N E W S
OF T H E N A T IO N A L A C A D E M Y OF SC IEN C ES OF THE R EPU BLIC OF K A Z A K H S T A N P H Y S IC O -M A T H E M A T IC A L S E R IE S
IS S N 1 9 9 1 -3 4 6 Х h ttp s://d o i.o r g /1 0 .3 2 0 1 4 /2 0 2 0 .2 5 1 8 -1 7 2 6 .3 4 V olu m e 3, N um ber 331 (2 0 2 0 ), 34 - 43
UDC 52-48, 524, 539.14, 539.17
A .V . D z h a z a ir o v -K a k h r a m a n o v 1,2, L .T . K a r ip b a y e v a 1,2, A .A . S te b ly a k o v a2 1V.G. Fesenkov Astrophysical Institute “NCSRT” NSA RK, Almaty, Kazakhstan;
2International Informatization Academy, Almaty, Kazakhstan.
E-mail: albert-j@yandex.ru, larisa_karipbaeva@mail.ru, a_steblyakova@mail.ru
PARTIAL POTENTIALS FOR THE 8Li(p,y)9Be CAPTURE AT ASTROPHYSICAL ENERGIES
Abstract. The total cross sections o f the radiative proton capture on 8Li at astrophysical energies are considered in the framework o f the modified potential cluster model with forbidden states, with the classification o f the orbital cluster states according to Young diagrams. The recalculation o f the total cross sections for 9Be(y,p0)8Li photodisintegration is used as experimental data. Parameters for Gaussian partial potentials were obtained for description the 8Li(p,y)9Be capture at astrophysical energies. Simultaneously, in work o f Shoda, & Tanaka (1999), different binary channels o f disintegration o f 9Be, namely, 9Be(y,p)8Li, 9Be(y,d)7Li, 9Be(y,t)6Li and also 9Be(y,3He)6He, were experimentally studied. It is evident that the processes o f two-body radiative capture connected with them by the detailed balancing principle lead to the synthesis o f 9Be and require the corresponding estimation contextually in the astrophysical supplements. Meanwhile, it should be noted that the first three reactions have the Coulomb barrier in channels p8Li, d7Li and t6Li lower than in 3H e6He along with 4He5He channels, namely, in the ratio o f 3:4. The cross section of 8Li(p,y)9Be is hard to obtain directly due to low 8Li beam intensity and the small cross section at astrophysical energies. In addition, the difficulty o f studying the reaction o f 8Li(p,y)9Be also lies in the fact that the direct experimental measurement o f cross sections is practically impossible due to the very short half-life o f 8Li - 838 ms. However, as in the case of the neutron capture on 8Li, some indirect methods o f extracting direct capture cross sections can be used with the help o f the radiative capture model and spectroscopic factor.
Keywords: Nuclear astrophysics; primordial nucleosynthesis; thermal and astrophysical energies; p8Li cluster system; radiative capture; total cross section.
1. In tr o d u c tio n
The study o f the form ation m echanism s o f 9B e directly concerns the problem o f the overlap o f the A = 8 m ass gap and the synthesis o f heavier elem ents in the early U niverse, as w e ll as the r-process nucleosyn th esis in supernovae (see, for exam ple, [1,2]). In the present tim e, it is considered that 9B e is form ed as a result o f a tw o-stage process: the radiative capture o f alpha particles a ( a ,y )8B e, leading to the synthesis o f the short-half-life isotope 8B e (tm = 6.7*10"17s), and radiative neutron capture 8B e(n, y)9B e [2,3]. In addition, there is the m ore d ifficu lt process o f the direct three-particle capture a a n ^ y9B e, (see, for exam ple, [4 -8 ]).
Sim ultaneously, in [8], different binary channels o f disintegration o f 9B e, nam ely, 9B e(y,p )8Li, 9B e(y,d )7Li, 9B e(y,t)6Li and also 9B e(y,3H e)6H e, w ere experim entally studied. It is evident that the p rocesses o f tw o-b od y radiative capture connected w ith them by the detailed balancing principle lead to the synthesis o f 9B e and require the corresponding estim ation contextually in the astrophysical supplem ents. M eanw hile, it should be noted that the first three reactions have the C oulom b barrier in channels p8Li, d7Li and t6Li low er than in 3H e6H e along w ith 4H e5H e channels, nam ely, in the ratio o f 3:4.
The cross section o f 8L i( p ,y 9B e is hard to obtain directly due to lo w 8Li beam intensity and the sm all cross section at astrophysical energies. In addition, the d ifficu lty o f studying the reaction o f 8L i( p ,y 9Be also lies in the fact that the direct experim ental m easurem ent o f cross section s is practically im possible due to the very short h alf-life o f 8Li - 838 m s [9]. H ow ever, as in the case o f the neutron capture on 8Li
[10], som e indirect m ethods o f extracting direct capture cross sections can be used w ith the help o f the radiative capture m od el and spectroscopic factor [9,11].
