The nuclear fusion reaction rate based on relativistic equilibrium velocity distribution

Transcription

1 he nulear fusion reation rate based on relativisti equilibriu veloity distribution Jian-Miin Liu* Departent of Physis, Nanjing University Nanjing, he People's Republi of China *On leave. E-ail address: ABSRAC he Coulob barrier is in general uh higher than theral energy. Nulear fusion reations our only aong few protons and nulei with higher relative energies than Coulob barrier. It is the equilibriu veloity distribution of these high-energy protons and nulei that partiipates in deterining the rate of nulear fusion reations. In the irustane it is inappropriate to use the Maxwellian veloity distribution for alulating the nulear fusion reation rate. We use the relativisti equilibriu veloity distribution for this purpose. he rate based on the relativisti equilibriu veloity distribution has a redution fator with respet to that based on the Maxwellian distribution, whih fator depends on the teperature, redued ass and atoi nubers of the studied nulear fusion reations. his signifies uh to the solar neutrino proble. PACS: 8.5, 05.0, 03.30, Introdution. o reate a nulear fusion reation, a proton or nuleus ust penetrate the repulsive Coulob barrier and be lose to another proton or nuleus so that the strong interation fore between the ats. he Coulob barrier is in general uh higher than theral energy. Nulear fusion reations an take plae only aong few protons and nulei with higher relative energies than Coulob barrier. If, under the onditions for nulear fusion reations, interating protons and nulei reah their equilibriu distribution in the period of tie that is infinitesial opared to the ean lifetie of nulear fusion reations, it is the equilibriu veloity distribution of these high-energy protons and nulei that partiipates in deterining the rate of nulear fusion reations. In this irustane, it is inappropriate to use the Maxwellian veloity distribution to alulate the nulear fusion reation rate. In our reent work, we ade the relativisti orretions to the Maxwellian veloity distribution [,] and explained the observed non-maxwellian high-energy tails in veloity distributions of astrophysial plasa partiles [3]. We nae the orreted distribution the relativisti equilibriu veloity distribution keeping in ind that the question whether suh a nae is suited to it is a atter to be deided in the light of experiene. In this letter, we use the relativisti equilibriu veloity distribution to alulate the nulear fusion reation rate. Veloity spae and relativisti equilibriu veloity distribution. Soe results in Refs.[,] are quoted here for use below. he veloity spae in the speial theory of relativity and the odified speial relativity theory [4] an be represented by dy =δ rs dy r dy s, r,s=,,3, () in the so-alled pried veloity-oordinates {y r }, r=,,3, or by dy =H rs (y)dy r dy s, r,s=,,3, (a) H rs (y)= δ rs /( -y )+ y r y s /( -y ), real y r and y<, (b) in the usual veloity-oordinates {y r }, r=,,3, where y r is alled the pried veloity, y r is the welldefined Newtonian veloity, y=(y r y r ) / is the radial oponent of y r that we siply all the radial veloity, and is the speed of light. he pried and the usual veloity-oordinates are onneted by dy r =A r s(y)dy s, r,s=,,3, (3a) A r s(y)=γδ rs +γ(γ-)y r y s /y, (3b) beause δ rs A r p(y)a s q(y)=h pq (y), where γ=/(-y / ) /. (4) In the veloity spae, the veloity-length between two pried veloities y r and y r or two Newtonian veloities y r and y r is Y(y r,y r )= [(y r -y r )(y r -y r )] /, (5)

2 or Y(y r,y r )= n b + a, b a (6a) b= -y r y r, r=,,3, (6b) a={( -y i y i )(y j -y j )(y j -y j )+[y k (y k -y k )] } /, i,j,k=,,3. (6) Eqs.(5) and (6a-6) iply y r =[ n + y y y, r=,,3, (7a) y = n + y y (7b) when (y, y, y 3 ) and (y, y, y 3 ) represent the sae point, where y =(y r y r ) /. Differentiating Eq.(7b), we get dy dy =. (8) ( y / ) In the veloity spae, the Galilean addition law of pried veloities links up to the Einstein addition law of their Newtonian veloities. he Eulidean struture of the veloity spae in the pried veloityoordinates onvines us of the Maxwellian distribution of pried veloities, P(y,y,y 3 )dy dy dy 3 =N( ) 3/ exp[- (y ) ]dy dy dy 3, (9a) πk B K B P(y )dy =4πN( ) 3/ (y ) exp[- (y ) ]dy. (9b) πk B K B where N is the nuber of partiles, their rest ass, the teperature, and K B the Boltzann onstant. Putting Eqs.(3a-3b), (7b) and (8) into Eqs.(9a-9b), we find the relativisti equilibriu distribution of Newtonian veloities, P(y,y,y 3 )dy dy dy 3 = N ( / ) 3/ πk B exp[- ( n + y ( y / ) 8K B y ) ]dy dy dy 3, (0a) P(y)dy= π N ( / ) 3/ πk B ( n + y ( y / ) y ) exp[- ( n + y 8K B y ) ]dy. (0b) his distribution is lose-fitting to the Maxwellian distribution for low-energy partiles (y<<) but substantially differs fro the Maxwellian distribution for high-energy partiles. As y goes to, it falls off to zero slower than any exponential deay and faster than any power-law deay [3]. ransforation of relative veloities. Let partile and partile ove with Newtonian veloities y r and y r, r=,,3, respetively. We denote the relative veloity of partile to partile with v r, r=,,3. If three orresponding pried veloities are respetively y r, y r, v r, r=,,3, the Galilean addition law aong the reads v r =y r -y r, r=,,3, () and the Einstein addition law aong the three Newtonian veloities is v r = y / {(y r -y r )+ ( -)y y s y s y s r ( ) y k y k }/[- ], r,s,k=,,3. y / y Due to Eqs.(5) and (6a-6), we have ()

