14.9 Draft: Mod­el­ing the Deuteron

This book largely lim­its it­self to rel­a­tively sim­ple, but ef­fec­tive, mod­els for nu­clei. How­ever, the deuteron, the deu­terium nu­cleus, is the most sim­ple non­triv­ial nu­cleus, as it con­sists of only a sin­gle pro­ton. So, to give a rough idea of what sort of more ad­vanced nu­clear the­o­ries there are out there, this one sec­tion will look at the deuteron in some more de­tail.

Ad­den­dum {A.41} ex­plores some of the nu­clear po­ten­tials that you can write down to model the deuteron. Sim­ple po­ten­tials such as those de­scribed there give a lot of in­sight in the deuteron. They give ball­park val­ues for the po­ten­tial and ki­netic en­er­gies. They also give an ex­pla­na­tion for the ob­ser­va­tions that the deuteron is only weakly bound and that there is no sec­ond bound state. They show the rel­a­tively large size of the deuteron: there is a good chance that the pro­ton and neu­tron can be found way apart. Sim­ple ad­di­tions to the po­ten­tial can de­scribe the spin de­pen­dence and the vi­o­la­tion of or­bital an­gu­lar mo­men­tum con­ser­va­tion.

How­ever, there is a prob­lem. These po­ten­tials de­scribe a deuteron con­sist­ing of a pro­ton and a neu­tron. But the pro­ton and neu­tron are not el­e­men­tary par­ti­cles: each con­sists of three quarks.

This in­ter­nal struc­ture should not be a much of a con­cern when the pro­ton and neu­tron are rel­a­tively far apart. But if the two get re­ally close? Surely in that case the physics should be de­scribed in terms of six quarks that in­ter­act through glu­ons and the Pauli ex­clu­sion prin­ci­ple, [5, p. 95]? Even­tu­ally the pro­ton and neu­tron must lose their iden­tity. Then there is no longer any rea­son­able jus­ti­fi­ca­tion for a pic­ture of a free-space pro­ton in­ter­act­ing with a free-space neu­tron.

A rough idea of the scales in­volved may be ob­tained by look­ing at charge radii. The charge ra­dius of a par­ti­cle is a mea­sure of the size of its charge dis­tri­b­u­tion. Now the pro­ton has a charge ra­dius of about 0.88 fm. The deuteron has a charge ra­dius of 2.14 fm. So at least the charge ra­dius of a pro­ton is not that much smaller than the size of the deuteron. And a quan­tum de­scrip­tion of the deuteron needs to con­sider all pos­si­ble nu­cleon spac­ings. Clearly for the small­est nu­cleon spac­ings, the two must in­trude non­triv­ially into each other’s space.

Con­sider an­other ex­am­ple of com­pound struc­tures, no­ble gas atoms. When such atoms are rel­a­tively far apart, they can be mod­eled well as point par­ti­cles form­ing an ideal gas. You might add some sim­ple Van der Waals forces to that pic­ture. How­ever, when the atoms get pushed close to­gether, the elec­tro­mag­netic in­ter­ac­tions be­tween them be­come much more com­plex. And if you try to push the atoms even closer to­gether, they re­sist that very strongly. The rea­son is the Pauli ex­clu­sion prin­ci­ple, chap­ter 5.10. More than two elec­trons can­not be pushed in the same spa­tial state.

The big dif­fer­ence is of course that the elec­tro­mag­netic in­ter­ac­tions of the elec­trons and nu­clei that make up atoms are well un­der­stood. There is as yet no good way to de­scribe the color force in­ter­ac­tions be­tween the quarks that make up nu­cle­ons. (Cer­tainly not at the rel­a­tively low en­er­gies of im­por­tance for nu­clear struc­ture.)

Physi­cists have de­vel­oped a model that is some­where in­ter­me­di­ate be­tween that of in­ter­act­ing free-space nu­cle­ons and in­ter­act­ing quarks. In the model, the forces be­tween nu­cle­ons are pro­duced by the ex­change of par­ti­cles called “pi­ons.” That is much like how in rel­a­tivis­tic quan­tum me­chan­ics, elec­tro­mag­netic forces are pro­duced by the ex­change of pho­tons. Or how the forces be­tween quarks are be­lieved to be pro­duced by the ex­change of glu­ons. These ex­changed par­ti­cles are vir­tual ones.

Roughly speak­ing, rel­a­tivis­tic mass-en­ergy equiv­a­lence com­bined with quan­tum un­cer­tainty in en­ergy al­lows some pos­si­bil­ity for these par­ti­cles to be found near the real par­ti­cles. (See ad­den­dum {A.42} for a more quan­ti­ta­tive de­scrip­tion of these ideas.)

