Facolt`a di Scienze Matematiche Fisiche e Naturali Corso di Laurea Magistrale in Fisica
Curriculum Fisica Teorica Piano di Studi Nucleare
Non-Symmetrized Hyperspherical
Harmonics Method for a three-body
system
Thesis for the Master’s Degree in Physics
Author:
Alessia Nannini
Advisor:
Prof.ssa Laura Elisa
Marcucci
Academic Year
The Hyperspherical Harmonics (HH) method has been widely applied in the study of the bound states of few-body systems, namely A = 3 and A = 4. Usually, the use of the HH basis is preceeded by a symmetrization procedure that takes into account the fact that protons and neutrons are fermions, and therefore the wave function has to be antisymmetric under exchange of any pair of these particles. However, this preliminary step is not strictly necessary, since after the diagonalization of the Hamiltonian, the eigenvectors turn out to have a well-defined symmetry under particle permutation. In this case, the method is known as Non-Symmetrized Hy-perspherical Harmonics (NSHH) method. In this work we present a generalization of the NSHH method for a three-body system, composed by two particles having equal masses, but di↵erent from the mass of the third particle. In particular we focus on the 3H,3He and 3⇤H systems.
In the first part of the thesis we present a complete description of the NSHH method. Then, we study the convergence of the method in order to estimate its accuracy. We also compare some selected cases, for the3H nucleus, treated as an
equal masses three-body system, to the results present in the literature, obtained with the (symmetrized) standard HH method. The agreement has been found in general very nice. We proceed studying the di↵erence of binding energy between
3H and3He due to the di↵erence between the proton and the neutron masses, and
we have obtained a nice agreement with the “standard” perturbative estimate. Finally, we study the3
⇤H system with several nucleon-nucleon (N N ) and
hyperon-nucleon (Y N ) potential models. Also in this case, the results are found in nice agreement with those present in the literature.
a three-body system. Our results, obtained with a variety of nuclear potential models, demonstrate the validity of the method, and are therefore very promising. In a near future, we plan to apply this method to other systems, which can be clustered as three-body systems, where one of the constituents is an4He particle, as the Borromean 6He (seen as 4He + n + n) and the 6Li (seen as 4He + n + p) nuclei. Furthermore, our method can also be generalized to the case of A = 3 scattering systems, taking advantage of the already existing expertise in the case of the standard HH method.
Abstract ii
List of Figures v
List of Tables vii
1 Introduction 1
2 The Hyperspherical Harmonics formalism 7
2.1 The Jacobi coordinates for a three-body system . . . 7
2.2 The Hyperspherical coordinates and the infinitesimal volume element . . . 10
2.3 The Hyperspherical Harmonics functions . . . 11
2.3.1 Spin part of the wave function . . . 14
2.4 The A=3 wave function . . . 15
3 The Non-Symmetrized Hyperspherical Harmonics Method 19 3.1 The Transformation Coefficients . . . 20
3.1.1 The transformation coefficients for A=3 . . . 21
3.2 Solution of the A=3 Schr¨odinger equation . . . 23
3.2.1 The norm matrix elements . . . 24
3.2.2 The kinetic matrix elements . . . 25
3.2.3 The potential matrix elements . . . 26 iii
4 Nuclear interactions 31
4.1 Central potentials for3H and 3He . . . 31
4.1.1 The Volkov potential . . . 32
4.1.2 The Afnan-Tang potential . . . 32
4.1.3 The Malfliet-Tjon potential . . . 33
4.1.4 The Minnesota potential . . . 34
4.1.5 The Argonne AV40 potential . . . 34
4.2 Central potentials for the hypernucleus3 ⇤H . . . 35
4.2.1 The Gaussian potential . . . 35
4.2.2 The Minnesota potential . . . 35
4.2.3 The USMANI potential . . . 36
5 Numerical results 39 5.1 Convergence study . . . 39
5.2 Results for the 3H and 3He systems . . . 41
5.3 Results for the 3 ⇤H hypernucleus . . . 47
6 Conclusions 55 6.1 Outlook . . . 57
A 59 A.1 Details of the calculation for A = 3 . . . 59
A.1.1 Simplification of the kinetic energy . . . 59
A.1.2 Calculation of the Raynal-Revai coefficients . . . 61
B 67 B.1 Numerical integration methods . . . 67
B.1.1 The Chebyshev-Gauss quadrature method . . . 67
B.1.2 The Legendre-Gauss quadrature method . . . 68
B.1.3 The Gauss-Laguerre quadrature method . . . 69
1.1 Big Bang Nucleosynthesis reactions network from Ref. [3], where p stands for proton, n for neutron, for photon, D for deuteron and T for3H. . . . . 3
1.2 Upper panels: the energy density as a function of the total baryon density (upper left panel) and the pressure versus the energy density (upper right panel). The upper (lower) curves refer to the case of purely neutron (hyperonic) matter. Lower panels: gravitational mass as a function of the stellar radius (lower left panel) and of the central baryon density (lower right panel) in the case of nucleon stars (upper curves) and hyperon stars (lower curves). The measured value of ⇠ 2M for the PSR J0348+0432 NS is given as a dashed horizontal line. Figure taken from Ref. [11]. . . 5 2.1 The three possible definitions of the Jacobi coordinates, listed
in Eqs. (2.8), (2.9) and (2.10), respectively. See text for more details. . . 9 5.1 In panel (a) we show the binding energy B (in MeV) as
func-tion of the parameter (in fm 1) for the Volkov potential
model [19], with Gmax = 20, j
max = 6 and Nmax = 16,
and using mn = mp. Panel (b) is a focus on panel (a) for
2 fm 1 7 fm 1. . . 43 v
5.2 In panel (a) we show the binding energy B (in MeV) as func-tion of the parameter Nmax for the AV40 potential model [23],
with Gmax = 20, j
max = 8 and = 4 fm 1, and using
2.1 List of the channels for3H and3He (J⇡ = 1/2+). ⇤ and ⌃ are
the total orbital angular momentum and the total spin of the nuclei, while Gmaxis the maximum value for the grandangular
momentum so that Eq. (2.28) is satisfied, to be chosen in order to reach the desired accuracy. See text for more details. . . 16 2.2 List of all the states st, obtained from the channel (⇤, ⌃, Gmax) =
(0, 1/2, 4). See text for more details. . . 17 4.1 List of the parameters for the Volkov potential [19]. . . 32 4.2 List of parameters for the Afnan-Tang potential [20]. . . 33 4.3 List of the parameters for the Malfliet-Tjon potential [21]. . . 33 4.4 List of the parameters for the Minnesota potential [22]. . . 34 4.5 List of the parameters for the central N N and ⇤N potentials
of Ref. [25]. . . 36 4.6 List of the parameters for the Usmani potential [28]. . . 37 5.1 The 3H binding energy B (in MeV) calculated with the
