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Coulomb polarization and nuclear reactions at astrophysically low energies
Nuclear Physics A491 (1989) 109-129 North-Holland. Amsterdam
COULOMB POLARIZATION AND NUCLEAR REACTIONS AT ASTROPHYSICALLY LOW ENERGIES
Received
IS July 1988
Abstract: The effects of electric
polarizability of particles are shown to play an insignificant role in the nucleosynthesis reactions at astrophysically low energies. The polarization corrections to the reaction S-factors are defined by the value of the polarization potential at the boundary of the nuclear force range and do not exceed a quantity of the order of 0.1%. It is shown that the conclusion of papers claiming the very strong influence of the particle polarirability on pp-, pdand dt-reaction cross sections is a consequence of an incorrect approximation of the initi~l-~h~nne~ wave function.
1. Introduction A charged structural particle placed in an electric field undergoes deformation. As a result of the electric polarizability of the particle, an additional long-range attractive interaction arises in the system. For a spherically symmetric charge distribution this polarization potential decreases as rm4 [ref. ‘)] when the radius r is large. In the elastic scattering of like-charged complex particles the electric polarizability results in a substantial distortion of the wave function at large distances and also in the divergence ‘,‘) of the standard Coulomb-modi~ed scattering length ‘.5). For the proton-deuteron system the divergence of the scattering length due to deuteron polarizability was noted in refs. ‘,‘,‘). In those papers it was pointed out, in particular, that to extract the pd scattering lengths experimentally, one should use the data for not too low energies, E 3 20 keV [ref. “f]. At higher energies the polarization potential contribution to the nuclear-polarization scattering length becomes negligibly small, and the contribution to the effective radius ranges from ~3% at 100 keV [ref. “)I to -0.1% at 400 keV [refs. “.“)]. The computation “I) performed by the phase function method reveals that the polarization potential does not practically affect the proton-deuteron scattering length at distances up to -300 fm. At larger distances the polarization potential contribution becomes dominant, resulting, in particular, in the above-cited divergence of the Coulomb modified nuclear-polarization scattering length. So, to obtain the required accuracy at low energies -2 keV one needs to integrate the polarization 03759474/89/$03.50 0 Elsevier Science Publishers (No~h-~oll~~ld Physics Publishing Division)
N.V.
I/. P. Leonshev / Coulomb
110
~o~ur~~u~~on
potential up to the radius -15rc104fm [refs. “‘,“)I, which considerably exceeds the Coulomb turning point value r(. = 700 fm. To describe the low-energy scattering of charged composite particles, it is necessary to make use of the modified effective-range theory 3*8*‘0)in which the nuclear phase shift is defined with respect to the phase shift from the Coulomb and polarization long-range potentials. The resultant scattering length and effective range modified by the polarization-Coulomb field are very close 8-‘0) to relevant quantities in the nuclear-Coulomb problem. The present paper is aimed at studying the role of the polarization effects in low-energy nuclear reactions. We investigate the possibility for the nucleosynthesis reaction cross sections to increase due to an additional attraction generated by the electric polarizability of colliding particles. Sect. 2 presents in brief the characteristics of the reactions concerned and also the information on the pp reaction considered here as a typical example. The parameters of the polarization potential are discussed in sect. 3. In sect. 4 we give the expressions for the astrophysical S-factor of the pp reaction, ignoring the polarization effects. The influence of the polarization potential on the wave function at small distances is studied for the low-energy proton-proton scattering in sect. 5. In sect. 6 the expressions for the polarization contributions to the pp reaction S-factor are obtained and it is shown that the polarization effects in nucleosynthesis reactions are very small at low energies. In conclusion, a brief summary of the main results of the paper is presented.
2. Nucleosynthesis
reactions. Proton-proton
reaction
We consider the reactions which are induced by a collision of two nuclei with charges Z,e and Z,e and result in a synthesis of more complex nuclear systems. The reactions pp + de+v, pd + y’He, dt -+ n4He, etc. typify the processes concerned. These reactions and similar ones including more complex nuclei play an important role in astrophysics, providing, in particular, the stellar energy generation. At low energies the totai cross sections ofthe reactions under consideration are characterized by an exponential decrease because of the Coulomb repulsion: a(E)
= S( E)E’
em’nq.
(2.1)
In eq. (2.1) E = (hk)‘/Zp is the kinetic energy of colliding particles in the center-ofmass system (k is the wave number, supposed to be small, k s 1, p is the reduced mass of the system), ‘7 = (ka,)-’ is the Sommerfeld
parameter, an = h’/&Z,Zz
denotes
(2.2)
the Bohr radius
of the system.
(2.3)
V. P. Leva,she~~ / Coulombpolarization
111
The values of the reaction S-factor in the experimentally unattained region of low energies are obtained either by extrapolating the available experimental data or by exploiting the knowledge of the mechanism of a relevant nuclear reaction. The approach, based on excluding the Coulomb penetration factor from the reaction cross section (2.1), does not take into account the distortion of the Coulomb field due to particle polarizability. The role of the polarization effects in low-energy nucleosynthesis reactions is studied in the present paper. In our discussion we use as an example the proton-proton reaction p+p+d+e++v,
(2.4)
whose mechanism is well known. This reaction is realised due to P-decay of one of the protons and has a very small probability. The interest in studying the pp reaction (2.4) is also caused by an existing significant (a factor 3) discrepancy between the predicted and measured solar neutrino fluxes I’). Theoretical estimates of the neutrino output are very sensitive to the rate of the reaction (2.4) opening the proton-proton cycle of hydrogen burning in the sun 13). In the stellar depth the reaction (2.4) proceeds at an effective energy determined by a maximum condition on the product of the Coulomb penetration factor and the high-energy tail of the Maxwell-Boltzman thermal distribution I”). For the temperature T = 1.5 x 10’ K that corresponds to the central part of the sun, the effective energy of the pp reaction equals 5.9 keV. Experimental data for the reaction (2.4) in this energy region are unavailable. The astrophysical S-factor of the pp reaction S(E)--n(E)Lt”(E) is determined
by the radial
matrix
element
(2.5)
“)
(2.6) In eq. (2.6) Gd and Q,, are the wave functions deuteron radius, y -’ = 4.3 fm, and
of a deuteron
and two protons,
y is
the inverse
7 C-denotes
the Coulomb
penetration
, C-(n)=2rrn(e-
1”‘) - 1))’
factor decreasing
C--hemrT’I,
?j>l.
exponentially
(2.7) at low energies, (2.8)
A detailed account of pp reaction theory involving earlier computations is given e.g. in ref. lb). In subsequent papers (see review I’)) the sensitivity of the cross section of this reaction to the choice of the nuclear potentials, the deuteron wave function, and to the consideration of meson exchange currents and relativistic effects was studied. We examine the possibility that the S-factor of the pp reaction is increased due to an attraction produced by the electric polarizability of protons.
V.P. Levashev / Coulomb polarization
112
In what follows the deuteron
wave function
$d involved
in eq. (2.6) is chosen as 16)
(2.9) where N = 1.3 is the normalizing forces,
factor, and Z+(Y) beyond
R, takes on the asymptotic
ud( r) = em” , The wave function to a unit amphtude
the range of the nuclear
form raR.
(2.10)
of the proton-proton elastic S-wave in an incident wave, has the form
scattering
I&,, normalized
(2.11) where w denotes the total phase shift. Substituting the expressions for the wave functions I,!J~and +,, (2.9) and (2.11) into eq. (2.6) and taking eq. (2.5) into account, we obtain A(E)=e’“l(E>,
(2.12)
S(E)-Z’(E),
(2.13)
Z(E)=2
-Q(P) u(k, r) dr.
u( k, r) entering the integrand interaction between protons
At low energies the values of the radial wave function (2.14) depend significantly on whether the polarization is taken into account or not. 3. Polarization We assume
the effective
of three parts: - the short-range
- the long-range
nuclear
repulsive
two-particle
(2.14)
potential
interaction
in the initial
channel
to consist
potential VN(r)=
V~(r)~(~-r);
Coulomb
interaction
(3.1)
(3.2) where uH is the Bohr radius - the attractive polarization of colliding particles
of the system (2.3); potential induced by the electric dipole
Vp(rf=-P2@(r--rp) r4
p2=z*
,
polarizability
(3.3) (3.4)
V.P. Levashev
/ Coulomb
polarizution
In eqs. (3.1) and (3.3) O(x) is the step function, (3.1), and rp is the radius
of the polarization
The quantity CY~in eq. (3.4) is defined colliding particles a(l) and (~(2):
113
R is the range of the nuclear potential
by the electric
dipole
polarizabilities
a,=rr(l)$+ru(2)$. 1 For the case of slow protons
the quantity
forces
cut-off at small distances. of
(3.5)
a(p)
can be represented
as a sum of two
terms: Q(P) = %(P) + %(P) ,
(3.6)
ah(p) = 1.13 . lo-’ fm3 [ref. ‘“)I
(3.7)
where
characterizes the “hadron” polarizability of a proton as a system consisting of a quark bag and a pion cloud, and N,(P) is the polarizability of a cloud of virtual electron-positron pairs e e+. Because of the small electron mass, the quantity cry,(p) is three orders of magnitude greater than the pion cloud polarizability that dominates in (Y,,(P). Indeed, the estimate of a,(p) [ref. “)I using the sum rules gives a,(p) 2 0.7 fm3 , whereas
the estimate
with a proton
electric
form factor results
in
(Y,(P) = 1.5 fm3. The mean of these values a,(p) = 1.1 fm’
(3.8)
exceeds by a factor 103, the hadron contribution a,,(p) (3.7). Therefore, in low-energy proton-proton scattering the dominant contribution to proton electric dipole polarizability comes from the nonlinear electrodynamic effects generating a,(p): Q(P) = a,(p). Taking
the identity
of protons
into account
(3.9)
in eq. (3.5) and using eq. (3.9), we have
cr,. =2&(p)
.
(3.10)
The description of the polarization pp interaction by the potential (3.3) with parameter /?’ (3.4) which corresponds to electric dipole polarizability (3.10), (3.8) is valid only at distances beyond the electron-positron cloud r,3
A,-,+
,
where A,
h e+
E-x
2m,c
193.1 fm
(3.11)
114
is the Compton
V./? Levashev
wavelength
/ Coulomb
pdurization
of the electron-positron
eq. (3.4) the value of the Bohr radius
pair divided
of the pp system
p2 = 3.82 * lo-” fm' At distances
smaller
variable r, decreasing at r=Xe-e+ down to
than
li,-,+ the quantities
with decreasing
.
,
(3.12)
CX~and j3” are dependent
r from the values
cyE= 2ffh(p) = 2.26 * 10-j fm'
by 27~. Using in
uH = 57.64 fm, we obtain
on the
in (3.10), (3.8) and (3.12)
f3’ = 3.92 . lo-” fm’
at distances of the order of the size of a nuclear system r - R, y-l. In this case the character of the radial dependence of the polarization potential may also change. It should, however, be noted that at distances R s r s Xc-,+ the distortion of the Coulomb forces because of vacuum polarization plays a more significant role as compared to electric polarizability. In the lowest order in the fine structure constant this effect is described by the Uehling potential ‘O). This additional electrostatic potential characterizes the contribution to the interaction of charged particles that comes from the diagram involving the decay of an exchange photon into an e-e+ pair followed by its annihilation into a photon. The Uehling potential decreases exponentially at large distances and has a finite range of the order of X,-,+. This range is very large relative to the range of nuclear forces R. It has been shown in ref. “) that for proton-proton scattering the Uehling potential results in a strong energy dependence of the standard Coulomb-modified scattering length already at energies lower than 100 keV, whereas the polarization potential VP, whose range is greater, dominates below 0.7 keV. To describe low-energy pp scattering, it is necessary to subtract from the total phase shift the contributions coming from the long-range Coulomb and polarization potentials and the Uehling potential. The remaining part of the phase shift will satisfy the modified effective range theory “) whose parameters are close in magnitude to the corresponding Coulomb-modified nuclear quantities. In the case of the pp reaction (2.4), taking the Uehling potential into. account results in the S-factor decreasing by a few percent “). It is assumed in the following that the cut-off radius rp in the polarization (3.3) exceeds slightly the characteristic nuclear dimensions, rpa R, y-l,
potential
(3.13)
and the strength of the polarization potential p’ (3.4) is specified by the value (3.12) in the range rpb rs X,-,4, too. The Uehling potential is not taken into account. Such an approximation of the polarization interaction inside the electron-positron cloud will, perhaps, enhance slightly the influence of the polarization potential VP on the reaction cross section. However, as it will be shown below, the effect is insignificant even in this case.
V.P. Levashev
4. The pp reaction Let us consider
S-factor
polarization
disregarding
the case when the interaction
the sum of nuclear The proton-proton
entering
/ Coulomb
115
polarization between
(V,)
and repulsive Coulomb (V,) elastic scattering wave function
eq. (2.11) is determined
by the solution
,
equation
$+k’-
V,,
>
effects
protons
is described
potentials
of the radial
by
(3.1) and (3.2).
S-wave SchrGdinger
uN<.(k, r)=O
(4.1)
with the potential v,, and the boundary
- vN+ v,
(4.2)
conditions*
u&k,
0) = 0,
uNc.( k, r) = cos 6,,C[ F( k, r) + G( k, r) tan 6,,c],
r3 R ;
-sin(kr-~In2kr+w,c). I-r
(4.3) (4.4)
The functions F(k, r) and G(k, r) in eq. (4.3) represent the regular and irregular Coulomb S-wave functions. They satisfy eq. (4.1) with the potential V, used instead of V,,. and the boundary conditions F(k,O)=O,
F(k,
G(k, r) -I-*
sin(kr-nln2kr+&.), r) -+ ,- A cos(kr-qIn2kr+6,),
(4.5)
where SC.= arg r( 1 + in) denotes
the pure coulombic
phase
the Sommerfeld parameter (2.2). The Coulomb-modified nuclear sin 6 N.c = -At low energies
the phase
1
shift, I‘(z)
F(k,
All quantities
function
“) and 77 is
in eq. (4.3) is given by
r)VN(r)uNC.(k,
r) dr.
(4.7)
k Io
shift 6,,(. decreases 6N,c.- C’k-em2;ir,,
l
is the gamma
phase shift 6,,, ’
(4.6)
labeled NC correspond
by the exponential
law,
7731.
to the nuclear-C’oulomb
interaction
(4.8) V,,.
(4.2).
V. P. Leuasheu / Coulomb ~~Iur~z~~if~~l
116
The total phase shift W=6JNC.=SC.+SN,(. defines
the asymptotic
The low-energy by the penetration
behaviour
dependence factor
of the function of the Coulomb
uN,-(k, P) (4.4) at large r. F and G is characterized
functions
C (2.7): G(k, r) = C-‘c?(k,
F( k, r) = CkF( k, r) , Here F(k, r) is an entire
(4.9)
of k’, and the function
function
c?(k,r) =
6(k,
where a(k, r) is an entire function is defined by
r)+:
h($F(k,
r) .
(4.10)
G(k, r) has the form “)
rl ,
of k’, too. The function
(4.11) h(v)
entering
(4.12)
~(~)=Re~(~~)-In~, where I/I(Z) is the digamma for h(v) is valid:
function
h(7)=-
“). At small
1
1 12772+ 12074 ’
k the following
r)=$a,xl,(x),
6(k,
(4.13) I3
r)-xK,(x),
I, and Kr are the modified Bessel functions Taking into account the relations (4.10) nuclear-Coulomb
wave function
denotes
the
values of r) the solutions of the
x2 = 8r,fa,,
of first order. the boundary
condition
(4.14)
(4.3) for the
uN(. takes the form
uNC.(k,r)=Ck~~sfiN,C.[F(k,
where uN,(. (k) length ‘“)
approximation
,=$-%I.
When the condition kr<2T is satisfied (low energies or small functions F and 6 are well approximated 22,23) by zero-energy Coulomb problem: F(k,
eq. (4.11)
r)-a,,,.(k)d(k,r)],
generalized
a,,.(k)z
(energy-dependent)
-(C'k)-'
At low energies the quantity a,,,.(k) range theory expansion “*5) 1 -----+lh(q)= a,.<,(k) ars
rZR,
Fermi
tan SN,C..
is p arametrized
-$+&.k’+O(k*j.
by the well-known
(4.15) scattering
(4.16) effective-
(4.17)
NC
According to eqs. (4.17) and (4.13), the generalized scattering length a,,,-(k) in a zero-energy limit goes into the constant A N,C which is the ordinary Coulombmodified nuclear scattering length. The experimental value of pp scattering length is A,,,. = -7.828 fm [ref. 25)].
V.P. Leva.sbev / Cou/omh p”lariralion
the case of the interaction
In
the overlap
integral
(4.2) the pp reaction
of the wave functions
117
S-factor
(2.13) is defined
by
ud and u = uN(. (2.14): *
I(E)=
uduN(. dr .
INC(E)=(Ck)-‘N$
For non-resonant reactions the S-factors are assumed the energy. In the case of the pp reaction, theoretical realistic nuclear forces give (see e.g. ref. I’))
(4.18)
to be weakly dependent on evaluations of I,,.(E) with
I& (0) = 7 . The quantity . . contrrbuttons region:
(4.19)
IN<.(E) can be represented as a sum of two terms characterizing the to I,, from the nuclear force range and from an exterior (asymptotic)
I,,.(E)
)
= IE’( E) + I;;)(E)
(4.20)
K ZF;‘(E)=(Ck)
‘Ny’ 50
ud”N<
(4.21)
dr,
, lk;J( E) = Ny’ cos SN,<
em”‘[F(k,r)-a.,,(k)G(k,r)]dr.
(4.22)
IK To obtain eq. (4.22),the asymptotic expressions for the functions ud and uN( (2.10) and (4.15) have been used. For our further treatment we also need the irregular solution of eq. (4.1) g,, (k, r) specified by the boundary condition g,, (k, r) = cos 6,,,.[ G( k, r) - F( k, r) tan 6,,,.] and the Green
function
of the nuclear-Coulomb
YN(.( r, r’; k) = -i where
r. (r .) is the lesser
5. The proton-proton
(larger)
uNc (k, r
of the values
wave function
allowing
,
r3 R
(4.23)
problem )gN,
(k
r
(4.24)
1,
of r and r’.
for the polarization
interaction
We now study the problem where, in addition to the above nuclear and Coulomb interactions, the attractive polarization potential VP (3.3), (3.12) is taken into account. In this case the radial part of S-wave pp scattering wave function (2.11) u(k, r) = UN&k, is determined
by a solution
of the Schrodinger d’ d’+ r
k’-
V,,,
>
r) equation
uN,,(.(k, r) =0
(5.1)
118
V.P. Lev&hev
/ Coulomb polarization
with the potential
v,+ vp+ v,
VNpC=
and with the boundary
(5.2)
conditions
uNPC(kr) = ~0s b,Nd~NC.(k r)+gNc(S r) tan&,N~l, --+ l-r
sin (kr-q
r3-rc;
(5.3)
In 2kr+wNPc).
The total phase shift w = wNPC occurring
(5.4)
in eq. (2.11) and (5.4) is given by the sum
where X sin6
---
P,NC‘ -
+dIk
r)
Vdr)u.dk
r)
dr
(5.6)
is the phase shift caused by the polarization potential and counted off from the total nuclear-Coulomb phase shift w NC. For the pp scattering at energies lower than a few keV the phase shift 6 ,,,Nc practically coincides with the Coulomb-modified polarization phase shift 6P,c that has the power energy dependence ‘) 6P,c-
k’,
k*l.
As a result, the quantity cos (sP.Nc.involved is actually coincident with unity.
(5.7)
in eq. (5.3) and in the following
formulae
Using the Green function Y?,, (4.24), the differential equation (5.1) with the boundary conditions (5.3) can be transformed into the Fredholm-type integral equation
UN&k,
r) = Cos s,,r@,,-(k,
r)+
YNC.(r, r’; k) VP(r’)uNPC.(k, r’) dr’.
(5.8)
I0
From definition (3.3), V,(r) = 0 at r < rip. This, with account of the inequality (3.13), R s rp, allows us to express the Green function YINc in eq. (5.8) in terms of asymptotes of the functions uNC and gNc (4.3) and (4.23). As a result, we obtain for uNPc. the representation UN&k,
r) =f(k)[u&k,
r)+ wNPc(k, r)@(r-
rP)l,
where @(a) is the step function and WNpf denotes an addition Coulomb function uNc. due to the polarization interaction.
(5.9) to the nuclear-
V.P. Leva.shetl / Cortlomh
The term
MJ~~(-in eq. (5.9) is defined
polarization
119
by the solution
of the Volterra
integral
equation N’
N&k, r) =
jl BNc,(r, r’; k) Vp(r’)[
uNc.(k, r’) + wNI,(.(k, r’)] dr'
,
(5.10)
whose kernel
%.dr, r’; k) =i [udk beyond
the range of the nuclear 9?&r,
r’; k) = F(k,
r)gNdk r’i - UN&k,r’)gdk
forces reduces r)c(k,
r’) - F(k,
r)l
to r’)G(k,
r) ,
r, r’s
R.
(5.11)
Upon using the expression (4.11) for the function G(k, r) in eq. (5.11) it is not difficult to represent the integral kernel gN,.( r, r’; k) as a linear combination of the entire functions of k’, F( k, r) and 6( k, r): SNc.(r,
r’; k)=l”‘(k,
r)~(k,r’)-~(k,
r’)i%(k,r),
r, r’> R.
(5.12)
At low energies the expansion of the integral kernel and functions in a power series in the energy may be used as an effective method of solving eq. (5.10). Because of the properties of the functions F and 19 (4.14), the kernel (5.12) in the domain kr<2q and kr’e-277 is, in fact, energy-independent and expressed in terms of the first-order Bessel functions of an imaginary argument I,(X) and K,(X). The factor j’(k) in the radial wave function uNr(. (5.9) has the form f(k)
= cos 6 rs.N<.(1-5(k))-’
(5.13)
,
where C(k)=
1”gNc (k,
-;
0
Since the polarization rz
potential
r)V,,(r)[u&k,
is a small
r)+wNr,t.(k,
correction
r)@(r-rrla)]dr.
to the Coulomb
(5.14)
interaction
r,,), it can be taken
into account with a high accuracy by (I VP/ Vcl = at/2r”, employing the Born approximation. The direct calculations “) show that within the energy range l-lo3 keV the Born approximation reproduces the Coulomb-modified polarization phase shift 6,.,(. up to 0.01%. A similar result has also been obtained in ref. ‘I). Let us evaluate the function c(k) in the Born approximation. Substituting into eq. (5.14) the expression for VP (3.3) and the asymptotes of the functions uN(. and gNc- (4.3) and (4.23) and also neglecting the small terms of the order of (Ck)’ and higher, we obtain 5(k) = h(k, co),
(5.15)
V.P. Levashev / Coulomb polarization
120 where
r
b(k, r)= Applying
F(k, r’)G(k, r’)Vp(r’) dr’.
(5.16)
i FP
in eq. (5.16) the inequality (F(k, r)C?(k, r)l
following
from inequality
(4.12) of ref. “),
jF(k, r)(G(k, r)+iF(k, we get an upper
bound
r))j<6kr,
on Ib(k, r)l: (5.17)
The dimensionless quantity M in eq. (5.17) is defined by the value of the polarization potential at the cut-off point, i.e. at the boundary of the nuclear force range:
M-rr’,lVP(rp)/=*. For our polarization pp interaction choosing, e.g., r,, = 5 fm
model
(5.18)
aHrF
(3.3),
(3.12)
and
M = 1.5 . lo-“. We note that the value of the parameter M for the proton-deuteron system. Indeed, using nuclear polarizability 24) ah(d) = 0.7 fm’ and corresponding to (3.8) and taking rp = 5 fm,
(3.13),
we have by
(5.19)
close to that in (5.19) is obtained also the experimental value of the deuteron also the values a,(p) = a,(d) = 1.1 fm’ we have
M = ah(d) + a,(d) + a,(p) = 2_9 , 1o-.’
(5.20)
a&d)6 Returning to the formula (5.13) and taking into account that 6P,NC.- k5 and due to eqs. (5.15), (5.17)-(5.19) the modulus of the function t(k) does not exceed the number of the order of 10w3, we conclude that at low energies the factor f(k)’ is close to unity: f(k)=
1+0(M)
>
M - 10-3.
(5.21)
We now study the polarization contribution wNPc. to the wave function uNpC (5.9). It should, first, be noted that at distances r < rp the function wNpc.(k, r) =0 and, therefore, the total wave function u Npc.(k, r) coincides with the nuclear-coulomb solution u &k, r) up to the factorJ(k) (5.21).
K P. Levashev
We represent
/ Coulomb
r) in the region
w,,,(k,
polarization
121
r 2 rP in a form similar
to the asymptote
of uNc. (4.15). As it has been noted above, at low energies the function wNPc. can be approximated with great accuracy by the Born solution of eq. (5.10): r) = w:,,, (k.r)=j’-
w,,,(k,
Using the expressions wipc(k,
r’) dr’.
(5.22)
for SN(. (5.11) and uNc. (4.15) in eq. (5.22), we get r)=-Ckco~S~,~.{F(k,
r)[b(k,
+ G(k, r)[a(k,
where the auxiliary
r’; k) V,(r’)u,,.(k,
S&T, II’
functions
r)-a.,,(k)d(k,
r)+a&k)b(k,
r)]
r)lI,
(5.23)
are introduced I
a(k, r)=
I
F”( k, r’) V,,( r’) dr’ ,
(5.24)
rl’ G’( k, r’) V,( r’) dr’ ,
d(k, r) = -
(5.25)
I,’ and.b(k, r) is defined by formula (5.16). It follows from a comparison of eqs. (5.23) and (4.15) that at distances rz rp the factors a(k, r), perturbation w:,,. differs from uNc. by the presence of additional h(k, Y) and d(k, r) for the functions F( k, r) and G(k, Y). The magnitudes of these factors just determine the deviation of the total wave function uNF(. from the unperturbed solution uNc.. It has been established above (eqs. (5.17)-(5.19)) that the modulus of the function b(k, r) is majorized uniformly in k and r by the small parameter M determined by the value of the potential VP at the cut-off point rp. We show that at low energies a similar bound exists also for the function d(k, r), whereas the factor a(k, r) is strongly dependent on the ratio between the variable r and the Coulomb turning point (5.26) Taking
into account
an explicit
form of the potential
VP (3.3), we get
I d(k, r)
G’(k,
r)F“
dr.
(5.27)
I VP We divide the integration path in (5.27) into three parts where the function is approximated by the expressions “) G( k, r) = xK,(x) G(k, r)=
,
C(2r))“(‘&
G(k,r)=Ccos(kr-qln2kr+&),
x2= 8r/a,,
,
r
Bi(z) ,
z-(27j)“‘(l-r/r,),
;
r>rr,..
G(k, r)
(5.28) r-rr,;
(5.29) (5.30)
% P. Leuashev
122
/ Cmilm7zh palariza~i~~n
Above the turmng point (r 3 rc ) the Airy function Bi(z) oscillates with an amplitude not exceeding the value Bi(0) = (3”“Z’(.f))-’ =0.62. In the case of r< r,. and large values of z s 1 the function Bi(z) increases exponentially leading to the expression for d in this region
G( k, r)
in the form
= C(-’ exp ($&“)
,
It follows from the approximations (5.29) and (5.30) and the behaviour of the Airy function Bi (z) that at low energies (n * 1) the contributions to d(k, ~0) from the due to a poor integration intervals r - rc and rc < rd CO are much suppressed penetrability
of the Coulomb
barrier
(2.X),
C-fie-“n,
7731.
Therefore, the quantity d(k, co) will be completely determined by the contribution coming from a small integration region below the Coulomb turning point:
To obtain the estimate (5.31), the constraint 22) K,(x)s 1.7 for x20.5 has been used, leading to the condition r,,’=- laH 32 that is easily met in our case. With account of eqs. (5.27) and (5.31), we obtain the desired bound on the function d(k, r): d(k, r)Sz Unlike the functions
M,
k
b(k, r) and d( k, r), the energy
(5.32) behaviour
of the factor a( k, r)
is substantially dependent on the values of the variable r. In the region using the approximation (4.14) for the function F(k, r), we get a(k,r)=-OZ
au
y Z~(Y)/dx’, Xi’
r%rc,
xZ=8r/a,,,
r =Crc on
(5.33)
so that la(k, r)/ < My$[R(x)
- R(xp)]
c MS
I’
R(x),
r
(5.34)
where R(x)=;x’[zf(x)-Z,,(X)Z,(X)])
xZ,= Xr,/aB .
Thus, in the region r < rc. the function a( k, r) is bounded by the energy-independent quantity (5.34) that contains the small parameter M = CY~/U~T$-- lo-“.
V.P. Levashev / Coulomb
In the vicinity is approximated
of the turning
point r,. and beyond
by the expressions
F( k, r) = ( Ck)m’&(2v)“h
polarization
zE(27p
Ai( z) ,
1-t
)
r-r<-
The Airy function Ai in eq. (5.35) (5.35) and (5.36) contain the energy As a result, the values of the factor r > r(.) depend strongly on the energy
;
(5.35)
>
r> r,. .
F(k,r)=(Ck)-‘sin(kr-nln2kr+&.),
(5.36)
is bounded as IAi( ~0.6. The expressions factor (Ck)-’ that diverges at the threshold. a(k, r) (5.24) at large distances (r - rc and increasing unboundedly with E + 0,
a(k,r)--~m’e7711),
the expressions
F(k, r)
it up to 00, the function
“)
(
Comparing
123
ra rc..
77>‘1,
for the wave functions
(5.37)
uNPC., uN(. and
wNPC. (5.9),
(4.15) and (5.22), (5.23) and taking into account the constraints (5.17) and (5.32) on the factors h( k, r) and d( k, r) and the properties of the factor a( k, r), we conclude: (i) at small distances below the Coulomb turning point (r =Sr(.) the polarization potential contribution wNP(. to the total wave function uNP(. is insignificant and determined by the parameter M - lo-’ (5.18), (5.19); in this case the wave function uNP(. contains (as in the short-range force case) the Coulomb penetration factor C; (ii) at large distances (r- r(. and r s r(.) the polarization factor a( k, r) involved in wNP( affects significantly the behaviour of the function uNPC(k, r) as compared to that of the undistorted solution u,,(k, r). The asymptotic value of the function a(k, r), * u(k, 00) =
F’(k, 1 F,>
coincides with the Coulomb-modified the Born approximation: a(k,m)=
r)V,,(r)
polarization
uIlc(k)-
-(C’k))‘(tan
dr,
scattering
(5.38)
length
S,,,)“.
calculated
in
(5.39)
According to eqs. (5.33) and (5.34), the integration region r,< r Q r,. gives a small contribution to the integral (5.38) of the order of A4 - 10 ‘. The value of the polarization phase shift fi,,,. (5.39), (5.38) IS, . in fact, determined by an integration in the regions r- rc and r,. < r s 00, which just produces the k’-dependence (5.7) obtained in ref. ‘). As a result, the phase shift SP,(. depends weakly on the choice of a specific value of the polarization cut-off parameter r,, *. The dependence (5.7) is a consequence of the long range of the polarization potential and differs much from the exponential decrease C’k - exp (-2~77) (4.8) typical of the Coulombmodified nuclear phase shift S,,c. For this reason the generalized scattering length u,,.(k) defined by eq. (5.39) diverges at the threshold 2,6,7): a,,,, (k) - -k’( C”k))’ l
The numerical
calculations
“.I’)
confirm
- -qP5 e2nV -CC. rJ+‘”
this, too
(5.40)
V.I? Leuasheu / Coulomb p~}~ari~ari~t?
124 6.
Influence of the polarization
potential on S-factor
In the case when the polarization interaction is taken into account, the S-factor of the pp-+de+v reaction is determined according to eqs. (2.13) and (2.14) by the overlap integral of the radial wave function ud and ‘u 5 uNPc.:
* r(E)~z.,,.(E)=(Ck)-‘Nr’ J 0
Substituting
the expression
(5.9) for the function
I NW(E)
%UNPC
dr.
(6.1)
aNP(. into eq. (6.1), we get
16.2)
=f(k)[I,,-(E)+f,,~(E)l,
Iu
where the addend
(6.3)
ff’
characterizes the contribution caused by the polarization potential VP. The augend I,,.(E) in eq. (6.2) is defined by the expression (4.18), while the function f(k), according,to eq. (5.21), equalsf(k) = 1 +O(lO-‘). Owing to the weakness of the potential VP as compared to the nuclear and Coulomb potentials, the contribution JNpC can be calculated with a great accuracy in the Born approximation
in V,: ~NI’C’(~)=~k’<
Jf&.(E) In formula
= -N
cos 6,,-[A,(k)
(6.5) the following
notations
(6.4)
cE),
- a iv.<-(k)(A,(k)
-A,(k))+
A,(k)1 .
(6Sj
are introduced:
(6.6) (6.7) Taking account of the relation R G rp (3.13) we notice that the expression for JNP(.( E) (6.4)-(6.7) differs from the asymptotic part of the nuclear-Coulomb matrix element (6.6) and I’,‘:)(E) (4.22) by the factors b(k, r), d(k, r ) and a(k, r) in the integrands (6.7). Using in eqs. (6.6) and (6.7) the bounds on the functions b(FE, r) and d(k, r) that are uniform in r and also the numerical value of INC.(O) (4.19), we conclude that the terms A,(k), i = 1,2,3 in eq. (6.5) give a small relative contribution of the order of M - 10e3 (.5.18), (5.19) to the matrix element INPC.(E) (6.2):
(6.8)
V.P. Lemsher
/ Coulomb polarizution
125
The term A,(k) is the only contribution to eq. (6.5) that may be thought to be appreciably affected by the polarization interaction. This is because the factor a(k, r) in A,(k) shows an exponential increase in the region of the variables k < 1, r 3 rc. = 2n/ k. We prove, however, that due to the rapidly decreasing factor exp (- yr) in the integrand
(6.7), only the small integration
region
r Q rc. contributes
to the matrix
element A,(k). By approximating the functions G( k, r) and a( k, r) by the expressions (5.28) and (5.34), respectively, the term A,(k) will then have the same smallness order (-M) as the terms A,(k), i = 1,2,3 (6.8). Using in eq. (6.7) the definition (5.24) for the function a( k, r), we represent Aj( k) as I A,(k) = (rP)’
, dr’ r’ ‘F’( k, r’) .
dr emY’G(k, r) I 1II
I ’ I’
(6.9)
The contribution A4( k) may be estimated by approximating the functions G( k, r) and F(k, r) in eq. (6.9) by the expressions (5.28)-(5.30) and (4.14), (5.35), (5.36), followed by calculations of the contributions from different parts of the twodimensional domain of the integration over r, r’. The dominant contribution to the integral (6.9) will then come from the integration region r’s r s r( , whereas the contributions from the regions r’s r, r - r(‘ and r’< r, r( < r G ~0 will be negligibly small at low energies. The obtained value of A,(k) will be proportional to the small parameter M (5.18), (5.19). A simpler way to estimate the quantity A,(k) is to employ in formula (6.9) the inequality “) lG(k, r)F(k,
With account
of f/r
rar’.
r’)I 4 (2mr’)“‘,
s 1 this gives
P,(k)1
I.:drre~“~~:,dr’r’~~,~(k,r’),.
s G(Pr)’
On interchanging the order (6.10) is simplified to
of the integrations
and integrating
lA,(k)l~~M(yr,)~~(l+yr,)B(k).
Here the function
B(k)
is defined
(6.10) over r the relation
(6.11)
as \
B(k) = y2
dr em”lF(k,
r)l ,
(6.12)
I I,’ M is the small the condition We estimate function F(k,
parameter
(5.18), (5.19), and the factor (1 + yrp)( yrp)m”s 2 because
-yrpa 1 (3.13).
the low-energy
value of B(k) using the expansion of the Coulomb of an imaginary argument Z,,(x) [ref. ‘I)]:
r) in the Bessel functions F(k,
r) =;a,
i
II I
b,,(k)($x)“l,(x)
,
x2 = 8r/a,.
(6.13)
V. P. Levushev
126
/ Coulomb
The coefficients h,(k) in (6.13) are determined ref. “) and are proportional to various powers entire
function
of k’. Substituting
polarization
by recurrent relations (14.4.3) of of k’. We remind that F(k, r) is
(6.13) into (6.12) and integrating
.X B(k)
over r, we get (6.14)
,
the small parameter (6.15)
4=2/Y%.
Under the condition k < y corresponding to low energies E G 100 keV, we can restrict ourselves to taking into account a few first terms on the right-hand side of the inequality (6.14). Substituting the explicit expressions for h, into eq. (6.14), we get
~~lf~~~Y~l+~~l~~‘~l+~~~+~~~~l~~4~1, k
the exponent B(k)<
in eq. (6.16) in powers of the small parameter
1i-q+(k/Y)‘+O(C4+(kly)‘12),
key.
(6.16) 4, we obtain (6.17)
It follows from (6.17) or (6.16) that at low energies (k< y) the values of the factor B(k) in eq. (6.1 I) do not exceed a quantity of the order of unity. As a result, the numerical value of the contribution A,(k) is completely defined by the small parameter M (5.18) having the order of 10m3 (5.19). The small magnitude of A,(k) follows directly from comparing the bound on A,(k) (6.11) with the bound, for example, on A,(k), whose smallness is a consequence of eq. (6.8). Indeed, majorizing in eq. (6.6) the function b(k, r) according to (5.17), we obtain for A,(k) the bound of the same form as eq. (6.11): ~A~(k){~~~M~(k~.
(6.18)
Applying in formulae (6.2)-(6.5) the above bounds on Ai( i = 1,2,3,4 and also using the numerical value of Z&O) (4.19), we conclude that the relative contribution of the polarization term JNPC‘(Ef to the matrix element lNpC defining the pp reaction S-factor (2.13) is a small quantity of the order of M - lo-” (5.18), (5.19). Such a small correction to the pp reaction cross section indicates that the solar neutrino problem mentioned in the second section is impossible to resolve by taking polarization effects into account. The method used here to estimate the polarization corrections to the pp+ de+ v reaction cross section is general and holds also for the reactions involving more complex nuclei in an initial channel. Thus, we come to the main result of the present paper that polarization effects play an insignificant role in the low-energy nucleosynthesis reactions. Physically this is due to that fact that the reactions concerned are realized in a small region of the configuration space (of the order of a synthesized nucleus size). In this region the polarization potential represents a small correction to the nuclear and Coulomb interactions. The value of the polarization potential at
V.P. Leuarhev J Coulomb polarization
its cut-off point rP (i.e., at the boundary the parameter M (5.18) characterizing cross section polarizability
of the nuclear the polarization
1’7
force range) just determines corrections to the reaction
and the wave function at small distances. Because of the weak electric of nuclei LYEand the large value of the Bohr radius a,, the parameter
M = a,/a,ri is small. For the reaction reaction, the parameter M (5.20) has the the above pp reaction. Our prediction that reaction cross section must be insignificant
induced by deuterons, e.g., pd+ y7He same order of smallness (- 10m3) as for the polarization contribution to the pd agrees well with the calculations of this
reaction using a quasi-classical approach “I.“). Concurrently to our studies of the problem presented in refs. 3’,31) the publications of the group of authors appeared “LX ), wherein the incorrect result was obtained claiming
that the polarization interaction affects very strongly low-energy cross sections of the pp + de+v, pd + y3He and dt + n”He reactions. The reaction S-factors calculated in refs. 33m’5) at the near threshold energy exceed by many orders the extrapolated experimental values of these S-factors. The mistake of the authors of refs. 33m35)consists in using an inadequate approximation for the factor a(k, r) (eq. (5.24)) determining the polarization contribution wN,&k, r) (eq. (5.23)) to the initial wave function u.,,,(k, r) (eq. (5.9)). As it was shown in sect. 5, the energy behaviour of the factor a(k, r) is crucially dependent on the variable r. At the left of the Coulomb turning point (r s r(.) the modulus of the function a(k, r) is bounded by an energy-independent small quantity (5.34), and whereas at large distances (r- r( and r > r( ) it is strongly energy-dependent has the threshold singularity (5.37). In refs. 31m35),the factor a(k, r) was approximated for all r by the asymptote a(k, ~0) having the threshold singularity in (5.39), (5.40). With such an approximation the polarization contribution to the wave function, w,,,.(k, r) (eq. (5.23)) at small distances was unjustifiably forced to be characterized by the threshold divergence k’/C, rather than by the exponential decrease Ck specific for it. Just this threshold divergence introduced “by hand” in refs. 3’m1’) has resulted in an incorrect description of the reaction S-factor (2.13), (6.1)-(6.3) at low energies. When the factor a(k, r) in the region r < r ( is approximated by the correct expression (5.33), the polarization contribution JN,,< (E) (6.3) to the S-factor is found to be insignificant - of the order of the quantity M = a, /a,,&-- lo->.
7. Conclusion In our paper we have studied the influence of the electric polarizability of particles on nucleosynthesis reaction cross sections at low energies. It has been found that the dominant contribution to the matrix element determining the reaction S-factor comes from the small integration region below Coulomb turning point. Taking account of the polarization interaction in this region has been shown not to lead to any noticeable distortion of the wave function of coliiding particles.
128
V.P. Lruashev
/ Cmlomb
polarization
The relative value of the polarization correction to the nucleosynthesis reaction S-factor is characterized by the quantity M determined by the value of the polarization potential at the boundary of the nuclear force range: M - r’,] V,( rp)l = a,/a,rE, r,>
R. Due to the weak polarizability
of the Bohr radius uB, the parameter
of nuclear
systems
(Ye and the large values
M takes on the small value - 10e3. As a result,
the role of the polarization effects in the nucleosynthesis reactions at low energies turns out to be insignificant. It has been established that allowance for the polarization interaction can not resolve the existing discrepancy between the predicted and measured solar neutrino fluxes. It has been shown that the conclusion of refs. 33-34) that the polarizability of particles affects strongly pp, pd and dt reaction cross sections is wrong and caused by an incorrect approximation of the initial-channel wave function at small distances. The author expresses his gratitude to V.F. Kharchenko for fruitful discussions and interest in the work and also to Ya.A. Smorodinsky, D.A. Kirznitz, V.A. Petrun’kin and J.L. Friar for discussions of the results obtained. Note added: After the present paper was completed a publication by Gy. Bencze appeared in Phys. Lett. B202 (1988) 289. In that publication the upper bound on the polarization effects in the deuteronwas obtained which is by five orders smaller as compared to our result induced nuclear reactions -lo-’ -lo-‘. The great underestimate of the polarization effects by Bencze is a consequence of the use of an insufficiently good approximation for the phase shift fi,,,(- (B4) (i.e. formula (4) of Bencze’s paper). By using in Bencze’s approach a more accurate expression for 6,,,, (V.P. Levashev, 1986, unpublished), different from eq. (84) by the presence of an extra factor (I + 2b( k) - d( k)A,,,-) for the addend in the numerator, we come to eq. (814) with an additional term f(k) = 26(k)-d(k) Re A,,, on the right-hand side. With allowance for our estimates (5.17), (5.18), (5.32) for the functions b(k)= b(k, CO) (5.16) and d(k)= d(k, 03) (5.25), we have IF(k 10-j so that Bencze’s term tan* 6,.,10m8 in eq. (814) may be neglected. Thus we obtain again the result of the present paper.
References 1) L.D. Landau and E.M. Lifshitz, Quantum mechanics (Nauka, Moscow, 2) K. Kalikstein, L. Rosenberg and L. Spruch, Nucl. Phys. 49 (1963) 257 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
1974)
R.O. Berger and L. Spruch, Phys. Rev. Bl38 (1965) 1106 L.D. Landau and Ya.A. Smorodinsky, Zh. Eksp. Teor. Fiz. 14 (1944) 269 H.A. Bethe, Phys. Rev. 76 (1949) 38 A.A. Kvitsinsky and S.P. Merkuriev, Yad. Fiz. 41 (1985) 647 Gy. Bencze and C. Chandler, Phys. Lett. B163 (1985) 21 V.P. Levashev, Contributions to the Int. Conf. on the theory of few-body and quark-hadronic systems (Dubna, 16-20 June, 1987), Dubna (1987) p. 121 A.I. L’vov, preprint FIAN-14, Moscow (1987) Gy. Bencze, C. Chandler, J.L. Friar, A.G. Gibson and G.L. Payne, Phys. Rev. C35 (1987) 1188 V.V. Pupyshev and O.P. Solovtsova, preprint JINR E4-87-467, Dubna (1987) J.N. Bahcall and R. Davis, Jr., in: Essays in nuclear astrophysics, ed. C.A. Barnes et al. (Cambridge University Press, Cambridge, 1982) ch. 12 R.W. Kavanagh, ibid., ch. 8 W.A. Fowler, Rev. Mod. Phys. 56 (1984) 149
V.P. Levashev / Coulomb
polarization
129
IS) E.E. Salpeter, Phys. Rev. 88 (1952) 547 16) J.N. Bahcall and R.M. May, Astrophys. J. 155 (1969) 501 17) J.N. Bahcall, W.F. Huebner, S.H. Lubow, P.D. Parker and R.K. Ulrich, Rev. Mod. Phys. 54 (1982) 767 IS) V.A. Petrun’kin, Physics of Elementary Particles and Atomic Nuclei 13 (1981) 692 19) Yu.A. Aleksandrov, Fundamental properties of a neutron (Energoatomizdat, Moscow, 1982) 20) E.A. Uehling, Phys. Rev. 48 (1935) 55 21) J.E. Brolley, Solar Phys. 20 (1971) 249 22) M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (US National Bureau of Standards, Washington, DC, 1972) 23) E. Lambert, Helv. Phys. Acta 42 ( 1969) 667 24) H.A. Bethe and P. Morrison, Elementary nuclear theory, sec. ed. (Wiley, New York, 1956) 25) J.P. Naisse, Nucl. Phys. A278 (1977) 506 26) Chr. Bargholtz, Astrophys. J. 233 (1979) L161 27) Gy. Bencze and C. Chandler, Phys. Lett. B182 (1986) 121 28) S. Klarsfeld, Nuovo Cim. 43A (1966) 1077 29) N.L. Rodning, L.D. Knutson, W.G. Lynch and M.B. Tsang, Phys. Rev. Lett. 49 (1982) 909 30) V.V. Pupyshev and O.P. Solovtsova, preprint JINR P4-86-346, Dubna (1986) 31) V.P. Levashev, Contributions to the Int. Conf. on the theory of few body and quark-hadronic systems (Dubna, 16-20 June, 19X7), Dubna (1987), p. 120; preprint ITP-87-165E, Kiev (1988) 32) V.P. Levashev, ihid., Dubna (1987) p. 125 33) V.B. Belyaev and V.E. Kuzmichev, Contributions to the Xlth IUPAP Conf. on few-body systems in particle and nuclear physics, eds. T. Sasakawa et ul. (Tohoku University, 1986) p. 388; preprint ITP-86.122P. Kiev (1986) 34) V.B. Belyaev, 0.1. Kartavtsev and V.E. Kuzmichev, preprint JINR E4-86-66, Dubna (1986) 35) V.B. Belyaev, V.E. Kurmichev, V.V. Peresypkin and M.L. Zepalova, preprint JlNR E4-87-35, Dubna (1987)
COULOMB POLARIZATION AND NUCLEAR REACTIONS AT ASTROPHYSICALLY LOW ENERGIES
Received
IS July 1988
Abstract: The effects of electric
polarizability of particles are shown to play an insignificant role in the nucleosynthesis reactions at astrophysically low energies. The polarization corrections to the reaction S-factors are defined by the value of the polarization potential at the boundary of the nuclear force range and do not exceed a quantity of the order of 0.1%. It is shown that the conclusion of papers claiming the very strong influence of the particle polarirability on pp-, pdand dt-reaction cross sections is a consequence of an incorrect approximation of the initi~l-~h~nne~ wave function.
1. Introduction A charged structural particle placed in an electric field undergoes deformation. As a result of the electric polarizability of the particle, an additional long-range attractive interaction arises in the system. For a spherically symmetric charge distribution this polarization potential decreases as rm4 [ref. ‘)] when the radius r is large. In the elastic scattering of like-charged complex particles the electric polarizability results in a substantial distortion of the wave function at large distances and also in the divergence ‘,‘) of the standard Coulomb-modi~ed scattering length ‘.5). For the proton-deuteron system the divergence of the scattering length due to deuteron polarizability was noted in refs. ‘,‘,‘). In those papers it was pointed out, in particular, that to extract the pd scattering lengths experimentally, one should use the data for not too low energies, E 3 20 keV [ref. “f]. At higher energies the polarization potential contribution to the nuclear-polarization scattering length becomes negligibly small, and the contribution to the effective radius ranges from ~3% at 100 keV [ref. “)I to -0.1% at 400 keV [refs. “.“)]. The computation “I) performed by the phase function method reveals that the polarization potential does not practically affect the proton-deuteron scattering length at distances up to -300 fm. At larger distances the polarization potential contribution becomes dominant, resulting, in particular, in the above-cited divergence of the Coulomb modified nuclear-polarization scattering length. So, to obtain the required accuracy at low energies -2 keV one needs to integrate the polarization 03759474/89/$03.50 0 Elsevier Science Publishers (No~h-~oll~~ld Physics Publishing Division)
N.V.
I/. P. Leonshev / Coulomb
110
~o~ur~~u~~on
potential up to the radius -15rc104fm [refs. “‘,“)I, which considerably exceeds the Coulomb turning point value r(. = 700 fm. To describe the low-energy scattering of charged composite particles, it is necessary to make use of the modified effective-range theory 3*8*‘0)in which the nuclear phase shift is defined with respect to the phase shift from the Coulomb and polarization long-range potentials. The resultant scattering length and effective range modified by the polarization-Coulomb field are very close 8-‘0) to relevant quantities in the nuclear-Coulomb problem. The present paper is aimed at studying the role of the polarization effects in low-energy nuclear reactions. We investigate the possibility for the nucleosynthesis reaction cross sections to increase due to an additional attraction generated by the electric polarizability of colliding particles. Sect. 2 presents in brief the characteristics of the reactions concerned and also the information on the pp reaction considered here as a typical example. The parameters of the polarization potential are discussed in sect. 3. In sect. 4 we give the expressions for the astrophysical S-factor of the pp reaction, ignoring the polarization effects. The influence of the polarization potential on the wave function at small distances is studied for the low-energy proton-proton scattering in sect. 5. In sect. 6 the expressions for the polarization contributions to the pp reaction S-factor are obtained and it is shown that the polarization effects in nucleosynthesis reactions are very small at low energies. In conclusion, a brief summary of the main results of the paper is presented.
2. Nucleosynthesis
reactions. Proton-proton
reaction
We consider the reactions which are induced by a collision of two nuclei with charges Z,e and Z,e and result in a synthesis of more complex nuclear systems. The reactions pp + de+v, pd + y’He, dt -+ n4He, etc. typify the processes concerned. These reactions and similar ones including more complex nuclei play an important role in astrophysics, providing, in particular, the stellar energy generation. At low energies the totai cross sections ofthe reactions under consideration are characterized by an exponential decrease because of the Coulomb repulsion: a(E)
= S( E)E’
em’nq.
(2.1)
In eq. (2.1) E = (hk)‘/Zp is the kinetic energy of colliding particles in the center-ofmass system (k is the wave number, supposed to be small, k s 1, p is the reduced mass of the system), ‘7 = (ka,)-’ is the Sommerfeld
parameter, an = h’/&Z,Zz
denotes
(2.2)
the Bohr radius
of the system.
(2.3)
V. P. Leva,she~~ / Coulombpolarization
111
The values of the reaction S-factor in the experimentally unattained region of low energies are obtained either by extrapolating the available experimental data or by exploiting the knowledge of the mechanism of a relevant nuclear reaction. The approach, based on excluding the Coulomb penetration factor from the reaction cross section (2.1), does not take into account the distortion of the Coulomb field due to particle polarizability. The role of the polarization effects in low-energy nucleosynthesis reactions is studied in the present paper. In our discussion we use as an example the proton-proton reaction p+p+d+e++v,
(2.4)
whose mechanism is well known. This reaction is realised due to P-decay of one of the protons and has a very small probability. The interest in studying the pp reaction (2.4) is also caused by an existing significant (a factor 3) discrepancy between the predicted and measured solar neutrino fluxes I’). Theoretical estimates of the neutrino output are very sensitive to the rate of the reaction (2.4) opening the proton-proton cycle of hydrogen burning in the sun 13). In the stellar depth the reaction (2.4) proceeds at an effective energy determined by a maximum condition on the product of the Coulomb penetration factor and the high-energy tail of the Maxwell-Boltzman thermal distribution I”). For the temperature T = 1.5 x 10’ K that corresponds to the central part of the sun, the effective energy of the pp reaction equals 5.9 keV. Experimental data for the reaction (2.4) in this energy region are unavailable. The astrophysical S-factor of the pp reaction S(E)--n(E)Lt”(E) is determined
by the radial
matrix
element
(2.5)
“)
(2.6) In eq. (2.6) Gd and Q,, are the wave functions deuteron radius, y -’ = 4.3 fm, and
of a deuteron
and two protons,
y is
the inverse
7 C-denotes
the Coulomb
penetration
, C-(n)=2rrn(e-
1”‘) - 1))’
factor decreasing
C--hemrT’I,
?j>l.
exponentially
(2.7) at low energies, (2.8)
A detailed account of pp reaction theory involving earlier computations is given e.g. in ref. lb). In subsequent papers (see review I’)) the sensitivity of the cross section of this reaction to the choice of the nuclear potentials, the deuteron wave function, and to the consideration of meson exchange currents and relativistic effects was studied. We examine the possibility that the S-factor of the pp reaction is increased due to an attraction produced by the electric polarizability of protons.
V.P. Levashev / Coulomb polarization
112
In what follows the deuteron
wave function
$d involved
in eq. (2.6) is chosen as 16)
(2.9) where N = 1.3 is the normalizing forces,
factor, and Z+(Y) beyond
R, takes on the asymptotic
ud( r) = em” , The wave function to a unit amphtude
the range of the nuclear
form raR.
(2.10)
of the proton-proton elastic S-wave in an incident wave, has the form
scattering
I&,, normalized
(2.11) where w denotes the total phase shift. Substituting the expressions for the wave functions I,!J~and +,, (2.9) and (2.11) into eq. (2.6) and taking eq. (2.5) into account, we obtain A(E)=e’“l(E>,
(2.12)
S(E)-Z’(E),
(2.13)
Z(E)=2
-Q(P) u(k, r) dr.
u( k, r) entering the integrand interaction between protons
At low energies the values of the radial wave function (2.14) depend significantly on whether the polarization is taken into account or not. 3. Polarization We assume
the effective
of three parts: - the short-range
- the long-range
nuclear
repulsive
two-particle
(2.14)
potential
interaction
in the initial
channel
to consist
potential VN(r)=
V~(r)~(~-r);
Coulomb
interaction
(3.1)
(3.2) where uH is the Bohr radius - the attractive polarization of colliding particles
of the system (2.3); potential induced by the electric dipole
Vp(rf=-P2@(r--rp) r4
p2=z*
,
polarizability
(3.3) (3.4)
V.P. Levashev
/ Coulomb
polarizution
In eqs. (3.1) and (3.3) O(x) is the step function, (3.1), and rp is the radius
of the polarization
The quantity CY~in eq. (3.4) is defined colliding particles a(l) and (~(2):
113
R is the range of the nuclear potential
by the electric
dipole
polarizabilities
a,=rr(l)$+ru(2)$. 1 For the case of slow protons
the quantity
forces
cut-off at small distances. of
(3.5)
a(p)
can be represented
as a sum of two
terms: Q(P) = %(P) + %(P) ,
(3.6)
ah(p) = 1.13 . lo-’ fm3 [ref. ‘“)I
(3.7)
where
characterizes the “hadron” polarizability of a proton as a system consisting of a quark bag and a pion cloud, and N,(P) is the polarizability of a cloud of virtual electron-positron pairs e e+. Because of the small electron mass, the quantity cry,(p) is three orders of magnitude greater than the pion cloud polarizability that dominates in (Y,,(P). Indeed, the estimate of a,(p) [ref. “)I using the sum rules gives a,(p) 2 0.7 fm3 , whereas
the estimate
with a proton
electric
form factor results
in
(Y,(P) = 1.5 fm3. The mean of these values a,(p) = 1.1 fm’
(3.8)
exceeds by a factor 103, the hadron contribution a,,(p) (3.7). Therefore, in low-energy proton-proton scattering the dominant contribution to proton electric dipole polarizability comes from the nonlinear electrodynamic effects generating a,(p): Q(P) = a,(p). Taking
the identity
of protons
into account
(3.9)
in eq. (3.5) and using eq. (3.9), we have
cr,. =2&(p)
.
(3.10)
The description of the polarization pp interaction by the potential (3.3) with parameter /?’ (3.4) which corresponds to electric dipole polarizability (3.10), (3.8) is valid only at distances beyond the electron-positron cloud r,3
A,-,+
,
where A,
h e+
E-x
2m,c
193.1 fm
(3.11)
114
is the Compton
V./? Levashev
wavelength
/ Coulomb
pdurization
of the electron-positron
eq. (3.4) the value of the Bohr radius
pair divided
of the pp system
p2 = 3.82 * lo-” fm' At distances
smaller
variable r, decreasing at r=Xe-e+ down to
than
li,-,+ the quantities
with decreasing
.
,
(3.12)
CX~and j3” are dependent
r from the values
cyE= 2ffh(p) = 2.26 * 10-j fm'
by 27~. Using in
uH = 57.64 fm, we obtain
on the
in (3.10), (3.8) and (3.12)
f3’ = 3.92 . lo-” fm’
at distances of the order of the size of a nuclear system r - R, y-l. In this case the character of the radial dependence of the polarization potential may also change. It should, however, be noted that at distances R s r s Xc-,+ the distortion of the Coulomb forces because of vacuum polarization plays a more significant role as compared to electric polarizability. In the lowest order in the fine structure constant this effect is described by the Uehling potential ‘O). This additional electrostatic potential characterizes the contribution to the interaction of charged particles that comes from the diagram involving the decay of an exchange photon into an e-e+ pair followed by its annihilation into a photon. The Uehling potential decreases exponentially at large distances and has a finite range of the order of X,-,+. This range is very large relative to the range of nuclear forces R. It has been shown in ref. “) that for proton-proton scattering the Uehling potential results in a strong energy dependence of the standard Coulomb-modified scattering length already at energies lower than 100 keV, whereas the polarization potential VP, whose range is greater, dominates below 0.7 keV. To describe low-energy pp scattering, it is necessary to subtract from the total phase shift the contributions coming from the long-range Coulomb and polarization potentials and the Uehling potential. The remaining part of the phase shift will satisfy the modified effective range theory “) whose parameters are close in magnitude to the corresponding Coulomb-modified nuclear quantities. In the case of the pp reaction (2.4), taking the Uehling potential into. account results in the S-factor decreasing by a few percent “). It is assumed in the following that the cut-off radius rp in the polarization (3.3) exceeds slightly the characteristic nuclear dimensions, rpa R, y-l,
potential
(3.13)
and the strength of the polarization potential p’ (3.4) is specified by the value (3.12) in the range rpb rs X,-,4, too. The Uehling potential is not taken into account. Such an approximation of the polarization interaction inside the electron-positron cloud will, perhaps, enhance slightly the influence of the polarization potential VP on the reaction cross section. However, as it will be shown below, the effect is insignificant even in this case.
V.P. Levashev
4. The pp reaction Let us consider
S-factor
polarization
disregarding
the case when the interaction
the sum of nuclear The proton-proton
entering
/ Coulomb
115
polarization between
(V,)
and repulsive Coulomb (V,) elastic scattering wave function
eq. (2.11) is determined
by the solution
,
equation
$+k’-
V,,
>
effects
protons
is described
potentials
of the radial
by
(3.1) and (3.2).
S-wave SchrGdinger
uN<.(k, r)=O
(4.1)
with the potential v,, and the boundary
- vN+ v,
(4.2)
conditions*
u&k,
0) = 0,
uNc.( k, r) = cos 6,,C[ F( k, r) + G( k, r) tan 6,,c],
r3 R ;
-sin(kr-~In2kr+w,c). I-r
(4.3) (4.4)
The functions F(k, r) and G(k, r) in eq. (4.3) represent the regular and irregular Coulomb S-wave functions. They satisfy eq. (4.1) with the potential V, used instead of V,,. and the boundary conditions F(k,O)=O,
F(k,
G(k, r) -I-*
sin(kr-nln2kr+&.), r) -+ ,- A cos(kr-qIn2kr+6,),
(4.5)
where SC.= arg r( 1 + in) denotes
the pure coulombic
phase
the Sommerfeld parameter (2.2). The Coulomb-modified nuclear sin 6 N.c = -At low energies
the phase
1
shift, I‘(z)
F(k,
All quantities
function
“) and 77 is
in eq. (4.3) is given by
r)VN(r)uNC.(k,
r) dr.
(4.7)
k Io
shift 6,,(. decreases 6N,c.- C’k-em2;ir,,
l
is the gamma
phase shift 6,,, ’
(4.6)
labeled NC correspond
by the exponential
law,
7731.
to the nuclear-C’oulomb
interaction
(4.8) V,,.
(4.2).
V. P. Leuasheu / Coulomb ~~Iur~z~~if~~l
116
The total phase shift W=6JNC.=SC.+SN,(. defines
the asymptotic
The low-energy by the penetration
behaviour
dependence factor
of the function of the Coulomb
uN,-(k, P) (4.4) at large r. F and G is characterized
functions
C (2.7): G(k, r) = C-‘c?(k,
F( k, r) = CkF( k, r) , Here F(k, r) is an entire
(4.9)
of k’, and the function
function
c?(k,r) =
6(k,
where a(k, r) is an entire function is defined by
r)+:
h($F(k,
r) .
(4.10)
G(k, r) has the form “)
rl ,
of k’, too. The function
(4.11) h(v)
entering
(4.12)
~(~)=Re~(~~)-In~, where I/I(Z) is the digamma for h(v) is valid:
function
h(7)=-
“). At small
1
1 12772+ 12074 ’
k the following
r)=$a,xl,(x),
6(k,
(4.13) I3
r)-xK,(x),
I, and Kr are the modified Bessel functions Taking into account the relations (4.10) nuclear-Coulomb
wave function
denotes
the
values of r) the solutions of the
x2 = 8r,fa,,
of first order. the boundary
condition
(4.14)
(4.3) for the
uN(. takes the form
uNC.(k,r)=Ck~~sfiN,C.[F(k,
where uN,(. (k) length ‘“)
approximation
,=$-%I.
When the condition kr<2T is satisfied (low energies or small functions F and 6 are well approximated 22,23) by zero-energy Coulomb problem: F(k,
eq. (4.11)
r)-a,,,.(k)d(k,r)],
generalized
a,,.(k)z
(energy-dependent)
-(C'k)-'
At low energies the quantity a,,,.(k) range theory expansion “*5) 1 -----+lh(q)= a,.<,(k) ars
rZR,
Fermi
tan SN,C..
is p arametrized
-$+&.k’+O(k*j.
by the well-known
(4.15) scattering
(4.16) effective-
(4.17)
NC
According to eqs. (4.17) and (4.13), the generalized scattering length a,,,-(k) in a zero-energy limit goes into the constant A N,C which is the ordinary Coulombmodified nuclear scattering length. The experimental value of pp scattering length is A,,,. = -7.828 fm [ref. 25)].
V.P. Leva.sbev / Cou/omh p”lariralion
the case of the interaction
In
the overlap
integral
(4.2) the pp reaction
of the wave functions
117
S-factor
(2.13) is defined
by
ud and u = uN(. (2.14): *
I(E)=
uduN(. dr .
INC(E)=(Ck)-‘N$
For non-resonant reactions the S-factors are assumed the energy. In the case of the pp reaction, theoretical realistic nuclear forces give (see e.g. ref. I’))
(4.18)
to be weakly dependent on evaluations of I,,.(E) with
I& (0) = 7 . The quantity . . contrrbuttons region:
(4.19)
IN<.(E) can be represented as a sum of two terms characterizing the to I,, from the nuclear force range and from an exterior (asymptotic)
I,,.(E)
)
= IE’( E) + I;;)(E)
(4.20)
K ZF;‘(E)=(Ck)
‘Ny’ 50
ud”N<
(4.21)
dr,
, lk;J( E) = Ny’ cos SN,<
em”‘[F(k,r)-a.,,(k)G(k,r)]dr.
(4.22)
IK To obtain eq. (4.22),the asymptotic expressions for the functions ud and uN( (2.10) and (4.15) have been used. For our further treatment we also need the irregular solution of eq. (4.1) g,, (k, r) specified by the boundary condition g,, (k, r) = cos 6,,,.[ G( k, r) - F( k, r) tan 6,,,.] and the Green
function
of the nuclear-Coulomb
YN(.( r, r’; k) = -i where
r. (r .) is the lesser
5. The proton-proton
(larger)
uNc (k, r
of the values
wave function
allowing
,
r3 R
(4.23)
problem )gN,
(k
r
(4.24)
1,
of r and r’.
for the polarization
interaction
We now study the problem where, in addition to the above nuclear and Coulomb interactions, the attractive polarization potential VP (3.3), (3.12) is taken into account. In this case the radial part of S-wave pp scattering wave function (2.11) u(k, r) = UN&k, is determined
by a solution
of the Schrodinger d’ d’+ r
k’-
V,,,
>
r) equation
uN,,(.(k, r) =0
(5.1)
118
V.P. Lev&hev
/ Coulomb polarization
with the potential
v,+ vp+ v,
VNpC=
and with the boundary
(5.2)
conditions
uNPC(kr) = ~0s b,Nd~NC.(k r)+gNc(S r) tan&,N~l, --+ l-r
sin (kr-q
r3-rc;
(5.3)
In 2kr+wNPc).
The total phase shift w = wNPC occurring
(5.4)
in eq. (2.11) and (5.4) is given by the sum
where X sin6
---
P,NC‘ -
+dIk
r)
Vdr)u.dk
r)
dr
(5.6)
is the phase shift caused by the polarization potential and counted off from the total nuclear-Coulomb phase shift w NC. For the pp scattering at energies lower than a few keV the phase shift 6 ,,,Nc practically coincides with the Coulomb-modified polarization phase shift 6P,c that has the power energy dependence ‘) 6P,c-
k’,
k*l.
As a result, the quantity cos (sP.Nc.involved is actually coincident with unity.
(5.7)
in eq. (5.3) and in the following
formulae
Using the Green function Y?,, (4.24), the differential equation (5.1) with the boundary conditions (5.3) can be transformed into the Fredholm-type integral equation
UN&k,
r) = Cos s,,r@,,-(k,
r)+
YNC.(r, r’; k) VP(r’)uNPC.(k, r’) dr’.
(5.8)
I0
From definition (3.3), V,(r) = 0 at r < rip. This, with account of the inequality (3.13), R s rp, allows us to express the Green function YINc in eq. (5.8) in terms of asymptotes of the functions uNC and gNc (4.3) and (4.23). As a result, we obtain for uNPc. the representation UN&k,
r) =f(k)[u&k,
r)+ wNPc(k, r)@(r-
rP)l,
where @(a) is the step function and WNpf denotes an addition Coulomb function uNc. due to the polarization interaction.
(5.9) to the nuclear-
V.P. Leva.shetl / Cortlomh
The term
MJ~~(-in eq. (5.9) is defined
polarization
119
by the solution
of the Volterra
integral
equation N’
N&k, r) =
jl BNc,(r, r’; k) Vp(r’)[
uNc.(k, r’) + wNI,(.(k, r’)] dr'
,
(5.10)
whose kernel
%.dr, r’; k) =i [udk beyond
the range of the nuclear 9?&r,
r’; k) = F(k,
r)gNdk r’i - UN&k,r’)gdk
forces reduces r)c(k,
r’) - F(k,
r)l
to r’)G(k,
r) ,
r, r’s
R.
(5.11)
Upon using the expression (4.11) for the function G(k, r) in eq. (5.11) it is not difficult to represent the integral kernel gN,.( r, r’; k) as a linear combination of the entire functions of k’, F( k, r) and 6( k, r): SNc.(r,
r’; k)=l”‘(k,
r)~(k,r’)-~(k,
r’)i%(k,r),
r, r’> R.
(5.12)
At low energies the expansion of the integral kernel and functions in a power series in the energy may be used as an effective method of solving eq. (5.10). Because of the properties of the functions F and 19 (4.14), the kernel (5.12) in the domain kr<2q and kr’e-277 is, in fact, energy-independent and expressed in terms of the first-order Bessel functions of an imaginary argument I,(X) and K,(X). The factor j’(k) in the radial wave function uNr(. (5.9) has the form f(k)
= cos 6 rs.N<.(1-5(k))-’
(5.13)
,
where C(k)=
1”gNc (k,
-;
0
Since the polarization rz
potential
r)V,,(r)[u&k,
is a small
r)+wNr,t.(k,
correction
r)@(r-rrla)]dr.
to the Coulomb
(5.14)
interaction
r,,), it can be taken
into account with a high accuracy by (I VP/ Vcl = at/2r”, employing the Born approximation. The direct calculations “) show that within the energy range l-lo3 keV the Born approximation reproduces the Coulomb-modified polarization phase shift 6,.,(. up to 0.01%. A similar result has also been obtained in ref. ‘I). Let us evaluate the function c(k) in the Born approximation. Substituting into eq. (5.14) the expression for VP (3.3) and the asymptotes of the functions uN(. and gNc- (4.3) and (4.23) and also neglecting the small terms of the order of (Ck)’ and higher, we obtain 5(k) = h(k, co),
(5.15)
V.P. Levashev / Coulomb polarization
120 where
r
b(k, r)= Applying
F(k, r’)G(k, r’)Vp(r’) dr’.
(5.16)
i FP
in eq. (5.16) the inequality (F(k, r)C?(k, r)l
following
from inequality
(4.12) of ref. “),
jF(k, r)(G(k, r)+iF(k, we get an upper
bound
r))j<6kr,
on Ib(k, r)l: (5.17)
The dimensionless quantity M in eq. (5.17) is defined by the value of the polarization potential at the cut-off point, i.e. at the boundary of the nuclear force range:
M-rr’,lVP(rp)/=*. For our polarization pp interaction choosing, e.g., r,, = 5 fm
model
(5.18)
aHrF
(3.3),
(3.12)
and
M = 1.5 . lo-“. We note that the value of the parameter M for the proton-deuteron system. Indeed, using nuclear polarizability 24) ah(d) = 0.7 fm’ and corresponding to (3.8) and taking rp = 5 fm,
(3.13),
we have by
(5.19)
close to that in (5.19) is obtained also the experimental value of the deuteron also the values a,(p) = a,(d) = 1.1 fm’ we have
M = ah(d) + a,(d) + a,(p) = 2_9 , 1o-.’
(5.20)
a&d)6 Returning to the formula (5.13) and taking into account that 6P,NC.- k5 and due to eqs. (5.15), (5.17)-(5.19) the modulus of the function t(k) does not exceed the number of the order of 10w3, we conclude that at low energies the factor f(k)’ is close to unity: f(k)=
1+0(M)
>
M - 10-3.
(5.21)
We now study the polarization contribution wNPc. to the wave function uNpC (5.9). It should, first, be noted that at distances r < rp the function wNpc.(k, r) =0 and, therefore, the total wave function u Npc.(k, r) coincides with the nuclear-coulomb solution u &k, r) up to the factorJ(k) (5.21).
K P. Levashev
We represent
/ Coulomb
r) in the region
w,,,(k,
polarization
121
r 2 rP in a form similar
to the asymptote
of uNc. (4.15). As it has been noted above, at low energies the function wNPc. can be approximated with great accuracy by the Born solution of eq. (5.10): r) = w:,,, (k.r)=j’-
w,,,(k,
Using the expressions wipc(k,
r’) dr’.
(5.22)
for SN(. (5.11) and uNc. (4.15) in eq. (5.22), we get r)=-Ckco~S~,~.{F(k,
r)[b(k,
+ G(k, r)[a(k,
where the auxiliary
r’; k) V,(r’)u,,.(k,
S&T, II’
functions
r)-a.,,(k)d(k,
r)+a&k)b(k,
r)]
r)lI,
(5.23)
are introduced I
a(k, r)=
I
F”( k, r’) V,,( r’) dr’ ,
(5.24)
rl’ G’( k, r’) V,( r’) dr’ ,
d(k, r) = -
(5.25)
I,’ and.b(k, r) is defined by formula (5.16). It follows from a comparison of eqs. (5.23) and (4.15) that at distances rz rp the factors a(k, r), perturbation w:,,. differs from uNc. by the presence of additional h(k, Y) and d(k, r) for the functions F( k, r) and G(k, Y). The magnitudes of these factors just determine the deviation of the total wave function uNF(. from the unperturbed solution uNc.. It has been established above (eqs. (5.17)-(5.19)) that the modulus of the function b(k, r) is majorized uniformly in k and r by the small parameter M determined by the value of the potential VP at the cut-off point rp. We show that at low energies a similar bound exists also for the function d(k, r), whereas the factor a(k, r) is strongly dependent on the ratio between the variable r and the Coulomb turning point (5.26) Taking
into account
an explicit
form of the potential
VP (3.3), we get
I d(k, r)
G’(k,
r)F“
dr.
(5.27)
I VP We divide the integration path in (5.27) into three parts where the function is approximated by the expressions “) G( k, r) = xK,(x) G(k, r)=
,
C(2r))“(‘&
G(k,r)=Ccos(kr-qln2kr+&),
x2= 8r/a,,
,
r
Bi(z) ,
z-(27j)“‘(l-r/r,),
;
r>rr,..
G(k, r)
(5.28) r-rr,;
(5.29) (5.30)
% P. Leuashev
122
/ Cmilm7zh palariza~i~~n
Above the turmng point (r 3 rc ) the Airy function Bi(z) oscillates with an amplitude not exceeding the value Bi(0) = (3”“Z’(.f))-’ =0.62. In the case of r< r,. and large values of z s 1 the function Bi(z) increases exponentially leading to the expression for d in this region
G( k, r)
in the form
= C(-’ exp ($&“)
,
It follows from the approximations (5.29) and (5.30) and the behaviour of the Airy function Bi (z) that at low energies (n * 1) the contributions to d(k, ~0) from the due to a poor integration intervals r - rc and rc < rd CO are much suppressed penetrability
of the Coulomb
barrier
(2.X),
C-fie-“n,
7731.
Therefore, the quantity d(k, co) will be completely determined by the contribution coming from a small integration region below the Coulomb turning point:
To obtain the estimate (5.31), the constraint 22) K,(x)s 1.7 for x20.5 has been used, leading to the condition r,,’=- laH 32 that is easily met in our case. With account of eqs. (5.27) and (5.31), we obtain the desired bound on the function d(k, r): d(k, r)Sz Unlike the functions
M,
k
b(k, r) and d( k, r), the energy
(5.32) behaviour
of the factor a( k, r)
is substantially dependent on the values of the variable r. In the region using the approximation (4.14) for the function F(k, r), we get a(k,r)=-OZ
au
y Z~(Y)/dx’, Xi’
r%rc,
xZ=8r/a,,,
r =Crc on
(5.33)
so that la(k, r)/ < My$[R(x)
- R(xp)]
c MS
I’
R(x),
r
(5.34)
where R(x)=;x’[zf(x)-Z,,(X)Z,(X)])
xZ,= Xr,/aB .
Thus, in the region r < rc. the function a( k, r) is bounded by the energy-independent quantity (5.34) that contains the small parameter M = CY~/U~T$-- lo-“.
V.P. Levashev / Coulomb
In the vicinity is approximated
of the turning
point r,. and beyond
by the expressions
F( k, r) = ( Ck)m’&(2v)“h
polarization
zE(27p
Ai( z) ,
1-t
)
r-r<-
The Airy function Ai in eq. (5.35) (5.35) and (5.36) contain the energy As a result, the values of the factor r > r(.) depend strongly on the energy
;
(5.35)
>
r> r,. .
F(k,r)=(Ck)-‘sin(kr-nln2kr+&.),
(5.36)
is bounded as IAi( ~0.6. The expressions factor (Ck)-’ that diverges at the threshold. a(k, r) (5.24) at large distances (r - rc and increasing unboundedly with E + 0,
a(k,r)--~m’e7711),
the expressions
F(k, r)
it up to 00, the function
“)
(
Comparing
123
ra rc..
77>‘1,
for the wave functions
(5.37)
uNPC., uN(. and
wNPC. (5.9),
(4.15) and (5.22), (5.23) and taking into account the constraints (5.17) and (5.32) on the factors h( k, r) and d( k, r) and the properties of the factor a( k, r), we conclude: (i) at small distances below the Coulomb turning point (r =Sr(.) the polarization potential contribution wNP(. to the total wave function uNP(. is insignificant and determined by the parameter M - lo-’ (5.18), (5.19); in this case the wave function uNP(. contains (as in the short-range force case) the Coulomb penetration factor C; (ii) at large distances (r- r(. and r s r(.) the polarization factor a( k, r) involved in wNP( affects significantly the behaviour of the function uNPC(k, r) as compared to that of the undistorted solution u,,(k, r). The asymptotic value of the function a(k, r), * u(k, 00) =
F’(k, 1 F,>
coincides with the Coulomb-modified the Born approximation: a(k,m)=
r)V,,(r)
polarization
uIlc(k)-
-(C’k))‘(tan
dr,
scattering
(5.38)
length
S,,,)“.
calculated
in
(5.39)
According to eqs. (5.33) and (5.34), the integration region r,< r Q r,. gives a small contribution to the integral (5.38) of the order of A4 - 10 ‘. The value of the polarization phase shift fi,,,. (5.39), (5.38) IS, . in fact, determined by an integration in the regions r- rc and r,. < r s 00, which just produces the k’-dependence (5.7) obtained in ref. ‘). As a result, the phase shift SP,(. depends weakly on the choice of a specific value of the polarization cut-off parameter r,, *. The dependence (5.7) is a consequence of the long range of the polarization potential and differs much from the exponential decrease C’k - exp (-2~77) (4.8) typical of the Coulombmodified nuclear phase shift S,,c. For this reason the generalized scattering length u,,.(k) defined by eq. (5.39) diverges at the threshold 2,6,7): a,,,, (k) - -k’( C”k))’ l
The numerical
calculations
“.I’)
confirm
- -qP5 e2nV -CC. rJ+‘”
this, too
(5.40)
V.I? Leuasheu / Coulomb p~}~ari~ari~t?
124 6.
Influence of the polarization
potential on S-factor
In the case when the polarization interaction is taken into account, the S-factor of the pp-+de+v reaction is determined according to eqs. (2.13) and (2.14) by the overlap integral of the radial wave function ud and ‘u 5 uNPc.:
* r(E)~z.,,.(E)=(Ck)-‘Nr’ J 0
Substituting
the expression
(5.9) for the function
I NW(E)
%UNPC
dr.
(6.1)
aNP(. into eq. (6.1), we get
16.2)
=f(k)[I,,-(E)+f,,~(E)l,
Iu
where the addend
(6.3)
ff’
characterizes the contribution caused by the polarization potential VP. The augend I,,.(E) in eq. (6.2) is defined by the expression (4.18), while the function f(k), according,to eq. (5.21), equalsf(k) = 1 +O(lO-‘). Owing to the weakness of the potential VP as compared to the nuclear and Coulomb potentials, the contribution JNpC can be calculated with a great accuracy in the Born approximation
in V,: ~NI’C’(~)=~k’<
Jf&.(E) In formula
= -N
cos 6,,-[A,(k)
(6.5) the following
notations
(6.4)
cE),
- a iv.<-(k)(A,(k)
-A,(k))+
A,(k)1 .
(6Sj
are introduced:
(6.6) (6.7) Taking account of the relation R G rp (3.13) we notice that the expression for JNP(.( E) (6.4)-(6.7) differs from the asymptotic part of the nuclear-Coulomb matrix element (6.6) and I’,‘:)(E) (4.22) by the factors b(k, r), d(k, r ) and a(k, r) in the integrands (6.7). Using in eqs. (6.6) and (6.7) the bounds on the functions b(FE, r) and d(k, r) that are uniform in r and also the numerical value of INC.(O) (4.19), we conclude that the terms A,(k), i = 1,2,3 in eq. (6.5) give a small relative contribution of the order of M - 10e3 (.5.18), (5.19) to the matrix element INPC.(E) (6.2):
(6.8)
V.P. Lemsher
/ Coulomb polarizution
125
The term A,(k) is the only contribution to eq. (6.5) that may be thought to be appreciably affected by the polarization interaction. This is because the factor a(k, r) in A,(k) shows an exponential increase in the region of the variables k < 1, r 3 rc. = 2n/ k. We prove, however, that due to the rapidly decreasing factor exp (- yr) in the integrand
(6.7), only the small integration
region
r Q rc. contributes
to the matrix
element A,(k). By approximating the functions G( k, r) and a( k, r) by the expressions (5.28) and (5.34), respectively, the term A,(k) will then have the same smallness order (-M) as the terms A,(k), i = 1,2,3 (6.8). Using in eq. (6.7) the definition (5.24) for the function a( k, r), we represent Aj( k) as I A,(k) = (rP)’
, dr’ r’ ‘F’( k, r’) .
dr emY’G(k, r) I 1II
I ’ I’
(6.9)
The contribution A4( k) may be estimated by approximating the functions G( k, r) and F(k, r) in eq. (6.9) by the expressions (5.28)-(5.30) and (4.14), (5.35), (5.36), followed by calculations of the contributions from different parts of the twodimensional domain of the integration over r, r’. The dominant contribution to the integral (6.9) will then come from the integration region r’s r s r( , whereas the contributions from the regions r’s r, r - r(‘ and r’< r, r( < r G ~0 will be negligibly small at low energies. The obtained value of A,(k) will be proportional to the small parameter M (5.18), (5.19). A simpler way to estimate the quantity A,(k) is to employ in formula (6.9) the inequality “) lG(k, r)F(k,
With account
of f/r
rar’.
r’)I 4 (2mr’)“‘,
s 1 this gives
P,(k)1
I.:drre~“~~:,dr’r’~~,~(k,r’),.
s G(Pr)’
On interchanging the order (6.10) is simplified to
of the integrations
and integrating
lA,(k)l~~M(yr,)~~(l+yr,)B(k).
Here the function
B(k)
is defined
(6.10) over r the relation
(6.11)
as \
B(k) = y2
dr em”lF(k,
r)l ,
(6.12)
I I,’ M is the small the condition We estimate function F(k,
parameter
(5.18), (5.19), and the factor (1 + yrp)( yrp)m”s 2 because
-yrpa 1 (3.13).
the low-energy
value of B(k) using the expansion of the Coulomb of an imaginary argument Z,,(x) [ref. ‘I)]:
r) in the Bessel functions F(k,
r) =;a,
i
II I
b,,(k)($x)“l,(x)
,
x2 = 8r/a,.
(6.13)
V. P. Levushev
126
/ Coulomb
The coefficients h,(k) in (6.13) are determined ref. “) and are proportional to various powers entire
function
of k’. Substituting
polarization
by recurrent relations (14.4.3) of of k’. We remind that F(k, r) is
(6.13) into (6.12) and integrating
.X B(k)
over r, we get (6.14)
,
the small parameter (6.15)
4=2/Y%.
Under the condition k < y corresponding to low energies E G 100 keV, we can restrict ourselves to taking into account a few first terms on the right-hand side of the inequality (6.14). Substituting the explicit expressions for h, into eq. (6.14), we get
~~lf~~~Y~l+~~l~~‘~l+~~~+~~~~l~~4~1, k
the exponent B(k)<
in eq. (6.16) in powers of the small parameter
1i-q+(k/Y)‘+O(C4+(kly)‘12),
key.
(6.16) 4, we obtain (6.17)
It follows from (6.17) or (6.16) that at low energies (k< y) the values of the factor B(k) in eq. (6.1 I) do not exceed a quantity of the order of unity. As a result, the numerical value of the contribution A,(k) is completely defined by the small parameter M (5.18) having the order of 10m3 (5.19). The small magnitude of A,(k) follows directly from comparing the bound on A,(k) (6.11) with the bound, for example, on A,(k), whose smallness is a consequence of eq. (6.8). Indeed, majorizing in eq. (6.6) the function b(k, r) according to (5.17), we obtain for A,(k) the bound of the same form as eq. (6.11): ~A~(k){~~~M~(k~.
(6.18)
Applying in formulae (6.2)-(6.5) the above bounds on Ai( i = 1,2,3,4 and also using the numerical value of Z&O) (4.19), we conclude that the relative contribution of the polarization term JNPC‘(Ef to the matrix element lNpC defining the pp reaction S-factor (2.13) is a small quantity of the order of M - lo-” (5.18), (5.19). Such a small correction to the pp reaction cross section indicates that the solar neutrino problem mentioned in the second section is impossible to resolve by taking polarization effects into account. The method used here to estimate the polarization corrections to the pp+ de+ v reaction cross section is general and holds also for the reactions involving more complex nuclei in an initial channel. Thus, we come to the main result of the present paper that polarization effects play an insignificant role in the low-energy nucleosynthesis reactions. Physically this is due to that fact that the reactions concerned are realized in a small region of the configuration space (of the order of a synthesized nucleus size). In this region the polarization potential represents a small correction to the nuclear and Coulomb interactions. The value of the polarization potential at
V.P. Leuarhev J Coulomb polarization
its cut-off point rP (i.e., at the boundary the parameter M (5.18) characterizing cross section polarizability
of the nuclear the polarization
1’7
force range) just determines corrections to the reaction
and the wave function at small distances. Because of the weak electric of nuclei LYEand the large value of the Bohr radius a,, the parameter
M = a,/a,ri is small. For the reaction reaction, the parameter M (5.20) has the the above pp reaction. Our prediction that reaction cross section must be insignificant
induced by deuterons, e.g., pd+ y7He same order of smallness (- 10m3) as for the polarization contribution to the pd agrees well with the calculations of this
reaction using a quasi-classical approach “I.“). Concurrently to our studies of the problem presented in refs. 3’,31) the publications of the group of authors appeared “LX ), wherein the incorrect result was obtained claiming
that the polarization interaction affects very strongly low-energy cross sections of the pp + de+v, pd + y3He and dt + n”He reactions. The reaction S-factors calculated in refs. 33m’5) at the near threshold energy exceed by many orders the extrapolated experimental values of these S-factors. The mistake of the authors of refs. 33m35)consists in using an inadequate approximation for the factor a(k, r) (eq. (5.24)) determining the polarization contribution wN,&k, r) (eq. (5.23)) to the initial wave function u.,,,(k, r) (eq. (5.9)). As it was shown in sect. 5, the energy behaviour of the factor a(k, r) is crucially dependent on the variable r. At the left of the Coulomb turning point (r s r(.) the modulus of the function a(k, r) is bounded by an energy-independent small quantity (5.34), and whereas at large distances (r- r( and r > r( ) it is strongly energy-dependent has the threshold singularity (5.37). In refs. 31m35),the factor a(k, r) was approximated for all r by the asymptote a(k, ~0) having the threshold singularity in (5.39), (5.40). With such an approximation the polarization contribution to the wave function, w,,,.(k, r) (eq. (5.23)) at small distances was unjustifiably forced to be characterized by the threshold divergence k’/C, rather than by the exponential decrease Ck specific for it. Just this threshold divergence introduced “by hand” in refs. 3’m1’) has resulted in an incorrect description of the reaction S-factor (2.13), (6.1)-(6.3) at low energies. When the factor a(k, r) in the region r < r ( is approximated by the correct expression (5.33), the polarization contribution JN,,< (E) (6.3) to the S-factor is found to be insignificant - of the order of the quantity M = a, /a,,&-- lo->.
7. Conclusion In our paper we have studied the influence of the electric polarizability of particles on nucleosynthesis reaction cross sections at low energies. It has been found that the dominant contribution to the matrix element determining the reaction S-factor comes from the small integration region below Coulomb turning point. Taking account of the polarization interaction in this region has been shown not to lead to any noticeable distortion of the wave function of coliiding particles.
128
V.P. Lruashev
/ Cmlomb
polarization
The relative value of the polarization correction to the nucleosynthesis reaction S-factor is characterized by the quantity M determined by the value of the polarization potential at the boundary of the nuclear force range: M - r’,] V,( rp)l = a,/a,rE, r,>
R. Due to the weak polarizability
of the Bohr radius uB, the parameter
of nuclear
systems
(Ye and the large values
M takes on the small value - 10e3. As a result,
the role of the polarization effects in the nucleosynthesis reactions at low energies turns out to be insignificant. It has been established that allowance for the polarization interaction can not resolve the existing discrepancy between the predicted and measured solar neutrino fluxes. It has been shown that the conclusion of refs. 33-34) that the polarizability of particles affects strongly pp, pd and dt reaction cross sections is wrong and caused by an incorrect approximation of the initial-channel wave function at small distances. The author expresses his gratitude to V.F. Kharchenko for fruitful discussions and interest in the work and also to Ya.A. Smorodinsky, D.A. Kirznitz, V.A. Petrun’kin and J.L. Friar for discussions of the results obtained. Note added: After the present paper was completed a publication by Gy. Bencze appeared in Phys. Lett. B202 (1988) 289. In that publication the upper bound on the polarization effects in the deuteronwas obtained which is by five orders smaller as compared to our result induced nuclear reactions -lo-’ -lo-‘. The great underestimate of the polarization effects by Bencze is a consequence of the use of an insufficiently good approximation for the phase shift fi,,,(- (B4) (i.e. formula (4) of Bencze’s paper). By using in Bencze’s approach a more accurate expression for 6,,,, (V.P. Levashev, 1986, unpublished), different from eq. (84) by the presence of an extra factor (I + 2b( k) - d( k)A,,,-) for the addend in the numerator, we come to eq. (814) with an additional term f(k) = 26(k)-d(k) Re A,,, on the right-hand side. With allowance for our estimates (5.17), (5.18), (5.32) for the functions b(k)= b(k, CO) (5.16) and d(k)= d(k, 03) (5.25), we have IF(k 10-j so that Bencze’s term tan* 6,.,10m8 in eq. (814) may be neglected. Thus we obtain again the result of the present paper.
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polarization
129
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