The eﬀective-range function in nuclear physics A method to parameterize phase shifts and extract ANCs Oscar Leonardo Ram´

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The eﬀective-range function in nuclear physics

A method to parameterize phase shifts and extract ANCs

Oscar Leonardo Ram´ırez Su´arez

Facult´e des Sciences

Universit´e libre de Bruxelles

Thèse de doctorat présentée en vue de l’obtention du grade de Docteur en Sciences Physiques

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Acknowledgements

Along this journey I have had the great opportunity of sharing ideas, opinions, ex-periences and much more with amazing people.

Jean-Marc Sparenberg, thank you for introducing the exciting world of the effective-range theory to me. Your support and enthusiasm were the key to give my first steps in this field and improve several aspects in my research. I would also like to thank you for drawing my attention to the Feldenkrais method, it was very relaxing.

Pierre Descouvemont, thank you for uncountable academic and logistic assistance. You were the ﬁrst who without almost knowing me gave me the opportunity to be part of the PNTPM group. Thank you for several discussions (especially, about the error analysis of experimental data), advises and hospitality along the PhD project.

Daniel Baye, thank you for crucial discussions, comments, corrections, advises and suggestions. You have shown me the other side of the coin; I cannot be more grateful. I am sure that all your help will play an important role in my future researches.

Jury committee members: Natasha Timofeyuk, Riccardo Raabe, Michele Sferrazza, Daniel Baye, Pierre Descouvemont and Jean-Marc Sparenberg. Thank you all for spend-ing part of your valuable time readspend-ing this thesis and for your comments.

PNTPM members and ex-members, thank you all for supporting me directly and indirectly. It was really nice to talk in several occasions with Horacio Olivares Pil´on, Jeremy Dohet-Eraly, Thomas Druet, Wouter Ryssens, Janina Grineviciute, Pierre Capel and Bikashkali Midya, and to know their points of view and experiences. I am also grateful to the PNTPM visitors, especially to Naoyuki Itagaki, his kindness makes every discussion a pleasurable time.

I am truly indebted to all my family and friends in Colombia, USA and Europe. Their exceptional support has always encouraged me to keep on the road.

Carolina, thank you for staying with me along this journey and supporting every single aspect of my research and life. You are my all.

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CHAPTER

1

Introduction

The partial-wave analysis of elastic cross sections in terms of phase shifts is one of the main methods to study nuclear collisions at low energies. Thus, a good phase shift description plays an important role to extract physical information like resonances properties [2, 3, 4], eﬀective range parameters [3, 5], ANCs [6], etc.

Several methods have been given to obtain a phase shift parametrization, each one with advantages and disadvantages. The most usual is the R-matrix method [7], which is well established and connected with the scattering matrix Sl [4, 2]. In this method, the R-matrix is expanded in terms of its poles and weights, which gives enough ﬂexibility to describe phase shifts, especially around resonances. As Ref. [8] and references therein point out, the indirect connection between “formal” and “observable” parameters can be seen as a drawback of the method, and for some authors [9] the introduction of a channel radius is also seen as a drawback. Another sophisticated proposal is given by J. Humblet and collaborators [10] (and several other publications). This proposal is based on an expansion of the modiﬁed K-matrix via its poles, it does not require a channel radius and, in principle, it agrees with the Sl properties (see discussions by R. Huby for details [11]). However, to describe correctly experimental data, it requires a large set of free parameters, hiding in some manner their physical meaning.

Thanks to its analytic properties the effective range function [12, 13] is another good candidate to parametrize phase shifts. This function, which does not require a channel radius, has shown a precise phase shift description at low energies via its Maclaurin expansion. However, such an expansion is limited generally at low energies, which re-stricts the analysis of experimental data. Thus, some proposals have been suggested to extend the description in larger energy ranges. This is the case of Ref. [14] where the effective range function of the neutron-deuteron doublet has been very well described at low energies by a Padé approximant of energy. Unfortunately, some parameters in that approximation do not have a clear physical meaning, but the good description suggests that Padé approximants can be a good choice if the meaning is clarified.

Phase shifts are important not only for elastic scattering processes, they also deﬁne the asymptotic behavior of the initial state (as a superposition of partial waves), for a more general reaction between a target A and a projectile B, i.e., A(B, �)� [boxes are

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determined by the ﬁnal state, e.g., A(B, B)A for elastic scattering]. Thus, wherever a nuclear reaction takes place, a phase shift description will provide a valuable piece of information to obtain for instance cross sections. This fact is particularly important in astrophysical scenarios where a wide spectrum of reactions can occur. Let us explore and connect the physics of these scenarios with the phase shifts as follows.

The life and the stage of a star depend on the nuclear reactions occurring inside of it [15]. For instance, reactions like 3_He(3_{He, 2p)}4_He, 3_He(4_{He, γ)}7_{Be and} 7_{Be(p, γ)}8_B can take place in stars with typical masses between 0.1M⊙ and 1.5M⊙ and surface temperatures between 0.6T⊙ and 1.7T⊙ [6] (M⊙ and T⊙ are solar values). Among these stars, the sun is a good example; its evolution is tightly related with the reactions in the p-p chain [6, 16] which plays a crucial role to explain the released energy by this star, and is the starting point to propose solar models and explain the emission of solar neutrinos [16, 17]. The overall result of the p-p chain can be summarized as [17]

4p_→4He + 2e++ 2νe,

which shows that the fuel of the star is hydrogen and the ash is helium.

Heavier elements can be formed after the hydrogen burning stage in stars with a central density _{∼ 0.1-100 kg/cm}3 and a central temperature _{∼ 1-2 ×10}8 K [18]. This stage is called the helium-burning or α-burning stage and the main product is12_{C which} is the fourth most abundant element in the universe (the ﬁrst one is hydrogen followed by helium and oxygen respectively).

The amount of carbon inside a star depends on the probability and the rate to “create” and “destroy” it. In particular, red giant stars1 _{can “destroy”}12_{C by “creating”} 16_{O after the α-burning stage via the radiative capture} 12_{C(α, γ)}16_{O which is expected} to be the main source for explaining the ratio12C/16O [18, 20, 8]. This reaction shows its eﬀects in a small energy range which is determined by the so-called Gamow peak [21, 6, 18]. This peak is located at ∼ 300-320 keV for temperatures around 2 × 108 K. Nowadays, laboratory conditions are not suﬃcient to measure precisely the capture cross section for this reaction around the Gamow peak (_{≈ 10}−17 b). This limitation is due to the wide Coulomb barrier in the collision12_{C+α as Fig. 1.1 illustrates.}

R = r0 � A1₁/3+ A1₂/3 � �� r �� VC � ��

Figure 1.1: Point-point Coulomb interaction for the collision 12C+α for internuclear separations r > R (r0= 1.2 fm, A1= 12 and A2= 4).

1_{For instance, Betelgeuse with a radius ∼ 950-1200 R}

⊙, a mass ∼ 15-20 M⊙and a surface temperature

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3 Thus, in order to explain the physics of this reaction at low energies, an optional way is to extrapolate the tendency of experimental cross section data. This option should be analyzed carefully because the extrapolation is very sensitive to any small change (uncertainties for instance) of the available experimental data, which can lead to inconsistent results if no constraint is known [22, 23]. To overcome this problem in the extrapolation, theory can provide valuable information and alternatives.

From the theoretical side, radiative-capture reactions are in general studied via the S-factor [S(E) = E exp(2πη)σ(E), with σ(E) the radiative capture cross section, E the energy in the center-of-mass frame and η the Sommerfeld parameter] [7, 6]. At low energies, the S-factor removes the strong energy dependence from the cross section leading to a much smoother function, and therefore, its extrapolation can be handled more accurately. Moreover, contributions of 12C(α, γ)16O for producing 16O can be separated via the S-factor (see for instance Ref. [23]). In particular, the direct capture to the 6.92 MeV (2+) excited state of16O (see Fig. 1.2) can be analyzed independently from the direct capture to the 7.12 MeV (1−) state.

16_O 6.92 (2+) 7.12 (1−) 12_C+α

7.16

Figure 1.2: 12C(α, γ)16O to the 2+ and 1− levels below the 12C+α threshold. Energy labels in MeV.