Ar-gonne AV40 potential model [23], using mn = mp, jmax = 8,
= 4 fm 1 and Gmax = 20, and varying N
max. . . 45
5.2 The3H binding energy B (in MeV) calculated with the Volkov
potential model [19], using mn = mp, Gmax = 20, jmax = 10
and = 4 fm 1. . . 45
5.3 The3H binding energy B (in MeV) calculated with the
Afnan-Tang potential model [20], using mn = mp, Gmax = 20, jmax =
10 and = 4 fm 1. . . 45
5.4 The3H binding energy B (in MeV) calculated with the
Malfliet-Tjon potential model [21], using mn= mp, Gmax = 20, jmax =
10 and = 4 fm 1. . . 45
5.5 The 3H binding energy B (in MeV) calculated with the
Min-nesota potential model [22], using mn = mp, Gmax = 20,
jmax = 10 and = 4 fm 1. . . 45
5.6 The3H binding energy B (in MeV) calculated with the Volkov
potential model [19], using mn = mp, Nmax = 16 and = 4
fm 1. . . 46
5.7 The 3H binding energy B (in MeV) calculated with the
Min-nesota potential model [22], using mn = mp, Nmax = 16 and
= 4 fm 1. . . 46
5.8 The3H binding energy B (in MeV) calculated with the
Afnan-Tang potential model [20], using mn = mp, Nmax = 16 and
= 4 fm 1. . . 47
5.9 The3H binding energy B (in MeV) calculated with the
Malfliet-Tjon potential model [21], using mn = mp, Nmax = 16 and
= 4 fm 1. . . 47
5.10 The 3H binding energy B (in MeV) calculated with the AV40
potential model [23], using mn = mp and = 4 fm 1. . . 48
5.11 The 3H binding energy B (in MeV) calculated with the AV40
potential model [23], using mn = mp, Gmax = 60, Nmax = 24
5.12 The 3H binding energy obtained in the present work (p.w.)
and in the literature, using mn= mp, the3H and3He binding
energy calculated taking into account the di↵erence of masses, B as defined in the text, and the 3He binding energy
calcu-lated including also the Coulomb interaction between the two protons. All the values are measured in MeV. . . 49 5.13 List of values for , Nmax, jmax, and Gmax of Eq. (2.42), used
in Table 5.12 for all the potential models. . . 49 5.14 Mean value for the kinetic energy operator, B estimated
with the perturbative theory (PT), and B calculated with the NSHH for the di↵erent potential models considered in this work. See text for more details. . . 50 5.15 The3
⇤H binding energy B (in MeV) calculated with the
Gaus-sian potential model of Ref. [25], using Nmax = 20 and = 4
fm 1. . . 50
5.16 The 3
⇤H binding energy B (in MeV) calculated with the the
Gaussian potential model of Ref. [25], using Gmax = 20, j max =
8 and = 4 fm 1. . . 51
5.17 The 3
⇤H binding energy B (in MeV) calculated with the MN9
potential model of Ref. [26], using = 4 fm 1. . . 52
5.18 The 3
⇤H binding energy B (in MeV) calculated with the AU
potential model of Ref. [28], using = 4 fm 1. . . 53
5.19 The3
⇤H binding energy (in MeV) obtained in the present work
(p.w.) is compared with the results present in the literature. . 54 5.20 List of values for , Nmax, jmax and Gmax of Eq. (2.42), used
in Table 5.19 for all the potential models. . . 54 B.1 List of the number of points for the di↵erent integrations
present in our calculations, in the case of spin-independent and Minnesota potentials (A) and AV40 potential (B). . . 69
Introduction
The aim of this thesis is to present and test a method applicable for a few-body system composed of particles of di↵erent masses. In particular, we study the case of a three-body system, composed by two particles having the same mass but di↵erent from the mass of the third particle. We have decided to use the so-called Non-Symmetrized Hyperspherical Harmonics (NSHH) method [1], which is a recent development of the “standard” Hyperspherical Harmonics (HH) method, where the HH basis is used without previous sym-metrization. This method was developed by Gattobigio et al. [1], and later extended by Barnea et al. [2]. With the HH method, the wave function is written as
= X
p
p , (1.1)
p = 1, 2, 3 corresponding to the di↵erent particle permutations we can have by coupling first particle i with particle j, and then with particle k. For instance, we indicate with p = 3 the permutation 12; 3. By introducing the `, Sij, Tij quantum numbers, namely the orbital angular momentum, the
spin and the isospin of the ij couple, the condition ( )`+Sij+Tij = odd must
be imposed in order to guarantee the antisymmetry of the wave function under the exchange of any pair of particles, which, in our case, are fermions.
This symmetrization procedure is not used in the NSHH. In fact, in Ref. [1] it has been demonstrated that the eigenvectors reflect the correct symmetry under particle exchange of the Hamiltonian matrix even by choosing one single permutation, and avoiding to antisymmetrize the HH basis after its diagonalization. The NSHH method requires though a larger number of the expansion elements with respect to the standard HH method. However, it has the advantage to reduce the computational e↵ort due to the symmetrization procedure, and, moreover, the same expansion can be easily re-arranged for di↵erent systems. In this work we exploit this last propriety to study the3H, 3He and 3
⇤H systems.
In order to test our method, we study the first two systems listed above with five di↵erent potentials, and the latter, the hypernucleus system, with three potential models. We start with simple central and spin-independent nucleon-nucleon (N N ) and hyperon-nucleon (Y N ) interactions, and then, we take into account more complex ones. None of the interactions considered is realistic. Therefore, we will not compare our results to the experimental data. However, these interactions are useful to test step by step the method and to compare with results obtained with other techniques. The interest in developing a method which allows the study of light nuclei, composed by di↵erent particles, is linked to some actual topics. In the field of nuclear reactions of astrophysical interest, we find the so-called “6Li puzzle”, while
the growing interest in hypernuclei is due to the so-called “hyperon puzzle” in neutron stars.
According to modern cosmology, about 180 s after the Big Bang (BB), it is possible to form stable light nuclei due to the decrease of temperature, and therefore of the average particles kinetic energy. This process is known as Big Bang Nucleosynthesis (BBN) and is believed to last for about 10 20 minutes. After this period, the temperature, and consequently the average kinetic energy, becomes too low to overcome the Coulomb barrier between nuclei: further nucleosynthesis is stopped. According to the BBN theory, the main products of this process are 2H, 3He, 4He and 7Li. In Fig.1.1 we
Figure 1.1: Big Bang Nucleosynthesis reactions network from Ref. [3], where p stands for proton, n for neutron, for photon, D for deuteron and T for
3H.
show a summary scheme of the main reactions involved [3]. To be noticed that among the primordial nuclei, 6Li is not considered. This is because 6Li
is a weakly bound nucleus, and therefore is supposed to be present only in small percentages. The only reaction which, according to BBN, should be responsible for primordial 6Li is
4He + d
! 6Li + . (1.2) The predicted ratio between the 6Li and 7Li primordial abundances should
be about 10 5. In 2006, Asplund et al. [4] performed high resolution
obser-vations of Li absorption lines in old stars, and they found a ratio of 10 2, i.e.
discrep-ancy is known as “6Li puzzle”. In order to solve this 6Li puzzle, there has
been a renewed interest for the reaction (1.2) [5–8]. In particular, theoretical studies [7] have been performed within a two-body framework, where6Li has
been considered as a compound system, made of a deuteron and a 4He. In
principle though, one should perform an ab-initio study of reaction (1.2), where the 6Li bound and 4He + d scattering states are seen as a six-body systems. This kind of study is still nowadays hardly used for systems with A > 4, and has been only partially performed in Ref. [9]. A good compro-mise, more realistic than the clusterization adopted in Ref. [7], but still more manageable than solving a six-body problem, consists in describing the 6Li
as an4He + n + p three-body system, that could be handled with the present
NSHH method. The first study of the reaction (1.2), using a three-body framework to solve the bound state, has been performed in Ref. [10].
The interest in hypernuclei is a consequence of the so-called “hyperon puz-zle” in neutron stars (NSs). A neutron star is the product of gravitational collapse of a star, which before collapsing had a mass between 10 and 29 solar masses (M ). In stars with these masses, nuclear fusion reactions, that are essential in order to contrast gravitational pressure, proceed until the formation of 56Fe. When in the core of the star all light nuclei have been
exhausted, the fusion ends and the core is supported by electron-degeneracy pressure only. Further deposits of mass from the burning outer shells cause the core to exceed the Chandrasekhar limit. At this point, the gravitational pressure overcomes the electron-degeneracy pressure, and the core collapses further. This contraction turns the temperature to values so high to allow electrons and protons to combine into neutrons, according to the reaction p + e ! n + ⌫e . When a density of about 4⇥ 1017 kg/m3 is reached,
neutron-degeneracy pressure halts the contraction. What is left is a NS. Note that if the remnant has a mass greater than about 3M , it collapses further to become a black hole. The interior of a NS is in fact not made of only neutrons. Since the neutron and the proton chemical potentials, are increasing functions of the density [11], at the high values of the stellar
cen-Figure 1.2: Upper panels: the energy density as a function of the total baryon density (upper left panel) and the pressure versus the energy density (upper right panel). The upper (lower) curves refer to the case of purely neutron (hyperonic) matter. Lower panels: gravitational mass as a function of the stellar radius (lower left panel) and of the central baryon density (lower right panel) in the case of nucleon stars (upper curves) and hyperon stars (lower curves). The measured value of⇠ 2M for the PSR J0348+0432 NS is given as a dashed horizontal line. Figure taken from Ref. [11].
tral density, they get large enough to trigger the formation of hyperons. This needs to be taken into account in the equation of state (EoS), which becomes
softer than in the case of purely neutron matter. This can be understood essentially considering that the presence of hyperons produces a significant decrease of the system pressure, with a consequence reduction of the stellar masses. This is shown in Fig. 1.2, from which we can conclude that if on the one hand the presence of hyperons is unavoidable, on the other hand it seems to be not compatible with the measured mass of the PSR J0348+0432 NS, which has been found to be about 2M . This discrepancy is known as the “hyperon puzzle”. Some solutions are proposed in Ref. [11], and are related to our poor knowledge of the hyperon-nucleon (Y N ) two-body interaction, as well as the three-body interactions of the type Y N N , Y Y N and Y Y Y . Therefore, the study of hypernuclei, in order to create realistic models for these interactions, becomes crucial. With our NSHH method applied to the
3
⇤H, we have moved the first step.
We conclude this Chapter by mentioning how the thesis is organized.
In Chapter 2, we introduce the adopted system of coordinates, i.e. the Jacobi and the hyperspherical coordinates, and we describe their properties. Then we define the Hyperspherical Harmonics functions.
In Chapter 3, we present the NSHH method in its generalized form, in order to treat systems with di↵erent species of particles and interactions. Some de-tails are given, as the discussion of the so-called transformation coefficients, and the solution of the Schr¨odinger equation.
In Chapter 4, we present all the di↵erent interactions that we have used for the 3H, 3He and 3
⇤H systems.
In Chapter 5, we show our results. First we consider the 3H and 3He
sys-tems, treating them as equal mass systems in order to compare our results with those present in the literature. Then we study these nuclei considering that neutron and proton have di↵erent masses. Finally, we consider the 3
⇤H
hypernucleus.
