The rate of breakup of helium nuclei by particle induced reactions is computed. It is shown that the rate is determined by the endothermic reaction p + 4He  3He + d, becoming effective at kT approx. few MeV. It is suggested that matter having been processed to these temperatures will be depleted in helium and in the elements C, N, O, and Ne

Gould, R. J.

English

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207
OG 8.2-20
BREAKDOWN OF HELI_YM NUCLEI IN MATTER PROCESSED
NEAR BLACK HOLES
Robert J. Gould
Physics Department B-019
Univ. of Calif., San Diego
La Jolia, CA 92093 USA
ABSTRACT
The rate of breakup of helium nuclei by particle-induced
reactions is computed. It is shown that the rate is
determined by the endothermic reaction p + 4He * 3He + d,
becoming effective at kT > few MeV. It is suggested
that matter having been processed to these temperatures
will be depleted in helium and in the elements C, N, O,
and Ne.
i. Introduction. In recent years considerable interest has developed
in relativistic plasmas in which the electron gas is highly relativis-
tic (0 = kT/mc 2 _ i0). At these temperatures the ions would be non-
relativistic but capable of undergoing endothermic reactions. In a
recent paper [i] devoted to nuclear-force effects in p-p scattering, it
was suggested that helium nuclei will break down in the following series
of endothermic reactions:
p + 4He + 18.353 MeV + 3He + d (i)
p + 3He + 5.493 MeV + d + p + p (2)
p + d + 2. 225 MeV + p + p + n (3)
Because of its significantly larger threshold, the reaction (i) would be
the slowest and therefore determine the rate of the overall helium-de-
struction process. Once the helium is broken down to hydrogen, any sub-
sequent buildup would be very slow because of the small cross section
for the fundamental p-p reaction p + p . d + e+ + re" The plasma would
then remain essentially pure hydrogen. Other elements would also be
broken down (see [i]), such as the abundant medium-weight C, N, O, Ne,
although large Coulomb barriers are involved, since the threshold ener-
gies for the breakup reactions are much larger than in reaction (i).
The most elementary estimate of the characteristic breakdown
temperature is obtained [i] bysetting kT _ O.iEb, where E. is the
binding energy difference (18,353 MeV) in reaction (i). The _actor 0.i
arises since the equation for the breakdown temperature will be of the
form, characteristic of ionization and dissociation phenomena,
Eb/kT
e _ N = large number . (4)
The logarithm of any large number is _ i0, and this yields for the
breakdown temperature:
Ob(He) _ 4 (5)
2. Calculation of Breakup Reaction. The calculation of the character-
https://ntrs.nasa.gov/search.jsp?R=19850026603 2020-03-20T16:48:09+00:00Z
208
OG 8.2-20
istic breakdown temperature is really a non-equilibrium problem requir-
ing a knowledge of the density (n) of the plasma and the availableP
time (T). If r = dN/dVdt is the number of reaction (i) per unit
volume per unit time, the breakdown temperature will be computed from
rT = np (6)
The rate r will be proportional to n 2 and an integral over the reac-
tion cross section: P
= ½ np > (7)r 2 <v r ,
where vr is the relative velocity of the reacting protons.
The problem is in the correct use of the reaction cross section
a and in the extrapolation of its experimental values to low energies.
For in this case of a reaction with a threshold energy and reactants
with a Maxwellian velocity distribution, the main contribution to
< ov r > will come from c.m. energies just above threshold. However,
then the Coulomb effects in the cross section will be very important,
especially for the outgoing particles in the reaction. To include these
effects the experimental cross section [2] has been fit to a form
a = a C2C '2 , (8)
nuc
where °nu is a "nuclear part" and C2 and C '2 are (incoming andc
outgoing) Coulomb factors of the form [3]
C2 = 2_n/(e2_-l) (9)
for the incoming and outgoing particles. Here n = ZlZ2(Ry_/E)i/2, in
terms of the charges zle and z2e of the particles involved and
Ry_ = (_/m)Ry is the Rydberg energy corresponding to the reduced mass
of the pair of nuclei. In terms of the c.m. energy (before reacting)
and the threshold energy Et for reaction (i), anu c is found to have
the form
Onu c = Onuc_ t [i- as(E-Et)/Et] (i0)
with o = 93mb and a = 0.148.
nuc-t s
Due primarily to the Coulomb factor C '2 for the outgoing
particles, the integrand in < avr > is found to have a "Gamow peak"
where most of the contribution is centered. An asymptotic evaluation
of the integral yields the following set of breakdown temperatures for
given values of Tnp:
0b(He) 2 3 4 5 6 7 8
_n (yr cm -3) 1.4xlO 15 3.8x1012 2..lxlO 11 3.8x1010 1.2xlO 10 5.4x10 9 3.0xlO 9
P
That is, the original estimate [i] Ob(He ) _ 4 is reasonable, although
the result does depend logarithmically on Tn .P
Although no details were given of the method of calculation,
Aharonian and Sunyaev [4] have also computed the rate of reaction (i)
in the high-temperature range. Their lowest T corresponds to the high-
est value in tile above table and the results agree. However the domain
2O9
OG 8.2-20
@ < 5 is probably most relevant, since it is especially important to
know when the process begins to be effective as the temperature rises.
Further details of the calculations described in the present work may be
found in a separate paper [5]. For example, competing reactions have
been investigated, such as
p + 4He + 20.578 MeV --+ 3He + p + n
(ii)
p + 4He + 19.814 MeV --+ 3H + p + p
These are found to contribute only about 1% as much as reaction (i), due
essentially to the smaller cross sections for the reactions (ii) at low
energies (just above threshold); this is a phase space effect.
2
3. Discussion. In a strong gravitational potential, say, $ _ 0.i c ,
it can be expected that a temperature kT _ 0.1 Mc 2 _ i00 MeV can be
attained (M = nucleon mass). This potential (and temperatures) could
occur in regions around black holes, for example, and is well beyond
that necessary to break up He and other nuclei. Matter that has
attained this temperature can then be expected to be pure hydrogen. How-
ever, it might not be observed until after it has cooled. We would then
see a recombination-cascade spectrum with only hydrogen lines after cool-
ing to _ 105K. While at _ 107K there would be a pure bremsstrahlung x-ray
spectrum without x-ray lines from excitations of one- and two-electron
ions from the K-shell.
4. Acknowledgement. This research was supported by NASA through
Grant NGR 05 005 004.
References
[i] R. J. Gould, Astrophys. J. 263, 879 (1982).
[2] J. P. Meyer, Astron. Astrophys. Suppl. 7, 417 (1972).
[3] N. F. Mott and H. S. W. Massey, Theory of Atomic Collisions, 3rd
Ed. (Oxford: Oxford Univ. Press, 1965).
[4] F. A. Aharonian and R. A. Sunyaev, Mon. Not. Roy. Astron. Soc.
210, 257 (1984).
[5] R. J. Gould, Breakup of Helium Nuclei in High-Temperature Plasmas
(Sub. to Nucl. Phys.)


Breakdown of helium nuclei in matter processed near black holes

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