Black Body Radiation Continuous Spectrum Atom

Cryogenic Technologies

Bahman Zohuri , in Physics of Cryogenics, 2018

1.4.2 Radiation

Blackbody radiation strongly and solely depends on the temperature of the emitting body, with the maximum of the power spectrum given by Wien's law

(1.9) ${\mathit{\lambda }}_{\max }T=2898\left[\text{μm}\text{K}\right]$

and the total power radiated is given by the Stefan–Boltzmann law as:

(1.10) $Q=\mathit{\sigma }A{T}^4$

with the Stefan–Boltzmann' constant σ   ≃   5.67   ×   10⁻⁸  W/m²  K⁴. The dependence of the radiative heat flux on the fourth power of temperature makes a strong plea for radiation shielding of low-temperature vessels with one or several shields cooled by liquid nitrogen or cold helium vapor. Conversely, it makes it very difficult to cool equipment down to low temperature by radiation only, despite the 2.7K background temperature of outer space and notwithstanding the Sun's radiation and the Earth's albedo. While they can be avoided by proper attitude control, satellites or interplanetary probes can make use of passive radiators to release heat down to only about 100K, and embarked active refrigerators are required to reach lower temperatures.

Technical radiating surfaces are usually described as "graybodies" and characterized by an emissivity ε  <   1.

(1.11) $Q=\mathit{\epsilon \sigma }A{T}^4$

The emissivity ε strictly depends on the material, surface finish, radiation wavelength, and angle of incidence. For materials of technical interest, measured average values are found in the literature,⁵ a subset of which is given in Table 1.8. As a general rule, emissivity decreases at low temperature for good electrical conductors and for polished surfaces. As Table 1.7 shows, a simple way to obtain this combination of properties is to wrap cold equipment with aluminum foil. Conversely, radiative thermal coupling requires emissivity as close as possible to that of a blackbody, which can be achieved in practice by special paint or adequate surface treatment, such as anodizing of aluminum.

Table 1.8. Emissivity of Some Technical Materials at Low Temperature

	Radiation From 290K Surface at 77K	Radiation From 77K Surface at 4.2K
Stainless steel, as found	0.34	0.12
Stainless steel, mesh polished	01.2	0.07
Stainless, electropolished	0.10	0.07
Stainless steel   plus   aluminum foil	0.05	0.01
Aluminum, black anodized	0.95	0.75
Aluminum, as found	0.12	0.07
Aluminum, mesh polished	0.10	0.06
Aluminum, electropolished	0.08	0.04
Copper, as found	0.12	0.06
Copper, mesh polished	0.06	0.02

The net heat flux between two "gray" surfaces at temperatures $T_1$ and $T_2$ is similarly given by

(1.12) $Q=E\mathit{\sigma }A\left(T_1^4-T_2^4\right)$

with the emissivity factor $E$ being a function of the emissivities ${\mathit{\epsilon }}_1$ and ${\mathit{\epsilon }}_2$ of the surfaces, of the geometrical configuration, and of the type of reflection (specular or diffuse) between the surfaces. Its precise determination can be quite tedious, apart from the few simple geometrical cases of flat plates, nested cylinders, and nested spheres.

If an uncooled shield with the same emissivity factor $E$ is inserted between the two surfaces, it will "float" at temperature $T_s$ given by the energy balance equation

(1.13) $Q_s=E\mathit{\sigma }A\left(T_1^4-T_s^4\right)=E\mathit{\sigma }A\left(T_s^4-T_2^4\right)$

Solving for $T_s$ yields the value of $Q_s=Q/2$ : the heat flux is halved in presence of the floating shield. More generally, if n floating shields of equal emissivity factor are inserted between the two surfaces, the radiative heat flux is divided by $\mathit{n}+1$ .

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Controlling thermal radiation from surfaces

C.G. Ribbing , in Optical Thin Films and Coatings, 2013

8.2 Blackbody radiation

Blackbody radiation is the upper limit on the thermal emission intensity from a solid surface ( Wolfe, 1989; Zalewski, 1995). It is based upon Planck's Law for oscillators, which in turn is derived by using the Bose-Einstein distribution for vibrations in a box (a 'holeraum') of macroscopic dimension. The spectral radiance emitted from a small hole in this 'box' in one unit of space angle is:

[8.2] $L_{\mathit{bb}}\left(v,T\right)=\frac{2h}{n{c}^2}\frac{v^3}{exp\left(\mathit{hv}/\mathit{kT}\right)-1}$

where ν is the frequency, c the vacuum velocity for light, h is the Planck constant, k the Bolzmann constant and T the absolute temperature. In most cases the refractive index of the medium, n  =   1. The SI-dimension of spectral radiance is W/m²,Hz,sr. The radiance from a blackbody is Lambertian, so the total emission into the half-sphere is given by multiplication with 2π.