In Chapter 6, we summarize the di↵erent steps of this work and the obtained results, and we suggest further applications of the method.
The Hyperspherical Harmonics
formalism
The aim of this chapter is to describe the so-called Hyperspherical-Harmonics (HH) method.
2.1
The Jacobi coordinates for a three-body
system
Let us consider a three-body system: we first have to define a convenient set of coordinates. Clearly, one of the coordinates should be that of the center of mass, in order to decouple the internal relative motion from the motion of the center of mass. By doing so, the kinetic energy operator can be expressed as a sum of two terms: one depending only on the center of mass coordinates, and one depending on the internal coordinates. In the case of a system of A particles, with mass mi, position ri, and momentum pi (i = 1, 2, . . . , A),
we can write the kinetic operator T as the sum of the single-particle kinetic operators [12]:
T = A X i=1 p2 i 2mi = ~ 2 2 A X i=1 r2 ri 1 mi . (2.1) We then define xi = pmi ri, so that
T = ~ 2 2 A X i=1 r2xi . (2.2)
Let us now define the center of mass coordinate yA as yA= A X i=1 p mi xi mtot , (2.3)
where mtotstands for the total mass of the system, and the Jacobi coordinates
yi, i = 1, . . . , N , with N = A 1, as a linear combination of xi, i.e.
yi =
A
X
j=1
cijxj . (2.4)
It is possible to demonstrate that (see Appendix A.1.1 for the details of this calculation), imposing the following conditions [12]
A X i=1 cjicji = 1 M (j = 1, . . . , N ) , (2.5) A X i=1 cjicki = 0 (j 6= k = 1, . . . , N) , (2.6)
M being a reference mass, the kinetic energy operator can be cast in the form T = ~ 2 2mtotr 2 yA ~2 2M N X i=1 r2yi . (2.7)
For a three-body problem A = 3 and N = 2, and we choose the Jacobi coordinates so that y2 is proportional to r2 r1, as in Fig. 2.1.
In this first case, we find the following set of coordinates: 8 > < > : y(3)2 = q m2 M (m1+m2)x1+ q m 1 M (m1+m2)x2 , y(3)1 = q m1m3 M mtot(m1+m2)x1 q m2m3 M mtot(m1+m2)x2+ q m1+m2 M mtotx3 . (2.8)
(a) (b) (c)
Figure 2.1: The three possible definitions of the Jacobi coordinates, listed in Eqs. (2.8), (2.9) and (2.10), respectively. See text for more details.
Note that with y(3)1 and y(3)2 we mean particle m1 coupled with particle
m2 and particle m3 coupled with the center of mass of the previous two.
Proceeding in an analogous way, but taking y2 proportional to r1 r3 or r3 r2, we find two other sets of coordinates:
8 > < > : y(2)2 =q m3 M (m1+m3)x1 q m 1 M (m1+m3)x3 , y(2)1 = q m1m2 M mtot(m1+m3)x1+ q m1+m3 M mtotx2 q m 2m3 M mtot(m1+m3)x3 . (2.9) 8 > < > : y(1)2 = q m3 M (m2+m3)x2+ q m 2 M (m2+m3)x3 , y(1)1 =qm2+m3 M mtotx1 q m 1m2 M mtot(m2+m3)x2 q m 1m3 M mtot(m2+m3)x3 . (2.10) Finally, the Jacobian of the transformation from the cartesian to the Jacobian coordiantes is provided by the determinant of the matrix A 1, where
A = 0 B B B @ q m1m3 M mtotm12 q m2m3 M mtotm12 + q m12 M mmtot q m2 M m12 + q m2 M m12 0 +q m1 mtot + q m2 mtot + q m3 mtot 1 C C C A .
The notation mij ⌘ mi+ mj is here introduced in order to simplify the
ex-pression. In the case of two equal mass particles m1 = m2, and the Jacobian
2.2
The Hyperspherical coordinates and the
infinitesimal volume element
Let us consider one set of Jacobi coordinates y1, y2, . . . , yN as discussed in
the previous section and let ˆyi ⌘ (✓i, 'i) be the polar angles of these Jacobi
coordinates. Therefore yi corresponds to (yi, ✓i, 'i), with yi ⌘ |yi|. We now
introduce the hyperradius r as r = v u u tXN i=1 y2 i , (2.11)
and N 1 so-called hyperangles 2, . . . , N, through the relations:
y1 = r sin( N) . . . sin( 2) ,
. . .
yi = r sin( N) . . . sin( i+1) . . . cos( i) ,
yN = r cos( N) , (2.12)
with i = 2, . . . , N 1, and 0 i ⇡/2.
Using as coordinates (r, i), with i = 2, . . . , N , the Laplace operator can be
cast in the form [12] r2 = N X i=1 r2yi = @2 @r2 + (3N 1) r @ @r + ⇤2 N(⌦N) r2 , (2.13)
where ⇤N(⌦N) is called the grandangular momentum operator, and is
ex-plicitly written as [12] ⇤2i(⌦i) =
@2
@ 2 i
+ [3(i 2) cot i+ 2(cot i tan i)]
@ @ i ˆl2 i(ˆyi) cos2 i +⇤ 2 i 1(⌦i 1) sin2 i . (2.14) Here ˆl2
i is the angular momentum operator associated with the i-th Jacobi
The volume element d⌧ = dy1. . . dyN, in terms of the hyperspherical
coor-dinates, becomes
d⌧ = dˆy1. . . dˆyN y21. . . y2N dy1. . . dyN
= J dˆy1. . . dˆyN y12. . . yN2 dr d 2. . . d N , (2.15)
where J is the Jacobian of the hyperspherical transformation J = y1 r y2 r y3 r . . . yN r
y1cot N y2cot N y3cot N . . . yNcot N
... ... ... . .. ... y1cot 2 y2tan 2 0 0 0
, and it is explicitly written as
J = 1 r 1 sin 2cos 2 . . . 1 sin Ncos N y1. . . yN . (2.16)
Finally, writing d⌧ as a function of r and ⌦N, we get
d⌧ = rN 1r2Ndr d⌦ N ,
where d⌦N encloses all the angular and hyperangular part, and is explicitly
written as d⌦N = dˆy1. . . dˆyN N Y j=2 (sin j)3j 4 cos2 jd j . (2.17)
2.3
The Hyperspherical Harmonics functions
The Hyperspherical Harmonics functions are the generalization to a 3N di-mensional space of the spherical harmonics functions, which are usually em-ployed in the study of three-dimensional two-body quantum mechanic prob-lems. In order to deduce their expression, we consider an homogeneous poly-nomial of order n defined as
⇣n = X (n=n1+···+nN) a(n)xn11. . . x nN N , (2.18)
where (n) stands for the set n1, . . . , nN such that
n1+· · · + nN = n . (2.19)
⇣n is called harmonic polynomial and denoted as hn if, when Eq. (2.19) is
satisfied, the coefficients a(n) are such that
r2h
n= 0 . (2.20)
If hGis an harmonic polynomial with degree G, the function YG(⌦) = r GhG
is independent from the hyperradius. Therefore, it is possible to demonstrate that [12] r2h G = @2 @r2 + (3N 1) r @ @r + ⇤2 N(⌦N) r2 r GY G(⌦N) = ⇥G(G + 3N 2) + ⇤2N(⌦N) ⇤ rG 2YG(⌦N) = 0 . (2.21)
The function YG(⌦N) is called Hyperspherical Harmonic (HH) function, and
it is an eigenfunction of the grandangular momentum operator ⇤2
N(⌦N), with
eigenvalue G(G + 3N 2). It is straightforward to demonstrate that in the case of N = 1, ⇤2(⌦
N) = l2, i.e. the orbital angular momentum, and the
previous expression reduces to
l2Ylm(ˆy) = l(l + 1)Ylm(ˆy) , (2.22)
where Ylm(ˆy) is the spherical harmonic function. On the other hand, for
a system of three particles, A = 3 and therefore N = 2. Then Eq. (2.21) becomes
⇤22(⌦2)Y[G](⌦2) = G(G + 4)Y[G](⌦2) , (2.23)
where with [G] we mean all the quantum numbers we used to construct YG(⌦2).
Let us consider the following expression:
where f (cos 2 2) is a function to be determined [12].
By using Eqs. (2.14) and (2.23), and calling z = cos 2 2, we obtain
(1 z2)f00+ (l2 l1 z(l1+ l2+ 3))f0+ ↵f = 0 , (2.25)
with
↵ = G(G + 4) (l1+ l2)(l1+ l2+ 4) . (2.26)
The functions f (cos 2 2) which satisfy Eq. (2.25) are the so-called Jacobi
polynomials, and are indicated as Pl1+12,l2+12
n (cos 2 2) [12].
