Is Blackbody Radiation Continuous Involved in the Operation of Lasers

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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Controlling thermal radiation from surfaces☆

Carl G. Ribbing , in Optical Thin Films and Coatings (Second Edition), 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 on 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) $L_{\mathit{bb}}\left(v,T\right)=\frac{2h}{{\mathit{nc}}^2}\frac{v^3}{exp\left(\mathit{hv}/\mathit{kT}\right)-1}$

where ν is the frequency, c the vacuum velocity for light, h is the Planck constant, k the Boltzmann constant, and T is 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 behavior in the infrared where the unit W/m², THz, sr is appropriate.

Fig. 8.2. The spectral radiance from a blackbody as a function of frequency in THz at the four temperatures indicated. Note 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 a 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) $L_{\mathit{bb}}\left(\lambda ,T\right)=\frac{2{\mathit{hc}}^2}{{\mathit{nc}}^2{\lambda }^5}\frac{1}{exp\left(\mathit{hv}/\mathit{kT}\right)-1}$

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

Fig. 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 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 a shorter wavelength when the temperature increases. The corresponding expression for the frequency version of Eq. (8.2) is

(8.5) $v_m=b_{v}T$

with the constant b _v   =   5.8786   ×   10¹⁰  (s   K)^{−   1}.

As expected, the maxima move to higher frequencies when the temperature increases. A comparison of Figs. 8.2 and 8.3, shows, 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 nonlinear 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 nonlinear 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. Eq. 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. Eq. (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 nonlinear 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.

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 favor 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. Eq. (8.6) is obtained, whether the variable is ν or λ.

This signals that we should recheck are the differences in curve shapes noted above. The area under each curve, that is, the total radiance, should be independent of variables. 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 Eq. (8.4) into Eq. (8.3) (Ribbing, 1999). In contrast, the peaks of the frequency curves only grow as T ³, which is found by inserting Eq. (8.5) into Eq. (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 noncoherent. This may be the reason for the widespread notion that thermal radiation in general is noncoherent. 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 the Lambertian emission requires a source with some degree of periodicity.

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https://www.sciencedirect.com/science/article/pii/B9780081020739000084

Energy Production

Ibrahim Dincer , Yusuf Bicer , in Comprehensive Energy Systems, 2018

3.17.7.2.4 Ideal p–n junction–dissipation process

Ideal p–n junction also dissipates heat because of dark current over the diode which can be calculated as

(58) ${\dot{E}}_9={\dot{Q}}_{\mathrm{dark}}\mathrm{where}{\dot{Q}}_{\mathrm{dark}}\mathrm{is}\mathrm{equal}\mathrm{to}{A}_{\mathrm{c}}{J}_{\mathrm{dark}}{V}_{\mathrm{D}}$

The blackbody radiation emitted from the p–n junction at state point 10 can be determined for the wavelengths between 280 and 4000 nm as

(59) ${\dot{E}}_{10}=A_{\mathrm{c}}\sigma T_{\mathrm{c}}^4$

The exergy rates at state points 9 and 10 are determined:

(60) ${\dot{\mathrm{E}}\mathrm{x}}_9={\dot{Q}}_{\mathrm{dark}}\left(1-T_0/T_{\mathrm{c}}\right)$

(61) ${\dot{\mathrm{E}}\mathrm{x}}_{10}={\dot{E}}_{10}\left(1-T_0/T_{10}\right)$

where

(62) $T_{10}=\frac{hc}{{\mathcal{S}}_{\mathrm{\lambda }}}\frac{{\int }_{{\mathrm{\lambda }}_{\mathrm{g}}}^{\infty }{I}_{\lambda ,\mathrm{b}}\left(T_{\mathrm{c}}\right)d\mathrm{\lambda }}{{\int }_{{\mathrm{\lambda }}_{\mathrm{g}}}^{\infty }\lambda I_{\lambda ,\mathrm{b}}\left(T_{\mathrm{c}}\right)d\mathrm{\lambda }}$

as used from Ref. [44] and ${\lambda }_{\mathrm{g}}=\frac{hc}{E_{\mathrm{g}}}.$

The overall exergy destruction rate in ideal p–n junction–dissipation process is calculated as follows:

(63) ${\dot{\mathrm{E}}\mathrm{x}}_{\mathrm{d},\mathrm{tot},\mathrm{dark}}={\dot{\mathrm{E}}\mathrm{x}}_8+{\dot{\mathrm{E}}\mathrm{x}}_{10}$

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Radiation heat transfer

Charles H. Forsberg , in Heat Transfer Principles and Applications, 2021

9.2 Blackbody emission

The surfaces of all objects emit thermal radiation by virtue of their temperature being above absolute zero. This radiation is emitted as electromagnetic radiation in the wavelength range of about 0.1–100   μm (1   μm   =   10⁻⁶  m   =   1 micron). Some other areas of the electromagnetic spectrum are visible light (about 0.4–0.8   μm), X-rays (about 10⁻¹¹  m to 2   ×   10⁻⁸  m), microwaves (about 1   mm–10   m), and radio waves (about 10   m–30   km). Unlike conduction and convection, which require a medium for their transport, radiation is transported unhindered through a vacuum. Air, oxygen, and nitrogen are also essentially transparent to radiation. Some other gases, however, (e.g., water vapor, carbon dioxide, and some hydrocarbon gases), have appreciable absorption of radiation in specific wavelength regions. We will not be considering radiation transport through these latter types of gases.

The wavelength $\lambda$ of the radiation is related to the frequency $\upsilon$ of the radiation by the speed of light $c_o$ :

(9.2) $c_o=\lambda \upsilon$

where $c_o=2.9979\times {10}^8\text{m}/\text{s in a vacuum}\text{.}$

A "blackbody" has a surface that emits radiation at the maximum possible rate. The emission is over all wavelengths and is given by Planck's law, which is

(9.3) $E_{b\lambda }(\lambda ,T)=\frac{2\pi h{c}_o^2{\lambda }^{-5}}{\exp \left(\frac{h{c}_o}{\lambda kT}\right)-1}$

where

$\begin{array}{l}h=\text{Planck's}\text{constant}=6\text{.}62607\times {10}^{-34}\text{J}·\text{s}\\ k=\text{Boltzmann's}\text{constant}=1\text{.}38065\times {10}^{-23}\text{J}/\text{K}\\ E_{b\lambda }=\text{spectral}\text{emissive}\text{power}\text{of}\text{a}\text{blackbody}\end{array}$

The spectral emissive power is the rate of thermal emission per unit surface area per unit wavelength. If the wavelength of the radiation is in $\mu \text{m}$ , then the units of the spectral emissive power are W/(m² μm).

Putting in the values for the various parameters, Eq. (9.3) can be expressed as

(9.4) $E_{b\lambda }(\lambda ,T)=\frac{C_1{\lambda }^{-5}}{\exp \left(C_2/\lambda T\right)-1}$

where $C_1=3.7417\times {10}^8{(\text{W/m}}^2)·{\left(\mu \text{m}\right)}^4\text{and}{C}_2=1.4388\times {10}^4\text{μm}·\text{K}$ .

The blackbody spectral emissive power is shown in Fig. 9.1 for blackbodies having temperatures of 300   K, 1000   K, and 5800   K. The 300   K curve is for bodies at typical room temperature and the 5800   K curve approximates the emission from the sun.

Figure 9.1. Blackbody spectral emissive power.

The spectral emissive power of a blackbody given in Eq. (9.4) is a function of both the wavelength of the radiation and the absolute temperature of the blackbody. The blackbody emits over all wavelengths. If we integrate spectral emissive power $E_{b\lambda }$ over all wavelengths, we get the total emissive power $E_b$ which is a function of only the temperature T of the blackbody.