In Fig. 8.2 we plot this spectral radiance for a few temperatures chosen to show the characteristic behaviour in the infrared where the unit W/m²,THz,sr is appropriate.

8.2. The spectral radiance from a blackbody as a function of frequency in THz at the four temperatures indicated. Notice that the diagram is lin-log.

We notice that the curves never intersect, that is, a curve for a higher temperature, is always above one for a lower temperature. In Fig. 8.2, frequency is the independent variable, which is directly linked to the Planck theory. In optics the corresponding expression as function of wavelength is often used. The coordinate transformation, λ  = c/ν, is nonlinear which has consequences for the Planck function. The wavelength version is

[8.3] $L_{\mathit{bb}}\left(\lambda ,T\right)=\frac{2h{c}^2}{n^2{\lambda }^5}\frac{1}{exp\left(\mathit{hc}/\mathit{kT}\mathit{\lambda }\right)-1}$

with dimension W/m³, sr. In Fig 8.3 radiance as a function of wavelength for the same temperatures are plotted per μm wavelength as the relevant unit in the infrared.

8.3. The spectral radiance from a blackbody as a function of wavelength in μm at the same temperatures as in the previous figure. As in Fig. 8.2 the y-axis is logarithmic. The peaks of the spectral curves for different temperatures are joined by the fat dash-dotted curve which illustrates the Wien's displacement law.

In this case we also use the diagram to illustrate the well-known Wien's displacement law. The thick dash-dotted curve joins the maxima λ _m , of the spectral curves. It is given by the expression:

[8.4] ${\lambda }_m=\frac{b_{\lambda }}{T}$

Where the constant b _λ   =   2.8978   ×   10⁻³ mK.

It shows that the maxima of the blackbody curves move to shorter wavelength when the temperature increases. The corresponding expression for the frequency version of Equation[8.2] is

[8.5] $v_m=b_{v}T$

with the constant b_v   =   5.8786   ·   10¹⁰ (sK)⁻¹

As expected, the maxima move to higher frequencies when the temperature increases. A comparison of Figs 8.2 and 8.3, reveals, however, that the positions of the maxima are not conserved in the coordinate transformation. The maximum of the 1000   K curve in Fig. 8.3 is ≈ 2.9   μm. If this wavelength is converted to frequency ν  = c/λ, we get 103 THz. Looking at Fig. 8.2 the maximum position of the 1000   K curve is considerably lower at ≈ 59   THz. The reason for this shift is the non-linear ν ⇔ λ coordinate transformation. Physically, it is a consequence of the Planck function being a distribution and having a dimension per frequency or per wavelength unit. It gives the power density in each infinitesimal frequency or wavelength interval. The non-linear transformation makes the corresponding infinitesimal steps unequal, which influences the shape of the curve. Comparing the diagrams above, we notice that the widths of the peaks increase with temperature in Fig. 8.2, while they decrease in Fig. 8.3.

As an example we choose the solar spectrum, which is on the short wavelength side of Fig. 8.3. It agrees roughly with that for a blackbody at 5800   K. The maximum is at λ  =   0.50   μm (cf. Equation[8.4]). This wavelength almost agrees with the peak of the sensitivity of the human eye – but this agreement is only in the wavelength version. Equation[8.5] gives the corresponding maximum on the frequency axis at 341 THz, which corresponds to 0.88   μm, that is, well beyond 0.50 and actually outside the visible range. The eye sensitivity curve is dimensionless and not affected by the transformation. Consequently, in the wavelength representation the peak of solar radiation is in the middle of the sensitivity of the human eye, but in frequency space the maximum is outside our range of vision. This and a few more consequences of the non-linear transformation of the Planck distribution function have been described in more detail by Soffer and Lynch (1999). Heald (2003) has also discussed the issue of the 'Wien peak' position.

The expressions [8.2] and [8.3] tend to 0 in both directions, that is, whether ny or λ  ⇒   0 or ∞. Analyzing the integral of the expression it turns out that they are both finite as long as the temperature is finite (Zalewski, 1995). This was a strong argument in favour of quantum mechanics when Planck made his derivation, because earlier classical attempts had indicated the opposite. The integral is required to calculate the total radiance M from the surface of a blackbody, that is, the Stefan-Boltzmann equation summing the contributions for all wavelengths into the solid angle 2π:

[8.6] $M\left(T\right)=\sigma T^4$

with the Stefan-Boltzmann constant σ   =   5.6693   ×   10⁻⁸  W/m²/K⁴. Equation[8.6] is obtained, whether the variable is ν or λ.

