Appendix A3 Derivation of the p + D

Transcription

1 Andix A Drivation of th + D H Raction Rat Writtn: Octobr 5, minor udats //9. Introduction W hav notd in Chatrs and 4 that th crucial slownss of th raction which initiats stllar burning of hydrogn, i.. D, is du in art to th nd for th rotons to ntrat th Coulomb barrir bfor thy can ract. This bing th cas, w hav qurid (Chatr 4) why th scond raction, i.. + D H +, should not also b slow, sinc a Coulomb barrir nds to b ntratd in this cas also. Yt ths two ractions diffr in rat by about a factor of 7. In this Andix w driv th rat of th + D H + raction from first rincils. W shall s that it is indd surssd by th Coulomb barrir. This is th rason for this raction rat rducing xtrmly stly, ffctivly switching off, blow tmraturs of ~ 7 K. W conclud that th 7 ordrs of magnitud diffrnc in th two raction rats must actually b du ntirly to th waknss of th wak nuclar forc, contrary to th imrssion givn by many oular hysics books.. Th Contributing Matrix Elmnts W ar daling with a systm of two rotons and on nutron. Th ground stat of hlium- will b an S-stat. Th two rotons must thrfor hav anti-alignd sins, by th xclusion rincil, so th ovrall sin of H must b ½, arising from th nutron sin. In th fr stat, th two roton sins may altrnativly b alignd, in which cas, to b consistnt with th xclusion rincil, thy must b in an odd L stat. Following th drivation of th roton-nutron matrix lmnts (Andix A) w considr two intraction Hamiltonians: th lctric diol intraction, H I E r cos ; and th magntic diol intraction, H I B Mag, whr M ag is th magntic momnt orator and E and B ar th lctric and magntic filds du to th gamma ray hoton. [In th cas of hotodisintgration, this is th incidnt radiation. In th cas of roton catur, this is th mittd hoton]. W firstly considr th magntic diol intraction. Rcall from Andix that th magntic diol matrix lmnt was non-zro for th n- systm only whn th fr stat was chosn to b a sin singlt stat, in contrast to th bound stat (dutron) which is a sin trilt. Th sin dndnc of th strong nuclar otntial thn lads to a non-zro matrix lmnt. For th n systm, considring just S-wav (L = ) stats, thr is just on ossibl total sin stat, sin ½, in which th two rotons form a singlt sub-stat. Th matrix lmnt btwn two such stats of diffring nrgy is zro, by orthogonality. Unlik th n- systm w do not hav th choic btwn two diffrnt sin stats with diffring nuclar otntials. On th othr hand, if w considr th fr stat to b a P-wav (L = ), and th sin stat to b /, thn th two stats will b ignfunctions corrsonding to diffrnt (sin dndnt) nuclar otntials. Howvr, th matrix lmnt is again zro du to th angular intgral (i.. S and P stats ar always orthogonal, irrsctiv of th radial dndnc). W conclud that th magntic diol matrix lmnt is idntically zro (at last in first NB: Th matrix lmnt of th orator Mag btwn sin ½ and sin / stats can b non-zro as long as th two stats hav th sam azimuthal sin comonnt.