(9.5) $E_b\left(T\right)={\int }_0^{\infty }{E}_{b\lambda }(\lambda ,T)d\lambda =\sigma T^4$

where $\sigma =5.670\times {10}^{-8}{\text{W/m}}^2{\text{K}}^4=\text{Stefan}-\text{Boltzmann constant}$

Eq. (9.5) is called the Stefan–Boltzmann law.

We have used the adjectives "spectral" and "total". "Spectral" is for a parameter that depends on the wavelength. A parameter with "total" is independent of wavelength.

Looking at Fig. 9.1, it is seen that the maximum (or peak) emission from a blackbody occurs at lower wavelengths for higher temperatures. For 5800   K, the peak emission is at about 0.5   μm. For 1000   K, it is at about 3   μm. And, for 300K, it is about 9   μm. This is Wien's displacement law, which is

(9.6) ${\lambda }_{\max }T=2898\text{μm}·\text{K}$

where ${\lambda }_{\max }$ is the wavelength of peak emission.

Using Eq. (9.6), we get the following wavelengths of peak emission that we previously estimated by looking at Fig. 9.1:

$\text{For}5800\text{K,}{\lambda }_{\max }=2898/5800=0.500\text{μm}$

$\text{For}1000\text{K,}{\lambda }_{\max }=2898/1000=2.90\text{μm}$

$\text{For}300\text{K,}{\lambda }_{\max }=2898/300=9.66\text{μm}$

This shifting of wavelengths with temperature causes the "greenhouse effect" We have all been in hot, uncomfortable cars and rooms on hot, sunny days. This is primarily due to the transmissivity property of window glass. Glass has high transmission at the low wavelengths of solar radiation (5800   K blackbody) and very low transmission at the higher wavelengths of the 300   K radiation inside the car or room. The solar radiation easily goes through the glass into the inside air space, but it cannot leave the space through the essentially opaque glass. The temperature of the space increases to a very uncomfortable level if air conditioning is not provided.

We have seen that blackbody emission depends on wavelength and temperature. Sometimes we are interested in finding the emissive power from a blackbody for a given wavelength band. This can be determined by integrating the spectral emissive power over the wavelength range. For example, if we want the total emissive power for a blackbody at temperature T for the wavelength band ${\lambda }_1\mathrm{to}{\lambda }_2$ , we have

(9.7) $E_b({\lambda }_1\to {\lambda }_2,T)={\int }_{{\lambda }_1}^{{\lambda }_2}{E}_{b\lambda }(\lambda ,T)d\lambda$

If we want the fraction of the radiation emitted by a blackbody that is in wavelength band ${\lambda }_1\mathrm{to}{\lambda }_2$ , we have the total emissive power for the wavelength band divided by the total emissive power for all wavelengths. That is,

(9.8) $F_{{\lambda }_1\to {\lambda }_2}=\frac{{\int }_{{\lambda }_1}^{{\lambda }_2}{E}_{b\lambda }(\lambda ,T)d\lambda }{{\int }_0^{\infty }{E}_{b\lambda }(\lambda ,T)d\lambda }$

where $F_{{\lambda }_1\to {\lambda }_2}$ is the fraction of blackbody emission that is in wavelength range ${\lambda }_1\mathrm{to}{\lambda }_2$ . From the Stefan–Boltzmann Law, the denominator is equal to $\sigma T^4$ , so Eq. (9.8) may be written as

(9.9) $F_{{\lambda }_1\to {\lambda }_2}=\frac{{\int }_{{\lambda }_1}^{{\lambda }_2}{E}_{b\lambda }(\lambda ,T)d\lambda }{\sigma T^4}$

Using Eq. (9.4), Eq. (9.9) becomes

(9.10) $F_{{\lambda }_1\to {\lambda }_2}=\frac{{\int }_{{\lambda }_1}^{{\lambda }_2}\frac{C_1{\lambda }^{-5}}{\exp \left(C_2/\lambda T\right)-1}d\lambda }{\sigma T^4}$

