Monochromatic (Line) Radiation

Abstract

Classical electron radius. Consider the orbital motion of an electron of mass, m, and charge, e, about a nucleus of charge Z e. Equating the Coulomb force of attraction to the force from centripetal acceleration we obtain $$\frac{{Z\,{e^2}}}{{{r^2}}} = \frac{{m\,{v^2}}}{r}$$ (2-1) where r is the radius of the orbit, and v is the velocity of the electron. Solving for r, $$r = {r_0}{\left( {\frac{c}{v}} \right)^2}Z$$ (2-2) where the classical electron radius $${r_0} = {{{e^2}} \mathord{\left/ {\vphantom {{{e^2}} {\left( {m{c^2}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {m{c^2}} \right)}} \approx 2.818 \times {10^{ - 13}}\,cm$$ (2-3) Radius of the first Bohr orbit. Following Bohr (1913) we may assume that the angular momentum of the electron is quantized so that $$m\,v\,r = \frac{{h\,n}}{{2\pi }}$$ (2-4) where h ≈ 6.625 × 10−27 erg sec is Planck’s constant and n is an integer. Using Eqs. (2-1) and (2-4) we obtain the radius r n of the nth Bohr orbit. $${r_n} = {a_0}\frac{{{n^2}}}{Z}$$ (2-5) where the radius of the first Bohr orbit of hydrogen (Z =1) is $${a_0} = \frac{{{h^2}}}{{4{\pi ^2}m\,{e^2}}} \approx 0.529 \times {10^{ - 8}}\,cm$$ (2-6) Line frequency. For atomic or molecular radiation resulting from the transition between two levels of energy, E m and E n, the frequency of radiation, v mn , is given by (Planck, 1910; Bohr, 1913) $${v_{mn}} = {{\left| {{E_m} - {E_n}} \right|} \mathord{\left/ {\vphantom {{\left| {{E_m} - {E_n}} \right|} h}} \right. \kern-\nulldelimiterspace} h}$$ (2-7) where | | denotes the absolute value, and h is Planck’s constant.