ISSN 1062-8738, Bulletin of the Russian Academy of Sciences: Physics, 2008, Vol. 72, No. 3, pp. 295–300. © Allerton Press, Inc., 2008.Original Russian Text © L.D. Blokhintsev, B.F. Irgaziev, A.M. Mukhamedzhanov, A.N. Safronov, A.A. Safronov, 2008, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fiz-icheskaya, 2008, Vol. 72, No. 3, pp. 321–326.

295

Determination of the Nuclear Vertex Constantsfor the

7

Be

↔

3

He

4

He Vertex using the

N

/

D

Equations and Calculation of the Astrophysical

S

Factorfor the

4

He(

3

He,

g

)

7

Be Reaction

L. D. Blokhintsev, B. F. Irgaziev, A. M. Mukhamedzhanov, A. N. Safronov, and A. A. Safronov

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991 Russia

Abstract

—

N

/

D

equations taking into account the Coulomb interaction effects are used to consider the phaseanalysis data on

3

He

4

He scattering and determine the nuclear vertex constants for the

7

Be

↔

3

He

4

He

vertex,where the beryllium nucleus is in the ground (

3/2

–

) or first excited (

1/2

–

) states. Information about the asymp-totic normalization coefficients of the wave functions of the

7

Be nucleus in the noted states is derived. The dataobtained are used to calculate the astrophysical

S

factor for the

4

He(

3

He,

γ

)

7

Be

reaction.

DOI:

10.3103/S1062873808030064

INTRODUCTIONDerivation of information about the characteristics of

the wave functions of bound states (in particular, aboutthe asymptotic normalization coefficients (ANCs) ofthese functions) from analysis of nucleus–nucleus scat-tering processes is of great interest; primarily, for nuclearastrophysics. It is known [1] that ANCs are related to themodel-independent characteristics of nuclear reactions—nuclear vertex constants, which determine the residues ofthe partial

S

matrix in the poles corresponding to boundstates. The purpose of this study is to derive informationabout the vertex constants of the processes of virtual two-cluster dissociation (synthesis),

7

Be

↔

3

He

4

He

, of the

7

Be nucleus in the ground or first excited states on thebasis of the existing phase analysis data on

3

He

4

He scat-tering [2, 3], using the

N

/

D

equations, which take intoaccount the Coulomb interaction effects [4–6]. With theuse of the found values of ANCs for the ground andexcited states of the

7

Be nucleus, the astrophysicalfactor

S

(

E

)

is calculated for the

4

He(

3

He,

γ

)

7

Be

reac-tion at ultralow energies. The rate of this reactionsignificantly affects the solar neutrino flux and deter-mines the abundance of the

7

Be isotope in nature.The results obtained are compared with the existingexperimental data.

1. ANALYTIC PROPERTIES OF THE REDUCED COULOMB–NUCLEAR AMPLITUDE

OF NUCLEUS–NUCLEUS SCATTERINGAND THEIR RELATIONSHIP

WITH THE CHARACTERISTICS OF BOUND STATES OF THE SYSTEM

In view of the recently published concepts, accord-ing to which the properties of multinucleon states can-

not be in principle related to phase shifts of nucleus–nucleus scattering processes [7], we will briefly presentour point of view on this problem. We proceed from theprinciples lying in basis of the analytical theory of the

S

matrix [8]. In accordance with these principles, thepartial scattering amplitude of two stable particles(such as hadrons, nuclei, etc.) is an analytic function inthe complex plane of the square of the relative momen-tum of colliding particles,

s

=

k

2

, with poles related tobound states and cuts generated by these poles, due tothe unitarity conditions in all possible scattering chan-nels. Thus, the characteristics of the bound states, i.e.,the positions of the poles corresponding to these statesand the values of residues of the scattering amplitude inthese poles, in view of the analyticity condition, areinevitably related to the phase shifts, which are deter-mined by the boundary values of the analytic function(scattering amplitude) on the upper edge of the right cutin the complex

s

plane. The analytic properties of thereduced Coulomb–nuclear partial scattering amplitude,

(

k

2

)

, which has the same analytical structure on aphysical sheet as the purely nuclear scattering ampli-tude, play a key role in the approach under consider-ation [1, 4–6]. In the physical region, this amplitude is

related to the Coulomb–nuclear phase shift through the equality

(1)

where

(

s

) = ( )

,

TlCN

δlCN

TlCN s( )

δlCN( ) iδl

CN( )expsin

ρlC s( )

-----------------------------------------------,=

ρlC sCl

2 s

296

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Vol. 72

No. 3

2008

BLOKHINTSEV et al.

(2)

is the Coulomb penetrability factor,

λ

=

α

C

Z

B

Z

C

m

BC

,

α

C

is the fine-structure constant,

Z

B

and

Z

C

are thecharges of colliding particles, and

m

BC

is their reducedmass. In the first (physical) sheet of the Riemann sur-

face in the complex

s

plane, the amplitude

(

k

2

)

hasa right (unitary) cut

C

R

; poles, corresponding to boundstates; and a left (dynamic) cut

C

L

, caused by exchangeprocesses. Above the inelastic-channel threshold, thisamplitude has also an “inelastic” cut. As calculationsshow, the effect of inelastic channels on the phase shiftsand other characteristics of partial amplitudes (seebelow) in the low-energy region is weak; therefore, theeffect of inelastic channels will be neglected in thisstudy. Thus, at the right cut

C

R

, the unitarity conditionhas the form

(3)

where

θ

(

s

)

is the Heaviside function.

When a system contains a bound state with the bind-ing energy

ε

=

κ

2

/2

m

BC

, the residue of the reduced par-

tial amplitude

(

s) in the pole at s = –κ2 can berelated to the vertex constant Gl of virtual dissociation(synthesis), A ↔ B + C, of the bound state A into frag-ments B and C; this constant is determined taking intoaccount the Coulomb interaction between the frag-ments [1, 4]:

(4)

The vertex constant Gl is related to the normaliza-tion coefficient Cl in the asymptotics of the wave func-tion of the bound state as follows:

(5)

2. N/D EQUATIONS FOR THE REDUCED COULOMB–NUCLEAR AMPLITUDE

AND A METHOD FOR THEIR SOLUTION

The amplitude (s) obeys the N/D representation

(6)

where the function Nl(s) has only left (dynamic) cut CLand the function Dl(s) has a right (unitary) cut CR in thecomplex s plane. The unitarity condition (3) on the rightcut leads to the relation

(7)

Cl2 k( ) πλ

k------ πλ

k------⎝ ⎠

⎛ ⎞coth 1– 1 λ2

n2k2----------+⎝ ⎠

⎛ ⎞n 1=

l

∏=

TlCN

ImTlCN s( ) ρl

C s( ) TlCN s( ) 2θ s( ),=

TlCN

s κ2+( )TlCN s( )[ ]

s κ2–→lim

mBC

π---------Gl

2.–=

Cl i–( )l mBC

πl!-----------Γ l 1 λ

κ---+ +⎝ ⎠

⎛ ⎞ Gl.–=

TlCN

TlCN s( )

Nl s( )Dl s( )-------------,=

ImDl s( ) ρlC s( )Nl s( )θ s( ).–=

On the left cut CL, we have the equality

(8)

where νl(s) = Im (s) on CL and sL is the point of thedynamic cut that is closest to the physical region. Under

the assumption that the amplitude (s) has noCastillejo–Dalitz–Dyson poles [9], to determine thefunctions Nl(s) and Dl(s) with the use of relations (7)and (8), we obtain the following system of equationstaking into account the effects of long-range Coulombinteraction:

(9)

(10)

It can be shown that the function Nl(s) satisfies theequation that explicitly takes into account its thresholdbehavior (Nl(s) ~ sl):

(11)

where Ll(s) = sl (s) is a potential function and (s) ina reduced potential function, determined by the integralover the left cut:

(12)

Equations (10) and (11) demonstrate that, to calculatethe scattering amplitude in the physical region, it is nec-essary to set the reduced potential function (12) on theright cut CR. We can calculate this function on CR withan arbitrary accuracy, using the quadrature formula andchoosing a sufficiently large number of nodal points NLon the left cut:

(13)

where si are the positions of the nodal points on the leftcut and λi are the values determined by jumps of thepartial amplitude at the nodal points νl(si) and the coef-ficients ci of the quadrature formula:

(14)

Note that application of the quadrature formula (13) tothe reduced potential function is equivalent to approxi-

ImNl s( ) νl s( )Dl s( )θ sL s–( ),=

TlCN

TlCN

Nl s( ) 1π---

νl s '( )Dl s '( )s ' s–

---------------------------- s 'd

∞–

sL

∫=

Dl s( ) 11π---

Nl s '( )s ' s–--------------ρl

C s '( ) s '.d

0

∞

∫–=

Nl s( ) Ll s( ) 1π---

s/s '( )lLl s '( ) Ll s( )–s ' s–

----------------------------------------------

0

∞

∫+=

× Nl s '( )ρlC s '( ) s ',d

L̃l L̃l

L̃l s( ) 1π---

νl s '( )s 'l s ' s–( )---------------------- s '.d

∞–

sL

∫=

L̃l s( )λi

sil si s–( )

--------------------,i 1=

NL

∑≅

λi1π---νl si( )ci.=

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS Vol. 72 No. 3 2008

DETERMINATION OF THE NUCLEAR VERTEX CONSTANTS 297

mation of the jump of the amplitude (s) on thedynamic cut νl(s) by a finite number of δ functions:

(15)

Thus, the dynamic cut of the amplitude (s) isapproximated in this case by poles at nodal points.Choosing a sufficiently large number of nodal points (inpractice, NL was taken to be 600), one can approximatethe amplitude with a high degree of accuracy not onlyon the right cut but also in the nonphysical region.

It follows from formulas (11) and (13) that the func-tion Nl(s) has the form

(16)

where the coefficients bi = λiDl(si) satisfy l equations (ifl > 0)

(17)

which ensure correct threshold behavior of the Cou-

lomb–nuclear amplitude (s). Thus, if the jump ofthe partial amplitude νl(si) at the nodal points on the leftcut si is known (i.e., the values of λi in formula (13) areknown), using formulas (10), (11), and (13), one canobtain the following system of linear equations fordetermining the coefficients bi:

(18)

where fi = fl(si) and hi = hl(si); the functions fl(s) andhl(s) are determined by the expressions

(19)

(20)

Note that, upon scattering, the coefficients λi are notarbitrary in states with nonzero angular momenta. Theyshould be such to satisfy Eqs. (17). It is known that, inthe case l ≠ 0, a solution to the N/D equations for an

TlCN

νl s( ) π λiδ s si–( ).i

NL

∑=

TlCN

Nl s( )bi

si s–-----------,

i 1=

NL

∑≈

b j

s jn

----j 1=

NL

∑ 0, n 1…l,= =

TlCN

bi λi 1 hibi

f j f i–s j si–---------------b j

j i≠

NL

∑+ + ,=

f l s( ) sπ---

ρlC s '( ) s 'd

s ' s ' s–( )----------------------,

0

∞

∫=

hl s( ) 1π---

ρlC s '( ) s 'd

s ' s–( )2----------------------.

0

∞

∫=

arbitrary partial amplitude jump on the dynamic cut isgenerally absent [8].

3. METHOD FOR TAKING INTO ACCOUNTTHE DYNAMIC CUTS OF THE PARTIAL

AMPLITUDE THAT ARE REMOTEFROM THE PHYSICAL REGION

Analyzing specific processes, one cannot exactlycalculate partial amplitude jumps on all dynamic cuts.However, in many cases, it is possible to calculate thecontributions of the dynamic singularities that are near-est to the physical region. In this section, we willassume that the dynamic cut consists of two portions: aportion between the points [sL, sA], which is nearest tothe physical region, and a portion between the points[sA, sB], which is at a larger distance from the physical

region. We will consider the jump (s) of the reducedpartial amplitude on the first portion of the cut to beknown. Let us assume that NA nodal points are locatedon this portion of the dynamic cut. Using relation (14),we find the first NA coefficients λi (i = 1, …NA). To

approximate the partial amplitude jump (s) on thecut between the points sA and sB, we will choose someset of basis functions (for example, polynomials orbasis functions of the Fourier expansion) ϕβ(s) (β =1…NB). Let us represent the coefficients bj (see (16)) atthe nodal points sj (at j > NA) as an expansion over thesebasis functions:

(21)

Note that setting the coefficients bj is equivalent to set-ting the partial amplitude jump on the dynamic cut atthe nodal points sj. The relationship between thesequantities is determined by formulas (14) and (18). Theexpansion coefficients aβ in formula (21) will be chosenwith regard to Eqs. (17); the required position of zerosof the function Dl(s), corresponding to bound states (ifsuch exist); and the condition of best description of thephase analysis data. For definiteness, we will assumethat there is one bound state in the system (the equa-tions can easily be generalized to the case of a largernumber of bound states). Let us consider in more detailthe equations that take into account all the above-men-tioned conditions.

Requirement of the presence of a bound state in thesystem with the binding energy εb = κ2/2mBC leads to theequation

(22)

νlA

νlB

b j aβϕβ s j( ).β 1=

NB

∑=

1f j f l sb( )–

s j sb–-------------------------b j

j 1=

NL

∑+ 0,=

298

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS Vol. 72 No. 3 2008

BLOKHINTSEV et al.

where sb = –κ2. Let us now formulate the equations opti-mizing the description of phase shifts for choosing thecoefficients aβ in expansion (21). The theoretical Cou-

lomb–nuclear scattering phase for a specified setof the coefficients bi, which are found by solvingEqs. (9) and (10), is determined by the expression

(23)

where

(24)

and Nl(s) is given by formula (16). It follows from for-mulas (16), (23), and (24) that, if the experimental Cou-

lomb–nuclear phase exactly coincides with thetheoretical phase at some point sn, the following equal-ity is satisfied:

(25)

where

(26)

fn = Refl(sn + iε).

Let us now assume that the phase shifts are knownfrom analysis of the experimental data for some set ofpoints sn (n = 1…NR) in the physical region. By defini-tion, we will introduce the function

(27)

where Zn are some weighting coefficients. It followsfrom the above-said that the function Φ is nonnegative;is can be equal to zero only if the theoretical phase

δlCN

δlCN s( )tan

ρlC s( )Nl s( )ReDl s( )

---------------------------,=

ReDl s( ) 1f j Re f l s iε+( )–

s j s–-----------------------------------------b j

j 1=

NL

∑+=

δlnCN

Q jnb j

j 1=

NL

∑ δlnCNtan+ 0,=

Q jn

ρlCN sn( ) f j f n–( ) δln

CNtan+s j sn–

---------------------------------------------------------------,=

Φ Zn2 Q jnb j

j 1=

NL

∑ δlnCNtan+

⎩ ⎭⎨ ⎬⎧ ⎫

2

,n 1=

NR

∑=

shifts at all points sn (n = 1…NR) coincide with theexperimental phase shifts for this set of the coeffi-cients bj .

To determine the coefficients bj at the nodal pointsfor j > NA (i.e., on the portion of the dynamic cutbetween the points sA and sB, which is far from the phys-ical region) in Eqs. (16), (17), (22), and (24) andexpression (27), we will use expansion (21). To calcu-late the Coulomb–nuclear amplitude (6), we have todetermine N = NA + NB values: NA coefficients bj (j =1…NA) and NB coefficients aβ. The first NA equations areobtained using relations (18) at i = 1, …, NA (recall thatthe coefficients λi at i = 1, …, NA are assumed to beknown). Using relations (17) and (22), we obtain l + 1equations that ensure the required position of the poleof the partial amplitude, which corresponds to a boundstate, as well as the correct threshold behavior of thisamplitude. Finally, the remaining NC = NB – l –1 equa-tions are obtained by minimizing the function Φ withrespect to variation in the parameters aβ at β = l +2, …, NB

(28)

4. RESULTS OF NUMERICALCALCULATIONS

During the 3He4He scattering, the dynamic cuts ofthe partial Coulomb–nuclear scattering amplitude

(s) that are nearest to the physical region are due tothe mechanism of neutron transfer (Fig. 1a). The partialamplitude jumps for scattering in the states Jp = 3/2–,1/2– were calculated taking into account the radiativecorrections due to the Coulomb interaction in the three-particle intermediate state (Fig. 1b) and in the two-par-ticle initial and final states. The jump of the P-waveamplitude, corresponding to the mechanism demon-strated in Fig. 1a, has the form

(29)

where Ga is the nuclear vertex constant of virtual disso-ciation (synthesis), 4ç ↔ 3ç + n, of the 4He nucleus,

mN is the nucleon mass, = 3εamN/2, εa is the dissoci-ation energy in the above-mentioned process, sL =

−16 /49, and sP = –16 . In accordance with the data

of [10], the constant was chosen to be 2.5 fm. The

factor of the Coulomb–nuclear renormalization ofthe amplitude of the diagram in Fig. 1a on the dynamiccut (which is due to the electromagnetic interaction inthe three-body intermediate state (Fig. 1b)) within the

∂Φ∂aβ-------- 0.=

TlCN

ν1 s( )25Ga

2mN2

112s-------------------- 1

16κa2

25s-----------+⎝ ⎠

⎛ ⎞ θ sL s–( )θ s sP–( ),=

κa2

κa2 κa

2

Ga2

Z p3CN

3He 4He 4He3He

3He4He3He4He

nnγ

(a) (b)

Fig. 1. Diagrams determining the dynamic cuts of the par-

tial Coulomb–nuclear amplitudes (s) of 3He4He scat-

tering that are nearest to the physical region.

TlCN

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS Vol. 72 No. 3 2008

DETERMINATION OF THE NUCLEAR VERTEX CONSTANTS 299

so-called polar approximation [4, 11] (the cosine of thescattering angle in this approximation is assumed to beequal to its value at the amplitude pole point correspond-ing to the diagram in Fig. 1a) has the form

(30)Z p3CN 3αCmN

2κay s( )------------------- 1 y s( )+

1 y s( )–-------------------⎝ ⎠

⎛ ⎞ln– ,exp=

where y(s) = .

Figure 2 shows the results of three versions, A, B,and C (solid, dotted, and dashed lines, respectively), ofthe analytical approximation of the phase shifts for the3He4He scattering in the 3/2– state, which were obtainedon the basis of the above-described method, and the

sL s–sL sP–---------------

180

160

140

120

100

δ–3/2, deg

0 2 4 6 8 10 12 14E, MeV

Fig. 2. Versions of analytical approximation of the Cou-lomb–nuclear phase shift for the 3He4He scattering in the3/2– state.

200

150

100

50

00 0.5 1.0 1.5 2.0 2.5 3.0

s, fm–2

L(s), fm3

Fig. 3. Reduced potential function (13) for the versions ofanalytical approximation of the Coulomb–nuclear phaseshift for the 3He4He scattering in the 3/2– state, shown inFig. 2.

