Resonant nuclear reaction ^{23}Mg (p,γ) ^{24}Al in strongly screening magnetized neutron star crust
aa r X i v : . [ nu c l - t h ] O c t Resonant nuclear reaction Mg ( p, γ ) Al in strongly screeningmagnetized neutron star crust
Jing-Jing Liu , and Dong-Mei, Liu College of Marine Science and Technology, Hainan Tropical Ocean University, Sanya,Hainan 572022, ChinaReceived ; acceptedNot to appear in Nonlearned J., 45. Corresponding author: [email protected] 2 –
ABSTRACT
Based on the relativistic theory of superstrong magnetic field (SMF), by usingthree models of Lai (LD), Fushiki (FGP), and ours (LJ), we investigate theinfluence of SMFs due to strong electron screening (SES) on the nuclear reaction Mg ( p, γ ) Al in magnetars. At relatively low density environment (e.g., ρ < .
01) and 1 < B < , our screening rates are in good agreement with those ofLD and FGP. However, in relatively high magnetic fields (e.g., B > ), ourreaction rates can be 1.58 times and about three orders of magnitude larger thanthose of FGP and LD, respectively ( B , ρ are in units of 10 G, 10 g cm − ).The significant increase of strongly screening rates can imply that more Mg willescape from the Ne-Na cycle due to SES in a SMF. As a consequence, the nextreaction Al ( β + , ν ) Mg will produce more Mg to participate in the Mg-Alcycle. Thus, it may lead to synthesize a large amount of production of
A >
Subject headings: dense matter— nuclear reactions, nucleosynthesis, abundances—stars: magnetic fields—stars: interiors
1. Introduction
In the dense sites of universe, such as novae, X-ray bursts and supernova, there areexplosive hydrogen burning process in high temperature and high hydrogen environments.This burning is called the rapid-proton (rp) process (Wallace et al. 1981). In the stage ofhydrogen burning, the proton capture reactions and β + -decays (rp-process) will be ignitedin the nuclei whose mass numbers A >
20. For example, the timescale of the proton capturereaction of Mg in the Ne-Na cycle at sufficient high temperature is shorter than that ofthe β + -decay. Therefore, some Mg will kindle and escape from the Ne-Na cycle by protoncapture. The Mg leaks from the Ne-Na cycle into the Mg-Al cycle synthesizing a largeamount of heavy nuclei. Thus the reaction Mg ( p, γ ) Al in stellar environment is animportant reaction for producing heavy nuclei. Wallace et al. (1981) firstly discussed thereaction rate of Mg ( p, γ ) Al. Then, Iliadis et al. (2001) also investigated this nuclearreaction rate. Kubono et al. (1995) reconsidered the rate by considering four resonancesand the structure of Al. Based on some new experimental information on Al excitationenergies, Herndl et al. (1998); Visser et al. (2007), and Lotay et al. (2008) carried out anestimation of the rate. However, they all seem to have overlooked the influence of electronscreening on nuclear reaction.In the pre-supernova stellar evolution and nucleosynthesis, the strong electron screening(SES) is always a challenging and interesting problem. Some works (Bahcall et al. 2002;Liu 2013, 2014, 2016; Liu et al. 2017,?) have been done on stellar weak-interaction ratesand thermonuclear reaction rates. In the high-density surrounding, some SES models havebeen widely investigated, such as Salpeter model (Salpeter 1954; Salpeter et al. 1969),Graboske model (Graboske et al. 1973), and Dewitt model (Dewitt et al. 1976). Recentlythese issues were discussed by Liolios et al. (2000, 2001), Kravchuk et al. (2014), and Liu(2013). However, they neglected the effects of SES on thermonuclear reaction rate in 4 –superstrong magnetic field (SMF).It is widely known that nuclear reaction rates at low energies play a key role in energygeneration in stars and the stellar nucleosynthesis. The bare reaction rates are modifiedin stars by the screening effects of free and bound electrons. The knowledge of the barenuclear reaction rates at low energies is important not only for the understanding of variousastrophysical nuclear problems, but also for assessing the effects of host material in lowenergy nuclear fusion reactions in matter.It is universally accepted that the surface dipole magnetic field strengths of magnetarsare in a range from 10 to 10 G (Peng et al. 2007; Gao et al. 2011, 2013, 2015, 2017,?;Li et al. 2016; Lai 2001). The momentum space of the electron gas is modified substantiallyby so intense magnetic fields. The electron Fermi energy and nuclear reaction are alsoaffected greatly by a SMF in magnetars.Anamalou x-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are conceived asmagnetars, which are a kind of special pulsars powered by their magnetic energy (Duncan1992). The Fermi energy of the electrons will increase with magnetic field and quantumeffects of electron gas will be very obvious in a SMF. As we all know, the positive energylevels of electrons must abide by Landau quantization. The distribution of the electron inthe momentum space will be strongly modified by a SMF. Some