The p sL i ^ 9B e y reaction presents significant astrophysical interest because it is included in the list o f p rocesses o f primordial n u cleosyn th esis o f the U niverse [12]. H ow ever, its experim ental investigation in the astrophysical range has so far been insufficient. Intrinsically, there is on ly one w ork [13] w here the astrophysical range is considered. In [14], it is also added, where m easurem ents w ere carried out at higher energies. H ow ever, these w orks w ere published in the 1960s and w e do not currently p o sse ss more m odern experim ental studies o f the total cross sections o f the considered reaction [15]. This is in spite o f the fact that studies o f spectra 9B e in the p 8Li channel are still continuing [16]. In addition, the available and considered further theoretical results differ so greatly that it is difficult to draw certain conclusions regarding the rate o f this reaction. Furthermore, these calculations do not take into account the existen ce o f resonances in the p 8Li system at lo w energies [16].
In the present study, w e consider the reaction o f the proton capture on 8Li in the frame o f the m odified potential cluster m odel (M PC M ) [17] and define h o w the criteria o f this m odel a llow us to correctly describe total cross sections and the astrophysical S-factor at astrophysical energies. The energy range o f 10 k eV to 7.0 M eV is considered but only taking into account the structure o f resonances up to 2.0 M eV , as discussed p reviously [16]. The rate o f the reaction is calculated at the temperature range o f 0.01 to 10 T9. The analysis o f the influence o f the location and m agnitude o f resonances to the value and shape o f the reaction rate is presented.
2. M o d e l an d c a lc u la tio n m eth o d s
F u r th e r w e use w e ll-k n o w n fo r m u la s f o r to ta l cro ss se c tio n s a n d m a trix ele m e n ts o f E1 tra n sitio n o p era to rs [1 8 ] (Si = Sf = Sj
a ( N J , J f ) = 8 ^Ke3--- M ‘ A j ( N J , K )---— ---2 X P j 2 (N J , J f , J i ) I J2( J f , J i), (1) c h q (2 S 1 + 1)(2S2 + 1) J [(2J + 1)!!]2 C t J f i J f i
where m atrix elem ents o f E J - transitions have a form
, , IL S J i ] 2
Pj2(E J , J f , J i) = 5SiSf [(2J + 1)(2Li + 1)(2J i + 1)(2Jf + 1 ) ] ( L J | L f0 )2j J j
f Z Z ^
A J (E J, K ) = K J ^ - J + ( - 1 ) J - J > IJ v ' J j ( J f , J i ) = { х У к / (2) v m 1 m 2 у
Here S i, Sf, Lf, L i, Jf, J i - are total spins and m om ents o f input (i) and output (f ) channel particles, m1, m2, Z1, Z2 are m asses and charges o f input channel particles, Ij is the integral over w ave functions o f the initial xi and final Xf states, as relative m otion functions o f clusters w ith intercluster distance r, ц - reduced m ass.
For the spin part o f the m agnetic process M 1(S ) at J = 1, the fo llo w in g expression is obtained in the m odel used (S i = Sf = S , L i = Lf = L )
, I S L Ji} 2
P 2(M 1,J f ,J i ) = 5SiSfSLiLf[S ( S + 1)(2S + 1)(2J i + 1)(2J f + 1 )]jj 1 ^ J , h K
A1( M 1, K ) = ---л/3 V-1 — - V-2 —m 2 m 1
m m
Ij(J f , J , ) = X f|rJ-1 \x) . (3)
Here, m is the m ass o f the nucleus, Ц1, Ц2 are m agnetic m om ents o f the clusters, and the rem aining notation, are g iv e n as in the previous expression.
Constant h 2/m0 is equal to 4 1 .4 6 8 6 M eV fm 2, w here m0 is the atom ic m ass unit (am u). The Coulom b potential at zero C oulom b radius Rcoul = 0 is written in the form Vcoul = 1.439975 • - —2/ r , where r is the relative distance b etw een particles o f the initial channel in fm and Z are charges o f particles in the elem entary charge “e ” units. Furthermore, the m agnetic m om ent o f proton = 2 .7 92847ц э and 8Li nucleus ^(8Li) = 1.65356ц э [18], w here Ц0 is the nuclear m agneton.
m 0c
35
3. C la ssific a tio n o f p 8L i sta tes a c c o r d in g to Y o u n g d ia g ra m s
W e take Y ou n g diagram {4 3 1 } for 8Li - it w as show n in [19] that exactly this diagram corresponds to the ground state (G S) o f 8Li, i f to consider it in the n7Li channel. For m any other cluster system s, their correspondence to certain Y ou n g diagrams w ere studied by us in [17].
Therefore, for the p 8Li system , w e have: {4 3 1 } + {1 } = {5 3 1 } + {4 4 1 } + {4 3 2 } + {4 3 1 1 } . The first diagram {5 3 1 } is forbidden because it cannot be five nucleons in the s-sh ell - it corresponds to orbital m om enta L = 1, 2, 3, determ ined by the E lliot rule [20]. The second diagram {4 4 1 } corresponds to L = 1, 2, 3, 4, the third {4 3 2 } has L = 1, 2, 3 and the fourth {4 3 1 1 } corresponds to L = 0, 2. For the second diagram {4 4 1 } , w e w ill consider allow ed - it corresponds to the ground state (G S) o f 9B e in the p 8Li channel [21]. D iagram s {4 3 2 } and {4 3 1 1 } are not considered because w ithout product tables o f Y oung diagrams [22], it is im possible to understand i f they are forbidden or allow ed.
