Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy
The cross section of the direct E0 pair emission has meaningful contribution to the total cross section of the ¹²C(α, γ)¹⁶O reaction at low energy ≤ 1:9MeV . E0 resonance emission and internal pair conversion have significant effect to the total cross section of the ¹²C(α, γ)¹⁶O reaction. In this pa...
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irk-123456789-1121232017-01-18T03:03:18Z Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy Tabassam, U. Mehboob, K. Ядерная физика и элементарные частицы The cross section of the direct E0 pair emission has meaningful contribution to the total cross section of the ¹²C(α, γ)¹⁶O reaction at low energy ≤ 1:9MeV . E0 resonance emission and internal pair conversion have significant effect to the total cross section of the ¹²C(α, γ)¹⁶O reaction. In this paper e⁺e⁻ paired emission has been focused on taking into account the angular correlation. E0 contribution is also significant in a presence of E1 and E2 transition, therefore e⁺e⁻ pair emission may not be neglected and has a signi cant effect on the total cross section in the case of the ¹²C(α, γ)¹⁶O reaction. Поперечний переріз прямої E0 емісії e⁺e⁻ - пар дає значний вклад у повний переріз для реакції ¹²C(α, γ)¹⁶O при низьких енергіях ≤ 1:9 МеВ. E0 резонансна емісія і внутрішня конверсія пар мають значний вплив на повний переріз реакції ¹²C(α, γ)¹⁶O. У даній роботі ми сконцентрувалися на врахуванні кутових кореляцій емісії e⁺e⁻ пар. E0 вклад також є суттевим поряд з E1- і E1 -переходами, і тому емісією e⁺e⁻- пар не можна нехтувати, вона має значний вплив на повний переріз у випадку ¹²C(α, γ)¹⁶O - реакції. Поперечное сечение прямой E0 эмиссии e⁺e⁻ - пар дает значительный вклад в полное сечение для реакции ¹²C(α, γ)¹⁶O при низких энергиях ≤ 1:9 МэВ. E0 резонансная эмиссия и внутренняя конверсия пар оказывают значительное влияние на полное сечение реакции ¹²C(α, γ)¹⁶O . В настоящей работе мы сконцентрировались на учете угловых корреляций эмиссии e⁺e⁻ пар. E0 вклад также является существенным наряду с E1 - и E2 -переходами, и поэтому эмиссия e⁺e⁻ - пар не может принебрегаться, она имеет значительное влияние на полное сечение в случае ¹²C(α, γ)¹⁶O - реакции. 2015 Article Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy / U. Tabassam, K.Mehboob // Вопросы атомной науки и техники. — 2015. — № 3. — С. 44-48. — Бібліогр.: 20 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk http://dspace.nbuv.gov.ua/handle/123456789/112123 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Tabassam, U. Mehboob, K. Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy Вопросы атомной науки и техники |
| description |
The cross section of the direct E0 pair emission has meaningful contribution to the total cross section of the ¹²C(α, γ)¹⁶O reaction at low energy ≤ 1:9MeV . E0 resonance emission and internal pair conversion have significant effect to the total cross section of the ¹²C(α, γ)¹⁶O reaction. In this paper e⁺e⁻ paired emission has been focused on taking into account the angular correlation. E0 contribution is also significant in a presence of E1 and E2 transition, therefore e⁺e⁻ pair emission may not be neglected and has a signi cant effect on the total cross section in the case of the ¹²C(α, γ)¹⁶O reaction. |
| format |
Article |
| author |
Tabassam, U. Mehboob, K. |
| author_facet |
Tabassam, U. Mehboob, K. |
| author_sort |
Tabassam, U. |
| title |
Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy |
| title_short |
Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy |
| title_full |
Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy |
| title_fullStr |
Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy |
| title_full_unstemmed |
Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy |
| title_sort |
discussion on importance of e⁺e⁻ pair emission in the ¹²c(α, γ)¹⁶o capture reaction below 1.9 mev energy |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2015 |
| topic_facet |
Ядерная физика и элементарные частицы |
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http://dspace.nbuv.gov.ua/handle/123456789/112123 |
| citation_txt |
Discussion on importance of e⁺e⁻ pair emission in the ¹²C(α, γ)¹⁶O capture reaction below 1.9 MeV energy / U. Tabassam, K.Mehboob // Вопросы атомной науки и техники. — 2015. — № 3. — С. 44-48. — Бібліогр.: 20 назв. — англ. |
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Вопросы атомной науки и техники |
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AT tabassamu discussiononimportanceofeepairemissioninthe12cag16ocapturereactionbelow19mevenergy AT mehboobk discussiononimportanceofeepairemissioninthe12cag16ocapturereactionbelow19mevenergy |
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2025-07-08T03:25:50Z |
