10.2 Ionic Solids

A typ­i­cal ex­am­ple of a ionic solid is or­di­nary salt, NaCl. There is lit­tle quan­ti­ta­tive quan­tum me­chan­ics re­quired to de­scribe ei­ther the salt mol­e­cule or solid salt. Still, there are some im­por­tant qual­i­ta­tive points, so it seems use­ful to in­clude a dis­cus­sion in this book. Both mol­e­cule and solid will be de­scribed in this sub­sec­tion, since the ideas are very sim­i­lar.

To form a NaCl salt mol­e­cule, a clorine atom takes the loosely bound lone 3s elec­tron away from a na­trium (sodium) atom and puts it in its sin­gle still va­cant 3p po­si­tion. That leaves a neg­a­tive chlo­rine ion with filled K, L, and M shells and a pos­i­tive na­trium ion with just filled K and L shells. Since the com­bined elec­tron dis­tri­b­u­tion of filled shells is spher­i­cally sym­met­ric, you can rea­son­ably think of the two ions as some­what soft bil­liard balls. Since they have op­po­site charge, they stick to­gether into a salt mol­e­cule as sketched in fig­ure 10.1. The na­trium ion is a bit less than two Å in di­am­e­ter, the clorine one a bit less than four.

Fig­ure 10.1: Bil­liard-ball model of the salt mol­e­cule.
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The en­er­get­ics of this process is rather in­ter­est­ing. As­sume that you start out with a neu­tral na­trium atom and a neu­tral chlo­rine atom that are far apart. To take the lone 2s elec­tron out of the na­trium atom, and leave it at rest at a po­si­tion far from ei­ther the na­trium or the chlo­rine atom, takes an amount of en­ergy called the “ion­iza­tion en­ergy” of na­trium. Its value is 5.14 eV (elec­tron volts).

To take that free elec­tron at rest and put it into the va­cant 3p po­si­tion of the chlo­rine ion gives back an amount of en­ergy called the “elec­tron affin­ity” of clorine. Its value is 3.62 eV.

(Elec­tron affin­ity, the will­ing­ness to take on free elec­trons, is not to be con­fused with “elec­troneg­a­tiv­ity,” the will­ing­ness to take on elec­trons in chem­i­cal bonds. Un­like elec­troneg­a­tiv­ity, elec­tron affin­ity varies wildly from el­e­ment to el­e­ment in the pe­ri­odic ta­ble. There is some sys­tem in it, still, es­pe­cially within sin­gle columns. It may also be noted that there seems to be some dis­agree­ment about the de­f­i­n­i­tion of elec­troneg­a­tiv­ity, in par­tic­u­lar for atoms or mol­e­cules that can­not sta­bly bind a free elec­tron, {N.19}.)

Any­way, since it takes 5.14 eV to take the elec­tron out of na­trium, and you get only 3.62 eV back by putting it into clorine, you may won­der how a salt mol­e­cule could ever be sta­ble. But the de­scribed pic­ture is very mis­lead­ing. It does not re­ally take 5.14 eV to take the elec­tron out of na­trium; most of that en­ergy is used to pull the lib­er­ated elec­tron and pos­i­tive ion far apart. In the NaCl mol­e­cule, they are not pulled far apart; the pos­i­tive na­trium ion and neg­a­tive chlo­rine ion stick to­gether as in fig­ure 10.1.

In other words, to cre­ate the widely sep­a­rated pos­i­tive na­trium ion and neg­a­tive chlo­rine ion took $5.14-3.62$ eV, but watch the en­ergy that is re­cov­ered when the two ions are brought to­gether to their cor­rect 2.36 Å sep­a­ra­tion dis­tance in the mol­e­cule. It is ap­prox­i­mately given by the Coulomb ex­pres­sion

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\frac{e^2}{4\pi\epsilon_0} \frac{1}{d}
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where $\epsilon_0$ $\vphantom0\raisebox{1.5pt}{$=$}$ 8.85 10$\POW9,{-12}$ C$\POW9,{2}$/J m is the per­mit­tiv­ity of space and $d$ is the 2.36 Å dis­tance be­tween the nu­clei. Putting in the num­bers, drop­ping an $e$ to get the re­sult in eV, this en­ergy is 6.1 eV. That gives the to­tal bind­ing en­ergy as $-5.14+3.62+6.1$, or 4.58 eV. That is not quite right, but it is close; the true value is 4.26 eV.

There are a few rea­sons why it is slightly off, but one is that the Coulomb ex­pres­sion above is only cor­rect if the ions were bil­liard balls that would move unim­peded to­wards each other un­til they hit. Ac­tu­ally, the atoms are some­what softer than bil­liard balls; their mu­tual re­pul­sion force ramps up quickly, but not in­stan­ta­neously. That means that the re­pul­sion force will do a small amount of neg­a­tive work dur­ing the fi­nal part of the ap­proach of the ions. Also, the un­cer­tainty prin­ci­ple does not al­low the lo­cal­ized ions to have ex­actly zero ki­netic en­ergy. But as you see, these are small ef­fects. It may also be noted that the re­pul­sion be­tween the ions is mostly Pauli re­pul­sion, as de­scribed in sec­tion 5.10.