Let us analyze the direct capture to the 6.92 MeV (2+_{) state. As one sees in Fig. 1.2,} this reaction starts from a scattering state between12C and α and ends in a bound state of 16O. This process can be understood, in a potential model picture, as a transition of the system12_{C+α from a scattering state to a weakly bound state at}_{∼ 0.24 MeV below} threshold. The fact that the binding energy of the final state is very low implies that the probability of finding this system in an extended region is high because the nuclear potential cannot confine strongly this state. Adding that the Coulomb interaction re-duces the probability of fusion between12_{C and α at low energies (see Fig. 1.1), we have} then that the probability of this capture reaction is dominated by the contribution at large separations between 12C and α. This means that the asymptotic behavior of the weakly bound state, which is determined by the bound state energy and the asymptotic normalization coefficient or ANC, plays a crucial role to compute the S-factor. Thus, a precise ANC estimation provides better and more support to quantify the contribu-tion of the direct capture 12_{C(α, γ)}16_{O to the 6.92 MeV (2}+_{) [7.12 MeV (1}−_{)] state for} producing oxygen inside of a star.

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of cascade transitions [25], R-matrix analyses of elastic scattering cross sections [26], an α+12_{C microscopic calculation in the framework of the generator coordinate method} (GCM) [27] and potential models based on supersymmetry [28]. However the results show that the precision and the accuracy of the ANC can be diﬀerent according to the method. For instance, the DWBA analysis by C. R. Brune et al. [24], the microscopic calculations by M. Dufour and P. Descouvemont [27] and the potential model study by J-M. Sparenberg [28] provide respectively 114(10), 126(5) and 145(9) (all of them in 103 _fm−1/2_{). On the other hand, R-matrix analyses provides a larger range for this} estimation, 102-272 (×103 _fm−1/2_{) [26, 25]. All these results indicate that an accurate} ANC value cannot be chosen without a disagreement among methods. However, the ANC is an unique value and therefore, the question “which is the most accurate method to determine this ANC?” is still opened at this point.

In this thesis we shall explore an alternative way to determine ANCs via the effective range theory. This is possible thanks to the analytic properties of the effective range function [3, 5, 12], which can be used to connect elastic scattering states with bound states [13]. Moreover, this function is well defined for each partial wave, which means that by analyzing elastic phase shifts of12C(α, α)12C (d-wave), according to the properties of the effective range function, the ANC required to compute the S-factor of12C(α, γ)16O to the 2+ _{state can be determined.}

To explore how the effective range function can determine an ANC once a correct phase shift description is obtained we divide the study in two parts. First, we show how the Maclaurin expansion of the effective range function (also known as the effective range expansion) describes accurately elastic phase shifts at low energies according to a potential model. To do this, a technique to compute the effective range parameters [29, 3, 5, 4] up to an arbitrary order is developed here. This technique is well supported and rigorously built in the framework of potential models and the effective range the-ory, and its application opens the possibility to improve the S-factor calculation at low energies [30]; nevertheless, the impact on the S-factor by the effective range parameters is expected to be small in comparison with that by the ANC [30]. Unfortunately, the effective range expansion requires the contribution of several orders to describe, at least partially, phase shifts in the energy range where experimental information is available. However, even in the case that the expansion requires a small number of parameters, the potential model must describe accurately the experimental data in order to extract reliable values.

To overcome the limitation for parametrizing elastic phase shifts via the eﬀective-range expansion (too many parameters in practice), a method is developed here. This method is constructed according to the analytic properties of the eﬀective range function, and correctly connected with the scattering matrix. Several important features are displayed by this method, for instance, it can describe elastic phase shifts in an extended energy range, it does not require a potential model (which is very important to avoid discrepancies between models and experiment), it allows us to determine the location and width of a resonance in a very simple way, and the ANC can be extracted by using experimental data as an input. We also explore the accuracy of the method to determine ANCs. To do this we use potential model phase shifts as an input of the method and compare the estimation with the exact result.

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5 information about this width by applying the method, which can be very useful to reduce the free parameter space in phenomenological R-matrix analyses. Another piece of information that we can deduce from the ANC is the nuclear vertex constant (NVC) [32].

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CHAPTER

2

Elastic scattering

In quantum mechanics, and especially in nuclear physics, several phenomena are studied via collisions, e.g., production and decay of nuclei, electromagnetic excitations, transfer of nucleons, etc. The simplest case is the collision between two spinless particles. This case is discussed here in the framework of nuclear physics by assuming nuclei as particles and their interactions depend on the separation between them only. This allows us to understand the basic concepts along this text and helps us to interpret the results from a physical point of view later.

2.1 Collision between two particles

Consider a collision between two spinless particles of masses m1 and m2, the interaction (potential) depends only on their separation r and is time-independent. If r is very large (or r _{→ ∞) the particles cannot feel each other, which allows us to assume that the} strength of their interaction should decrease up to vanish. This assumption implies that for any positive energy the potential cannot confine the system in a particular region indefinitely and therefore we observe an elastic scattering process. This view classifies the states in two groups: elastic-scattering states (E > 0) and bound states (E < 0).

The previous assumptions involve almost all the information required to start with the formal description of elastic collisions. It just remains to decide which kind of theory is needed for such a description. Here the non-relativistic formalism, which allows a simpler mathematical treatment, will be adopted. Moreover, relativistic corrections will not be necessary in our study because the energies of interest are much lower than the rest energies (E _{≪ m}ic2).

With the previous conditions the Hamiltonian is H =− � 2 2m1∇ 2 1− �2 2m2∇ 2 2+ V (�r2− �r1), (2.1)

where the two-particle system and the corresponding coordinates are shown in Fig. 2.1.

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� � m1 m2 O �r1 �r2 �r � R center of mass

Figure 2.1: Coordinates �r1 and �r2 in Eq. (2.1). Or equivalently, �R and �r in Eq. (2.2).

The Hamiltonian (2.1) is time-independent which allows us to separate the solution of the Schr¨odinger equation in two, one time dependent and another spatial dependent1_. Moreover, by deﬁning

�r = �r2− �r1 and R =�

m1�r1+ m2�r2 m1+ m2

, we can write the time-independent Schr¨odinger equation as

� − � 2 2M∇ 2 � R− �2 2µ∇ 2 �r+ V (�r) � φ( �R, �r) = ETφ( �R, �r), (2.2) with M = m1 + m2 the total mass, µ = m1m2/M the reduced mass and ET the total energy. Equation (2.2) shows that the wave function φ( �R, �r) can be decoupled as φ( �R, �r) = Φ( �R)Ψ(�r) which leads to − � 2 2M∇ 2 � RΦ( �R) = EMΦ( �R), (2.3) and _� −� 2 2µ∇ 2 �r+ V (�r) � Ψ(�r) = EµΨ(�r), (2.4)

with ET = EM + Eµ. The ﬁrst of the previous two equations describes a free particle of mass M with energy EM, and the second one represents a particle of mass µ with energy Eµ in presence of a potential V (�r). The contribution of Eq. (2.3) can be eliminated by choosing the center-of-mass system as the reference frame, which leads to sum up all the dynamic of the original two-particle collision in

� −� 2 2µ∇ 2 � r+ V (�r) � Ψ(�r) = EΨ(�r), (2.5)

where the subscript in Eµ is dropped to stress that the reference frame is located in the center of mass.

2.2 Method of partial waves

In order to solve Eq. (2.5), we need more information about the potential V (�r). For our purposes, we shall study central potentials, i.e., V (�r) = V (r). This kind of potentials

1_{Remember, the Schr¨}_{odinger equation reads i�}∂ϕ

∂t = Hϕ, with ϕ = ϕ(t, �r1, �r2). If H is

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2.2. Method of partial waves 9 provides useful models to explain several quantum systems satisfactorily. For example, hydrogen-like atoms (Coulomb potential) [34], diatomic molecules (Morse potential) [35, 36], two-fermion scattering by exchanging a meson (Yukawa potential) [37], etc. In nuclear physics, models are built usually by assuming Gaussian, Wood-Saxon or square well potentials, among others [38]. All of them are central potentials and, depending on the nuclei, a particular shape can provide a better description than another.