The pic­ture is now that the pro­ton and neu­tron are el­e­men­tary par­ti­cles but dressed in a coat of vir­tual pi­ons. Pi­ons con­sist of two quarks. More pre­cisely, they con­sist of a quark and an an­ti­quark. There are three pi­ons; the pos­i­tively charged $\pi^+$, the neu­tral $\pi^0$, and the neg­a­tively charged $\pi^-$. Pi­ons have no spin. They have neg­a­tive in­trin­sic par­ity. The charged pi­ons have a mass of 140 MeV, while the un­charged one has a slightly smaller mass of 135 MeV.

When the neu­tron and pro­ton ex­change a pion, they also ex­change its mo­men­tum. That pro­duces a force be­tween them.

There are a num­ber of re­deem­ing fea­tures to this model:

It ex­plains the short range of the nu­clear force. That is worked out in ad­den­dum {A.42}.

How­ever, the same re­sult is of­ten de­rived much quicker and eas­ier in lit­er­a­ture. That de­riva­tion does not re­quire any Hamil­to­ni­ans to be writ­ten down, or even any math­e­mat­ics above the el­e­men­tary school level. It uses the pop­u­lar so-called “en­ergy-time un­cer­tainty equal­ity,” chap­ter 7.2.2,

\mbox{any energy difference you want}
\mbox{any time difference you want}
= {\textstyle\frac{1}{2}} \hbar

To ap­ply it to pion ex­change, re­place “any en­ergy dif­fer­ence you want” with the rest mass en­ergy of the vir­tual pion that sup­pos­edly pops up, about 138 MeV on av­er­age. Re­place “any time dif­fer­ence you want” with the time that the pion ex­ists. (Model the pion here as a clas­si­cal par­ti­cle with def­i­nite val­ues of po­si­tion ver­sus time.) From the time that the pion ex­ists, you can com­pute how far it trav­els. That is be­cause clearly it must travel with about half the speed of light: it can­not travel with a speed less than zero nor more than the speed of light. Put in the num­bers, ig­nore the stu­pid fac­tors $\frac12$ be­cause you do not need to be that ac­cu­rate, and it shows that the pion would travel about 1.4 fm if it had a po­si­tion to be­gin with. That range of the pion is con­sis­tent with the ex­per­i­men­tal data on the range of the nu­clear force.

(OK, some­one might ob­ject the pi­ons do most de­cid­edly not pop up and dis­ap­pear again. The ground state of a nu­cleus is an en­ergy eigen­state and those are sta­tion­ary, chap­ter 7.1.4. But why worry about such mi­nor de­tails?)

It gives a rea­son­able ex­pla­na­tion of the anom­alous mag­netic mo­ments of the pro­ton and neu­tron. The mag­netic mo­ment of the pro­ton can be writ­ten as

\mu_{\rm {p}} = g_{\rm {p}} {\textstyle\frac{1}{2}} \mu_{\r...
...\hbar}{2m_{\rm p}}
\approx \mbox{5.051~10$\POW9,{-27}$\ J/T}

Here $\mu_{\rm N}$ is called the nu­clear mag­ne­ton. It is formed with the charge and mass of the pro­ton. Now, if the pro­ton was an el­e­men­tary par­ti­cle with spin one-half, the fac­tor $g_{\rm {p}}$ should be two, chap­ter 13.4. Or at least close to it. The elec­tron is an el­e­men­tary par­ti­cle, and its sim­i­larly de­fined $g$-​fac­tor is 2.002. (Of course that uses the elec­tron mass in­stead of the pro­ton one.)

The mag­netic mo­ment of the neu­tron can sim­i­larly be writ­ten

\mu_{\rm {n}} = g_{\rm {n}} {\textstyle\frac{1}{2}} \mu_{\r...
...\hbar}{2m_{\rm p}}
\approx \mbox{5.051~10$\POW9,{-27}$\ J/T}

Note that the pro­ton charge and en­ergy are used here. In fact, if you con­sider the neu­tron as an el­e­men­tary par­ti­cle with no charge, it should not have a mag­netic mo­ment in the first place.

