10.3 Met­als

Met­als are unique in the sense that there is no true mol­e­c­u­lar equiv­a­lent to the way the atoms are bound to­gether in met­als. In a metal, the va­lence elec­trons are shared on crys­tal scales, rather than be­tween pairs of atoms. This and sub­se­quent sec­tions will dis­cuss what this re­ally means in terms of quan­tum me­chan­ics.

10.3.1 Lithium

The sim­plest metal is lithium. Be­fore ex­am­in­ing solid lithium, first con­sider once more the free lithium atom. Fig­ure 10.4 gives a more re­al­is­tic pic­ture of the atom than the sim­plis­tic analy­sis of chap­ter 5.9 did. The atom is re­ally made up of two tightly bound elec­trons in ${\left\vert\rm {1s}\right\rangle}$ states very close to the nu­cleus, plus a loosely bound third va­lence elec­tron in an ex­pan­sive ${\left\vert\rm {2s}\right\rangle}$ state. The core, con­sist­ing of the nu­cleus and the two closely bound 1s elec­trons, re­sem­bles an he­lium atom that has picked up an ad­di­tional pro­ton in its nu­cleus. It will be re­ferred to as the atom core. As far as the 2s elec­tron is con­cerned, this en­tire atom core is not that much dif­fer­ent from an hy­dro­gen nu­cleus: it is com­pact and has a net charge equiv­a­lent to one pro­ton.

Fig­ure 10.4: The lithium atom, scaled more cor­rectly than be­fore.

One ob­vi­ous ques­tion is then why un­der nor­mal cir­cum­stances lithium is a solid metal and hy­dro­gen is a thin gas. The quan­ti­ta­tive dif­fer­ence is that a sin­gle-charge core has a fa­vorite dis­tance at which it would like to hold its elec­tron, the Bohr ra­dius. In the hy­dro­gen atom, the elec­tron is about at the Bohr ra­dius, and hy­dro­gen holds onto it tightly. It is will­ing to share elec­trons with one other hy­dro­gen atom, but af­ter that, it is sat­is­fied. It is not look­ing for any other hy­dro­gen mol­e­cules to share elec­trons with; that would weaken the bond it al­ready has. On the other hand, the 2s elec­tron in the lithium atom is only loosely at­tached and read­ily given up or shared among mul­ti­ple atoms.

Fig­ure 10.5: Body-cen­tered-cu­bic (BCC) struc­ture of lithium.

Now con­sider solid lithium. A per­fect lithium crys­tal would look as sketched in fig­ure 10.5. The atom cores arrange them­selves in a reg­u­lar, re­peat­ing, pat­tern called the “ crys­tal struc­ture.” As in­di­cated in the fig­ure by the thick red lines, you can think of the to­tal crys­tal vol­ume as con­sist­ing of many iden­ti­cal lit­tle cubes called “(unit) cells.”. There are atom cores at all eight cor­ners of these cubes and there is an ad­di­tional core in the cen­ter of the cu­bic cell. In solid me­chan­ics, this arrange­ment of po­si­tions is re­ferred to as the “body-cen­tered cu­bic” (BCC) lat­tice. The crys­tal “ba­sis” for lithium is a sin­gle lithium atom, (or atom core, re­ally); if you put a sin­gle lithium atom at every point of the BCC lat­tice, you get the com­plete lithium crys­tal.

Around the atom cores, the 2s elec­trons form a fairly ho­mo­ge­neous elec­tron den­sity dis­tri­b­u­tion. In fact, the atom cores get close enough to­gether that a typ­i­cal 2s elec­tron is no closer to the atom core to which it sup­pos­edly be­longs than to the sur­round­ing atom cores. Un­der such con­di­tions, the model of the 2s elec­trons be­ing as­so­ci­ated with any par­tic­u­lar atom core is no longer re­ally mean­ing­ful. It is bet­ter to think of them as be­long­ing to the solid as a whole, mov­ing freely through it like an elec­tron gas.

Un­der nor­mal con­di­tions, bulk lithium is “poly-crys­talline,” mean­ing that it con­sists of many mi­cro­scop­i­cally small crys­tals, or “grains,“ each with the above BCC struc­ture. The “grain bound­aries“ where dif­fer­ent crys­tals meet are cru­cial to un­der­stand the me­chan­i­cal prop­er­ties of the ma­te­r­ial, but not so much to un­der­stand its elec­tri­cal or heat prop­er­ties, and their ef­fects will be ig­nored. Only per­fect crys­tals will be dis­cussed.

Key Points

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Lithium can mean­ing­fully be thought of as an atom core, with a net charge of one pro­ton, and a 2s va­lence elec­tron around it.

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In the solid, the cores arrange them­selves into a body-cen­tered cu­bic (BCC) lat­tice.

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The 2s elec­trons form an elec­tron gas around the cores.

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Nor­mally the solid, like other solids, does not have the same crys­tal lat­tice through­out, but con­sists of mi­cro­scopic grains, each crys­talline, (i.e. with its lat­tice ori­ented its own way).

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The grain struc­ture is crit­i­cal for me­chan­i­cal prop­er­ties like strength and plas­tic­ity. But that is an­other book.

10.3.2 One-di­men­sional crys­tals

Even the quan­tum me­chan­ics of a per­fect crys­tal like the lithium one de­scribed above is not very sim­ple. So it is a good idea to start with an even sim­pler crys­tal. The eas­i­est ex­am­ple would be a crys­tal con­sist­ing of only two atoms, but two lithium atoms do not make a lithium crys­tal, they make a lithium mol­e­cule.

Fig­ure 10.6: Fully pe­ri­odic wave func­tion of a two-atom lithium crys­tal.

For­tu­nately, there is a dirty trick to get a crys­tal with only two atoms: as­sume that na­ture keeps re­peat­ing it­self as in­di­cated in fig­ure 10.6. Math­e­mat­i­cally, this is called us­ing pe­ri­odic bound­ary con­di­tions. It as­sumes that af­ter mov­ing to­wards the left over a dis­tance called the pe­riod, you are back at the same point as you started, as if you are walk­ing around in a cir­cle and the pe­riod is the cir­cum­fer­ence.