It is important to remark that a central potential is spherically symmetric, which suggests that spherical polar coordinates (r, θ, φ) (hereafter spherical coordinates) are convenient to solve the time-independent Schr¨odinger equation (Eq. (2.5) for a central potential) � −� 2 2µ � 1 r2 ∂ ∂r � r2 ∂ ∂r � +L 2 Ω r2 � + V (r) � Ψ(�r) = EΨ(�r), (2.6)

where the Laplacian has been written in spherical coordinates (see Box 2.1). In spherical coordinates the L2

Ω-operator reads [39] L2_Ω = 1 sin(θ) ∂ ∂θ � sin(θ) ∂ ∂θ � + 1 sin2(θ) ∂2 ∂φ2, and the orthogonality of the spherical harmonics is given by

� _π 0

� _2π 0

Ylm(θ, φ)Yl∗′_m′(θ, φ) sin(θ)dθdφ = δll′δ_mm′.

Notice that if l = l′ and m = m′ we obtain the normalization property. Box 2.1: L2_Ω-operator and orthogonality of the spherical harmonics. As usual, one can deﬁne L2 = −�2_L2

Ω in Eq. (2.6). In this manner, a natural connection arises between the formalism and the orbital angular momentum. In fact, L2 _{is known as the operator “square of the orbital angular momentum” and satisﬁes}

L2Ylm(θ, φ) = �2l(l + 1)Ylm(θ, φ), (2.7) with Ylm(θ, φ) the spherical harmonics, l the orbital quantum number and m the pro-jection quantum number.

Because the spherical harmonics depend on the angles θ and φ only, we can decouple the radial and the angular dependence of Ψ(�r) by proposing

Ψ(�r) = ∞ � l=0 l � m=−l Ψl(k, r)Ylm(θ, φ), (2.8)

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By replacing Eq. (2.8) in (2.6) and using the orthogonality property of the spherical harmonics (see Box 2.1) one ﬁnds

� −� 2 2µ � 1 r2 d dr � r2 d dr � −l(l + 1) r2 � + V (r) � Ψl(k, r) = EΨl(k, r), (2.9) or equivalently � d2 dr2 − l(l + 1) r2 + k 2_{− U(r)} � ul(k, r) = 0, (2.10)

with ul(k, r) = krΨl(k, r) (known as the modiﬁed radial wave function) and U (r) = 2µV (r)/�2. Thus, we have reduced the initial problem, or Eq. (2.2), to a much simpler diﬀerential equation, or Eq. (2.10).

2.2.1 Modiﬁed radial wave function for a free particle

It is instructive to solve Eq. (2.10) for a free particle. Deﬁning ρ = kr, Eq. (2.10) can be rewritten as [V (r) = 0] � ρ2 d 2 dρ2 − l(l + 1) + ρ 2 � ufree_l (k, ρ) = 0, (2.11)

which is a Riccati diﬀerential equation and its solution, ufree_l (k, ρ), can be expressed by a combination of the Riccati-Bessel functions [40] ˆjl(ρ) (ﬁrst kind) and ˆnl(ρ) (second kind)2 like

ufree_l (k, ρ) = A(k)ˆjl(ρ) + B(k)ˆnl(ρ), (2.12) or a combination of the Riccati-Hankel functions ˆh(±)_l (ρ) = ˆnl(ρ)± iˆjl(ρ) like

ufree_l (k, ρ) = C(k)ˆh(+)_l (ρ) + D(k)ˆh(−)_l (ρ). (2.13) It is important to stress that both Eqs. (2.12) and (2.13) are equivalent. In practice one can choose one of them rather than the other one depending on what kind of analysis or procedure is carried out. For instance, Eq. (2.13) gives a direct connection with the most usual form of the wave function of a free particle when ρ → ∞ (incoming and outgoing waves) which makes this form more convenient for a physical argumentation. On the other hand, Eq. (2.12) is more useful for computational calculations because for real arguments the Riccati-Bessel functions are also real.

2.2.2 Modiﬁed radial wave function for a short-range potential

We can now analyze how a short-range potential aﬀects the solution of Eq. (2.10) in comparison with ufree_l (k, ρ) shown previously.

First of all, we have to define short-range potentials. As we shall see, a closed defini-tion could be “meaningless” in general; this depends on what kind of descripdefini-tion, approx-imation or assumption one adopts. As starting point, we can understand a short-range potential as a potential such that its contribution at large distances can be neglected in

2_{In notation of Ref. [40] ˆj}

l(ρ) = ρjl(ρ) and ˆnl(ρ) = −ρyl(ρ) with jl(ρ) and yl(ρ) the spherical Bessel

functions of the ﬁrst and second kind respectively. The notation ˆjl(ρ) and ˆnl(ρ) is used for instance in

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2.2. Method of partial waves 11 comparison to the centrifugal potential plus the Coulomb interaction. This looks like a closed deﬁnition, but for theoretical analyses it will be necessary to provide more details about the behavior of the short-range potential.

Fortunately, there is a particular class of short-range potentials that allows to solve Eq. (2.10) without any mathematical limitation. Moreover, the results for this class are valid for a wider kind of potentials as we shall see later. Here this class is named as ﬁnite-range potentials with representative member Va(r) which is a potential that strictly vanishes at r > a and is less singular than r−2 at the origin (see Box 2.2 for details), as Fig. 2.2 illustrates.

Va(r)

a 0

Figure 2.2: Example of a ﬁnite-range potential Va(r).

(i) Va(r > a) = 0: This guaranties that the solution of the Schrödinger equation has the same functional form of ufree_l (k, ρ) for all r > a. This condition is assumed in general. For instance, the R-matrix method divides the configuration space in two, the internal (r < a) and the external (r > a) regions. In that case a is called the channel radius which allows us to define a region where the nuclear potential cannot be neglected.

(ii) limr→0r2Va(r)→ L: This condition classiﬁes the potential as a regular or singular one (L = 0 or L _{�= 0 respectively) in the sense of diﬀerential equations. Singular} potentials require a special mathematical treatment and therefore a new physical interpretation. For this reason, scattering theory is usually focused on the study of regular potentials, leaving the singular ones as a special case.

In general, several authors avoid to point out explicitly that the potential is regular, and adopt a less abstract language by saying the short range potential satisfies the usual conditions. Other authors do not mention it at all. In those cases, regular potentials are usually assumed, which can be confirmed via the expressions in the for-malism. For example, regular potentials are assumed to deduce the Levinson theorem [5, 3] which differs with the generalized Levinson theorem for singular potentials [42].

Box 2.2: Conditions of a ﬁnite-range potential.

In the following, V (r) will be assumed as a ﬁnite-range potential. This will provide general results and, if it is necessary, when a result is restricted by the type of short-range potential, a comment will be included together with the corresponding restriction.

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energies (E_{≥ 0), i.e.,} � d2 dr2 − l(l + 1) r2 + k 2 − Ua(r) � ul(k, r) = 0, for r≤ a, (2.14) � d2 dr2 − l(l + 1) r2 + k 2 � ul(k, r) = 0, for r > a. (2.15)

Comparing Eqs. (2.15) and (2.11) we note that both equations are equivalent, which leads to ul(k, r > a) = A1(k) � ˆjl(kr) + A2(k)ˆnl(kr) � , (2.16)

where we have assumed ul(k, r > a) similarly to Eq. (2.12).

Now we have to solve Eq. (2.14) such that ul(k, r ≤ a) matches correctly with ul(k, r > a) (see Box 2.3 for details).

For a ﬁnite potential V (r) the Schr¨odinger equation implies that the wave function and its derivative should be continuous functions. In particular, these conditions make that the solutions of Eqs. (2.14) and (2.15) match correctly at r = a. This can be summarized as

ul(k, r → a−) = ul(k, r→ a+) = ul(k, a), u′_l(k, r → a−) = u′_l(k, r→ a+) = u′_l(k, a),

where the primes mean derivative with respect to r. In order to cancel the normal-ization constant, it is convenient to divide both equations as follows

u′_l(k, r_{→ a}−) ul(k, r→ a−) = u ′ l(k, r→ a+) ul(k, r→ a+) = u ′ l(k, a) ul(k, a) ,

which is nothing but the match of the logarithmic derivative of ul(k, r) for r→ a. Box 2.3: Continuity of the modiﬁed wave function.

