Multipotential analysis of the radiative capture alpha + d -> 6Li + gamma

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UNIVERSIT `

A DEGLI STUDI DI PISA

DIPARTIMENTO DI FISICA “E. FERMI”

CORSO DI LAUREA MAGISTRALE IN FISICA

MULTIPOTENTIAL ANALYSIS OF THE RADIATIVE

CAPTURE REACTION α + d →

6

Li + γ

TESI DI LAUREA MAGISTRALE

Candidato:

ALESSANDRO GRASSI

Relatrice:

Prof.ssa

LAURA ELISA MARCUCCI

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Dedicated to my parents, my brother and my fianc´e. My life would have never been the same without you.

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List of Figures

1.1 Big Bang Nucleosynthesis reaction network from Ref. [1]. Here we in-dicated with p the proton, n the neutron, γ the photon, D the deuteron and T the3H nucleus. . . 2

1.2 Predicted abundances for the 2_H/H, 3_He/H, 7_{Li/H ratio and for the}

Yp=4He/H mass ratio are shown as functions of the η parameter in the

lower axes, or the baryon density of the universe Ωbh2in the upper axis.

The yellow squares show the experimental data. The shaded bound corresponds to the BBN and cosmic microwave background (CMB) prediction for Ωbh2. The figure and the relative notation is taken from

Ref. [2], where the “p” subscripts stands for “primordial”, and 2_{H≡D. .} ₃

1.3 The data for the astrophysical S-factor, taken from Ref. [4] (blue tri-angles), Ref. [5](black circles), Ref. [6] (green open circles), Ref. [7] (magenta circle), Ref. [8] (cyan diamonds) and Ref. [9] (red squares). The data from Refs. [5, 7] are upper limits to the S-factor. . . 5

1.4 Experimental setup for the LUNA measurements of Ref. [9]. . . 6

3.1 The Hammache et al. potential for each channel 2S+1_`

J, with S = 1.

The 3_P

1 and 3D2 lines are indistinguishable. . . 25

3.2 Same as Fig. 3.1, but for the Tursunov et al. potential. . . 26

3.3 The Mukhamedzhanov et al. (red) potential for the 3_S

1 channel in

comparison with the Hammache et al. (black) model. . . 28

3.4 The Dubovichenko potential for the coupled3_S

1and3D1channels. The

potentials for the 3_S

1 → 3D1 and for the 3D1 → 3S1 transitions are

equal, since we must have h3S1|V |3D1i = h3D1|V†|3S1i = h3D1|V |3S1i.

This is due to the fact that V = V†. . . 29

3.5 The Dubovichenko potential for the (uncoupled)2S+1_`

J →2S+1`J

tran-sitions. . . 30

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viii LIST OF FIGURES 3.6 The percentage error ε%

r as defined in Eq. (3.116) obtained within the

Numerov’s method varying the three parameters h (black line), R (red line) and rc(green line). The solid (dashed) lines are used for the single

channel (coupled channels) case. The diamonds (circles) indicate the initial values for the input parameters for the single channel (coupled channels) case. . . 44

3.7 Same as Fig. 3.6 but whithin the variational method. The parameters varied are NL(black line), h (red line), R (green line), γα(blue line) and

a1 (magenta line). Solid (dashed) lines are used for the single channel

(coupled channels) case. The diamonds for the single channel and the circles for the coupled channels correspond to the references values of Eqs. (3.117x), where x = a, b, c, d, e. . . 45

3.8 The 6_{Li wave function calculated with the V}

H potential from Ref. [11]. 47

3.9 The modulus for the 6Li wave function in logarithmic scale obtained with the variational (black) and the Numerov’s (red dashed) methods. . 49

3.10 Same as Fig.3.8, but for the VT potential. . . 50

3.11 Same as Fig.3.9, but for the VT potential. . . 51

3.12 The6_{Li wave function calculated using the Numerov’s algorithm with}

the potentials VM (solid black) and VH (dashed red). . . 52

3.13 Same as Fig.3.9, but for the VM potential. . . 53

3.14 The ANCs C0

0, C0W 0 and C0W as functions of r. . . 55

3.15 The 6Li S- (black line) and D- (red line) state radial wave functions, obtained with the VD potential. . . 56

3.16 Same as Fig. 3.13, but for the VD potential. Here we show both the

` = 0 and ` = 2 waves with the following notation: the ` = 0 variational (Numerov) wave function is shown as a black solid (red dashed) line, while the ` = 2 variational (Numerov) wave function is shows as a blue solid (green dashed) line. . . 57

3.17 The 6_{Li S- (black line) and D- (red line) state radial wave functions,}

obtained with the VN D potential (solid lines) and with the VD potential

(dashed lines). . . 60

3.18 The behaviour of C0, the ANC for the S-state wave function, with

respect to r. The value for C0 has been calculated with the VN D (VD)

potential and has been shown with a black (red) line. In the inner box, we show a detail of the region between 5 and 20 fm. . . 60

3.19 Comparison between the 6_{Li radial wave functions obtained with the}

various potentials. We show here the results for VH (black line), VT

(red line), VM (green line), VD (orange lines) and VN D (blue lines). The

solid (dashed) lines indicate the results for the 3_S

1 (3D1) state wave

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LIST OF FIGURES ix 3.20 Comparison between the S-state wave function tails obtained with the

potential models VH (black line), VT (red line), VM (green line), VD

(orange lines) and VN D (blue line). . . 62

4.1 The experimental data for the scattering phase shifts in deg as function of the CM relative energy in MeV. The color indicates the channel, the shape indicates the article from which the data were taken. In particular for the colors we used black (3_S

1), blue (3P0), dark green

(3P1), maroon (3P2), red (3D1), cyan (3D2), green (3D3), cyan+magenta

(ε) and black+red (coupled 3_S

1 and 3D1). Here ε is the mixing angle

of the coupled channel Jπ _{= 1}+_{. For the shape we used circles [}₂₅_],

triangles down [26], diamonds [27], squares [28] and triangles up [29]. On the right y-axis the channels are also denoted as δJ

l, being l and J

the orbital and the total angular momentum of the scattering partial wave, respectively. Note that δ1₀ and δ1₂ (marked in red together with the mixing angle ε) refer to the coupled channel Jπ _{= 1}+_{. The data}

of Ref. [28] for δ1

0 and δ21 are not included here, since they agree with

those of Refs. [26] and [29], but they would have made the figure too crowded. . . 70

4.2 Convergences for the h parameter in the NM case. The lines are the percentage errors for every channel phase shift (see legend at the side of the picture) with respect to the same value calculated with h = 0.01. The dashed lines indicates the results for E = 0.01 MeV, while the solid lines for E = 1.50 MeV. . . 72

4.3 Same as Fig.4.2, but for the parameters NL (black line) and γ (fm−1,

red line), with NL 0 = 50 and γ0 = 3 fm−1. The convergences for the

phase shift have been shown only for the channel 3D3 at E = 1 MeV. . 73

4.4 The phase shift δJ

` for every partial wave3`J, where ` = {0, 1, 2}. The

data have been taken from Refs. [25, 26, 27, 28, 29]. The notation for the shape of the experimental data points is the same as in Fig. 4.1, while the lines are the results for VH and VM (black solid line), VT (blue

dash-dotted line) and VD and VN D (red dashed line). . . 74

4.5 Same as Fig.4.4, but only the δ3₂ is shown. We can see here the 3+ res-onance at E = 0.711 MeV. In the inner box a detail after the resres-onance region is shown. . . 75

4.6 Same as Fig. 4.4, but for the coupled channels in the Jπ = 1+ state. The phase shift results evaluated with the VD and the VN D potentials

for the coupled channels 3_S

1 and 3D1 are displayed with the two red

dashed lines, the results for the mixing angle ε are displayed with the blue dashed line. Note that here we have shown also the data of Ref. [28] for δ1+

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x LIST OF FIGURES 5.1 The percentage error defined in Eq. (5.29), where X = h (fm) or α1.

