text{y}),$

where

(51) $\text{A}(\text{x},\text{x}\prime ,\text{y})={\displaystyle \sum _{\text{n}}}{\psi }_{\text{n}}(\text{x}){\psi }_{\text{n}}^{*}(\text{x}\prime )\delta ({\text{E}}_{\text{n}}-\text{y})+\int \text{<mi mathvariant="italic" is="true">dE</mi>}\prime {\psi }_{\text{E}\prime }(\text{x}){\psi }_{\text{E}\prime }^{*}(\text{x}\prime )\delta (\text{E}\prime -\text{y}).$

The function A(x, x′, y) is called the spectral function of the Green's function.

Equation (50) provides a definition for the Green's function as a function of the complex variable E. It is analytic throughout the complex plane except at the eigenvalues of the Hermitian operator ℋ. The singular points of G(x, x′, E) are therefore these eigenvalues.

If an eigenvalue E _n is nondegenerate, the residue of G(x, x, E) at E _n is ψ* _n (r)ψ _n (r). Since nondegenerate eigenfunctions can be chosen to be real, this determines ψ _n (r). In fact, all the physical properties can be determined once G(x, x′, E) is known. Thus, it is possible to work entirely with Green's functions rather than with eigenfunctions and eigenvalues. This is particularly valuable in problems involving many degrees of freedom.

When the eigenvalues have a continuous spectrum, the Green's function defined in Eq. (50) is ambiguous. We have already seen this in the special cases discussed in Section III. In fact, the most general inverse of (E  − y) is

(52) $\frac{1}{\text{E}-\text{y}}=\frac{\text{P}}{\text{E}-\text{y}}+\alpha (\text{y})\delta (\text{E}-\text{y}),$

where α(y) is an arbitrary function and where the symbol P indicates that the principal value of any integral including this term is to be taken. Thus, every function α(y) defines a different solution of Eq. (26). The actual function α(y) required is determined by the boundary conditions as illustrated by the examples given in Section III.

In problems of quantum statistical mechanics, it turns out to be easiest to calculate the Green's function, Eq. (50), at the discrete imaginary points E  = i2πlkT (l  =   0, ±1, ±2,…) if the Green's function describes the behavior of particles that obey Bose statistics, or the points E  = i(2l  +   1)πkT if it describes particles that obey Fermi–Dirac statistics. By analytic continuation, the complete Green's function can be determined once these values are known.

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