The Number of Resonators per unit Volume lying in Wavelength and Frequency

The Number of Resonators per unit Volume lying in the Wavelength Range \(\lambda \) to \(\lambda + d\lambda\) or Frequency Range \(\nu\) and \(\nu + d\nu \)

Let us consider the radiation to be enclosed in a hollow cubic enclosure, the walls of which are perfectly reflecting. The radiation is supposed to consist of the number of waves. These waves travel in all possible directions in the enclosure and undergo multiple reflections from the various walls of the enclosure. As a result of the superposition of the superposition of the reflected waves with the corresponding incident waves, stationary vibrations are formed with the walls as nodal planes. The formation of stationary vibrations can be understood as the vibrations of a stretched string with both end points fixed. We know that the end points of the string are two nodes of stationary vibrations and only certain discrete frequencies of vibrations are allowed. If l is the length of the string, the allowed wavelengths are given by

\begin{align*} \lambda &= \frac {2l}{n}: n= 1, 2, 3, 4, \dots \infty \\ \text {The corresponding allowed frequencies are} \\ \nu &= \frac {c}{\lambda } = \frac {nc}{2l}:\: n= 1, 2, 3, 4, \dots \infty \end{align*}

Every allowed frequency is called a mode of vibration. The allowed modes of vibration inside the cube can be calculated as of string but in this case the waves are confined to a three dimensional space.

Let us suppose each side of the cube be ‘a’ and three interesting edges of the cube as x, y, z axes of a Cartesian co-ordinate system. The direction cosines of the direction of propagation of the particular wave is \(\cos \alpha , \cos \beta \:\text {and} \cos \gamma \) and \(a\cos \alpha , a\cos \beta \:\text {and}\: a\cos \gamma \) be the direction projection of the edges of the cube on the direction of the propagation of waves. In this case only those waves will be allowed wavelength has to satisfy three conditions of the type of equation (1) in which l is replaced by \(a\cos \alpha , a\cos \beta \:\text {and} \: a\cos \gamma \) respectively. Thus an allowed wavelength \(\lambda \) must satisfy.

\(\lambda = \frac {2a\cos \alpha }{n_1},\: \lambda = \frac {2a\cos \beta }{n_2}\: \lambda = \frac {2a\cos \gamma }{n_3} \dots (2) \)

where \(n_1, n_2 \:\text {and}\: n_3\) are positive integers.

Now, according to trigonometric condition of direction cosines

$$\cos^\alpha + \cos ^2\beta + \cos ^\gamma = 1 \dots (3)$$

Substituting values of \(\cos \alpha , \cos \beta \: \text {and} \cos \gamma \) from equation (2) in (3), we get

\begin{align*} \frac {n_1^2\lambda ^2}{4a^2}+ \frac {n_2^2\lambda ^2}{4a^2}+ \frac {n_3^2\lambda ^2}{4a^2}+&= 1 \\ \text {or,} n_1^2 + n_2^2 + n_3^2 &= \frac {4a^2}{\lambda ^2} =\left (\frac {2a}{\lambda } \right )^2 \\ \text {But}\: \lambda = \frac {c}{\nu}, \text {we have} \\ n_1^2 + n_2^2 + n_3^2 &= \left (\frac {2a\nu}{c} \right )^2 \dots (4)\end{align*}

This equation gives the allowed frequencies inside the cube. According to this only those frequencies are allowed for which the equation (4) holds with various \(n_1, n_2, \text {and}\: n_3\). Each choice of the value of \(n_1, n_2 \text {and}\: n_3\) corresponds to a frequency. Total number of modes of vibration are total number of possible sets of \(n_1, n_2, n_3\).