The black line (red line) is the percentage error for the h (α1) parameter. 83

5.2 Same as Fig. 5.1. In addition, we have the green line representing the percentage error for the Nα parameter. . . 85

5.3 The E1 and E2 multipole operator contributions to the astrophysical

S-factor are shown in the first two panels, respectively. Here the black, red, green and blue lines indicate the J = 0, 1, 2 and 3 cases, respec-tively. In the last panel, the cyan line (violet line) is the total E1 (E2)

contribution to the S-factor. The results are obtained using the VH

potential. . . 87

5.4 Same as Fig. 5.3, but for the VT potential. . . 88

5.5 Same as Fig. 5.3, but for the VM potential. . . 89

5.6 Same as Fig. 5.3, but for the VD potential. In addition, we have two

new line styles and one new color. The line styles indicate the different ì → `f transitions. Dashed lines (dotted lines) indicate ì → 0 (ì → 2)

transitions. The new color, magenta, indicates the transition 0 → `f,

while we keep the same color for the transition 2 → 0, i.e. red. The dashed lines in the first two panels are indistinguishable from the solid lines, due to the smallness of the corresponding `i → 2 transitions. . . . 90

5.7 Same as Fig. 5.6, but for the VN D potential. . . 91

5.8 The S-factors for all the different J_iπ initial states S_`(Λ)

iJi as defined in

Eq. (5.26) with the VH (black line), VT (red line), VM (green line), VD

(orange line) and VN D (blue line) potentials. . . 92

5.9 The S-factors for both E1 and E2 transitions. The colors are the same

as in Fig. 5.8. . . 93

5.10 The total astrophysical S-factor is compared with the data of Ref. [4] (blue triangles), Ref. [5](black circles), Ref. [6] (green circles), Ref. [7] (magenta circle), Ref. [8] (cyan diamonds) and Ref. [9] (red squares). The data from Refs. [5,7] are upper limits to the S-factor. In the insert we show the tails of the S-factor in the range 10-50 keV. The colors of the solid lines are the same as in Fig. 5.8. . . 94

A.1 The test function (A.8). . . 103

A.2 Percentage errors for the first (black) and second (red) derivatives of the test function of Fig. A.1. In the upper panel, ε% is plotted versus

the number of differentiation points. In the lower panel the integration points are fixed to 9, and ε% is plotted versus the x position. . . 104

C.1 Scattering phase shifts δJ_l as function of the CM energy. Our results are shown as solid lines, while the original ones of Ref. [11] as dashed lines. . . 112

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LIST OF FIGURES xi C.2 The value of I(R) defined in Eq. (C.1) for the S_1J(1)

i (S

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2Ji) S-factors of

Eq. (5.26) in the left (right) panels. The Jivalues are Ji = 0, 1, 2 (1, 2, 3)

from the top to the bottom panels. Our results are displayed with black solid lines, while the results of Ref. [12] are displayed with red dashed lines. . . 113

C.3 The wave functions with the two potentials VH and VM, compared to

the ones on the left panel of Fig. 2 of Ref. [13]. The solid black line (blue dashed line) is our result for the reduced radial wave function of

6_{Li with the V}

H (VM) potential, while the green dotted line (red dotted

line) has been obtained by Mukhamedzhanov et al. in Ref. [13]. . . 114

C.4 The ratio defined in Eq. (C.2) between the the reduced radial wave function calculated with the VM potential and the same wave function

calculated with the VH potential, compared to the ones on the right

panel of Fig. 2 of Ref. [13]. The solid black line is our result, while the red dashed line has been obtained by Mukhamedzhanov et al. in Ref. [13]. . . 114

C.5 The reduced radial wave functions of6_{Li calculated with the two set of}

parameters for the VD potential, compared with the ones in Ref. [15].

The solid black line (solid red line) is the 3_S

1 (3D1) reduced radial

wave function of 6_{Li, calculated with the first set of parameters from}

Ref. [15], while the dashed black line (red dashed line) is the 3S1 (3D1)

reduced radial wave function of 6_{Li, calculated with the second set of}

parameters from the same article. The orange circles (magenta circles) and the blue circles (dark green circles) are the results from Ref. [15] for the first (second) set of parameters for the3_S

1 and3D1 wave functions,

respectively. . . 115

C.6 The scattering phase shifts for the 1+ channel compared with the ones from Ref. [15]. The black solid line and blue solid line are the phase shift δ1

0 + δ21 and the admixing angle ε, respectively. The red dashed

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List of Tables

3.1 Experimental data for the6Li, α and d nuclei. We list, besides the Jπ of each nucleus, its mass in 10−3 atomic mass unit (u = 931.5 MeV), the binding energy per nucleon B/A in keV, the magnetic dipole and electric quadrupole moments in µN ≡ e2~/2mp = nuclear magneton

and fm2, respectively. . . 20

3.2 Set of constants present in the potential models taken in consideration, labelled as VH, VT, VM, VD, and VN D, taken from Refs. .[11,12,13,15],

respectively. Aα (Ad) is the mass numbers of the α (d) particle, mu is

the mass unit, equal to the atomic mass unit for VH and VM, and to the

average nucleon mass for VT, VD and VN D. Furthermore, µ is the α + d

system reduced mass, htm is given in Eq. (3.16), α is the fine-structure constant and α~c is the product of α defined before, the reduced Plank constant ~ and the speed of light c. The underlined results are deduced from other data given by the authors. . . 24

3.3 Parameters for the Tursunov et al. potential of Ref. [12]: a`J are given

in fm−2, while V`J

0 in MeV. . . 27

3.4 Parameters for the present version of the Dubovichenko potential model. a, b and V1 have been kept the same as the original model. V0 is in MeV. 31

3.5 Comparison between our results for the average values hT i, hV i and hHi defined in the text, the electric quadrupole and magnetic dipole moments Q6 and µ6, the mean radius phR2i and for the observables of

Eqs. (3.119a)-(3.119f), obtained using the Numerov’s (NM), the varia-tional (Var) and the renormalized Numerov’s method (RNM). We also list the corresponding results from Ref. [15] and the experimental data from Refs. [35, 36, 37,38, 39]. . . 59

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xiv LIST OF TABLES 5.1 The allowed channel transitions for our reaction in spectroscopic

nota-tion, up to `i = 2 initial states. On the first and fourth rows there are

the initial partial wave states, on the second and fifth rows there are the allowed final state for the 6_{Li particle, and on the third and last}

rows there are the allowed electric multipole operators responsible for the transition from the initial to the final states. . . 82

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List of Outputs

3.1 Output of the NM with the Hammache et al. potential . . . 47

3.2 Same as Out. 3.1 but for the VT potential. . . 50

3.3 Same as Out. 3.1 but for the VM potential. . . 52

3.4 Same as Out. 3.1 but for the VD potential. . . 54

3.5 Output of the RNM with the VD potential. . . 58

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Chapter 1 Introduction

The aim of this thesis is to study the electromagnetic capture reaction between a deuteron d and a 4_{He nucleus (the α particle), which create a} 6_{Li particle, i.e.}

α + d → 6Li + γ . (1.1)

To do so, we evaluated the reaction astrophysical S-factor S(E), E being the total energy in the center of mass (CM) reference frame. The astrophysical S-factor is related to the reaction cross section and, as we will see, it is defined as

S(E) ≡ σ(E) E e √

EG/E_, _(1.2)

where EG is the so-called Gamow energy, the most probable energy at which the

re-action takes place, which will be defined in Chapter 2. This definition is useful to remove from the reaction cross section the nuclear contribution, factorizing out the Coulomb and the 1/E dependences from it.

The interest in studying this reaction is due to the so-called 6_{Li problem, a}

discrep-ancy between the measured and predicted primordial abundance of 6Li. The theory on which this prediction is based is known as the theory of Big Bang Nucleosynthesis (BBN). According to modern cosmology, the universe was born from the expansion of a singularity, a point with infinite density and temperature. This phenomena is called Big Bang (BB). Immediately after the BB, at about 10−37 s, the universe begun to expand itself with an exponential growth. This phase is known as inflation. After inflation, the cosmos consisted in a quark-gluon plasma, with also other elementary particles, as electrons and neutrinos. At about 10−9 s after the BB, baryogenesis started, with the formation of neutrons and protons. The continuing slow expansion of the universe lowered its temperature and therefore the particles kinetic energy, al-lowing to create, about after 180 s, the right conditions to build some first stable light nuclei: BBN began. This phase is believed to last about 10-20 minutes, since after

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2 CHAPTER 1. INTRODUCTION

Figure 1.1: Big Bang Nucleosynthesis reaction network from Ref. [1]. Here we indi-cated with p the proton, n the neutron, γ the photon, D the deuteron and T the 3_H

nucleus.

this period the average kinetic energy is too small to overcome the Coulomb repulsion among the different nuclei and therefore to allow for further nucleosynthesis. The the-ory of BBN is based on a reaction network, which is summarized in Fig.1.1. Through this network of reactions, the BBN phase results in the synthesis of (primordial) 2H,

3_He, 4_He, 7_{Li. Small amounts of} 3_{H and} 7_{Be, do not last long, since they soon}

trans-form to 3_{He and} 7_{Li, respectively. In order to quantify these primordial abundances,}

BBN needs some crucial inputs, as the rates of the reactions of the network (and therefore their astrophysical S-factors), the mean number of initial neutron and pro-tons (and therefore the neutron mean lifetime), the number of initial neutrinos (also needed to set up the initial conditions of BBN) and the parameter η, which is defined as the ratio between the number of baryons and the number of photons, and therefore is strictly related to the baryon density of the universe. The results are summarized

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3

Figure 1.2: Predicted abundances for the 2_H/H, 3_He/H, 7_{Li/H ratio and for the}

Yp=4He/H mass ratio are shown as functions of the η parameter in the lower axes, or

the baryon density of the universe Ωbh2 in the upper axis. The yellow squares show

the experimental data. The shaded bound corresponds to the BBN and cosmic mi-crowave background (CMB) prediction for Ωbh2. The figure and the relative notation

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4 CHAPTER 1. INTRODUCTION in Fig. 1.2, where the abundances for the ratio 2_H/H, 3_He/H, 4_{He/H and} 7_{Li/H are}

shown. Note that for the4He its mass ratio Y = MHe/MHis shown, while for the other

elements the ratio is defined as the number of X nuclei over the number of H nuclei, where X=(2_H, 3_He, 7_{Li). As we can see from the figure, there is a large discrepancy}

between the predicted and experimental abundance for 7Li. This disagreement goes under the name of first Lithium problem.