authors discussed this issuein detail in strong magnetic fields of magnetars. For instance, Gao et al. (2015, 2017,?)investigated not only the spin-down and magnetic field evolutions, but also the electronLandau level effects on emission properties of magnetars.In this paper, according to the relativistic theory in a SMF (Peng et al. 2007; Gao et al.2011, 2013, 2015, 2017), we discuss the problem of SES and then investigate the effect ofSES on the thermonuclear reaction within three different models (i.e., our model (LJ), Laimodel (LD)(Lai et al. 1991; Lai 2001), and Fushiki model (FGP)(Fushiki et al. 1989)) on 5 –the surface of magnetars.Our work differs from previous work of Liu (2016) about the discussion of nuclearreaction rates. Firstly, in Liu (2016), though we cited several work from Gao et al., but it isnot familiar to the calculations involved in electron Fermi energy in a superhigh magneticfield, a non-relativistic electron cyclotron solution was applied when calculating the rates.Secondly, our previous work Liu (2016) did not give a comparison among LJ, LD, and FGPmodels in the case with a SMF. Finally, we analyze the nuclear reaction rates in a SMF andalso give a comparison for our model with Dewitt model(Dewitt et al. 1976), and Lioliosmodel(Liolios et al. 2000), in which the SMF were not taken into consideration. Maybe SESuniversally occur in pulsars, and the screening rate calculations in a SMF is of importanceto the future studies on cooling, nucleosynthesis,and emission properties of magnetars.In this paper, following the works of Peng et al. (2007), and Gao et al. (2011, 2013,2015, 2017), we calculate the resonant reaction rates in the case with SMF and withoutSMF in several screening models. In the case of the former, the results from LD and FGPmodels will be compared with those of our model, while in the latter case, the results fromDewitti and Liolio models also will be compared. We derive new results for SES theory andthe screening rates for nuclear reaction in relativistic strong magnetic fields.The article is organized as follows. In the next Section, we analyse three SES modelsin a SMF of magnetars. In Section 3 we discuss the effects of SES on the proton capturereaction rate of Mg, in which the four resonances contributions will also be considered.The results and discussions will be shown in Section 4. The article is closed with someconclusions in Section 5. 6 –
2. The SES in SMF
In astrophysical systems, the SMF may have significant influence on the quantumprocesses. In this Section, we will study three models of the electron screening potential(ESP) in SMF, i.e., LJ model, LD model, and FGP model.
The rate of nuclear reaction in high density matter is affected by the fact that theclouds of the electrons surrounding nuclei alter the interactions among nuclei. The positiveenergy levels of electrons in SMF are given by (Landau et al. 1977) ε n m e c = [( p z m e c ) + 1 + 2( n + 12 + σ ) b ] / = ( p + Θ) / , (1)where Θ = 1 + 2( n + + σ ) b , n = 0 , , , .... , b = BB cr = 0 . B , B is the magneticfields in units of 10 G, i.e., B ≡ B/ G, B cr = m c e ~ = 4 . × G is the electronquantum critical magnetic field, and p z is the electron momentum along the field, σ is thespin quantum number of an electron, when n = 0, σ = 1 /
2, and when n ≥ σ = ± / B ≫ B cr ), the Landau column becomes avery long and narrow cylinder along the magnetic field. According to the Pauli exclusionprinciple, the electron number density should be equal to its microscopic state density. Byintroducing the electron Landau level stability coefficient, the Fermi energy of the electronis given by (Gao et al. 2013; zhu et al. 2018) U F = 5 . × ( BB cr ) / ( ρY e ρ × . / = 5 . × ( BB cr ) / ( n e . × ρ N A ) / keV , (2)where ρ = 2 . × g / cm is the standard nuclear density. 7 –In order to evaluate the Thomas-Fermi screening wave-number K LJTF , we defined aparameter D LJ ( U e ) and according to Eq.(3), we have n e = 0 . ρ N A ( U F . × b / ) (3) D LJ ( U F ) = ∂n e ∂U F = ∂∂U F (0 . ρ N A ( U F . × b / ) )= 4 . × n / b − / cm − KeV − . (4)According to Eq.(5), the Thomas-Fermi screening wave-number K LJTF is given by(Ashcroft et al. 1976) ( K LJTF ) = 4 πe D LJ ( U F ) = 4 πe ∂n e ∂U F = 6 . × e ( n e ) / b − / cm − . (5)By using the uniform electron gas model (Kadomtsev 1971), the binding energy of themagnetized condensed matter at zero pressure can be estimated. The energy per cell canbe written as E total = E k + E latt = 3 π e z j b r i + 9 e z r e MeV , (6)where the first term is the kinetic energy and the second term is the lattice energy. r i = z / r e a is the Wigner-Seitz cell radius, a = 0 . × − cm is the Bohr radius, and r e = (3 / πn e ) / is the mean electron spacing. z j is the charge number of the species j . b = B/B = 425 . B = 1 . × b and B = m ce / ~ = 2 . × − G is the natural(atomic) unit for the field strength (Lai 2001). For the zero-pressure condensed matter, werequire dE total /dr