Thus, it fo llo w s from the g iv en classification that for the p 8Li system (it is know n that J " ,T = 2+,1 for 8Li [21]) in potentials o f S w a v es there is no forbidden state (F S), in P w a v es for diagram {4 4 1 } , there are FS and allow ed state (A S ), in D and F w a v es there is FS for the sam e diagram, w h ich can be considered as bound. The state in the P3/2 w ave corresponds to the G S o f 9B e w ith J K,T = 3/2",3/2 and lays at the binding energy o f p 8Li system o f -1 6 .8 8 8 2 M eV [21]. Som e p 8Li scattering states and bound states (B S s) can be m ixed by spin w ith S = 3/2 and 5/2. H ow ever, the sam e as for the n 8Li system [10], here w e w ill consider that the G S o f 9B e p 8Li channel is m ost probably is the 4P3/2 lev el (in spectroscopic notations 2S+1Lj).
4. S tr u c tu r e o f th e p 8L i r e so n a n ce states
Let us consider n o w the state structure o f 9B e, w here w e consider further the G S and six lo w -ly in g resonance states. The GS o f 9B e w ith J = 3/2" is at the energy o f -1 6 .8 8 8 M eV (round o f f energy dow n to
1 k eV ) [16] relatively to the threshold o f the p 8Li channel and w ill considered only as the 4P3/2 state.
Table 1 - Comparison of data on the 9Be spectrum from different works. The threshold of the p8Li channel in 9Be equals 16.888 MeV [21]. The results coinciding for all three works are marked by bold
[16] 2018 year [23] 2016 year [21] 2004 year
Er, MeV J" Г c.m., keV Er, MeV J" rc.m., keV Er, MeV J" Г c.m., keV
- - - 0.087(1) 1/2- 0.39(1) 0.0868(8) 1/2- 0.389(1)
0.420(7) 5/2- 210(20) 0.410(7) 5/2- 195 0.410(7) 5/2- 200
0.610(7) 7/2+ 47(7) 0.605(7) 7/2+ 47 0.605(7) 7/2+ 47
1.100(30) 3/2+ 50(22) 1.132(50) - - 1.132(50) - -
1.650(40) 7/2- 495(34) 1.688(30) 5/2+ 432(50) 1.692(40) - -
1.800(40) 5/2- 79(17) 1.758(40) 7/2+ 490(81) 1.762(50) 5/2- 300(100)
- - - 2.352(50) 3/2+ 310(80) 2.312(50) - 310(80)
H ere, the E1 capture is possib le from 4S3/2 scattering w ave to the 4P3/2 G S o f 9B e, that is, the main contribution is g iv en due to the transition 4S3/2^ 4P3/2. The spectrum o f resonance states o f 9B e in the cluster p 8Li channel is listed in Table 1.
W e n o w discuss this spectrum m ore clo sely . G iven b elo w , the system atization allow s a p r io r i estim ate o f the m ost significant contribution o f resonance states in the capture cross section (the resonance states considered in this w ork are marked by bold italics).
1. The f i r s t r e s o n a n c e s ta te (1st R S ) is located at 16.975(8) M eV relative to the G S or 0 .0 8 6 8 (8 ) M eV [21] (0 .0 8 7 (1 ) M eV [23]) in the center o f m ass (c.m .) relative to the threshold o f the p 8Li channel.
J = 1/2- is g iven for this lev e l [21,23] that allow s us to take L = 1 for it, that is, to consider it quartet 4P m resonance. The le v e l w idth o f r c.m. = 0 .3 9 (1 ) k eV is g iv en in [21,23]. It is p ossib le to construct absolutely unam biguous 4P m potential o f the elastic scattering according these data. A m biguity o f its parameters, w ith the bound FS at the basic variant o f state classification by Y ou n g diagrams, w ill be caused only by the error o f w idth o f this resonance [17]. The M 1 transition 4P1/2^ 4P3/2 to the GS is p ossib le for this state.
2. The s e c o n d r e s o n a n c e s ta te (2 nd R S ) is located at 17.298(7) M eV relative to the G S or 0 .4 1 0 (7 ) M eV in the c.m . relative to the threshold o f the p 8Li channel [21,23]. J = 5 /2 - is g iv en for this level [21,23] that allow s us to take L = 1 for it, that is, to consider it quartet 4P5/2 resonance. The le v e l w idth o f
Г , т . = 2 0 0 k eV is g iv en in [21] and 195 k eV in [23]. The energy 4 2 0 (7 ) k eV at the w idth 2 1 0 (2 0 ) k eV is g iv en in n ew w ork [16]. The M 1 transition 4P5/2^ 4P3/2 to the G S is also p ossib le for this state.