2 ordr rturbation thory), ssntially as a consqunc of th xclusion rincil. Sinc th magntic trm is dominant in th n- systm (accounting for 99+% of th ractions blow ~ 8 K, s Andix ) w can thrfor xct th + D H + raction rat to b much slowr than th catur of rotons by nutrons (vn bfor accounting for th Coulomb barrir). Th lctric diol matrix lmnt is non-zro, howvr. In viw of th rsnc of th cos factor in th intraction Hamiltonian, th only non-zro matrix lmnt will b that in which th fr stat is a P-wav. Th two roton sins must thn b alignd, by th xclusion rincil. According to whthr th nutron sin is alignd or antialignd with th roton sins, th total sin will b / or ½. Howvr, bcaus th bound stat ( H) is sin ½, and bcaus th lctric diol intraction dos not affct th sin, th sin / fr stat will giv a zro matrix lmnt. Consquntly, th only non-zro matrix lmnt contributing in first ordr rturbation thory is, matrix lmnt = sin /, S (bound) E r cos sin /, P(fr) (Equ.) noting that th factor of r in th intraction Hamiltonian is what rvnts orthogonality making th radial intgration zro. Not that Equ. is th matrix lmnt factor r roton, i.. a charg of just has bn assumd. Sinc thr ar two rotons, this matrix lmnt is dulicatd for th othr roton. Th fact that th sin stat is th sam for th fr stat as for th bound stat is fortunat. This is bcaus w do not yt know what th ffctiv nuclar otntial is for H. W can, howvr, dtrmin th ffctiv otntial for th sin ½ stat (or, rathr, w can dtrmin ossibl airs of V and a) from th masurd binding nrgy of H. This sam otntial & rang combination will aly also for th fr stat bcaus it is also sin ½. Had th sin / fr stat bn involvd, w would hav had a difficulty dtrmining th ffctiv otntial and rang for this sin stat.. Effctiv Nuclar Potntial For H Th radial Schrodingr quation is, m r m L(L r ) V(r) E u(r) whr, u r (Equ.) Now, to writ a Schrodingr quation in just on (radial) coordinat, w must b assuming a two-body roblm. Strictly, w ar now daling with a thr-body roblm, two rotons and a nutron. Howvr, w intnd to trat th dutron as if it wr a singl articl in its own right. Th two-body systm in qustion is thrfor th -D systm. Sinc ths two articls ar of mass M and M (aroximatly), th rducd mass m is just ( /)M. A ositiv otntial is rulsiv, so th cntrifugal forc is sn to b rulsiv, as it should b. In th siml squar-wll modl for th nuclar forc w hav, V(r) for r < a (Equ.a) V

3 V(r) Z(Z ) ~ r for r > a (Equ.b) For th dutron, Z = and th Coulomb otntial dos not contribut. For hlium, Z = and th Coulomb otntial is ~ / r. W hav ~. 44 MVfm. Th known binding nrgy of hlium- is B = 7.78MV. W hav numrically intgratd th Schrodingr quation (EXCEL macro NSWa) to find combinations of V and a which rroduc this binding nrgy to at last 4 dcimal lacs. Ths ar givn in th Tabl blow. Th Tabl also shows th valus of a cot(a ) and a, whr mb and m(v B). Ths two quantitis would b qual if it wr not for th influnc of th Coulomb otntial. It can b sn that thy ar quit clos, which shows that th Coulomb otntial dos not hav a dramatic ffct on th binding nrgy, though it dos hav a noticabl ffct. Th numrical intgration alid th boundary condition nar th origin at r =. fm and usd an intgration st siz ( r) of. fm. Snsitivity to a chang to a st siz ( r) of.5 fm causd a chang in th rquird V to rroduc th rquird binding nrgy of only.8%. Effctiv Nuclar Potntial And Rang For H a (fm) V (MV) a cot(a ) a Not that thr is no siml combination of V and a (.g. a V ) which is invariant. Thr was no othr bound stat of hlium- for th abov otntials (i.. no xcitd stats of hlium-, which I bliv is in fact th cas). If th abov rscrition for th nuclar -body roblm wr corrct, w would xct to b abl to driv th binding nrgy of tritium (nn) from th sam otntial & rang combinations just by switching off th Coulomb otntial. Th actual binding nrgy of tritium is MV, som.768 MV largr than that of hlium-. Th sign of this binding nrgy diffrnc is as w would xct, bing du to th rulsiv natur of th Coulomb forc causing hlium- to b lss tightly bound. Running NSWa with th Coulomb otntial switchd off, and for nuclar forc rangs of 4,, and.6 fm, givs incrass in binding nrgy of.56,.7,.57 and.676 MV rsctivly. W s that, for th shortst rangs w ar indd aroaching th corrct binding nrgy diffrnc (.768 MV). Th lsson from this obsrvation is that for nucli largr than th dutron, th absolut siz (a) is significant in dtrmining nuclar rortis.