The 6_{Li nucleus is not considered as one of the main BBN products, because it is}

believed to appear in very small percentages, being a weakly bound nucleus. The only reaction in the BBN network (not shown in Fig. 1.1) which is believed to contribute to the 6_{Li creation is the one of Eq. (}_1.1_{). In 2006 Asplund et al. performed high}

resolution observations of Li absorption lines in old halo stars [3]. They saw a6Li/7Li ratio of about ∼ 5 × 10−2, three orders of magnitude larger than the BBN expected one, of about ∼ 10−5. Since the analysis is performed on old stars, the quantity of the present 6Li can be considered to be essentially the same as the one at the star formation, i.e. the same after BBN. This great discrepancy between the predicted and observed 6_{Li abundance has been called the second Lithium problem.}

In order to solve this second Li problem, the reaction of Eq. (1.1) has been extensively studied both theoretically and experimentally. The BBN energy window is located between 50 and 400 keV. Experimental studies of this reaction at BBN energies are very difficult, due to the exponential drop of the reaction cross section at low energies as a consequence of the Coulomb barrier. Furthermore, this reaction is affected by the isotopic suppression of the electric dipole operator. As we will see in Chapter 2, the reaction cross section can be expanded in multipole terms, the higher the order, the less the contribution to the cross section. In most reactions, the most important term is due to the electric dipole, which depends from the quantity (A1Z2− A2Z1)2, where

1 and 2 are the two colliding particles. This quantity turns out to be very close to zero for our reaction. The reaction (1.1) was first studied experimentally in the early 1980s [4] and then thorough the 1990s [5, 6, 7, 8]. However the data in the BBN energy range were affected by large uncertainties. The latest measurement is that performed by the LUNA Collaboration [9]. We summarize in Fig. 1.3 all the experimental data for the astrophysical S-factor [4, 5, 6,7, 8,9].

The LUNA measurements for the process were done deep underground, at the Gran Sasso National Laboratories (LNGS), in Italy, which is covered by 1400 m of rocks. This allows to reduce the background noise brought by the cosmic rays interacting with the atmosphere. In Fig.1.4we show the experimental setup schematically for the LUNA experiment. The experiment proceeds as follows: a4_He+ _{beam passes through}

several narrow apertures and then enters a windowless chamber, filled with 0.3 mbar deuterium gas. In the chamber lies a large high-purity germanium (HPGe) detector, placed perpendicular to the beam direction. On the opposite side respect to the beam

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5 10-2 10-1 100 E (MeV) 10-9 10-8 10-7 10-6 S (MeV b)

Figure 1.3: The data for the astrophysical S-factor, taken from Ref. [4] (blue triangles), Ref. [5](black circles), Ref. [6] (green open circles), Ref. [7] (magenta circle), Ref. [8] (cyan diamonds) and Ref. [9] (red squares). The data from Refs. [5,7] are upper limits to the S-factor.

entrance, there is a copper beam dump which works as a constant temperature gra-dient calorimeter. The natural background arising from the environment (especially considering the rocks of the mountain) is then reduced by a lead shield, enclosed in a antiradon box. This box with N2 flushing allows to remove the background from the

radioactive Radon, which is present in the environment of the experiment as a gas. This quite complex setup demonstrate how difficult is the experimental study of this reaction. Reliable theoretical calculations become then essential.

The theoretical study of this reaction is also very difficult, since, in principle, we should solve a six-body problem, i.e. we should consider the six nucleons contained in the α + d and6_{Li particles, and their interaction with the photon. Such an approach}

is known as ab-initio method, and it has been used only by Nollett et al. in Ref. [10]. The numerical techniques used in Ref. [10] to solve the six-body problem, however, provide solutions for the initial and final state wave functions with uncertainties at

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6 CHAPTER 1. INTRODUCTION

Figure 1.4: Experimental setup for the LUNA measurements of Ref. [9].

the 10-20 % level. Since ab-initio methods are still nowadays hardly implemented for A>4 systems, the study of the reaction has been done using models to simplify the problem. In the literature, most of the theoretical calculations consider the α and d particles as point-like and structureless, and reduce the problem to a two-body problem, for which a crucial input is the potential model, which describes the α-d in-teraction. This approximation, for example, has been used in the works of Hammache et al. [11], Tursunov et al. [12], Mukhamedzhanov et al. [13] and Dubovichenko et al. [14].

In this thesis we also worked in the two-body framework and analysed the astro-physical S-factor of the reaction (1.1) with different potential models, taken from Refs. [11,12, 13, 15]. In addition, we created a totally new potential model, starting from the one in Ref. [15] and then modifying it in order to reproduce the asymptotic normalization coefficient (ANC). The ANC is defined as the ratio between the α-d relative radial wave function in 6Li and the Whittaker function for large distances, and it describes the bound state wave function in the asymptotic region (see Chapter

3 for a detailed explanation). We first reproduced the results of the original articles with our own techniques, and then we made a complete comparison among the results obtained with each potential. Our calculations have been performed developing differ-ent methods to solve the Schr¨odinger equation, in order to verify that our results were not affected by significant numerical uncertainties. The potentials of Hammache et al. [11], Tursunov et al. [12] and Mukhamedzhanov et al. [13] are central potentials, fitted to reproduce the 6_{Li binding energy and the scattering α-d phase shifts up to}

` = 2 (` being the relative orbital angular momentum). The last two potentials used in this thesis, modiefied versions of the one derived by Dubovichenko [15], have also a tensor component. In fact, the original potential of Ref. [15] is fitted to reproduce

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7 only the 6_{Li bound state characteristics and not all the α + d phase shifts. We have}

modified this potential in order to reproduce also the data for the α − d phase shifts up to ` = 2. A further modification has turned out to be necessary in order to repro-duce the ANC, and we have shown through this thesis how strongly the astrophysical S-factor at low energies is affected by the ANC. The use of these two non-central potentials to study the astrophysical S-factor have been done here for the first time. The thesis is organized as follows: in Chapter 2 we derive the cross section of the reaction, starting from perturbation theory in quantum mechanics; in Chapter 3 we show several methods used to calculate the6_{Li wave function and the resulting bound}

state wave functions; in Chapter 4 we evaluate the initial α-d scattering wave func-tion, and we check that the used potentials fit the scattering data for the phase shifts; in Chapter 5 we derive the astrophysical S-factor from the cross section formula, we evaluate it for every potential model and we compare the results with the experimen-tal data. Finally, in Chapter 6, we summarize the different steps of this work and the obtained results and we suggest how the study of this reaction can be further improved.

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Chapter 2 Radiative capture reactions

We go into the details of the calculations of the astrophysical factor S(E). We start with a generic capture cross section of two bodies. Each body has Ai nucleon and

thus the final body has A = A1+ A2 nucleons and the reaction can be written as

A1 + A2 → A + γ . (2.1)

What we observe is the astrophysical factor of this reaction, defined as S(E) ≡ σ(E)E e

√

EG/E _, _(2.2)

where E is the energy of the incoming particles in the CM reference frame, σ is the total cross section of the process and EG is the Gamow energy

EG ≡ 2 µ (παZ1Z2)2, (2.3)

Z1 and Z2 being the charges of the nuclei, µ the reduced mass of the system, and α

the fine structure constant. The reason we use this quantity is because it has a slowly varying dependence in energy, so it can be extrapolated easily at astrophysical BBN energies which are prohibitive in laboratory experiments.