To solve Eq. (2.14) we have to know explicitly the potential. Although analytic solu-tions can be found for a few potentials, these solusolu-tions provide very useful information to check methods and theorems, to deduce precision and efficiency of some approxima-tions or techniques, etc. In practice, potentials are built with the aim of reproducing experimental data. For these kinds of potentials, in general, Eq. (2.14) is very difficult to solve analytically, and therefore numerical algorithms, such as the Numerov algorithm [43], should be implemented. Defining Rl= � a ∂ ln(ul(k, r)) ∂r � � � � r=a �−1 , (2.17)

which is the single channel R-matrix [7] and supposing that Eq. (2.14) has been solved analytically or numerically, one obtains [according to Eq. (2.16)]

A2(k) =−

ˆjl(ka)− aRlˆjl′(ka) ˆ

nl(ka)− aRlnˆ′l(ka)

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2.2. Method of partial waves 13 where the primes mean derivative with respect to r. On the other hand, the asymptotic behavior of ul(k, r) according to Eq. (2.16) is given by

ul(k, r→ ∞) = A1(k) [sin(kr− lπ/2) + A2(k) cos(kr− lπ/2)] , (2.19) = A1(k)

cos(δl)

sin(kr− lπ/2 + δl), (2.20)

with δl = arctan(A2(k)), which means that Eq. (2.18) can be rewritten as

tan(δl) =−

nl(ka)− aRlnˆ′l(ka)

. (2.21)

This equation is crucial for our study. It provides a connection between the phase shift, δl, and the potential through Dloga . Moreover, by choosing a adequately, Eq. (2.21) can provide a good phase-shift approximation for short-range potentials satisfying [3]:

(i) V (r→ ∞) = O(r−3−ǫ1_{) with ǫ} 1> 0. (ii) V (r→ 0) = O(r−2+ǫ2_{) with ǫ}

2> 0.

(iii) V (r) is continuous for 0 < r < _{∞, except perhaps at a ﬁnite number of ﬁnite} discontinuities.

Hereafter, these three conditions will be named as the usual conditions of short-range potentials [3].

If exist ǫ > 0 such that � ∞

0

r_{|V (r)| exp(2ǫr)dr < ∞,} (2.22)

(for details about this condition see for instance Refs. [5, 3]) we have a stronger constraint than condition (i). Because of the upper limit in the integral, this condition demands that _{|V (r → ∞)| < O(e}−2ǫr) which means that the potential is exponentially bounded at large distances [3]. Thus for simplicity, if Eq. (2.22) is satisﬁed we will say that the potential is exponentially bounded.

Notice that V (r) makes reference to the nuclear potential only, i.e., it does not include the centrifugal term. This also holds when the Coulomb interaction is present. As we shall see later, in that case the potential is written as V (r) = VC(r) + VN(r) (subscripts C and N for Coulomb and nuclear respectively), which means that V (r) in Eq. (2.22) should be replaced by VN(r).

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2.2.3 Mathematical aspects of phase shifts

At this point we have found an equation [Eq. (2.21)] to compute a quantity δl called the phase shift. We do not have a physical meaning for it yet, but after a few more mathematical analyses we can understand it in much simpler way.

Let us ﬁrst replace A2(k) = tan(δl) in Eq. (2.16) and rewrite it in terms of the Riccati-Hankel functions as ul(k, r→ ∞) = N(k) � ˆ h(−)_l (kr)_{− e}2iδlˆ_h(+) l (kr) � , (2.23)

with N (k) = A1(k)ie−iδl

2 cos(δl) . Equation (2.23) is then our new starting point.

S-matrix and absolute phase shifts

The term e2iδl _{in Eq. (2.23) is usually deﬁned as S}

land denotes the scattering matrix or S-matrix. In this case, Sl is a 1-dimensional matrix associated with the elastic channel. As it is evident the S-matrix satisﬁes _|Sl| = 1 at positive energies (it is unitary) and S_l−1= S_l∗.

Notice also that the S-matrix can be determined by δl. Moreover, redeﬁning δl → ˜

δl= δl+ nπ, with n∈ Z, one ﬁnds that Sl(˜δl) = Sl(δl). As Fig. 2.3 shows, this property is useful to deﬁne the absolute phase shifts [2, 41, 4], δabs_l , which is a continuous function on energy satisfying Sl(δabsl ) = Sl(δl).

Do not confuse δabs

l with the absolute value|δl| (δlabs �= |δl| in general).

δl0 π −π E δ a b s l0 π −π E 0 1 −1 Re(Sl) Im(Sl) E

Figure 2.3: Phase shifts (left), absolute phase shifts (center) and, real and imaginary part of Sl= Sl(δabsl ) = Sl(δl) (right).

The Jost function

The Jost function [3], Fl(k), plays an important role in the formalism of scattering theory. It deﬁnes the scattering matrix and the phase shifts as follows.

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2.2. Method of partial waves 15 we can write both solutions explicitly as

ui→ Ni � Fl,i(k)ˆh(−)l (kr)− Fl,i∗(k)ˆh(+)l (kr) � , (2.24) uj → Nj � Fl,j(k)ˆh(−)_l (kr)− Fl,j∗ (k)ˆh (+) l (kr) � , (2.25)

where_Ni and Nj will be deﬁned shortly.

Clearly, the Riccati-Hankel functions do not depend on the potential and their values are the same in Eqs. (2.24) and (2.25) (same l and k). Thus, the information of the potential should be in _Fl(k), the Jost function. One can still factorize Fl(k) in both equations as (for all k such as Fl(k)�= 0)

ui → Ni � ˆ h(−)_l (kr)₋F ∗ l,i(k) Fl,i(k) ˆ h(+)_l (kr) � , (2.26) uj → Nj � ˆ h(−)_l (kr)₋F ∗ l,j(k) Fl,j(k) ˆ h(+)_l (kr) � , (2.27)

where, omitting the subscripts i or j and comparing with Eq. (2.23), one ﬁnds N = N

Fl(k) =

A(k)ie−iδl

2 cos(δl) and Sl = e

2iδl ₌ Fl∗(k)

Fl(k). This means that for each potential in the

family_{V1, V2,· · · } there is a Jost function that deﬁnes its corresponding S-matrix and phase shifts. This shows that, in general, all the properties of Sl and δl come from the Jost function and not vice versa [5, 3].

We have introduced exponentially bounded potentials as those satisfying Eq. (2.22). For these potentials, the Jost function _Fl(k), in the complex k-plane, is analytic for Im(k) >−ǫ [5, 44].

Scattering amplitude and diﬀerential cross section

Recalling Eq. (2.8) we can explore how each partial wave modiﬁes the asymptotic be-havior of the radial wave function. We just need to replace Ψl(k, r) in terms of ul(k, r) and separate the scattered contribution from the non-scattered one as follows.

The asymptotic behavior of Eq. (2.8) is given by

Ψ(�r)_−−−→ r→∞ ∞ � l=0 l � m=−l 1 krul(k, r→ ∞)Ylm(θ, φ), (2.28)

which in term of phase shifts is [see Eq. (2.23) and take into account that, for r_{→ ∞,} ˆ h(±)_l (kr)_{∼ e}±i(kr−lπ/2)] Ψ(�r)_−−−→ r→∞ ∞ � l=0 l � m=−l 1 krNl(k) �

e−i(kr−lπ/2)_{− e}2iδl_ei(kr−lπ/2)

�

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By deﬁning Cl(k, θ, φ) =−N_ill(k)_k

�l

m=−lYlm(θ, φ), one can rewrite this expression as

Ψ(�r)−−−→ r→∞ �_∞ � l=0 Cl(k, θ, φ)il ei(kr−lπ/2)− e−i(kr−lπ/2) r � + �_∞ � l=0 Cl(k, θ, φ)(e2iδl− 1) � eikr r . (2.30) Now we have to compare this expansion with that for

Ψ(�r)_−−−→ r→∞ N (k) � ei�k·�r+ fsa(k, θ, φ) eikr r � , (2.31)

which defines the scattering amplitude fsa(k, θ, φ). The full procedure to expand Eq. (2.31) in partial waves is shown in several books of scattering theory (it usually covers two or more pages of mathematical steps, see for instance Ref. [4]). The main point in such a deduction, is that the first (second) square brackets in Eq. (2.30) is associated with the incident plane wave (outgoing spherical wave) in Eq. (2.31). The final result connects the coefficient Cl(k, θ, φ) with the scattering amplitude and leads to fsa(k, θ, φ) = fsa(k, θ) when �k = kˆz, with fsa(k, θ) = 1 k ∞ � l=0

(2l + 1)Pl(cos(θ))eiδlsin(δl). (2.32)

Equation (2.32) is very useful to compute and analyze the partial wave contribution on the diﬀerential cross section

dσ

dΩ(k, θ) = |fsa(k, θ)|

2_. _(2.33)

An extended discussion about cross sections can be found in several textbooks. Here, we wish to recall the definition in Ref. [4] to keep in mind the connection with the experiment (other textbooks show equivalent definitions by considering specific labo-ratory conditions, see for instance [41]).

