Photon energy mode density

The photon energy mode density can be expressed as the product of Equations 3.28 and 3.32 giving the well-known Planck s formula (Planck s black body radiation distribution law) ... 

Note that p(v) denotes the photon mode density in units of m whereas pe(v) depicts the energy density in units of J m . ... 

Graphite exhibits strong second-order Raman-active features. These features are expected and observed in carbon tubules, as well. Momentum and energy conservation, and the phonon density of states determine, to a large extent, the second-order spectra. By conservation of energy hut = huty + hbi2, where bi and ill) (/ = 1,2) are, respectively, the frequencies of the incoming photon and those of the simultaneously excited normal modes. There is also a crystal momentum selection rule hV. = -I- q, where k and q/... 

A small dose of a soluble fast-decay positron-emitting artificial radioisotope (produced as needed not too far from the PET instrument 6C11, 8015, 9F18 or 37Rb82) is put into human tissue (e.g., blood) the positron typically travels about 1 mm, meets an electron from within the human body, and the pair decays into two y photons of energy 0.51 MeV each, within microseconds to nanoseconds. Two spin states are possible for the positron— electron ion pair before their annihilation singlet and triplet. The annihilation rate for the triplet state depends sensitively on the electron density of the body tissue. Two y counters are set in coincidence mode, and several hundred thousand coincidence events are used to provide valuable tissue information (in addition to a CT scan). 

Now consider the density p of 1-photon states. Because each of these states is characterized by one mode being in the first excited state while all others are in the ground state, the number of states is the same as the munber of modes and the required density of states per unit energy is given by pE of Eq. (3,20). Using this, together with (3.30) in (3.29) leads to... 

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