Energy density of 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. 

Its fundamental importance derives from the fact that the energy density of radiation in the cavity does not depend on the properties of the cavity, i.e., properties of the walls or the cavity size, provided it is large compared to the wavelength of the radiation. In the following we will discuss radiation not as a function of wavelength but as a function of photon energy, because it is the photons (light quanta) that are absorbed and excite electrons in matter. 

By considering the radiation as such a gas, the principles of quantum-statistical thermodynamics can be applied to derive an expression for the energy density of radiation per unit volume and per unit wavelength ast... 

To find the effect of radiation on thermal problems, however, we need to interpret the optical rays diverging in all directions in terms of thermal concepts such as energy density and heat flux. First, introduce the definition of the energy density of radiation,... 

What is the energy density of radiation at 500 nm emitted by a black-body source at the surface temperature of the sun (6,600 K) ... 

From equations (9.16) and (9.17) we obtain an expression for the energy density of radiation in the cavity ... 

In contrast the light emitted by a laser is generated by stimulated emission in an optical resonator. As a result the radiation has high spatial and temporal coherence which is not found in light emitted by any other source. In addition, the energy densities of radiation in laser beams... 

The energy densities of laser beams which are conventionally used in the production of thin films is about 10 — 10 Jcm s and a typical subsU ate in the semiconductor industry is a material having a low drermal conductivity, and drerefore dre radiation which is absorbed by dre substrate is retained near to dre surface. Table 2.8 shows dre relevant physical properties of some typical substrate materials, which can be used in dre solution of Fourier s equation given above as a first approximation to dre real situation. 

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 ... 

The basic parameters of this problem are the lifetime of the neutron (887 seconds) and the number of neutrino species (three), both given by modern microphysics. At the time which interests us here, i.e. r = 1 s, the energy density of electromagnetic radiation was greater than that of matter. This is therefore referred to as the radiation era. 

Let us assume that the system is in the lower state n and exposed to radiation of density p (v) defined as the energy of radiation per unit volun between the frequencies v and v + dv. The probability per unit time th it will absorb the radiation and will thereby be raised to the upper sta m, is proportional to the number of particles Nn in the state n and tl density of radiation of frequency vm . Hence... 

To date, neither PAH emission nor absorption has been detected in the circumstellar envelope around a cool carbon star PAH emission has only been seen in carbon-rich environments where there is substantial energy density of ultraviolet radiation. This correlation could simply be an excitation effect the carbon features are only excited by the presence of ultraviolet radiation. However, it could also be that carbon particles are eroded into PAHs in the environment where ultraviolet penetrates either directly by the ultraviolet radiation or indirectly by shocks that accompany the radiation. 

It can easily be shown that if is the number of negative muons present in a D level of a helium muonic ion (ii"4 He)p and E/V is the energy density of the radiation at the site of the stopping muon, then the fraction of transitions is given by (v = u/2it is the radiation frequency) ... 

For the radiation density p(v u), instead of the Planck formula, we use the energy density of the electromagnetic field ... 

P+ and P are the probabilities for absorption and emission, respectively B+ and B are the coefficients of absorption and of induced emission, respectively A- is the coefficient of spontaneous emission and p v) is the density of radiation at the frequency that induces the transition. Einstein showed that B+ = B, while A frequency dependence, spontaneous emission (fluorescence), which usually dominates in the visible region of the spectrum, is an extremely improbable process in the rf region and may be disregarded. Thus the net probability of absorption of rf energy, which is proportional to the strength of the NMR signal, is... 

The above relation is known as the Stefan-Boltzmann Law (1879) the constant is empirically determined to be ct = 5.67 x 10 W/K m. One should note the very strong dependence of the energy density of the electromagnetic radiation on the temperature. 

Consider a Prevost chamber of volume V (whose walls are impermeable to heat). Let its temperature be T and the energy density of the radiation in its interior be u. Let the chapiber be furnished with a movable and frictionless piston (also impermeable to heat) which separates the interior of the chamber from the external (empty) space. 

Let us now push forward the piston so as to diminish the volume of the chamber from 7 to V. If it were possible to perform this operation without doing work, the total energy of the chamber would be the same as before, but the energy density of the radiation would have increased, since... 

We are now in a position to deduce the equation of state of a Prevost chamber, and to determine the change in the energy density of the radiation produced by an adiabatic change in volume. The work done in compressing the radiation by dV... 

Thus the energy density of the radiation in a Prevost chamber and the emissivity of a perfect black body are both proportional to the fourth power of the absolute temperature. The constant (T is of universal significance, and applies to all black bodies of whatever materials they may be composed. 

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