We now define the four vectors p1, p2, P and q, where pi (i=1, 2) are the initial

momenta of the i-th particle, while P and q are the momenta of the final nucleus and the photon, respectively. In the CM-reference frame p1+ p2 = 0 and P + q = 0. It

follows that p1 = −p2 ≡ p and P = −q.

The conservation of energy for the process (2.1) can be written as (c = 1) p2

2µ + m1+ m2 = M + q2

2M + q , (2.4)

where mi (i = 1, 2) are the masses of the two initial nuclei, and M is the mass of

the A nucleus. The binding energy of the final nucleus with respect to the two initial particles is

B = m1+ m2− M . (2.5)

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10 CHAPTER 2. RADIATIVE CAPTURE REACTIONS Therefore, if we call E the initial kinetic energy E = p2_{/2µ, we have}

E + B = q

2

2M + q . (2.6)

From this we obtain that the momentum of the emitted photon is q = "r 1 + 2E + B M − 1 # M . (2.7)

2.1 Perturbation theory in quantum mechanics and

transition probabilities

We use perturbation theory to calculate the transition probability from the initial to the final state. We can write the Hamiltonian operator as

H = H0+ HI, (2.8)

where H0 is the unperturbed Hamiltonian operator and HI the perturbation

Hamil-tonian operator. Let us write the eigenvectors of the unperturbed HamilHamil-tonian as

H0|ni = En|ni . (2.9)

This wave function which satisfies the time dependent Schr¨odinger equation (here ~ = 1),

i∂t|Ψ(t)i = H|Ψ(t)i . (2.10)

can be expanded in terms of the eigenvectors |ni as |Ψ(t)i =

N

X

n=0

cn(t) e−iEnt|ni . (2.11)

Therefore, Eq. (2.10) becomes

i X n ˙cne−iEnt|ni − X n iEncne−iEnt|ni ! =X n Encne−iEnt|ni + HI X n cne−iEnt|ni , (2.12)

where ˙cn≡ ∂tcn(t), and we have dropped the t-dependence in cn. Multiplying for hk|

and using the orthonormality of the eigenvectors, i.e. hk|ni = δkn, we obtain that

˙ck = −i

X

n

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2.2. THE RADIATIVE CAPTURE 11 where ωkn = Ek− En.

Let us expand the coefficients with respect to the perturbation, i.e. cn(t) = c(0)n + c (1) n + ... = X i c(i)_n (t) . (2.14)

For the 0-th order term we get

˙c(0)_k = 0 ⇒ c(0)_k = δik, (2.15)

where the label i indicates the initial state.

We now write the first order term. From Eq. (2.13) we get c(1)_k (T ) = −i

Z T

0

dt eiωkit_hk|H

I|ii , (2.16)

where T is the time during which the interaction HI is acting. The total transition

probability for the system to switch from the initial state |ii to the final state |ki, is, at first order Pi→k = c (1) k (T ) 2 . (2.17)

If our final state |ki is in the continuum we must multiply Eq. (2.17) for the density number dn of available states. Furthermore, if we want the transition probability per unit time dwik, we must divide the transition probability for the time T , and therefore

dwik = 1 T|c (1) k (T )| 2_{dn .} _(2.18)

With a time-dependent interaction term that oscillates as e±iωt, the integration over t in Eq. (2.16) gives the energy conservation delta function and dwik becomes

dwik = 2πhk|HI|iiδ(Ek− Ei± ω)dn (2.19)

This the famous Fermi’s Golden Rule.

2.2 The radiative capture

We start defining our unperturbed Hamiltonian operator H0. It is made up by a

nuclear term HN and by an electromagnetic term He. The nuclear term HN is written

as the sum of the kinetic and potential energy. It is convenient to include in the potential term V12 also the electromagnetic Coulomb interaction between the two

charges Z1 and Z2. Our unperturbed Hamiltonian can be written then as

H0 = p2₁ 2m1 + p 2 2 2m2 + V12(r1, r2) + X k,λ ωk ˆ a†λ_k ˆaλ_k+ 1 2 . (2.20)

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12 CHAPTER 2. RADIATIVE CAPTURE REACTIONS To use the perturbation theory developed in the previous section we must find the eigenfunctions of H0, and then we can add the perturbation HI to calculate the

transition probability.

Our perturbation is the matter-electromagnetic field coupling, written as HI = −e

Z

dx J(x) · A(x) , (2.21)

where A(x) is the vector potential of the electromagnetic field and J(x) is the nuclear density current. The former term in second quantization is

A(x) =X λ Z dk √ 2ωk ˆ λ_kaλ_keik·x + h.c. . (2.22)

Thanks to the commutation relation [aλ_k, a†λ_k00] = δ3(k−k0)δλλ0, we can write the matrix

element between the electromagnetic vacuum |0i_γ and the one-photon final state hγλ q|

as

hγ_qλ|A(x)|0i_γ = 1 p2ωq

ˆ†λ_q e−iq·x. (2.23) We can now calculate the matrix element between the initial and final quantum states in Eq. (2.19), reducing our problem to evaluate the nuclear term. In fact

where J(x) is the hadronic current and J†(q) is the Hermitian of its Fourier transform. Note that this matrix element depends on the initial and final particles spins J1, J2

and J ; such dependence will be made explicit later.

Once we have the reaction rate per unit time we can calculate the process cross section. It is defined as the ratio between the transition probability per unit time and the incoming particle flux, which is simply the relative velocity vr between the two

colliding particles. Therefore

dσ = dwf i vr

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2.2. THE RADIATIVE CAPTURE 13

Combining this equation with Eqs. (2.19) and (2.27), and remembering that the pho-ton carries energy away from the system, we obtain

dσ = 2πe 2 2q ˆ†λ_q · hΨA|J†(q)|ΨA1A2i 2 δ(Ef − Ei+ ω) dn . (2.29)

Remembering that dn is the number of final allowed states, we can write it as

dn = dP (2π)3 dq (2π)3 = dq (2π)3 = q2dq dΩq (2π)3 . (2.30)

The second equality follows from the fact that P is completely defined by momentum conservation.

We can now write Eq. (2.29) as dσ = e 2 8π2 |ˆ †λ q · hΨA|J†(q)|ΨA1A2i| 2_δ(E f − Ei+ ω) q dq dΩq. (2.31)

We can then integrate over q using the general property of the delta function Z

dx δ(f (x)) g(x) = g(x0) |f0_(x

0)|

, (2.32)

where f (x) is a given function and x0is the abscissas of the root of f (x) = 0 contained

in the integration path. In our case

δ(f (q)) = δ(Ef − Ei+ ω) ; (2.33)

from Eqs. (2.6), remembering that q = ω and changing M with mA we obtain

f0(q) = 1 + q mA

. (2.34)

Finally, for unpolarized cross section, we sum over the polarization of the final photon (λ) and the spin projection of the final particle (M ), and average over the initial spins (indicated with J1, M1 and J2, M2. Therefore

dσ dΩq = e 2 8π2 q 1 + q/mA 1 (2J1+ 1)(2J2+ 1) X M1,M2,λ,M |ˆ†λ_q · hΨA|J†(q)|ΨA1A2i| 2 _(2.35)

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14 CHAPTER 2. RADIATIVE CAPTURE REACTIONS

2.3 Multipole expansion

We want now to simplify the formula for the matrix element ˆ†λ_q · hΨA|J†(q)|ΨA1A2i

appearing in Eq. (2.35). We first introduce the following mathematical identity ˆ λ_q e−iq·x = −√2π ×X Λ≥1 (−i)Λ√2Λ + 1 λ jΛ(qx) YλΛΛ1(ˆx) + 1 q ∇ × jΛ(qx)YΛΛ1λ (ˆx) , (2.36) where jΛ(z) is the spherical Bessel function of the first kind and YλΛΛ1(ˆx) is the vector

spherical harmonic, defined as

YM_Λ`1(ˆx) =X

mµ

h`m, 1µ|Λ λi Y`m(ˆx) ˆµq. (2.37)

We define the electric (E) and magnetic (M ) multipole operators as EΛλ(k) = 1 k Z dx∇ × jΛ(kx)YΛΛ1λ (ˆx) · J(x) , (2.38) MΛλ(k) = Z dx jΛ(kx)YΛΛ1λ (ˆx) · J(x) . (2.39)

Thanks to the properties of the current operator J, it can be shown that EΛλ and

MΛλ are irreducible tensors in the nuclear Hilbert space [16] and that their parity is

respectively

P+EΛλP = (−)ΛEΛλ, (2.40)

P+MΛλP = (−)Λ+1MΛλ. (2.41)

Using these operators, the scalar product ˆ†λ_q · J†_{(q) can be written as}

ˆ†λ_q · J†(q) = −√2πX Λ≥1 (−i)Λ√2Λ + 1 EΛλ(q) + λ MΛλ(q) , (2.42)

apart from an overall phase.

