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Photonuclear Sum Rules for 6He. 2011
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Title | Photonuclear Sum Rules for 6He. |
Creator |
Goerke, Raymond |
Date Created | 2011-09-14T20:59:09Z |
Date Issued | 2011 |
Description | We perform the ab-initio calculation of photonuclear sum rules for 4He and 6He using semi-realistic potentials. Our results include the first ab-initio calculation of the electric polarizabililty of 6He. We perform the calculation by expanding the the wavefunctions in a Hyperspherical Harmonic basis and solving the Schr¨odinger equation exactly. Model space trunction effects are estimated by incrementing the size of the Hilbert space, and evaluating the convergence pattern. We also compute sum rules by integrating the theoretically computed cross section available from the literature. Full convergence is not yet achieved. A discussion of the preliminary results is presented. Future work, taking advantage of negligible symmetries in the hyperspherical harmonic basis, should be able to achieve full convergence. |
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Graduating Project |
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Text |
Language | Eng |
Collection |
Physics & Astronomy Undergraduate Honours Theses |
Series | Undergraduate Honours Theses |
Date Available | 2011-09-14T20:59:09Z |
DOI | 10.14288/1.0085969 |
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Science, Faculty of Physics and Astronomy, Department of |
Citation | Goerke, Raymond. 2011. Photonuclear Sum Rules for 6He. Undergraduate Honours Thesis. Department of Physics and Astronomy. University of British Columbia. |
Peer Review Status | Unreviewed |
Scholarly Level | Undergraduate |
Copyright Holder | Goerke, Raymond |
URI | http://hdl.handle.net/2429/37311 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/547/items/1.0085969/source |
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Photonuclear Sum Rules for 6He by Raymond Goerke A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Bachelor of Science in THE FACULTY OF SCIENCE (Combined Honours Physics and Mathematics) The University Of British Columbia (Vancouver) April 2011 c© Raymond Goerke, 2011 Abstract We perform the ab-initio calculation of photonuclear sum rules for 4He and 6He using semi-realistic potentials. Our results include the first ab-initio calculation of the electric polarizabililty of 6He. We perform the calculation by expanding the the wavefunctions in a Hyperspherical Harmonic basis and solving the Schrödinger equation exactly. Model space trunction effects are estimated by incrementing the size of the Hilbert space, and evaluating the convergence pattern. We also compute sum rules by integrating the theoretically computed cross section available from the literature. Full convergence is not yet achieved. A discussion of the preliminary results is presented. Future work, taking advantage of negligible symmetries in the hyperspherical harmonic basis, should be able to achieve full convergence. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Electric Polarizability . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Photoabsorption Cross Sections . . . . . . . . . . . . . . . . . . 6 2.2.1 Photonuclear Moments and Sum Rules . . . . . . . . . . 8 2.3 Sum Rules and Nuclear Observables . . . . . . . . . . . . . . . . 8 2.3.1 Polarizability Sum Rule . . . . . . . . . . . . . . . . . . 9 2.3.2 Thomas-Reiche-Kuhn Sum Rule . . . . . . . . . . . . . . 9 2.3.3 Bremstrahlungs Sum Rule . . . . . . . . . . . . . . . . . 10 3 Computational Techniques . . . . . . . . . . . . . . . . . . . . . . . 12 3.1 Hyperspherical Basis . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Evaluation of Matrix elements . . . . . . . . . . . . . . . . . . . 13 3.3 The Lanczos Algorithm . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4.1 Calculating the Sum Rules . . . . . . . . . . . . . . . . . 18 iii 4 Calculating the Full Continuum Response . . . . . . . . . . . . . . . 21 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.1 Testing the Sum Rule Code . . . . . . . . . . . . . . . . . . . . . 24 5.2 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.4 Photoabsorption Cross Section . . . . . . . . . . . . . . . . . . . 32 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A Complete Derivations of Sum Rule Formulas . . . . . . . . . . . . . 38 A.1 Thomas-Reiche-Kuhn Sum Rule . . . . . . . . . . . . . . . . . . 39 A.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.2 Bremstrahlung Sum Rule . . . . . . . . . . . . . . . . . . . . . . 45 A.