Question

Ruby lasers have chromium atoms doped in an aluminum oxide crystal. The energy level diagram for chromium in a ruby is shown in Figure 30.64. What wavelength is emitted by a ruby laser?

Figure 30.64 Chromium atoms in an aluminum oxide crystal have these energy levels, one of which is metastable. This is the basis of a ruby laser. Visible light can pump the atom into an excited state above the metastable state to achieve a population inversion.

Short answer

The wavelength of the ruby laser is 691.3 nm.

Step-by-step solution

Determine the formulas:

Consider the formula for the energy of X ray photons as follows:

\({\bf{E = }}\frac{{{\bf{hc}}}}{{\bf{\lambda }}}\)

Here,

λ = Wavelength

E = Energy of the x-ray photons

h = Planck's constant

c = Speed of light

Consider the formula for the change in energy from one level to other as follows:

\({\bf{\Delta E = }}{{\bf{E}}_{\bf{i}}}{\bf{ - }}{{\bf{E}}_{\bf{f}}}\)

Here,

\(\Delta E = \)Change in energy

\(\begin{array}{c}{E_i} = {\rm{Initial Energy }}\\{E_f} = {\rm{Final Energy}}\end{array}\)

Calculate the energy released during transition

From energy level diagram,

\(\begin{array}{l}{E_m} = 1.79\;{\rm{eV}}\\{E_g} = 0\;{\rm{eV}}\end{array}\)

Energy released during transition is calculated:

\(\begin{array}{c}{E_m} - {E_g} = 1.79\;{\rm{eV - }}0\;{\rm{eV}}\\ = 1.79\;{\rm{eV}}\end{array}\)

Therefore, the energy released during transition is\(\Delta E = 1.79\;{\rm{eV}}\).

Calculate the wavelength emitted by ruby laser

Now, wavelength emitted by ruby laser is calculated using Planck's equation

Substituting the values in equation of energy and solve:

\(\begin{array}{l}E = \frac{{hc}}{\lambda }\\E = \frac{{\left( {6.626 \times {{10}^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}} \right)\left( {3 \times {{10}^8}\;\frac{{\rm{m}}}{{{\rm{sec}}}}} \right)}}{\lambda }\\\lambda = \frac{{\left( {6.626 \times {{10}^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}} \right)\left( {3 \times {{10}^8}\;\frac{{\rm{m}}}{{{\rm{sec}}}}} \right)}}{{1.79\;{\rm{eV}}}}\\\lambda = 691.3 \times {10^{ - 9}}\;{\rm{m}}\end{array}\)

Rewrite as:

\(\lambda = 691.3\;{\rm{nm}}\)

Therefore, the ruby laser's wavelength is 691.3nm.