2.3.1 Partial wave decomposition of the initial scattering wave

function

For the initial scattering wave, we quantize the angular momentum with the z-axis along the relative momentum p. Thus, we can write the initial scattering wave func-tion |Ψi(M )i as |Ψi(M )i = √ 4π X `SJ i`√2` + 1 h`0, SM |J M i |Ψ`SJ M(p)i , (2.43)

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2.4. LONG WAVELENGTH APPROXIMATION 15 where ` is the total orbital angular momentum, S the total spin and ` and S are coupled to the total angular momentum J and its third component M .

It is now useful to rewrite the transition matrix and simplify it, rotating the wave functions to quantize them along q. It follows from Eqs. (2.42) and (2.43) that

ˆ†λ_q · hΨf|J†(q)|Ψii = √ 4πX `SJ i`√2` + 1 h`0, SM |J M i ×ˆqhΨf|e−iθJy " −√2πX Λ≥1 (−i)Λ√2Λ + 1 EΛλ(q) + λMΛλ(q) # eiθJy_|Ψ`SJ M i (p)iqˆ, (2.44) where θ is the angle between q and p and the notation | i_q means that the state is quantized along q. We can use this formula in Eq. (2.35) and, if we are not interested in the angular distribution, we can integrate over Ωq to get the final formula for the

total cross section of the process as

σ(E) = 32π 2 (2J1+ 1)(2J2+ 1) α vr q 1 + q/mA X Λ≥1 X `SJ E_Λ`SJ 2 + M_Λ`SJ 2 . (2.45) With ElSJ

Λ and MΛ`SJ we indicate the reduced matrix element of the electric and

magnet Λ-th multipole operators between the final state and the initial scattering wave with orbital angular momentum `, total spin S and total angular momentum J . This elements are given by

T_Λ`SJ = hΨf(JfMf)|TΛλ|Ψ`SJ Mi (p)i

p2Jf + 1

hJM, Λλ|JfMfi

, (2.46)

with T`SJ

Λ = EΛ`SJ or MΛ`SJ, and we have indicated with Mf the third component of

the A nucleus angular momentum.

2.4 Long wavelength approximation

If the energy is low, the wavelength is large, and we can perform the so-called long wavelength approximation (LWA) of the electric and magnetic multipole operators. This approximation is useful when qR 1, where R = r0A1/3 with r0 ∼ 1.25 fm is of

the order of the nuclear diameter of the compound system. From this condition and from Eq. (2.7), it follows the approximate condition

E " 1 mNr0A4/3 + 1 2 − 1 # AmN 2 − B , (2.47)

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16 CHAPTER 2. RADIATIVE CAPTURE REACTIONS where B is the binding energy of the final nucleus and mN is the nucleon mass.

If this condition holds we can take Eqs. (2.38) and (2.39) and expand them in series of q, using the useful identity

jΛ(z) = zΛ +∞ X k=0 1 k! (2k + 2Λ + 1)!! −z 2 2 k . (2.48)

It can be shown [16, 17] through some useful mathematical identities and the conti-nuity equation

∇ · J = −i [ρ, H] , (2.49)

ρ being the charge operator, that the electric multipoles at leading order in q can be written as EΛλ ≈ qΛ (2Λ + 1)!! r Λ + 1 Λ Z dx xΛYΛλ(ˆx) ρ(x) . (2.50) with ρ given by ρ(x) = X i=1,2 Ziδ(3)(x − xi) . (2.51)

In the CM-reference frame, we can rewrite EΛλ of Eq. (2.50) as

EΛλ = Ze(Λ) (2Λ + 1)!! r Λ + 1 Λ (qr) Λ_Y Λλ(ˆr) , (2.52)

where r is the relative distance between the two particles and Ze(Λ) is the so-called

effective charge defined as Z_e(Λ) ≡ Z1 m2 m1+ m2 Λ + Z2 _−m 1 m1+ m2 Λ . (2.53)

2.5 The α + d →

6

Li + γ reaction

We developed the formalism to calculate the cross section of radiative captures of the form (2.1). It has come the time to go into the details of the α + d →6 _{Li + γ reaction.}

The old generic parameters will be replaced with the ones we will use in the next chapters. We label the physical quantities of the Lithium particle with a subscript 6, for the initial particles we use subscripts α and d for Helium and Deuteron, respec-tively. The intrinsic angular momenta and parity of the three particles are J_απ = 0+, Jπ

d = 1+ and J6π = 1+. The former subscripts become 1 → α and 2 → d. The total

spin is therefore S = J1+ J2 = 1.

We can write Eq. (2.45) as

σ(E) = 32π 2 (2Jα+ 1)(2Jd+ 1) α vr q 1 + q/m6 X Λ≥1 X `SJ E_Λ`SJ 2 +M_Λ`SJ 2 (2.54)

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2.5. THE α + D →6_{LI + γ REACTION} ₁₇

where the reduced operators in Eq. (2.46) become T_Λ`SJ = hΨ6(JfMf)|TΛλ|Ψ`αdiSJiMi(p)i

p2Jf + 1

hJiMi, Λλ|JfMfi

. (2.55)

In conclusion, to obtain the reaction cross section (and consequently the astrophysical S-factor), we are left with the calculation of the E- and M - multipole operators, and therefore we need to construct the wave functions of the initial scattering and the final bound states. This will be done using several two-body nuclear potential models, as it will be discussed in the next chapters.

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Chapter 3 The

6

Li wave function

In this chapter we will first analyse the properties of the 6_{Li nucleus and then we}

will solve the Schr¨odinger equation to obtain the6Li wave function. This requires the definition of the nuclear potential and the development of a numerical technique to calculate the6_{Li wave function.}

3.1

6

Li properties

The nucleus of6

3Li3 is composed by three neutron and three protons. Its ground state

is a stable isotope with Jπ _{= 1}+ _{and binding energy per nucleon of B/A = 5332.331}

keV. Its first exited state is a Jπ = 3+ state with excitation energy of 711 keV with respect to the ground state energy.

The 6_{Li ground state as a non null electric quadrupole moment and magnetic dipole}

moment. The presence of a quadrupole moment indicates that the nucleus has a non perfect spherical shape. The quadrupole is very small with respect, for example, to

7_{Li: the} 6_{Li and} 7_{Li quadrupole moments are -0.082 fm}2 _{and ∼ 4.0 fm}2_{, respectively.}

To obtain such a value, one needs a very good understanding of the tensor interaction between the constituents of the nucleus.

In this thesis we will consider 6Li as a two-body system, i.e. 6Li ≡ α + d. This can be understood considering the strong binding energy of the double-magic α nu-cleus. It has also been shown in Ref. [18] that the probability of a αd clusterization is of the order of 60-80%.

We summarize the main properties of the three particles involved in the reaction in Table 3.1. From this table we can calculate the binding energy of the α + d system.

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20 CHAPTER 3. THE 6_{LI WAVE FUNCTION} Jπ _{m (10}−3_u) _{B/A (keV)} _{µ (µ} N) Q (fm2) 6_Li ₁+ _6015.122 _5332.331 _0.822 _-0.082 4_He ₀+ _4002.603 _7073.915 ₀ ₀ 2_H ₁+ _2014.102 _1112.283 _0.857 _0.286

Table 3.1: Experimental data for the 6Li, α and d nuclei. We list, besides the Jπ of each nucleus, its mass in 10−3 atomic mass unit (u = 931.5 MeV), the binding energy per nucleon B/A in keV, the magnetic dipole and electric quadrupole moments in µN ≡ e2~/2mp = nuclear magneton and fm2, respectively.

Using the data from the table we have

B6(αd) = mα+ md− m6 (3.1a)

= B6− (Bα+ Bd) (3.1b)

= 1473.76 keV , (3.1c)

where Bα and Bd are the α particle and the deuteron nuclear binding energies. This

result is in agreement with data from Ref. [19]. We now go into the details of the calculation of the wave function of the α + d bound state.

3.2 The Schr¨

odinger equation for the

6

Li = α + d

bound state

We start with the basic Schr¨odinger equation for the nuclear α + d bound system. The Hamiltonian operator H can be written as

H = p 2 α 2mα + p 2 d 2m2 d + Vαd (3.2a) = P 2 2M + p2 2µ + Vαd, (3.2b)

where in the second line we separate the CM motion from the relative motion, M is the total mass and µ is the reduced mass of the αd system. Thus we define

(

r = rα− rd

M R = mαrα+ mdrd,

(3.3) and the conjugate momenta are

(

p = (mdpα− mαpd)/M

P = pα+ pd.