Sup­pose how­ever that the neu­tron oc­ca­sion­ally briefly flips out a neg­a­tively charged vir­tual $\pi^-$. Be­cause of charge con­ser­va­tion, that will leave the neu­tron as a pos­i­tively charged pro­ton. But the pion is much lighter than the pro­ton. Lighter par­ti­cles pro­duce much greater mag­netic mo­ments, all else be­ing the same, chap­ter 13.4. In terms of clas­si­cal physics, lighter par­ti­cles cir­cle around a lot faster for the same an­gu­lar mo­men­tum. To be sure, as a spin­less par­ti­cle the pion has no in­trin­sic mag­netic mo­ment. How­ever, be­cause of par­ity con­ser­va­tion, the pion should have at least one unit of or­bital an­gu­lar mo­men­tum to com­pen­sate for its in­trin­sic neg­a­tive par­ity. So the neu­tral neu­tron ac­quires a big chunk of un­ex­pected neg­a­tive mag­netic mo­ment.

Sim­i­lar ideas ap­ply for the pro­ton. The pro­ton may tem­porar­ily flip out a pos­i­tively charged $\pi^+$, leav­ing it­self a neu­tron. Be­cause of the mass dif­fer­ence, this can be ex­pected to sig­nif­i­cantly in­crease the pro­ton mag­netic mo­ment.

Ap­par­ently then, the vir­tual $\pi^+$ pi­ons in­crease the $g$-​fac­tor of the pro­ton from 2 to 5.586, an in­crease of 3.586. So you would ex­pect that the vir­tual $\pi^-$ pi­ons de­crease the $g$-​fac­tor of the neu­tron from 0 to $\vphantom{0}\raisebox{1.5pt}{$-$}$3.586. That is roughly right, the ac­tual $g$-​fac­tor of the neu­tron is $\vphantom{0}\raisebox{1.5pt}{$-$}$3.826.

The slightly larger value seems log­i­cal enough too. The fact that the pro­ton turns oc­ca­sion­ally into a neu­tron should de­crease its to­tal mag­netic mo­ment. Con­versely, the neu­tron oc­ca­sion­ally turns into a pro­ton. As­sume that the neu­tron has half a unit of spin in the pos­i­tive cho­sen $z$-​di­rec­tion. That is con­sis­tent with one unit of or­bital an­gu­lar mo­men­tum for the $\pi^-$ and mi­nus half a unit of spin for the pro­ton that is left be­hind. So the pro­ton spins in the op­po­site di­rec­tion of the neu­tron, which means that it in­creases the mag­ni­tude of the neg­a­tive neu­tron mag­netic mo­ment. (The other pos­si­bil­ity, that the $\pi^{-1}$ has no $z$-​mo­men­tum and the pro­ton has half a pos­i­tive unit, would de­crease the mag­ni­tude. But this pos­si­bil­ity has only half the prob­a­bil­ity, ac­cord­ing to the Cleb­sch-Gor­dan co­ef­fi­cients fig­ure 12.6.)

The same ideas also pro­vide an ex­pla­na­tion for a prob­lem with the mag­netic mo­ments of heav­ier nu­clei. These do not fit the­o­ret­i­cal data that well, fig­ure 14.42. A closer ex­am­i­na­tion of these data sug­gests that the in­trin­sic mag­netic mo­ments of nu­cle­ons are smaller in­side nu­clei than they are in free space. That can rea­son­ably be ex­plained by as­sum­ing that the prox­im­ity of other nu­cle­ons af­fects the coats of vir­tual pi­ons.

It ex­plains a puz­zling ob­ser­va­tion when high-en­ergy neu­trons are scat­tered off high-en­ergy pro­tons go­ing the op­po­site way. Be­cause of the high en­er­gies, you would ex­pect that only a few pro­tons and neu­trons would be sig­nif­i­cantly de­flected from their path. Those would be caused by the few col­li­sions that hap­pen to be al­most head-on. And in­deed rel­a­tively few are scat­tered to say about 90$\POW9,{\circ}$ an­gles. But an un­ex­pect­edly large num­ber seems to get scat­tered al­most 180$\POW9,{\circ}$, straight back to where they came from. That does not make much sense.

The more rea­son­able ex­pla­na­tion is that the pro­ton catches a vir­tual $\pi^-$ from the neu­tron. Or the neu­tron catches a vir­tual $\pi^+$ from the pro­ton. Ei­ther process turns the pro­ton into a neu­tron and vice-versa. So an ap­par­ent re­flected neu­tron is re­ally a pro­ton that kept go­ing straight but changed into a neu­tron. And the same way for an ap­par­ent re­flected pro­ton.