4 4. Numrical Evaluation Of Matrix Elmnts In th cas of th n- systm w usd th zro-rang aroximation to find algbraic xrssions for th matrix lmnts (s Andix ). This was ossibl bcaus w hav siml, xact, analytic solutions for th wavfunctions in th absnc of a Coulomb otntial. Unfortunatly, th rsnc of th Coulomb otntial maks it rfrabl to rsort to numrical solutions. Th EXCEL macro NSWb has bn usd. Bfor rocding to th + D raction, w carry out chcks on th numrical rocdur by rroducing th analytical rsults for th n- systm. 4. Chck of Program NSWb Against n- Matrix Elmnts For alication to th -D systm w shall only nd th lctric diol matrix lmnt btwn th bound S stat and a fr P stat, both of th sam sin stat (hnc th sam nuclar otntial). Howvr, for comltnss, th rogram NSWb is also chckd hr for th cas of th magntic diol matrix lmnt btwn two S stats of diffrnt sin (i.. diffrnt nuclar otntials). W start with th lattr. 4.. Chck of S(bound) S(fr) For Th n- Potntials Rcall first of all that matrix lmnts involving fr stats ar dfind only if a normalisation volum is scifid. Th matrix lmnt has bn dtrmind in Andix A to b, whr, ~ S (bound) S(fr) n k B ~ (Equ.) / E rmax mb ~ ~, B is th dutron binding nrgy (.45MV), mb ~ and also / a, whr a is th n- singlt scattring lngth, i...69fm. Hnc, ~.4fm, and B ~.745MV. E is th total nrgy of th two nuclons in th cms systm (th ignvalu of th Schrodingr quation), and k me. Finally, r max is th radius of th normalisation volum. Th numrical valu of th matrix lmnt is maninglss without r max bing scifid. It should b rcalld that Equ. was drivd on th assumtion that k(r max a) n for som intgr n. Numrical valuation for combinations of nrgy and r max not obying this rlationshi may b xctd to diffr from Equ.(), i. Equ.() will not b corrct in such cass. It should also b rcalld that Equ. was drivd using th zro-rang aroximation. W may xct, thrfor, that numrical valuations will agr bttr with Equ. if combinations of V and a ar mloyd in which a is small. Bcaus of this w hav usd otntials with two diffrnt rangs, as follows, Effctiv n- Nuclar Potntials Singlt Trilt V (MV) a (fm) V (MV) a (fm) B xrimntal rducd a ,

5 Th first lin givs valus (from Evans) drivd from xrimntal data, and hnc ar bst stimats. Th scond lin givs quivalnt otntials with th rang rducd to fm. For th trilt stat, V was first found from a = fm using th xact solution for a squar wll otntial to rroduc th binding nrgy of.45mv. Th last column of th Tabl givs th binding nrgis drivd from th rogram NSWa for th statd trilt otntials. Thy ar clos nough to th tru valu for our uross. Not that thr is no bound singlt stat and B ~ is not a binding nrgy. Finally, th singlt otntial for a = fm was found using th nrgy quantisation xrssion for a ~ squar wll, i.. a ~ cot(a ~ ) a, whr ~ m(v B ~ ~ ) and mb ~, whilst noting that this is not a bound stat bcaus ~ is ngativ. Th following Tabl comars th numrical rsults of NSWb with Equ. for th a = fm otntials. Equ. is valuatd using th binding nrgy comatibl with th rogram, i.. B =.98MV. Rsults ar rsntd for r max = 57.fm. W hav not rsctd th rquirmnt k(r max a) n, so this may introduc som of th diffrncs obsrvd. For th lowst nrgis calculatd, th wavlngth of th fr stat ( /k) is far gratr than r max, and this would introduc srious rror in th numrical rsults if scial rcautions wr not takn. For th low nrgy cass (in italics in th Tabl) NSWb alis a corrction to th normalisation of th fr stat. Dtails ar givn in a not at th nd of this Andix. This was nvr addd. Matrix Elmnts S(bound) S(fr) For n- Systm (r max = 57.fm, No Coulomb Potntial, Nuclar Potntials with a = fm) E (MV) M.E. (from NSWb) M.E. (Equ.) Not that at small nrgis th matrix lmnt dros to zro roortional to E. At larg nrgis th matrix lmnt also dros to zro, roortional to /E. Grahical lots of ths rsults follow