Then Eq. (2.24) becomes
Y[G](⌦2) = Nnl2,⌫2(cos 2)l2(sin 2)l1Yl1m1(ˆy1)Yl2m2(ˆy2) ⇥ Pl1+12,l2+12 n (cos 2 2) , (2.27) with G = 2n + l1+ l2 , n = 0, 1, . . . , (2.28) and ⌫2 = 2 + 2n + l1+ l2 . (2.29) Nl2,⌫2
n2 is a normalization factor given by [12]
Nl2,⌫2 n2 = s 2⌫2 (2 + n2 + l1 + l2)n2! (n2+ l1+ 3/2) (n2+ l2+ 3/2) , (2.30) and rGY
[G](⌦2) is an harmonic polynomial of order G. Finally, we notice that
we have indicated with [G] the quantum numbers [G] = l1, l2, n. The HH
functions just derived for A = 3, can be generalized also for larger values of A, but this generalization is beyond the scope of this thesis.
It is useful to combine the HH functions in order to construct the eigenfunc-tions of the total orbital angular momentum ⇤. Using the Clebsch-Gordan coefficients, we can define for A = 3
H{G}= X m1,m2 Y[G](⌦2)(l1m1l2m2|⇤⇤z) ⌘ [Yl1(ˆy1)Yl2(ˆy2)]⇤,⇤zP l1,l2 n2 ( 2) , (2.31)
where {G} stands for {l1, l2, ⇤, n2} and we have defined Pl1,l2 n2 ( 2) = N l2,⌫2 n2 (cos 2) l2(sin 2)l1P l1+12,l2+12 n (cos 2 2) . (2.32)
In a system of A = 3 fermions we also need to consider the spin part of the wave function, that will be discussed in the next subsection. On the other hand, when we study3H and3He we are considering protons and neutrons as
di↵erent particles and therefore, we do not use the isospin formalism. In the case of the 3
⇤H hypernucleus, neutrons and protons will be again considered
as the same particle, and the third di↵erent particle will be the ⇤-hyperon.
2.3.1
Spin part of the wave function
In this work we use the LS-coupling scheme, and therefore the total spin ⌃ and the total orbital angular momentum ⇤ need to be combined so that
~
J = ~⇤+~⌃, J being the total angular momentum, and Jzits third component.
Therefore, including the spin part, Eq. (2.31) can be written as BHJ{G}(⌦2) = 8 < :[Yl1(ˆy1)Yl2(ˆy2)]⇤,⇤z " 1 2 ⌦ 1 2 S,s⌦ 1 2 # ⌃,⌃z 9 = ; J,Jz ⇥ Pl1,l2 n2 ( 2) , (2.33)
where S is the spin of the first couple, ⌃ is the total spin of the system and ⌃z its third component. By indicating [[12⌦12]S,s⌦12]⌃,⌃z with S ,⌃,⌃z(i, j, k),
it can be demonstrated by direct calculations that
S0,⌃0,⌃0z(jk, i)† S ,⌃,⌃z(ij, k) = ( )S p (2S0 + 1)(2S + 1) ⇥ ( 1/2 ⌃ S 1/2 1/2 S0 ) ⌃0⌃ ⌃0z⌃z , (2.34)
where {...} indicates the 6j Wigner coefficients.
2.4
The A=3 wave function
We now consider our system made of A = 3 particles, and we indicate with (i, j, k) a permutation p of the indices (1, 2, 3). In the following, we will refer to the pair ij as a reference pair, i.e. the pair of particles which are coupled first, and such that their relative position is proportional to the Jacobi coordinate y2. From now on we call
H{G}(i, j, k)⌘ H{G}(⌦(p)2 ) , (2.35) and equivalently BH{G}J (i, j, k)⌘ BHJ {G}(⌦ (p) 2 ) , (2.36)
where p = 1, 2, 3, and (⌦(p)2 ) specifies the hyperangular variables constructed with the permutation p of the particles. For instance, if p = 3 (i, j, k = 1, 2, 3) the reference-pair is (ij) = (12), as in panel (a) of Fig. 2.1. In our approach, the two equal-mass particles are chosen as particles 12, while the third particle with di↵erent mass is set as particle 3. Therefore, we expand the wave function only with p = 3. The wave function that describes our system can now be cast in the form
= X
{G}
BH{G}J (⌦(3)) u{G}(r) , (2.37) where BHJ
{G}(⌦(3)) is given by Eq. (2.33) (we drop the subscript “2” from
⌦2), and u{G}(r) is a function of only the hyperradius r. Furthermore, {G}
stands for all the quantum numbers useful to describe our system, as outlined in the previous section, i.e. (we drop the subscript “2” from n2)
{G} = {l1, l2, ⇤, n, S, ⌃} .
Therefore, we have found convenient to classify the BHJ
{G} functions as
fol-lows:
• we call “channel” (ch) a set of possible values for the total spin of the system ⌃, the total orbital angular momenta ⇤, so that ~J = ~⇤ + ~⌃, and the maximum value of the grandangular momentum Gmax for ⌃ and ⇤;
ch ⇤ ⌃ Gmax
1 0 1/2 -2 1 1/2 -3 1 3/2 -4 2 3/2
-Table 2.1: List of the channels for3H and3He (J⇡ = 1/2+). ⇤ and ⌃ are the
total orbital angular momentum and the total spin of the nuclei, while Gmax
is the maximum value for the grandangular momentum so that Eq. (2.28) is satisfied, to be chosen in order to reach the desired accuracy. See text for more details.
• we call “state” (st) the set of all possible momenta l1, l2, all possible spin
quantum numbers and all possible values of n, so that Eq. (2.28) is satisfied for G that runs from Gmin = l
1+ l2 to Gmax;
• we call “sub-channel” ↵ the pair channel-state (ch, st) but excluding n. To be more explicit we will have
ch⌘ (⇤, ⌃, Gmax) , st⌘ l1, l2, S, n ,
↵⌘ (⇤↵, ⌃↵, Gmax↵ , l↵,1, l↵,2, S↵) .
The channels used in this work are listed in Table 2.1, and some examples of states are given in Table 2.2. In particular, since in this work we use central potentials, only the first channel of Table 2.1 is necessary. Thus, the particular configuration that we want to analyze will be completely described by giving the pair (ch, st), or equivalently (↵, n↵). Here the index ↵, labeling
n, was introduced to indicate all the possible values for n compatible with the ↵ sub-channel according to Eq. (2.28). Furthermore, in Table 2.2 we have imposed G = l1 + l2 + 2n↵, with G 6 Gmax, ~l1 + ~l2 = ~⇤, l1 + l2 =
even, since 3H and 3He have positive parity, and l
2 6 jmax+ S, jmax being
st l1 l2 S n 1 0 0 0 0 2 0 0 1 0 3 0 0 0 1 4 0 0 1 1 5 1 1 0 0 6 1 1 1 0 7 0 0 0 2 8 0 0 1 2 9 1 1 0 1 10 1 1 1 1 11 2 2 0 0 12 2 2 1 0
Table 2.2: List of all the states st, obtained from the channel (⇤, ⌃, Gmax) =
(0, 1/2, 4). See text for more details. reached.
In conclusion, we rewrite Eq. (2.37) as = X
↵,n↵
BH↵,nJ ↵(12, 3) u↵,n↵(r) . (2.38)
In the present work, the hyperradial function is itself expanded on a suitable basis, i.e. a set of generalized Laguerre polynomials. Therefore, following Ref. [13], we can write
u↵,n↵(r) =
X
l
cl,↵,n↵fl(r) , (2.39)
where cl,↵,n↵ are unknown coefficients, and
fl(r) = s l! (l + 5)! 3 (5)L l( r)e 2r . (2.40)
Here(5)L
l( ⇢) are Laguerre polynomials, and the first factor is a
normaliza-tion term so that Z
dr r5fl(r)fm(r) = lm . (2.41)
In Eq. (2.40) is a non-linear parameter, whose typical values are in the range [2 5] fm 1. With this assumptions, the set of functions f
l(r) satisfies
two criteria, i.e. (i) fl(r) goes to zero for r that goes to infinity; (ii) fl(r)
constitutes an orthonormal basis. Note that the sum of Eq. (2.39) will be truncated up to a maximum value Nmax, to be chosen to reach the desired
accuracy.
By using Eq. (2.39), the wave function can be cast in the form = ↵Xmax ↵=1 nXmax n↵=0 NXmax l=0 cl,↵,n↵BH J ↵,n↵(⌦ (3))f l(r) . (2.42)
The sum over ↵ is truncated after the inclusion of ↵max most important
sub-channels. The values of ↵max, nmax, Nmax are determined through the study
The Non-Symmetrized
Hyperspherical Harmonics
Method
The use of the HH basis in the study of an A-body system of nucleons, treated as indistinguishable particles, is usually preceeded by a preliminary symmetrization procedure, which takes into account the fermionic nature of these particles. In fact protons and neutrons, having very similar masses, are often considered as identical spin-1/2 particles, and therefore, the wave func-tion needs to be antisymmetric under the exchange of a nucleon-nucleon pair. Several schemes to construct hyperspherical functions, with an arbitrary per-mutational symmetry, have been developed, like those in Refs. [2, 14–16]. However, this preliminary procedure is not strictly necessary. In fact, as shown in Ref. [1], after the diagonalization of the Hamiltonian matrix, the eigenvectors reflect anyway the symmetries present in it. Moreover, they have well defined symmetry under particles permutations, respecting the antisym-metry of the wave function. In the present work, the HH functions are used without any previous symmetrization, in order to describe the bound states of three-particles systems. This Non-Symmetrized Hyperspherical Harmonics method (NSHH) requires to push the expansion to large angular momentum
values, and convergence is obtained including a large number of expansion terms. On the contrary, some advantages are present and are here consid-ered. First, the computational e↵ort due to the symmetrization procedure of the HH functions is not needed. Then, the same expansion can be applied to di↵erent physical systems without any significant change. Lastly, the in-clusion of particles with di↵erent mass is straight forward.
