Study of reaction and decay using densities from relativistic mean field theory

0.28 MB·English

by  G. Gangopadhyay

Study of reaction and decay using densities from
relativistic mean field theory
G. Gangopadhyay∗
2 Department of Physics, University of Calcutta, Kolkata - 700009, INDIA
1
0 Relativisticmeanfieldcalculationshavebeenperformedtoobtainnucleardensitypro-
2 file. Microscopic interactions have been folded with the calculated densities of finite
nucleitoobtainasemi-microscopicpotential. Lifetimevaluesfortheemissionofproton,
n
alphaparticles andcomplexclustershavebeencalculated intheWKBapproachassum-
a
J ing a tunneling process through the potential barrier. Elastic scattering cross sections
have been estimated for proton-nucleus scattering in light neutron rich nuclei. Low en-
2
ergy proton reactions have been studied and their astrophysical implications have been

discussed. The success of the semi-microscopic potentials obtained in the folding model
]
h with RMF densities in explaining nuclear decays and reactions has been emphasized.
t
-
l
c I. INTRODUCTION are scarce and difficult to obtain in near fu-
u
ture. For example, low energy reactions are
n
veryimportantfromtheastrophysicalpointof
[
Relativisticmeanfield(RMF)approachhas
view. In astrophysical environments, neutron
1 proved to be very successful in explaining and proton reactions are the keys to nucle-
v different features of stable and exotic nuclei
osynthesis of heavy elements. However, very
1 suchasgroundstatebindingenergy,deforma-
often,thetargetnucleiarenotavailableinter-
1 tion, radius, excited states, spin-orbit split-
4 restrial laboratories and one needs to depend
ting, neutron halo, etc[1]. Particularly, the
0 on theoretical inputs.
radius and the nuclear density are known to
. The presentation has been arranged in the
1 bewellreproducedinthismodel. Innucleifar
0 following manner. In Section 2, the method
away from the stability valley, the single par-
2 of constructing the potential is outlined. The
ticlelevelstructureundergoescertainchanges
1 next section presents a few theoretical results
in which the spin-orbit splitting plays an im-
: for density, followedby selected results of cal-
v portant role. Being based on the Dirac La-
i culation for decays involving emission of pro-
X grangian density, and thus naturally incorpo-
ton, alpha and complex clusters. We also
ratingthespindegreeoffreedom,RMFispar-
r briefly discuss the results of elastic proton
a ticularly suited to investigate these nuclei.
scattering in neutron rich nuclei. A few low
RMF calculations have been found to pro- energyprotonreactionsin A=60−80 region
vide good description of densities, particu- have also been studied and astrophysical im-
larly at large radii. Processes, such as pro- portanceofsuchreactionshasbeendiscussed,
ton and alpha decay, low energy scattering Finally, we summarize our results.
and reaction, probe this part of the nuclear
volume. Thus, a good description of such
processes may be expected if we use micro-
II. METHOD
scopic nucleon-nucleon (NN) interactions and
densities from RMF calculations to construct
nucleon(nucleus)-nucleuspotentials. This will There is a large body of work for the top-
beveryusefultoextendanycalculationtoar- ics that have been presented in this talk. In
eas far from the stability valley, where data thissection,werestrictourselvestothesalient
points of the method of calculation that have
been followed to obtain the semi-microscopic
potentials used in the present work.
∗Electronicaddress: [email protected] Theoretical density profiles have been ex-
tracted from RMF calculations. In this ap- as C = 2.07 and β = 1.624 fm2. We have
proach, nucleons interact via exchange of a adopted all the standard parameters in our
number of mesons. There are different vari- calculation without any modification. This
ations of the Lagrangian density as well as interaction has been employed widely in the
a number of different parametrizations. In study of nucleon nucleus as well as nucleus
the present work we have employed the FSU nucleus scattering and radioactivity.
Gold[2]Lagrangiandensity. Itcontains,apart In the JLM potential[6], finite range of the
from the usual component describing a sys- interaction has been introduced by including
tem of nucleons interacting via exchange of a Gaussian form factor[7]. Here, we have
mesons, nonlinear terms involving self cou- adopted the global parameters for the inter-
pling of the scalar-isoscalar and the vector- action and the default normalizations[7].
isoscalar meson, as well as coupling between The semi-microscopic nuclear potentials
the vector-isoscalar meson and the vector- have been obtained by folding the interac-
isovector meson. tions with the microscopic densities obtained
In the conventional RMF+BCS approach in the RMF calculation. The Coulomb po-
foreven-evennuclei,theEuler-Lagrangeequa- tential has similarly been obtained by folding
tionsaresolvedundertheassumptionsofclas- the Coulombinteractionwith the microscopic
sicalmesonfields,timereversalsymmetry,no- protondensities. Thespin-orbitpotentialwas
sea contribution, etc. Pairing is introduced chosen from the Scheerbaum prescription[8].
undertheBCSapproximation. Sinceaccuracy The total potential consists of the nuclear
ofthenucleardensityisveryimportantinour part, the Coulomb potential as well as the
calculation, we have solved the equations in centrifugal potential. We have not included
co-ordinate space. The strength of the zero the contributionofisovector(Lane)potential.
range pairing force is taken as 300 MeV-fm However, we expect its effect to be small.
for both protons and neutrons. These values
have been chosen to represent a good fit for
the binding energy values. In nuclei contain- III. RESULTS
ing odd number of neutrons or protons, the
taggingapproximationhasbeen usedto spec- A. Density
ify the level occupied by the last odd nucleon
of either type. We have observed that mod- The charge density has been obtained from
eratevariationsofthe pairingstrengthdo not the point proton density ρ by convoluting
p
influence the results to any great extent. with a Gaussian form factor to account the
Effective NN interactions, such as density finite sizeofthe proton. We plotinFig. 1the
dependent M3Y (DDM3Y)[3–5], or that of calculated charge density for 62Ni and 66Zn
Jeukenne,LejeuneandMahaux(JLM)[6]may as representatives of our results. One can see
be used to construct the nucleon (nucleus)- that the theoretical and experimental values
nucleus potential. Both the interactions men- agreeverywell,particularlyatlargerradiival-
tioned above have been derived from nuclear ues,whichistheregionexpectedtocontribute
matter calculation and have been applied in to the optical potential at low energy. Other
finite nuclei with success. nuclei also show similar agreement.
The DDM3Y interaction[3, 4] is obtained
from a finite range energy independent M3Y
interaction by adding a zero range energy de- B. Decay
pendent pseudopotential and introducing a
density dependent factor. The density depen- Lifetimevalueshavebeencalculatedfordif-
dencemaybechosenasexponential[3]orbeof ferent types of decays, i.e. proton, alpha and
the form C(1−βρ2/3)[4]. The constants were cluster radioactivity. We employ the super-
obtained from nuclear matter calculation[5] asymmetricfissionmodel(SAFM).The decay
3) 10-1
-m TABLE I: Proton decay half lives (T) for spher-
ef 10-2 62Ni 66Zn ical proton emitters with even neutron number.
y ( Results are for the DDM3Y interaction. The an-
nsit 10-3 gular momentum of the proton involved is given
e
e d 10-4 by l.
arg Nucleus l log10T(s)
Ch 10-5 (~) Exp. Present Ref[11]
0 4 8 0 4 8 105Sb 2 2.049+0.058 2.27(46) 1.97(46)
−0.067
Radius (fm) Radius (fm) 109I 2 −3.987+0.020 −4.03(4) −4.25
−0.022
113Cs 2 −4.777+0.018 −5.34(4) −5.53
−0.019
FIG. 1: Calculated charge density in 62Ni and 145Tm 5 −5.4097+0.109 −5.20(6) −5.14(6)
−0.146
66Zn (solid lines) compared with experimental 147Tm 5 0.591+0.125 0.98(4) 0.98(4)
−0.175
measurements (filled circles). 147Tm* 2 −3.444+0.046 −3.26(6) −3.39(5)
−0.051
151Lu 5 −0.896+0.011 −0.65(3) −0.67(3)
−0.012
151Lu* 2 −4.796+0.026 −4.72(10) −4.88(9)
−0.027
155Ta 5 −4.921+0.125 −4.67(6) −4.65(6)
productisassumedtotunnelthroughthebar- −0.125
157Ta 0 −0.523+0.135 −0.21(11) −0.43(11)
riercreatedbytherepulsiveCoulomb(andthe −0.198
161Re 0 −3.432+0.045 −3.28(7) −3.46(7)
centrifugal) potential and attractive nuclear −0.049
161Re* 5 −0.488+0.056 −0.57(7) −0.60(7)
potential, constructed by folding. The prob- −0.065
165Ir* 5 −3.469+0.082 −3.52(5) −3.51(5)
ability of barrier penetration has been calcu- 171Au 0 −4.770+−00..118050 −4.84(15) −5.02(15)
latedintheWKBapproximation. Theassault 171Au* 5 −2.654+−00..015541 −3.03(4) −3.03(4)
frequency is obtained from the zero-point vi- 177Tl 0 −1.174+−00..109610 −1.17(25) −1.36(25)
−0.349
bration energy[9] that, in turn, has been cal- 177Tl* 5 −3.347+0.095 −4.52(5) −4.49(6)
−0.122
culated from the Q-values. Unlike some other 185Bi 0 −4.229+0.068 −5.33(13) −5.44(13)
−0.081
works,wehavenotnormalizedtheopticalpo- 112Cs 2 −3.301+0.079 −2.93(11) −3.13
−0.097
tential, and have used the values that were 150Lu 5 −1.180+0.055 −0.59(4) −0.58(4)
−0.064
obtainedfromnuclearmattercalculations,for 150Lu* 2 −4.523+0.620 −4.24(15) −4.38(15)
−0.301
the decay studies. 156Ta 2 −0.620+0.082 −0.22(7) −0.38(7)
−0.101
Many calculations exist for proton 156Ta* 5 0.949+−00..110209 1.66(10) 1.66(10)
radioactivity[10]. Our results for the proton 160Re 2 −3.046+−00..007556 −2.86(6) −3.00(6)
164Ir 5 −3.959+0.190 −3.95(5) −3.92(5)
radioactivity half life calculation, in nuclei −0.139
166Ir 2 −0.824+0.166 −0.96(10) −1.11(10)
where such decay has been observed from −0.273
166Ir* 5 −0.076+0.125 0.22(8) 0.21(8)
low energy states, are tabulated in Table I. −0.176
For comparison, we have also presented the
results for the DDM3Y interaction of Basu et
al.[11], who have used a simple phenomeno- 112,113Cscouldnotbereproducedwithoutthe
logical form of the densities. We have also inclusion of deformation effects. For example,
shown the uncertainties in the calculated half a deformation of the order of β ∼0.05−0.15
life values corresponding to the errors in the was judged to be essential to explain the ob-
measured Q-values within parentheses. Our served data. However, our calculation repro-
results deviate from experimental measure- ducesthedatafor109Iwithconsiderableaccu-
ments in 147Tm, 150Lu and 156Ta, and more racy. In the deformed calculations, it is usu-
prominently in 185Bi and 177Tl∗, the last two ally assumed that the deformationof the par-
results being off by an order of magnitude. entandthedaughternucleiareidentical. The
One can show that the last two discrepancies daughter in this particular case is 108Te. Te
canbe explainedas the effect ofconfiguration nuclei are well known vibrational nuclei with
mixing[10]. very small deformation. One possibility may
It was often suggested that the half life bethatthedeficiencyoftheWoodsSaxonpo-
values for proton radioactivity of 109I and tential far away from the stability valley was
responsible for the failure of the earlier cal- study emission of complex clusters [15, 16].
culations. We also stress that our results are Basu [17] has studied the nuclear cluster ra-
nearly identical for both the NN interactions. dioactivityalsointheframeworkofSAFMus-
Theresultsfor113Csandtosomeextent112Cs ing a phenomenological density and the real-
are not reproduced so well, which may be an istic M3Y interaction. Bhagwat and Gamb-
effect of deformation. However, in none of hir [15] have used densities from RMF calcu-
them do we have an order of magnitude dis- lations. It has been suggested[18] that in the
agreement between theory and experiment as case of decay of heavy clusters, the spectro-
obtained in certain other calculations. scopic factor may scale as
AlphadecayhavebeenstudiedinRefs. [12–
14]. Foremissionofcomplexproducts,thereis
S =(S )(A−1)/3 (1)
anotherfactor,thespectroscopicfactor,intro- α
duced to incorporate the preformation prob-
where A is the mass of the heavy cluster and
ability. It contains the nuclear structure ef-
S is the spectroscopic factorfor the α-decay.
α
fects, and may be thought as the overlap be-
Thus a plot of log S against A should be
tween the actual ground state configuration 10
a straight line. In Fig. 3, we have plotted
of the parent and the configuration described
the negative of log S for the decays where
by the complex decay product coupled to the 10
both the parent and the daughter are even-
ground state of the daughter. Obviously, it is
even nuclei against the mass number of the
expected to be much less than unity as there
cluster and plotted a best fit line. One can
are contributions from many other configura-
see that the points fall nearly on a straight
tions other than the one mentioned above.
line.
In a small mass region, we do not expect
the spectroscopic factor to vary to any large
20
extent. For example,in the superheavynuclei
presented in Fig. 2, we have taken a constant
16
value1.4×10−2forallthedecaysfromafitof
the half life values[12]. In some other calcula- S 12
tions, the spectroscopicfactors have been cal- 10
g
culated from theory or phenomenological for- -lo 8
mulas have been obtained for them[13].
4
Exp 0
1.5 Calc 0 5 10 15 20 25 30 35
A
0.5
T FIG. 3: Negative of logarithm of spectroscopic
10 -0.5 factors (S) as a function of cluster mass number
g
lo -1.5 A for even-evenparentsand daughters.
-2.5
-3.5
282 284 286 288 290 292 294 C. Elastic scattering
A
Elastic proton scattering in inverse kine-
FIG.2: Calculated andexperimentalhalflifeval- matics provides a test for the calculated den-
ues in superheavynuclei sities, particularlyin absence ofelectronscat-
tering data. Density information is not avail-
The above method may be extended to able in exotic neutron richnucleiin very light
mass region. Elastic scattering cross sections varied the rates of certain reactions by a fac-
for scattering of these nuclei from proton tar- torofonehundred. Obviously,this makesthe
get have been calculated[19] with the opti- results uncertain to some extent and affects
cal model potential generated in the semi- the final abundance.
microscopic approach. In Fig. 4, the mea- Afullymicroscopiccalculationmaybeused
suredpartialcrosssectionsforprotonscatter- to estimate the rates to reduce the above un-
ing of 6,8He in inverse kinematics have been certainty. A consistent framework for calcu-
comparedwiththeory. Therealandtheimag- lation may be constructed based on micro-
inary parts of the potential have been chosen scopic densities and may be extended to un-
as0.8times and0.2times the foldedvalue for known mass regions with confidence. In the
the DDM3Y potential. One can see that the presentwork,wehavetriedtocalculatethere-
experimental values are nicely reproduced. actionratestakingdensitiesfromapurelymi-
croscopic model, i.e. RMF. We have already
mentioned that RMF is particularly suitable
D. Low energy proton reactions of to describe nuclei far away from the stability
astrophysical significance valleywhereexperimentalknowledgeisscarce.
Asemi-microscopicopticalpotentialobtained
by folding an appropriate microscopic NN in-
Capture and charge exchange reactions at
teractionis expectedto be more accurateand
very low energy play a very important role
may do away with the necessity of any arbi-
in nucleosynthesis. Particularly, rapid pro-
trary variation in the reaction rates.
ton capture (rp) process in explosive nucle-
osynthesis is a basic ingredient in driving the However,eveninasemi-microscopicoptical
abundance along the N = Z line[20]. As this potential, there often remain certain parame-
processhastoovercomealargeCoulombbar- ters that can be fixed only after comparison
rier, it can occur only at a higher tempera- with experiment. We have compared the re-
ture range. For example, X-ray bursts pro- sultsforanumberof(p,γ)and(p,n)reactions
vide large fluxes of protons at peak tempera- in mass region A = 60−80 for which exper-
turesaround1-2GKandareexpectedtoplay imental cross sections are available. This has
a significant role in the creation of nuclei up helped us in determining a set of parameters
to mass 110. for this mass region.
The rp-process proceeds along the N = Z The reaction calculations have been per-
line in mass 60-80 region. In nature, the formedwiththecomputercodeTALYS1.2[22]
important proton capture reactions may in- assuming spherical symmetry for the target
volve certain nuclei as targets which are not nuclei. Therealpartofthepotentialhasbeen
available to us. Hence, experimental infor- obtained by normalizing the folded DDM3Y
mation is difficult, if not impossible, to ob- potential by a factor of 0.7, while the imag-
tain, at least in near future. In such a sit- inary part, by a factor of 0.1, so as to ex-
uation, one has to rely on theory for the re- plain the S-factors obtained in the above ex-
action rates. Cross sections were calculated periments. WehaveemployedthefullHauser-
for proton capture reactions in mass 60-80 Feshbachcalculationwith transmissioncoeffi-
region using semi-microscopic optical poten- cients averagedovertotalangularmomentum
tials in the local density approximation with values andwith correctionsdue to width fluc-
phenomenologicaldensityprescriptions. How- tuations. The gamma ray strength has been
ever, far from the stability valley, these pre- calculated in the HFB and HF+BCS model.
scriptions may notrepresentthe actualdensi- In Figs. 5 and 6, we plot the results of our
ties very well, leading to considerable uncer- calculations for S-factors and compare them
tainties in the reaction rates. Very often, the with experimental values for (p,γ) reactions
reactions rates are varied by a large factor to for various targets in mass 60-70 region. The
study their effects. For example, Schatz[21] energy relevant to the rp-process in this mass
1000
mb/sr) 100 DDMJL3MY DDMJL3MY
n( p(6He,6He)p p(8He,8He)p
o
cti 10
e
oss s 1
cr
al 0.1
arti
P 0.01
15 30 45 60 75 15 30 45 60
CM Angle in degrees CM Angle in degrees
FIG.4: Partialcrosssectionforelasticprotonscatteringininversekinematics. Theprojectileenergies
of 6He and 8Heare 71 MeV/A and 72 MeV/A, respectively.
region lies between 1.1 to 3.5 MeV. As the Once the parameters have been fixed, we
cross-section varies very rapidly at such low employ them to calculate the rates of a num-
energy, a comparison between theory and ex- ber of astrophysically important proton re-
perimentisratherdifficult. Theusualpractice actions. Some very important N = Z nu-
in low energy nuclear reaction is to compare clei, which have the highest abundance in an
anotherkeyobservable,viz. theS-factor. Itis equilibrium in a chain, are termed as wait-
given by ing points[20] for the chain. These nuclei
have negative or small positive Q-values for
S(E)=Eσ(E)e2πη (2) proton capture. An equilibrium between the
(p,γ)/(γ,p) processes is established and the
where E is the energyin centre ofmassframe
rp processmayhaveto waitfor beta-decayor
in keV, σ(E) indicates reaction cross-section
α-capture to proceed to heavier nuclei. Cer-
in barn and η is the Sommerfeld parameter
tain N =Z waiting point nuclei with A<80,
with
viz. 64Zn, 68Se, 72Kr, and 76Sr have long half
µ lives,theirtotallifetimebeinglargecompared
2πη =31.29Z Z (3)
p trE to the time scale of typical X-ray bursts (10-
100 sec). Thus, they may produce a bottle-
Here, Z and Z are the charge numbers of neck in the rp-process that would slow down
p t
the projectileandthe target,respectivelyand the rate of hydrogen burning and necessitate
µ is the reducedmass (in amu). The quantity extended burst tails unless two proton cap-
S-factor varies much more slowly than reac- turecanreducethesehalflivesandbridgethe
tion cross-sections as the exponential energy waiting points. X-ray burst model calcula-
dependence of cross-section is not present in tionsarethereforeparticularlysensitivetothe
it. For this reason, we calculate this quantity rates of proton capture for these nuclei. We
andcompareitwithexperimentallyextracted have used the microscopic approach, outlined
values. inthepresentwork,tocalculatetherateswith
In Fig. 7, we plot the results for cross sec- an aim to study the bridging of the waiting
tions to populate the ground state and the point nuclei.
first two excited states for 63,65Cu(p,γ) reac-
tions. Resultsforprotoncapturebyothernu- For calculation of proton capture at wait-
clei as well as charge exchange reactions have ing points, a small network, which includes
also been calculated in this mass region and the following processes, has been employed.
found to agree reasonably well with experi- ThewaitingpointnucleuswithZ =N,which
mental observations[23–27]. acts as a seed, may capture a proton. The re-
100 62Ni 64Ni 63Cu 65Cu
b)
V-
Me 10-1
60
1
ctor ( 10-2
a
S-f
10-3
1.0 2.0 3.0 2.0 3.0 2.0 3.0 2.0 3.0
E (MeV) E (MeV) E (MeV) E (MeV)
cm cm cm cm
FIG. 5: S-factors extracted from experimental measurements compared with theory for 62,64Ni and
63,65Cu targets. Solid and dashed lines indicate respectively the results of the HF+BCS and HFB
approaches for level density and E1 gamma strength.
101
b) 64Zn 66Zn 67Zn 68Zn
eV- 100
M
60
1
ctor ( 10-1
a
S-f
10-2
1.0 2.0 3.0 2.0 3.0 2.0 3.0 2.0 3.0
E (MeV) E (MeV) E (MeV) E (MeV)
cm cm cm cm
FIG.6: S-factorsextractedfromexperimentalmeasurementscomparedwiththeoryfor64,66−68Zn. See
caption of Fig. 5 for details.
sulting nucleus, with Z = N +1, may either Rauscher et al.[28] by dash-dotted lines. The
capture another proton or undergo photodis- effects ofthe uncertainties onthe halflife val-
integration emitting a proton to go back to uesintheQ-valueshavebeenindicatedinthe
theseednucleus. ThenucleuswithZ =N+2 figuresby dotted lines. The half life decreases
may also undergo photodisintegration. In ad- and possibly goes to a value substantially less
dition, all the three nuclei mentioned above thantenseconds,theminimumdurationofan
may undergo β-decay. The photodisintegra- X-ray burst. However, one sees that the un-
tionratesatdifferenttemperatureshavebeen certainty in mass measurement prevents one
calculatedfromtheprotoncaptureratesusing from reaching any firm conclusion. Depend-
the principle of detailed balance. The density ing on the actual value of the masses, it may
has been taken as 106 gm/cm3 unless other- even be possible that a burst of the order of
wisementioned. Theprotonfractionhasbeen ten seconds cannot bridge this waiting point
assumed to be 0.7. effectively.
Fig. 8showsthechangeintheeffectivehalf A larger network, starting from 56Ni seed
life of 64Ge in explosive hydrogen rich envi- and consisting of (p,γ), (γ,p) reactions and
ronment. For comparison, we have also plot- β-decay, has been employed to study the the
ted the results calculated from the rates in timeevolutionofabundance. InFig. 9weplot
Zr(40)
n) 10000 Y(39)
bar 1000 Sr(38)
o
cr Rb(37) 40
mi 100
n( Kr(36) 39
ctio 10 Br(35) 38
e
s-s 1 Se(34) 37
s
o As(33) 36
cr 0.1
1 2 3 1 2 3 Ge(32)
Energy (MeV) Energy (MeV) Ga(31)
Zn(30)
FIG. 7: Partial cross sections for (p,γ) reactions Cu(29) 34
to thelow-lying states in Zn for 63Cu (left panel)
Ni(28)
and 65Cu (right panel) targets. Squares, circles
28 30 32
and triangles represent data for transition to the
ground state, the first excited state (multiplied
by 10) and the second excited state (multiplied FIG. 9: The rp-process path for 1.2 GK.
by 100), respectively.
indicated by black lines while gray lines show
2
10 thecorrespondingpathsforfluxbetween1%-
10%. One can see that 64Ge is easily bridged
ec) by two proton capture at 1.2 GK. However,
S 1
e ( 10 we have found that at a higher temperature,
Ge Half lif 100 svcieigzs.nsi1fis.oc5atGnhtKalyt,.ttAhhitestwβh-aeditenicneaxgytpowofian6itt4iGndgeelapcyoosinnttthr,ieb6pu8Srteoes-,
4
6 the photodisintegration is sufficiently strong
-1 so that, independent of the temperature, β-
10
0.8 1.0 1.2 1.4 1.6 1.8 decay is practically the only available path.
This delays the nucleosynthesis significantly.
Temperature (GK)
At 72Kr, the inverse process predominates in
FIG.8: Effectivehalflifevaluesof64Geasafunc- higher temperature driving the flux through
β-decay. Thus, here also, lower temperature
tionoftemperature. Thesolid linerepresentsthe
helps nucleosynthesis speed up. The waiting
results of our calculation while the dotted lines
mark the two extremes for the errors in the Q- pointat76Srpresentsadifferentpicturewhere
values of the reactions involved. The dash dot- the path essentially does not depend on the
tedlineshowstheresultsobtainedusingtherates temperature and principally flows along de-
from [28]. cay. It is clear that the actual process is sig-
nificantlydependentonthemodeloftheburst
processwherethetemperatureandtheproton
the path followed by the rp-process between fraction are functions of time.
A=56andA=80forT =1.2GK.Thedark The abundance at two different tempera-
boxes indicate the waiting point nuclei. The tures are plotted as a function of time in Fig.
elements are indicated at the left of the dia- 10. We find that at 1.2 GK, at the end of 100
gramwhiletheneutronnumbersareshownat seconds, the population that reaches A = 80
the bottom. The lines indicate the path fol- or beyond is more than 1.5 times than the
lowed by nucleosynthesis. The paths through corresponding quantity at the higher temper-
whichmorethan10%ofthetotalfluxflowsare ature of 1.5 GK. In both the cases, the popu-
lation beyond A=76 is significant. It should densitiesusingRMFandeffectiveNNinterac-
be noted that these numbers are strongly de- tions have been described. Life time for emis-
pendent on the masses of the nuclei near the sion of protons, neutrons and complex nuclei
waiting point. Masses of these nuclei have ei- from nuclei have been successfully calculated
thernotbeenmeasured,ormeasuredveryim- intheWKBapproximationusingtherealpart
precisely. Thus one has to depend on theory. of the abovepotential. The potentialhas also
In fact, the final abundance at variousmasses been used to describe elastic scattering and
change by a largefactor as one uses estimates proton reactions at low energy. Astrophysical
from different formulas. This conclusion is in implications of the results of proton capture
agreement with the variation in effective half reactionshavebeendiscussed. Representative
life values seen in Fig. 8. In the present in- results for above calculations have been pre-
stance, the Duflo-Zuker formula[29] has been sented. It is possible to conclude that semi-
used. microscopic optical potentials obtained in the
folding model with RMF densities are very
100 much successful in explaining nuclear decays
D-Z T=1.2 GK 64
and reactions over the entire mass region.
68
80 72
ce 76
n
a
nd 60
bu Acknowledgments
a
ve 40
elati The work presented is actually a joint
R 20 effort with contributions from Madhubrata
Bhattacharya, Chirashree Lahiri, Partha
0
Roy Chowdhury, Abhijit Bhattacharyya and
0 10 20 30 40 50 60 70 80 90 100
Subinit Roy. Financial assistances provided
Time (Sec)
by the DAE-BRNS and the UGC spon-
100
D-Z T=1.5 GK 64 sored DRS Programme of the Department
68 of Physics of the University of Calcutta are
80 72
ce 76 gratefully acknowledged.
n
a
nd 60
u
b
a
ve 40 References
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