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| fulltext |
DISCUSSION ON IMPORTANCE OF e+e− PAIR EMISSION IN
THE 12C(α, γ)16O CAPTURE REACTION BELOW 1.9MeV
ENERGY
U.Tabassam∗, K.Mehboob
Department of Physics, COMSATS Institute of Information Technology,
Chak Shahzad, Park Road, Islamabad, 4400, Pakistan
(Received June 2, 2014)
The cross section of the direct E0 pair emission has meaningful contribution to the total cross section of the
12C(α, γ)16O reaction at low energy ≤ 1.9MeV . E0 resonance emission and internal pair conversion have sig-
nificant effect to the total cross section of the 12C(α, γ)16O reaction. In this paper e+e− paired emission has been
focused on taking into account the angular correlation. E0 contribution is also significant in a presence of E1 and E2
transition, therefore e+e− pair emission may not be neglected and has a significant effect on the total cross section
in the case of the 12C(α, γ)16O reaction.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
Study of heavy nuclei reactions at higher energies
≥ 20MeV leads to the information about shape tran-
sitions and nuclear models. While, light nuclei in-
teractions at low energies ≤ 5MeV gives informa-
tion about the reaction mechanisms for astrophysical
purposes. The 12C(α, γ)16O is very important reac-
tion and is considered as the ’Holy Grail’ process in
the nuclear astrophysics [1-3]. Many experimental at-
tempts have been carried out to get better determina-
tion of 12C(α, γ)16O rate, but required precision has
not been achieved yet due to the lack of statistics [4].
Matei Catalin [5] has studied the cascade transition
through 6.05MeV state of the 16O nucleus, which has
been ignored in previous studies of the 12C(α, γ)16O
reaction. He has observed the first excited state in
16O over a wide range of energies and made the sub-
sequent fits for the cross section (S-factor). In his
experiment the 0+ → 0+ transition in the 16O nu-
cleus was analyzed for the first time in connection
with the 12C(α, γ)16O reaction cross section at he-
lium burning energies. But the statistics was very
weak and could not meet the required purpose [5].
They summed up the E1 and E2 transitions to get
the estimation of 6.05MeV transition. The experi-
mental set up consisted of the BGO array which does
not allow for the separation of primary transitions
to 6.05 and 6.13MeV states in 16O. The reason for
this is the fact that the energy resolution of these de-
tectors was about 100...150 keV . Fig.1 is excitation
function, where error bars are the data points that
start from 2MeV , below this there is only extrapo-
lation that was done using the R-matrix calculation.
They account a systematic error of 30%. This 30%
error is not accepted well in astrophysical relevance.
Also the S-factor, that they calculated, was very high.
Due to the moderate energy resolution of
BGO crystals and limited acceptance of the
separator, ERNA experiment was performed.
Fig.1. The R-matrix fit of the excitation function
for the E = 6.05MeV transition (Error bar). Dot
and line – E1 transition, small dots – E2 transition
[5]
D. Schuermann et al. [6] studied the radiative capture
reaction 12C(α, γ)16O in the energy range between
3.3MeV to 4.5MeV . This experiment focused in
particular on the cascade transition to 0+ state at
E = 6.05MeV in 16O. In this experiment γ-rays
were detected. The 6.05MeV transition has been
considered as a component accounting for up to 15%
∗Corresponding author E-mail address: uzma.tabassam@comsats.edu.pk Phone: +92 335 9145354
44 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2015, N3(97).
Series: Nuclear Physics Investigations (64), p.44-48.
of the 12C(α, γ)16O total cross section at astrophys-
ical energies. The experiment did not go beyond the
5MeV to avoid the high energy background. The cal-
culated S-factor was less than 1 keV b (Fig.2). Also
they concluded that 6.05MeV cascade transition
in 12C(α, γ)16O has no astrophysical relevance at
the stellar burning temperatures. Sum of 6.05MeV
and 6.13MeV amplitudes resulted in S-factor val-
ues, which were very close to the DRAGON data.
Fig.2. R-matrix analysis of 6.05MeV transition.
Circles are 6.05MeV data (E2) only, Black solid and
dashed lines are two best fits; dotted line is R-Matrix
calculation of E1 component [6]
These two discrepancies can be removed while study-
ing the emission of e+e− pairs at energies less than
2MeV . Snover and Hurd [7] estimated the direct
capture cross section for the E0 pair emission of re-
action at low bombarding energies. They compared
this transition with the E2 photon emission. By com-
paring them it was shown that E0 pair emission and
E2 photon emission are relating each with other. In
this case the pair emission cross section is of the or-
der of 10−6 of the total cross section at low energies.