6 Figur: Magntic Diol S-S Matrix Elmnt for n- Systm (Chck of Numrical Procdur).6.5 Numrical rsult Analytic rsult (Equ.).4... Enrg y, E ( M V ) Figur: Magntic Diol S-S Matrix Elmnt for n- Systm (Chck of Numrical Procdur).6.5 Numrical rsult Analytic rsult (Equ.).4... Enrg y, E ( M V ) Chck of S(bound) r cos P(fr) For Th n- Potntials Andix A has also drivd an analytic xrssion for this lctric diol matrix lmnt in th zro-rang aroximation, namly, 4k 8 S (bound) r cos P(fr) a (Equ.4) n rmax k Numrical and grahical rsults from th numrical intgration rogram NSWb ar comard with Equ.4 as follows, for V =.7MV and a = fm,

7 Matrix Elmnts S(bound) r cos P(fr) For n- Systm (r max = 49.6fm, No Coulomb Potntial, Nuclar Potntials with a = fm) E (MV) M.E. (from NSWb) M.E. (Equ.4) Figur: Elctric Diol S-P Matrix Elmnt for n- Systm: Chck of Numrical Procdur Numrical rsults Analytic rsult, Equ En r gy, 8 E ( M V)

8 Matrix Elmnt (fm) Figur: Elctric Diol S-P Matrix Elmnt for n- Systm: Chck of Numrical Procdur Numrical rsults Analytic rsult, Equ Enrgy, E (MV) Th matrix lmnt again falls off as /E at larg nrgy, but at small nrgis th dro-off is lss st than for th S-S matrix lmnt, falling roortionally to E. 4.. Snsitivity of n- Matrix Elmnts to Coulomb Potntial To gain an imrssion for th snsitivity of a matrix lmnt to switching on th Coulomb otntial, w considr th n- systm as if a Coulomb otntial alid. This may b thought of as bing what th - matrix lmnts would b if th xclusion rincil did not forbid thir xistnc S(bound) S(fr) With Coulomb Potntial For comarison, w us th sam singlt and trilt otntials as in Sction 4.., namly V = 99. MV, V =.7 MV rsctivly, with a = fm. Th Coulomb otntial is switchd on only for r > a and is ~ ~ V(r) / r, whr. 44 MV.fm. Th switching-on of th Coulomb otntial changs th bound stat binding nrgy from.98 MV to.6854 MV. [I found it surrising, initially, that th Coulomb otntial causs th binding nrgy to dcras rathr than incras. This was bcaus th Coulomb otntial causs th aarnt dth of th otntial wll to incras from V to V + V c, whr ~ Vc / a. Howvr, it must b rmmbrd that th bulk of th wavfunction lis outsid th rgion of th nuclar otntial, i.. it is far mor likly to find th nuclons sacd by r > a. Th ffctiv forc of attraction is thrfor th nuclar forc minus th rulsiv Coulomb otntial. Clarly this will wakn th binding]. In th Tabl blow w comar th numrical rsults from rogram NSWb with th analytic rsult of Equ.(). Th lattr dos not account for th Coulomb otntial. Howvr, w us th shiftd binding nrgy (B =.6854 MV) in Equ.(). This roughly accounts for th Coulomb ffct on th bound stat in Equ.(), but thr is no such allowanc for th fr stat. In th numrical rsults from NSWb w hav usd r max = 57. fm. Howvr, for small nrgis, NSWb normaliss th fr stats by using an xtndd radial intgration, u to a radius which is givn by,

9 ~ rnorm N norm (Equ.5) E me whr N norm is som ositiv intgr, a valu of 4 having bn gnrally usd hr. Although NSWb uss this incrasd intgration rang, th rsulting normalisation constant is rscald as if th normalisation had bn to radius r max, i.. th wavfunction is multilid by r / r. Hnc, r max may b dloyd hraftr as norm max if it had bn th normalisation radius, and th matrix lmnts tc for small nrgis can b comard dirctly with thos at highr nrgis. E (MV) Equ.(): Coulomb Barrir Not Prsnt NSWb With Coulomb Barrir E E-7.6 Figur: Effct of Coulom b Barrir on S-S Matrix Elmnt.5 Equ.() No Coulomb Barrir NSWb With Coulomb Barrir S-S Matrix Elmnt.4... Enrgy, E (MV)

