PASJ: Publ. Astron. Soc. Japan 48, 613-618 (1996)
Thermodynamic Enhancement of Nuclear Reactions
in Dense Stellar M a t t e r
Setsuo ICHIMARU and Hikaru KlTAMURA
Department of Physics, School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113
E-mail (SI):
[email protected]
(Received 1995 October 6; accepted 1996 May 14)
Abstract
An enhancement in the rates of nuclear reactions in dense matter is approached through calculations
of increments between the Coulombic chemical potentials before and after the reactions. The formalism
is applied to specific cases of the p-p reactions in the solar interior and of the 1 2 C- 1 2 C reactions in a
white-dwarf progenitor of a supernova; the dependence of the resultant enhancement factors on the relative
abundances of the elements in the outgoing channels is thereby illustrated. The effects of plasma screening
in the p-p reactions on the solar 8 B neutrinos are examined.
Key words: Dense matter — Nuclear reactions — Sun: interior — Supernovae: general — White
dwarfs
1.
Introduction
where pm denotes the mass density and TTIN = 1.6605
x 10~ 24 g refers to the average mass per nucleon. We
The rates of nuclear reactions in dense stellar mat- expect that those electrons are all in the metallic (i.e.,
ter, being proportional to the contact probabilities or fully ionized) states if the Fermi (or average thermal)
the two-particle joint-probability densities at the nuclear- energy is greater than the largest atomic binding energies
force ranges, depend sensitively on the thermodynamic of the nuclei (e.g., Ichimaru 1994).
states of the matter (e.g., Ichimaru 1993). When the inIn the present study, we confine ourselves to the cases
ternuclear separations are significantly greater than ther- of exoergic reactions, Qij > 0, and assume that Qij and
mal de Broglie wavelengths of the reacting nuclei, we the nuclear binding energies are greater than the Fermi
may regard the nuclei as being particles obeying classi- energy of the electrons. The nuclei themselves are thus
cal statistics (Salpeter 1954; Jancovici 1977; for a review, stable against (3 captures; we need not consider such a
see Ichimaru 1994). The electrons (and positrons), on the j3 process in these circumstances.
other hand, may retain their quantum-statistical characIn dense stellar matter, multiparticle-correlation efter as fermions.
fects act to enhance the reaction rates through the screenWe consider the nuclear reactions between elements ing potentials, which are closely related to the thermo"z" and uj" yielding elements ukn and "/" and energy dynamic functions of the dense plasmas (e.g., Ichimaru
Qij, which may be expressed as
1994). The resultant enhancement factors (Salpeter
i + j=>k + l + Qi:i.
(1) 1954; Salpeter, Van Horn 1969; Ogata et al. 1991;
Ichimaru 1993) of the thermonuclear rates (Gamow,
These pre- and apre-reaction elements may include
Teller 1938; for a review, see e.g., Ichimaru 1993) are senatomic nuclei (i.e., ions), electrons (or positrons), and/or
sitive functions of the increments between the Coulombic
photons (i.e., 7-rays); when electrons or positrons particchemical potentials before and after the reactions. When
ipate, we call such reactions /3 processes.
elements uk" and "/" of the outgoing channels involve
The stellar matter under consideration may consist electrons or positrons, the thermodynamic functions asof nuclear species (subscript, i) with charge number Zj, sociated with the electrons can affect the conditions for
mass number A{, and molar fraction xi as well as elec- nuclear statistical equilibrium in the reaction yields.
trons (—) and positrons (+). Since the positron ultiIn this paper, we explicitly formulate and calculate
mately annihilates an electron in the stellar matter (e.g.,
these increments in the thermodynamic functions, and
Bethe, Critchfield 1938), the number density of the nuclei
thereby elucidate the quantum-statistical effects of the
and that of the electrons are expressed as
relative abundances of the elements in the outgoing channe = n_ — n+ = 2_^ Zini,
(2) nels as well as those in the incoming channels on the enUi =
hancement factors for the nuclear reaction rates in dense
raN / v ZjAj
© Astronomical Society of Japan • Provided by the NASA Astrophysics Data System
S. Ichimaru and H. Kitamura
614
matter; the former effects have not been appropriatelytaken into consideration in the literature (e.g., Ichimaru
1993). In so doing, we distinguish between the cases of
nuclear reactions with and without f3 processes; for example, the p-p reactions involve (3 processes in the form
of positron emission, while 1 2 C- 1 2 C reactions, significant
in a white-dwarf progenitor of a supernova (Barkat et al.
