6.9 Ground State of a Sys­tem of Elec­trons

So far, only the physics of bosons has been dis­cussed. How­ever, by far the most im­por­tant par­ti­cles in physics are elec­trons, and elec­trons are fermi­ons. The elec­tronic struc­ture of mat­ter de­ter­mines al­most all en­gi­neer­ing physics: the strength of ma­te­ri­als, all chem­istry, elec­tri­cal con­duc­tion and much of heat con­duc­tion, power sys­tems, elec­tron­ics, etcetera. It might seem that nu­clear en­gi­neer­ing is an ex­cep­tion be­cause it pri­mar­ily deals with nu­clei. How­ever, nu­clei con­sist of pro­tons and neu­trons, and these are spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ fermi­ons just like elec­trons. The analy­sis be­low ap­plies to them too.

Non­in­ter­act­ing elec­trons in a box form what is called a free-elec­tron gas. The va­lence elec­trons in a block of metal are of­ten mod­eled as such a free-elec­tron gas. These elec­trons can move rel­a­tively freely through the block. As long as they do not try to get off the block, that is. Sure, a va­lence elec­tron ex­pe­ri­ences re­pul­sions from the sur­round­ing elec­trons, and at­trac­tions from the nu­clei. How­ever, in the in­te­rior of the block these forces come from all di­rec­tions and so they tend to av­er­age away.

Of course, the elec­trons of a free elec­tron gas are con­fined. Since the term “non­in­ter­act­ing-elec­tron gas” would be cor­rect and un­der­stand­able, there were few pos­si­ble names left. So free-elec­tron gas it was.

At ab­solute zero tem­per­a­ture, a sys­tem of fermi­ons will be in the ground state, just like a sys­tem of bosons. How­ever, the ground state of a macro­scopic sys­tem of elec­trons, or any other type of fermi­ons, is dra­mat­i­cally dif­fer­ent from that of a sys­tem of bosons. For a sys­tem of bosons, in the ground state all bosons crowd to­gether in the sin­gle-par­ti­cle state of low­est en­ergy. That was il­lus­trated in fig­ure 6.2. Not so for elec­trons. The Pauli ex­clu­sion prin­ci­ple al­lows only two elec­trons to go into the low­est en­ergy state; one with spin up and the other with spin down. A sys­tem of $I$ elec­trons needs at least $I$$\raisebox{.5pt}{$/$}$​2 spa­tial states to oc­cupy. Since for a macro­scopic sys­tem $I$ is a some gi­gan­tic num­ber like 10$\POW9,{20}$, that means that a gi­gan­tic num­ber of states needs to be oc­cu­pied.

Fig­ure 6.11: Ground state of a sys­tem of non­in­ter­act­ing elec­trons, or other fermi­ons, in a box.
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In the sys­tem ground state, the elec­trons crowd into the $I$$\raisebox{.5pt}{$/$}$​2 spa­tial states of low­est en­ergy. Now the en­ergy of the spa­tial states in­creases with the dis­tance from the ori­gin in wave num­ber space. There­fore, the elec­trons oc­cupy the $I$$\raisebox{.5pt}{$/$}$​2 states clos­est to the ori­gin in this space. That is shown to the left in fig­ure 6.11. Every red spa­tial state is oc­cu­pied by 2 elec­trons, while the black states are un­oc­cu­pied. The oc­cu­pied states form an oc­tant of a sphere. Of course, in a real macro­scopic sys­tem, there would be many more states than a fig­ure could show.

The spec­trum to the right in fig­ure 6.11 shows the oc­cu­pied en­ergy lev­els in red. The width of the spec­trum in­di­cates the den­sity of states, the num­ber of sin­gle-par­ti­cle states per unit en­ergy range.


Key Points
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Non­in­ter­act­ing elec­trons in a box are called a free-elec­tron gas.

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In the ground state, the $I$$\raisebox{.5pt}{$/$}$​2 spa­tial states of low­est en­ergy are oc­cu­pied by two elec­trons each. The re­main­ing states are empty.

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The ground state ap­plies at ab­solute zero tem­per­a­ture.