“The cross section of a certain type of event in a given collision is the ratio of the number of events of this type per unit time and per unit scatter, to the relative ﬂux of the incident particles with respect to the target.”

2.2.4 Physical aspects of phase shifts

The usual way to ﬁnd the physical meaning of δlis via the wave function (2.20) and Eq. (2.21). First of all, we can explore the case δl= 0 for all E. In this case, the hypothetical short-range potential should reduce Eq. (2.21) to

nl(ka)− aRlˆn′l(ka)

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2.2. Method of partial waves 17 which should be valid for all E. This implies that the short-range potential should vanish at every point to ensure that Dalog = ˆj_l′(ka)/ˆjl(ka), and therefore the wave function is [see Eq. (2.20) for δl= 0]

ul(k, r→ ∞) ∝ sin(kr − lπ/2). (2.35)

Now, we can suppose that the short-range potential does not vanish for r < a. Thus, except perhaps for a ﬁnite number of energies, Eq. (2.21) is not null. This means that the eﬀect of the short-range potential on the asymptotic wave function is to shift it a quantity δl in comparison with the wave function for V (r) = 0 (see Fig. 2.4).

ul r → ∞ δl No pote ntia l Shor t-ra nge pote ntia l

Figure 2.4: Eﬀect of the short-range potential on the wave function.

The connection between the short-range potential and the phase shift can also be understood from the scattering amplitude and the differential cross section. Suppose for the moment that there is no scattering center [V (r) = 0] and there is a plane wave incoming. Clearly, there is nothing affecting this wave, and therefore, the differential cross section should be zero for all angles and energies. To satisfy this, Eq. (2.32) should vanish (for all E and θ), which means that δl = 0. Similarly, starting from δl �= 0 one can deduce that a scattering center is present.

At this point, we have seen how to obtain phase shifts for a given short-range poten-tial, and consequently, this information can be used to compute scattering amplitudes and diﬀerential cross sections. Now, it is instructive to explore the other perspective, i.e., how to obtain the phase shifts when the diﬀerential cross section is known. In fact, this is the experimental case. Unfortunately, δl cannot be measured experimentally, but it can be extracted from the scattering amplitude which can be “determined” from the experimental dσ/dΩ (see Eq. (2.33)). It is important to remark that, experimental phase shifts should be understood as δl deduced (not measured) from experimental data.

Let me clarify what I mean with: fsa(k, θ) can be “determined” from dσexp/dΩ. In practice, one cannot handle all the terms in Eq. (2.32), but one can analyze the contri-bution up to l = lmax. Thus, if the contribution of the terms with l > lmaxis negligible, one gets fsa(k, θ)≈ fsa(k, θ, lmax) = 1 k l_�max l=0

(2l + 1)Pl(cos(θ))eiδlsin(δl) (2.36) and

dσexp

dΩ (k, θ)≈ |fsa(k, θ, lmax)|

2_. _(2.37)

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For a system with �2

2µ = 1 MeV fm2 and the square-well potential of depth 5 MeV and range 1 fm, the previous approximations [Eqs. (2.36) and (2.37)] are illustrated in Fig. 2.5.

l or lmax ∆l �rad� Re�fsa� �fm� �—�,

Im�fsa� �fm� �� fsa 2 �fm2�sr� 0 �Π₂ 0 Π 2 �1.6 0 1.6 0 2 4 1 �Π₂ 0 Π 2 �1.6 0 1.6 0 2 4 2 �Π₂ 0 Π 2 �1.6 0 1.6 0 2 4 3 �Π₂ 0 Π 2 �1.6 0 1.6 0 2 4 4 0 3 6 9 �Π₂ 0 Π 2 0 3 6 9 �1.6 0 1.6 0 3 6 9 0 2 4

E �MeV� E �MeV� E �MeV�

Figure 2.5: Phase shifts and partial wave contributions for the scattering amplitude [fsa = fsa(k, θ, lmax)] and the diﬀerential cross section (|fsa|2). Conditions: square-well potential of depth 5 MeV and range 1 fm, �2

2µ = 1 MeV fm2 and θ = 18◦.

For energies below 9 MeV, Fig. 2.5 shows good approximations of the scattering amplitude and the diﬀerential cross section when the partial wave contribution at high orbital momentum (lmax≈ 3) is negligible. This agrees naturally with the low values of the phase shifts for l > llmax and E < 9 MeV.

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2.2. Method of partial waves 19 previous example, l _{∈ {3, 4, · · · } can be adopted as l}max for E < 9 MeV. Notice the agreement with lmax�ka = (3 fm−1)× (1 fm).

To understand why lmax can be determined by lmax � ka, one can explore a semi-classical perspective. Consider the semi-classical collisions between a projectile of mass µ and a rigid sphere of inﬁnity mass. In this scenario, the magnitude of the classical-angular-momentum is L = bp (in the sphere frame).

� � Case 1 � p1 � p2 Case 2 Rigid sphere of radius a b1 b2

As one sees only projectiles in the case 2 can be scattered by the sphere.

From a semiclassical perspective one assumes L = l� and p = �k, which means that the expression l = bk deﬁnes the quantum number l. Adding that the critical impact parameter is a, i.e., for b < a the projectile is scattered and for b > a it is not, one sees that only those partial waves with l _{≤ ka play a relevant role in the scattering} process at E = �2

2µk2.

Box 2.4: Connection between lmax and energy in a semiclassical perspective.

2.2.5 Coulomb plus short-range potential

In the previous section we saw how to solve the Schr¨odinger equation for the neutral case (e.g., a neutron colliding with a nucleus). For that case, we also explored the properties of phase shifts, scattering amplitudes and diﬀerential cross sections among others, by assuming a short-range interaction. Now, we shall consider that both particles have positive electric charges, Z1e and Z2e, with Zi a positive integer (the atomic number). In this case or charged case, the deduction of Eq. (2.10) is still valid with V (r) = VC(r) + VN(r), i.e., � d2 dr2 − l(l + 1) r2 + k 2_{− U} C(r)− UN(r) � ul(k, r) = 0, (2.38) where UC,N(r) = 2µVC,N(r)/�2, VC(r) is the Coulomb interaction and VN(r) the nuclear interaction satisfying the usual conditions of short-range potentials (for more details see for instance Refs. [5, 3, 41]).

At very short distances the centrifugal term is dominant and VC does not introduce any mathematical limitation. In fact, it is usual to redeﬁne the Coulomb potential for short distances to consider geometrical eﬀects (a typical example is to adopt a point-sphere- instead of a point-point- Coulomb potential). On the other hand a very large distances, the Coulomb interaction is dominant and its behavior,

VC(r) = 1 4πǫ0

Z1Z2e2

r , (2.39)

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At this point, it is convenient to deﬁne the nuclear Bohr radius (aN), the nuclear Rydberg energy (EN) and the Sommerfeld parameter (η) as

aN = �2 µ 4πǫ0 Z1Z2e2 , EN = �2 2µ 1 a2 N and η = � EN E . (2.40)

This will helps us to understand and interpret, in a simpler way, some results and will simplify mathematical procedures. For instance Eq. (2.39) adopts a much simpler form

VC(r) = EN aN

r . (2.41)

In the framework of nuclear physics, the interpretation of EN and aN is not direct as in atomic physics. For the latter, the meaning is more evident: for a hydrogen-like atom in its ground state, “EN” is the binding energy of the electron, and “aN” is the radius of the classical trajectory of that electron. In nuclear physics, EN and aN do not involve any information about the nuclear interaction, thus one cannot relate them with bound-state energies or trajectories (at least in a direct way).

A more convenient form of VC in the context of nuclear physics is perhaps VC(r) = EN aN ¯ r ¯ r r = EB ¯ r r,

with ¯r deﬁned by considering the nuclear interaction. For example, for a ﬁnite-range potential (see Fig. 2.2) ¯r = a is can be a good choice, which spontaneously relates EB with the Coulomb barrier.