This small factor of cross section is unimportant for
the determination of S34(0). They concluded that E0
pair emission and internal conversion are very negli-
gible relatively the direct single photon emission [7].
There is a theoretical analogy of 6.05MeV transition
to ground state given by G.Baur et al. [8]. They did
not support the E0 emission. The E0 emission is not
seen directly up to now in the 12C(α, γ)16O reaction,
there are theoretical evidences that do not support
the E0 emission.
In parallel some experimental and theoretical
studies have been carried out for capturing rays in
the 12C(α, γ)16O reaction at low energy. Gialanella
et al. [9] obtained the excitation function while ob-
serving E1 capture amplitude in the energy range be-
tween 1.32...2.99MeV . SchÄurmann et al. [10] have
measured the total cross section of 12C(α, γ)16O re-
action in energy range between 1.9...4.9MeV . They
also studied the ground state emission E1 and E2.
In this paper, a discussion on the 0+ → 0+ tran-
sition via e+e− pair transition at energy < 1.9MeV
has been carried out. A miniscule work has been
carried out at low energy level because experimental
facilities have not been introduced yet to carry out
the experimental studies. Therefore it is significant
to focus on the interactions at low energy level. This
paper aims to describe the importance of 0+ → 0+
transitions via e+e− pair transition. This transition
still has been neglected but it has worth in total cross
section measurements of 12C(α, γ)16O reaction. So
it is desirable to explain the transition at low energy
below 1.9MeV that may be helpful to avoid extrap-
olation.
2. E0 TRANSITION
E0 transitions in light nuclei might have a large
impact in nuclear astrophysics, since they may con-
nect 0+ states that cannot be connected by radiative
transitions. E0 transitions occur between two states
of the nucleus both of which have spin I = 0. As
we know from electromagnetism, there are no multi-
poles of order l = 0 in the radiation field. Hence no
radiative transition can take place. The transition
0+ → 0+ is strictly forbidden for electromagnetic
radiation. However, a transition is possible in which
the K electrons take over the energy [11, 12]. In this
case we must take into account the region of con-
figuration space for which the electron is within the
nucleus. Its contribution is negligible in general, but
it is the most important in the case considered here.
Since a transition of the 0+ → 0+ type does not pro-
duce any electromagnetic field outside the nucleus,
the energy transfer must take place inside. If the
energy difference between the nuclear states is larger
than 2mec
2, a new type of internal conversion can
occur. The energy can be transmitted to electrons in
the negative-energy states near the nucleus. These
states occur in the relativistic wave equation of the
electron. It is well known that Dirac’s theory of the
positron assumes that the states of negative kinetic
energy are occupied by electrons. The lifting of one of
these negative-energy electrons into a positive-energy
state appears as the creation of an electron-positron
pair. This process has been suggested by Oppen-
heimer et al. [13] and calculations have been done
by various authors [14, 15]. The probability of inter-
nal pair formation is larger than the probability of
ejection of a K shell electron if the available energy
is appreciably larger than 2mec
2 Internal pair for-
mation supplements ordinary internal conversion in
that the pair formation rate is the largest where the
internal conversion rate is the smallest, namely in
the region of low atomic number and high transition
energies.
3. THE CAPTURE REACTION 12C(α, γ)16O
The capture reaction 12C(α, γ)16O (Q = 7.16MeV )
takes place in the helium burning stage of red giant
stars and represents a key reaction of nuclear astro-
physics as it strongly influences the production of all
45
elements heavier than A = 16 as well as the stellar
evolution from the helium burning phase to the late
explosive stages. It is the regulator of C/O abun-
dance in the universe and influences the composition
of CO White dwarfs [10, 6]. The cross section of this
reaction at the Gamow energy E0 ∼ 300KeV deter-
mines the He burning time scale together with the
convection mechanism. The carbon abundance at
that stage has important consequences for the sub-
sequent evolution of various astrophysical scenarios,
e.g. a direct influence on type II supernova nucle-
osynthesis, the maximum luminosity, kinetic energy
of type I supernova nucleosynthesis and the cooling
sequence of CO White dwarfs [4, 8]. The possible role
of E0 emission has already been addressed by Baur-
Snover, as E0 emission is important in 12C(α, γ)16O
capture measurement since they are made by detect-
ing the emitted e+e− pairs. The major factor that
enhances the importance of E0 emission is that it
occurs by s-wave capture [8].