3 [(y r -y r )(y r -y r )] / = n b + a b a with b= -y r y r, r=,,3, a={( -y i y i )(y j -y j )(y j -y j )+[y k (y k -y k )] } /, i,j,k=,,3, or equivalently tanh {[(y r -y r )(y r -y r )] / /}= {( -y i y i )(y j -y j )(y j -y j )+[y k (y k -y k )] }/( -y r y r ). (3) We separately use Eqs.() and () on the left-hand and the right-hand sides of Eq.(3) and obtain tanh {v /}=v, (4) where v =(v r v r ) / and v=(v r v r ) /. Eq.(4) speifies the transforation between relative Newtonian veloity v r and its relative pried veloity v r, v = n + v, (5a) v v r = ( v n + v v )vr, r=,,3. (5b) Differentiating Eq.(5a) iediately yields dv dv =. (6) ( v / ) Equilibriu distribution of relative veloities. Suppose we have two types of partiles, type and type, and the type i partiles have ass i, i=,. It has been proved [5] that when veloities of the type partiles, as well as those of the type partiles, obey the Maxwellian distribution, the relative veloities of the type partiles to the type partiles obey the like, in whih provided we take redued ass = /( + ). In Eqs.(9a-9b) we see that the pried veloities of the type partiles and of the type partiles obey the Maxwellian distribution. he distribution of the relative pried veloities of the type partiles to the type partiles ust be in aordane with P*(v,v,v 3 )dv dv dv 3 =N( ) 3/ exp[- (v ) ]dv dv dv 3, (7a) πk B K B P*(v )dv =4πΝ( ) 3/ (v ) exp[- K B (v )]dv. (7b) πk B Inserting Eqs.(5a) and (6) in Eq.(7b), we obtain the relativisti equilibriu distribution for radial relative (Newtonian) veloities of the type partiles to the type partiles, P*(v)dv=π N ( / ) / 3 πk B ( n v + ( v / ) v ) exp[- ( n + v 8K B v ) ]dv. (8) Nulear ross setion. Now we onsider a kind of nulear fusion reations ourring between protons or nulei of type and protons or nulei of type, where protons or nulei of type i have density N i, ass i and atoi nuber z i, i=,. he rate of these nulear fusion reations is R= N N vσ ( v) = N N vσ( v) f ( v) dv, (9) ( + δ ) ( + δ ) where v denotes the radial relative veloities of the type protons or nulei to the type ones, σ(v) is a ross setion of nulear fusion reations of the kind, eans the therodynai-equilibriu average over v, 0 v<. he noralized equilibriu distribution f(v), 0 0 f ( v) dv =, an be piked out in Eq.(8). 3

4 S As for the ross setion σ(v), its reognized for of exp[- πzz e v ] is no longer suitable / v here. he ross setion σ(v) is a produt of three fators: the penetration probability fator, the slowly varying fator S and the fator of the squared de Broglie wavelength. he penetration probability fator desribes a distribution of the probability for an inoing proton or nuleus with atoi nuber z to penetrate through the repulsive Coulob barrier of a target proton or nuleus with atoi nuber z due to quantu tunnel effet [6,7], as well as f(v) is a distribution of the probability for the inoing proton or nuleus to have radial veloity v relative to the target proton or nuleus. he penetration probability fator is also a funtion of radial relative veloity. We ought to treat it in the sae way that we did for f(v). In other words, we take exp[- πzze ] for the penetration probability fator in the pried veloityoordinates and exp[-πz z e / ( v' n + v )] in the usual veloity-oordinates, where Eq.(5a) is used. v he other two fators are together with the penetration probability fator to for the ross setion as a whole, so we deal with these two fators onsistently keeping their original fors in the pried veloityoordinates and looking for their fors in the usual veloity-oordinates with the aid of Eq.(5a). otally, for the ross setion, we have S exp[- πzze ] (0) v' / v' in the pried veloity-oordinates and {S/ ( n + v v ) }exp[-πz z e / ( n + v )] () v in the usual veloity-oordinates. Readers ay refer to Ref.[8] for ore explanations of this treatent. Nulear fusion reation rate. o alulate the nulear fusion reation rate, we take P*(v)/N in Eq.(8) for f(v) and put it and Eq.() into Eq.(9). Introduing a new variable, x= n + v, we find v R= 8π ( x x ) 3/ N N S[tanh( )] exp[- - πk B ( + δ ) πzze ]dx. () K 0 B x Calling tanh( x n n )= ( ) n+ ( ) Bn x n ( ), n= ( n)! in a region, where B n, n=,,3,------, are the Bernoulli nubers, B =/6, B =/30, B 3 =/4, , and using the ethod of the steepest desents, we further find R= R n, (3) n= 4 R n = N N S eff exp[-λ] ( ) ( + δ ) 3K B where S eff is the average of slowly varying funtion S and λ= 3 K Bπzze ( K B ) /3 is n-independent. he ost effetive energy is n+ n n ( ) ( n)! B n ( πz z K B e ) n 3 (4), 4