1972; Graboske 1973; Couch, Arnett 1975), do not involve such a f3 process. In connection with the p-p reactions, we exhibit the extents of strong Coulomb-coupling
effects in the electron-screened hydrogen-helium-mixture
plasmas and examine the effects of plasma screening on
the solar 8 B neutrinos.
2.
1994). Quantum-mechanical corrections to the enhancement factors arise in equation (4) due to the finiteness
of r x p / a . The first term on the right-hand side of equation (4) dominates over the remaining terms representing the quantum corrections, however, since 7"TP <C a
in many cases. The evaluation of H(0) depends on the
thermodynamic states of the electron system as well as
on those of the ion system. It is these thermodynamic
effects on nuclear reactions that we intend to elucidate
in the sequel.
3.
p-p Reactions in t h e Solar Interior
Consider, for example, the p-p reactions (Bethe,
Critchfield 1938; Bethe 1939; Salpeter 1952),
Thermodynamic Enhancement Factor
The enhancement factors for the thermonuclear rates
are expressed compactly as (Ogata et al. 1991; Ichimaru
1993
[Vol. 48,
)
P + P ^ d + e + + i/ + 1.442 MeV.
T h e s e reactions m a y b e split into t w o stages via
2
(8)
b
l
e
unsta
He as
A( / 0 m ,T) = e x p ( 0 ,
(3)
where
X
H + XB ^
2
H(0)
kBT
—r
f TPV
32 s \ a )
(^X
(4)
with d = 1.1858, C2 = -0.2472, and C3 = -0.07009.
Here, kBT is the temperature in energy units; H(0) is the
value of the screening potential (Salpeter 1954; Jancovici
1977; Ichimaru 1994) at zero separation;
1/3"
1
2
\4:irne)
\4irne)
(5)
is the effective ion-sphere radius between the reacting
nuclei, i and j ;
rs =
ZjZje2
akBT exp
U)
(6)
defines the effective Coulomb-coupling parameter
(Ichimaru, Ogata 1991; Ichimaru 1993); rxp refers to
the radius of the classical turning point for the reacting
nuclei; and Ds denotes the short-range screening length
of the electrons (e.g., Ichimaru 1994), calculated with the
knowledge of the wave-number and frequency dependent
dielectric function e(k,u) of the electrons as
J__2
f°
dk
D3 7T J0
He,
(9a)
He => 2 H + e+ + v + 1.442 MeV.
(9b)
r
1 + ( d + C 2 l n r s ) ^ + C3
a=
2
e(fc,0).
(7)
In the calculations of the enhancement factors (3)
and expression (4), it has been assumed that the ions
obey classical statistics; the screening potential H(r) has
been evaluated under such assumptions (e.g., Ichimaru
The former reactions are nuclear fusion, whose enhancement factors may be calculated according to equations (3) and (4). The latter are (3 processes, since the
outgoing channels contain positrons. Due to the involvement of these (3 processes the nuclear cross-section factors take on extremely small values for the p-p reactions
(Fowler et al. 1967).
The enhancement factor (3) for nuclear fusion (9a)
depends on H(0).
Statistical-mechanical arguments
(Jancovici 1977; for a review, see Ichimaru 1994) rigorously prove that H(0) may be evaluated as the balance
in the excess free energies, F e x = F — F 0 , before and after the reactions, where F and Fo denote, respectively,
the total and ideal-gas free energies of the system. Taking
hydrogen as " 1 " and helium as "2," we may thus express
/f(0) = F ex (Ar 1 ,Ar 2 ;7Ve)-F ex (7V 1 -2,iV 2 + l;7Ve),(10)
where N{ = riiV and 7Ve = neV. No isotope effects
arise between the protons and deuterons or between 4 He
and 2 He as far as the Coulombic chemical potentials are
concerned, since the atomic nuclei are here assumed to
obey classical statistics. A derivation of equation (10) is
given in appendix 1.
Since TVi, N2 ^> 1, equation (10) may be rewritten in
the form
H(p)
kBT '
=
v
[[ ni+n2
drii -d^) -
where
/ex(7ll,
n2\
Tl e ) =
F«(NuNnN.)
{Ni + N2)kBT
(12)
© Astronomical Society of Japan • Provided by the NASA Astrophysics Data System
Nuclear Reactions in Dense Matter
defines the Coulombic
Coulor
chemical potential (per ion) in
^ units of the average
averag thermal energy. Since ne = Zri\ +
g 2Zn2 with Z =
= 1, we
^ may regard ne as being a constant
S under the differential
different operation of equation (11).