This signals that we should recheck are the differences in curve shape noted above. The area under each curve, that is, the total radiance, should be independent of variable. A formal verification requires integration of the two versions [8.2] and [8.3]. The following comment is only a hint: the peak heights in the wavelength version increase as T ⁵, which is easily shown by inserting Equation[8.4] in Equation[8.3] (Ribbing, 1999). In contrast, the peaks of the frequency curves only grow as T ³, which is found by inserting Equation[8.5] in Equation[8.2]. This is compensated for by the changes in peak widths noted above. In both versions therefore the Stefan-Boltzmann integrals grow as T ⁴.

Emission from a small hole through a large enclosure is virtually non-coherent. This may be the reason for a widespread notion that thermal radiation in general is non-coherent. It is often a reasonable first assumption that radiation from thermal sources has a very short coherence length. Nevertheless, microscopic features on a thermally emitting surface cause spectral and directional interference variations (Carter and Wolf, 1975; Wolf and Carter, 1975). In particular it was proved that Lambertian emission requires a source with some degree of periodicity.

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Blackbody Radiation, Image Plane Intensity, and Units

Robert H. Kingston , in Optical Sources, Detectors, and Systems, 1995

1.1 Planck's Law

By convention and definition blackbody radiation describes the intensity and spectral distribution of the optical and infrared power emitted by an ideal black or completely absorbing material at a uniform temperature T. The radiation laws are derived by considering a completely enclosed container whose walls are uniformly maintained at temperature T, then calculating the internal energy density and spectral distribution using thermal statistics. Consideration of the equilibrium interaction of the radiation with the chamber walls then leads to a general expression for the emission from a "gray" or "colored" material with nonzero reflectance. The treatment yields not only the spectral but the angular distribution of the emitted radiation.

Although we usually refer to blackbody radiation as "classical", its mathematical formulation is based on the quantum properties of electromagnetic radiation. We call it classical since the form and the general behavior were well known long before the correct physics was available to explain the phenomenon. We derive the formulas using Planck's original hypothesis, and it is in this derivation, known as Planck's law, that the quantum nature of radiation first became apparent. We start by considering a large enclosure containing electromagnetic radiation and calculating the energy density of the contained radiation as a function of the optical frequency v. To perform this calculation we assume that the radiation is in equilibrium with the walls of the chamber, that there are a calculable number of "modes" or standing-wave resonances of the electromagnetic field, and that the energy per mode is determined by thermal statistics, in particular by the Boltzmann relation

(1.1) $p\left(U\right)=A{e}^{-U/kT}$

where p(U) is the probability of finding a mode with energy, U; k is the Boltzmann constant; T, the absolute temperature; and A is a normalization constant.

Example:

The Boltzmann distribution will be used frequently in this text since it has such universal application in thermal statistics. As an interesting example, let us consider the variation of atmospheric pressure with altitude under the assumption of constant temperature. The pressure, at constant temperature, is proportional to the density and thus to the probability of finding an air molecule at the energy U associated with altitude h, given by U = mgh, with m the molecular mass and g the acceleration of gravity. Thus the variation of pressure with altitude may be written

$P\left(h\right)=P\left(0\right)e^{-mgh/kT}$

and the atmospheric pressure should drop to 1/e or 37% at an altitude of h = kT/mg. Using 28 as the molecular weight of nitrogen, the principal constituent, yields

$\begin{array}{l}mg=28\left(1.66\times {10}^{-27}\right)9.8=45\times {10}^{-25}newtons\\ kT=1.38\times {10}^{-23}\left(300\right)=4.1\times {10}^{-21}joules\\ h\left(37\%\right)=9\times {10}^{3}meters=9\text{km}\text{or}30,000\text{feet}.\end{array}$

This is quite close to the nominal observed value of 8 km, determined by the more complicated true molecular distribution and a significant negative temperature gradient. We discuss a simpler way of calculating energies in section 1.5.

Returning to the chamber, each mode corresponds to a resonant frequency determined by the cavity dimensions. In the original treatments, each mode was considered to be a "harmonic oscillator" having, as we shall see, an average thermal energy kT. Before we start counting the number of these modes versus optical frequency, let us first verify this average energy of a single mode according to Boltzmann's formula. First of all, we know that an ensemble of identical modes, either in time or over many systems, must have a total probability distribution over all energies U, which adds to unity, i.e.,

(1.2) $\begin{array}{cc}{\displaystyle {\int }_0^{\infty }p\left(U\right)dU}={\displaystyle {\int }_0^{\infty }A{e}^{-U/kT}dU=1}& \therefore A=\frac{1}{{\displaystyle {\int }_0^{\infty }{e}^{-U/kT}}dU}\end{array}$

The average energy of the mode is the integral over the product of the energy and the probability of that energy and is