4. EXPLANATION
Many experimental works have been done on the cap-
ture cross section of 12C(α, γ)16O reaction consider-
ing the γ-ray emission but at the energies more than
1.9MeV for the contribution in the total cross sec-
tion. We will show theoretically the importance of
e+e− pair emission at low energy. A comparison of
the E0 and E2 emissions for the transitions between
individual levels of finite spin is given in [7]. While
the ratio of cross sections is given as [8]:
σE0
σE2
=
4π
5
fE0
fE2
|R00|2
|R02|2
, (1)
where fE0 and fE2 are given by [14, 15]:
fE0(E) =
e4
27(h̄c)6
b(S)(E −mc2)3(E +mc2)2 ,
fE2(E) =
4πe2
27(h̄c)5
E5 . (2)
Estimation on |R00|2
|R02|2 ratio tells that the capture takes
place at the nuclear radius, so at low interaction ener-
gies the effective radius is large. Thus to get the max-
imum advantage from the E0 transition we must add
a factor obtained in the following manner to approach
to the angular correlation term. From the above esti-
mates and working on the potential model calculation
of Woods-Saxon potential the SE0(0.3) factor calcu-
lated was:
SE0(0.3) = 0.02 keV b . (3)
This factor may be improved by taking into account
the angular correlation of emitted particles. The E0
direct capture occurs between the identical quantum
numbers j = 0 and even parity. According to the
selection rule there is no E0 transition, when l = 0
as given by the following relation [16]:
HL,M,i =
qi
mi
E0√
2iω
⟨
ΨL
∣∣∣e−ik⃗r⃗iε · pi
∣∣∣ΨH
⟩
−
−gi
qi
2mi
E0√
2c
⟨
ΨL
∣∣∣e−ik⃗r⃗iB · Si
∣∣∣ΨH
⟩
. (4)
Solving (4) only for the electric component gives us:
HEl
L,M,i = −qiE0√
2
(−ik)l−1
(l − 1)!
⟨
ΨL
1
l
∣∣rl−1
ik riε
∣∣ΨH
⟩
.
(5)
Here ri,k is the component of position of particle in
the direction of motion, ri,ε is the component of par-
ticle in the direction of electric field and angular mo-
mentum components are in the direction of magnetic
field. In order to estimate the E0 transition proba-
bility, a convenient formulation has been proposed in
[17]. E0 is possible through the internal conversion
process [18, 19]. The electromagnetic processes also
contribute to the astrophysical rate for 12C + α cap-
ture reaction at low energy. The phase space factor
for the emission of e+e− pairs is small but we can
improve the results by working under angular cor-
relation to get the better contribution of the cross
section at low energy. If the angular distribution be-
tween electron and positron is isotropic then, for a
high energy pair transition in a nuclide of low atomic
number, the energy distribution is given by:
N(E)dE ∝ E2(E0 − E)2dE , (6)
where N(E)dE is the number of electrons (positrons)
emitted with some kinetic energy between E and
E + dE and E0 = ET − 2mec
2, where ET is the
nuclear transition energy [20]. If the transition is, in
addition, 0+ → 0+ decay, then angular correlation is
given by:
W (θ) ∝ 1 + cos(θ) . (7)
Equations (6) and (7) tell us that the probability of
pair emission is maximum for an equal sharing of en-
ergy between electron and positron. By using the
Dirac equation and Coulomb distorted wave approxi-
mation we get the angular correlation for E0 conver-
sion as
d2η
dk d cos(θ)
=
1
2
dη
dk
[1 + ε cos(θ)] , (8)
where ε the anisotropy factor, and k is the kinetic
energy. Simplification gives us:
dη
d cos(θ)
= [1 + cos(θ)] . (9)
This relation shows that the emission of electron and
positron pairs can be enhanced by working under
angular correlation and this is possible at smaller
values which favor the 6.05MeV e+e− pairs at low
energy. If the interaction plane is set in such a way
as to get the jet of particles at some lower angle
rather than transverse to the beam axis, then the
probability may be increased to favor the pair emis-
sion Fig.1. For this reason a strong experimental
setup that surrounds the 4π angle of the detection
system is required to get the better contribution of
electron and positron pairs. To increase the S factor
46
for E0 pair emission at Gamow energy, it will be rea-
sonable to take into account the angular correlation
factor, so as to increase the pair emission probability.