3. The th ird reso n a n c e sta te (3rd R S) is located at 17.493(7) relative to the G S or 0 .6 0 5 (7 ) M eV in the c.m . relative to the threshold o f the p 8Li channel w ith the w idth o f 47 k eV [21,23]. J = 7/2+ is g iven for this lev e l [21,23] that us allow s to take L = 2 for it, that is, to consider it quartet 4D7/2 resonance. The energy 6 10(7) k eV at the w idth 4 7 (7 ) k eV is g iv en in n ew w ork [16]. Since, such resonance state corresponds to the w ave, and then on ly E 2 transition to the G S o f 9B e is p ossib le here, w h ich w e w ill not consider due to its sm all value.
4. The f o u r t h r e s o n a n c e s ta te (4th R S ) at energy 1.100 M eV w ith the w idth 5 0(22) k eV and m om entum 3/2+, w h ich is in data [16], can be considered as refinem ent o f data from [21,23]. N either its w idth nor its m om entum is not g iv en in them , on ly the energy o f 1.132(50) M eV that approxim ately coin cid es w ith n ew results from [16] is given . I f to take the D 3/2 state for it, the E1 transition 4D3/2^ 4P3/2 to the G S turn out to be p ossible. W e failed to obtain the resonance in the 4S3/2 w ave w ith such characteristics; therefore, w e consider this w ave nonresonance.
5. The f i f t h reso n a n c e sta te (5th R S) according to [21] is located at the energy o f 1.692(40) M eV relative to the threshold, but the m om entum and the w idth are not g iv en for them . The sim ilar state at the energy o f 1.688(30) M eV w ith the m om entum 5/2+ and the w idth o f 4 3 2 (5 0 ) k eV is g iv en in [23]. In new w ork [16], the energy is equal to 1.650(40) M eV and the w idth o f 4 9 5 (3 4 ) k eV that coin cid e w ith data from [23], but the value 7 /2 - is g iv e n for m om entum . I f to assum e that the last m om entum corresponds to 5th R S, so it can refer to the 4F m scattering state, and then only E 2 transition to the G S is p ossible, w hich w e w ill not consider.
6. The six th re so n a n c e sta te (6th R S) according to [21] is located at the excitation energy o f 18.65(5) M eV or 1.762(50) M eV in the c.m . relatively to the threshold o f the p 8Li channel w ith the w idth 3 0 0 (1 0 0 ) keV . J = 5 /2 - is g iv en for this lev el [21] that allow s us to take L = 1 for it, that is, to consider it quartet 4P5/2 resonance. The M 1 transition 4P5/2^ 4P3/2 to the G S is p ossib le for this resonance. H ow ever, the energy o f 1.758(40) M eV w ith the w idth o f 4 9 0 (8 1 ) k eV and other m om entum 7/2+ are g iv en in [23].
A t the same tim e, in n ew w ork [16], the energy o f 1.800(40) M eV w ith the w idth o f 7 9 (1 7 ) k eV and m om entum 5 /2 -, coincided w ith primary data o f w ork [21], are given . I f to take for it the last m om entum , it can refer to the 4P5/2 scattering state and then the M 1 transition 4P5/2^ 4P3/2 to the G S is possible.
H ow ever, w e do not succeed to obtain characteristics o f the P w ave o f continuous spectrum that were noted in Table 1. Therefore, the F scattering w ave is com pared to it, w h ich g iv e s the E 2 transition to the GS, but since their contribution is sm all, w e w ill not consider them.
7. The se v e n th reso n a n c e sta te (7th R S) w ith the energy o f 2 .3 1 2 (5 0 ) w ith the w idth 3 1 0 (8 0 ) k eV and unknow n m om entum are g iv en in [21]. The energy o f 2 .3 5 2 (5 0 ) w ith the w idth the sam e w idth and m om entum 3/2+ is g iv en in [23]. In n ew w ork [16], this and higher states, unfortunately, are not considered. A s w e have seen above, the results o f [23] on tw o previous lev els differ from n ew data [16];
therefore, the data for this lev el m ost probably should be sp ecified , and n o w w e w ill not consider this resonance.
Slightly higher at excitation energy o f 19.420(50) M eV , there is other resonance w ith the w idth o f 6 0 0 (1 0 0 ) keV , but it has a presum able m om entum 9/2+ [23] and cannot lead to E1 or M 1 transitions and w e therefore do not consider it. Furthermore, w e w ill base on spectra g iv en in n ew w ork [16], including first resonance at 87 k eV from [21,23] and lim iting by energies not higher the threshold o f 2 M eV .
Thus, selected by us, the basic transitions to the G S that are considered here and also P2 coefficien ts for total cross sections, g iv en in [17], are presented in Table 2. The obtained values o f isotop ic spin 2Ti for resonance states w ith J * are g iv e n here.
