\(k=\cfrac{{m}_{e}c^2}{{r}^2_{e}}\)
It is natural then to consider damped forced motion and the associated resonance. But what would be the driving function? Photons on passing, blob the electrons up and down, but there is another driving force, an increase in temperature, \(T\). Increasing temperature increases random motions across a wide range of frequency. As a driving force, it acts over a wide range of frequency at the same time. The resulting response under such a driving force is also simultaneously over the same range of frequency. So, we have, based on classical mechanics,
\(x^{''}+2px^{'}+\omega^2_ox={T}cos(\omega t)\)
where \(\omega_o =\sqrt{\cfrac{k}{m_e}}\) and \(p=\sqrt{\cfrac{k_1}{2m_e}}\), \(k_1\) the damping factor is to be to be determined and given physical interpretation later. We will used a solution of the form,
\(x=\cfrac{T}{\sqrt{(2\omega p)^2+(\omega^2_o-\omega^2)^2}}cos(\omega t-\gamma)\)
where \(tan(\gamma)=\cfrac{2\omega p}{(\omega^2_o-\omega^2)}\)
The electrons at increase temperature are driven to an amplitude above it normal orbital position. It has a maximum potential energy of (\(\cfrac{1}{2}kx^2\)),
\(PE_{ e }=\cfrac { 1 }{ 2 } \cfrac { { m }_{ e }c^{ 2 } }{ { r }^{ 2 }_{ e } } .\left\{ \cfrac { T }{ \sqrt { (2\omega p)^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } } } \right\} ^{ 2 }\)
\(PE_{ e }=\cfrac { 1 }{ 2 } \cfrac { m_{ e }T^{ 2 }c^{ 2 } }{{ r }^{ 2 }_{ e }} .\cfrac { 1 }{ (2\omega p)^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } } \)
They fall from such amplitude and emits heavy light of energy (\({m}_{e}{v}^2\)) equals to this amount. Therefore,
\({m}_{e}v^2=PE_{ e }=\cfrac { 1 }{ 2 } \cfrac { m_{ e }T^{ 2 }c^{ 2 } }{{ r }^{ 2 }_{ e } } .\cfrac { 1 }{ (2\omega p)^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } } \)
\({ v }^{ 2 }=\cfrac { 1 }{ 2 } \cfrac { T^{ 2 }c^{ 2 } }{{ r }^{ 2 }_{ e } } .\cfrac { 1 }{ (2\omega p)^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } } \)
A plot \(\cfrac { 1 }{ (2\omega p)^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } }\) is shown below on the left. The actual parameter used are \({r}_{e}\) = 53e-12 m for hydrogen, \(c\) = 299792458 ms-1,
The actual graph plotted is, 10^7*1/((2*(x*100)*a)^2+(299.792458^2/53-(x*100)^2)^2) on the left, where \(a\) range from 0.5 to 20.5 in increment of 4. The y-axis is proportional to energy \(\) and the x-axis, \(\omega\). \(p\) is the damping factor per unit mass, which is related to the \(T\), equivalently the velocity of the electrons, and the damping force.
We can form the refractive index,
\(\cfrac { c }{ v } ={ n }_{ T }=\sqrt { 2 }\cfrac{1 }{ T } r_{ e } .\sqrt { \{ (2\omega p)^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 }\} } \)
A plot of \(n_T\) vs \(\omega\), ((2*(x*1000)*20)^2+(299.792458^2/53-(x*1000)^2)^2)^(1/2)*10^(-15) is shown below. The value of \(n_T\) increases with \(\omega\). This is the reverse of the case for spectra line where \(n_n\) decreases with increasing \(n\).
The amount of deflection, \(\theta\) at one medium interface \(n_T | n_p\) is
\(n_T sin(\theta_i)=n_p sin(\theta_r)\)and at the exit interface \(n_p | n_T\),
\(n_p sin(\Phi-\theta_r)=n_T sin(\theta_e)\),
where \(\Phi\) is the angle between the two interfaces. As \(n_T\) increases \(\theta_e\) decreases and the spectrum is sweep from right to left towards the normal. This is the reverse of spectra lines.
And at resonance, we have,
\(\omega=\sqrt{\omega^2_o-2p^2}\)
\(f=\cfrac{1}{2\pi}\sqrt{\cfrac{{c^2}}{{r}^2_{e}}-2p^2}\)
This frequency is however, not the frequency of the emitted light spectrum. The spectrum spread after the prism is the direct result of increasing speed of the electrons with increasing \(\omega\). The relative speed of the electrons just before and after the prism determines the amount of deflection and the direction of deflection relative to the normal at the point of incidence. High temperature drive the system over a wide spread of frequencies, one of which is this resonance value that corresponds to a high energy output (max. amplitude). The spread of velocities after the prism is related to the driving frequency due to the raised temperature, but they are not equivalent. One driving frequency gives raise to a particular amplitude of oscillation and so a value of velocity and a particular deflection point on the screen. All such driving frequencies are present at the same time with raised temperature. This driving frequency, \(\omega\) is not unlimited and neither is the \(x\), the amplitude of the oscillation. Both are valid up to point when the electron is pull from the nucleus, that the electron has gain enough energy to escape the attraction of the nucleus.
However, the resonance frequency, undamped, provide information to obtain \({r}_{e}\), the atomic radius of the element concerned, because,
\(\omega_o =\sqrt{\cfrac{k}{m_e}}\)
\(\omega^2_o =\cfrac{c^2}{r_e}\)
\({r}_{e}=\left (\cfrac{c}{\omega_o}\right)^2\)
In conclusion, the x-axis of all black body plot is wrong. The observed spectrum spread and the driving force frequencies and observed color of the spectrum are not the same thing.
The damping factor in relation to temperature needs to be further investigated. It seems \(p\) is inversely proportional to \(T\).
It should be noted that \(n_p\), the refractive index of the prism used should be high. When \(n_T\) is bigger than \(n_p\) the direction of deflection reverses, and the spectrum is refracted away from the normal at the point of incidence and towards the normal at the point of exit.
Have a nice day.