(1.3) $\overline{U}={\displaystyle {\int }_0^{\infty }UA{e}^{-U/kT}dU}=\frac{{\displaystyle {\int }_0^{\infty }U{e}^{-U/kT}dU}}{{\displaystyle {\int }_0^{\infty }{e}^{-U/kT}dU}}=\frac{{\left(kT\right)}^2{\displaystyle {\int }_0^{\infty }x{e}^{-x}dx}}{kT{\displaystyle {\int }_0^{\infty }{e}^{-x}dx}}=kT$

where we have used the mathematical relationship,

(1.4) ${\displaystyle {\int }_0^{\infty }{x}^n{e}^{-x}dx}=n!$

We have thus obtained the standard classical result, which says that the energy per mode or degree of freedom for a system in thermal equilibrium has an average value of kT, the thermal energy. Soon we shall find that the number of allowed electromagnetic modes of a rectangular enclosure, or any enclosure for that matter, increases indefinitely with frequency. If each of these modes had energy kT, then the total energy would increase to infinity as the frequency approached infinity or the wavelength went to zero. This "ultraviolet catastrophe" as it was called, led to the proposal by Planck that at frequency, v, a mode was only allowed discrete energies separated by the energy increment, ΔU = hv. The value of the quantity, h, Planck's constant, was determined by fitting this modified theory to experimental measurements of thermal radiation.

Figure 1.1 shows the difference between the classical continuous Boltzmann distribution, (a), and a discrete or "quantized" distribution, (b). In the continuous distribution the area under the probability curve p(U) is equal to unity. In the discrete or quantized case the allowed energies as shown by the bars are separated by ΔU = hv and the sum of the heights of all bars becomes unity. We may state this mathematically by writing the energy of the nth state as

Figure 1.1. (a) Continuous and (b) discrete Boltzmann distribution with ΔU = hv = kT/4.

$\begin{array}{ccc}{U}_n=nhv& & n=0,1,2,etc.\end{array}$

with

(1.5) $p\begin{array}{cc}\left(U_n\right)=A{e}^{-U_n/kT}=A{e}^{-nhv/kT}& \therefore {\displaystyle \sum _{n=0}^{\infty }A{e}^{-nhv/kT}}\end{array}=1$

In a similar manner we may calculate the average energy, U(v), by summing the products of the nth state energy and its probability of occupation. Then

(1.6) $\begin{array}{cc}U\left(v\right)=\frac{{\displaystyle \sum _0^{\infty }nhv{e}^{-nhv/kT}}}{{\displaystyle \sum _0^{\infty }{e}^{-nhv/kT}}}=\frac{hv{\displaystyle \sum _0^{\infty }n{x}^n}}{{\displaystyle \sum _0^{\infty }{x}^n}};& x=e^{-hv/kT}\end{array}$

and using the identities,

(1.7) $\begin{array}{cc}{\displaystyle \sum _0^{\infty }{x}^n}=\frac{1}{1-x};& {\displaystyle \sum _0^{\infty }n{x}^n}=x\frac{d}{dx}{\displaystyle \sum _0^{\infty }{x}^n}=\frac{x}{{\left(1-x\right)}^2}\end{array}$

we finally obtain:

(1.8) $U\left(v\right)=hv\frac{x}{\left(1-x\right)}=hv\frac{e^{-hv/kT}}{\left(1-e^{-hv/kT}\right)}=\frac{hv}{\left(e^{hv/kT}-1\right)}$

This average energy for an electromagnetic mode at a single specific frequency, v, now has a markedly different behavior from the classical result of Eq. (1.3) when the energy hv becomes comparable to or greater than the thermal energy kT. In the two frequency limits, Eq. (1.8) goes to kT for low frequencies while it becomes hve ^-hv/kT as the frequency becomes very large. Of major significance is that the ratio of hv to kT for visible radiation at room temperature is of the order of one hundred, as we will see when we discuss the values of the various constants. As a result, the average energy per mode at visible frequencies is much less than kT.

The behavior of U(v) can be understood by examination of Figure 1.1. As the spacing of the discrete energies becomes smaller and smaller, the distribution of energies approaches the classical form, while as the spacing increases, the probability of the mode being in the zero-energy state approaches unity, and the occupancy of the next state, n = 1 or larger, becomes negligibly small, and thus U (v) goes to zero.

Given the expected energy for a single cavity mode at frequency, v, we may calculate the energy density in an enclosed cavity by counting the number of available electromagnetic modes as a function of the frequency. We start with the rectangular chamber of Figure 1.2, of dimensions, a by b by d, which has walls at temperature, T. We then write the equation for the allowed electromagnetic standing wave modes subject to the condition that the electric field, E, goes to zero at the walls. This is

Figure 1.2. Rectangular box for calculation of mode densities.