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3.2. THE SCHR ¨ODINGER EQUATION FOR THE6_{LI = α + D BOUND STATE21}

The Schr¨odinger equation

H|Ψi = E |Ψi (3.5)

can be separated in two terms depending on r and R, respectively. Therefore, with

Ψ(R, r) = ϕ(r) φ(R) , (3.6)

the φ function is given by

φ(R) = √1 4π e

iP·R_. _(3.7)

The Schr¨odinger equation then reduces to

(Hr+ B)|ϕi = 0 , (3.8)

where

Hr =

p2

2µ + Vαd, (3.9)

and B = −E is the binding energy of the6Li with respect to the clusters α and d (see Eq. (3.1c)).

Being made up of a spinless α particle and by a spin-1 deuteron we can conclude that in order to have a Jπ _{= 1}+ 6_{Li ground state, we must have an admixture of S−}

and D−waves. We know that the percentage of D-wave ηD must be non zero, because

otherwise the quadrupole moment would be zero. Therefore, we can write ϕ in the Schr¨odinger representation as ϕM(r) = X `=0,2 u`(r) X σ h`m, 1σ|1M i Y`m(ˆr )χ1σ, (3.10)

where u`(r) is the radial dependence of ϕ and χ1σ is the spinor function with third

component σ. The spinor components satisfy the following normalization relation

χ†_1mχ_1m0 = δ_mm0 . (3.11)

Note that in Eq. (3.10) we must have m = M − σ.

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22 CHAPTER 3. THE 6_{LI WAVE FUNCTION} therefore hϕM|ϕM0i =X ``0 X σσ0 Z dr r2u∗_`0u_` Z dΩrY`∗0_m0Y_`mχ † 1σ0χ1σh`m, 1σ|1M ih`m0, 1σ0|1M0i (3.12a) =X ``0 X σσ0 Z dr r2u∗_`u`δ``0δ_mm0δ_σσ0h`m, 1σ|1M ih`m0, 1σ0|1M0i _(3.12b) = δM M0 X `σ Z dr r2u∗_`u`h`m, 1σ|1M i2 (3.12c) = δM M0 X ` Z dr r2|u`(r)|2. (3.12d)

Therefore we must impose X

`

Z

dr r2|u`(r)|2 = 1 . (3.13)

3.3 The relative wave function

We discuss here some properties of the α + d bound-state wave function ϕM(r) which can be deduced from some of the 6Li observables.

Forbidden bound states The 6Li S-state, seen as an α + d cluster, must have a non-trivial node in the radial wave function. This is due to the configuration of the six nucleons which in the end must be grouped in the two particles α and d. The nodeless state, which with our potential models has a binding energy of ∼ 25 − 30 MeV, can be seen as a condensate of the two particles in a six nucleons core, and thus must be discarded.

D -state component The present of a D−state component in the wave function requires the inclusion of a tensor potential in the Schr¨odinger equation, which couples the ` = 0 and ` = 2 states. This was first done for the α + d system by Frick et al. in an optical potential approach [20]. In Ref. [21], Merchant and Rowley derived the tensor component of the α + d potential by folding the nucleon-nucleon potential and they also demonstrated that the basic characteristic of the 6Li bound state could be reproduced. All these works focused either on the scattering problem or on the bound state problem. It was Dubovichenko who first tried to reproduce both scattering and bound-state properties at the same time including the tensor potential [15].

Note that the D−state must be nodeless. In fact the diagram {42} is consistent with the orbital-angular-momentum value of 2.

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3.4. POTENTIAL MODELS 23 The ANCs Another important observable of 6_{Li is the asymptotic normalization}

coefficient (ANC), as pointed out by Mukhamedzhanov et al. in Ref. [13].

The ANC C_αd(`) of the bound-state wave function, with orbital angular momentum `, is defined as C_αd(`) ≡ lim r→+∞ r u`(r) W−η,1/2+`(2kr) , (3.14)

where W−η,1/2+`(2kr) is the Whittaker function, k is the wave number of the relative

momentum and η is the Coulomb parameter, i.e. η = ZαZd α

√ µ

p2|E| . (3.15)

In our study, we use the experimental value for the S-wave ANC of Ref. [22], C_αd(0) = [2.30 ± 0.12] fm−1/2. As observed in Ref. [13], the ANCs are not uniquely determined by the potential which reproduces the α + d scattering phase shifts and the bound state properties. In fact, there are a whole family of potentials which reproduces these data and gives different values for the ANCs. This ambiguity is understood using the Gel’fand-Levitan and Marchenko theorem of the inverse scattering problem: there is an infinite number of phase-equivalent potentials, which provide different binding energies and ANCs [23]. To find the correct unique two-body potential, we must add the constraints on both the binding energy and the ANCs.

It was observed in Refs. [13, 24] that the role of the ANCs is very important for the calculation of the S-factor at low energies, where only the tail of the bound-state wave function mostly contributes to the calculation of the different multipoles. In fact, since the S-factor at astrophysical energies depends approximately on the square of the ANCs, an incorrect value of these can bring great changes in the S(E) value. A correct value for the ANCs is fundamental to have a good understanding of the reaction.

3.4 Potential models

In this thesis we calculate the astrophysical factor using several potential models. Each one of these models has its own set of constants, summarized in Table 3.2. Among these constants we have defined htm as

~tm = ~

2

2µ , (3.16)

where µ is the reduced mass. Note that Dubovichenko and Tursunov et al. set ~tm equal to

~tm = ~tmN

Aα+ Ad

AαAd

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24 CHAPTER 3. THE 6_{LI WAVE FUNCTION}

units VH and VM VT, VD and VN D

Ad - 2.01411 2 Aα - 4.00260 4 mu MeV 931.494043 938,973881 µ MeV 1248.09137 1251.96518 ~tm MeV fm2 15.5989911176 15.5507250000 α - 7.297352568×103 _{7.297405999×10}3 α~c MeV fm 1.4399644567 1.4399750000

Table 3.2: Set of constants present in the potential models taken in consideration, labelled as VH, VT, VM, VD, and VN D, taken from Refs. .[11, 12, 13, 15], respectively.

Aα (Ad) is the mass numbers of the α (d) particle, mu is the mass unit, equal to the

atomic mass unit for VH and VM, and to the average nucleon mass for VT, VD and

VN D. Furthermore, µ is the α + d system reduced mass, htm is given in Eq. (3.16), α

is the fine-structure constant and α~c is the product of α defined before, the reduced Plank constant ~ and the speed of light c. The underlined results are deduced from other data given by the authors.

where

~tmN ≡ ~ 2

2mN

= 20.7343 MeV fm2. (3.18)

We now go in the details of each potential model. They are the model of Hammache et al. of Ref. [11] (VH), Tursunov et al. of Ref. [12] (VT), Mukhamedzhanov et al. of

Ref. [13] (VM), and Dubovichenko [15] (VD) and a modified version of this potential

(VN D). Note that all the potential models, except VD and VN D, do not include a

tensor component.

3.4.1 The Hammache et al. model

The Hammache et al. potential model is derived in Ref. [11]. The nuclear term includes a central Wood-Saxon term and a spin-orbit term, to which we add a modified Coulomb potential. The potential can be written as

VH(r) = Vc(r) + VSO(r) + Ve(r) , (3.19) where Vc(r) = −V0` 1 + er−Ra −1 , (3.20a) VSO(r) = V1 λ2_{L · ˆ}_ˆ _S r d dr 1 + er−Ra −1 , (3.20b)

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3.4. POTENTIAL MODELS 25 0 2 4 6 8 r (fm) -70 -60 -50 -40 -30 -20 -10 0 10 V H (MeV) 3 S₁ 3 P 0 3 P 1 3 P₂ 3 D 1 3 D 2 3 D 3

Figure 3.1: The Hammache et al. potential for each channel2S+1`J, with S = 1. The 3_P

1 and 3D2 lines are indistinguishable.

Ve(r) = ZαZdα

3 − (r/R)2 (2R)−1 if r ≤ R

r−1 if r > R , (3.20c)

and the constants are: R = r0 A1/3, r0 = 1.25 fm, a = 0.65 fm, λ = 2 fm, V1 =

2.4 MeV, V0

0 = 60.712 MeV and V `6=0

0 = 56.7 MeV. The potential is shown for the

different channels in Fig.3.1.

The choice of the constants has been done by Hammache et al. to reproduce the experimental values for the 6_{Li binding energy (in this case chosen to be B = 1.474}

MeV) and the 3+ resonance energy with respect to the α + d threshold. These parameters also nicely reproduce the scattering phase shifts for energies up to 1.5 MeV in the CM-reference frame (see Fig. 1 of Ref. [11]).

The potential in Eq. (3.19) does not contain a tensor term and therefore the 6Li bound-state is a pure S−state.