i = 0, so we have r i = r i = 0 . z / j b − / a cm . (7)By using linear response theory, the energy correction per cell due to non-uniformity is 8 –given by (Lattimer et al. 1985) E LJTF ( r i , z j ) = − K LJTF r i ) ( z j e ) r i = − . × − e ( n e ) / z / j b / MeV . (8)For the relativistic electrons, the influence from exchange free energy were discussedby Refs.Stolzmann et al. (1996); Yakovlev et al. (1989). Their works showed that thecorrelation correction is very small. Therefore, in this paper we have neglected the correctionof Coulomb exchange free energy interaction in the electron gas model. By taking intoconsideration of the Coulomb energy and Thomas-Fermi correction due to non-uniformityof the electron gas, the energy per cell should be corrected as E LJs ( r i , z j ) = E k ( r i , z j ) − U coul ( r i , z j ) − E LJTF ( r i , z j ) . (9)For two interaction nuclides, the energy required to bring two nuclei with nuclearcharge numbers z and z so close together that they essentially coincide differs from thebare Coulomb energy by an amount which in the Wigner-Seitz approximation is U sc = E s ( r i , z ) − E s ( r i , z ) − E s ( r i , z ) , (10)where z = z + z . If the electron distribution is rigid, the contribution to from E s thebulk electron energy cancel in expression (11), and the screening potential is simply givenas U sc = E coul ( r i , z ) − E coul ( r i , z ) − E coul ( r i , z )= 6 . × b / ( z / − z / − z / )MeV , (11)where we assume the electron density is uniform, and the screening potential is independentof the magnetic field. 9 –From expression (9), the change of the screening potential due to the compressibilityof the electrons in the zero-pressure magnetized condensed matter can obtained as δE LJTF = − K LJTF r i ) e ( z − z − z ) r i = − . × − e n / ( z / − z / − z / ) b / . (12)In accordance with the above discussions, the total screening potential is the sum ofthe screening potential with a uniformity distribution and a corrected screening potentialwith a non-uniformity distribution. The screening potential in SMF is given by U LJsc = U sc + δE LJTF . (13) Lai (2001) and Lai et al. (1991) discussed the equation of state and the electron energyin a SMF. In a SMF the electron number density n e is related to the chemical potential U e by n e = 1(2 π b ρ ) ~ ∞ X g n Z + ∞−∞ f dp z = 1(2 π b ρ ) ~ ∞ X g n Z + ∞−∞ [1 + exp( E − U e kT )] − dp z , (14)where b ρ = ( ~ c/eB ) / = 2 . × − B / cm is the electron cyclotron radius (thecharacteristic size of the wave packet), and E = [ c p z + m e c (1 + nb )] / is the free electronenergy, g n is the spin degeneracy of the Landau level, g = 1 and g n = 2 for n >
1, andthe Fermi-Dirac distribution is given by f = [1 + exp( E − U e kT )] − . (15) 10 –The electron Fermi energy including the electron rest mass is given by n e = 12 π / λ Te b ρ ∞ X ( n =0) g n I − / ( U e − n ~ ω ce kT ) , (16)where the thermal wavelength of the electron is λ Te = (2 π ~ /m e kT ) , and the Fermi integralis written as I n ( y ) = Z ∞ x n exp( x − y ) + 1 dx. (17)The binding energy of the magnetized condensed matter at zero pressure can beestimated using the uniform electron gas model. Under the condition of super-strongmagnetic field, the Fermi energy U F is less than the cyclotron energy ~ ω ce , the electronsonly occupy the ground Landau level. According to their viewpoint of (Lai 2001), theThomas-Fermi screening wave-number is given by( K LDTF ) = 4 πe D LD ( ε F ) = 4 πe ∂n e ∂ε F = 4 πe ∂n e ∂U F , (18)where ∂n e /∂ε F is the density of states per unit volume at the Fermi surface. ε F = P / m e .From Eq.(6.16) of Lai (2001), so we have D LD = ∂n e ∂ε F = 3 . × b r e . (19)The Thomas-Fermi screening wave-number will be given by K LDTF = ( 43 π ) / b r / = 6 . × br / . (20)Using the linear response theory, the energy correction (in atomic units) per cell dueto non-uniformity can be calculated and gives by (Lai 2001) E LDTF ( r i , z j ) = − K LDTF r i ) e z j r i = − . b r i z j . (21)The uniform electron gas model can be improved by taking into consideration of theCoulomb energy and Thomas-Fermi correction due to non-uniformity of the electron gas. 11 –When the electron density is assumed to be uniform, the screening potential is independentof the magnetic field. The change of the screening potential due to the compressibility ofthe electrons for the zero-pressure magnetized condensed matter can obtained δE LDTF = − . × − b / ( z / − z / − z / ) . (22)When we summed of a screening potential with a uniformity distribution and acorrected screening potential with a non-uniformity distribution, the screening potential ina SMF is given by U LDs = U sc + δE LDTF . (23) The influence of SES in a SMF on nuclear reaction was also discussed in detail byFushiki et al. (1989) (hereafter FGP). The electron Coulomb energy by an amount whichin the Wigner-Seitz approximation in a SMF was given by U FGPsc = E atm ( r i , z ) − E atm ( r i , z ) − E atm ( r