H
The Coulombic chemical potential may be expressed
as a sum of the separate contributions (e.g., Ichimaru
1994): ion-ion, ion-electron, and electron-electron; that
is,
/e X (ni,n 2 ;ne)
= fuiriun2\rie)
+ / i e ( r a i , n 2 ; n e ) + fee(ne).
(13)
Since me <$C m^, we may adopt a Born-Oppenheimer
approximation and regard the electron-electron interaction term / e e as being a function of only ne. This term
therefore does not contribute to the calculations of equation (11).
As we observe in equations (11) and (13), the enhancement factor depends strongly on the ion densities, n\ and
7i2, as well as on the electron density, ne. In this study,
we are particularly concerned with the screening effects
in equation (11), brought about by the pre-existing density ri2 (^ 0) in the outgoing channels.
In dense matter, the stage (9b) likewise produces an
increment, which amounts to
AF = Fex(N1-2,N2
+
l;Ne)
1/3
reff
2/3
(Z?m + Z2n2)2
(ney/2}.
„2
(15)
with
kBT \dneJTV
_
=
__
l n e +
+-
ln
(16)
—
(17)
1 + AS~b
with A = 0.25954, B = 0.072, and b = 0.858, as a function of the degeneracy parameter of the electrons,
2mekBT
(18)
e = h2(37r2ne)2/3'
This formula (17) is applicable for an electron gas in the
nonrelativistic density regime.
An accurate representation of / e x for such a weakly
coupled plasma may be obtained by going beyond a
random-phase approximation (RPA) evaluation (e.g.,
Ichimaru 1992), such as the Debye-Huckel approximation or the Thomas-Fermi approximation, and thus by
proceeding to the Abe formula (Abe 1959; for a review,
see Ichimaru 1994), which reads
-p 3/2
-p 3
leff
_ leff
y/3
2
3 , /OT , x
-]n(3Tes)+y--
(14)
where FQ denotes the ideal-gas free energy for the electrons, including the exchange effects. Such an increment
should generally produce an effect on the nuclear statistical equilibrium resulting from these ft processes, since
it has not been taken into account in the usual /3-decay
factors (e.g., Bethe, Critchfield 1938). The effects of the
increment (14) on the /3 processes will be deferred to a
future study.
Since T*TP <C a for the p-p reactions in the solar interior, the exponent of the enhancement factor (3) may
be given by equation (11). For a weakly Coulombcoupled plasma appropriate to the solar interior, the effective Coulomb coupling parameter Teff (calculated in
appendix 2),
kBT
takes on a value in the vicinity of or smaller than 0.1.
Here, /IQ refers to the ideal-gas chemical potential of the
electrons, and has been expressed (Ichimaru 1992) in a
useful analytic formula,
/ e x (ni,7i 2 ;ne) = -
- i U t f i - 1 , J V 2 ; AT,-1)
+F0e(JVe) - F$(Ne - 1),
. (ni + n 2 )[(Cn e ) 1 /2 + (Z2nx + Z2n2 +
615
11
(19)
where 7 = 0.57721.. is Euler's constant. The first term
on the right-hand side of equation (19) represents the
RPA contribution, and usually overestimates the screening effects.
In the solar interior, at the mass density p m =
148 g c m - 3 and core temperature T c = 1.56 x 107 K, con^
sisting of hydrogen (with a mass fraction X\ = 0.3411)
and helium (with a mass fraction X 2 = 1 — X\ — 0.0202,
the mass fraction of the heavy elements with Z» > 3),
the effective Coulomb-coupling parameter (15) and the
enhancement factor (3) are calculated to be
r e f f = 0.0737,
A = 1.021.
(20)
This value of reff is significantly greater than the standard evaluation for hydrogen under the solar-interior
conditions (e.g., in table VII of Ichimaru 1993), owing
primarily to the participation of higher-Z helium ions
(n2 z£ 0) in the effective Coulomb coupling. The enhancement of the p-p reactions by 2.1% thus predicted
in (20), however, is smaller than 4.7%, an RPA enhancement listed as A&\—1 in table VII of Ichimaru (1993); this
reduction resulted from the non-RPA correction terms
contained in equation (19).