Of course, this is an out­ra­geous as­sump­tion. If na­ture re­peats it­self at all, and that is doubt­ful at the time of this writ­ing, it would be on a cos­mo­log­i­cal scale, not on the scale of two atoms. But the fact re­mains that if you make the as­sump­tion that na­ture re­peats, the two-atom model gives a much bet­ter de­scrip­tion of the math­e­mat­ics of a true crys­tal than a two-atom mol­e­cule would. And if you add more and more atoms, the point where na­ture re­peats it­self moves fur­ther and fur­ther away from the typ­i­cal atom, mak­ing it less and less of an is­sue for the lo­cal quan­tum me­chan­ics.

Key Points

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Pe­ri­odic bound­ary con­di­tions are very ar­ti­fi­cial.

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Still, for crys­tal lat­tices, pe­ri­odic bound­ary con­di­tions of­ten work very well.

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And no­body is go­ing to put any real grain bound­aries into any ba­sic model of solids any­way.

10.3.3 Wave func­tions of one-di­men­sional crys­tals

To de­scribe the en­ergy eigen­states of the elec­trons in one-di­men­sion­al crys­tals in sim­ple terms, a fur­ther as­sump­tion must be made: that the de­tailed in­ter­ac­tions be­tween the elec­trons can be ig­nored, ex­cept for the ex­clu­sion prin­ci­ple. Try­ing to cor­rectly de­scribe the com­plex in­ter­ac­tions be­tween the large num­bers of elec­trons found in a macro­scopic solid is sim­ply im­pos­si­ble. And it is not re­ally such a bad as­sump­tion as it may ap­pear. In a metal, elec­tron wave func­tions over­lap greatly, and when they do, elec­trons see other elec­trons in all di­rec­tions, and ef­fects tend to can­cel out. The equiv­a­lent in clas­si­cal grav­ity is where you go down far be­low the sur­face of the earth. You would ex­pect that grav­ity would be­come much more im­por­tant now that you are sur­rounded by big amounts of mass at all sides. But they tend to can­cel each other out, and grav­ity is ac­tu­ally re­duced. Lit­tle grav­ity is left at the cen­ter of the earth. It is not rec­om­mended as a va­ca­tion spot any­way due to ex­ces­sive pres­sure and tem­per­a­ture.

In any case, it will be as­sumed that for any sin­gle elec­tron, the net ef­fect of the atom cores and smeared-out sur­round­ing 2s elec­trons pro­duces a pe­ri­odic po­ten­tial that near every core re­sem­bles that of an iso­lated core. In par­tic­u­lar, if the atoms are spaced far apart, the po­ten­tial near each core is ex­actly the one of a free lithium atom core. For an elec­tron in this two atom crys­tal, the in­tu­itive eigen­func­tions would then be where it is around ei­ther the first or the sec­ond core in the 2s state, (or rather, tak­ing the pe­ri­od­ic­ity into ac­count, around every first or every sec­ond core in each pe­riod.) Al­ter­na­tively, since these two states are equiv­a­lent, quan­tum me­chan­ics al­lows the elec­tron to hedge its bets and to be about each of the two cores at the same time with some prob­a­bil­ity.

Fig­ure 10.7: Flip-flop wave func­tion of a two-atom lithium crys­tal.

But as soon as the atoms are close enough to start no­tice­ably af­fect­ing each other, only two true en­ergy eigen­func­tions re­main, and they are ones in which the elec­tron is around both cores with equal prob­a­bil­ity. There is one eigen­func­tion that is ex­actly the same around both of the atom cores. This eigen­func­tion is sketched in fig­ure 10.6; it is pe­ri­odic from core to core, rather than merely from pair of cores to pair of cores. The sec­ond eigen­func­tion is the same from core to core ex­cept for a change of sign, call it a flip-flop eigen­func­tion. It is shown in fig­ure 10.7. Since the grey-scale elec­tron prob­a­bil­ity dis­tri­b­u­tion only shows the mag­ni­tude of the wave func­tion, it looks pe­ri­odic from atom to atom, but the ac­tual wave func­tion is only the same af­ter mov­ing along two atoms.

To avoid the grey fad­ing away, the shown wave func­tions have not been nor­mal­ized; the dark­ness level is as if the 2s elec­trons of both the atoms are in that state.

As long as the atoms are far apart, the wave func­tions around each atom closely re­sem­ble the iso­lated-atom ${\left\vert\rm {2s}\right\rangle}$ state. But when the atoms get closer to­gether, dif­fer­ences start to show up. Note for ex­am­ple that the flip-flop wave func­tion is ex­actly zero half way in be­tween two cores, while the fully pe­ri­odic one is not. To in­di­cate the de­vi­a­tions from the true free-atom ${\left\vert\rm {2s}\right\rangle}$ wave func­tion, par­en­thet­i­cal su­per­scripts will be used.

Fig­ure 10.8: Wave func­tions of a four-atom lithium crys­tal. The ac­tual pic­ture is that of the fully pe­ri­odic mode.

A one-di­men­sion­al crys­tal made up from four atoms is shown in fig­ure 10.8. Now there are four en­ergy eigen­states. The en­ergy eigen­state that is the same from atom to atom is still there, as is the flip-flop one. But there is now also an en­ergy eigen­state that changes by a fac­tor ${\rm i}$ from atom to atom, and one that changes by a fac­tor $\vphantom{0}\raisebox{1.5pt}{$-$}$${\rm i}$. They change more slowly from atom to atom than the flip-flop one: it takes two atom dis­tances for them to change sign. There­fore it takes a dis­tance of four atoms, rather than two, for them to re­turn to the same val­ues.

Key Points

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The elec­tron en­ergy eigen­func­tions in a metal like lithium ex­tend over the en­tire crys­tal.

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If the cores are rel­a­tively far apart, near each core the en­ergy eigen­func­tion of an elec­tron still re­sem­bles the 2s state of the free lithium atom.

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How­ever, the mag­ni­tude near each core is of course much less, since the elec­tron is spread out over the en­tire crys­tal.

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Also, from core to core, the wave func­tion changes by a fac­tor of mag­ni­tude one.

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The ex­treme cases are the fully pe­ri­odic wave func­tion that changes by a fac­tor one (stays the same) from core to core, ver­sus the flip-flop mode that changes sign com­pletely from one core to the next.