In this chapter, we will use the Rayleigh-Ritz variational principle, in order to determine the eigenvalues and the eigenfunctions of the bound-state Ref. [13]. To be more explicit, we will discuss how the norm, the kinetic energy and the potential matrix elements are calculated. It will be shown that in order to speed up the calculation, it is necessary a good choice of the reference pair (see Fig. 2.1). Therefore, we first discuss the transformation coefficients needed to switch from a given H{G}(⌦(p)) to a di↵erent H
{G}(⌦(p
0)
).
3.1
The Transformation Coefficients
The HH functions, as shown in Chapter 2, are built from the Jacobi vectors, and di↵erent permutations of the particles can be used. It can be shown that [13] H{G}(⌦(p)N ) = X {G0} a(p{G},{G!p0),G,⇤0} H{G0}(⌦(p 0) N ) , (3.1)
where the grandangular momentum is conserved, i.e. G = G0, but we have
{G} 6= {G0}, since all possible combinations of l
1, l2, ⇤, n2 are allowed. Note
that also ⇤ = ⇤0, and we neglected the spin-part, already discussed in Chap-ter 2 (see Eq. (2.34)).
The transformation coefficients (TC) a(p{G},{G!p0),G0} can be calculated, for A = 3,
through the Raynal-Revai recurrence relations [17]. Alternately we can use the orthonormality of the HH basis [13]
a(p{G},{G!p0),G,⇤0} =
Z
d⌦(pN0) [H{G0}(⌦(p 0)
3.1.1
The transformation coefficients for A=3
Let us write Eq. (3.2) in the case of A = 3 as a(p!p`1,`2,n,`0),G,⇤0 1,`02,n0 = Z d⌦(p2 0) [H{`0 1,`02,n0,⇤⇤z}(⌦ (p0) 2 )]† H{`1,`2,n,⇤⇤z}(⌦ (p) 2 ) , (3.3)
where `i is the orbital angular momentum, associated with the Jacobi
coor-dinate y(p)i , and `0i is associated with y (p0)
i .
The Jacobi vectors constructed with the p permutation are linearly related to the Jacobi vectors for the p0 permutation as
y(p)i = 2 X j=1 ↵(p)ij(p0)y (p0) j , i = 1, 2 , (3.4)
where ↵(p)ij(p0)are numerical coefficients that can be determined from Eqs. (2.8),
(2.9) and (2.10).
Furthermore, we observe that the following identity holds for spherical har-monics functions [13]: c`Yl,m(ˆc) = X `a+`b=` a`ab`bp4⇡D `,`a,`b[Y`a(ˆa)Y`b(ˆb)]`,m , (3.5)
where a, b and c are moduli of vectors such that c = a + b, and D`,`a,`b =
s
(2` + 1)! (2`a+ 1)!(2`b + 1)!
. (3.6) It can be demonstrated (the details of the calculation are in Appendix A.1.2) that (sin (p)2 )`1(cos (p) 2 )`2[Y`1(ˆy (p) 1 )Y`2(ˆy (p) 2 )]⇤,⇤z = X `0 1,`02 C`(p),(p1,`2,`00) 1,`02(sin 2 (p0) , cos 2(p 0) ) ⇥ [Y`0 1(ˆy (p0) 1 )Y`0 2(ˆy (p0) 2 )]⇤,⇤z , (3.7)
with C`(p),(p1,`2,`00) 1,`02(sin 2 (p0) , cos 2(p 0) ) = X 1+ 2=`1 X 0 1+ 02=`2 (sin 2(p 0) ) 1+ 01(cos 2(p 0) ) 2+ 02 ⇥ (↵(p)11(p0))(↵ (p) 12(p0))(↵ (p) 21(p0))(↵ (p) 22(p0)) ⇥ ( ) 1+ 2+ 01+ 02 D `1, 1, 2D`1, 1, 2 ⇥ ˆ`1`ˆ2`ˆ01`ˆ02ˆ1ˆ2 ˆ01ˆ02 1 01 `01 0 0 0 ! ⇥ 2 0 2 `02 0 0 0 !8>>< > > : 1 2 `1 0 1 02 `2 `01 `02 ⇤ 9 > > = > > ; . (3.8) Here ˆ` ⌘ p2` + 1, and the round (curly) brackets denote 3j (9j) Wigner coefficients.
The HH functions that appear in Eq. (3.3) depend on z = cos (2 (p)2 ) through the Jacobi polynomials. Therefore, we can indicate with µ the angle included between the two Jacobi vectors y(p)1 and y(p)2 (µ = y(p)1 · y(p)2 ), obtaining
cos (2 (p)2 ) = 2[(↵(p)11(p0)) 2(sin (p0) 2 )2+ (↵ (p) 22(p0)) 2(cos (p0) 2 )2 + 2(↵(p)12(p0))(↵ (p) 21(p0))µ sin (p0) 2 cos (p0) 2 ] 1 . (3.9) Moreover, Z dˆy1dˆy2 [Y`0 1(ˆy1)Y`02(ˆy1)] † ⇤,⇤z[Y`1(ˆy1)Y`2(ˆy1)]⇤,⇤z = 1 2`ˆ02`ˆ1`ˆ2`ˆ01 X (2 + 1) ⇥ ( )⇤+`2+`02 ( `01 `02 ⇤ `2 `1 ) ⇥ `01 `1 0 0 0 ! `0 2 `2 0 0 0 ! ⇥ Z dµ P (µ) . (3.10)
Finally, using Eq. (3.3) and Eq. (3.10), we obtain a(p`1!p,`2,n,`0),G,L0 1,`02,n0 = N `0 2,⌫2 n0 2 N `2,⌫2 n2 1 2 Z ⇡ 2 0 d 2 Z 1 1 dµ (cos 2(p 0) )2+`02(sin 2(p 0) )2+`01 ⇥ P`01+1/2,`02+1/2 n02 (cos 2 2 (p0) )P`1+1/2,`2+1/2 n2 (cos 2 2 (p)) ⇥ X , 1, 2 C`(p),(p1,`2,0)1, 2(sin 2(p 0) , cos 2(p 0) )P (µ) ⇥ ( )⇤+ 2+`02(2 + 1)ˆ`0 1`ˆ02ˆ1ˆ2 ⇥ ( `0 1 `02 ⇤ 2 1 ) `0 1 1 0 0 0 ! `0 2 2 0 0 0 ! . (3.11) Here the curly brackets indicate the 6j Wigner coefficients. The integra-tions over 2 and µ has been performed by mean of the Chebyshev-Gauss
and Legendre-Gauss quadrature methods respectively, as explained in Ap-pendix B.1.1 and B.1.2. In Eq. (3.11), ⌫2 = G + 2, and Nn`22,⌫2 is the
normal-ization given by Eq. (2.30). More details of the calculation are provided in Appendix A.1.2.
3.2
Solution of the A=3 Schr¨
odinger
equa-tion
In the center of mass reference frame, the Schr¨odinger equation can be written with the help of Eq. (2.7) as
[H E] = " ~2 2M N X i=1 r2 yi + V (y1, y2) E # = 0 . (3.12) The wave function is expanded as in Eq. (2.42) and can be cast in the form
= X
⇠
c⇠ ⇠ , (3.13)
where ⇠ is a complete set of states, and ⇠ is the index that labels all the
quantum numbers defining the basis elements. The sum is limited as indi-cated in Eq. (2.42). The expansion coefficients can be determined using the
Rayleigh-Ritz variational principle [13], which states that
h c |H E| i = 0 , (3.14)
where c denotes the variation of the wave function with respect to the
coefficients c⇠. By doing the di↵erentiation, the problem is then reduced to
a generalized eigenvalue-eigenvector problem of the form X
⇠0
h ⇠|H E| ⇠0ic⇠0 = 0 . (3.15)
The Lanczos algorithm [18] is used to solve this problem. The use of this algorithm is required by the large dimension of the calculated matrices, as it will be discuss in Chapter 5.
Finally, we can write X
⇠0
[h ⇠|T + V | ⇠0i Eh ⇠| ⇠0i] c⇠0 = 0 , (3.16)
and therefore we need to calculate the norm, the kinetic and potential energy matrix elements.
3.2.1
The norm matrix elements
The norm matrix elements can be written as
N{G0},k,{G},l ⌘ h ⇠0| ⇠i ⌘ h⇠0|⇠i = h n , k|↵ n↵, li , (3.17)
where ↵ and denote the sub-channels, n↵ and n are linked to the
grandan-gular momentum by Eq. (2.28), and l and k are the indexes of the Laguerre polynomials.