10 / E ^. 5 (M V ) Equ.() No Coulomb Barrir NSWb With Coulomb Barrir..... Figur: Effct of Coulomb Barrir on S-S Matrix Elmnt W hav alrady sn that th matrix lmnt without a Coulomb barrir rducs to zro at low nrgis roortionally to E. Th abov logarithmic lot shows that at b / E th lowst nrgis, th matrix lmnt with th Coulomb barrir varis as. Th grah shows that b.69 MV. This nrgy dndnc can b drivd simly as follows. Firstly, w not that th bound stat wavfunction is littl changd by th rsnc of th Coulomb otntial:- wavfunction (fm^-/) Wavfunction vrsus Radial Distanc Bound, No Coulomb Bound, With Coulomb Fr (.5MV), No Coulomb Fr (.5MV), With Coulomb.4. r (fm) 4 5 6

11 In contrast, th fr stat wavfunction is markdly rducd by th Coulomb otntial. (Th abov illustration is for a fr stat nrgy of E =.5 MV). Th rason for th markd chang in th fr stat wavfunction is simly that, with th Coulomb otntial switchd on, th fr stat has ngativ kintic nrgy (i.. has xonntial radial dndnc rathr than oscillatory) for r < 8.8 fm. This lads to th fr stat wavfunction bing far smallr in th rgion whr th bound stat wavfunction is significant (i.. r < fm). In th abov xaml with E =.5 MV, it is smallr by a factor of ~4. It is simly this which rsults in th matrix lmnt bing smallr by a similar factor. Th abov grah also illustrats clarly that most of th contribution to th matrix lmnt coms from outsid th rang of th nuclar forcs (i.. from r > a = fm in this cas). Having mad this obsrvation, w now ralis that nrgy dndnc of th matrix lmnt must driv from that of th fr stat in th rgion r < fm, and hnc dnd ssntial on how much of th fr stat manags to ntrat th Coulomb barrir from outsid. Not, though, that this dos not man how much of th fr stat xists in th intrior rgion r < a. W hav sn that vry littl dos, at any nrgy, and indd this is th basis of th zro-rang aroximation. It is actually th siz of th fr wavfunction within th rgion whr th Coulomb otntial is significant (a < r < ~ re / E ) that mattrs. This is th rgion within which th fr stat has ngativ kintic nrgy, th xonntial dcay lngth bing (th rcirocal of) m(v(r) E) /, noting that this varis with r. Roughly saking, w may xct th magnitud of th wavfunction to diminish from r = r E to r = a by a factor of r E a dr X. Th intgral may b valuatd xactly to giv a factor, x ~ m E sin whr, tan - V c E, (Equ.6) ~ Vc a For nrgis sufficintly small comard with V c, / and hnc th sin trm is zro and w gt, X x b / E whr, b ~ m.5 MV (for - ) (Equ.7) hnc driving th functional dndnc on nrgy sn in th numrical rsults, and also confirming that th numrical valu for b givn abov is vry clos to th xctd valu. This validats th numrical rocdur of rogram NSWb S(bound) r cos P(fr) With Coulomb Potntial Sinc th -D cross sction will rquir th S(sin ½)-P(sin ½) matrix lmnt, w also xamin th snsitivity of th n- systm matrix lmnt to turning on th Coulomb otntial. [Th sam rmarks aly as in Sction 4..., namly that this

12 dos not corrsond to any actual hysical rocss]. In this cas w us th bst stimat trilt otntial (for both fr and bound stats), i.. V = 5.8 MV and a =.54 fm and r max = 57. fm. Program NSWa givs th bound stat binding nrgy whn th Coulomb otntial is switchd on to b.947 MV. Th comarison btwn th numrical rsults of rogram NSWb, with th Coulomb otntial on, and Equ.4 (which dos not account for th Coulomb barrir although w us th amndd binding nrgy) is givn by th following Tabl and grahs, E (MV) Equ.(): Coulomb Barrir Not Prsnt NSWb With Coulomb Barrir E E E-8 Again it can b sn, from th scond grah blow, that th matrix lmnt varis b / E with nrgy, at sufficintly small nrgis, roortionally to and th numrical rsults giv b ~.69 from th last two data oints, in good agrmnt with th thortical stimat drivd abov.