Fig­ure 10.2: Bil­liard-ball model of a salt crys­tal.
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Now the elec­tro­sta­tic force that keeps the two ions to­gether in the mol­e­cule is omni-​di­rec­tional. That means that if you bring a lot of salt mol­e­cules to­gether, the clorine ions will also at­tract the na­trium ions of other mol­e­cules and vice versa. As a re­sult, un­der nor­mal con­di­tions, salt mol­e­cules pack to­gether into solid salt crys­tals, as shown in fig­ure 10.2. The ions arrange them­selves very neatly into a pat­tern that al­lows each ion to be sur­rounded by as many at­tract­ing ions of the op­po­site kind as pos­si­ble. In fact, as fig­ure 10.2 in­di­cates, each ion is sur­rounded by six ions of the op­po­site kind: four in the same ver­ti­cal plane, a fifth be­hind it, and a sixth in front of it. A more de­tailed de­scrip­tion of the crys­tal struc­ture will be given next, but first con­sider what it means for the en­ergy.

Since when the mol­e­cules pack into a solid, each ion gets next to six ions of the op­po­site type, the sim­plest guess would be that the 6.1 eV Coulomb at­trac­tion of the ions in the mol­e­cule would in­crease by a fac­tor 6 in the solid. But that is a bad ap­prox­i­ma­tion: in the solid, each ion is not just sur­rounded by six at­tract­ing ions of the op­po­site kind, but also by twelve re­pelling ions of the same kind that are only slightly fur­ther away, then again eight at­tract­ing ions still a bit fur­ther away, etcetera. The net ef­fect is that the Coulomb at­trac­tion is only 1.75 times higher in the solid than the lone mol­e­cules would have. The fac­tor 1.75 is called the “Madelung con­stant. So, all else be­ing the same, by form­ing a salt crys­tal the salt mol­e­cules would raise their Coulomb at­trac­tion to $1.75\times6.1$ or 10.7 eV.

That is still not quite right, be­cause in the solid, the ions are far­ther apart than in the mol­e­cule. Re­call that in the solid, each at­tract­ing ion is sur­rounded by re­pelling ions of the op­po­site kind, re­duc­ing the at­trac­tion be­tween pairs. In the solid, op­po­site ions are 2.82 Å apart in­stead of 2.36, so the Coulomb en­ergy re­duces to 10.7 $\times$ 2.36/2.82 or 8.93 eV. Still, the bot­tom line is that the mol­e­cules pick up about 2.8 eV more Coulomb en­ergy by pack­ing to­gether into salt crys­tals, and that is quite a bit of en­ergy. So it should not come as a sur­prise that salt must be heated as high as 801 $\POW9,{\circ}$C to melt it, and as high as 1 465 $\POW9,{\circ}$C to boil it.

Fi­nally, con­sider the crys­tal struc­ture that the mol­e­cules com­bine into. One way of think­ing of it is as a three-di­men­sion­al chess board struc­ture. In fig­ure 10.2, think of the frontal plane as a chess board of black and white cubes, with a na­trium nu­cleus in the cen­ter of each white cube and a clorine nu­cleus in the cen­ter of each black one. The next plane of atoms can sim­i­larly be con­sid­ered to con­sists of black and white cubes, where the back cubes are be­hind the white cubes of the frontal plane and vice-versa. And the same way for fur­ther planes.

Fig­ure 10.3: The salt crys­tal dis­as­sem­bled to show its struc­ture.
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How­ever, this is not how a ma­te­r­ial sci­en­tist would think about the struc­ture. A ma­te­r­ial sci­en­tist likes to de­scribe a crys­tal in terms copies of a sim­ple unit, called the “ba­sis,” that are stacked to­gether in a reg­u­lar man­ner. One pos­si­ble choice for the ba­sis in salt is a sin­gle na­trium ion plus a sin­gle clorine ion to the right of it, like the mol­e­cule of fig­ure 10.1. In fig­ure 10.3 the ions of the salt crys­tal have been moved far apart to make the ac­tual struc­ture vis­i­ble, and the two atoms of the ba­sis units have been joined by a blue line. Note that the en­tire struc­ture con­sists of these ba­sis units.

But also note that the mol­e­cules lose their iden­tity in a ionic solid. You could just as well build up the crys­tal from ver­ti­cal mol­e­cules, say, in­stead of hor­i­zon­tal ones. In fact, there are six rea­son­able choices of ba­sis, de­pend­ing on which of its six sur­round­ing chlo­rine ions you want to as­so­ciate each na­trium ion with. There are of course al­ways count­less un­rea­son­able ones...

The reg­u­lar way in which the bases are stacked to­gether to form the com­plete crys­tal struc­ture is called the “lat­tice.” You can think of the vol­ume of the salt crys­tal as con­sist­ing of lit­tle cubes called “unit cells” in­di­cated by the red frames in fig­ure 10.3. There are clorine atoms at the cor­ners of the cubes as well as at the cen­ter points of the faces of the cubes. That is the rea­son the salt lat­tice is called the “face cen­tered cu­bic” (FCC) lat­tice. Also note that if you shift the unit cells half a cell to the left, it will be the na­trium ions that are at the cor­ners and face cen­ters of the cubes. In gen­eral, every point of a ba­sis is arranged in the crys­tal ac­cord­ing to the same lat­tice.

You will agree that it sounds much more pro­fes­sional to say that you have stud­ied the face-cen­tered cu­bic arrange­ment of the ba­sis in a NaCl crys­tal than to say that you have stud­ied the three-di­men­sion­al chess board struc­ture of salt.


Key Points
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In a fully ionic bond like NaCl, one atom takes an elec­tron away from an­other.

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The pos­i­tive and neg­a­tive ions stick to­gether by elec­tro­sta­tic force, cre­at­ing a mol­e­cule.

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Be­cause of the same elec­tro­sta­tic force, mol­e­cules clump to­gether into strong ionic solids.

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The crys­tal struc­ture of NaCl con­sists of copies of a two-atom NaCL ba­sis arranged in a face-cen­tered cu­bic lat­tice.