i , z ) , (24)where E atm ( r i , z j ) is the total energy of Wigner-Seitz cell. If the electron distribution isrigid, the contribution to E atm ( r i , z j ) from the bulk electron energy cancel, the electronscreening potential at high density can be expressed as U FGPsc = E latt ( r i , z ) − E latt ( r i , z ) − E latt ( r i , z ) , (25)where E latt ( r i , z j ) is the electrostatic energy of Wigner-Seitz cell and E atm ( r i , z j ) = − . z / j e /r e . Due to the influence of the compressibility of the electron, the change in thescreening potential is given by (Fushiki et al. 1989) δU FGPs = − e r e ) 1 n e ∂n e ∂U e [( z ) / − ( z ) / − ( z ) / ]= − e r e ) 1 n e D FGP [( z ) / − ( z ) / − ( z ) / ] , (26) 12 –where D FGP = 823 . r e n e e ( Az ) / ρ − / B . (27)The Thomas-Fermi screening wave-number will be given by( K FGPTF ) = 1 . × r e n e ( Az ) / ρ − / B . (28)Thus, the corresponding result for the changes in the screening potential in a SMF is δU FGPs = − . Az ) / ρ − / B [( z ) / − ( z ) / − ( z ) / ]= − . Az ) / ρ − / b [( z ) / − ( z ) / − ( z ) / ]MeV , (29)where ( A/z ) is the average ratio of
A/z , which corresponding to the mean molecular weighper electron. Thus the electron screening potential in a SMF of FGP model is given by U FGPs = U sc + δE FGPTF = U sc + δU FGPs . (30)
3. Resonant reaction process and rates3.1. Calculations of resonant reaction rates with and without SES
The reaction rates are summed of contribution from the resonant reaction andnon-resonant reaction. In the case of a narrow resonance, the resonant cross section σ r isapproximated by a Breit-Wigner expression (Fowler et al. 1967) σ r ( E ) = πωκ Λ i ( E )Λ f ( E )( E − E ) + Λ ( E )4 , (31)where κ is the wave number, the entrance and exit channel partial widths are Λ i ( E ) andΛ f ( E ) , respectively. Λ total ( E ) is the total width, and the statistical factor, ω is given by ω = (1 + δ ) 2 J + 1(2 J + 1)(2 J + 1) , (32) 13 –where the spins of the interacting nuclei and the resonance are J , and J , respectively, δ is the Kronecker symbol.The partial widths is dependent on the energy, and can be written as(Lane et al. 1958)Λ i,f = 2 ϑ i,f ψ l ( E, a ) = Λ i,f ψ l ( E, a ) ψ l ( E f , a ) . (33)The penetration factor ψ l is associated with l and a , which are the relative angularmomentum and the channel radius, respectively. a = 1 . A / + A / ) fm. Λ i,f is the partialenergy widths at the resonance process. E r and ϑ i,f is the reduced widths, given by ϑ i,f = 0 . ϑ = 0 . ~ Aa . (34)Based on the above analysis, in the phases of explosive stellar burning, the narrowresonance reaction rates without SES are determined by (Schatz et al. 1998; Herndl et al.1998) λ = N A h σv i r = 1 . × ( AT ) − / × X i ωγ i exp( − . E r i /T ) cm mol − s − , (35)where N A is Avogadro’s constant, A is the reduced mass of the two collision partners, E r i isthe resonance energies and T is the temperature in unit of 10 K. The ωγ i is the strengthof resonance in units of MeV and given by ωγ i = (1 + δ ) 2 J + 1(2 J + 1)(2 J + 1) Λ i Λ f Λ total . (36)On the other hand, due to SES the reaction rates of narrow resonance is given by λ s r = F r N A h σv i r ′ = 1 . × ( AT ) − / X i ωγ i exp( − . E ′ r i /T )= 1 . × F r ( AT ) − / × X i ωγ i exp( − . E r i /T ) cm mol − s − , (37) 14 –where F r is the screening enhancement factor (hereafter SEF). The values of E ′ r i shouldbe measured by experiment, but it is too hard to provide sufficient data. In general andapproximate analysis, we have E ′ r i = E r i − U = E r i − U s . Dewitt et al. (1976) discussed the problem of thermonuclear ion-electron screening atsome densities. Based on a statistical mechanical theory for the screening function, theinfluence of the electron screening on the nuclear reaction process also was investigatedin their paper. The strong electron screening potential function is given by (Dewitt et al.1976) H sc12 = e r e kT { . z ) / ( z / − z / − z / )+ c ( z ) / ( z / − z / − z / ) } +[ c ( z ) − / ( z / − z / − z / )] , (38)where c = 0 . c = 0 . z , the average charge of ionic, is given by z = X i z i f i = X i z i n i n I , (39)where n i and n I are the ion densities of nuclear species i and I of the total system,respectively.The screening enhancement factor (hereafter SEF) in Dewitt model is written as F r (Dew) = exp(H sc12 ) . (40) 15 – At astrophysical energies the electron-screening acceleration in laboratory fusionreactions always play a key role and is an interesting problem for astrophysics. Basedon a mean-field model, Liolios et al. (2000) studied the screened nuclear reactions atastrophysical energies. The electron screening potential in Liolios screened Coulomb modelis given as (Liolios et al. 2000) U Lios0 = 158 z z e Ξ , (41)where Ξ = ( 158 πz i ) / a = 0 . a ( z / + z / ) / , (42)The SEF for the resonant reaction in Liolios model is F (Lios) = exp( 11 . Lios0 T ) . (43) In this Subsection, we will discuss the screening potential in the strong screening limit.The dimensionless parameter (Γ), which determines whether or not correlations betweentwo species of nuclei ( z , z ) are important, is given byΓ = z z e ( z / + z / ) r e kT , (44)Under the conditions of Γ ≫