Table 2 - Characteristics of taken into account transitions at the 8Li(p,y)9Be capture No. [2S+1Lj]1 Resonance energy,
MeV J 1П, 2Ti Transition [2S+'Lj]f ,
2Tf = 1 P2
1. 4S3/2 - 3/2+ (?) E1 4P3/2 4
2. 4D3/2 1.100 (4th RS) 3/2+, (3?) E1 4P3/2 64/25
3. 4P1/2 0.087 (1st RS) 1/2-, 3 M1 4P3/2 10/3
4. 4P 5/2 0.410 (2nd RS) 5/2-, 3 M1 4P3/2 18/5
37
A s seen in Table 2, for the 1st R S (0 .0 8 7 ) and the 2nd R S (0 .4 1 0 ), the isosp in T = 3/2. This m eans that for these states unam biguously there are no channel m ixin g p + 8Li, d + 7Li, and also a + 5H e, w h ich have the isosp in T = 1/2. Problem s o f channels coupling are discussed in [24,25]. Thus, the single-channel approach, w h ich w e use here, is quite ju stified , esp ecially allow in g for the fact that capture from 1st R S (0 .0 8 7 ) is highest (see further).
5. C r ite r ia o f th e p o te n tia l co n stru c tio n
The potentials o f resonance w a v es are constructed in order to correctly describe the location o f resonance Er and its w idth rc.m., therefore their parameters are obtained quite unam biguously. GS potentials w ill be constructed in such a form that allow s one to correctly describe the channel binding energy, the charge radius o f 9B e and its asym ptotic constant in the p 8Li channel. Since, all know n values o f the asym ptotic norm alization co efficien t (A N C ) and the spectroscopic factor Sf, according to w h ich the asym ptotic constant (A C ) is obtained, have enough large error. The G S potentials also can have few options w ith different parameters o f w idth. H ow ever, at the g iv en values o f A C and binding energy, its parameters are constructed absolutely unam biguously.
The radius o f 8Li equals 2 .3 2 7 ± 0 .0 2 9 8 fm, w h ich is g iven in a database [26], w as used in further calculations. The radius o f 2 .5 1 8 ± 0 .0 1 1 9 fm for 9B e also is know n from the database [26]. In addition, for exam ple, the value o f 2 .2 9 9 (3 2 ) fm for 8Li radius w as found in [27]. In w ork [28], for these radiuses, values o f 2 .3 0 (4 ) and 2 .5 1 9 (1 2 ) fm w ere obtained, correspondingly. A ll these data agree w e ll am ong th em selves w ithin the lim its o f errors. The accurate values o f m (8Li) = 8 .0 2 2 4 8 7 amu [29] and mp = 1.0 0 7 2 7 6 4 6 6 9 amu [30] w ere used for the m asses o f nucleus and proton. The charge and m ass radius o f the proton is equal to 0 .8 7 7 5 (5 1 ) fm [30]. For 1 amu energy equivalent o f 9 3 1 .4 9 4 1 0 2 4 M eV w as used [30].
The spectroscopic factor Sf o f the GS and Anc A N C are connected by the next w ay [31]:
ANc = S f X C 2 , (4)
where C is the dim ensioned asym ptotic constant in fm -1/2, w h ich connects w ith d im en sion less A C Cw [32]
by C = -J 2 k 0 C w, and the d im ensionless constant Cw defined from the relation [32]:
XL(r) = 4 l k aCwW_4h+m (2k0r ) , (5)
where %b(r) is the num erical B S radial w avefunction, obtained as the solution o f the Schrodinger equation norm alized to unit size, W-,L+1/2(2k0r) is the W hittaker function o f the bound state, determ ining the asym ptotic behavior o f the w avefunction and obtained as the solution o f the sam e equation w ithout nuclear potential, k0 is a w avenum ber related to the channel binding energy E w here k 0 = ^ 2 p E / %2 in fm -1, n is the Coulom b parameter ц = цZ lZ 2e 2/(%2k0) = 3.44476 -10 2 y Z xZ 2/ k 0 , Z \ and Z2 are the particle charges, L is the orbital m om entum o f the bound state.
N ote that the spectroscopic factor Sf is used b y us on ly for the standard procedure o f the obtaining p ossib le C w range from the obtained in the experim ent A NC value [3 1 ,3 2 ].
6. P o te n tia ls o f th e p 8L i in tera ctio n
A s in our previous w orks [1 7 ] for other nuclear system s, w e use the potential o f the G aussian form w ith the point-like C oulom b term w ith the g iv en orbital m om entum L in each partial w ave as the p 8Li interaction:
V (r,L) = -VLexp(-yLr2). (6)
The 4P3/2 lev el w e consider as the ground state o f 9B e in the p 8Li and such potential should correctly describe the A C for this channel. In order to extract this constant C in form (4) or Cw (5) from the available experim ental data, let us consider inform ation regarding the spectroscopic factors Sf and asym ptotic norm alization coefficien ts A NC. For exam ple, in [3 3 ], excep t for their o w n results, the authors add data o f previous works. I f to separate from these sim ilar results, that is, w ith c lo se ly spaced valu es o f spectroscopic factors that can be presented in the form o f table 3.