The integral in Eq. (9.10) may be broken into two integrals, each starting at $\lambda =0.$ That is,

(9.11) $F_{{\lambda }_1\to {\lambda }_2}=F_{0\to {\lambda }_2}-F_{0\to {\lambda }_1}=\frac{{\int }_0^{{\lambda }_2}\frac{C_1{\lambda }^{-5}}{\exp \left(C_2/\lambda T\right)-1}d\lambda }{\sigma T^4}-\frac{{\int }_0^{{\lambda }_1}\frac{C_1{\lambda }^{-5}}{\exp \left(C_2/\lambda T\right)-1}d\lambda }{\sigma T^4}$

The integrals in Eqs. (9.10) and (9.11) are over wavelengths $\lambda$ for a given blackbody temperature T. The variable of integration can be changed from $\lambda \text{ to}\lambda T$ , which modifies Eq. (9.11) to

(9.12) $F_{{\lambda }_{1}T\to {\lambda }_{2}T}=F_{0\to {\lambda }_{2}T}-F_{0\to {\lambda }_{1}T}={\int }_0^{{\lambda }_{2}T}\frac{C_1/\sigma }{{\left(\lambda T\right)}^5[\exp \left(C_2/\lambda T\right)-1]}d\left(\lambda T\right)-{\int }_0^{{\lambda }_{1}T}\frac{C_1/\sigma }{{\left(\lambda T\right)}^5[\exp \left(C_2/\lambda T\right)-1]}d\left(\lambda T\right)$

Table 9.1 gives the blackbody radiation function $F_{0\to \lambda T}={\int }_0^{\lambda T}\frac{C_1/\sigma }{{\left(\lambda T\right)}^5[\exp \left(C_2/\lambda T\right)-1]}d\left(\lambda T\right)$ for different values of $\lambda T$ . The use of Table 9.1 in determining the fraction of blackbody radiation in the wavelength band of ${\lambda }_1\mathrm{to}{\lambda }_2$ is illustrated in Example 9.1.

Table 9.1. Blackbody radiation function.

λT (μm·K)	F 0-λT	λT (μm·K)	F 0-λT	λT (μm·K)	F 0-λT
1000	0.0003	5000	0.6338	10,500	0.9238
1200	0.0021	5200	0.6580	11,000	0.9320
1400	0.0078	5400	0.6804	11,500	0.9390
1600	0.0197	5600	0.7011	12,000	0.9452
1800	0.0393	5800	0.7202	13,000	0.9552
2000	0.0667	6000	0.7379	14,000	0.9630
2200	0.1009	6200	0.7542	15,000	0.9691
2400	0.1403	6400	0.7692	16,000	0.9739
2600	0.1831	6600	0.7833	18,000	0.9809
2800	0.2279	6800	0.7972	20,000	0.9857
3000	0.2733	7000	0.8082	25,000	0.9923
3200	0.3181	7200	0.8193	30,000	0.9954
3400	0.3618	7400	0.8296	40,000	0.9981
3600	0.4036	7600	0.8392	50,000	0.9998
3800	0.4434	7800	0.8481	75,000	0.9998
4000	0.4809	8000	0.8563	100,000	1.0000
4200	0.5161	8500	0.8747
4400	0.5488	9000	0.8901
4600	0.5793	9500	0.9031
4800	0.6076	10,000	0.9143

Example 9.1

Blackbody emission in a wavelength band

Problem

What fraction of the emission from a blackbody at 5800   K lies in the visible range of the electromagnetic spectrum, i.e., in the range from 0.4   to 0.8   μm?

Solution

There are different ways to solve this problem. We will first solve it using Table 9.1.