(1.9) $E=E_0\sin \left(k_{x}x\right)\sin \left(k_{y}y\right)\sin \left(k_{z}z\right)\sin 2\pi vt$

with each k taking only positive values. Using Maxwell's wave equation,

$\frac{{\partial }^{2}E}{\partial x^2}+\frac{{\partial }^{2}E}{\partial y^2}+\frac{{\partial }^{2}E}{\partial z^2}=\frac{1}{c^2}\frac{{\partial }^{2}E}{\partial t^2}$

we obtain

(1.10) $k_x^2+k_y^2+k_z^2=\frac{4{\pi }^2{v}^2}{c^2}={\left(\frac{2\pi }{\lambda }\right)}^2=k^2$

where c is the velocity of light and λ is the wavelength of the radiation. The quantity k = 2π/λ is the the magnitude of the total wave vector for the particular mode. To determine the mode density versus frequency we use Figure 1.3, which is a representation of the allowed modes in k-space. These allowed modes occur at those values of k which cause the field to become zero at x = a, y = b, and z = d, since the sine function already produces a zero at x = 0, y = 0, and z = 0. The requisite values of k are respectively mπ/a, nπ/b, and pπ/d, where m, n, and p are integers. The allowed modes thus form a rectangular lattice of points in k-space with spacing as shown in Figure 1.3. We now assume that the box dimensions are much greater than the wavelength λ and the distribution of points is then effectively continuous, since π/a for example is much less than 2π/λ, the magnitude of the k-vector in Eq. (1.10).

Figure 1.3. k-space showing discrete values of k _x , k _y , and k _z.

We now determine the number of modes dN in a thin octant (or eighth of a sphere) shell of thickness dk by multiplying the density of modes by the volume of the shell. Since the radius of the shell is k = 2πv/c, all modes on its surface are at the same frequency v. In addition, each point representing a mode lies on the corner of a rectangular volume with dimensions, π/a by π/b by π/d. Therefore the density of points is the inverse of this volume or abd/π ³ = V/π ³ , where V is the volume of the box. The volume of the octant shell is one eighth of 4π k ² dk so that

(1.11) $dN=\frac{V}{{\pi }^3}·\frac{4\pi k^{2}dk}{8}=\frac{4\pi V{v}^{2}dv}{c^3}$

using the relation between k and v from Eq. (1.10). Finally, we use the average energy per mode from Eq. (1.8) to calculate the energy density per unit frequency range, u _v = du/dv, where u = U/V, the electromagnetic energy per unit volume in a blackbody equilibrium cavity at temperature, T. In counting the modes we must take into account the two possible polarizations of the electric field, thus doubling the result of Eq. (1.11) and yielding

(1.12) $\begin{array}{c}du=2dN\frac{U\left(v\right)}{V}=\frac{8\pi v^{2}dv}{c^3}·\frac{hv}{\left(e^{hv/kT}-1\right)}=\frac{8\pi h{v}^{3}dv}{c^3\left(e^{hv/kT}-1\right)}\\ =\begin{array}{ccc}\frac{8\pi {\left(kT\right)}^4}{h^3{c}^3}·\frac{x^{3}dx}{\left(e^x-1\right)}& with& x=\frac{hv}{kT}\end{array}\end{array}$

This is the fundamental Planck equation, which we have written in terms of the universal function F(x) = x³/(e^x − 1) sketched in Figure 1.4. The energy density reaches a maximum at x = hv/kT = 2.8 and then the curve falls exponentially to zero.

Figure 1.4. The Planck energy density function, F(x).

Before we continue with our manipulations of Planck's law, we should discuss briefly the concept of the "photon," which is after all the heart of our topic. As we have reviewed, blackbody radiation was explained by Planck in terms of allowed discrete energies of an electromagnetic mode. In that context, a photon is a discrete step or quantum of energy of magnitude hv. An alternative concept of the photon is that of a particle and the average energy of Eq. (1.8) may be written as the product of hv, the photon energy, and 1/(e^hv/kT − 1), the occupation probability of the mode or the number of photons per mode. This probability factor is known in a more general form as the Bose-Einstein factor and is applicable in quantum mechanical treatments to "bosons" or particles with spin unity. Even though we soon speak of photon detectors, we shall use the term photon in the sense of a discrete energy gain or loss by the electromagnetic field, never as the description of a localized particle.

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Interaction of Radiation with Matter: Absorption, Emission, and Lasers

Robert H. Kingston , in Optical Sources, Detectors, and Systems, 1995

2.1 The Einstein A and B Coefficients and Stimulated Emission

We consider the interaction of blackbody radiation with a simple assemblage of particles each of which has two possible energy levels, U ₁ and U _2. These particles might be, for example, ammonia molecules, NH₃, which were used in the first demonstration of stimulated emission, the maser, a microwave device. As shown in Figure 2.1, the particles are distributed such that N ₂ are in the upper state and N ₁ in the lower state. The particles are contained in a blackbody chamber at temperature T, and the occupancy probability is given by Eq. (1.1), the Boltzmann relation. Thus the ratio N ₂ /N ₁ becomes e ^−hv/kT , since energy absorbing or emitting transitions between the two levels occur with a change in electromagnetic field energy of (U ₂ - U ₁ ) = hv. Defining the transition probabilities per unit time as W ₁₂ and W ₂₁ , we may write that N ₁ W ₁₂ = N ₂ W ₂₁ since the total rates must be equal in thermal equilibrium.