1, the nuclear reaction rates will be influenced appreciablyby SES. According to the above three SES models (LD, FGP, LJ) in SMFs, the threeenhancement factors for resonant reaction process in SMFs can be expressed as follows F Br (LD) = exp( 11 . LDs T ) , (45) F Br (FGP) = exp( 11 . FGPs T ) , (46) 16 – F Br (LJ) = exp( 11 . LJs T ) . (47)
4. Numerical results4.1. Analysis of the results on a SEF
The strong magnetic fields modify significantly the properties of the matter and alwaysplay a critical role in astronomical conditions. Figure 1 presents the variations of ESP asa function of B for our SES model. The SMF has only a slight influence on ESP when B > × and ρ <
1. But the ESP increases greatly when B < . × and ρ < B , ρ are in units of 10 G, 10 g cm − , respectively). Numerical results in our modelshow that the maximum value of ESP reaches to 0 . B . The ESP increases rapidly and reaches the maximum valueof 0 . B = 80, then decreases with increasing of a SMF.Based on the Thomas-Fermi and Thomas-Fermi-Dirac approximations, Fushiki et al.(1989) analyzed the electron Fermi energy, electron Landau level, and SES problem in aSMF. The results show that, as a consequence of the field dependence of the screeningpotential, magnetic fields can significantly increase nuclear reaction rates (Fushiki et al.1989). According to electron screening model of Ref.(Fushiki et al. 1989) (hereafter FGPmodel ) in a SMF, Figure 2 (b) shows the ESP as a function of B under some typicalastrophysical conditions. The ESP increases greatly when B < and gets to themaximum value of 0 . B = 580 . ρ = 0 .
1. Then the ESP decreasesaround two orders of magnitude when 10 < B < × at ρ = 0 . .
632 for B = 78 .
17 and T = 0 .
2, asshown in Figure 3, where T is the temperature in units of 10 K. But for B > . . B of FGP and LJ models.From sub-figures 4(a) and 4 (b), one find that the shifty trend of SEF in FGP model is ingood agreement with those of LD at low density (e.g. ρ = 0 . .
66 for B = 84 .
18 and ρ = 0 .
01. On the contrary, theSEF increases with increasing of B at relatively high density (e.g. ρ = 1), then gets tothe maximum value of 5 .
166 at T = 0 .
2. Sub-figures 4(c) and 4(d) show that in LJ modelshow that the SEF increases with increasing of B , and the maximum value will reach upto 5 .
056 for B = 1000, T = 0 . ρ = 0 . ρ . B and T to compare withthose of LD and FGP.The SES problem always plays important roles in stellar evolution process. Based ona statistical mechanical theory for the screening function, Dewitt et al. (1976) investigatedthe influence of the electron screening on nuclear reaction. Based on a mean-field model,Liolios et al. (2000) also studied the effect about screened nuclear reactions. However, theyneglected the influence of SMFs on SES. We compare the SEF of the two models (Dewitt,and Liolios model) with those of LD, FGP, and LJ. One can conclude that the SEF ofDewitt model is larger than those of other three SES models for B < ρ = 0 . T < .
17, shown as in Figure 6. However, when T < . ρ = 0 .
01, the resultsof our model are larger than those of Dewitt and Liolios. At a relatively high density(e.g., ρ = 0 . B = 10 . It is because that the ESP increases very rapidlyas SMF increases. The higher the ESP, the larger the influence on SES becomes. On thecontrary, the SEF of LD decreases with increasing of magnetic fields because ESP is reduced.The SEF of FGP model gets to the maximum of 1 .
929 when B = 10 , ρ = 1 , T = 0 . K TF is a very key parameter, whichstrongly depends on the electron number density and ESP. In consequence the electronnumber density and ESP will play important roles in a SMF. Lai et al. (1991), analyzed indetail the electron Fermi energy and electron number density in a SMF based on the worksof Canuto et al. (1968, 1971); Kubo. (1965), and Pathria (2003). By using the uniformelectron gas model and linear response theory, Lai (2001)discussed the electron energy (percell) corrections due to non-uniformity in a SMF. According to their theory, we study theESP and the SES model (i.e., LD model). The results show that the ESP decreases as themagnetic fields increase due to the diminution of electron chemical potential. The LD modelis valid only in the condition of K TF r i ≪ U F = U e = ∂w ex ∂n e = r cyc πa ~ ω nI ( n ) , (48)where w ex is the exchange energy and I ( n ) can be found in Ref. (Fushiki et al. 1989). Byusing the linear response theory, Fushiki et al. (1989) discussed the exchange energy andelectron chemical potential in the lowest Landau level for non-uniformity electron gas in aSMF. They analyzed the SES problem in a SMF and their results shown that a SMF onlythe lowest Landau level is occupied by electrons on the condition of r e > (3 π/ / r cyc orequivalently ρ < . × B / (A/z)g / cm . The cyclotron radius in the lowest Landaulevel orbital is give by r cyc = (2 ~ c/eB ) / ≃ . × − B − / . FGP used the expression of n e ∂n e /∂U F = (3 / n e /U F in dealing with ∂n e /∂U F . In FGP model, they thought at highdensity the exchange correction is very small, thus they neglected the exchange correctionto ∂n e /∂U F and had n e ∂n e /∂U F = (1 / n e /U F in a SMF. Due to different ways of dealingwith exchange correction under this condition, the SEF of FGP model has some differencecompared with other SES models.According to statistical physics the microscopic state number dxdydzdp x dp y dp z can begiven by dxdydzdp x dp y dp z /h in a 6-dimension phase-space. The number of states occupiedby completely degenerate relativistic electrons per volume is calculated by (Canuto et al.1968, 1971) N phase = P p x P p y P p z = h R ∞−∞ R ∞−∞ R ∞−∞ dp x dp y dp z = h R p F dp z R ∞ p ⊥ dp ⊥ R π dθ = πp F h R ∞ dp ⊥ , (49)where θ = tan − p y /p x , p ⊥ → m c BB cr n , So, R ∞ dp ⊥ → P ∞ n =0 ω n , and the ω n is thedegeneracy of the n -th electron Landau level in relativistic magnetic field, and can becalculated by (Canuto et al. 1971; Kubo. 1965; Pathria 2003) ω n = 1 h Z π dφ Z k
In the explosive hydrogen burning stellar environments, the nuclear reaction Mg( p, γ ) Al plays a key role because of breaking out the Ne-Na cycle to heavy nuclearspecies (i.e., Mg-Al cycle). Therefore, it is very important to accurate determinate the ratesfor the reaction Mg( p, γ ) Al. However, the resonance energy has a large uncertainty dueto the inconsistent Mg( He,t) Al measurements mentioned. So it may lead to a factorof 5 variation in the reaction rate at T = 0 .
25 because of its exponential dependence on E r (Visser et al. 2007). Some authors discussed the contributions from several importantresonance states, such as (Wallace et al. 1981; Wiescher et al. 1986; Kubono et al. 1995;Visser et al. 2007). In order to reduce the uncertainty of the reaction rates in this paper,we reference some information about this reaction and the values of the E r i , E x andcorresponding to ωγ i and some average values of ωγ i are adopted and listed in Table 2.According to these information, we analysis the total rates for these five SES models.Tables 3 and 4 give a brief description of the factor S i ( i = 1 , ,
3) for LD, FGP, andLJ models when B = 10 , , respectively. As the density and temperature increase, the 21 –Table 1: The comparisons of the resonant SEFs for Dewitt, Liolios, LD, FGPand LJ models in several typical astronnomical conditions. The former twomodels are in the case without SES and SMFs, while the latter three modelsare in the case with SES and SMFs. B = 10 B = 10 ρ T F (Lios) F (Dew) F Br (LD) F Br (FGP) F Br (LJ) F Br (LD) F Br (FGP) F Br (LJ)0.01 0.1 1.7475 3.8973 1.6956 1.6964 0.1749 1.0725e-15 1.3472e-13 25.56800.05 0.1 1.7475 10.9451 1.6956 1.7013 8.4513e-4 1.0725e-15 0.6045 19.57170.1 0.2 1.3219 4.3605 1.3021 1.3045 0.0022 3.2750e-8 2.4894 3.88480.1 0.3 1.2051 2.5873 1.1922 1.1941 0.0174 1.0221e-5 1.8371 2.47130.2 0.3 1.2045 3.3934 1.1924 1.1939 9.1604e-4 1.0236e-5 2.4990 2.13610.3 0.4 1.1497 2.8170 1.1411 1.1422 7.7822e-4 1.8097e-4 2.1178 1.60551.0 0.5 1.1181 3.5124 1.1113 1.1122 6.2630e-7 0.0011 1.9290 0.951210 0.7 1.0830 7.2692 1.0780 1.0791 6.2012e-9 0.0072 1.6142 0.0894 results of LD model are in good agreement with those of FGP, but disagreement with ourresults at B = 10. This is because that the electron Fermi energy of our model is lowerthan those of LD and FGP in relatively low magnetic fields. As the magnetic fields increasefrom B = 10 to 10 , the factor S increases about 2 ∼ ρ = 0 . , T = 0 . ρ = 0 . , T = 0 . B = 10 the factor S is about 39.74, 5.69, 1.56 times larger than S (FGP model) at ρ = 0 . , T = 0 . ρ = 0 . , T = 0 . ρ = 0 . , T = 0 .
2, respectively.From what has been discussed above, the LD model maybe only adapts to the relatively lowmagnetic field and low density surroundings. The FGP and LD models are both unadaptedto relatively low density, and high magnetic field surroundings (e.g. ρ < . , B > ).However, our model can be well adapted to relatively high magnetic field and low densitysurroundings (e.g. B > , ρ < . Mg ( p, γ ) Al. E x (MeV) a E x (MeV) b J π E r i (MeV) c Γ p Γ γ ωγ i (meV) d ωγ i (meV) e ωγ i (meV) f ± ± + ± ± + ± ± + ± ± + a is adopted from Ref. (Endt 1998) b from Ref.(Visser et al. 2007) c from Ref.(Audi et al. 1995) d from Ref.(Herndl et al. 1998) e from Ref.(Wiescher et al. 1986) f is adopted in this paper Summing up the above discussions, our calculations show that this SES effect in a SMFcan increase nuclear reaction rates of Mg ( p, γ ) Al by several orders magnitude. A moreprecise thermonuclear rates of Mg ( p, γ ) Al will help us to constrain the determinationof nuclear flow out of the Ne-Na cycle, and production of A ≥
20 nuclides, in explosivehydrogen burning over a temperature range of 0 . ≤ T ≤ .