With some calculations it can be demonstrated that h n , k|↵ n↵, li = J ↵ n↵n
Z
dr r(5)fl(r) fk(r)
= J ↵ n↵n lk , (3.18)
where fl(r) is the hyperradial function given by Eq. (2.40) and J is the
Jacobian calculated in Eq. (2.16) of Chapter 2. Therefore, the norm matrix is basically the identity matrix in the NSHH method.
3.2.2
The kinetic matrix elements
Di↵erently from Eq. (3.17), where N{G0},k,{G},l is simply a numerical quantity,
the calculation of the kinetic energy matrix elements is a little bit more complex. We define
T{G0},k,{G},l ⌘ h ⇠|T | ⇠0i = h n , k|T |↵ n↵, li , (3.19)
and substituting Eq. (2.21) in Eq. (3.19), we obtain T{G0},k,{G},l = ~ 2 2M J ↵ n↵n Z dr r5fk(r) ⇥ G(G + 4)fl(r) r2 + 5 f0 l(r) r + f 00 l (r) . (3.20)
The first derivative of the hyperradial functions fl(r) is
fl0(r) = Nl d dr ⇣ e 2r (5)Ll( r) ⌘ = Nl 1 2e x 2 (5)Ll(x) + e x 2 d dx (5)L l(x) = 2 fl(r) + Nle x 2 d dx (5)L l(x) , (3.21)
while the second derivative is fl00(r) = 2 f 0 l(r) + Nl 2 d dxe x 2 d dx (5)L l(x) = 2 f 0 l(r) Nl 2 2 e x 2 d dx (5)L l(x) + Nl 2e x 2 d 2 dx2 (5)L l(x) , (3.22)
where Nl stands for
q
l! (l+5)
3 of Eq. (2.40). Finally, we can write
T{G0},k,{G},l = ~ 2 2M J ↵ n↵n G(G + 4) Z dr r4fk(r)fl(r) + 5 Z dr r4fk(r)fl0(r) + Z dr r5fk(r)fl00(r) = ~ 2 2M J ↵ n↵n 2A G(G + 4) Z dx x3 e xF1(x) + 5 Z dx x4e xF2(x) + Z dx x5e xF3(x) . (3.23)
In Eq. (3.23) we have defined A = s k! l! (6 + k) (6 + l) , (3.24) F1(x) =(5)Lk(x)(5)Ll(x) , (3.25) F2(x) = (5)L5(x) ✓ (5)L l(x) 1 2+ (5)L0 l(x) ◆ , (3.26) F3(x) = (5)Lk(x) ✓ 1 4 (5)L l(x) 3 2 (5)L0 l(x) +(5)L00l(x) ◆ , (3.27) and (5)L0
l(x) and (5)L00l(x) denote the first and the second derivative of the
Laguerre polynomial (5)L
l(x) respectively.
The change of variable
x = r 2 ,
applied in Eqs. (3.21), (3.22) and (3.23), has been used in order to simplify the calculations of the integrals in r. In fact, the one-dimensional integration in x can be easily performed with the Gauss-Laguerre quadrature method, according to Eq. (B.11), as shown in Appendix B.1.3.
3.2.3
The potential matrix elements
The potential matrix elements in Eq. (3.12) are given by V{G0},k,{G},l ⌘ h ⇠|V (y1, y2)| ⇠0i
=h ⇠|V (1, 2) + V (2, 3) + V (1, 3)| ⇠0i , (3.28)
with V (i, j), indicating the two-body interaction between particle i and par-ticle j.
Since it is easier to evaluate the matrix elements when the Jacobi coordi-nate y2 is proportional to ri rj, we will make use of the TC as defined
in Eq. (3.11). We remind that, as explained in Chapter 2, we indicate with p = 1 the permutation (23,1), with p = 2 the permutation (13,2) and finally
with p = 3, the permutation (12,3) (see Fig. 2.1). Then, we can write h n , k|V (1, 2)|↵ n↵, li = J Z dr r5d⌦(3)2 fk(r)fl(r) ⇥ BH†n (⌦ (3) 2 )V (1, 2)BH↵n↵(⌦ (3) 2 ) . (3.29)
Using Eq. (2.33) we obtain
h n , k|V (1, 2)|↵ n↵, li = JN ` ,⌫ n Nn`↵↵,⌫↵ L↵L 1 2 Z dr r5 Z d 2 ⇥ X I X K (2⇤ + 1)(2⌃ + 1)(2I + 1)(2K + 1) ⇥ 8 > > < > > : ` S I L 1/2 K ⇤ ⌃ J 9 > > = > > ; 8 > > < > > : `↵ S↵ I L↵ 1/2 K ⇤ ⌃ J 9 > > = > > ; ⇥ fk(r)fl(r)V` ,S `I,(12)↵,S↵(r, cos 2 2) ⇥ (cos 2)`↵+` +1/2(sin 2)L↵+L +1/2 ⇥ PL↵+1/2,`↵+1/2 n↵ (cos 2 2) ⇥ PnL +1/2,` +1/2(cos 2 2) . (3.30)
Note that here we have indicated with Laand `athe orbital angular momenta
associated with the Jacobi vectors y2 and y1, and they correspond to `2 and
`1 as used in Chapter 2. However, the notation of Chapter 2 would be here
quite heavy. The 9j Wigner coefficients are due to the recoupling from LS-scheme to JJ-LS-scheme, and we have indicated with ~I = ~`+ ~S↵ and ~K = ~L +~12.
For the termh ⇠0|V (2, 3)| ⇠i, we can write
h n , k|V (2, 3)|↵ n↵, li = J Z dr r5d⌦(3)2 fk(r)fl(r) ⇥ BH†n (⌦ (3) 2 )V (2, 3)BH↵n↵(⌦ (3) 2 ) = J Z dr r5d⌦(1)2 f n ,k(r)f↵n↵,l(r) ⇥ X `0,L0,S0 a(3` ,L ,n ,`!1),G,⇤0,L0,n0 BH†0n0(⌦ (1) 2 )V (2, 3) ⇥ X `0↵,L0↵,S0↵ a(3!1),G,⇤`↵,L↵,n↵,`0↵,L↵0,n0↵BH↵0n0↵(⌦(1)2 ) . (3.31)
The integration is performed as in the previous case, obtaining h ⇠0|V (2, 3)| ⇠i = J L0↵L0 S0 ↵12S012 1 2 Z dr r5 fk(r)fl(r) Z d (1)2 ⇥ X `0,L0,S0 a(3` ,L ,n ,`!1),G,⇤0,L0,n0 X `0↵,L0↵,S0↵ a(3`↵!1),G,⇤,L↵,n↵,`0↵,L0↵,n0↵ ⇥ Nn`00,⌫ N `0↵,⌫↵ n0↵ BH†0n0(⌦ (1) 2 )BH↵0n0↵(⌦(1)2 ) ⇥ X I X K (2⇤ + 1)(2⌃ + 1)(2I + 1)(2K + 1) ⇥ 8 > > < > > : `0 S0 I L0 1/2 K ⇤ ⌃ J 9 > > = > > ; 8 > > < > > : `0↵ S↵0 I L0↵ 1/2 K ⇤ ⌃ J 9 > > = > > ; V`I,(23)0,S0`0↵,S0↵(r, cos 2 2) ⇥ (cos 2)` 0 ↵+`0+1/2(sin 2)L 0 ↵+L0+1/2 ⇥ PL0↵+1/2,`0↵+1/2 n0↵ (cos 2 2)P L0+1/2,`0+1/2 n0 (cos 2 2) . (3.32)
We proceed similary for V (1, 3), and in conclusion, the final expression for the potential matrix elements is
h ⇠0|V (y1, y1)| ⇠i = J L0↵L0 S0 ↵12S012 1 2 Z dr r5fk(r)fl(r) ⇥ X `0,L0,S0 X `0↵,L0↵,S0↵ Nn`00,⌫ N `0 ↵,⌫↵ n0↵ ⇥ X I X K (2⇤ + 1)(2⌃ + 1)(2I + 1)(2K + 1) ⇥ 8 > > < > > : `0 S0 I L0 1/2 K ⇤ ⌃ J 9 > > = > > ; 8 > > < > > : `0↵ S↵0 I L0 ↵ 1/2 K ⇤ ⌃ J 9 > > = > > ; ⇥ 0, ↵,↵0V120↵0(r) + a(3`↵!1),G,⇤,L↵,n↵,`0↵,L0↵,n0↵a (3!1),G,⇤ ` ,L ,n ,`0,L0,n0V230↵0(r) + a(3!2),G,⇤`↵,L↵,n↵,`0↵,L0↵,n0↵a(3!2),G,⇤` ,L ,n ,`0,L0,n0V130↵0(r) ,(3.33)
where Vij0↵0(r) = Z d 2 (cos 2)` 0 ↵+`0+1/2(sin 2)L 0 ↵+L0+1/2VI,(ij) `0,S0`0↵,S↵0(r, cos 2 2) ⇥ PL0↵+1/2,`0↵+1/2 n0↵ (cos 2 2)P L0+1/2,`0+1/2 n0 (cos 2 2) , (3.34)
and V`I,(ij)0,S0`0↵,S↵0(r, cos 2 2) is the nuclear interaction. All the nuclear
interac-tions used in this work will be presented in the next Chapter.