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The other eigen­func­tions change by an amount in be­tween these two ex­tremes from core to core.

10.3.4 Analy­sis of the wave func­tions

There is a pat­tern to the wave func­tions of one-di­men­sion­al crys­tals as dis­cussed in the pre­vi­ous sub­sec­tion. First of all, while the spa­tial en­ergy eigen­func­tions of the crys­tal are dif­fer­ent from those of the in­di­vid­ual atoms, their num­ber is the same. Four free lithium atoms would each have one ${\left\vert\rm {2s}\right\rangle}$ spa­tial state to put their one 2s elec­tron in. Put them in a crys­tal, and there are still four spa­tial states to put the four 2s elec­trons in. But the four spa­tial states in the crys­tal are no longer sin­gle atom states; each now ex­tends over the en­tire crys­tal. The atoms share all the elec­trons. If there were eight atoms, the eight atoms would share the eight 2s elec­trons in eight pos­si­ble crys­tal-wide states. And so on.

To be very pre­cise, a sim­i­lar thing is true of the in­ner 1s elec­trons. But since the ${\left\vert\rm {1s}\right\rangle}$ states re­main well apart, the ef­fects of shar­ing the elec­trons are triv­ial, and de­scrib­ing the 1s elec­trons as be­long­ing pair-wise to a sin­gle lithium nu­cleus is fine. In fact, you may re­call that the an­ti­sym­metriza­tion re­quire­ment of elec­trons re­quires every elec­tron in the uni­verse to be slightly present in every oc­cu­pied state around every atom. Ob­vi­ously, you would not want to con­sider that in the ab­sence of a non­triv­ial need.

The rea­son that the en­ergy eigen­func­tions take the form shown in fig­ure 10.8 is rel­a­tively sim­ple. It fol­lows from the fact that the Hamil­ton­ian com­mutes with the “trans­la­tion op­er­a­tor” that shifts the en­tire wave func­tion over one atom spac­ing $\vec{d}$. Af­ter all, be­cause the po­ten­tial en­ergy is ex­actly the same af­ter such a trans­la­tion, it does not make a dif­fer­ence whether you eval­u­ate the en­ergy be­fore or af­ter you shift the wave func­tion over.

Now com­mut­ing op­er­a­tors have a com­mon set of eigen­func­tions, so the en­ergy eigen­func­tions can be taken to be also eigen­func­tions of the trans­la­tion op­er­a­tor. The eigen­value must have mag­ni­tude one, since pe­ri­odic wave func­tions can­not change in over­all mag­ni­tude when trans­lated. So the eigen­value de­scrib­ing the ef­fect of an atom-spac­ing trans­la­tion on an en­ergy eigen­func­tion can be writ­ten as $e^{{\rm i}2\pi\nu}$ with $\nu$ a real num­ber. (The fac­tor $2\pi$ does noth­ing ex­cept rescale the value of $\nu$. Ap­par­ently, crys­tal­lo­g­ra­phers do not even put it in. This book does so that you do not feel short-changed be­cause other books have fac­tors $2\pi$ and yours does not.)

This can be ver­i­fied for the ex­am­ple en­ergy eigen­func­tions shown in fig­ure 10.8. For the fully pe­ri­odic eigen­func­tion $\nu$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, mak­ing the trans­la­tion eigen­value $e^{{\rm i}2\pi\nu}$ equal to one. So this eigen­func­tion is mul­ti­plied by one un­der a trans­la­tion by one atom spac­ing $d$: it is the same af­ter such a trans­la­tion. For the flip-flop mode, $\nu$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12$; this mode changes by $e^{{\rm i}\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$1 un­der a trans­la­tion over an atom spac­ing $d$. That means that it changes sign when trans­lated over an atom spac­ing $d$. For the two in­ter­me­di­ate eigen­func­tions $\nu$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pm\frac14$, so, us­ing the Euler for­mula (2.5), they change by fac­tors $e^{\pm{\rm i}\pi/2}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pm{\rm i}$ for each trans­la­tion over a dis­tance $d$.

In gen­eral, for an $J$-​atom pe­ri­odic crys­tal, there will be $J$ val­ues of $\nu$ in the range $-\frac12$ $\raisebox{.3pt}{$<$}$ $\nu$ $\raisebox{-.3pt}{$\leqslant$}$ $\frac12$. In par­tic­u­lar for an even num­ber of atoms $J$:

$$\nu = \frac{j}{J} \quad\mbox{for}\quad j = -\frac{J}{2}+1,\... ...+2,\; -\frac{J}{2}+3,\; \ldots,\; \frac{J}{2}-1,\; \frac{J}{2}$$

Note that for these val­ues of $\nu$, if you move over $J$ atom spac­ings, $e^{{\rm i}2\pi\nu{J}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 as it should; ac­cord­ing to the im­posed pe­ri­odic bound­ary con­di­tions, the wave func­tions must be the same af­ter $J$ atoms. Also note that it suf­fices for $j$ to be re­stricted to the range $\vphantom{0}\raisebox{1.5pt}{$-$}$$J$$\raisebox{.5pt}{$/$}$​2 $\raisebox{.3pt}{$<$}$ $j$ $\raisebox{-.3pt}{$\leqslant$}$ $J$$\raisebox{.5pt}{$/$}$​2, hence $-\frac12$ $\raisebox{.3pt}{$<$}$ $\nu$ $\raisebox{-.3pt}{$\leqslant$}$ $\frac12$: if $j$ is out­side that range, you can al­ways add or sub­tract a whole mul­ti­ple of $J$ to bring it back in that range. And chang­ing $j$ by a whole mul­ti­ple of $J$ does ab­solutely noth­ing to the eigen­value $e^{{\rm i}2\pi\nu}$ since $e^{{\rm i}2\pi{J}/J}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^{{\rm i}2\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.

10.3.5 Flo­quet (Bloch) the­ory

Math­e­mat­i­cally it is awk­ward to de­scribe the en­ergy eigen­func­tions piece­wise, as fig­ure 10.8 does. To ar­rive at a bet­ter way, it is help­ful first to re­place the ax­ial Carte­sian co­or­di­nate $z$ by a new crys­tal co­or­di­nate $u$ de­fined by

$$\fbox{$\displaystyle z {\hat k}= u \vec d $}$$ (10.2)

where $\vec{d}$ is the vec­tor shown in fig­ure 10.8 that has the length of one atom spac­ing $d$. Ma­te­r­ial sci­en­tists call this vec­tor the “prim­i­tive trans­la­tion vec­tor” of the crys­tal lat­tice. Prim­i­tive vec­tor for short.