The fraction in the band from ${\lambda }_1=0.4\text{μm}\text{to}{\lambda }_2=0.8\text{μm is}{F}_{0\to {\lambda }_{2}T}-F_{0\to {\lambda }_{1}T}$

$\begin{array}{l}{\lambda }_{1}T=\left(0.4\right)\left(5800\right)=2320\\ {\lambda }_{2}T=\left(0.8\right)\left(5800\right)=4640\\ \text{From Table}9\text{.}1\text{,}{F}_{0\to {\lambda }_{2}T}=0.5850\text{ and}{F}_{0\to {\lambda }_{1}T}=0.1245\end{array}$

$\text{Fraction}=F_{0\to {\lambda }_{2}T}-F_{0\to {\lambda }_{1}T}=0.5850-0.1245=0.4605$

46.1% of the radiation from a blackbody at 5800   K lies in the wavelength band from 0.4 to 0.8   μm.

This problem can also be solved without using Table 9.1. For example, the quad function of Matlab can be used to determine the integral in Eq. (9.10) directly. For this problem, we have

(9.10) $F_{{\lambda }_1\to {\lambda }_2}=\frac{{\int }_{{\lambda }_1}^{{\lambda }_2}\frac{C_1{\lambda }^{-5}}{\exp \left(C_2/\lambda T\right)-1}d\lambda }{\sigma T^4}$

Putting in the values for the limits and constants for this problem, we have

(9.13) $F_{0.4\to 0.8\text{μm}}=\frac{{\int }_{0.4}^{0.8}\frac{3.7417\times {10}^8{\lambda }^{-5}}{\exp (1.4388\times {10}^4/\lambda \left(5800\right))-1}d\lambda }{(5.67\times {10}^{-8}){\left(5800\right)}^4}$

Eq. (9.13) simplifies to

(9.14) $F_{0.4\to 0.8\text{μm}}=5.8314{\int }_{0.4}^{0.8}\frac{{\lambda }^{-5}}{\exp \left(2.4807/\lambda \right)-1}d\lambda$

The integral in Eq. (9.14) can be determined using the following Matlab statement interactively:

$\gg S=\text{quad}({}^{\prime }x.^(-5)./{(\exp (\mathrm{2.4807.}/x)-1)}^{\prime },0.4,0.8)$

(Note that the periods for element-by-element operations are needed in the Matlab statement.)

The result is S  =   0.07907, which, using Eq. (9.14), gives the same result as we got using Table 9.1.

$F_{0.4\to 0.8\text{μm}}=\left(5.8314\right)\left(0.07907\right)=0.461$

Excel could also have been used to determine the integral in Eq. (9.14). Indeed, the spreadsheet could have been programmed to determine the integral using, for example, Simpson's rule. Such programming, however, is a bit involved, and this is a study of heat transfer, not computer programming. Therefore, we will determine the integral in an alternative way, as follows:

Open an Excel spreadsheet. Our wavelength limits are 0.4–0.8. Using an increment of 0.02, we put the wavelength values in Column A, cells A1 through A21. In Column B, we put the values of the integrand in Eq. (9.14) corresponding to the wavelength values of Column A. The integrand values are in cells B1 through B21. We highlight Columns A and B and do Insert of a scatter chart. Clicking on the plot area, using the Layout tab, we Insert a trendline of a polynomial of order 3. We click the appropriate boxes to display the trendline equation and the R-squared value on the chart.

Doing this, we got the trendline equation $y=3.5663x^3-7.3632x^2+4.7057x-0.7328$ and an R-squared value of 0.9991, which showed that the third order polynomial was an excellent fit for the integrand. We chose the polynomial form as a polynomial is easy to integrate by hand. Integrating the integrand function, we get

$\underset{0.4}{\overset{0.8}{\int }}ydx=\frac{3.5663}{4}({0.8}^4-{0.4}^4)-\frac{7.3632}{3}({0.8}^3-{0.4}^3)+\frac{4.7057}{2}({0.8}^2-{0.4}^2)-0.7328(0.8-0.4)=\text{0}\text{.}0791$

From Eq. (9.14), $F_{0.4\to 0.8\text{μm}}=\left(5.8314\right)\left(0.0791\right)=0.461$ .