Figure 2.1. Transition rates and occupancy for states in equilibrium with blackbody field.

Now Einstein hypothesized that there were both spontaneous and induced or stimulated transitions. The spontaneous transitions occurred from the upper to the lower state with a probability per unit time of A, characteristic of the particle. In contrast the induced transition probability was assumed to be proportional to the electromagnetic spectral energy density, u _v , of Eq. (1.12), and transitions were induced in both directions. It was this latter conjecture that was most surprising since absorption had always been treated as the single process of excitation from the lower to the higher energy state. As we will see, it is essential to include the induced downward transitions to satisfy the rate equations, which we write as

(2.1) $N_1{W}_{12}=N_1{B}_{12}{u}_V=N_2{W}_{21}=N_2\left(A+B_{21}{u}_V\right)$

with B ₁₂ and B ₂₁ the proportionality constants for the upward and downward induced or stimulated transitions. Manipulation of the second and fourth terms of Eq. (2.1) yields

(2.2) $u_V=\frac{A}{\frac{N_1}{N_2}{B}_{12}-B_{21}}=\frac{A}{B_{12}{e}^{hv/kT}-B_{21}}$

But we know from Eq. (1.12) that

$u_v=\frac{8\pi h{v}^3}{c^3\left(e^{hv/kT}-1\right)}$

and, therefore, to satisfy the relationship of Eq. (2.1) for all frequencies and all temperatures, B ₁₂ = B ₂₁ = B, and A/B = 8πhv ³/c ³. Since we have established that the upward and downward induced rates are equal, we shall use the single constant B, which can be written

(2.3) $B=\frac{A{c}^3}{8\pi h{v}^3}=\frac{c^3}{8\pi h{v}^3{t}_s}=\frac{{\lambda }^3}{8\pi h{t}_s}$

where t _s is defined as the spontaneous emission time and is equal to 1/A. The actual values of the A and B coefficients are determined by the specific system considered. Obviously the larger the thermal equilibrium absorption, as determined by the B coefficient, the smaller the radiative or spontaneous emission time. Also, the higher the optical frequency, the shorter is t _s for the same value of B or absorption coefficient.

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Quantum Mechanics: Validity of Classical Molecular Dynamics

T. Prevenslik , in Reference Module in Materials Science and Materials Engineering, 2016

Abstract

Over a century ago, Planck to explain blackbody radiation abandoned classical physics in favor of frequency dependent quanta of energy to give birth to the field of quantum mechanics. No longer was the heat capacity of atom independent of the resonant frequency of the confining structure. Today, nanotechnology has renewed interest in quantum mechanics because the validity of molecular dynamics used in the design analysis of nanostructures is based on the questionable assumption of classical physics that the atom has heat capacity in contradiction to quantum mechanics that requires the heat capacity of the atom to vanish at the nanoscale. Classical molecular dynamics modified to be consistent with quantum mechanics is illustrated for the stiffening of nanowires in tensile tests.

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Introduction to Quantum Mechanics

Warren S. Warren , in The Physical Basis of Chemistry (Second Edition), 2001

5.2.2 Applications of Blackbody Radiation

Planck's law is universally accepted today, and blackbody radiation is a tremendously important concept in physics, chemistry, and biology. The blackbody distribution is graphed on a log scale for a variety of temperatures in Figure 5.2.

FIGURE 5.2. Blackbody radiation distribution for a variety of different temperatures. Notice that the curves shift with increasing temperature to shorter wavelengths and higher intensities, but otherwise they look identical.

We know that the surface temperature of the sun is approximately 5800K, because the spectrum of sunlight observed from outer space matches the distribution from a 5800K blackbody. At that temperature λ _max ≈500 nm, which is blue-green light; perhaps coincidentally (but more likely not) the sensitivity of animal vision peaks at about the same wavelength. Unfortunately, this temperature is well above the melting point of any known material. The only practical way to sustain such temperatures is to generate sparks or electrical discharges. In fact one of the dangers of "arc welding" to join metals is the extremely high temperature of the arc, which shifts much of the radiated energy into the ultraviolet. The light can be intense enough to damage your eyes even if it does not appear particularly bright.

Tungsten filaments are the light source in incandescent light bulbs. The efficiency of such a bulb increases dramatically with increasing temperature, because of the shift in λ_max. Figure 5.3 illustrates this efficiency using a historical (but intuitive) unit of brightness—the candle.