Table 3 - Data on spectroscopic factors Sf for the GS of 9Be in the p8Li channel from works [9,11,33,34].
is the average value on data interval Reaction from what Sf
was determined
Sf for the p 8LiGS
channel Ref.
8Li(d,n)9Be 0.64(21) [33]
12C(9Be,8Li)14N 0.73(15) [11]
Average value 0.69, that is, = 0 83
Average value on data interval S f = 0.66(22), 0.43-0.88 = 0.81(14)
9Be(8Li,9Be)8Li 1.50(28) [34]
Potential model 1.50(27) [9]
Average value on all results 1.09, = 1.05
Data interval on all results
Sf = 1.11(68), 0.43-1.78
^ . = 1.05(28)
Furthermore, the A N C 4P3/2 G S in the p 8Li channel w as g iv en in [35], w here Anc = 10.75(12) fm"1/2 w as obtained. The constant for the 6P3/2 state is m uch less - 0 .2 5 (1 0 ) fm"1/2 [35]. Therefore, w e consider here only one spin channel w ith S = 3/2. On the basis o f expression (4) and average value on interval
= 0 .6 6 (2 2 ) from Table 2 according w orks [11,33] for the A C G S, the value C = 13.7 1 (2 .5 2 ) fm-1/2 w as obtained, and because ^ 2 k0 = 1.307, so d im en sion less A C (5) is equal to Cw = 10.49(1.93).
H ow ever, i f to use the average value on all w orks from table 3, equals 1.05(28), then for constant C, w e obtain a w id er data interval 11.06(3.06) fm"1/2 and Cw = 8 .4 6 (2 .3 4 ). C onsequently, the p ossib le interval o f Cw valu es on tw o data groups from Table 3 is approxim ately from 6 (from the latest data) to 12.5 (from the previous results).
Furthermore, tw o options o f the GS potentials w ith FS, w h ich a llow us to obtain the d im ensionless asym ptotic constant Cw in the g iv en above lim its, w ere obtained. The parameters o f these potentials Vl and Yl, and also m ain characteristics o f the nucleus, obtained w ith them (binding energy Eb, asym ptotic constant Cw, m ass radius <R>m and charge radius <R>ch) are listed in table 4.
Table 4 - GS potential parameters and main characteristics of 9Be
No. Vl, MeV ,YL fji Eb, MeV Cw <R>m, fm <R>ch, fm
1 212.151135 0.17 -16.88820 10.2(1) 2.40 2.46
2 286.178045 0.25 -16.88820 6.6(1) 2.36 2.38
For exam ple, potential N o . 1 has the FS and leads to the binding energy -1 6 .8 8 8 2 0 M eV . The definition o f the calculation expressions used here for the radii is g iven , for exam ple, in [17]. The above A C errors are defined by their averaging over the distance interval from 6 - 8 to 1 0 -1 2 fm , w h ich is the A C stabilization range. The phase shift o f the elastic scattering o f this potential for the G S 4P3/2 sm oothly decreases dow n to zero and at 5.0 M eV has the value o f ~ 3 3 0 °. H ere, w e consider that in the presence o f tw o bound FS and A S , the phase shift according to generalized L evinson theorem starting from 360° [20].
Furthermore, another option o f the G S potential that leads to a sm aller A C g iven in table 4.
39
Table 5 - Options of potential parameters with FS for resonance states of nuclear and some characteristics obtained with them.
In the two last columns, the experimental values listed above in Table 1 are shown.
No. 2s+1Lj Vj, MeV yj, fm-2 Er (c.m.), keV Г c.m,, keV Er (c.m.), keV rc.m,, keV
1. %/2 -5 0.1 - - - -
2. 4D3/2 269.242 0.2 1100 56 1100(30) 50(22)
3. 4P1/2 66.69121 0.075 87 ~0.4 87(1) 0.39(1)
4. 4P 5/2 34.0399 0.04 410 203 420(7) 210(20)
Potential N o . 3 from table 5 leads to the 4P1/2 scattering phase shift, plotted in figure 1 by the black solid curve, w h ich is show n at energies up to 5.0 M eV and has the resonance at 87 keV . Scattering potential N o . 4 has the phase shift 4P5/2 presented in figure 1 by the red dashed curve and at the considered energies has the resonance at 4 1 0 keV . Potential N o . 2 leads to the 4D3/2 phase shift show n in figure 1 by the green solid curve. A ll resonance potentials have the phase shift o f 9 0 .0 °(1 ) at the resonance energy and the bound FS.
7. C o n clu sio n
It is possib le to construct tw o-b od y potentials o f the p 8Li interaction, w h ich allow us to correctly describe the available data on characteristics o f the bound state o f 9B e in the p 8Li channel in the frame o f the M PCM . Suggested options o f the G S potentials o f 9B e in the p 8Li channel allow one to obtain A C w ithin lim its o f errors available for it and lead to the reasonable description o f 9B e radii. Such potentials generally a llow the available experim ental data for total cross sections o f the radiative proton capture on 8Li at lo w and ultralow energies. Obtained results for total cross sections and static characteristics o f 9Be strongly depend o f G S potential parameters o f this nuclear in the p 8Li channel.