The ad­van­tage of the crys­tal co­or­di­nate $u$ is that if it changes by one unit, it changes the $z$-​po­si­tion by ex­actly one atom spac­ing. As noted in the pre­vi­ous sub­sec­tion, such a trans­la­tion should mul­ti­ply an en­ergy eigen­func­tion by a fac­tor $e^{{\rm i}2\pi\nu}$. A con­tin­u­ous func­tion that does that is the ex­po­nen­tial $e^{{\rm i}2\pi\nu{u}}$. And that means that if you fac­tor out that ex­po­nen­tial from the en­ergy eigen­func­tion, what is left does not change un­der the trans­la­tion; it will be pe­ri­odic on atom scale. In other words, the en­ergy eigen­func­tions can be writ­ten in the form

$$\pp//// = e^{{\rm i}2\pi\nu u} \pp{\rm p}////$$

where $\pp{\rm {p}}////$ is a func­tion that is pe­ri­odic on the atom scale $d$; it is the same in each suc­ces­sive in­ter­val $d$.

This re­sult is part of what is called “Flo­quet the­ory:”

If the Hamil­ton­ian is pe­ri­odic of pe­riod $d$, the en­ergy eigen­func­tions are not in gen­eral pe­ri­odic of pe­riod $d$, but they do take the form of ex­po­nen­tials times func­tions that are pe­ri­odic of pe­riod $d$.

In physics, this re­sult is known as “Bloch’s the­o­rem,” and the Flo­quet-type wave func­tion so­lu­tions are called “Bloch func­tions” or Bloch waves, be­cause Flo­quet was just a math­e­mati­cian, and the physi­cists’ hero is Bloch, the physi­cist who suc­ceeded in do­ing it too, half a cen­tury later. {N.20}.

The pe­ri­odic part $\pp{\rm {p}}////$ of the en­ergy eigen­func­tions is not the same as the ${\left\vert\rm {2s}\right\rangle}^{(.)}$ states of fig­ure 10.8, be­cause $e^{{\rm i}2\pi\nu{u}}$ varies con­tin­u­ously with the crys­tal po­si­tion $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $ud$, un­like the fac­tors shown in fig­ure 10.8. How­ever, since the mag­ni­tude of $e^{{\rm i}2\pi\nu{u}}$ is one, the mag­ni­tudes of $\pp{\rm {p}}////$ and the ${\left\vert\rm {2s}\right\rangle}^{(.)}$ states are the same, and there­fore, so are their grey scale elec­tron prob­a­bil­ity pic­tures.

It is of­ten more con­ve­nient to have the en­ergy eigen­func­tions in terms of the Carte­sian co­or­di­nate $z$ in­stead of the crys­tal co­or­di­nate $u$, writ­ing them in the form

$$\fbox{$\displaystyle \pp{k}//// = e^{{\rm i}k z}\pp{{\rm p... ...th $\pp{{\rm p},k}////$ periodic on the atom scale $d$} $} %$$ (10.3)

The con­stant $k$ in the ex­po­nen­tial is called the wave num­ber, and sub­scripts $k$ have been added to $\pp////$ and $\pp{\rm {p}}////$ just to in­di­cate that they will be dif­fer­ent for dif­fer­ent val­ues of this wave num­ber. Since the ex­po­nen­tial must still equal $e^{{\rm i}2\pi\nu{u}}$, clearly the wave num­ber $k$ is pro­por­tional to $\nu$. In­deed, sub­sti­tut­ing $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $ud$ into $e^{{\rm i}{k}z}$, $k$ can be traced back to be

$$\fbox{$\displaystyle k = \nu D \qquad D = \frac{2\pi}{d} \... ...el{\raisebox{-.7pt}{$\leqslant$}}{\textstyle\frac{1}{2}} $} %$$ (10.4)

10.3.6 Fourier analy­sis

As the pre­vi­ous sub­sec­tion ex­plained, the en­ergy eigen­func­tions in a crys­tal take the form of a Flo­quet ex­po­nen­tial times a pe­ri­odic func­tion $\pp{{\rm {p}},k}////$. This pe­ri­odic part is not nor­mally an ex­po­nen­tial. How­ever, it is gen­er­ally pos­si­ble to write it as an in­fi­nite sum of ex­po­nen­tials:

$$\fbox{$\displaystyle \pp{{\rm p},k}//// = \sum_{m=-\infty}... ...m i}k_m z} \qquad k_m = m D \mbox{ for $m$ an integer} $} %$$ (10.5)

where the $c_{km}$ are con­stants whose val­ues will de­pend on $x$ and $y$, as well as on $k$ and the in­te­ger $m$.

Writ­ing the pe­ri­odic func­tion $\pp{{\rm {p}},k}////$ as such a sum of ex­po­nen­tials is called “Fourier analy­sis,” af­ter an­other French math­e­mati­cian. That it is pos­si­ble fol­lows from the fact that these ex­po­nen­tials are the atom-scale-pe­ri­odic eigen­func­tions of the $z$-​mo­men­tum op­er­a­tor $p_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\hbar\partial$$\raisebox{.5pt}{$/$}$​${\rm i}\partial{z}$, as is eas­ily ver­i­fied by straight sub­sti­tu­tion. Since the eigen­func­tions of an Her­mit­ian op­er­a­tor like $p_z$ are com­plete, any atom-scale-pe­ri­odic func­tion, in­clud­ing $\pp{{\rm {p}},k}////$, can be writ­ten as a sum of them. See also {D.8}.

10.3.7 The rec­i­p­ro­cal lat­tice

As the pre­vi­ous two sub­sec­tions dis­cussed, the en­ergy eigen­func­tions in a one-di­men­sion­al crys­tal take the form of a Flo­quet ex­po­nen­tial $e^{{\rm i}{k}z}$ times a pe­ri­odic func­tion $\pp{{\rm {p}},k}////$. That pe­ri­odic func­tion can be writ­ten as a sum of Fourier ex­po­nen­tials $e^{{\rm i}{k_m}z}$. It is a good idea to de­pict all those $k$-​val­ues graph­i­cally, to keep them apart. That is done in fig­ure 10.9.