13 Figur: Com arison of S-P Matrix Elmnt With and Without Coulomb Barrir Equ.(4), Without Coulomb Barrir NSWb, With Coulomb Barrir... E nr gy, E ( M V ) / E^.5 (MV^-.5) Matrix Elmnt (fm) Equ.(4), Without Coulomb Barrir NSWb, With Coulomb Barrir Figur: Comarison of S-P Matrix Elm nt With and Without Coulomb Barrir

14 4. Drivation of th -D Matrix Elmnt Having stablishd th tchniqu, NSWb is now usd togthr with th ffctiv nuclar otntials of Sction to valuat th matrix lmnt for th -D systm. W choos (arbitrarily) th a = fm otntial with V = 4.4 MV. Th Tabl blow givs th matrix lmnt (in fm) as a function of nrgy. Equ.4 is also shown for comarison, using th hlium- binding nrgy (7.78 MV). Th rducd mass is m = MV, and th rfrnc normalisation radius is r max = 57. fm, as usd abov. (Not that th actual normalisation is carrid out ovr a gratr radius for low nrgy fr stats, s abov discussion). Valus of sin /,S(bound) r cos sin /,P(fr) E (MV) matrix lmnt (fm) from NSWb (with Coulomb barrir) for th -D Systm (fm) matrix lmnt (fm) from Equ.4 (no Coulomb barrir) (B) (V c ) E E-7...6E E-.76

15 Figur: -D <S-P> Matrix Elmnt (Comarison with n- matrix lmnt and Equ.4) NSWb With Coulomb Barrir Equ.4 (for comarison only, no Coulomb barrir) Equ.4 for n- matrix lmnt Matrix Elmnt (fm) Enrgy, E (MV) / E^.5 (MV^-.5). E-4 E-5 E-6 Figur: -D <S-P> Matrix Elmnt (Comarison with n- matrix lmnt and Equ.4) E-7 E-8 E-9 E- NSWb With Coulomb Barrir Equ.4 (for comarison only, no Coulomb barrir) Equ.4 for n- matrix lmnt E- Th grahs show that thr is a band of nrgis (roughly to 5 MV) whr th n- and -D matrix lmnts ar of similar magnitud, but that for othr nrgis th -D matrix lmnt tnds to b much smallr. Its largst valu is about half that for n-. At sufficintly larg and sufficintly small nrgis, th -D matrix lmnt is ordrs of magnitud smallr than that for n-. At low nrgis, this is th rsult of th Coulomb barrir. Th last air of data oints givs a valu b =.4 for th aramtr b / E in th nrgy factor. This comars with th thortical stimat of.45 from Equ.7 (rcalling that -D has a largr rducd mass than n-). Hnc, th low nrgy bhaviour conforms to xctation.

16 Th rason for th str rduction in th matrix lmnt at high nrgis is unclar. 5. Calculation of th + D H + Raction Rat at a Givn Tmratur Following Andix A, th raction rat for th hotodisintgration raction H + + D is, mr max w (E ) S(bound) r cos P(fr) (Equ.8) k and w is th robability of disintgration r scond r hlium- nuclus. Th scond trm in brackts in Equ.8 is th dnsity of final stats. For th numrical valus of th matrix lmnt givn abov w must us, for consistncy, r max = 57. fm. Th lading factor of two accounts for th fact that th hoton may coul to ithr of th two rotons, and hnc doubls th raction rat. [Thr is a roblm of rincil hr. If th hlium- is rgardd as a articl of charg, thn th matrix lmnt involvs (E ), and this rsults in doubl th raction rat givn by Equ.8. Why is it not corrct to do th calculation this way? W not that at th hoton nrgis involvd, i.. ~5.5MV or abov, th hoton wavlngth is ~fm, and so th hoton cannot rsolv th two rotons. It may b that Equ.8 is wrong and should b multilid by ]. Rcall from Andix A that th cross-sction, hoton flux and fin structur constant ar givn by, dis w N J and J N E c and c 4 (Equ.9) whr is th hoton nrgy. Hnc, Equ.8 givs, m r 8 (Equ.) k max dis S(bound) r cos P(fr) W notd in Andix A that our drivation of th lctric diol cross-sction was too larg by a mystrious factor of 4 which w faild to trac. Anticiating that th sam rror occurs hr, w r-writ Equ. as, m r (Equ.) k max dis S(bound) r cos P(fr) Insrting th valus of th constants into Equ. givs, dis. S(bound) r cos P(fr) 8 (Equ.) kc

