Solid angle element

The photoelectric cross-section o is defined as the one-electron transition probability per unit-time, with a unit incident photon flux per area and time unit from the state to the state T en of Eq. (2). If the direction of electron emission relative to the direction of photon propagation and polarization are specified, then the differential cross-section do/dQ can be defined, given the emission probability within a solid angle element dQ into which the electron emission occurs. Emission is dependent on the angular properties of T in and Wfin therefore, in photoelectron spectrometers for which an experimental set-up exists by which the angular distribution of emission can be scanned (ARPES, see Fig. 2), important information may be collected on the angular properties of the two states. In this case, recorded emission spectra show intensities which are determined by the differential cross-section do/dQ. The total cross-section a (which is important when most of the emission in all direction is collected), is... 

The scattering probability into the solid angle element dQ about the direction of the final momentum p is... 

Since we have scattering from wave packets uniformly distributed over 6, the total scattering probability into the solid angle element dCl is / dbP(dil <— b). If this number is multiplied by the (relative) flux density of molecules in the beam, we get the flux of molecules that show up in dfl. Thus the cross-section is simply... 

The flux of the radiation scattered, 7(0, X), into a solid angle element AT2 may be expressed in general terms in the following way ... 

A certain direction in space is determined by two angular coordinates ft and p, Fig. 5.3. ft is the polar anglemeasured outwards from the surface normal ft = 0) and is the circumferential angle with an arbitrarily assumed position for p = 0. The radiative flux, that falls on a small area d.4n at a distance r from the surface element dA, perpendicular to the radiation direction, Fig. 5.4, is proportional to the solid angle element... 

The small area d/ln in Fig. 5.4 and with that the solid angle element duj result... 

Fig. 5.4 Radiative flux d24> into a solid angle element do) in the direction of the polar angle ft and the circumferential angle p...

Here d2radiation flow emitted by the surface element into the solid angle element dee in the direction of the angle /3 and total intensity L has units W/m2sr it belongs to the directional total quantities and represents the part of the emissive power falling into a certain solid angle element. 

The description of the direction and wavelength distribution of the radiation flow is provided by radiation quantities that are defined analogous to those for the emission of radiation. For the radiation flow d3in, from a solid angle element dee in the direction of the angles / and

surface element d/1, and which only contains the radiation in a wavelength interval dA, we can make a statement analogous to (5.4)... 

The distribution function Kx(X,/3,incident spectral intensity, is defined by this. It describes the wavelength and directional distribution of the radiation flow falling onto the irradiated surface element. Like the corresponding quantity Lx for the emission of radiation, Kx is defined with the projection d 4p = cos/SdAl of the irradiated surface element perpendicular to the direction of the incident radiation, Fig. 5.12. The SI units of Kx are W/(m2pmsr) the relationship to the wavelength interval dA and the solid angle element dw is also clear from this. 

Just as in 5.1.3 we will consider a radiation flow d3surface element d/1 and only... 

If, however, (5.33) is integrated over all wavelengths then the absorbed portion of the total radiative power from a solid angle element dw is obtained. This gives... 

We will now consider an enclosure with a body that has any radiation properties, Fig. 5.21. Thermodynamic equilibrium means that this body must also emit exactly the same amount of energy in every solid angle element and in every wavelength interval as it absorbs from the hollow enclosure radiation. It therefore holds for the emitted radiative power that... 

This is the law from G.R. Kirchhoff [5.5] Any body at a given temperature T emits, in every solid angle element and in every wavelength interval, the same radiative power as it absorbs there from the radiation of a black body (= hollow enclosure radiation) having the same temperature. Therefore, a close relationship exists between the emission and absorption capabilities. This can be more simply expressed using this sentence A good absorber of thermal radiation is also a good emitter. 

With e x G, the radiation flow d3wavelength interval dA, from the solid angle element dtp highlighted in Fig. 5.72, can be calculated. According to (5.4), the defining equation for the spectral intensity, it holds for this that... 

Fig. 5.72 Gas space with surface element di4 and associated solid angle element do)...

The spectral emissivity e)(G from (5.187) covers the radiation coming from the entire gas space, which is incident on a surface element dA = dA% in Fig. 5.77. The solid angle element daii which appears in... 

Fig. 5.77 Gas space with surface element dAi, which receives radiation from the solid angle element duu which is bounded by the surface element dAi...

Fa the number of molecules scattered into the solid angle element df2 about Cl per unit time (s),... 

The total scattering cross section is another statistical quantity that can be derived from the differential cross section, defined as the integral of aA ] g) over all solid angle elements, hence ... 

The measurable intensity /(i ), or differential cross section da/dfi, scattered into the solid angle element dO, with scattering angle , is then given by... 

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