5 K πz z e B E 0 =( ) /3, whih is also n-independent. In a opat for, we an rewrite R as 4 B R= N N S eff exp[-λ]tanh{( π ( + δ ) 3K B or z z K e ) /3 } (5) R= tanhq Q R M, (6a) Q=( πz z K B e ) /3, (6b) 4 R M = N N S eff exp[-λ]( πz z K B e ( + δ ) ) /3, (7) 3K B where R M is the nulear fusion reation rate based on the Maxwellian veloity distribution. Conluding rearks. he nulear fusion reation rate based on the relativisti equilibriu veloity distribution has a redution fator with respet to that based on the Maxwellian veloity distribution: tanhq/q. Sine 0<Q<, the redution fator satisfies 0<tanhQ/Q<. hat gives rise to 0<R<R M. (8) he redution fator depends on the teperature, redued ass and atoi nubers of the studied nulear fusion reations. For a statistial alulation relevant to equilibriu veloity distribution, in two ases we have to substitute the relativisti equilibriu veloity distribution for the Maxwellian distribution. One ase is where ost partiles rowd in the high-energy region. he other ase is in whih ost partiles rowd in the low-energy region but these partiles in the low-energy region are not involved in the statistial alulation. he alulation for the nulear fusion reation rate belongs here. When ost partiles rowd in the low-energy region and, onurrently, these partiles in the low-energy region are involved in the statistial alulation, substitution for the Maxwellian distribution is not so iportant. In solar interior, the height of Coulob barrier is far above solar theral energy: their ratio is typially greater than a thousand [9,0], and the interating protons and nulei reah their equilibriu distribution in a very short tie whih is infinitesial opared to the ean lifetie of a nulear fusion reation [0]. he obtained results in the ontext are appliable to all kinds of solar nulear fusion reations. As seen in Eq.(8), the relativisti equilibriu veloity distribution lowers the rates of solar nulear fusion reations. It will hene lower the solar neutrino fluxes. It will also hange the solar neutrino energy spetra. On the other hand, sine ost solar ions rowd in the low-energy region, at teperatures and densities in the Sun, and sine these ions are involved in the alulation of solar sound speeds, substituting the relativisti equilibriu veloity distribution for the Maxwellian veloity distribution is not iportant to the alulation of solar sound speeds. he relativisti equilibriu veloity distribution, one adopted in standard solar odels, will lower solar neutrino fluxes and hange solar neutrino energy spetra but aintain solar sound speeds. his work does not inlude any sreening orretion to the nulear fusion reation rate [-3]. ACKNOWLEDGMEN he author greatly appreiates the teahings of Prof. Wo-e Shen. he author thanks Prof. Gerhard Muller for useful suggestions. REFERENCES [] Jian-Miin Liu, Chaos Solitons&Fratals,, 49 (00) 5

6 [] Jian-Miin Liu, ond-at/ [3] Jian-Miin Liu, ond-at/0084 [4] Jian-Miin Liu, Chaos Solitons&Fratals,, (00) [5] D. D. Clayton, Priniples of Stellar Evolution and Nuleosythesis, University of Chiago Press, Chiago (983) [6] R. D E. Atkinson and F. G. Houterans, Zeits. Physik, 54, 656 (99) [7] G. Gaow, Phys. Rev., 53, 595 (938) [8] Jian-Miin Liu, Modifiation of speial relativity and its ipliations for the divergene proble in quantu field theory and the solar neutrino proble in standard solar odels, to be published [9] J. N. Bahall, Neutrino Astrophysis, Cabridge University Press, Cabridge (989) [0] S. urk-chieze et al, Phys. Rep., 30, 57 (993) [] J. N. Bahall et al, astro-ph/ [] G. Shaviv, astro-ph/0005 [3] G. Fiorentini, B. Rii and F. L. Villante, astro-ph/0030 6