Fig­ure 10.9: Rec­i­p­ro­cal lat­tice of a one-di­men­sional crys­tal.

The Fourier $k$ val­ues, $k_m$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mD$ with $m$ an in­te­ger, form a lat­tice of points spaced a dis­tance $D$ apart. This lat­tice is called the rec­i­p­ro­cal lat­tice. The spac­ing of the rec­i­p­ro­cal lat­tice, $D$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\pi$$\raisebox{.5pt}{$/$}$​$d$, is pro­por­tional to the rec­i­p­ro­cal of the atom spac­ing $d$ in the phys­i­cal lat­tice. Since on a macro­scopic scale the atom spac­ing $d$ is very small, the spac­ing of the rec­i­p­ro­cal lat­tice is very large.

The Flo­quet $k$ value, $k$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\nu{D}$ with $-\frac12$ $\raisebox{.3pt}{$<$}$ $\nu$ $\raisebox{-.3pt}{$\leqslant$}$ $\frac12$, is some­where in the grey range in fig­ure 10.9. This range is called the first “Bril­louin zone.’ It is an in­ter­val, a unit cell if you want, of length $D$ around the ori­gin. The first Bril­louin zone is par­tic­u­larly im­por­tant in the the­ory of solids. The fact that the Flo­quet $k$ value may be as­sumed to be in it is but one rea­son.

To be pre­cise, the Flo­quet $k$ value could in prin­ci­ple be in an in­ter­val of length $D$ around any wave num­ber $k_m$, not just the ori­gin, but if it is, you can shift it to the first Bril­louin zone by split­ting off a fac­tor $e^{{\rm i}{k_m}z}$ from the Flo­quet ex­po­nen­tial $e^{{\rm i}{k}z}$. The $e^{{\rm i}{k_m}z}$ can be ab­sorbed in a re­de­f­i­n­i­tion of the Fourier se­ries for the pe­ri­odic part $\pp{{\rm {p}},k}////$ of the wave func­tion, and what is left of the Flo­quet $k$ value is in the first zone. Of­ten it is good to do so, but not al­ways. For ex­am­ple, in the analy­sis of the free-elec­tron gas done later, it is crit­i­cal not to shift the $k$ value to the first zone be­cause you want to keep the (there triv­ial) Fourier se­ries in­tact.

The first Bril­louin zone are the points that are clos­est to the ori­gin on the $k$-​axis, and sim­i­larly the sec­ond zone are the points that are sec­ond clos­est to the ori­gin. The points in the in­ter­val of length $D$$\raisebox{.5pt}{$/$}$​2 in be­tween $k_{-1}$ and the first Bril­louin zone make up half of the sec­ond Bril­louin zone: they are clos­est to $k_{-1}$, but sec­ond clos­est to the ori­gin. Sim­i­larly, the other half of the sec­ond Bril­louin zone is given by the points in be­tween $k_1$ and the first Bril­louin zone. In one di­men­sion, the bound­aries of the Bril­louin zone frag­ments are called the “Bragg points.” They are ei­ther rec­i­p­ro­cal lat­tice points or points half way in be­tween those.

10.3.8 The en­ergy lev­els

Va­lence band. Con­duc­tion band. Band gap. Crys­tal. Lat­tice. Ba­sis. Unit cell. Prim­i­tive vec­tor. Bloch wave. Fourier analy­sis. Rec­i­p­ro­cal lat­tice. Bril­louin zones. These are the jar­gon of solid me­chan­ics; now they have all been de­fined. (Though cer­tainly not fully dis­cussed.) But jar­gon is not physics. The phys­i­cally in­ter­est­ing ques­tion is what are the en­ergy lev­els of the en­ergy eigen­func­tions.

Fig­ure 10.10: Schematic of en­ergy bands.

For the two-atom crys­tal of fig­ures 10.6 and 10.7, the an­swer is much like that for the hy­dro­gen mol­e­c­u­lar ion of chap­ter 4.6 and hy­dro­gen mol­e­cule of chap­ter 5.2. In par­tic­u­lar, when the atom cores are far apart, the ${\left\vert\rm {2s}\right\rangle}^{(.)}$ states are the same as the free lithium atom wave func­tion ${\left\vert\rm {2s}\right\rangle}$. In ei­ther the fully pe­ri­odic or the flip-flop mode, the elec­tron is with 50% prob­a­bil­ity in that state around each of the two cores. That means that at large spac­ing $d$ be­tween the cores, the en­ergy is the 2s free lithium atom en­ergy, whether it is the fully pe­ri­odic or flip-flop mode. That is shown in the left graph of fig­ure 10.10.

When the dis­tance $d$ be­tween the atoms de­creases so that the 2s wave func­tions start to no­tice­ably over­lap, things change. As the same left graph in fig­ure 10.10 shows, the en­ergy of the flip-flop state in­creases, but that of the fully pe­ri­odic state ini­tially de­creases. The rea­sons for the lat­ter are sim­i­lar to those that gave the sym­met­ric hy­dro­gen mol­e­c­u­lar ion and hy­dro­gen mol­e­cule states lower en­ergy. In par­tic­u­lar, the elec­trons pick up more ef­fec­tive space to move in, de­creas­ing their un­cer­tainty-prin­ci­ple de­manded ki­netic en­ergy. Also, when the elec­tron clouds start to merge, the re­pul­sion be­tween elec­trons is re­duced, al­low­ing the elec­trons to lose po­ten­tial en­ergy by get­ting closer to the nu­clei of the neigh­bor­ing atoms. (Note how­ever that the sim­ple model used here would not faith­fully re­pro­duce that since the re­pul­sion be­tween the elec­trons is not cor­rectly mod­eled.)