For the number of modes of vibration within the frequency interval \(\nu \) and \(\nu + d\nu \) let us count the modes of vibration in an analogous two dimensional problem. The two dimensional analogue of equation (4) is

$$n_1^2 + n_2^2 = \left (\frac {2av}{c}\right )^2$$

If we plot a graph \(n_1\) along x-axis and \(n_2 \) along y-axis for a given value of \(\nu \) we get the point lying on the circle sine equation (5) represents a circle of radius \frac {2a\nu }{c} \). In the graph the abscissa and ordinates have been drawn corresponding to all possible integral value of \(n_1\) and \(n_2\) and hence a mode of vibration. The point on the circle correspond to frequency \(\nu \), while those lying inside the circle correspond to frequency less than \(\nu \). As only positive values of \(n_1\) and \(n_2\) allowed, we have to consider the intersections only in the positive quadrant. The lines divide the quadrant in a number of unit squares. As the area of every square is unity, therefore, the number of squares is equal to the area inside the quadrant of the circle. Thus the number of modes of vibration with frequencies less than n is equal to the area of quadrant of a circle represented by equation (5).

This area is \(\frac 14 \pi r^2 = \frac 14\pi \left ( \frac {2a\nu}{c} \right ) = \frac {\pi a^2 \nu ^2}{c^2}\).

Similarly, we can state that the number of modes of vibration with frequency \(nu \: \text {and} \: d\nu\) is equal to the area in the positive quadrant lying between the two circles radii \(\frac {2a\nu }{c}\) and \(\frac {2a(\nu + d\nu}{c}\). The required area is

\begin{align*} &= \frac 14\pi \left [ \left (\frac {2a(\nu + d\nu )}{c}\right )^2 - \left (\frac {2a\nu }{c}\right )^2\right] \\ &= \frac 14 \pi \frac {8a^2\nu d\nu}{c^2} \\ &= \frac {2\pi a^2\nu d\nu}{c^2} \\ \end{align*}

Now considering the case of three dimensions, we see that the circle is replaced by a sphere and a mesh of squares is replaced by mesh of cubes. The number of modes of vibrations within frequency range \(\nu \) and \(\nu + d\nu \) is now equal to the value of \(\frac18 th \) of a spherical shell with the radii \(\frac {2a\nu }{a} \: \text {and}\: \frac {2a(\nu + d\nu)}{c} \).

The required volume is

\begin{align*} &= \frac 18.\frac 43 \pi \left [\left (\frac {2a(\nu + d\nu)}{c}\right )^3 - \left (\frac {2a\nu}{c} \right )^3\right ] \\ &= \frac 18 (4\pi )\left (\frac {2a\nu }{c} \right )^2.\frac {2ad\nu}{c} \\ &= \frac {4\pi \nu ^2}{c^3}d\nu .a^3 \\ &= \frac {4\pi \nu ^2 d\nu}{c^3}V \\ \end{align*}

Since a³ = V = volume of the cube. Thus the numbers of modes of vibration per unit volume with in frequency range \(\nu \) and \(\nu + d\nu \) =\(frac {4\pi \nu ^2 d\nu}{c^3}\)

Now the black body radiations travel with the velocity of light and are transverse in character unlike sound waves in the string with are longitudinal waves.

In the case of transverse waves there are two possible polarizations for each wave. Therefore, the modes of vibrations of transverse waves are double as for longitudinal waves. Therefore for black body radiations of electromagnetic waves, the modes of vibration within the frequency range \(\nu \) and \(\nu + d\nu \) are

\begin{align*} &= \frac {8\pi }{c^3}. \left (\frac {c^2}{\lambda ^2} \right )\left (- \frac {c}{\lambda ^2} d\lambda \right ) \:\:\: [\text {since}\: \nu = \frac {c}{\lambda }\: \text {and} d\nu = \frac {c}{\lambda ^2d m\lambda }\\ &= \frac {8\pi }{\lambda ^4}d\lambda \end{align*}

According to Plank each resonator the number of resonators per unit volume in the frequency range \(\nu \) and \(\nu + d\nu \), from equation (6), is

$$\frac {8\pi \nu^2}{c^3}d\nu $$

In the form of wavelength is

$$ \frac {8\pi}{\lambda ^4}d\lambda $$

Bibliography

S.S. Singhal, J.P. Agarwal, Satya Prakash. heat and thermodynamics and statistical physics. pragati prakashan, 2010.

—. Heat and Thermodynamics and Statistical Physics. Pragati Prakashan, 2010.

Vatsyayan, Dr. Rakesh Ranjan. Refresher Course in Physics. kathmandu: Surya Book Traders, 2015.