The enhancement factor A then induces a change in
the core-temperature estimate by an amount ATC, since
© Astronomical Society of Japan • Provided by the NASA Astrophysics Data System
[Vol. 48,
S. Ichimaru and H. Kitamura
616
the reaction rate (i.e., the luminosity) should be kept
invariant; this change is thus calculated according to
T 2 (T c )exp[-T(T c )]
= AT\TC
- AT c )exp[-T(T c - AT C )],
(21)
where
' e4mp \
,h2kBTj
(22)
ae/Ds
with m p denoting the mass of a proton. We thus find
ATC = 8.21 x 104 K.
According to Bahcall and Ulrich (1988), the total rate
of the 8 B neutrino production rate may be written in the
following suggestive form:
c/?(8B) « const x T77,
(23)
with r] « 18. The 8 B neutrino flux with this enhancement
factor is thus predicted to change from that in a standard
model calculation by the amount,
^( 8 B)
¥>( B),standard
8
(Tc - ATC) 18
T
18
These are important processes for carbon ignition in
white dwarfs. The electron density does not change
through the reactions in these cases.
In such ultradense matter, the electrons are relativistically degenerate, and their short-range screening
length (7), calculated with the relativistic RPA dielectric function (Jancovici 1962), has been parametrized as
(Ichimaru, Utsumi 1983)
= 0.909.
(24)
= 0.1718 + 0.09283# + 1.591# 2
+3.706# 4 - 1.311# 5 ,
3.800iT
(27)
with R = 10r s . Here,
ae = (3/47rn e ) 1 / 3 ,
rs = aemee '•it?
(28)
the formula (27) is applicable for the density domain,
0 < r s < 0.1.
Ion-ion and ion-electron contributions to the Coulombic chemical potentials for dense binary-ionic-mixture
plasmas (e.g., Ichimaru 1994) are given, to good accuracy, by a linear-mixing formula (see, however, Ogata et
al. 1993),
(29)
fii(ni,ri2]ne) = ^ Z i / o c p Q ^ ) ,
A reduction by 9% in the theoretically estimated 8 B neutrino flux, due to the plasma-screening effects on the process (9a), is thus predicted. The whole picture of densematter effects on the p-p reactions may be revealed,
(30)
/ie(ni,n2;ne) = - / J l
2DskBT'
however, only after the effects of the free-energy increments (14) on the /3-decay factors have been assessed.
To see the effects of the relative abundance x<i of he- where (Slattery et al. 1982; Ichimaru 1994)
lium on the p-p reaction rates, we repeated the calcu/ o c p ( r ) = -0.898004r + 3.87144r 1/4
lations described above at the same mass density and
-0.882812r~ 1 / 4 - 0.86097 In T-2.52692,
temperature, however, for the case of a pure hydrogen
(31)
matter (i.e., X\ = 1), to find
Teff = 0.0438,
A = 1.019,
(25)
a tiny reduction in the enhancement factor as compared
with that in (20). (The net p-p reaction rate would increase slightly, however, because of an increase of X 2 in
the forefactor.) As we remarked earlier, larger Z elements (e.g., helium) in the outgoing channels of fusion
reactions have a better efficiency in the Coulomb coupling, resulting in an increase of Teff.
12
4.
C- 1 2 C Reactions in W h i t e Dwarfs
As another example, we consider the case of the 1 2 C12
C reactions (Fowler et al. 1975) in dense matter consisting of carbon (suffix 1) and magnesium (suffix 2), which
may proceed as
12
C+12C
.24
Mg + 13.931 MeV.
(26)
r< =
aikBT exp
Note that Tj depend only on ne and do not contain #j.
Substituting equations (29) and (30) in equation (11),
we obtain
H(0)
kBT
2/ocp(ri)
/OCP^)
(Zief
DskBT
(Z2e)2
2DskBT'
+'
(33)
This expression is independent of Xi, owing to the adoption of the linear-mixing law (29) in the equation of state
for the binary-ionic mixture. For p m = 2 x 109 g c m - 3
and T = 5 x 107 K, the quantity H(0)/kBT takes on the
value 85.08, irrespective of the relative abundance x\. In
principle, however, the value of this quantity should generally depend on the relative abundance of the elements
in the outgoing channels as well, owing to a departure
from the linear mixing law.
© Astronomical Society of Japan • Provided by the NASA Astrophysics Data System
No. 4]
5.