In conclusion, the same result was obtained using Table 9.1, Matlab, and Excel.

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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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Photovoltaic Solar Energy

V. Badescu , in Comprehensive Renewable Energy, 2012

1.15.2.6 Diluted Thermal Radiation

Diffuse solar radiation may be treated as diluted blackbody radiation [3, 8, 18]. A full account on this subject may be found in the work of Landsberg and Tonge [19]. The most significant results are reviewed in Reference 8.

Entropy and energy fluxes of diluted radiation may be written as

[49] $\varphi =A(\Omega )\epsilon \sigma T_{\text{R}}^4$

[50] $\psi =\frac{4}{3}A(\Omega )\sigma \epsilon \chi (\epsilon )T_{\text{R}}^3$

where ε ≤   1 is the so-called dilution factor, T _R refers to the undiluted blackbody radiation (ε  =   1), and χ(ε) is a function exactly calculated in Reference 19, which can be approximated for small ε (i.e., less than 0.1) by

[51] $\chi (\epsilon )\cong 0.9652-0.2777\ln \epsilon +0.0511\epsilon$

and such that χ(1)   =   1. Another useful approximation is χ(ε)   ≅   1   −   45/(4π⁴)   ×   (2.336   −   0.260   ε)   ×   ln ε, which can be used for 0.005   < ε  <   1 [20]. The function A(Ω) from eqns [49] and [50] refers to a geometrical factor that depends on the solid angle Ω subtended by the source of radiation by

[52] $A(\Omega )\equiv \frac{B(\Omega )}{\pi }=\frac{\Omega }{\pi }\left(1-\frac{\Omega }{4\pi }\right)$

By using eqns [49] and [50], the effective temperature T _e of the diluted radiation can be derived from the usual definition:

[53] $T_{\text{e}}\equiv {\left(\frac{\partial \varphi }{\partial \psi }\right)}_{\epsilon \Omega }=\frac{4}{3}\frac{\varphi }{\psi }=\frac{T_{\text{R}}}{\chi (\epsilon )}$

Although T _e is not an equilibrium temperature, with its help the thermodynamics of diluted radiation is formally coincident with the thermodynamics of blackbody radiation. Indeed, by using eqns [49] and [53], we derive another form of the energy flux of diluted radiation [19]:

[54] $\varphi =\beta T_{\text{e}}^4$

with

[55] $\beta \equiv A(\Omega )\epsilon \sigma {\chi }^4(\epsilon )$

After dilution, the thermal radiation may be considered as undiluted with respect to the blackbody radiation of temperature T _e .

Two papers approached the theoretical maximum efficiency of diffuse solar radiation [8, 21]. Very different results were reported. In Reference 8, one proved that the maximum efficiency is 0.573. A significantly smaller value (i.e., 0.096) was obtained in Reference 21. In Reference 18, we showed that both results are consistent if the more general model presented next is used.

Now, one assumes the Sun at zenith. Also, one assumes that the Earth's atmosphere does not absorb solar radiation and radiation scattering is forward (in other words, there is no backscattered radiation). Then, we define a perfectly forward diffuser, that is, a finite thin body situated between the Sun and the observer whose surface elastically scatters into a 2π solid angle any narrow pencil of radiation incident on it. We assume this diffuser as subtending a solid angle Ω₁ when viewed from the Earth. Consider first the direct sunlight (dilution factor ε ₀) incident on a point M placed on the diffuser surface. Then, ε ₀  =   1 and

[56] $A({\Omega }_0)=\frac{{\Omega }_0}{\pi }\left(1-\frac{{\Omega }_0}{4\pi }\right)\approx \frac{{\Omega }_0}{\pi }$

where Ω₀  =   6.835   ×   10⁻⁵ sr is the solid angle subtended by the Sun (see eqn [48b]). Consequently, the flux of beam radiation is given by eqn [4]

[57] ${\varphi }_0=\frac{{\Omega }_0}{\pi }\epsilon \sigma T_{\text{s}}^4$

where T _s is Sun temperature (5760   K).

