Question

what will be the Fermi temperature .if the number density is increased eight times ?

Answer 1

Solution:

Fermi temperature by definition

=

ℏ

2

2

(3

2)

2/3

Thus

~

2/3

When the number density increased to eight times

~(8)

2/3 = 4

2/3 = 40

Answer: Fermi temperature increased to four times.

Answer 2

The definition of Fermi temperature is,

             ${ T }_{ f }\quad =\quad \left( \frac { { h }_{ 2 } }{ 2m{ K }_{ b } }\right) { \left( 3{ \pi}^{ 2 }n \right)}^{ \frac { 2 }{ 3 }}\quad \longrightarrow \left( 1 \right)$

Where n = number density

When number density is increased 8 times which means n = 8n

Substitute in (1),

              ${ T }_{ f }\quad =\quad \left( \frac { { h }_{ 2 } }{ 2m{ K}_{ B } }\right){\left( 3{ \pi}^{ 2 }8n \right)}^{ 2/3 }$

Let  $A\quad =\quad \left( \frac { { h }_{ 2 } }{ 2m{ K }_{ B } } \right) { \left( 3{ { \pi}^{ 2 } } \right)}^{ 2/3 }$

              ${ T }_{ f }\quad =\quad A{ \left( 8n \right)}^{ 2/3 }$

              ${ T }_{ f }\quad =\quad A\left( { 8 }^{ 2/3 }{ n }^{ 2/3 }\right)$

              ${ T }_{ f }\quad =\quad A4{ \left( n \right)}^{ 2/3 }$

              ${ T }_{ f }\quad =\quad 4A{ \left( n \right)}^{ 2/3 }$

Hence if number density is increased by eight times, then the Fermi temperature increases by 4 times.