Box 2.5: Another form to express Eq. (2.39).

In order to deﬁne phase shifts in the charged case, we should explore the eﬀect of VN on the radial wave function, i.e., we have to compare the solutions of Eq. (2.38) when VN = 0 and VN �= 0. The procedure is very similar to that for the neutral case, which allow us to infer directly the result [2]

tan(δl) =−

Fl(ka)− aRlFl′(ka) Gl(ka)− aRlG′_l(ka)

, (2.42)

with Fl(ka) and Gl(ka) the regular and irregular Coulomb functions [40]. Note that the only two diﬀerences between Eqs. (2.21) and (2.42) are ˆj_{→ F and ˆn → G.}

Analogously, the scattering amplitude and the diﬀerential cross section can be written as fsa(k, θ) = fsaC(k, θ) + fsaN(k, θ) (2.43) and dσ dΩ(k, θ) = |fsa(k, θ)| 2_, _(2.44) with f_saC(k, θ) =− η 2k sin2(θ/2)e −iη ln(sin2_(θ/2)) e2iσC0 _, _(2.45) f_saN(k, θ) = 1 k ∞ � l=0

(2l + 1)Pl(cos(θ))eiδlsin(δl)e2iσ C

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2.2. Method of partial waves 21 and σ_lC = arg(Γ(1 + l + iη)) the Coulomb phase shift.

Although fN

sa(k, θ) in Eq. (2.46) looks similar to the scattering amplitude for a short-range interaction [Eq. (2.32), except for the term e2iσC

l ], the total scattering amplitude

has explicitly a Coulomb contribution [f_saC(k, θ) in Eq. (2.43)]. This implies that the meaning of δl in Eq. (2.46) is “a little bit” different. In this case, one should understand δl as a modification of the Coulomb phase shift due to the nuclear interaction, i.e., the total phase shifts is in fact, δ_ltot = σ_lC+ δl, which defines the S-matrix as

Sl = e2i(σ C

l +δl)₌ Γ(1 + l + iη)

Γ(1 + l_{− iη)}e

2iδl_. _(2.47)

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CHAPTER

3

Eﬀective-range function

Effective-range function (ERF), effective-range expansion (ERE) and effective-range pa-rameters (ERPs) are the three topics to review in this chapter. The last one has a special treatment in the next chapter, thus regarding to the ERPs, the terminology and some comments will be provided to help the reader in the understanding and to make a smooth connection between these two chapters.

3.1 Origins of the ERF

Since the 1940s, the interest of describing phase shifts at low energies has led to pro-pose several approaches, methods and techniques. Among them, the semi-empirical ap-proach by Landau-Smorodinsky [46]1 _{opened the perspective to go beyond a phase-shift} parametrization and prepared the ﬁeld to introduce a formal deduction and interpreta-tion. In 1947, Schwinger [47] developed a variational method which provides a solid basis for the Landau-Smorodinsky approach. Since then, several ways to derive Schwinger re-sults have been introduced (by Bethe, Peierls, Hatcher, Arfken and Breit among others; see references in Ref. [46]), which reinforces its formal deduction.

From the physical and formal point of view, the Landau-Smorodinsky approach is the origin of the ERF, the ERE and the ERPs. Nowadays, this approach together with its theoretical support is known as the eﬀective range theory.

At this point and only as an “aperitif”, let us introduce the most usual expression of this theory, Kl=− 1 al +1 2rlk 2_{− P} lr3lk4+ ∞ � n=3 Ql,nk2n, (3.1)

which sums up the concepts: ERF (the term in l.h.s), ERE (the full r.h.s), and ERPs (the set_{al, rl, Pl, Ql,n}).

At positive energies the eﬀective range function (Kl) can be written in terms of the phase shifts (δl) [3, 5, 4, 12, 48]. From this fact one can infer that the explicit energy 1_{In fact, the original references are J. Phys. U.S.S.R 8, 154 (1944), J. Phys. U.S.S.R 8, 219 (1944)}

and J. Phys. U.S.S.R 11, 195 (1947).

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dependence of the ERF changes if the Coulomb interaction is present or not [compare for instance δl in Eqs. (2.21) and (2.42)]. Thus, it is not strange, and rather convenient, to analyze separately Kl for the neutral case from Kl for the charged case.

3.2 ERF for the neutral case

The starting point of the eﬀective range theory is to deﬁne Kl for the neutral case as

Kl= k2l+1cot(δl), (3.2)

which is analytic at zero energy when the potential is exponentially bound2. For instance, for a ﬁnite-range potential one can check this property by using Eq. (2.21) in Eq. (3.2), i.e., Kl =−k2l+1 ˆ nl(ka)− aRlnˆ′l(ka) ˆjl(ka)− aRlˆj_l′(ka) , (3.3)

and evaluating it at low energies as follows.

Around k = 0, the behaviors of ˆjl(ka) and ˆnl(ka) are [40] (see footnote on page 10) ˆjl(ka) = (ka)l+1 (2l + 1)!! � 1− (ka) 2 1!2(2l + 3) + (ka)4 2!22_{(2l + 3)(2l + 5)}− · · · � (3.4) and ˆ nl(ka) = (2l + 1)!! (ka)l � 1₋ (ka) 2 1!2(1− 2l) + (ka)4 2!22₍₁_{− 2l)(3 − 2l)} − · · · � , (3.5)

and therefore ˆj_l′(ka) and ˆn′_l(ka) behave as (remember, primes mean ∂/∂r)

ˆj′ l(ka) = (ka)l+1 (2l + 1)!! 1 a � (l + 1) � 1− (ka) 2 1!2(2l + 3) + (ka)4 2!22_{(2l + 3)(2l + 5)} − · · · � + � − 2(ka) 2 1!2(2l + 3)+ 4(ka)4 2!22_{(2l + 3)(2l + 5)} − · · · � � (3.6) and ˆ n′_l(ka) = (2l + 1)!! (ka)l 1 a � − l � 1₋ (ka) 2 1!2(1_{− 2l)} + (ka)4 2!22₍₁_{− 2l)(3 − 2l)} − · · · � + � − 2(ka) 2 1!2(1_{− 2l)} + 4(ka)4 2!22₍₁_{− 2l)(3 − 2l)} − · · · � � . (3.7) Replacing Eqs. (3.4)-(3.7) in Eq. (3.3) and simplifying one obtains

Kl=− (2l + 1)!!2 a2l+1 {(3.5)} − Rl{(3.7)} {(3.4)} − Rl{(3.6)} , (3.8)

where_{{(3.X)} makes reference to the curly brackets in Eq. (3.X).}

2_{For exponentially bound potentials the analyticity of the Jost function at k = 0 is warranted [3],}

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3.2. ERF for the neutral case 25 Note that Eq. (3.8) is a smooth function of k2 and therefore of energy3, which means that it can be expanded around zero energy as

Kl= ∞ � n=0

cl,nEn. (3.9)

Of course, a particular case appears when the denominator in the r.h.s of Eq. (3.3) vanishes at zero energy, and therefore, Kl diverges. This effect defines the “transition” from a weakly bound state to a resonance or virtual state if the interaction is slightly modified (for instance, changing a little bit the potential depth) [3, 5].

The neutron-proton system (n-p system) is perhaps the preferred system to analyze in the frame of the eﬀective range theory (see for instance Refs. [50, 51, 5, 2]). For this system, the phase shifts (δ0 at E < 10 MeV) have been well described via the ERF by using the ﬁrst two terms in the sum of Eq. (3.9).

This fact was already seen before 1949 (right in the decade when the ﬁrst formal sup-ports of the eﬀective range theory were published) but at that time the contribution of higher orders in such an expansion could not be analyzed because of experimental uncertainties [46]. Nowadays thanks to the development of realistic nucleon-nucleon interactions, it can be checked much easily.

This simple description has been one of the points to encourage the study and the development of the eﬀective range theory.

3.2.1 The eﬀective-range expansion (ERE)

As we have seen in Chapter 2, to determine experimentally elastic phase shifts we need to analyze the collision in Fig. 2.1 in a given energy range [Emin, Emax]. Thus, we always ﬁnd an energy gap between 0 and Emin (large or small depending on the collision and the laboratory conditions) where no experimental information is available. To bridge this gap, the ERF provides an interesting alternative thanks to its smoothness at low energies.