5. Conclusions
In this paper, based on the relativistic theory in a SMF, we investigate the problem ofSES, and the SES influence on the nuclear reaction of Mg ( p, γ ) Al by LD, FGP, and LJstrong screening models in a SMF. The results show that the SES thermonuclear reactionrates have a remarkable increase in a SMF. The rates can increase by around three orders ofmagnitude. For example, when B increases from 10 to 10 , the rates increase from 0.1749to 25.5680 at ρ = 0 . , T = 0 .
1, and from 0.0022 to 3.8848 at ρ = 0 . , T = 0 .
2. The 23 –Table 3: Comparisons of the rates of λ r , which are in the case without SES with those ofthe LD ( λ scBr (LD)), FGP ( λ scBr (FGP)) and our calculations λ scBr (LJ) in the case with SESfor some typical astronomical conditions at B = 10, respectively. S i = λ scBri /λ , i = 1 , , B = 10 ρ T λ λ scBr (LD) λ scBr (FGP) λ scBr (LJ) S S S considerable increase in the reaction rates for Mg ( p, γ ) Al implies that more Mg willescape the Ne-Na cycle due to SES in a SMF. Then it will make the next reaction convertmore Al ( β + , ν ) Mg to participate in the Mg-Al cycle. It may lead to synthesizing alarge amount of heavy elements at the crust of magnetars. These heavy elements, whichare produced from the nucleosynthesis process, may be thrown out due to the compactbinary mergers of double neutron star (NS-NS) or black hole and neutron star (BH and NS)systems. On the other hand, our model for the rates is in good agreement with those of LDand FGP models at relatively low density (e.g., ρ < .
01) and B < . In relatively lowmagnetic fields (e.g., B < λ , which are in the case without SES and SMFs withthose of the LD ( λ scBr (LD)), FGP ( λ scBr (FGP)) and our calculations λ scBr (LJ) in the case withSES for some typical astronomical conditions at B = 10 , respectively. The S i is the sameas in Table 3. B = 10 ρ T λ λ scBr (LD) λ scBr (FGP) λ scBr (LJ) S S S and three orders magnitude higher than those of FGP and LD in relatively high magneticfields and low density surroundings (e.g., B ≥ , ρ < . REFERENCES
R. K. Wallace, & S. E. Woosley, ApJS., : 389 (1981).C. Iliadis, J. M. D’Auria, S. Starrfield, et al., ApJS, : 151 (2001)S. Kubono, T. Kajino, & S. Kato, Nucl. Phys. A., : 521(1995)H. Herndl, , M. Fantini, C. Iliadis, P. M. Endt, & H. Oberhummer, Phys. Rev. C., : 1798(1998)D. W. Visser, , Wrede, C., J. A. Caggiano, et al., Phys. Rev. C., : 5803 (2007)G. Lotay, P. J. Wood, D. Seweryniak, et al., Phys. Rev. C., : 2802 (2008)J. N. Bahcall, L. Brown, A. Gruzinov, & R. Sawer, A&A : 291 (2002)J. J. Liu, MNRAS, : 1108 (2013)J. J. Liu, MNRAS, : 930 (2014)J. J. Liu, RAA, : 83 (2016)J. J. Liu, W. M. Gu., ApJS, : 29 (2016)J. J. Liu, et al., RAA, : 107 (2017)J. J. Liu, et al., ChPhC, :095101 (2017)E. E. Salpeter, & H. M. van Horn, ApJ, : 183 (1969)E. E. Salpeter, AuJPh., : 373 (1954)H. C. Graboske, & H. E. DeWitt, ApJ, : 457 (1973)H. E. Dewitt, Phys. Rev. A., : 1290 (1976) 27 –T. E. Liolios, EPJA., : 287 (2000)T. E. Liolios, Phys. Rev. C., : 8801 (2001)P. A. Kravchuk, & D. G. Yakovlev, Phys. Rev. C., : 5802 (2014)Q. H. Peng, & H. Tong, MNRAS, : 159 (2007)Z. F. Gao., N. Wang, J. P. Yuan, L. Jiang, D. L. Song, Ap&SS, : 129(2011)Z. F. Gao, N. Wang, Q. H. Peng, X. D. Li, & Y. J. Du, Mod. Phys. Lett. A., : 50138(2013)Z. F. Gao., N. Wang, Y. Xu, H. Shan, X. D. Li., AN, : 866(2015)Z. F. Gao., N. Wang, H. Shan, X. D. Li, W. Wang, ApJ, eprint