180

160

140

120

100

δ–1/2, deg

0 2 4 6 8 10 12 14E, MeV

80

Fig. 4. The same as in Fig. 2 for the 3He4He scattering in the1/2– state.

120

80

40

20

00 0.5 1.0 1.5 2.0 2.5 3.0

s, fm–2

L(s), fm3

60

100

Fig. 5. Reduced potential function (13) for the versions ofanalytical approximation of the Coulomb–nuclear phaseshift for the 3He4He scattering in the 1/2– state, shown inFig. 4.

Results of the calculation of the vertex constants G for the dissociation (synthesis) 7Be ↔ 3He4He of the 7Be nucleus in the ground(3/2–) or first excited (1/2–) states and asymptotic normalization coefficients C of the nuclear wave functions of these states

Jp 3/2– 1/2–

Version A B C A B

G2, fm –0.316 –0.261 –0.188 –0.154 –0.120

C, fm–1/2 3.867 3.515 2.983 2.950 2.612

300

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS Vol. 72 No. 3 2008

BLOKHINTSEV et al.

phase analysis data [2, 3]. Figure 3 shows the corre-sponding reduced potential functions (13). Similarresults for the 3He4He scattering in the 1/2– state areshown in Figs. 4 and 5.

The astrophysical S factor of the 4çÂ(3çÂ, γ)7Bereaction was calculated on the basis of the R-matrix for-malism, taking into account the transitions to theground and first excited states of the 7Be nucleus. Aswas shown previously, this process has a peripheralcharacter, and the region r < 4 fm makes only a smallcontribution [12]. The channel radius r0 (the radius of arigid sphere in the R-matrix approach) was chosen to be3.1 fm. This value leads to the ratio of the S(0) factorsof the transitions to the first excited and ground statesequal to 0.38, which is in good agreement with the the-oretical and experimental data in the literature. Assum-ing the asymptotic normalization coefficients for thetransitions to the 3/2– and 1/2– states to be, respectively,3.867 and 2.950 fm–1.2 (table), we obtain S(0) =0.47 keV b.

CONCLUSIONS

Information about the nuclear vertex constants forthe 7Be ↔ 3He4He vertex, where the beryllium nucleusis in the ground (3/2–) or first excited (1/2–) states, hasbeen derived on the basis of the analytical approxima-tion of the phase analysis data on the 3çÂ4ç scatteringin the P-wave states. The approximation was performedusing the N/D equations, which takes into account theCoulomb interaction effects. Information about theasymptotic normalization coefficients of the wave func-tions of the 7Ç nucleus in the noted states has also beenderived. On the basis of the analysis performed, one canconclude that the most preferred ANC values are in therange 3.86–3.51 fm–1/2 for the ground state of the 7Benucleus and in the range 2.95–2.61 fm–1/2 for the firstexcited state.

The data obtained by us are in good agreement withthe results of variational calculations by the MonteCarlo method [13] (C = 3.55 ± 0.15 fm–1/2 for the groundstate of the 7Be nucleus and C = 2.90 ± 0.10 fm–1/2 forthe first excited state). These values correspond to theastrophysical factor S(0) = 0.40 keV b. Analysis of thephase shifts for 3He4He scattering, performed in [14] on

the basis of the potential approach, gave S(0) =0.516 keV b. However, this result was obtained by mul-tiplying S(0) by the spectroscopic factor SF = 1.175. Webelieve this procedure to be inconsistent. Exclusion ofSF leads to S(0) = 0.44 keV b; this value is in goodagreement with the data of variational calculations. Toconclude, we should note that the last measurements ofS(0) gave a value of 0.507 keV b [15].

ACKNOWLEDGMENTS

This study was supported by the Russian Founda-tion for Basic Research, project no. 07-02-00609, andGrant NSh-5365.2006.2 of the President of the RussianFederation for Support of Leading Scientific Schools.

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