Next con­sider the case of a four-atom crys­tal, as shown in the sec­ond graph of fig­ure 10.10. The fully pe­ri­odic and flip flop states are un­changed, and so are their en­er­gies. But there are now two ad­di­tional states. Un­like the fully pe­ri­odic state, these new states vary from atom, but less rapidly than the flip flop mode. As you would then guess, their en­ergy is some­where in be­tween that of the fully pe­ri­odic and flip-flop states. Since the two new states have equal en­ergy, it is shown as a dou­ble line in 10.10. The third graph in that fig­ure shows the en­ergy lev­els of an 8 atom crys­tal, and the fi­nal graph that of a 24 atom crys­tal. When the num­ber of atoms in­creases, the en­ergy lev­els be­come denser and denser. By the time you reach a one hun­dredth of an inch, one-mil­lion atom one-di­men­sion­al crys­tal, you can safely as­sume that the en­ergy lev­els within the band have a con­tin­u­ous, rather than dis­crete dis­tri­b­u­tion.

Now re­call that the Pauli ex­clu­sion prin­ci­ple al­lows up to two elec­trons in a sin­gle spa­tial en­ergy state. Since there are an equal num­ber of spa­tial states and elec­trons, that means that the elec­trons can pair up in the low­est half of the states. The up­per states will then be un­oc­cu­pied. Fur­ther, the ac­tual sep­a­ra­tion dis­tance be­tween the atoms will be the one for which the to­tal en­ergy of the crys­tal is small­est. The en­ergy spec­trum at this ac­tual sep­a­ra­tion dis­tance is found in­side the van­ish­ingly nar­row ver­ti­cal frame in the right­most graph of fig­ure 10.10. It shows that lithium forms a metal with a par­tially-filled band.

The par­tially filled band means that lithium con­ducts elec­tric­ity well. As was al­ready dis­cussed ear­lier in chap­ter 6.20, an ap­plied volt­age does not af­fect the band struc­ture at a given lo­ca­tion. For an ap­plied volt­age to do that, it would have to drop an amount com­pa­ra­ble to volts per atom. The cur­rent that would flow in a metal un­der such a volt­age would va­por­ize the metal in­stantly. Cur­rent oc­curs be­cause elec­trons get ex­cited to states of slightly higher en­ergy that pro­duce mo­tion in a pref­er­en­tial di­rec­tion.

10.3.9 Merg­ing and split­ting bands

The ex­pla­na­tion of elec­tri­cal con­duc­tion in met­als given in the pre­vi­ous sub­sec­tion is in­com­plete. It in­cor­rectly seems to show that beryl­lium, (and sim­i­larly other met­als of va­lence two,) is an in­su­la­tor. Two va­lence elec­trons per atom will com­pletely fill up all 2s states. With all states filled, there would be no pos­si­bil­ity to ex­cite elec­trons to states of slightly higher en­ergy with a pref­er­en­tial di­rec­tion of mo­tion. There would be no such states. All states would be red in fig­ure 10.10, so noth­ing could change.

What is miss­ing is con­sid­er­a­tion of the 2p atom states. When the atoms are far enough apart not to af­fect each other, the 2p en­ergy lev­els are a bit higher than the 2s ones and not in­volved. How­ever, as fig­ure 10.11 shows, when the atom spac­ing de­creases to the ac­tual one in a crys­tal, the widen­ing bands merge to­gether. With this in­flux of 300% more states, va­lence-two met­als have plenty of free states to ex­cite elec­trons to. Beryl­lium is ac­tu­ally a bet­ter con­duc­tor than lithium.

Fig­ure 10.11: Schematic of merg­ing bands.

Hy­dro­gen is a more com­pli­cated story. Solid hy­dro­gen con­sists of mol­e­cules and the at­trac­tions be­tween dif­fer­ent mol­e­cules are weak. The proper model of hy­dro­gen is not a se­ries of equally spaced atoms, but a se­ries of pairs of atoms joined into mol­e­cules, and with wide gaps be­tween the mol­e­cules. When the two atoms in a sin­gle mol­e­cule are brought to­gether, the en­ergy varies with dis­tance be­tween the atoms much like the left graph in fig­ure 10.10. The wave func­tion that is the same for the two atoms in the cur­rent sim­ple model cor­re­sponds to the nor­mal co­va­lent bond in which the elec­trons are sym­met­ri­cally shared; the flip-flop func­tion that changes sign de­scribes the anti-bond­ing state in which the two elec­trons are an­ti­sym­met­ri­cally shared. In the ground state, both elec­trons go into the state cor­re­spond­ing to the co­va­lent bond, and the anti-bond­ing state stays empty. For mul­ti­ple mol­e­cules, each of the two states turns into a band, but since the in­ter­ac­tions be­tween the mol­e­cules are weak, these two bands do not fan out much. So the en­ergy spec­trum of solid hy­dro­gen re­mains much like the left graph in fig­ure 10.10, with the bot­tom curve be­com­ing a filled band and the top curve an empty one. An equiv­a­lent way to think of this is that the 1s en­ergy level of hy­dro­gen does not fan out into a sin­gle band like the 2s level of lithium, but into two half bands, since there are two spac­ings in­volved; the spac­ing be­tween the atoms in a mol­e­cule and the spac­ing be­tween mol­e­cules. In any case, be­cause of the band gap en­ergy re­quired to reach the empty up­per half 1s band, hy­dro­gen is an in­su­la­tor.

10.3.10 Three-di­men­sional met­als

The ideas of the pre­vi­ous sub­sec­tions gen­er­al­ize to­wards three-di­men­sion­al crys­tals in a rel­a­tively straight­for­ward way.

Fig­ure 10.12: A prim­i­tive cell and prim­i­tive trans­la­tion vec­tors of lithium.

As the lithium crys­tal of fig­ure 10.12 il­lus­trates, in a three-di­men­sion­al crys­tal there are three “prim­i­tive trans­la­tion vec­tors.” The three-di­men­sion­al Carte­sian po­si­tion ${\skew0\vec r}$ can be writ­ten as

$$\fbox{$\displaystyle {\skew0\vec r}= u_1 \vec d_1 + u_2 \vec d_2 + u_3 \vec d_3 $} %$$ (10.6)

where if any of the crys­tal co­or­di­nates $u_1$, $u_2$, or $u_3$ changes by ex­actly one unit, it pro­duces a phys­i­cally com­pletely equiv­a­lent po­si­tion.