arXiv:1709.03459Z. F. Gao., Y. Xu, H. Shan, X. D. Li., H. Shan, W. Wang, N. Wang., AN, eprintarXiv:1709.02186 (2017)X. H. Li, Z. F. Gao, X. D. li, et al., IJMPD, : 1650002(2016)D. Lai, & S. L. Shapiro, ApJ, : 745 (1991)D. Lai, Rev. Mod. Phys., : 629 (2001)R. C. Duncan, & C. Thompson, ApJ, 392, 9 (1992)I. Fushiki, E. H. Gudmundsson, & C. J. Pethick, ApJ, : 958 (1989)L. D. Landau, & E. M. Lifshitiz, Quantium mechanics , (3rd ed., Oxford: Pergamon Press1977), p.457C. Zhu, Z. F. Gao., X. D. Li.et al., Mod. Phys. Lett. A., : 50070(2016) 28 –N. W. Ashcroft, & N. D. Mermin, Solid State Physics , (Saunders College: Philadelphia1976), p.123B. B. Kadomtsev, O. P. Pogutse, Phys. Rev. L. : 1155 (1971)J. M. Lattimer, C. J. Pethick, D. G. Ravenhall, & D. Q. Lamb, Nucl. Phys. A., : 646(1985)W. Stolzmann, & T. Bloecker, A&A, : 1024 (1996)D. G. Yakovlev, & D. A. Shalybkov, Astrophys. Space. Phys. Rev., : 311 (1989)W. A. Fowler, G. R. Caughlan, & B. A. Zimmerman, ARA&A., : 525 (1967)A. M. Lane, & R. G. Thomas, Rev. Mod. Phys., : 257 (1958)H. Schatz, A. Aprahamian, J. Goerres, et al.,Phys. Rep., : 167 (1998)Canuto, V., & H. Y. Chiu, Phys. Rev., : 1210 (1968)Canuto, V., & H. Y. Chiu, Space. Sci. Rev., : 3 (1971)R. Kubo, Statistics Mechanics , (Amsterdam: North-Holland Publishing Co.1965) p.278R. K. Pathria,
Statistics Mechanics , (2nd. Singapore: Isevier 2003), p.280M. Wiescher, J. Gorres, F.-K. Thielemann, & H. Ritter, A&A, : 56 (1986)P. M. Endt, Nucl. Phys. A., : 1 (1998)G. Audi, & A. H. Wapstra, Nucl. Phys. A., : 409 (1995)This manuscript was prepared with the AAS L A TEX macros v5.2. 29 – −6 −5 −4 −3 −2 −1 B U sc L J ( M e V ) ρ =0.01 ρ =0.1 ρ =1 ρ =10 Fig. 1.— The electron screening potential as a function of B of LJ model for some typicalastronomical condition. −4 −3 −2 B U sc L D ( M e V ) (a) −5 −4 −3 −2 −1 B U sc F G P ( M e V ) ρ =0.01 ρ =0.1 ρ =1 ρ =10 (b) Fig. 2.— The electron screening potential as a function of B in LD, and FGP models forsome typical astronomical condition. 30 – B SE F (r) T =0.2T =0.4T =0.6T =0.8T =1 F rB (LD) Fig. 3.— The resonant SEF for LD model as a function of B in the case with SES andSMF. 31 – B SE F (r) T =0.2T =0.4T =0.6T =0.8T =1 ρ =0.01F rB (FGP)(a) B SE F (r) T =0.2T =0.4T =0.6T =0.8T =1 F rB (FGP) ρ =1(b) −1 B SE F (r) T =0.2T =0.4T =0.6T =0.8T =1 ρ =0.01F rB (LJ)(c) B SE F (r) T =0.2T =0.4T =0.6T =0.8T =1 ρ =1F rB (LJ) (d) Fig. 4.— The resonant SEF for FGP and LJ models as a function of B in the case withSES and SMF. 32 – −1 B SE F (r) ≤ T ≤ F rB (LD) F rB (LJ) ρ =0.01 ρ =0.1 ρ =1 ρ =10 ρ =100 −1 T SE F (r) F rB (LD) F rB (LJ) ρ =0.01 ρ =0.1 ρ =1 ρ =10 ρ =10 ≤ B ≤ −1 B SE F (r) F rB (LD) F rB (FGP) ≤ T ≤ ρ =0.01 ρ =0.1 ρ =1 ρ =10 ρ =100 −1 T SE F (r) F rB (FGP)F rB (LD) ρ =0.01 ρ =0.1 ρ =1 ρ =10 ρ =100 ≤ B ≤ −1 B SE F (r) ≤ T ≤ F rB (FGP) F rB (LJ) ρ =0.01 ρ =0.1 ρ =1 ρ =10 −1 T SE F (r) F rB (LJ) F rB (FGP) ρ =0.01 ρ =0.1 ρ =1 ρ =10 ≤ B ≤ Fig. 5.— The comparisons are plotted for some typical astronomical condition of the resonantSEF among the three models of LJ, LD, and FGP in the case with SES and SMF. 33 – −1 B SE F (r) F r0 (Lios)F r0 (Dew)F rB (LD)F rB (BGP)F rB (LJ) ρ =0.010.1 ≤ T ≤ −1 B SE F (r) F r0 (Lios)F r0 (Dew)F rB (LD)F rB (BGP)F rB (LJ) ≤ T ≤ ρ =0.1 −1 T SE F (r) F r0 (Lios)F r0 (Dew)F rB (LD)F rB (BGP)F rB (LJ) ρ =0.010.1 ≤ B ≤ −1 T SE F (r) F r0 (Lios)F r0 (Dew)F rB (LD)F rB (BGP)F rB (LJ) ≤ B ≤ ρ =0.1=0.1