FIGURE 5.3. Efficiency of an ideal blackbody radiator for generating visible light (expressed as brightness per radiated watt). As the temperature increases, so does the fraction of light emitted in the visible. Thus the efficiency rises as well.

The vapor pressure of tungsten also rises dramatically as the temperature increases, so increasing the temperature shortens the bulb life. In a standard light bulb, the operating temperature is held to about 2500K to make the lifetime reasonable (≈ 1000 hours). Halogen lamps, which have recently become widely available, incorporate a very elegant improvement. The filament is still tungsten, but a small amount of iodine is added.

The chemical reaction

(5.10) ${\text{W(g)\hspace{0.17em}+\hspace{0.17em}2I(g)}}_{\to }^{\leftarrow }{\text{WI}}_2$

shifts back towards the reactants as the temperature increases. Near the walls of the quartz bulb the temperature is relatively cool, and tungsten atoms emitted by the filament react with the iodine to form WI2 and other tungsten compounds. As these molecules migrate through the bulb they encounter the much hotter filament, which causes them to decompose—redepositing the tungsten on the filament and regenerating the iodine vapor. So halogen bulbs can run hotter (3000–3300K), yet still have a long life.

At still lower temperatures little of the emission is in the visible, but the effects of blackbody radiation can still be very important. The Sun's light warms the Earth to a mean temperature of approximately 290K; the Earth, in turn, radiates energy out into space. For the Earth λ_max≈10 μm, far out in the infrared. If this radiation is trapped (for example, by molecular absorptions) the Earth cannot radiate as efficiently and must warm. This is the origin of the greenhouse effect; as we will discuss in chapter 8, carbon dioxide and other common gases can absorb at these wavelengths, so combustion products lead directly to global warming.

Even the near-vacuum of outer space is not at absolute zero. The widely accepted "Big Bang" theory held that the universe was created approximately 15 billion years ago, starting with all matter in a region smaller than the size of an atom. Remnants of energy from the initial "Big Bang" fill the space around us with blackbody radiation corresponding to a temperature of 2.73K. Detection of this "cosmic background" garnered the 1965 Nobel Prize in Physics for Penzias and Wilson. Measurements from the Cosmic Background Explorer satellite showed "warm" and "cool" spots from regions of space 15 billion light-years away. These temperature variations (less than 10⁻⁴degrees!) reflect structures which were formed shortly after the "Big Bang," and which by now have long since evolved into groups of galaxies. Very recent measurements are forcing some modifications to the conventional framework. Not only is the universe still expanding; the expansion is accelerating!

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Quantum Theory

David W. Cohen , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.A Difficulties with Planck's Theory

Some of the difficulties with Planck's theory of blackbody radiation were obvious immediately; others were quite subtle and were not discovered until several years after the presentation of the theory in 1900.

Consider, first, the fuzzy relationship between Planck's use of probabilism and Boltzmann's. As we mentioned in Section I.F, after Boltzmann partitioned the energy interval into finitely many subintervals, he later allowed the number of subintervals to approach infinity and the size of each subinterval to approach zero. That was required to apply the classical continuity assumptions he needed in applications of his theory. Planck carefully noted in his address to the Physikalische Gesellschaft in 1900 that his energy quanta ɛ   = hν must not be allowed to tend toward zero. The finiteness of the number of oscillators N _ν makes it essential to maintain the finiteness of the number of energy quanta in order to apply the combinatoric procedure associated with Eq. (27).

There was another discrepancy between Planck's theory and Boltzmann's statistical mechanics. It had been a generally accepted principle of statistical mechanics that, in an aggregate of oscillators in thermal equilibrium, all with the same number of degrees of freedom, the toal energy of each oscillator must, on average (over time), be distributed equally among its degrees of freedom. This principle was a consequence of what was called the equipartition theorem. If Planck had applied the theorem to his oscillators, then instead of Eq. (34) he would have obtained E(ν, T)   = kT and would have arrived at an incorrect radiation law. Planck's theory violated the principle of equipartition. It is not completely clear whether Planck was even aware of this principle in 1900.

A more fundamental difficulty, a logical inconsistency, was recognized by Albert Einstein in 1905. Planck had originally thought of his partitioning of the total energy into discrete quantities as a mathematical device to obtain numbers to treat with probabilistic arguments. He did not realize until it was pointed out by Einstein that, for his derivation to be consistent, each of his oscillators had to be assumed to be able to absorb and emit energy only over a discrete range of values. On the other hand, Planck's derivation of Eq. (22) requires that the oscillators be able to absorb and emit energy over a continuum of values. It is therefore inconsistent to put Eq. (32) together with Eq. (22) to arrive at a radiation law.