Note that the vec­tors $\vec{d}_1$ and $\vec{d}_2$ are two bot­tom sides of the cu­bic unit cell de­fined ear­lier in fig­ure 10.5. How­ever, $\vec{d}_3$ is not the ver­ti­cal side of the cube. The rea­son is that prim­i­tive trans­la­tion vec­tors must be cho­sen to al­low you to reach any point of the crys­tal from any equiv­a­lent point in whole steps. Now $\vec{d}_1$ and $\vec{d}_2$ al­low you to step from any point in a hor­i­zon­tal plane to any equiv­a­lent point in the same plane. But if $\vec{d}_3$ was ver­ti­cally up­wards like the side of the cu­bic unit cell, step­ping with $\vec{d}_3$ would miss every sec­ond hor­i­zon­tal plane. With $\vec{d}_1$ and $\vec{d}_2$ de­fined as in fig­ure 10.12, $\vec{d}_3$ must point to an equiv­a­lent point in an im­me­di­ately ad­ja­cent hor­i­zon­tal plane, not a hor­i­zon­tal plane far­ther away.

De­spite this re­quire­ment, there are still many ways of choos­ing the prim­i­tive trans­la­tion vec­tors other than the one shown in fig­ure 10.12. The usual way is to choose all three to ex­tend to­wards ad­ja­cent cube cen­ters. How­ever, then it gets more dif­fi­cult to see that no lat­tice point is missed when step­ping around with them.

The par­al­lelepiped shown in fig­ure 10.12, with sides given by the prim­i­tive trans­la­tion vec­tors, is called the “prim­i­tive cell.” It is the small­est build­ing block that can be stacked to­gether to form the to­tal crys­tal. The cu­bic unit cell from fig­ure 10.5 is not a prim­i­tive cell since it has twice the vol­ume. The cu­bic unit cell is in­stead called the “con­ven­tional cell.”

Since the prim­i­tive vec­tors are not unique, the prim­i­tive cell they de­fine is not ei­ther. These prim­i­tive cells are purely math­e­mat­i­cal quan­ti­ties; an ar­bi­trary choice for the small­est sin­gle vol­ume el­e­ment from which the to­tal crys­tal vol­ume can be build up. The ques­tion sug­gests it­self whether it would not be pos­si­ble to de­fine a prim­i­tive cell that has some phys­i­cal mean­ing; whose de­f­i­n­i­tion is unique, rather than ar­bi­trary. The an­swer is yes, and the un­am­bigu­ously de­fined prim­i­tive cell is called the “Wigner-Seitz cell.” The Wigner-Seitz cell around a lat­tice point is the vicin­ity of lo­ca­tions that are closer to that lat­tice point than to any other lat­tice point.

Fig­ure 10.13: Wigner-Seitz cell of the BCC lat­tice.

Fig­ure 10.13 shows the Wigner-Seitz cell of the BCC lat­tice. To the left, it is shown as a wire frame, and to the right as an opaque vol­ume el­e­ment. To put it within con­text, the atom around which this Wigner-Seitz cell is cen­tered was also put in the cen­ter of a con­ven­tional cu­bic unit cell. Note how the Wigner-Seitz prim­i­tive cell is much more spher­i­cal than the par­al­lelepiped-shaped prim­i­tive cell shown in fig­ure 10.12. The out­side sur­face of the Wigner-Seitz cell con­sists of hexag­o­nal planes on which the points are just on the verge of get­ting closer to a cor­ner atom of the con­ven­tional unit cell than to the cen­ter atom, and of squares on which the points are just on the verge of get­ting closer to the cen­ter atom of an ad­ja­cent con­ven­tional unit cell. The squares are lo­cated within the faces of the con­ven­tional unit cell.

The rea­son that the en­tire crys­tal vol­ume can be build up from Wigner-Seitz cells is sim­ple: every point must be clos­est to some lat­tice point, so it must be in some Wigner-Seitz cell. When a point is equally close to two near­est lat­tice points, it is on the bound­ary where ad­ja­cent Wigner-Seitz cells meet.

Turn­ing to the en­ergy eigen­func­tions, they can now be taken to be eigen­func­tions of three trans­la­tion op­er­a­tors; they will change by some fac­tor $e^{{\rm i}2\pi\nu_1}$ when trans­lated over $\vec{d}_1$, by $e^{{\rm i}2\pi\nu_2}$ when trans­lated over $\vec{d}_2$, and by $e^{{\rm i}2\pi\nu_3}$ when trans­lated over $\vec{d}_3$. All that just means that they must take the Flo­quet (Bloch) func­tion form

$$\pp//// = e^{{\rm i}2\pi(\nu_1u_1+\nu_2u_2+\nu_3u_3)} \pp{\rm p}////,$$

where $\pp{\rm {p}}////$ is pe­ri­odic on atom scales, ex­actly the same af­ter one unit change in any of the crys­tal co­or­di­nates $u_1$, $u_2$ or $u_3$.

It is again of­ten con­ve­nient to write the Flo­quet ex­po­nen­tial in terms of nor­mal Carte­sian co­or­di­nates. To do so, note that the re­la­tion giv­ing the phys­i­cal po­si­tion ${\skew0\vec r}$ in terms of the crys­tal co­or­di­nates $u_1$, $u_2$, and $u_3$,

$${\skew0\vec r}= u_1 \vec d_1 + u_2 \vec d_2 + u_3 \vec d_3$$

can be in­verted to give the crys­tal co­or­di­nates in terms of the phys­i­cal po­si­tion, as fol­lows:

$$\fbox{$\displaystyle u_1 = \frac1{2\pi} \vec D_1 \cdot {\s... ...}\quad u_3 = \frac1{2\pi} \vec D_3 \cdot {\skew0\vec r} $} %$$ (10.7)

(Again, fac­tors $2\pi$ have been thrown in merely to fully sat­isfy even the most de­mand­ing quan­tum me­chan­ics reader.) To find the vec­tors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$, sim­ply solve the ex­pres­sion for ${\skew0\vec r}$ in terms of $u_1$, $u_2$, and $u_3$ us­ing lin­ear al­ge­bra pro­ce­dures. In par­tic­u­lar, they turn out to be the rows of the in­verse of ma­trix $(\vec{d}_1,\vec{d}_2,\vec{d}_3)$.