It can ex­plain why charge in­de­pen­dence is less per­fect than charge sym­me­try. A pair of neu­trons can only ex­change the neu­tral $\pi^0$. Ex­change of a charged pion would cre­ate a pro­ton and a nu­cleon with charge $\vphantom{0}\raisebox{1.5pt}{$-$}$1. The lat­ter does not ex­ist. (At least not for mea­sur­able times, nor at the en­er­gies of most in­ter­est here.) Sim­i­larly, a pair of pro­tons can nor­mally only ex­change a $\pi^0$. But a neu­tron and a pro­ton can also ex­change charged pi­ons. So pion in­ter­ac­tions are not charge in­de­pen­dent.

The nu­clear po­ten­tial that can be writ­ten down an­a­lyt­i­cally based on pion ex­change works very well at large nu­cleon spac­ings. This po­ten­tial is called the OPEP, for One Pion Ex­change Po­ten­tial, {A.42}.

There are also draw­backs to the pion ex­change ap­proach.

For one, the mag­netic mo­ments of the neu­tron and pro­ton can be rea­son­ably ex­plained by sim­ply adding those of the con­stituent quarks, [30, pp. 74, 745]. To be sure, that does not di­rectly af­fect the ques­tion whether the pion ex­change model is use­ful. But it does make the dressed nu­cleon pic­ture look quite con­trived.

A big­ger prob­lem is nu­cleon spac­ings that are not so large. One-pion ex­change is gen­er­ally be­lieved to be dom­i­nant for nu­cleon spac­ings above 3 fm, and rea­son­able for spac­ings above 2 fm, [35, p. 135, 159], [5, p. 86, 91]. How­ever, things get much more messy for spac­ings shorter than that. That in­cludes the vi­tal range of spac­ings of the pri­mary nu­cleon at­trac­tions and re­pul­sions. For these, two-pion ex­change must be con­sid­ered. In ad­di­tion, ex­cited pion states and an ex­cited nu­cleon state need to be in­cluded. That is much more com­pli­cated. See ad­den­dum {A.42} for a brief in­tro to some of the is­sues in­volved.

And for still shorter nu­cleon spac­ings, things get very messy in­deed, in­clud­ing multi-pion ex­changes and a zoo of other par­ti­cles. Even­tu­ally the ques­tion must be at what spac­ing nu­cle­ons lose their dis­tinc­tive char­ac­ter and a model of quarks ex­chang­ing glu­ons be­comes un­avoid­able. For­tu­nately, very close spac­ings cor­re­spond to very high en­er­gies since the nu­cle­ons strongly re­pel each other at close range. So very close spac­ings may not be that im­por­tant for most nu­clear physics.

Be­cause of the above and other is­sues, many physi­cists use a less the­o­ret­i­cal ap­proach. The OPEP is still used at large nu­cleon spac­ings. But at shorter spac­ings, rel­a­tively sim­ple cho­sen po­ten­tials are used. The pa­ra­me­ters of those “phe­nom­e­no­log­i­cal” po­ten­tials are ad­justed to match the ex­per­i­men­tal data.

It makes things a lot sim­pler. And it is not clear whether the the­o­ret­i­cal mod­els used at smaller nu­cleon spac­ings are re­ally that much more jus­ti­fied. How­ever, phe­nom­e­no­log­i­cal po­ten­tials do re­quire that large num­bers of pa­ra­me­ters are fit to ex­per­i­men­tal data. And they have a nasty habit of not work­ing that well for ex­per­i­men­tal data dif­fer­ent from that used to de­fine their pa­ra­me­ters, [31].

Re­gard­less of po­ten­tial used, it is dif­fi­cult to come up with an un­am­bigu­ous prob­a­bil­ity for the $l$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 or­bital an­gu­lar mo­men­tum. Es­ti­mates hover around the 5% value, but a clear value has never been es­tab­lished. This is not en­cour­ag­ing, since this prob­a­bil­ity is an in­te­gral quan­tity. If it varies non­triv­ially from one model to the next, then there is no real con­ver­gence on a sin­gle deuteron model. Of course, if the pro­ton and neu­tron are mod­eled as in­ter­act­ing clouds of par­ti­cles, it may not even be ob­vi­ous how to de­fine their or­bital an­gu­lar mo­men­tum in the first place, [Phys. Rev. C 19,20 (1979) 1473,325]. And that in turn raises ques­tions in what sense these mod­els are re­ally well-de­fined two-par­ti­cle ones.

Then there is the very big prob­lem of gen­er­al­iz­ing all this to sys­tems of three or more nu­cle­ons. One cur­rent hope is that closer ex­am­i­na­tion of the un­der­ly­ing quark model may pro­duce a more the­o­ret­i­cally jus­ti­fied model in terms of nu­cle­ons and mesons, [31].