A c k n o w le d g m e n ts
This w ork w as supported by the M inistry o f Education and S cience o f the R epublic o f Kazakhstan (Grant N o . A P 0 5 1 3 0 1 0 4 ) entitled “Study o f R eactions o f the R adiative Capture in Stars and in the Controlled Therm onuclear F usion” through the F esen k ov A strophysical Institute o f the N ational Center for Space R esearch and T ech n ology o f the M inistry o f D igital D evelopm ent, Innovation and A erospace Industry o f the R epublic o f Kazakhstan (RK).
А .В . Д ж азаи р ов -К ахр ам ан ов 12, Л .Т . К ари п баев а12, А .А .С тебл я к ов а2 1 В.Г. Фесенков атындагы Астрофизика институты ЕЖШС, Алматы, Казахстан;
2Халыкаралык Акпараттандыру Академиясы, Алматы, Казакстан А С Т Р О Ф И ЗИ К А Л Ь Щ Э Н Е Р Г И Я Л А Р Д А Г Ы 8L i(p ,y 9Be
Ц А М Т У Г А А Р Н А Л Г А Н Y Л Е С Т IК П О Т Е Н Ц И А Л Д А Р
А н н отац и я . Юнга сызбасы бойынша орбиталык кластерлш K Y ^i жштеумен катар, тыйым салынган ^ й д е п модификацияланган потенциалды кластерлi модель аясында, астрофизикалык энергияларда протондарды радиациялык кармауды толык кесу карастырылган. Экспериментпк мэлiметтердi алу уш ш 9Be(y,p0)8Li фотокирату реакциясын толык кесудi кайта есептеу колданылады.
Астрофизикалык энергияларда 8Li(p, y)9B e радиациялык кармау Yшiн гаусс потенциалдарынын кeрсеткiштерi аныкталды. Сонымен катар, [Shoda, Tanaka (1999)] 9Be: 9Be(y,p)8Li, 9Be(y,d)7Li, 9Be(y,?)6Li, сондай-ак 9Be(y,3He)6He эртYрлi екiлiк тарату арналары эксперименталды зерттелдi. Олардыц егжей- тегжейлi тепе-тендiгiмен байланысты еш бел ш ек т радиациялык басып алу процестерi 9Be синтезiне алып келедi ж эне астрофизикалык косымшалар контекстшде тиiстi багалауды талап етедi. Бул жагдайда, бiрiншi Yш реакция p8Li, d7Li ж эне t6L i арналарында 3He6He-ге караганда тем ен кулон к ед е р п с ш ц болуына, ягни 3:4 катынасында 4He5He арналарында, атап айтканда 3:4 катынасында назар аудару керек. 8Li(p,y)9B e келденен кимасы астрофизикалык кызыгушылык тудыратын энергия кезiндегi
екiншi 8Li шогырдыц ж эне шагын киманын тем ен каркындылыгынан тiкелей аныктау киын. Сонымен катар, 8Li(p,y)9Be реакциясын зерттеу мэселесi 8Li-838 мс ядросыныц жартылай ыдырауынын ете аз кезещ нде кималарды тшелей эксперименталды елш еу мYмкiн емес болып табылады. Алайда, n8Li- кармау жагдайында да радиациялык кармау моделiн ж эне спектроскопиялык факторды пайдалана отырып, тшелей кармау кимасын алу Yшiн кейбiр жанама эдiстер пайдаланылуы мYмкiн. 9Be механизмдердiн пайда болуын зерттеу А = 8 массалы санылауды енсеру ж эне ерте Э л ем деп ауыр элементтердiн синтезiне, сонымен катар supernovae r-процесс нюклеосинтез мэселесiне тж елей катынасы бар. Казiргi танда 9Be еш сатылы YPДiс нэтижесiнде пайда болады деген ой калыптасты:
o (a ,y )8B e альфа бeлш ектердiн радиациалык кармауы кыска гумырлы изотоптын 8Be (?ш = 6 .7 * 1 0-17 s) синтезiне, одан кейiн 8Be(n,y)9Be нейтронынын радиациалык кармауына экелiп согады.Аталган макалада бiз тeменгi астрофизикалык энергиядагы p 8Li ^ 9Bey реакциясын карастырамыз, себебi жарияланганына 2 0 жыл уакыт болганына карамастан «тэжiрибелiк зеттерулер жоспары» ретш де рел аткаратын Terasawa et al. фундаменталды жумысындагы ауыр элементтердщ синтезiне алып келетiн мэндi YPдiстiц тiзбегiне косылган. Leistenschneider et al. (2018) жаца жумысында келтiрiлген 2 М эВ-ке деш н п резонанс курылымын ескере отырып 10 кэВ-тен 7.0 М эВ-ге д е т н л энергиянын аймагы карастырылган. Бул реакцияныц жылдамдыгы 0.01-ден 10 Т9 температура аймагына гана есептелген.
Резонанстар шамалары мен орындарынын реакция жылдамдыгыныц формасы мен шамасына эсерiнiн анализi усынылган.