17 It is th matrix lmnt in Equ. which carris th dimnsions. Thus if th matrix lmnt is valuatd in fm (as in th abov numrical data) th cross-sction is in fm =. barn. To calculat th catur cross-sction w us, D H H D numbr of numbr of H sin stats D sin stats (Equ.) In th cntr of mass systm, th roton and dutron momnta ar qual and oosit and both qual in magnitud to k me, whr m is th rducd mass and E is th sum of th roton and dutron kintic nrgis in th CoM systm. For th low hoton nrgis in qustion, i.. not much mor than th minimum to caus hotodisintgration, th kintic nrgy of th hlium- nuclus is ngligibl (about.% of that of th hoton). Thus, th nrgy of th hoton rlasd by roton catur is givn accuratly by E lus th diffrnc of th binding nrgis of hlium- and th dutron, i.. B MV. Hnc th factors which occur in Equs. and ar, kc E B mc E (Equ.) So Equs.,, giv, E B 5 (Equ.4) mc E ca.55 S(bound) r cos P(fr) Thus, if w considr thr to b a dnsity of dutrons of mol r cm (i.. N 9 D 6.x m - ), thn th numbr of catur ractions r m r scond r roton is, Raction Rat 6.x 9 v (Equ.5) ca whr v is th rlativ closing sd of th roton and dutron. Thus Equ.5, togthr with Equ.4, suffic to find th raction rat if w know th rlativ sd (v) of th ractants in th laboratory fram of rfrnc, and w know th total kintic nrgy (E) in th CoM fram. Unfortunatly, for a gas of rotons and dutrons in thrmal quilibrium thr is a continuous distribution of ths variabls. For th nonrlativistic nrgis of intrst, th nrgis ar givn by th Maxwll distribution, P [ ]d d (Equ.6) whr E / kt and th distribution alis to both th roton and th dutron. W dnot th raction rat at tmratur T by R[T]. Convrsly, if all th rotons had sd v in th lab fram, and all th dutrons sd v D, and th two vlocitis wr at an angl, th rsulting raction rat is dnotd R[v, v D, ]. If w know th lattr w can find R[T] from,

18 R [T] P[ d(cos ]d P[ D ]d D R[v, v D, ] ) (Equ.7) M v M M v Dv D D whr, and D. Bcaus th random motions ar kt kt kt isotroic, all solid angls ar qually likly, hnc th uniform robability dnsity in cos. Now R[v, v D, ] will b givn by Equs.4, 5 rovidd w can find th lab fram rlativ sd (v) and th CoM systm total nrgy (E) in trms of v, v D and. In th (vry good) aroximation that th dutron is twic th mass of th roton, it is siml to show that, v v v D v v D cos and E M v (Equ.8) Thus, th raction rat at tmratur T can b found by numrically intgrating Equ.7 using Equs.4, 5, 6 and 8. Th rsults ar comard with th raction rats givn by Hoffman t al in th Tabl blow, Tmratur (K) Raction Rat, Equ.7 s - r mol/cm Raction Rat, Hoffman t al s - r mol/cm x () x x x x x x () Unrliabl xtraolation Although th agrmnt at individual tmraturs is not always imrssiv, this is artly bcaus th raction rat is so snsitiv to tmratur. Ovrall th agrmnt is rmarkably good givn th simlicity of our tratmnt, as th following grah shows,