3.4.2 The Tursunov et al. model

The Tursunov et al. potential model of Ref. [12] uses Gaussian functions to parametrize the α + d interaction. The nuclear term includes the Coulomb interaction, which is

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26 CHAPTER 3. THE 6_{LI WAVE FUNCTION} 0 2 4 6 8 r (fm) -80 -70 -60 -50 -40 -30 -20 -10 0 V T (MeV) 3 S₁ 3 P 0 3 P 1 3 P₂ 3 D 1 3 D 2 3 D 3

Figure 3.2: Same as Fig. 3.1, but for the Tursunov et al. potential. taken to be the one of two point-like sources. The potential can be written as

VT(r) = Vc(r) + Ve(r) , (3.21) where Vc(r) = −V0`J exp −a`J r2 (3.22) and Ve(r) = ZαZd α r . (3.23)

The parameters are chosen to reproduce the scattering phase shifts of Refs. [25,26,27], and are listed in Table 3.3. This potential has no tensor components. It is shown for different channels in Fig. 3.2.

3.4.3 The Mukhamedzhanov et al. model

The Mukhamedzhanov et al. potential has been constructed in Ref. [13], where it was stressed the importance of the ANCs to get the right astrophysical factor at low energies. The potential is also in this case central, and thus there are no D-state components in the bound-state wave function.

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3.4. POTENTIAL MODELS 27 ` 0 1 2 J 1 0 1 2 1 2 3 2S+1_` J 3S1 3P0 3P1 3P2 3D1 3D2 3D3 V`J 0 92.44 68.0 79.0 85.0 63.0 69.0 80.88 a`J 0.25 0.22 0.22 0.22 0.19 0.19 0.19

Table 3.3: Parameters for the Tursunov et al. potential of Ref. [12]: a`J are given in

fm−2, while V₀`J in MeV.

the S-state ANC calculated with VH is C0 = 2.7 fm−1/2 instead of C0exp = (2.30 ± 0.12)

fm−1/2 [13].

To obtain a phase-equivalent potential which reproduces C₀exp, Mukhamedzhanov et al. followed the procedure of Ref. [23], of which we give some basic information. Let us start with a Hamiltonian H0 which reproduces the scattering phase shifts. It is

possible to add a new term in the Hamiltonian which leaves the bound-state binding energy and the scattering phase shifts unchanged. This new potential depends on the old wave function ϕ(r) which satisfied the Schr¨odinger equation

Hϕ(r) = Eϕ(r) . (3.24)

It has been shown in Ref. [23], and we have also re-derived that the new potential must have the form

V1(r) = −2~ 2 2µ d2 dr2 log 1 + (τ − 1) Z r 0 dx x2ϕ2(x) , (3.25)

where τ is a generic constant. With this new term, the new wave function ϕ1(r) can

be rewritten in terms of the old one ϕ(r) as ϕ1(r) =

√

τ ϕ(r)

1 + (τ − 1)R₀rdx x2_ϕ2_(x) . (3.26)

Note that for r → +∞,

lim r→+∞ ϕ1(r) = ϕ(r) √ τ , (3.27) and C0 = Cold 0 √ τ . (3.28)

We see that, in order to get the correct ANC, it is sufficient to adjust τ . For example, to change the ANC of Ref. [11] to the correct experimental value, we need τ = 1.378

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28 CHAPTER 3. THE 6_{LI WAVE FUNCTION} 0 2 4 6 8 r (fm) -60 -50 -40 -30 -20 -10 0 V M (MeV) Mukhamedzhanov et al. Hammache et al.

Figure 3.3: The Mukhamedzhanov et al. (red) potential for the 3S1 channel in

com-parison with the Hammache et al. (black) model.

[13]. We have also demonstrated that the new wave function ϕ1(r) is normalized to

one, if the old one ϕ(r) was normalized to one. This can be proved easily as follows: Z +∞ 0 dr r2ϕ2₁(r) = Z +∞ 0 dr r2 _√ τ ϕ(r) 1 + (τ − 1)R₀rdx x2_ϕ2_(x) 2 (3.29a) = τ τ − 1 Z +∞ 0 dr (τ − 1)r 2_ϕ2_(r) 1 + (τ − 1) R₀rdx x2_ϕ2_(x)2 (3.29b) = − τ τ − 1 Z +∞ 0 dr d dr 1 1 + (τ − 1)R₀rdx x2_ϕ2_(x) (3.29c) = − τ τ − 1 1 1 + (τ − 1)Rr 0 dx x 2_ϕ2_(x) +∞ 0 (3.29d) = − τ τ − 1 1 τ − 1 = 1 . (3.29e)

The Mukhamedzhanov et al. model, in conclusion, has the same form of Hammache et al. of Ref. [11], but adds the new term V1(r) of Eq. (3.25) with τ = 1.378 and ϕ(r)

of Ref. [11]. The potential in the3_S

1 channel is shown in Fig.3.3, and compared with

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3.4. POTENTIAL MODELS 29

3.4.4 The Dubovichenko non-central potential model

This potential is the only one analysed in this thesis which involves a tensor com-ponent, and thus tries to reproduce the magnetic dipole and the electric quadrupole moments of 6Li. The potential has been introduced in Ref. [15] and can be written as VD(r) = Vc(r) + Vt(r)S12+ Ve(r) , (3.30)

where Vc(r) is the central term, Vt(r) is the tensor term and Ve(r) is the Coulomb

term written as in Eq.(3.23). In turn, Vc(r), Vt(r) and S12 are given by

Vc(r) = V0exp −a r2 , (3.31a)

Vt(r) = V1exp −b r2 (3.31b) and S12 ≡ S12(ˆr) = 6 (S · r)2 r2 − 2S(S + 1) , (3.31c)

S being the spin of the two-body system. Note that the presence of the tensor oper-ator implies that ` and S are not any more “good” quantum numbers and therefore

0 2 4 6 8 r (fm) -80 -70 -60 -50 -40 -30 -20 -10 0 V D 1+ (MeV) 3 S₁→3S 1 3 S 1→ 3 D 1 3 D 1→ 3 D 1 3 D₁→3S 1

Figure 3.4: The Dubovichenko potential for the coupled 3_S

1 and 3D1 channels. The

potentials for the 3S1 → 3D1 and for the 3D1 → 3S1 transitions are equal, since we

must have h3_S

1|V |3D1i = h3D1|V†|3S1i = h3D1|V |3S1i. This is due to the fact that

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30 CHAPTER 3. THE 6_{LI WAVE FUNCTION} 0 2 4 6 8 r (fm) -70 -60 -50 -40 -30 -20 -10 0 10 V D (MeV) 3 P 0 3 P₁ 3 P 2 3 D 2 3 D₃

Figure 3.5: The Dubovichenko potential for the (uncoupled) 2S+1`J →2S+1`J

transi-tions.

channel with the same J but different ` (or S) become coupled. This is the case of

6_{Li, which becomes a} 3_S

1-3D1 state.

The parameters used to fit the binding energy, the quadrupole moment and the D−state percentage are V0 = 71.979 MeV, a = 0.2 fm−2, V1 = 27.0 MeV and b = 1.12

fm−2 [15]. These are also the ones used in this thesis for the 3S1 and 3D1 channels,

while in Ref. [15] a second set of parameters has also been derived. The original po-tential of Ref. [15] has been fitted to reproduce also the elastic scattering phase shifts and mixing parameter in the Jπ = 1+ channel (see Fig. 4 of Ref. [15]). Nonetheless fails to reproduce the other channels phase shifts and the 3+ _{resonance at 0.711 MeV.}

Therefore, in this thesis we have modified it in order to reproduce also these observ-ables by slightly changing only V0. The new parameters are obtained with a χ2 test

for all the channels except for the 3+_{, in which the attention has been focused on}

reproducing the resonance. They are listed in Table. 3.4. The scattering phase shifts experimental data has been taken from Refs. [26,28, 29], where the scattering matrix is in the Blatt-Biedenham parametrization. In Fig. 3.4 we plot the potential for the coupled channels, while in Fig. 3.5 we plot the potential for the uncoupled channels. To implement the potential in the code it is useful to project it on the initial and final channel. From Ref. [30], we can write the potential for the transition 1+ _{to 1}+

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3.4. POTENTIAL MODELS 31

Jπ ₀− ₁− ₂− ₁+ ₂+ ₃+

V0 77.4 73.08 78.42 72.979 73.8 86.139

Table 3.4: Parameters for the present version of the Dubovichenko potential model. a, b and V1 have been kept the same as the original model. V0 is in MeV.