If you do not know lin­ear al­ge­bra, it can be done geo­met­ri­cally: if you dot the ex­pres­sion for ${\skew0\vec r}$ above with $\vec{D}_1$$\raisebox{.5pt}{$/$}$​$2\pi$, you must get $u_1$; for that to be true, the first three con­di­tions be­low are re­quired:

$$\fbox{$\displaystyle \begin{array}{lll} \vec d_1 \cdot \v... ... D_3 = 0, & \vec d_3 \cdot \vec D_3 = 2\pi. \end{array} $} %$$ (10.8)

The sec­ond set of three equa­tions is ob­tained by dot­ting with $\vec{D}_2$$\raisebox{.5pt}{$/$}$​$2\pi$ to get $u_2$ and the third by dot­ting with $\vec{D}_3$$\raisebox{.5pt}{$/$}$​$2\pi$ to get $u_3$. From the last two equa­tions in the first row, it fol­lows that vec­tor $\vec{D}_1$ must be or­thog­o­nal to both $\vec{d}_2$ and $\vec{d}_3$. That means that you can get $\vec{D}_1$ by first find­ing the vec­to­r­ial cross prod­uct of vec­tors $\vec{d}_2$ and $\vec{d}_3$ and then ad­just­ing the length so that $\vec{d}_1\cdot\vec{D}_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\pi$. In sim­i­lar ways, $\vec{D}_2$ and $\vec{D}_3$ may be found.

If the ex­pres­sions for the crys­tal co­or­di­nates are sub­sti­tuted into the ex­po­nen­tial part of the Bloch func­tions, the re­sult is

$$\fbox{$\displaystyle \pp{\vec k}//// = e^{{\rm i}{\vec k}\... ...vec k}= \nu_1 \vec D_1 + \nu_2 \vec D_2 + \nu_3 \vec D_3 $} %$$ (10.9)

So, in three di­men­sions, a wave num­ber $k$ be­comes a “wave num­ber vec­tor” ${\vec k}$.

Just like for the one-di­men­sion­al case, the pe­ri­odic func­tion $\pp{{\rm {p}},{\vec k}}////$ too can be writ­ten in terms of ex­po­nen­tials. Con­verted from crys­tal to phys­i­cal co­or­di­nates, it gives:

$$\fbox{$\displaystyle \begin{array}[b]{c} \displaystyle \... ..., $m_2$, and $m_3$ integers}\strut^{\strut} \end{array} $} %$$ (10.10)

If these wave num­ber vec­tors ${\vec k}_{\vec{m}}$ are plot­ted three-di­men­sion­ally, it again forms a lat­tice called the “rec­i­p­ro­cal lat­tice,” and its prim­i­tive vec­tors are $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$. Re­mark­ably, the rec­i­p­ro­cal lat­tice to lithium’s BCC phys­i­cal lat­tice turns out to be the FCC lat­tice of NaCl fame!

And now note the beau­ti­ful sym­me­try in the re­la­tions (10.8) be­tween the prim­i­tive vec­tors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$ of the rec­i­p­ro­cal lat­tice and the prim­i­tive vec­tors $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$ of the phys­i­cal lat­tice. Be­cause these re­la­tions in­volve both sets of prim­i­tive vec­tors in ex­actly the same way, if a phys­i­cal lat­tice with prim­i­tive vec­tors $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$ has a rec­i­p­ro­cal lat­tice with prim­i­tive vec­tors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$, then a phys­i­cal lat­tice with prim­i­tive vec­tors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$ has a rec­i­p­ro­cal lat­tice with prim­i­tive vec­tors $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$. Which means that since NaCl’s FCC lat­tice is the rec­i­p­ro­cal to lithium’s BCC lat­tice, lithium’s BCC lat­tice is the rec­i­p­ro­cal to NaCl’s FCC lat­tice. You now see where the word rec­i­p­ro­cal in rec­i­p­ro­cal lat­tice comes from. Lithium and NaCl bor­row each other’s lat­tice to serve as their lat­tice of wave num­ber vec­tors.

Fi­nally, how about the de­f­i­n­i­tion of the “Bril­louin zones” in three di­men­sions? In par­tic­u­lar, how about the first Bril­louin zone to which you of­ten pre­fer to move the Flo­quet wave num­ber vec­tor ${\vec k}$? Well, it is the mag­ni­tude of the wave num­ber vec­tor that is im­por­tant, so the first Bril­louin zone is de­fined to be the Wigner-Seitz cell around the ori­gin in the rec­i­p­ro­cal lat­tice. Note that this means that in the first Bril­louin zone, $\nu_1$, $\nu_2$, and $\nu_3$ are not sim­ply num­bers in the range from $-\frac12$ to $\frac12$ as in one di­men­sion; that would give a par­al­lelepiped-shaped prim­i­tive cell in­stead.

Solid state physi­cists may tell you that the other Bril­louin zones are also rec­i­p­ro­cal lat­tice Wigner-Seitz cells, [29, p. 38], but if you look closer at what they are ac­tu­ally do­ing, the higher zones con­sist of frag­ments of rec­i­p­ro­cal lat­tice Wigner-Seitz cells that can be as­sem­bled to­gether to pro­duce a Wigner-Seitz cell shape. Like for the one-di­men­sion­al crys­tal, the sec­ond zone are again the points that are sec­ond clos­est to the ori­gin, etcetera.

The bound­aries of the Bril­louin zone frag­ments are now planes called “Bragg planes.” Each is a per­pen­dic­u­lar bi­sec­tor of a lat­tice point and the ori­gin. That is so be­cause the lo­ca­tions where points stop be­ing first/, sec­ond/, third/, ...clos­est to the ori­gin and be­come first/, sec­ond/, third/, ...clos­est to some other rec­i­p­ro­cal lat­tice point must be on the bi­sec­tor be­tween that lat­tice point and the ori­gin. Sec­tions 10.5.1 and 10.6 will give Bragg planes and Bril­louin zones for a sim­ple cu­bic lat­tice.

The qual­i­ta­tive story for the va­lence elec­tron en­ergy lev­els is the same in three di­men­sions as in one. Sec­tions 10.5 and 10.6 will look a bit closer at them quan­ti­ta­tively.