Nuclear Reactions in Dense Matter
Concluding Remarks
-kBT
Through an explicit formulation of the Coulombic
chemical potentials for electron-screened binary-ionicmixture plasmas, we have evaluated the thermodynamic
enhancement factors for the nuclear-fusion reactions in
dense matter, and have thereby illustrated the dependence of the resultant enhancement factors on the relative abundances of the elements in the outgoing channels
for the cases of the p-p reactions in the solar interior and
of the 1 2 C- 1 2 C reactions in a white-dwarf progenitor of a
supernova.
It has been remarked that the free-energy increments,
such as equation (14), should analogously influence the
rates of the /3 processes in dense matter. The whole picture concerning the thermodynamic effects on the p-p reactions may therefore be revealed only after such effects
on the /?-decay factors have been elucidated.
\n
+
( & ) - «
Fex(N1,N2;Ne),
(A1.3)
where FQ denotes the ideal-gas free energy for the electrons.
The screening potential for the reacting nuclei, associated with equation (Al.l), is defined as (Z = Z\)
H(r) =
^?-+kBTln[gu(r)}.
(A1.4)
We then calculate
lim exp
T- ab ->o
{Zef 1
rahkBT
(
V2
1\
(A1.5)
with
Q'
N-i = lim
CN-1~-
rab^0
The authors thank Dr. K. Langanke, Dr. J.W. Truran,
and Dr. H.M. Van Horn for useful discussions on this
and related subjects. The final parts of this research
by S.I. were carried out at the Aspen Center for Physics.
617
N
/n
f
driTreexp
~ ^
W
+
H
o\
i(^a,b)
(A1.6)
where
W(ri,---,rN;{re})
(Ze)2
rahkBT'
= Hint
Appendix 1. Derivation of Equation (10)
(A1.7)
Since
Since mi ^> rae, the thermal de Broglie wavelengths of
the reacting nuclei may be looked upon as being significantly smaller than the average interionic spacings ~ a in
the dense matter of astrophysical interest (e.g., Ichimaru
1994), so that N[= N\ + N2) ions in the system with volume V obey classical statistics, while quantum statistics
applies to Ne electrons. Assuming that the ions, a and 6,
belong to species " 1 " (i.e., the reacting nuclei) in equation (10), we calculate the radial distribution function for
the " 1 " particles as
""->-H)£/ino
xdriTr e exp [ - ^ ( #
i n
t + # 0 e ) | • (Al-1)
-fc B Tln (y^Ek)
= F o + ^ ( J V i - 2, N2 + 1; Ne),
(A1.8)
equation (10) follows from equations (A1.3)-(A1.5).
Appendix 2. Effective Coulomb-Coupling Parameter in Equation (15)
We calculate the effective Coulomb-coupling parameter Teff appropriate to the electron-screened hydrogenhelium plasmas in the solar interior, in such a way that
the expression for the interaction energy, E-int = Ea+Eie,
calculated in the RPA (see e.g., Ichimaru 1992) agrees
with that in the Debye-Hiickel approximation,
Here, r a b = | r a - r b | ,
r
JN
:
/
3/2
(N1 + N2)kBT
N
Y[drjTreexp
kBT
(Hint + HQ) ,
(A1.2)
is the configurational integral for a system of N ions
(with Ne electrons), Tr e designates the state sum for
the electrons, and H{nt and HQ refer to the Coulombinteraction Hamiltonian between the N + Ne particles
and the kinetic-energy Hamiltonian for the electrons, respectively. In the notation of equation (10), one obtains
the relation
2
eff
'
(A2.1)
in the limit of weak coupling.
We begin with the general expression for the interaction energy,
E»
<s--\ C dk
27rZiZje2y/nirij
[bij(k) — bijl,
k2 + k2e
(A2.2)
where Sij refers to Kronecker's delta and
© Astronomical Society of Japan • Provided by the NASA Astrophysics Data System
S. Ichimaxu and H. Kitamura
618
References
47re2ne(
kBT '
K(x —
(A2.3)
Assuming that the ions obey classical statistics, we calculate the structure factors in terms of the static densitydensity responses, Xij(fc,0), as (see e.g., Ichimaru 1992)
Sij{k) — -
kBT
(A2.4)
XijfaO)
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X*PA(M:) =
Ui
Sij k T^
B
47re2 ZiZjUiiij
k e0(k,0)(kBT)2
2
(A2.5)
where
eo(k,0) =
+
k2
k2
(A2.6)
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kBT
kr> =
(A2.7)
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Snt —
V
27r(Z 2 m + Z|yi 2 ) 2 e 4
kD + ke
kBT
(A2.8)
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