6.11 De­gen­er­acy Pres­sure

Ac­cord­ing to the pre­vi­ous sec­tions, elec­trons, be­ing fermi­ons, be­have in a way very dif­fer­ently from bosons. A sys­tem of bosons has very lit­tle en­ergy in its ground state, as all bosons col­lect in the spa­tial state of low­est en­ergy. Elec­trons can­not do so. At most two elec­trons can go into a sin­gle spa­tial state. A macro­scopic sys­tem of elec­trons must oc­cupy a gi­gan­tic num­ber of states, rang­ing from the low­est en­ergy state to states with many or­ders of mag­ni­tude more en­ergy.

As a re­sult, a free-elec­tron gas of $I$ non­in­ter­act­ing elec­trons ends up with an av­er­age en­ergy per elec­tron that is larger than of a cor­re­spond­ing sys­tem of bosons by a gi­gan­tic fac­tor of or­der $I^{2/3}$. That is all ki­netic en­ergy; all forces on the elec­trons are ig­nored in the in­te­rior of a free-elec­tron gas, so the po­ten­tial en­ergy can be taken to be zero.

Hav­ing so much ki­netic en­ergy, the elec­trons ex­ert a tremen­dous pres­sure on the walls of the con­tainer that holds them. This pres­sure is called de­gen­er­acy pres­sure. It ex­plains qual­i­ta­tively why the vol­ume of a solid or liq­uid does not col­lapse un­der nor­mally ap­plied pres­sures.

Of course, de­gen­er­acy pres­sure is a poorly cho­sen name. It is re­ally due to the fact that the en­ergy dis­tri­b­u­tion of elec­trons is not de­gen­er­ate, un­like that of bosons. Terms like ex­clu­sion-prin­ci­ple pres­sure or “Pauli pres­sure” would cap­ture the essence of the idea. So they are not ac­cept­able.

The mag­ni­tude of the de­gen­er­acy pres­sure for a free-elec­tron gas is

$$P_{\rm {d}} = {\textstyle\frac{2}{5}} \left(3\pi^2\right)^{2/3} \frac{\hbar^2}{2m_e} \left(\frac{I}{{\cal V}}\right)^{5/3}$$ (6.18)

This may be ver­i­fied by equat­ing the work $-P_{\rm {d}}{ \rm d}{\cal V}$ done when com­press­ing the vol­ume a bit to the in­crease in the to­tal ki­netic en­ergy ${\vphantom' E}^{\rm S}$ of the elec­trons:

$$- P_{\rm {d}} { \rm d}{\cal V}= {\rm d}{\vphantom' E}^{\rm S}$$

The en­ergy ${\vphantom' E}^{\rm S}$ is $I$ times the av­er­age en­ergy per elec­tron. Ac­cord­ing to sec­tion 6.10, that is $\frac35I$ times the Fermi en­ergy (6.16).

A ball­park num­ber for the de­gen­er­acy pres­sure is very in­struc­tive. Con­sider once again the ex­am­ple of a block of cop­per, with its va­lence elec­trons mod­eled as a free-elec­tron gas, Us­ing the same num­bers as in the pre­vi­ous sec­tion, the de­gen­er­acy pres­sure ex­erted by these va­lence elec­trons is found to be 40 10$\POW9,{9}$ Pa, or 40 GPa.

This tremen­dous out­ward pres­sure is bal­anced by the nu­clei that pull on elec­trons that try to leave the block. The de­tails are not that sim­ple, but elec­trons that try to es­cape re­pel other, eas­ily dis­placed, elec­trons that might aid in their es­cape, leav­ing the nu­clei un­op­posed to pull them back. Ob­vi­ously, elec­trons are not very smart.

It should be em­pha­sized that it is not mu­tual re­pul­sion of the elec­trons that causes the de­gen­er­acy pres­sure; all forces on the elec­trons are ig­nored in the in­te­rior of the block. It is the un­cer­tainty re­la­tion­ship that re­quires spa­tially con­fined elec­trons to have mo­men­tum, and the ex­clu­sion prin­ci­ple that ex­plodes the re­sult­ing amount of ki­netic en­ergy, cre­at­ing fast elec­trons that are as hard to con­tain as stu­dents on the day be­fore Thanks­giv­ing.

Com­pared to a 10$\POW9,{10}$ Pa de­gen­er­acy pres­sure, the nor­mal at­mos­pheric pres­sure of 10$\POW9,{5}$ Pa can­not add any no­tice­able fur­ther com­pres­sion. Pauli’s ex­clu­sion prin­ci­ple makes liq­uids and solids quite in­com­press­ible un­der nor­mal pres­sures.

How­ever, un­der ex­tremely high pres­sures, the elec­tron pres­sure can lose out. In par­tic­u­lar, for neu­tron stars the spa­tial elec­tron states col­lapse un­der the very weight of the mas­sive star. This is re­lated to the fact that the de­gen­er­acy pres­sure grows less quickly with com­pres­sion when the ve­loc­ity of the elec­trons be­comes rel­a­tivis­tic. (For very highly rel­a­tivis­tic par­ti­cles, the ki­netic en­ergy is not given in terms of the mo­men­tum $p$ by the New­ton­ian value ${\vphantom' E}^{\rm p}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $p^2$$\raisebox{.5pt}{$/$}$​$2m$, but by the Planck-Ein­stein re­la­tion­ship ${\vphantom' E}^{\rm p}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $pc$ like for pho­tons.) That makes a dif­fer­ence since grav­ity too in­creases with com­pres­sion. If grav­ity in­creases more quickly, all is lost for the elec­trons. For neu­tron stars, the col­lapsed elec­trons com­bine with the pro­tons in the star to form neu­trons. It is the de­gen­er­acy pres­sure of the neu­trons, also spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ fermi­ons but 2 000 times heav­ier, that car­ries the weight of a neu­tron star.

Key Points

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Be­cause typ­i­cal con­fined elec­trons have so much ki­netic en­ergy, they ex­ert a great de­gen­er­acy pres­sure on what is hold­ing them.

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This pres­sure makes it very hard to com­press liq­uids and solids sig­nif­i­cantly in vol­ume.

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Dif­fer­ently put, liq­uids and solids are al­most in­com­press­ible un­der typ­i­cal con­di­tions.