Despite the difficulties, Planck's theory of radiation was acknowledged for the accuracy of the formula resulting from it, and history shows that the theory itself revolutionized physics. The "discontinuity" (more accurately, the "discreteness") of the energy variable and the statistical nature of the behavior of discrete energy quanta were ideas that were to become the foundations of a new and controversial view of the universe.

Albert Einstein, of course, was as important to quantum theory as he was to nearly every other development of physics in the early 20th century. Sometimes a friend and sometimes a foe of the rapidly evolving quantum theory, he made important contributions to it and, merely by paying attention to it, helped to spur the interest of the scientific community. Let us now discuss two ideas of Einstein that were instrumental in placing the "quantum" in the forefront of physics.

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THERMODYNAMICS OF SOLIDS

Milton Ohring , in Engineering Materials Science, 1995

5.3.4.2 Blackbody Radiation

Another phenomenon based on absorption of thermal energy is blackbody radiation. If enough heat is absorbed by a solid and it gets sufficiently hot, it begins to emit electromagnetic energy from the surface, usually in the infrared and visible regions of the spectrum. According to the formula given by Planck, the power density (P) radiated in a given wavelength (λ) range varies as

(5-15) $P(\lambda )=C_1{\lambda }^{-5}/\left\{\exp (C_2/\lambda T)-1\right\}{\text{W/m}}^2,$

where C ₁ and C ₂ are constants. The mathematical similarity between Eqs. 5-14 and 5-15 is a reason for introducing this phenomenon here. As the temperature is increased the maximum value of P shifts to lower wavelengths. This accounts for the fact that starting at 500°C, a heated body begins to assume a dull red coloration. As the temperature rises it becomes progressively red, orange, yellow, and white. The total amount of heat power emitted from a surface, integrated over all directions and wavelengths, depends on temperature as T ⁴. This dependence is known as the Stefan-Boltzmann law.

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Radiative properties of metals

In Smithells Metals Reference Book (Eighth Edition), 2004

TEMPERATURE MEASUREMENT AND EMITTANCE

Radiation pyrometers, both spectral and total, are usually calibrated in terms of blackbody radiation and, thus, measure what is known as radiance temperature. The radiance temperature, T_r , measured by a total radiation pyrometer is related to the true temperature, T, by the formula

$T=T_r/{\epsilon }^{1/4}$

where ε is the total emittance of the surface. True temperature always exceeds radiance temperature.

For an optical pyrometer which measures irradiance from a narrow spectral interval only, the radiance temperature T_r is related to the true temperature by the equation

$\frac{1}{T}=\frac{1}{T_r}-\frac{\lambda }{C_2}\ln \frac{1}{{\epsilon }_{\lambda }}$

where ε_λ is the spectral emittance and C ₂ = 1.438 8 cm K is the constant in Wien's approximation for the blackbody emissive power. For the same emittances the correction is considerably greater for total radiation than for spectral radiation. For metallic surfaces the difference in correction is even greater, since the spectral emittance in the visible region for a given temperature is always greater than the total emittance.

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Radiometric Temperature Measurements: II. Applications

Sergey N. Mekhontsev , ... Leonard M. Hanssen , in Experimental Methods in the Physical Sciences, 2010

5 CONCLUSIONS

In this chapter, the methods and measuring techniques for the experimental characterization of blackbody radiation sources were reviewed, with emphasis on recent experimental work. A review of the most recent 20 years has revealed the following major trends: (1) extensive use of both laser-based and broadband reflectometers to evaluate and monitor cavity emissivity, including built-in ones for spaceborne applications; (2) use of multiple and complementary techniques for blackbody characterization, including spectral comparison of different blackbodies; (3) proliferation of detector-based techniques, which are presently dominating in the visible spectrum and are being extended to the near- and long-wave infrared regions; (4) construction of several dedicated facilities for blackbody characterization; (5) the ubiquitous use of Monte-Carlo modeling techniques in nonisothermal and nondiffuse approximations, along with understanding of the need for validation of these results; and, finally, (6) an emerging trend to acquire variable angle reflectance and BRDF parameters of materials and coatings and use them for the prediction of blackbody performance.

The material reviewed here reflects a multitude of methods and experimental approaches, which are being used for the characterization of the nonideality of real-life blackbody sources, as well as the increasing attention being paid to this problem by the international community as the accuracy requirements continue to increase and the number of commercially available products continues to grow. At the same time, it is difficult not to notice the absence of a definitive and internationally accepted set of recommendations for commercial blackbody specification and evaluation procedures. This has led to the presence of commercial blackbodies with unclear and contradictory manufacturer specifications.

It is hoped that this review will facilitate further development of this area of optical radiation metrology and offer some practical help in the navigation of the multitude of available publications on this subject matter.

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