19 Figur: -D Catur Raction Rat Calculatd Hr Comard With Hoffman t al Raction rat, /s r mol/cm^ Calculatd Hr Hoffman t al Tmratur (K).E+7.E+8.E+9.E Discussion of th Tmratur Dndnc of th Raction Rat If w xamin th tmratur dndnc of th rdictd raction rats givn in th abov tabl/grah (using Equ.7 rathr than Hoffman, for dfinitnss) w s that fitting to th functional form Rat x{ b / E }, whr th man nrgy is.5kt, w gt numrical valus for b of about.85 at ~ 7 K, rising to ~.8 at ~4 x 8 K. It is imortant to ralis that th raction rat at a givn tmratur dos not driv only from th nrgy dndnc of th matrix lmnt. Th lattr has a b aramtr of ~.45, and bcaus it is squard in th xrssion for cross-sction, would imly a raction rat nrgy dndnc with b ~.8 if thr wr no othr sourcs of nrgy dndnc. In fact thr ar thr othr sourcs of tmratur dndnc. Th first two ar dislayd in Equs.(4) and (5). Thus, if w considr low nrgis, E << B, thn Equ.(4) has a factor of E / in th dnominator, in addition to th nrgy dndnt matrix lmnt. Euq.(5) introducs furthr nrgy dndnc via th vlocity factor. Ignoring th comlication of dndnc, th vlocity varis ssntially as E. Hnc, combining Equs(4) and (5) w hav a nt factor of /E. Howvr, th third sourc of tmratur dndnc is th most imortant, namly th Maxwll distribution of nrgis at a givn tmratur. This is dislayd xlicitly in Equs.(6, 7, 8). In th numrical valuation, abov, w hav corrctly accountd for all thr of ths tmratur dndncis. Thy ar th rason why th fittd b aramtr is not simly twic that for th matrix lmnt. It is instructiv to attmt an aroximat analytic drivation of th tmratur dndnc of th raction rat. From th abov discussion, ignoring th comlication of th dndnc, and assuming th nrgy dndnc of th matrix lmnt can b

20 aroximatd as x{ b / E}, w xct th tmratur dndnc of th raction rat at tmratur T to b roortional to, Raction rat de E E de b E E x (kt) E E E E kt b y x y dy y kt y x dx (Equ.9) (kt) y y whr, y = E + E = E, and x = E E. Not that th rang of intgration E [, ] and E [, ] mas into y [, ] and x [-y, +y], sinc E = (x + y)/ and E = (y x)/. Also not that th aramtr b in th abov is twic that alying for th matrix lmnt, and hnc is ~.8 for sufficintly small E. Th scond intgral is just y /, so (9) bcoms, Raction rat dy (kt) yx b y y kt (Equ.) W can stimat () by xanding th xonnt as a owr sris, noting that it attains a minimum whn, E.5bkT y (Equ.) this bing th nrgy which contributs most to th ovrall raction rat. [NB: At lowr nrgis th matrix lmnt and hnc th raction rat rducs raidly, whras at highr nrgis th Maxwll distribution nsurs that thr ar a raidly rducing numbr of articls with such nrgis]. Thus, f (y) b y y kt f min.5f (y y )... (Equ.a) whr, b 5 f min and f by (Equ.b) kt 4 Thus, () is aroximatd as, Raction rat y fmin x {.5f (y y ) } dy (Equ.) (kt)

21 y (kt) y y (kt) 9 4 (kt) kt (kt) fmin (kt) fmin fmin fmin 9 4 y fmin 5 f (Equ.4) Th dominant dndnc is th xonntial, but from Equ.(b) w s that this is not a x / E factor but has a diffrnt nrgy dndnc in th xonnt, namly f min b kt. If w quat Equ.(4) to an ffctiv trm of th form x / E, assuming b =.8, thn th nrgy dndnt aramtr varis as follows, T ( 6 K) E =.5kT (MV) f min dimnsionlss f min * (kt) MV * Normalisd by its valu at T = million K. Thus, th ffctiv b aramtr is ~. for low tmraturs, which comars with th numrical drivation abov which imlis that b is around for tmraturs of to million K. Th agrmnt is rasonabl if not scially imrssiv. Th main oint, howvr, is that th abov considrations xlain why th ffctiv b diffrs markdly from twic th matrix lmnt valu for b, i.. ~.8. At highr tmraturs th numrical drivation shows that b incrass, but our stimat, abov, would not thn b xctd to b accurat sinc th matrix lmnt thn cass to vary in th assumd mannr.

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