In the previous section we have checked that the ERF for a ﬁnite-range potential admits the expansion in Eq. (3.9). This series is also valid for potentials exponentially bounded [3] and usually it is rewritten in terms of k2 _as

K_l= ∞ � n=0 Q_l,nk2n, (3.10) with Ql,n= cl,n � �2 2µ �n

. Notice that Eq. (3.10) is the compact form of Eq. (3.1) which is known as the eﬀective range expansion (ERE).

As is discussed in Refs. [52, 3], Eq. (3.10) is not necessarily valid for other asymptotic conditions of VN. In particular, for potential tails in the class O(1/rν) with ν > 1, Eq. (3.10) is not valid. Thus, our study will be restricted to potentials exponentially bounded hereafter.

3_{The R-matrix can be written as R} l=

�_∞

λ=1 γ2

λ

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For E _{≈ 0, one expects that the ﬁrst few terms on the r.h.s. of} Kl=− 1 al +1 2rlk 2_{− P} lrl3k4+ ∞ � n=3 Ql,nk2n, (3.11)

describe correctly both Kl and δl behaviors. For instance, the contribution of the sum over n will be less important for k2 _{→ 0 and only the contribution of the first three, two} or one term(s) will be enough. For this reason, some works are especially focused on evaluating the first few coefficients (see for instance Refs. [49, 53]).

As one sees, the first three coefficients of the ERE have adopted a particular con-vention. The parameters al, rl and Pl are known as the scattering length, the effective range and the shape parameter respectively. These three parameters together with all Ql,n are named as the effective-range parameters (ERPs).

Special case

Up to now, we have found a phase shift description at low energies via the ERF and the ERPs. Before discussing the interpretation or meaning of the ERPs, let us show an example to illustrate how useful they are.

Consider the Bargmann potential [5] VN(r) =−8bβ2

exp(_−2βr) [1 + b exp(_−2βr)]2

�2

2µ, (3.12)

with b = β−α_β+α. If α = 0.04 fm−1 and β = 0.81 fm−1, this potential gives us a good description of the n-p system (s-wave) [51]. Suppose that we perform an experiment to determine δ0 for energies in 1-2 MeV as Fig. 3.1a shows (suppose also that the uncer-tainties are very small). Then we repeat the experiment to include extra data in 0.4-1 MeV (Fig. 3.1b), and ﬁnally, we repeat the experiment once more to include data in 0.1-0.4 MeV (Fig. 3.1c). �a� �b� �c� �d� ∆0 �rad � 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2

E �MeV� E �MeV� E �MeV� E �MeV�

Figure 3.1: Phase shifts for the s-wave of the n-p system according to the potential Eq. (3.12). For (a), (b) and (c): points represent experimental phase shifts, and dotted lines give the tendency at zero energy according to the points by ﬁtting polynomials up to second, fourth and ﬁfth order in energy respectively. For (d) the solid line gives the exact phase shifts.

As one sees, from the experimental phase shifts a good δ0-tendency at very low energies is only achieved when experimental data close to zero energy are available.

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3.2. ERF for the neutral case 27 �a� �b� �c� �d� K0 �fm � 1�10 � 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 ∆0 �rad � 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2

E �MeV� E �MeV� E �MeV� E �MeV�

Figure 3.2: ERF for the s-wave of the n-p system according to the potential Eq. (3.12). For (a), (b) and (c): points represent experimental data as in Fig. 3.1. Dashed lines in the top panels are obtained via linear ﬁts in energy. Dashed lines in the bottom panels are obtained from ﬁts in the top panels via δ0= arctan(k/K0). For (d), solid lines give the exact ERF or δ0.

To summarize, from the previous results we can understand the ERF as a transfor-mation of δl, such that its parametrization is much simpler of ﬁnding and more accurate thanks to its analytic properties at low energies.

Interpretation or “meaning” of the eﬀective-range parameters

Although the ERPs have been used in the literature to describe phase shifts at low ener-gies, there is not a clear and general physical meaning for them. This is not unexpected if we pay attention to their units as Table 3.1 displays.

Table 3.1: Units of the eﬀective-range parameters.

l al rl Pl Ql,n 0 fm fm - fm2n−1 1 fm3 fm−1 fm4 fm2n−3 2 fm5 fm−3 fm8 fm2n−5 .. . ... ... ... ... l fm2l+1 _fm−2l+1 _fm4l _fm2n−1−2l

Clearly, the dimensions of the scattering length and the eﬀective range agree with their names for l = 0 only, otherwise the names stand as a convention. This indicates that a direct physical meaning cannot be given for the ERPs, except perhaps if l = 0.

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via the Bargmann potential in Eq. (3.12). The ERF in this case is [5] K0=− 1 (α− β)/(αβ) + 1 2 2 β− αk 2_, _(3.13)

where a0 and r0 are easily obtaining by comparing Eqs. (3.13) and (3.11).

Note that K0 in Eq. (3.13) coincides with its Maclaurin series for all energy. In fact, the first two terms define this series making it ideal to find the meaning of the ERPs (especially, a0 and r0).

As one realizes by simple inspection of Eq. (3.13), if αβ > 0 then a0r0 < 0. This means that at least one of them (a0 or r0) is not a physical distance. For instance, for the n-p system (α = 0.04 fm−1 and β = 0.81 fm−1), the scattering length does not mean a physical distance (a0 = −23.8 fm) because of the sign, but its absolute value can be interpreted physically as we shall see later.

On the other hand, by choosing a set _{{α, β}, Fig. 3.3 shows a ﬁrst approach to why} r0 is called the eﬀective range.

-98.4 fm

3.3 fm

-23.4 fm

3.5 fm

-12.6 fm

3.7 fm

-98.8 fm

2.5 fm

-23.8 fm

2.6 fm

-13.1 fm

2.7 fm

-99.0 fm

2.0 fm

-24.0 fm

2.1 fm

-13.3 fm

2.1 fm

Α � Β 0.61 0.81 1.01 0.01 0 1 2 3 4 5 0 �1 �2 �3 0 1 2 3 4 5 0 �2 �4 �6 0 1 2 3 4 5 0 �2 �4 �6 �8 �10 0.04 0 1 2 3 4 5 0 �1 �2 �3 0 1 2 3 4 5 0 �2 �4 �6 0 1 2 3 4 5 0 �2 �4 �6 �8 �10 0.07 0 1 2 3 4 5 0 �1 �2 �3 0 1 2 3 4 5 0 �2 �4 �6 0 1 2 3 4 5 0 �2 �4 �6 �8 �10 r �fm� r �fm� r �fm�

Figure 3.3: Solid line: Bargmann potential or Eq. (3.12) (in MeV) with µ the neutron-proton reduced mass, according to the couple α and β (both in fm−1). Point: the eﬀective range, r0. Inset values: (negative) a0 and (positive) r0.

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3.3. ERF for the charged case 29 Regarding the scattering length, it is useful to recall Eq. (2.37) at very low energies, i.e.,

dσexp

dΩ ≈ |fas(k, θ, lmax= 0)|

2_{, for E} _{→ 0.} _(3.14)

By using Eq. (2.36) one can rewrite the previous equation as dσexp

dΩ ≈ 4π

cos2(δ0) [k cot(δ0)]2

. (3.15)

Note that the term k cot(δ0) is the ERF K0, and thanks to the condition E → 0, one can approximate it by the ERE. This leads us to

dσexp dΩ ≈ 4π cos2_(δ 0) � −a10 + 1 2r0k2− P0r30k4+ �∞ n=3Q0,nk2n �2. (3.16)

By expanding Eq. (3.16) in Maclaurin series we obtain a second interpretation of the ERPs, i.e., these parameters describe the curvature of the diﬀerential cross section at very low energies (from there the name shape parameter for Pl). Moreover, at zero energy the diﬀerential cross section is determined by

dσexp

dΩ = 4πa 2

0. (3.17)

This means that the scattering length deﬁnes a sphere of radius|a0| such that its surface area is the diﬀerential cross section at zero energy.

3.3 ERF for the charged case

A simple way to understand the ERF at low energies (E_{≈ 0) is to answer the question:} how can one transform δl(E) in a smooth function around zero energy? As we saw previously, the answer for the neutral case is simple and can be divided in three steps: (i) ﬁnd δl = δl(E), (ii) compute cot(δl) and (iii) multiply by k2l+1. The result is Eq. (3.2) which smoothness is seen in Eq. (3.9).

