Physics Asked by catalogue_number on August 5, 2020

Consider (for example) non-relativistic electron-impact ionisation, in which an effectively stationary atom in state $I$ is struck by an electron of momentum ${p_1}$, producing an atom in state $J$ with two additional electrons, momenta ${p_2}, {p’_2}$. The electron-impact ionisation transition rate is then expressed as $dw(final|initial) = dw(p_2, p_2′; B| p_1; A)$

The total rate is then given, in the presence of an electron distribution $f(p)$, as

$$Gamma_{eii} = N_e n_A int d^3p_1 f(p_1) dw(p_2, p_2′; B| p_1; A)$$

where $N_e$ is electron population and $n_A$ the number density of atoms in state $A$. Here, the interpretation $dw = frac{p_1}{m_e}sigma(p_1)$ makes clear, intuitive sense – $sigma(p)$ is the effective area of an atom as seen by an electron of momentum $p$. However, I’m confused by the detailed-balance equation for this process, for which the relation reads

$$ dw(p_2, p_2′; B| p_1; A) d^3p_1 g(p_1) = dw(p_1; A | p_2, p_2′; B) d^3p_2 g(p_2) d^3p’_2 g(p’_2)$$

I’ve seen the three-body rate expressed as

$$Gamma_{eii} = N^2_e n_A int d^3p’_2d^3p_2 f(p_2)f(p’_2) dw(p_1; A | p_2, p_2′; B)\

= N^2_e n_A int d^3p_1 f(p_2)f(p’_2) dw(p_2, p_2′; B | p_1; A ) frac{g(p_1)}{g(p_2)g(p’_2)} $$

The second line of this looks almost nonsensical – the integration in the non-$p_1$ variables must somehow be implicily carried out via $dw$, which conceals a number of Dirac deltas.

What exactly does one mean by $dsigma$ and $dw$ – are they differential forms on the manifold of all possible scattering parameters, and in this context, where do the density of states arise from in detailed balance? (i.e. why are they present in the detailed-balance equation, but absent from the total scattering rate integral?)

Get help from others!

Recent Answers

- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP