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Radiative capture and photodisintegration reactions in the _1hn4He system [Elektronische Ressource] / vorgelegt von Martin Trini

102 pages
Radiative capture and photodisintegrationreactions4in the He systemDen Naturwissenschaftlichen Fakult atender Friedrich-Alexander-Universit at Erlangen-Nurn bergzurErlangung des Doktorgradesvorgelegt vonMartin Triniaus Nurn bergAls Dissertation genehmigt von den Naturwissenschaftlichen Fakult atender Universit at Erlangen-Nurn bergTag der mundlic hen Prufung: 21. Juli 2006Vorsitzender derPromotionskommission: Prof. Dr. D.-P. H aderErstberichterstatter: Prof. Dr. H.M. HofmannZweitberich Prof. Dr. W. LeidemannContents1 Introduction 12 Re ned Resonating Group Model 52.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Scattering states . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Distortion channels . . . . . . . . . . . . . . . . . . . . . . . . . 103 Interactions 113.1 Two-nucleon interaction . . . . . . . . . . . . . . . . . . . . . . 113.1.1 Argonne v . . . . . . . . . . . . . . . . . . . . . . . . . 11183.2 Three-nucleon interaction . . . . . . . . . . . . . . . . . . . . . 123.2.1 Urbana IX . . . . . . . . . . . . . . . . . . . . . . . . . 134 Radiative capture 154.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Electromagnetic operators . . . . . . . . . . . . . . . . . . . . . 164.3 Long wave length approximation . . .
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Radiative capture and photodisintegration
reactions
4in the He system
Den Naturwissenschaftlichen Fakult aten
der Friedrich-Alexander-Universit at Erlangen-Nurn berg
zur
Erlangung des Doktorgrades
vorgelegt von
Martin Trini
aus Nurn bergAls Dissertation genehmigt von den Naturwissenschaftlichen Fakult aten
der Universit at Erlangen-Nurn berg
Tag der mundlic hen Prufung: 21. Juli 2006
Vorsitzender der
Promotionskommission: Prof. Dr. D.-P. H ader
Erstberichterstatter: Prof. Dr. H.M. Hofmann
Zweitberich Prof. Dr. W. LeidemannContents
1 Introduction 1
2 Re ned Resonating Group Model 5
2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Scattering states . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Distortion channels . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Interactions 11
3.1 Two-nucleon interaction . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Argonne v . . . . . . . . . . . . . . . . . . . . . . . . . 1118
3.2 Three-nucleon interaction . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Urbana IX . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Radiative capture 15
4.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Electromagnetic operators . . . . . . . . . . . . . . . . . . . . . 16
4.3 Long wave length approximation . . . . . . . . . . . . . . . . . 17
5 Binding and scattering calculation 21
5.1 Binding calculation . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Phase shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.1 Positive parities . . . . . . . . . . . . . . . . . . . . . . . 28
5.2.2 Negative parities . . . . . . . . . . . . . . . . . . . . . . 31CONTENTS
2 46 The H(d,) He reaction 33
6.1 Transition matrix elements and phases . . . . . . . . . . . . . . 33
6.2 Analyzing powers and di eren tial cross sections . . . . . . . . . 37
6.3 Astrophysical S-factor . . . . . . . . . . . . . . . . . . . . . . . 46
47 Radiative capture and photodisintegration of He 51
3 47.1 The H(p,) He radiative capture reaction . . . . . . . . . . . . 51
4 37.2 The He(,p) H photodisintegration reaction . . . . . . . . . . . 53
4 37.3 The He(,n) He photodisin . . . . . . . . . . 57
7.4 Total [3+1] photodisintegration reaction . . . . . . . . . . . . . 62
8 Summary and outlook 69
9 Zusammenfassung 73
A Partial waves 79
B Deuteron models 80
C Energy level diagram 82
D Detailed balance 83
E Astrophysical S-factor 85
F Analyzing powers 87
Bibliography 891
Chapter 1
Introduction
4The He system is one of the best examined nuclear systems. It is the lightest
system with two-body channels and resonances and has some typical features
of heavier systems (e.g. the binding energy per nucleus). Hence it acts as
an important link between few-body systems and more complex nuclei. In
4addition the He system can serve as a testing ground for two-body (NN) and
three-body (NNN) interactions.
The increased computational power made it possible to improve the results
4of the calculations of the He system. Although a lot of theoretical and ex-
perimental work has been done in the last decades there are still unsolved and
interesting problems that have to be clari ed both theoretically and experi-
mentally.
2 4One example is the H(d,) He radiative capture reaction, which is im-
portant for understanding the nucleosynthesis in the early universe and which
has some very interesting features. One of them is the suppression of the E1
and M1 electromagnetic transitions at very low energies by spin and isospin
selection rules and the dominance of the E2 transition. This domination is
4only possible if the He ground state contains a D-state component that al-
4lows for S-wave capture to the ground state of He via E2 transition at low
energies. Thus one gets a signi can t enhancement in the extrapolated value
of the astrophysical S-factor.
4Another interesting reaction is the [3+1] photodisintegration of the He.
In the last decades there have been a lot of theoretical calculations and ex-2 Introduction
perimental measurements to clarify the question of the giant dipole resonance.
The situation, however, is very unsatisfactory. Still today there is large dis-
agreement in the peak. Measurements and calculations showed both a strongly
pronounced and a rather suppressed peak. Even the latest experimental data
do not yield the same results ([Shi05], [Nil05] and [Nil06]). So it is very im-
portant to improve the existing theoretical calculations to be able to arrive at
a conclusion.
One step forward is to use realistic NN and NNN interactions in ab initio
calculations. It is an open question what e ects the usage of realistic NN and
NN plus NNN interaction might cause. At the moment only one ab initio
calculation using the Lorentz integral transform and realistic interactions is
available ([Gaz06]).
4In this thesis we will study the He system in the framework of the re-
ned resonating group model. This model is brie y discussed in Chapter 2.
Then a review of the realistic Argonnev NN interaction and the UrbanaIX18
NNN interaction that were used in our calculations is given in Chapter 3. In
order to decrease the computing time necessary, all our electromagnetic calcu-
lations were done in the long wave length approximation. The corresponding
electromagnetic operators are brie y discussed in Chapter 4.
The results of our binding and scattering calculations are presented and
discussed in Chapter 5. We used two di eren t deuteron wave functions and
the NN and the NN plus NNN interaction in our calculation to obtain the
e ects of using di eren t realistic potentials. It is the rst parameter free cal-
culation with realistic NN potential and NN plus NNN potential which treats
the scattering channels explicitly. At rst the results of our binding and our
scattering calculations that act as a basis for all electromagnetic calculations
are shown. Special care was taken for the channels relevant to the photonu-
+ 4clear processes, namely the 0 which includes the ground state of the He, the
1 state which causes the E1 transition that is dominant in the [3+1] photo-
+disintegration reactions and the 2 state which causes the E2 transition that
is dominant in the d-d radiative capture reaction.
In the following Chapters 6 and 7 we will study these radiative capture
and photodisintegration reactions of our ab initio calculation. We compare
our calculations with both, other theoretical calculations and experimental3
data to check up on the agreement. Furthermore we study the e ects of using
NN interaction alone or NN plus NNN interaction.
In the end we will give a brief summary of our conclusions concerning our
parameter free calculation.4 Introduction5
Chapter 2
Re ned Resonating Group
Model
The re ned resonating group model (RRGM) is a microscopic cluster model
for few-body systems which o ers the possibility for multi channel calculations.
4In this thesis we use it for nuclear systems especially the He system.
Within this model the nuclei are split into clusters and fragments. In-
side the clusters there are no orbital angular momenta. According to Pauli’s
principle four nucleons at maximum build up one cluster and the number of
fragments must be two due to properties of the RRGM.
For detailed descriptions of the RRGM see [Hof84], [Hof87], [Hof05] and
[Hof05-2].
2.1 Hamiltonian
The for the Schr odinger equation of an N-nucleon system is given
by
N NX X X1
H(1;:::;N) = T + V + V , (2.1)i ij ijk
2
i=1 i=j cycl:
whereT denotes the kinetic energy of one nucleon,V denotes the two nucleoni ij
interaction and V denotes an additional three nucleon interaction. The twoijk
66 Re ned Resonating Group Model
and three body interactions that we use will be explained in detail in chapter
3.
Separating o the centre of mass energy T and assuming all masses m ofcm
the nucleons to be equal yields
N NX X1 2T =T + (p p ) . (2.2)i CM i j
2mN
i=1 i<j
The RRGM allows only two fragment channels because of the boundary
conditions of the wave function. So the translationally invariant Hamiltonian
can be separated into the two fragment parts (H (1;:::;N ) and H (N +1 1 2 1
1;:::;N)) and the relative motion part T :rel X X
0H (1;:::;N) =H (1;:::;N )+H (N +1;:::;N)+T + V + V .1 1 2 1 rel ij ijk
i2f1;:::;N g1 cycl:
j2fN +1;:::;Ng1
(2.3)
In order to force the potential term to be short ranged we add and subtract
2the point Coulomb interaction between the two fragments Z Z e =R . That1 2 rel
yields:
0 2H (1;:::;N) = H (1;:::;N ) +H (N + 1;:::;N) +T +Z Z e =R1 1 2 1 rel 1 2 rel
P P
2+ V + V Z Z e =R ,i2f1;:::;N g1 ij ijk 1 2 relcycl:
j2fN +1;:::;Ng1
where R denotes the relative coordinate between the centres of mass of therel
two fragments. V contains terms from both fragments because i < j < kijk
and thus i must be from the rst fragment and k must be from the second
fragment.
2.2 Wave function
We use the following ansatz for the total wave function :‘( )
nkX
k J ‘k =A [ ] [ (R)] . (2.4)‘ chan rel
k=1

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