The flux φ ₀ is scattered over a Ω₁  =   2π solid angle around the point M. After scattering, solar radiation has a dilution factor ε ₁  <   1 and the effective temperature T _e,1. For an observer situated at the point M, the scattered radiation has a geometrical factor A(Ω₁)   =   1. The flux φ ₁ of dilute radiation is equal to the incoming flux φ ₀. By using eqns [49] and [57], we obtain

[58] ${\varphi }_1=A({\Omega }_1){\epsilon }_1\sigma T_{\text{s}}^4=\frac{{\Omega }_0}{\pi }\sigma T_{\text{s}}^4={\varphi }_0$

Consequently, the dilution factor is

[59] ${\epsilon }_1=\frac{{\Omega }_0}{\pi }$

and the effective temperature of the scattered radiation can be derived by using eqn [53]:

[60] $T_{\text{e},1}=T_{\text{s}}/\chi ({\epsilon }_1)=1459.5\text{K}$

This result was first reported in Reference 8.

Until now, we have analyzed the scattered radiation from the point of view of an observer situated on the surface of the diffuser. For an observer placed on the Earth's surface, the source of singly scattered radiation may be formally described as a blackbody of temperature T _e,1 (dilution factor ε ₁′   =   1), which subtends the solid angle Ω₁. To determine Ω₁, we use eqn [49] and the assumption that on the Earth's surface the flux φ ₁ of scattered radiation equates the flux φ ₀

[61] ${\varphi }_1=A({\Omega }_1){\epsilon }_{\text{e},1}\sigma T_{\text{e},1}^4=\frac{{\Omega }_0}{\pi }\sigma T_{\text{s}}^4={\varphi }_0$

By using eqn [52], we obtain

[62] ${\Omega }_1=2\pi \left[1-{\left(1-\frac{{\Omega }_0}{\pi }{\chi }^4\left({\epsilon }_1\right)\right)}^{1/2}\right]=0.01665\text{sr}$

The solid angle Ω₁ is enveloped by a cone symmetrically disposed around the nadir–zenith direction and whose half-angle δ ₁ can be derived from the relation:

[63] ${\Omega }_1=2\pi (1-\cos {\delta }_0)$

We obtain

[64] ${\delta }_1=2.083°$

We conclude that after a single scattering solar radiation is still strongly anisotropic (compare δ ₁ with the half-angle δ ₀  =   0.265° of the cone subtended by the Sun).

The second scattering is described next. This case implies the existence of a second perfectly forward diffuser. The incoming radiation consists in the flux φ ₁ of singly scattered solar radiation (dilution factor ε ₁  =   Ω₀/π) or, which is equivalent, blackbody radiation of temperature T _e,1 (dilution factor ε′₁  =   1). The flux φ ₁ is again dispersed over a solid angle Ω₂  =   2π, the scattered radiation having a geometrical factor A(Ω₂)   =   1 and a dilution factor ε ₂  <   1.

The flux φ ₂ of doubly scattered radiation equates to the incoming flux φ ₁. By using eqn [58], we obtain

[65] ${\varphi }_2=A({\Omega }_2){\epsilon }_2\sigma T_{\text{e},1}^4={\varphi }_1={\varphi }_0=\frac{{\Omega }_0}{\pi }\sigma T_{\text{s}}^4$

From eqn [65], we find

[66] ${\epsilon }_2=\frac{{\Omega }_0}{\pi }{\chi }^4({\epsilon }_1)=5.281\times {10}^{-3}$

and the effective temperature of the doubly scattered radiation is given by eqn [53]:

[67] $T_{\text{e},2}=\frac{T_{\text{e},1}}{\chi ({\epsilon }_2)}=\frac{T_{\text{s}}}{\chi ({\epsilon }_1)\chi ({\epsilon }_2)}$

For an observer placed on the Earth's surface, the source of thermal radiation is a blackbody of temperature T _e,2 (dilution factor ε ₂′   =   1) which subtends a solid angle Ω₂. To determine Ω₂, we use eqn [49] and the assumption that the energy flux φ ₂ incident on the Earth's surface equates to the flux φ ₀:

[68] ${\phi }_2=A({\Omega }_2){\epsilon \prime }_2\sigma T_{e,2}^4={\phi }_0=\frac{{\Omega }_0}{\pi }\sigma T_s^4$

By using eqn [52], we obtain

[69] ${\Omega }_2=2\pi \left[1-{\left(1-\frac{{\Omega }_0}{\pi }{\chi }^4({\epsilon }_1){\chi }^4({\epsilon }_2)\right)}^{1/2}\right]=0.599\text{sr}$

The half-angle δ ₂ of the cone that envelops the solid angle Ω₂ can be determined by a relation similar to eqn [63] and is

[70] ${\delta }_2=12.609°$

As we see, the doubly scattered solar radiation is still anisotropic.

The above procedure can be repeated for three and four scatterings with the following results:

[71] ${\epsilon }_i=\frac{{\Omega }_0}{\pi }{\displaystyle \prod _{j=1}^{i-1}{\chi }^4({\epsilon }_j)}$

[72] $T_{\text{e},i}=\frac{T_{\text{s}}}{{\displaystyle \prod _{j=1}^i\chi ({\epsilon }_j)}}$

[73] ${\Omega }_i=2\pi \left[1-{\left(1-\frac{{\Omega }_0}{\pi }{\displaystyle \prod _{j=1}^i{\chi }^4({\epsilon }_j)}\right)}^{1/2}\right](i=3,4)$

where we noted T _e,0  ≡ T _s. Table 1 shows the results. After four scatterings, solar radiation is completely isotropic and an observer at the ground would see a uniformly brilliant sky. This case was first analyzed in Reference 21. There are some indications that for a clear sky most of the diffuse solar radiation is received within a cone of half-angle δ _real  =   20   −   30° [22, 23]. This implies a mean number of three scatterings.

Table 1. Maximum efficiency of singly or multiply scattered solar radiation

Number of scatterings, i	Te,j (K)	δ i (deg)	Cmax
0	5760	0.265	45963
1	1459.5	2.083	189.5
2	602.8	12.609	5.5
3	418.9	30.961	1.3
4	393.4 (389.9) a	90	1

T _e, the effective temperature of scattered radiation; δ, the half-angle of the cone that subtends a blackbody at temperature T _e; C _max, the maximum concentration ratio.

a

Value computed from eqn [51] which is a good approximation only for small values of the dilution factor ε.

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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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Physical environmental factors and their properties

Genhua Niu , ... Nadia Sabeh , in Plant Factory (Second Edition), 2020

11.2.2 Radiation

Radiation in the far-infrared waveband is essentially blackbody radiation emitted by surrounding objects. Objects of higher temperature emit larger quantities of far-infrared radiation than objects at a lower temperature.

The primary sources of radiant energy in PFALs are lamps and reflectors. Traditional lamps for growth chambers and greenhouses, such as high-pressure sodium lamps and metal halide lamps, have surface temperatures of over 100°C and emit large quantities of far-infrared radiation. This radiation is absorbed by plants, causing increased plant temperature regardless of the surrounding air temperature, thereby impeding control over plant physiological activity. In PFALs, this challenge is compounded by the short distance between lamps and plants that is desirable for maximizing space use efficiency and plant productivity. Therefore, it is preferable to use light sources that emit much less far-infrared radiation, such as LEDs (light-emitting diodes; surface temperature: about 30°C) and fluorescent lamps (surface temperature: about 40°C).

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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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