4.36 - An Introduction to Thermal Physics

Question

An apparent limit on the temperature achievable by laser cooling is reached when an atom's recoil energy from absorbing or emitting a single photon is comparable to its total kinetic energy. Make a rough estimate of this limiting temperature for rubidium atoms that are cooled using laser light with a wavelength of 780 nm.

Step-by-Step Solution

Answer

Limiting temp is $1.2\times {10}^{-7} \mathrm{K}$

Step 1: Given Information

Wavelength is $\lambda =780 \mathrm{nm}$.
Mass of atom is $85\mathrm{u}$.

Step2: Explanation

de Broglie wavelength formula is written as
$p=\frac{h}{\lambda }$
Kinetic energy in terms of momentum , can be written as:
$K=\frac{p^2}{2m}$
Kinetic energy in terms of temperature can be written as:
$K=\frac{3}{2}kT$

Where 

K=kinetic Energy
P=momentum
M= mass
k= Boltzmann constant

T=Temperature

Now substitute values and calculate 

$$\begin{gathered} K=\frac{p^2}{2m}=\frac{3}{2}kT \\ \Rightarrow \frac{p^2}{2m}=\frac{3}{2}kT \\ \Rightarrow \frac{p^2}{3m}=kT \end{gathered}$$

$$\begin{gathered} \because p=\frac{h}{\lambda } \\ \Rightarrow \frac{{\left(\frac{h}{\lambda }\right)}^2}{3m}=kT \\ \Rightarrow T=\frac{h^2}{3m{\lambda }^2} \end{gathered}$$

$\Rightarrow T=\frac{{\left(6.63\times {10}^{-34}\mathrm{Js}\right)}^2}{3\times 85\times \left(1.66\times {10}^{-27} \mathrm{kg}\right){\left(780\times {10}^{-9} \mathrm{m}\right)}^2\left(1.38064852\times {10}^{-23}{ \mathrm{m}}^2 \mathrm{kg}{ \mathrm{s}}^{-2}{ \mathrm{K}}^{-1}\right)}$

So T= 1.2 x 10 ^-7 K