Fig.3. Schematic diagram to achieve enhanced prob-
ability of e+e− pairs under angular correlation
Fig.3 explains that the cone of particles in transverse
direction to the plane of interaction gives less prob-
ability to get an event as represented by Gaussian
distribution. While if the interaction plane is inclined
to certain angle no matter what the geometry is, then
the probability to get events has increased as clearly
indicating by asymmetrical shape. In this asymmet-
rical shape the points 1 and 2 show the coincidence
to get the desired events. It means if we count one
event transverse to beam axis then it may increase
to two when interaction is at some angle.
5. CONCLUSIONS
So we can conclude that the electromagnetic pro-
cesses do contribute significantly to the astrophysi-
cal rate for the 12C(α, γ)16O reaction. Extrapolation
of data to lower c.m. energies for the reaction may
be avoided using relation (9), which shows the sig-
nificance of the e+e− pair emission below 1.9MeV
energy.
ACKNOWLEDGEMENTS
We would like to acknowledge the COMSATS Insti-
tute of Information Technology, Islamabad, Pakistan,
which provided us the environment to write this ar-
ticle.
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PHYSICAL REVIEW C. 1990, v.42, N6, p.2632.
47
ÄÈÑÊÓÑÑÈß Î ÂÀÆÍÎÑÒÈ ÝÌÈÑÑÈÈ e+e− - ÏÀÐ Â ÐÅÀÊÖÈÈ ÇÀÕÂÀÒÀ
12C(α, γ)16O ÏÐÈ ÝÍÅÐÃÈßÕ ≤ 1.9ÌýÂ
Ó.Òàáàññàì, Ê.Ìåõáóá
Ïîïåðå÷íîå ñå÷åíèå ïðÿìîé E0 ýìèññèè e+e− - ïàð äàåò çíà÷èòåëüíûé âêëàä â ïîëíîå ñå÷åíèå äëÿ ðå-
àêöèè 12C(α, γ)16O ïðè íèçêèõ ýíåðãèÿõ ≤ 1.9ÌýÂ. E0 ðåçîíàíñíàÿ ýìèññèÿ è âíóòðåííÿÿ êîíâåðñèÿ
ïàð îêàçûâàþò çíà÷èòåëüíîå âëèÿíèå íà ïîëíîå ñå÷åíèå ðåàêöèè 12C(α, γ)16O.  íàñòîÿùåé ðàáîòå
ìû ñêîíöåíòðèðîâàëèñü íà ó÷åòå óãëîâûõ êîððåëÿöèé ýìèññèè e+e− - ïàð. E0 âêëàä òàêæå ÿâëÿåòñÿ
ñóùåñòâåííûì íàðÿäó ñ E1- è E2-ïåðåõîäàìè, è ïîýòîìó ýìèññèÿ e+e− - ïàð íå ìîæåò ïðèíåáðåãàòüñÿ,
îíà èìååò çíà÷èòåëüíîå âëèÿíèå íà ïîëíîå ñå÷åíèå â ñëó÷àå 12C(α, γ)16O - ðåàêöèè.
ÄÈÑÊÓÑIß ÏÐÎ ÂÀÆËÈÂIÑÒÜ ÅÌIÑI� e+e− - ÏÀÐ Â ÐÅÀÊÖI� ÇÀÕÂÀÒÓ
12C(α, γ)16O ÏÐÈ ÅÍÅÐÃIßÕ ≤ 1.9ÌåÂ
Ó. Òàáàññàì, Ê.Ìåõáóá
Ïîïåðå÷íèé ïåðåðiç ïðÿìî¨ E0 åìiñi¨ e+e− - ïàð ä๠çíà÷íèé âêëàä ó ïîâíèé ïåðåðiç äëÿ ðåàêöi¨
12C(α, γ)16O ïðè íèçüêèõ åíåðãiÿõ ≤ 1.9ÌåÂ. E0 ðåçîíàíñíà åìiñiÿ i âíóòðiøíÿ êîíâåðñiÿ ïàð ìàþòü
çíà÷íèé âïëèâ íà ïîâíèé ïåðåðiç ðåàêöi¨ 12C(α, γ)16O. Ó äàíié ðîáîòi ìè ñêîíöåíòðóâàëèñÿ íà âðàõó-
âàííi êóòîâèõ êîðåëÿöié åìiñi¨ e+e− - ïàð. E0 âêëàä òàêîæ ¹ ñóòòåâèì ïîðÿä ç E1- i E2-ïåðåõîäàìè,
i òîìó åìiñi¹þ e+e− - ïàð íå ìîæíà íåõòóâàòè, âîíà ì๠çíà÷íèé âïëèâ íà ïîâíèé ïåðåðiç ó âèïàäêó
12C(α, γ)16O - ðåàêöi¨.
48
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