as V11 = Vc(r) + Ve(r) + A11Vt(r) , (3.32a) V12 = V21= A12Vt(r) , (3.32b) V22 = Vc(r) + Ve(r) + A22Vt(r) , (3.32c) with Amn= −2(J − 1)/(2J + 1) 6pJ(J + 1)/(2J + 1) 6pJ(J + 1)/(2J + 1) −2(J + 2)/(2J + 1) . (3.33)

This projection works for two coupled-channels `1 = |J − 1| and `2 = J + 1. For the

other channels we have

3_P 0 →3P0 ⇒ Vc(r) + Ve(r) + Vt(r)A22, (3.34a) 3 P1 →3P1 ⇒ Vc(r) + Ve(r) − Vt(r)(A11+ A22) , (3.34b) 3 P2 →3P2 ⇒ Vc(r) + Ve(r) + Vt(r)A11, (3.34c) 3_D 2 →3D2 ⇒ Vc(r) + Ve(r) + Vt(r)(A11+ A22) , (3.34d) 3_D 3 →3D3 ⇒ Vc(r) + Ve(r) + Vt(r)A11. (3.34e)

3.4.5 The new non-central potential model

This potential model is a modified version of the Dubovichenko potential described in the previous section. The aim of this modification is to obtain a potential model with a tensor component and, in addition, able to reproduce the S-state ANC, C0 = 2.30

fm−1/2. We start from Ref. [13] and search for a potential for which the Hamiltonian operator has the solution

~ ϕ1(r) = √ τ ϕ(r)~ 1 + (τ − 1)R₀rdx x2 P i[ϕi(x)]2 , (3.35)

where ~ϕ(r) is the vector containing ϕ1(r) and ϕ2(r), the two solutions with different

` values, and τ is equal to

τ = C0 CN 0

2

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being C0and C0N the old and the new ANC for the S-state wave function, respectively.

To get the correct ANC for the S-state, i.e. C₀N = 2.30 fm−1/2, we need τ = 1.181, because, as we will see in Sec. 3.9.4, the value for the ANC obtained with the VD

potential is C0 = 2.5 fm−1/2.

The function of Eq. (3.35) is the solution of the Hamiltonian if the potential has the form VN D(r) = VD(r) + VN(r) , (3.37) with VN(r) defined as VN(r) = −2 ~ 2 2µ d2 dr2 log 1 + (τ − 1) Z r 0 dx x2ϕ(x) · ~~ ϕ(x) 1 0 0 1 + τ − 1 1 + (τ − 1)R₀rdx x2_{ϕ(x) · ~}_~ _ϕ(x) ϕ2 2(r) 0 0 −ϕ2 1(r) d drlog ϕ1(r) ϕ2(r) . (3.38) The demonstration of this result is given in Appendix B.

The new potential VN Dtherefore reproduces the6Li binding energy and S-state ANC,

and implements the D-state contribution to the bound wave function. For the ini-tial scattering state, VN D coincides with VD, since VD and VN D are phase equivalent

potentials.

3.5 The Numerov’s method

The first numerical approach to solve the Schr¨odinger equation (3.8) that we are going to analyse is the so-called Numerov’s method (NM). This method was developed by the astronomer Boris Vasil’evich Numerov and its goal is to solve second order differential equations which do not contain the first derivative of the function we are looking for. This is a two step method with an error at the fourth order; this means that to evaluate the function at the n-th step we need to know the function at the (n − 1) and (n − 2) steps and the function will be evaluated with a precision of h4_{, with h the}

step we are using.

The grid in the independent variable x is divided thus in equal length steps. Therefore, if we want to obtain the function over a range R, we need to evaluate it at N points, where

N = R

h . (3.39)

Then we define

xn≡ h n (3.40)

as the n-th step in the independent variable. We can also define a general zn quantity as

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3.5. THE NUMEROV’S METHOD 33 where z = z(x) is a function of x.

In this thesis we use the Numerov’s method in the following two versions:

• NM: this is the original Numerov’s method, used to solve equations in the form

y00(x) + A(x)y(x) = S(x) , (3.42)

where y(x) is our unknown function, while A(x) and S(x) are known functions. We have also developed a way to use this method to solve two coupled differential equations of the form

y₁00(x) + A1(x)y1(x) = S1(x)y2(x) ,

y₂00(x) + A2(x)y2(x) = S2(x)y1(x) ,

(3.43)

where yi(x), with i = {1, 2}, are our unknown functions, while Ai(x) and Si(x)

are known functions.

• RNM: this is the so-called the Renormalized Numerov’s method [31]. It is a generalization of the NM and can be used to solve coupled differential equations in the form I d 2 dx2 + A(x) y(x) = 0 , (3.44)

where y(x) is our unknown vector of functions of dimension K, A(x) is a known matrix of functions and I is the identity matrix, both of dimension (K × K). This method is therefore more general than the one used in Eq. (3.43), but it will be used for K = 2.

We now go into the details of these two versions of the method.

3.5.1 Numerov’s method

This method is used to solve equations as in (3.42). The recursive NM solution of this equation is yn+1 1 + 1 12An+1 − yn 2 −10 12h 2_A n + yn−1 1 + 1 12h 2_A n−1 = 1 12h 2_(S n+1+ 10Sn+ Sn−1) + O(h6) . (3.45)

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This can be easily derived considering the Taylor series of the y(x) function, i.e.

y(x ± h) = y(x) ± h y(1)(x) + 1 2!h 2_y(2)_{(x) ±} 1 3!h 3_y(3)_(x) + 1 4!h 4_y(4)_{(x) ±} 1 5!h 5_y(5)_{(x) + O(h}6_{) , (3.46)}

where y(n)_{(x) is the n-th derivative of y(x) respect to x.}

Summing y(x + h) and y(x − h) and using that

h2y_n(4) = −An+1 yn+1+ Sn+1+ 2An yn− 2Sn− An−1 yn−1+ Sn−1+ O(h4) (3.47)

we get Eq. (3.45). This equation can be used to evaluate yn+1 from yn and yn−1, or,

alternatively, yn−1 knowing yn and yn+1.

Application of the NM to the Schr¨odinger equation

The method discussed above is very useful to solve equations in one variable. Our time-independent Schr¨odinger equation for ϕ(r) contains three variables: r, θ and φ and in spherical coordinates has the form

" −~ 2 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + ˆ `2 2µr2 + V (r) # ϕ(r) = E ϕ(r) . (3.48)

Writing ϕ as in Eq. (3.10), multiplying by χ†_1σ0 Y_l∗0_m0(Ω) and integrating over dΩ =

dφ d cos θ we get Z dΩ χ†_1σ0Y_l∗0_m0(Ω) −h 2 2 1 r2 ∂ ∂r r2 ∂ ∂r +`(` + 1) 2µr2 + V (r) − E X ` u`(r) X σ h`m, 1σ|1M i Y`m(Ω)χ1σ = 0 (3.49) If we define V_m``00_σ0(r) ≡ X σ h`m, 1σ|1M i h`0_m0_{, 1σ}0_{|1M i}χ † 1σ0 Z dΩ Y_`∗0_m0(Ω)V (r)Y_`m(Ω) χ1σ, (3.50) it follows that −~ 2 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + `(` + 1) 2µr2 − E u`+ X `0 V_m``00_σ0u_`0 = 0 (3.51)

Multipotential analysis of the radiative capture alpha + d -> 6Li + gamma

UNIVERSIT `

A DEGLI STUDI DI PISA

DIPARTIMENTO DI FISICA “E. FERMI”

CORSO DI LAUREA MAGISTRALE IN FISICA

MULTIPOTENTIAL ANALYSIS OF THE RADIATIVE

CAPTURE REACTION α + d →

Li + γ

TESI DI LAUREA MAGISTRALE

Candidato:

ALESSANDRO GRASSI

Relatrice:

Prof.ssa

LAURA ELISA MARCUCCI

Contents

List of Figures

List of Tables

List of Outputs

Chapter 1

Introduction

Chapter 2

Radiative capture reactions

2.1

Perturbation theory in quantum mechanics and

transition probabilities

2.2

The radiative capture

2.3

Multipole expansion

2.3.1

Partial wave decomposition of the initial scattering wave

function

2.4

Long wavelength approximation

2.5

The α + d →

Li + γ reaction

Chapter 3

The

6

Li wave function

3.1

Li properties

3.2

The Schr¨

odinger equation for the

Li = α + d

bound state

3.3

The relative wave function

3.4

Potential models

3.4.1

The Hammache et al. model

3.4.2

The Tursunov et al. model

3.4.3

The Mukhamedzhanov et al. model

3.4.4

The Dubovichenko non-central potential model

3.4.5

The new non-central potential model

3.5

The Numerov’s method

3.5.1

Numerov’s method