91Ó°ÊÓ

Equation for Bose-Einstein condensation :

$N=\underset{\text{all}s}{∑}\frac{1}{e^{\frac{\left({\mathrm{Î\mu }}_1-\mathrm{μ}\right)}{kT}}-1}$ (Equation-1)

Sound waves behave almost like light waves. Each made of oscillation has a set of equally spaced energy levels, with the limit of energy equal to:

$\mathrm{Î\mu }=nhf$

The relation between Fermi temperatures and Fermi energy is.

$t{\mathrm{Î\mu }}_F=kT$

Here, $k$is the Boltzmann constant:

The relation between the chemical potential and Fermi energy is,

$\mathrm{μ}=c{\mathrm{Î\mu }}_f$

we get,

$N=\underset{\text{alls}}{∑}\frac{1}{e^{\frac{\left(nhf-c{c}_F\right)}{t{t}_F}}}\left(\text{here,}{\mathrm{Î\mu }}_F=hf\right)$

$=\underset{\text{alls}s}{∑}\frac{1}{e^{\frac{\left(n{e}_p-c_{r_r}\right)}{t{\mathrm{Î\mu }}_{r_r}}-1}}$

$=\underset{\text{alls}s}{∑}\frac{1}{e^{\frac{(n-c)}{t}-1}}$

Degeneracy of a level'$n\text{'}$is

$g\left(n\right)dn=\frac{1}{2}(n+1)(n+2)$

Thus the Bose-Einstein distribution function:

$N=\underset{0}{\overset{∞}{∑}}\frac{1}{2}(n+1)(n+2)\frac{d\mathrm{Î\mu }}{e^{\frac{(\mathrm{Î\mu }-\mathrm{μ})}{kT}-1}}$

we have,

$h\mathrm{Ó\neg }=1\left(\text{(}h\text{bar*omega *}\right)$

$kt=15\left(\text{(*Boltzmann constant*temperature *}\right)$

$N=2000(*\text{number of atoms}*)$

$\text{normsum}\left[N_{-}\right]:=sum\left[dist\left[c\right]*\frac{(n+1)*(n+2)}{2},\{m,0,sumlim\}\right]$

In an isotropic harmonic trap of 3-dimensional box, occupational number of particles can be attained by using this mathematic relation.

$\text{Listplot}\left[\text{occnum,plot range->}\right\{0,300\left\}\right];\left({}^{*}\text{plot occupation numbers}*\right)$

$\text{Listplot}\left[\text{Table}\right\{n,occ[n,-10.536,15]\},\{n,0,175\left\}\right]$

From the graph we can see that the occupancy peak is at height of. At this temperature the number of particles in the ground state is unity.

The following mathematical instruction accustomed plot the occupancy graph for $t=14$. At this temperature the occupancy within the state is tiny and roughly less than $1.5$.