ТYЙiн сездер: Ядролык астрофизика; бастапкы нюклеосинтез; жылулык ж эне астрофизикалык энергиялар; p 8Li кластерлi жYЙесi; радиациялык кармау; толык кесу.
А .В . Д ж азаи р ов -К ахр ам ан ов 1,2, Л .Т . К ари п баев а1,2, А .А .С тебл я к ов а2
А строф изический институт В.Г. Фесенкова “НЦКИТ” АКК МИР РК, Алматы, Казахстан;
2М еждународная Академия Информатизации, пр. Абылай Хана 79, Алматы, Казахстан П А Р Ц И А Л Ь Н Ы Е П О Т Е Н Ц И А Л Ы Д Л Я 8Li(p,y)9B e ЗА Х В А Т А
П Р И А С Т Р О Ф И ЗИ Ч Е С К И Х Э Н Е Р Г И Я Х
А н н отац и я . В рамках модифицированной потенциальной кластерной модели с запрещенными состояниями, с классификацией орбитальных кластерных состояний по схемам Юнга рассмотрены полные сечения радиационного захвата протонов на 8Li при астрофизических энергиях. Перерасчет полных сечений реакции фоторазвала 9Be(y,p0) 8Li используется для получения экспериментальных данных. Были определены параметры гауссовых потенциалов для радиационного захвата 8Li(p,y)9Be при астрофизических энергиях. В то же время в [Shoda, Tanaka (1999)] экспериментально исследованы различные бинарные каналы фоторасщепления 9Be: 9Be(y,p)8Li, 9Be(y,d)7Li, 9Be(y,?)6Li, а также 9Be(y,3H e)6He. Очевидно, что связанные с ними детальным равновесием процессы двухчастичного радиационного захвата приводят к синтезу 9Be и требую т соответствующей оценки в контексте астрофизических приложений. При этом, следует обратить внимание на то, что первые три реакции имеют кулоновский барьер в каналах p8Li, d7Li и t6L i ниже, чем в 3He6He равно как и 4He5He каналах, а именно в соотнош ении 3:4. Поперечное сечение 8Li(p,y)9Be трудно определить непосредственно из-за низкой интенсивности вторичного 8Li пучка и малого сечения при энергиях, представляющих астрофизический интерес. Кроме того, проблема изучения реакции 8Li(p,y)9Be заключается еще и в том, что прямое экспериментальное измерение сечений оказывается практически невозможным из-за очень малого период полураспада ядра 8Li - 838 мс. Однако, как и в случае n8Li-захвата, могут быть использованы некоторые косвенные методы для извлечения сечения прямого захвата с использованием модели радиационного захвата и спектроскопического фактора. Исследование механизмов образования 9Be имеет прямое отношение к проблеме преодоления массовой щели с A = 8 и синтезу более тяжелых элементов в ранней Вселенной, а также в r-процесс нюклеосинтеза в суперновых. В настоящее время сложилось мнение, что 9B e образуется в результате двухступенчатого процесса: радиационный захват альфа-частиц a(a,Y)8B e приводит к синтезу короткоживущего изотопа 8Be (?1/2 = 6 .7 * 1 0-17 s), и далее радиационный захват нейтрона 8Be(n,y)9Be. В данной статье мы рассматриваем реакцию p8L i ^ 9Bey при низких астрофизических энергиях в связи с тем, что она включена в цепочку значимых процессов, которые приводят к синтезу более тяжелых элементов в фундаментальной значимой работе Terasawa et al., которая с момента ее опубликования следующ ие почти 20 лет играет роль некоторого «плана
41
практических исследований». Рассмотрена область энергий от 10 кэВ до 7.0 М эВ, но с учетом структуры резонансов только до 2 М эВ, которая была приведена в новой работе Leistenschneider et al.
(2018). Скорость этой реакции рассчитана в области температур от 0.01 до 10 T9. Представлен анализ влияния положения и величины резонансов на величину и форму скорости реакции.
К л ю ч ев ы е слова: Ядерная астрофизика; первичный нюклеосинтез; тепловые и астрофизические энергии; кластерная система p8Li; радиационный захват; полное сечение.
Information about authors:
Dzhazairov-Kakhramanov A.V., Candidate of Physical and Mathematical Sciences, Fesenkov Astrophysical Institute.
Academician of the European Academy of Natural Sciences (EANS), academician of the International Informatization Academy (IIA), awarded of the European Carl Friedrich Gauss medal, Professor of the Russian Academy of Natural Sciences. albert- j@yandex.ru, https://orcid.org/0000-0003-2845-6889;
Karipbayeva L.T., Researcher, Fesenkov Astrophysical Institute, academician of the International Informatization Academy.
larisa_karipbaeva@mail.ru, https://orcid.org/0000-0002-6238-2441;
Steblyakova A.A., Researcher, Corresponding member of the International Informatization Academy, awarded of the National Order of Enbek Yzdigi. a_steblyakova@mail.ru, https://orcid.org/0000-0003-2730-922X
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