For the charged case one can follow a similar way, although more diﬃculties appear because of the Coulomb interaction (in our case repulsive Coulomb interaction). The Coulomb potential between two particles is not exponentially bounded, in fact it behaves as O(1/r) for r _{→ ∞. With this obstacle on the way, J. Hamilton, I. Øverb¨o and B.} Tromborg [12] show an interesting proposal to overcome it. They modify the Coulomb interaction (VC) by adopting a Yukawa form for it (∝ e−λr/r, which is exponentially bounded) and of course, they keep the nuclear potential satisfying _|VN| < VC at large distances. After separating the contribution of the nuclear part, they evaluate the limit λ→ 0 in order to recover the correct asymptotic behavior of VC. This procedure leads them to the ERF for the charged case

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where the functions wl and h read wl= l � j=0 � 1 + j 2 η2 � , (3.19) h = ψ(iη)− ln(iη) + 1 2iη − iπ e2πη_{− 1}, (3.20)

with ψ the digamma function [40].

Thus, the ERF deﬁned by Eq. (3.18) shows a smooth behavior around zero energy in the sense of Eq. (3.9) [48, 54, 12]. In terms of k2, Eq. (3.11) is adopted to keep the conventions for the ERPs.

This chapter has been devoted to provide the basis of the ERF. In Chapter 5 we shall introduce and analyze the function Δl= π cot(δ_e2πη₋₁l) [see Eq. (5.10)]. In this thesis, it is

found that Δl has useful properties to analyze phase shifts in extended energy ranges. For this reason we shall postpone this discussion up to Chapter 5.

3.4 Usual potential models according to the ERE

In nuclear physics protons and neutrons are considered as fundamental particles, which is in contrast with particle physics, where they are formed by quarks. Thus, the interaction among nucleons (nuclear interaction) can be understood as the residual strong interaction between quarks. Since 1934, when H. Yukawa provided the ﬁrst formal explanation about the nuclear force (by introducing the so-called Yukawa potential), the development of several nuclear potentials capable of explaining some features as bound-state spectra, scattering collisions, nuclear shapes, level transitions, etc, have been widely studied. Moreover, the idea of simplifying the nuclear system from a set of nucleons to a set of clusters (in analogy to a nucleon as a cluster of quarks) introduces a huge advantage for theoretical calculations. In fact, this is the idea behind Fig. 2.1, where m1 and m2 represent phenomenologically clusters. Notice that a system of A = N + Z particles (the nuclear system) is reduced to one of two bodies (two clusters).

Potential models as Bargmann, Wood-Saxon, Gaussian and square well are usually assumed to describe the central interaction between nuclei. All these potentials are exponentially bounded, which means that the ERF, the ERE and the ERPs are well established by Eqs. (3.2), (3.18) and (3.1). Thus. in the following chapters we shall pay special attention to these four potentials because (i ) they cover the standard potential shapes, (ii ) some of them can be analyzed analytically in the frame of the eﬀective-range theory (Bargmann and square well), (iii ) others let us to explore eﬀects introduced by small changes in the shape (Wood-Saxon especially), and (iv ) for some nuclear systems, potential models are suggested in the literature (Bargmann, Gaussian and Wood-Saxon).

3.5 Recovering phase shifts

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3.5. Recovering phase shifts 31 This is nothing but solving Eqs (3.2) and (3.18) for δl, i.e., for the neutral case

δl = arctan� k 2l+1 Kl

�

, (3.21)

and for the charged case

δl = arctan � 2wl l!2_a2l+1 N Kl− 2wlh π e2πη_{− 1} � . (3.22)

At low energies (E ≈ 0) the phase shifts can be computed by replacing Kl by its Maclaurin expansion [Eq. (3.11)] in Eqs. (3.21) and (3.22).

In this chapter we have reviewed the connection between δl and Kl (including the ERPs) for neutral and charged cases. In the previous one, we showed a similar connection between theory an experiment via the diﬀerential cross section and phase shifts. Thus, summing up all these results, “the neuron diagram”4 _{in Fig. 3.4 illustrates how phase} shifts are the meeting point among the experiment, the nuclear interaction and the ERF.

Exp dσ dΩ fsa δl Kl VN ERPs Eq. (2.33) or (2.44) Eq. (2.32) or (2.46) Eq. (2.21) or (2.42)

See next chapter or Refs. [29, 49] Eq. (3.1)

Eq. (3.21) or (3.22)

Figure 3.4: Phase shifts (δl) as a meeting point among the experiment (Exp), the nuclear interaction (VN) and the effective range function (Kl). Connections with the differential cross section (dσ/dΩ), the scattering amplitude (fsa) and the effective range parameters (ERPs) are also shown.

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CHAPTER

4

Precise computation of eﬀective-range parameters

In this chapter we shall see a detailed discussion of Ref. [29] which is the first contribution of the PhD project. The main objective is to find accurate values for the ERPs in neutral and charged cases. Following the proposal by D. Baye et al., [49] (where the scattering length, the effective range and the shape parameter are analyzed), a systematic technique is developed here to calculate ERPs up to an arbitrary order.

Keep in mind the following acronyms and conventions:

ERF: effective-range function, Kl. ERE: effective-range expansion. ERPs: effective-range parameters. al: scattering length.

rl: eﬀective range. Pl: shape parameter. Ql,n: ERP of order n > 2.

4.1 Motivation

Let us recall the ERE and the ERF for the neutral and charged case [Eqs. (3.1), (3.2) and (3.18)] Kl =− 1 al +1 2rlk 2_{− P} lr3lk4+ ∞ � n=3 Ql,nk2n, ERE, (4.1)

Kl = k2l+1cot(δl), ERF for the neutral case, (4.2) Kl = 2wl l!2_a2l+1 N � π cot(δl) e2πη_{− 1} + h �

, ERF for the charged case, (4.3) As we have discussed in Chapter 3, by truncating the r.h.s. of Eq. (4.1) one can expect a good approximation of the ERF at low energies. Thus, this approximation can be used to parameterize the phase shifts, δl in Eq. (4.2) and (4.3), in this energy regime. This is logical from the mathematical point of view; nevertheless, from the

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physical one, we need to know if this truncation can cover (at least partially) the energy interval where experimental information is available. This fact is illustrated in Figure 4.1 which shows schematically a set of 8 energies where experimental information is available (phase shifts), and how the truncation of the ERE describes progressively a larger energy interval.

n = 3 n = 2

n = 1

E

E = 0 E1 E2 E3 E4 E5 E6 E7 E8

Figure 4.1: Illustration about how the ERE provides a good description in an energy range determined by the order of truncation n (double arrows). The points E1, E2, etc, are hypothetical energies where experimental phase shifts are known.

In the example of Fig. 4.1, we need at least three ERPs to provide a correct descrip-tion of the ﬁrst three experimental data points, four to include points at E4, E5 and E6, and more than four to cover E7 and E8. Of course, this is not a rule and depends on the system, i.e., for some systems less ERPs will be needed to describe the same energy interval and for others more ERPs will be required. Thus, for a speciﬁc system, this study attempts to answer two questions: how many ERPs are needed to describe phase shifts at energies below Emax? and how can we compute ERPs precisely?

In this chapter, the previous two questions (or motivations) are explored in the frame of potential models. This will lead us to a solid technique to compute ERPs, and will help us to provide extra support in future studies to describe experimental data via potential models.

4.2 Conventions and re-normalization

Although neutral and charged cases will be analyzed separately, it is useful to deﬁne some conventions to keep the structure between them as close as possible.

Let us introduce the following convention:

• H1l= H1l(k, a) is the Ricatti-Bessel, ˆjl (regular Coulomb, Fl) function at r = a for the neutral (charged) case.

• H2l= H2l(k, a) is the Ricatti-Neumann, ˆnl (irregular Coulomb Gl) function at r = a for the neutral (charged) case.

• Prime represents the partial derivative with respect to r at r = a (e.g., H′ il = ∂H_il(k, r)/∂r_|_r=a).

With the previous conventions, the phase-shift δl is obtained through [Eqs. (2.21) and (2.42) in the previous notation]

tan(δl) =−

H1l− aRlH1l′ H_2l_{− aR}lH_2l′

The eﬀective-range function in nuclear physics A method to parameterize phase shifts and extract ANCs Oscar Leonardo Ram´