The integrations over r and 2 have been performed by mean of the
Gauss-Laguerre and the Chebyshev-Gauss quadrature methods respectively (see Appendix B.1.3 and B.1.1). The number of points used for the integration depends on the potential; in Table B.1 the number of points is shown for some of the potentials used, in relation to the di↵erent quadrature methods.
Nuclear interactions
In this chapter we discuss all the nuclear interaction models used in the present work. It is to be noticed that we have considered only the two-body interactions, i.e. VN N for nuclear systems, and also V⇤N for the
hypernu-cleus 3
⇤H. We present first the central spin-independent and spin-dependent
potential models, used to study 3H and 3He, and then the models used in
the study of the 3
⇤H hypernucleus. Note that the nucleus of 3He was studied
including the Coulomb interaction, acting between the two protons.
4.1
Central potentials for
3H and
3He
The first three models for VN N that we have considered are spin-independent.
They are the so-called Volkov, Afnan-Tang and Malfliet-Tjon potentials. The last two are spin-dependent potentials, and are the so-called Minnesota and Argonne AV40.
Constant Value Unit V1 144.86 MeV
V2 83.34 MeV 1 0.82 fm2 2 1.6 fm2
Table 4.1: List of the parameters for the Volkov potential [19].
4.1.1
The Volkov potential
The first spin-independent potential analyzed in this work is the one elabo-rated by Volkov [19]. Its explicit expression is
V (r) = V1e ⇣ r 1 ⌘2 + V2e ⇣ r 2 ⌘2 , (4.1) and therefore the potential is written as sum of Gaussian functions. The convergence of the expansion on the HH basis is, in this case, rather fast, as we will see in Chapter 5. All the parameters present in Eq. (4.1) are listed in Table 4.1.
4.1.2
The Afnan-Tang potential
The Afnan-Tang interaction of Ref. [20] is written again as sum of Gaussian functions, i.e. V (r) = V1e µ1r 2 + V2e µ2r 2 + V3e µ3r 2 + V4e µ4r 2 + V5e µ2r 5 , (4.2) and the parameters Vi and µi are listed in Table 4.2. Since the Afnan-Tang
potential is the sum of more terms compared with the Volkov potential, it is assumed to be a “more realistic” potential model. However it should be noticed that, being just central and spin-independent models, none of the models presented so far is realistic. They are typically used to test the methods, comparing the results with those obtained with other techniques.
Constant Value Unit V1 1000 MeV V2 163.35 MeV V3 21.50 MeV V4 83.0 MeV V5 11.50 MeV µ1 3.00 fm 2 µ2 1.05 fm 2 µ3 0.6 fm 2 µ4 0.8 fm 2 µ5 0.4 fm 2
Table 4.2: List of parameters for the Afnan-Tang potential [20].
4.1.3
The Malfliet-Tjon potential
The Malfliet-Tjon potential [21] is a central potential, written as sum of Yukawa functions. It can be cast in the form
V (r) = 1 r V1e
µ1r+ V
2e µ2r , (4.3)
where all the parameters are listed in Table 4.3. Constant Value Unit
V1 1458.047 MeV
V2 578.089 MeV
µ1 3.11 fm 1
µ2 1.55 fm 1
Table 4.3: List of the parameters for the Malfliet-Tjon potential [21]. As it will be shown in the next Chapter, this potential implies a slow con-vergence in the expansion of the HH basis.
4.1.4
The Minnesota potential
The first spin-dependent potential studied in this work is the so-called Min-nesota potential model [22]. This model has the advantage to be more real-istic than the previous ones, but still has a fast convergence. The interaction can be written as VS=0(r) = V1e µ1r 2 + V2e µ2r 2 , (4.4) VS=1(r) = V1e µ1r 2 + V3e µ3r 2 , (4.5) S being the spin of the N N system. The values for the di↵erent parameters are listed in Table 4.4.
Constant Value Unit V1 200 MeV V2 91 MeV V3 178 MeV µ1 1.487 fm 2 µ2 0.465 fm 2 µ3 0.639 fm 2
Table 4.4: List of the parameters for the Minnesota potential [22].
4.1.5
The Argonne AV4
0potential
A more realistic interaction is the Argonne AV40 of Ref. [23]. This is a
reprojection of the realistic Argonne AV18 [24], which is written as the sum of 18 operators, including both isospin-independent and isospin-dependent tensor, and spin-orbit operators.
The expression for the AV40 interaction is
V (r) = Vc(r) + V⌧(r) ⌧1· ⌧2+ V (r) 1· 2+ V ⌧(r) ( 1· 2)(⌧1· ⌧2) , (4.6)
where i (⌧i) are the spin (isospin) Pauli matrices, and Vc(r), V⌧(r), V (r)
potential model is the “most realistic” one considered in this work, and indeed the most challenging for our method, as we will discuss in the next Chapter.
4.2
Central potentials for the hypernucleus
3 ⇤
H
The hypernucleus 3⇤H is a bound system composed by a neutron, a proton
and the ⇤-hyperon. In this study we consider the proton and neutron as indistinguishable particles, denoted as N and having mass mN = 938.9187
MeV. They will be set on the pair 12, while particle 3 is the ⇤-hyperon. Therefore, we have to reintroduce the isospin formalism for the two nucleons, and we have to impose ` + S + T to be odd, to ensure the antisymmetry of the wave function under the exchange of particle 1 and 2 (` is the orbital angular momentum, S and T the pair spin and isospin, respectively). The ⇤ particle has been taken with mass M⇤ = 6/5 mNin the case of the Gaussian
potential, and with M⇤ = 1115.683 MeV in the case of the Minnesota and
the Usmani potential models. We review these models in the following.
4.2.1
The Gaussian potential
For both N N and ⇤N interactions, we use spin-independent Gaussian po-tentials [25] of the form
V↵↵ = V0↵↵e (r↵↵/ ↵↵)2 , (4.7)
where r↵↵ is the ↵ ↵ distance, ↵ being N or ⇤.
The values for the di↵erent parameters are listed in Table 4.5.
4.2.2
The Minnesota potential
The Minnesota potential is used also for the3
⇤H, to be able to compare our
re-sults ith those of Ref. [26]. In particular, for the N N pair we recall Eqs. (4.4) and (4.5), while as ⇤N interaction, we have used VS=1(r) of Eq. (4.5) reduced
Constant Value Unit VN N 0 72.5 MeV VN ⇤ 0 58.4 MeV N N 1.470 fm N ⇤ 1.034 fm
Table 4.5: List of the parameters for the central N N and ⇤N potentials of Ref. [25].
by a factor 0.9 [26]. In order to simplify the notation we will refer to this potential as MN9.
4.2.3
The USMANI potential
This is the most realistic model often used in our work and it is often consid-ered in the literature. For the ⇤N interaction, the Usmani potential model is an Argonne-like potential [27], which includes both two-body ⇤N and three-body ⇤N N components. In this work, however, we consider only the two-body ⇤N part. The expression for the potential is then written as
V⇤N(r) = V0(r) + V 4 T 2 ⇡(r) ⇤· N , (4.8) where V0(r) = Wc 1 + era¯r ¯ V T⇡2(r) , (4.9) and T⇡(r) = 1 + 3 µ⇡r + 3 (µ⇡r)2 e µ⇡r µ⇡r (1 e cr2) , (4.10) r being the distance of the ⇤N pair, and ⇤( N) in Eq. (4.8) being the Pauli
spin matrix for the ⇤(N ) particle. All the parameters present in Eqs. (4.8)-(4.10) were chosen as in Refs. [26] and [28], and are listed in Table 4.6. This ⇤N interaction model is used in conjunction with the Argonne AV40 of
Ref. [23], given in Eq. (4.6), for the N N component. We will refer to this potential model as AU.
Constant Value Unit Wc 2137 MeV ¯ r 0.5 fm a 0.2 fm ¯ V 6.15 MeV V 0.24 MeV c 2.0 fm 2 1/µ⇡ 1.429504 fm
Numerical results
In this Chapter we show the results obtained for all the potential models listed in Chapter 4, and we compare them with the results existing in the literature, in order to verify the validity of our method. Before doing this, we present the study of the convergence of the method for the 3H system.
The other considered nuclei present similar features.
5.1
Convergence study
We recall that the wave function is written as in Eq. (2.42) = ↵Xmax ↵=1 nXmax n↵=0 NXmax l=0 cl,↵,n↵BH J ↵,n↵(⌦ (3))f l(r) , (5.1)
and therefore we need to study the convergence on ↵max, nmax, and Nmax.
