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3.2 Osnovnye ponyatiya teorii ravnovesnogo izlucheniya

Pust' -- chislo fotonov v kletke fazovogo prostranstva, t.e. chislo zapolneniya, kotoroe bezrazmerno. Dlya ``yashika'' dannogo ob'ema mozhno govorit' o chisle sostoyanii. Esli sostoyanii mnogo, to chislo urovnei v ob'eme i v impul'snom ob'eme est'

$\displaystyle V\,d^3p/{(2\pi\hbar)}^3.$

Togda plotnost' chisla fotonov

$\displaystyle N={2\over {(2\pi\hbar)}^3}\,\int\limits_0^\infty nd^3p\,[$ sm $\displaystyle ^{-3}],$

(3.1)

gde koefficient 2 uchityvaet dva vozmozhnyh sostoyaniya s raznymi polyarizaciyami.

Izvestno, chto impul's fotona $\vec p$ svyazan s volnovym vektorom $\vec k$ formuloi De Broilya

$\displaystyle \vec p=\hbar\vec k.$

Podstavlyaya eto vyrazhenie v formulu (3.1), poluchim bolee naglyadnoe vyrazhenie dlya plotnosti chisla fotonov

$\displaystyle N=2\int nd^3k/{(2\pi)}^3.$

Poskol'ku $\vert k\vert=2\pi/\lambda$ , ochevidna razmernost' $N\sim 1/\lambda^3\sim$ $\mbox {sm}^{-3}$ i, krome togo, ischezaet postoyannaya Planka $\hbar$ , kotoraya byla vazhna tol'ko pri podschete chisla sostoyanii.

V sfericheskoi sisteme koordinat

$\displaystyle d^3p=p^2dpd\,\Omega\;\;\;(p=\vert\vec p\vert).$

V sluchae sfericheskoi simmetrii $\int d\Omega=4\pi$ i

$\displaystyle N={1\over {\pi^2 \hbar^3}}\,\int\limits_0^\infty np^2dp,$

gde

-- funkciya tol'ko ot $p:\;n=n(p)$ .

Poskol'ku energiya odnogo fotona

$\displaystyle E_{ph}=\hbar\,\omega=h\,\nu=cp,$

to plotnost' energii izlucheniya (fotonnogo gaza)

$\displaystyle \varepsilon_r={c\over {\pi^2\hbar^3}}\,\int\limits_0^\infty np^3dp.$

Krome plotnosti energii $\varepsilon_r$ vazhnoi harakteristikoi yavlyaetsya spektral'naya intensivnost' izlucheniya $F_\nu\;([F_\nu]=$ ergsm s Gc ster -- velichina, kotoraya pri otsutstvii poglosheniya ili rasseivaniya ne zavisit ot rasstoyaniya do istochnika. Iz nablyudenii vsegda poluchaetsya $F_\nu$ , a v fiziku dela vhodit chislo zapolneniya .

Poluchim svyaz' mezhdu $F_\nu$ i :

$\displaystyle F_\nu={2c\over {(2\pi\,\hbar)}^3}\,np^2\,{dp\over d\,\nu}\,cp={2np^3c\over {(2 \pi\,\hbar)}^2}={2nh\,\nu\over \lambda^2}$

(pri etom my ispol'zovali, chto $dp/d\,\nu=2\pi\hbar/c$ ). $F_\nu$ imeet smysl energii, prohodyashei cherez ploshadku $\lambda^2$ v edinicu vremeni v edinichnom intervale chastot, poetomu $[F_\nu]=$ erg

sm $^2\,$ c $\,$ Gc.

V obshem sluchae (i pri otsutstvii ravnovesiya) svyaz' mezhdu $\varepsilon_r$ i $F_\nu$ daetsya formuloi

$\displaystyle \varepsilon_r={1\over c}\,\int F_\nu\,d\,\nu\,d\Omega.$

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