Radiation, blackbody

Planck s radiation law determines the power emitted by a small aperture in a cavity, which is at a given equilibrium temperature. The spectral flux emitted by an isotropic blackbody source into a solid angle 2 = 2rr sin 0r (where 9r is the angular radius of the first optical element of the spectrometer) is  

To estimate the flux in more familiar units (e.g. Watts per wavenumbers), Equation (3.4) becomes  

Several questions present themselves immediately How good does the initial guess have to be How do we know that the procedure leads to better guesses, not worse How many steps (how long) will the procedure take How do we know when to stop These questions and others like them will play an important role in this book. You will not be surprised to leam that answers to questions like these vary from one problem to another and cannot be set down once and for all. Let us start with a famous problem in quantum mechanics blackbody radiation. 

We can sample the energy density of radiation p(v, T) within a chamber at a fixed temperature T (essentially an oven or furnace) by opening a tiny transparent window in the chamber wall so as to let a little radiation out. The amount of radiation sampled must be very small so as not to disturb the equilibrium condition inside the chamber. When this is done at many different frequencies v, the blackbody spectrum is obtained. When the temperature is changed, the area under the spechal curve is greater or smaller and the curve is displaced on the frequency axis but its shape remains essentially the same. The chamber is called a blackbody because, from the point of view of an observer within the chamber, radiation lost through the aperture to the universe is perfectly absorbed the probability of a photon finding its way from the universe back through the aperture into the chamber is zero. 

Many physicists of the late nineteenth century tried to derive expressions consistent with the experimental intensity-versus-frequency curves for several temperatures, as shown in Fig. 3.19, but without success. In fact, the Rayleigh-Jeans law gives the expression that is derived according to the laws of nineteenth century physics as  

McQuarrie, Quantum Chemistry, pp. 5 6, University Science Books, Mill Valley, CA (1983) M.I. Sobel, Light, pp. 74 75, University of Chicago Press, Chicago (1987). 

It can be seen that the Rayleigh-Jeans law reproduces the experimental data at low frequencies fairly well. However, at high frequencies, the Rayleigh-Jeans law diverges as v. Since the frequency increases in the ultraviolet region of the spectrum, this divergence was called the ultraviolet catastrophe, a phenomenon that classical physics was unable to explain theoretically. This was the first such phenomenon to be observed in physics and did in fact mark a major milestone in the annals of physics. 

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An excellent and readable discnssion of all aspects of the interaction of light with matter, from blackbody radiation to lasers and nonlinear optics. 

Unlike the typical laser source, the zero-point blackbody field is spectrally white , providing all colours, CO2, that seek out all co - CO2 = coj resonances available in a given sample. Thus all possible Raman lines can be seen with a single incident source at tOp Such multiplex capability is now found in the Class II spectroscopies where broadband excitation is obtained either by using modeless lasers, or a femtosecond pulse, which on first principles must be spectrally broad [32]. Another distinction between a coherent laser source and the blackbody radiation is that the zero-point field is spatially isotropic. By perfonuing the simple wavevector algebra for SR, we find that the scattered radiation is isotropic as well. This concept of spatial incoherence will be used to explain a certain stimulated Raman scattering event in a subsequent section. 

In practice, the NEP of a room-temperature THz spectrometer is usually limited by fluctuations (shot-noise) in the ambient blackbody radiation. Usmg an optical bandwidth Av = 3 THz (limited by, for example, a polyethylene/diamond dust window), a field of view (at nomial incidence) 0 = 9 and a detecting diameter (using a so-called Winston cone, which condenses the incident radiation onto the detecting element) laboratory applications, the background-limited NEP of a bolometer is given by... 

Dunbar R C and McMahon T B 1998 Activation of unimolecular reactions by ambient blackbody radiation Science 279 194-7... 

If we think in terms of the particulate nature of light (wave-particle duality), the number of particles of light or other electi omagnetic radiation (photons) in a unit of frequency space constitutes a number density. The blackbody radiation curve in Fig. 1-1, a plot of radiation energy density p on the vertical axis as a function of frequency v on the horizontal axis, is essentially a plot of the number densities of light particles in small intervals of frequency space. 

Figure 1-1 The Blackbody Radiation Spectrum. The short curve on the left is a Rayleigh function of frequency.

In the late nineteenth century, Wien analyzed experimental data on blackbody radiation and found that the maximum of the blackbody radiation specti um shifts with the temperature according to the equation... 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

In principle, emission spectroscopy can be applied to both atoms and molecules. Molecular infrared emission, or blackbody radiation played an important role in the early development of quantum mechanics and has been used for the analysis of hot gases generated by flames and rocket exhausts. Although the availability of FT-IR instrumentation extended the application of IR emission spectroscopy to a wider array of samples, its applications remain limited. For this reason IR emission is not considered further in this text. Molecular UV/Vis emission spectroscopy is of little importance since the thermal energies needed for excitation generally result in the sample s decomposition. 

For both aqueous and nonaqueous liquids, MBSL is caused by chemical reactions of high energy species formed duriag cavitation by bubble coUapse, and its principal source is most probably not blackbody radiation or electrical discharge. MBSL is predominandy a form of chemiluminescence. 

Fig. 2. Blackbody radiated photon flux interval distribution from ambient temperature up to 2000 K. Ambient objects do not have detectable flux for...

E = hemispherical emissive power of a blackbody. f = fraction of blackbody radiation lying below X. 

Blackbody Radiation Engineering calculations of thermal radiation from surfaces are best keyed to the radiation characteristics of the blackbody, or ideal radiator. The characteristic properties of a blackbody are that it absorbs all the radiation incident on its surface and that the quality and intensity of the radiation it emits are completely determined by its temperature. The total radiative fliix throughout a hemisphere from a black surface of area A and absolute temperature T is given by the Stefan-Boltzmann law ... 

Water Vapor The contribution to the emissivity of a gas containing H9O depends on Tc andp L and on total pressure P and partial pressure p . Table 5-8 gives constants for use in evaluating . Allowance for departure from the special pressure conditions is made by multiplying by a correction factor C read from Fig. 5-21 as a function of (p + P) and p ,L. The absorptivity 0t of water vapor for blackbody radiation is evaluated from Table 5-8 but at T instead of Tc and at p LT /Tc instead of p, h. Multiply by (Tc/Ti)° . ... 

Figure 4.24. The Planck distribution law spectral radiance of blackbody radiation as a function of temperature and wavelength. (After Touloukian and DeWitt (1972). Plenum Press.)...

FIGURE 4.33 Heat transfer factor representing blackbody radiation for various mean temperatures and temperature differences. 

A hundred years ago it was generally supposed that all the properties of light could be explained in terms of its wave nature. A series of investigations carried out between 1900 and 1910 by Max Planck (1858-1947) (blackbody radiation) and Albert Einstein (1879-1955) (photoelectric effect) discredited that notion. Today we consider light to be generated as a stream of particles called photons, whose energy E is given by the equation... 

The international temperature scale is based upon the assignment of temperatures to a relatively small number of fixed points , conditions where three phases, or two phases at a specified pressure, are in equilibrium, and thus are required by the Gibbs phase rule to be at constant temperature. Different types of thermometers (for example, He vapor pressure thermometers, platinum resistance thermometers, platinum/rhodium thermocouples, blackbody radiators) and interpolation equations have been developed to reproduce temperatures between the fixed points and to generate temperature scales that are continuous through the intersections at the fixed points. 

When Planck used this relationship to calculate the spectrum of blackbody radiation, he came up with a result that agreed perfectly with experiment. More importantly, he had discovered quantum mechanics. Energy emitted by a blackbody is not continuous. Instead, it comes in tiny, irreducible packets or quanta (a word coined by Planck himself) that are proportional to the frequency of the oscillator that generated the radiation. 

The constant h and the hypothesis that energy is quantized in integral multiples of hv had previously been introduced by M. Planck (1900) in his study of blackbody radiation. In terms of the angular frequency a> deflned in equation (1.2), the energy E of a photon is... 

Principles and Characteristics The term luminescence describes the radiative evolution of energy other than blackbody radiation which may accompany the decay of a population of electronically excited chro-mophores as it relaxes to that of the thermally equilibrated ground state of the system. The frequency of the... 

Variations in the temperature of a blackbody used as the source in a spectrometer. The energy density of blackbody radiation is given by the well-known formula ... 

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