Furthermore, the radial function written as in Eq. (2.40), presents a non-linear parameter , for which we need to find a range of values such that the binding energy is stable. Finally we have introduced in Section 2.4 the value jmax, i.e. the value of the total angular momentum of the reference
pair. Note that in these convergence studies we have used mn = mp.
We start by considering the parameter . The behavior of the binding energy as a function of is shown for the Volkov potential in Fig. 5.1. We mention here that for all the other potential models we have treated, the results are similar. The other parameters were kept constant: Gmax = 20, N
max = 16
and jmax = 6. This particular shape, that increases for low values of , is
constant for some central values and decreases again for large values of , allows to determine a so-called plateau, and the optimal value for has to be chosen on this plateau. In this work we have chosen = 4 fm 1 for all
the potentials.
In Fig. 5.2 we fix jmax = 8, = 4 fm 1 and Gmax = 20, and we show the
pattern of convergence for the binding energy B with respect to Nmax, in the
case of the Argonne AV40 model. Here convergence is reached for Nmax = 24,
i.e. for higher Nmax value, B changes by less than 1 keV. Some of the data
points of Fig. 5.2 are listed in Table 5.1. To be noticed that for the other potentials, convergence is already reached for Nmax = 16 20, as shown in
Tables 5.2, 5.3, 5.4 and 5.5.
The variation of the binding energy as a function of jmax and Gmax depends
significantly on the adopted potential model. Therefore, we need to analyze every single case. As we can see from the data of Tables 5.6 and 5.7, the convergence for the Volkov and the Minnesota potentials is really quick and we can reach an accuracy better than 2 keV for jmax = 10 and Gmax = 40.
Instead, for the Afnan-Tang we need to go up to jmax = 14 and Gmax = 50 in
order to get a total accuracy of about 2 keV (1 keV is due to the dependence on Nmax summarized in Table 5.3), as shown in Table 5.8. The Malfliet-Tjon
potential model implies a convergence even slower in the expansion on the HH basis, and we have to go up to Gmax = 70, N
max = 16 and jmax = 16, to
get an uncertainty of about 10 keV, as shown in Table 5.9. The Malfliet-Tjon potential, being a sum of Yukawa functions, is more difficult to be treated also with the symmetrized HH method [13]. Furthermore, since it is not a realistic potential model, we have prefered to concentrate our e↵orts on the next potential model, the AV40. In Tables 5.10 and 5.11, we show the
con-vergence study for the AV40, which is the most realistic potential model we have used for the 3H and 3He systems. Here, in order to reach an accuracy
of about 3 keV, we have to push Gmax up to 80, N
max to 24 and jmax to
20. Our results, though, agree well with those obtained with the standard symmetrized-HH method for which we obtain B = 8.992 MeV [29]. To be noticed that this latter study is performed with jmax = 6, while we have to
take jmax = 20 with the NSHH. Therefore, we need many more terms of the
expansion on the HH basis, if we use the NSHH method. In Table 5.12 the
3H binding energy, obtained in the present work using m
p = mn, is shown
for all the potentials used, and is compared with the results present in the literature. The corresponding values for , Nmax, jmaxand Gmax are given in
Table 5.13. The agreement is really nice in the cases of the Volkov and the Minnesota potentials. However, also for all the other models we find a nice agreement within our quoted uncertainty.
5.2
Results for the
3H and
3He systems
We now turn our attention to the 3H and 3He nuclei. We have first applied
the NSHH method and we have reached convergence with mn = mp, and
without the Coulomb potential. Then, we have switched on the di↵erence of mass between proton and neutron, to calculate the di↵erence of the binding energy, i.e.
B = B3H B3He . (5.2)
We would like to point out that, before our work, B was estimated within the standard symmetrized HH method only perturbatively [30]. To get a perturbative estimate of B, we proceed as follows: since m = mn mp =
1.2934 MeV, is about three orders of magnitude smaller than the mean mass of the neutron-proton system m, i.e. m = (mn+ mp)/2 = 938.9187 MeV, we
can assume also B to be small. Furthermore, we assume the potential to be insensitive to m, and we consider only the kinetic energy. In the center
of mass frame it can be written as T = 3 X i=1 p2 i 2mi = p 2 1+ p22 2me + p 2 3 2md , (5.3) according to Eq. (2.1). In Eq. (5.3), me stands for the mass of the two
equal particles, i.e. mn for the 3H nucleus and mp for the 3He nucleus, and
md for the mass of the third particle, di↵erent from the previous ones. By
defining E =hHi = hT + V i, where hHi stands for the average value of the Hamiltonian H, we obtain @E @me = ⌧ @H @me = ⌧ @T @me = h2Tei me , (5.4) @E @md = ⌧ @H @md = ⌧ @T @md = hTdi md . (5.5) Moreover, in terms of m and m, we define
mp ⌘ mp m = m 2 , (5.6) mn ⌘ mn m = m 2 . (5.7) Finally, using Eqs. (5.4), (5.5), (5.6) and (5.7), we get for the3He nucleus
B3He ⌘ Bmn=mp B3He ⇡ @E @me mp+ @E @md mn= h2Tei me m 2 hTdi md m 2 ⇡ h2Te Tdi m 2m . (5.8) Proceeding in a similar way for the 3H, we get
B3H= Bmn=mp B3H ⇡ @E @me mn+ @E @md mp = h2Tei me m 2 + hTdi md m 2 ⇡ h2Te Tdi m 2m . (5.9) In conclusion BP T ⌘ B3H B3He ⇡ h2Te Tdi m m . (5.10)
Figure 5.1: In panel (a) we show the binding energy B (in MeV) as function of the parameter (in fm 1) for the Volkov potential model [19], with Gmax =
20, jmax = 6 and Nmax = 16, and using mn = mp. Panel (b) is a focus on
Figure 5.2: In panel (a) we show the binding energy B (in MeV) as function of the parameter Nmax for the AV40 potential model [23], with Gmax = 20,
jmax = 8 and = 4 fm 1, and using mn= mp. Panel (b) is a focus on panel
(a) for 8 Nmax 30.
By noticing that both 3H and 3He nuclei present a high S-wave percentage
(about 90%), we can assume that hTei ' hTdi ' hT i /3, where hT i is the
average value of the kinetic energy. Therefore Eq. (5.10) becomes BP T ⇡ hT i
m
Nmax 4 8 12 16 20 24 28
B 7.579 8.595 8.666 8.676 8.680 8.682 8.682
Table 5.1: The 3H binding energy B (in MeV) calculated with the Argonne
AV40 potential model [23], using m
n = mp, jmax = 8, = 4 fm 1 and
Gmax = 20, and varying N max.
Nmax 4 8 12 16
B 8.100 8.461 8.462 8.462
Table 5.2: The 3H binding energy B (in MeV) calculated with the Volkov
potential model [19], using mn= mp, Gmax = 20, jmax = 10 and = 4 fm 1.
Nmax 8 12 16 20
B 6.579 6.615 6.619 6.620
Table 5.3: The 3H binding energy B (in MeV) calculated with the
Afnan-Tang potential model [20], using mn = mp, Gmax = 20, jmax = 10 and = 4
fm 1.
Nmax 8 12 16
B 7.938 7.943 7.943
Table 5.4: The 3H binding energy B (in MeV) calculated with the
Malfliet-Tjon potential model [21], using mn = mp, Gmax= 20, jmax = 10 and = 4
fm 1.
Nmax 8 12 16 20
B 8.377 8.380 8.381 8.381
Table 5.5: The3H binding energy B (in MeV) calculated with the Minnesota
jmax = 6 jmax = 8 jmax = 10
Gmax B Gmax B Gmax B
20 8.460 20 8.462 20 8.462 30 8.461 30 8.464 30 8.464 40 8.461 40 8.464 40 8.465
Table 5.6: The 3H binding energy B (in MeV) calculated with the Volkov
potential model [19], using mn= mp, Nmax = 16 and = 4 fm 1.
jmax = 6 jmax = 8 jmax = 10
Gmax B Gmax B Gmax B
20 8.376 20 8.381 20 8.381 30 8.378 30 8.383 30 8.385 40 8.378 40 8.383 40 8.385
Table 5.7: The3H binding energy B (in MeV) calculated with the Minnesota
potential model [22], using mn= mp, Nmax = 16 and = 4 fm 1.
In Table 5.14 we show the values for hT i, obtained using the considered potential models, and the estimate of BP T obtained using Eq. (5.11). As
we can see, the values for BP T are in nice agreement with those calculated
in this work with the NSHH. The small di↵erence for the Volkov potential is within the numerical accuracy of our method. In the cases of the Minnesota and AV40 potentials B
P T is 4 and 3 keV larger than BN SHH, respectively.
To be noticed that the approximation used in Eq. (5.11), represents an upper limit for B. Furthermore, we notice from Table 5.14, that B is not the same for all the potentials. In fact, while for the spin-independent Afnan-Tang and Malfliet-Tjon central potentials, and for the AV40 potential, B =
14 keV (from now we drop the subscript N SHH), for the Volkov and the Minnesota potential we find a smaller B. In all cases, though, we have verified that B is always equally distributed, i.e.
Bmn=mp = B3H
B
2 = B3He+ B