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 A.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 A.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 A.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.3 Polarizability Sum Rule . . . . . . . . . . . . . . . . . . . . . . . 50 A.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 B Hyperspherical Formalism . . . . . . . . . . . . . . . . . . . . . . . 51 B.1 Jacobi Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 51 B.2 Hyperspherical Coordinates and the Laplace Operator . . . . . . . 52 B.3 Hyperspherical Harmonics . . . . . . . . . . . . . . . . . . . . . 55 B.4 Hyperspherical Basis . . . . . . . . . . . . . . . . . . . . . . . . 57 iv List of Tables Table 1.1 Ab inito calculations of electric dipole polarizability of various nuclei. Note there is no value at all for 6He. . . . . . . . . . . 3 Table 5.1 Previous calculations from Gazit et al. [1] of the sum rules using the Lanzcos coefficients are compared with our calculations us- ing the same Lanzcos coefficients to test the validity of the sum rule code. We compared our results for largest model spaces denoted by Kmax. . . . . . . . . . . . . . . . . . . . . . . . . . 24 Table 5.2 Convergence error in the polarizability sum rule for 4He with the simple potential models. The value is estimated as the per- centage change from the Kmax = 10 to Kmax = 12 model space size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Table 5.3 Theoretical and experimental results for the polarizability of 4He. The two other theoretical values were computed using more realistic two-body potential models and included three- body forces. The experimental values were obtained through Compton scattering. . . . . . . . . . . . . . . . . . . . . . . . 26 Table 5.4 Convergence error estimate for sum rules. Computed as the percentage change in the value from the second-largest to the largest model space. . . . . . . . . . . . . . . . . . . . . . . . 30 Table 5.5 Estimated model-dependent error. Computed as the standard deviation from the mean value among potential models used. . 32 Table 5.6 Binding energies and root mean square radii for 6He and 4He using Kmax = 12. The error in the last digit is taken to be the difference between the Kmax = 12 and Kmax = 10 values. . . . . 32 v Table 5.7 PSR of 6He calculated by integrating the cross section com- pared to sum rule approach. . . . . . . . . . . . . . . . . . . . 34 vi List of Figures Figure 1.1 A nuclear chart showing the relative sizes of light. Highlighted are 4He, 6He, and some other halo nuclei. . . . . . . . . . . . 3 Figure 5.1 Polarizability of 4He computed using different sized model spaces denoted by maximum grandangular momentum Kmax. The experimental points were obtained by Compton scattering with experimental error indicated by the shaded regions. . . . 25 Figure 5.2 A cartoon of how the matrix elements in a hermitian matrix can be allocated efficiently between four processors. The colour of each matrix element corresponds to the processor it is assigned to. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Figure 5.3 TRK sum rule computations for different sizes of model space characterized by max grandangular momentum Kmax. . . . . . 29 Figure 5.4 BSR sum rule computations for different size of model space characterized by max grandangular momentum Kmax. . . . . . 30 Figure 5.5 Polarizability sum rule computations for different sizes of model space characterized by max grandangular momentum Kmax. The experimental point is from Pachucki and Moro [2] using Coulomb breakup, the shaded region is the experimental error. 31 Figure 5.6 The photoabsorption cross section computed by Bacca et al. [3] using the AV4’ potential. The inverse-square energy-weighted cross section is included (scaled to be visible) to show how the lack of data going to infinity does not effect the value of the PSR. The model space size used to compute the cross section was Kmax = 12. . . . . . . . . . . . . . . . . . . . . . . . . . 33 vii Figure B.1 The (unnormalized) Jacobi coordinates for four particles. The η̃ i vectors point along the direction of the normalized η i given by equation (B.1). . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure B.2 Geometric interpretation of the recursive generation of hyper- spherical coordinates . . . . . . . . . . . . . . . . . . . . . . 53 viii Acknowledgements I would like to thank TRIUMF for their summer student program where I learnt much of the background and theory involved in this project. I’m grateful to the UBC Faculty of Science, Department of Physics and Astronomy, and Undergrad- uate Physics Society for funding me to present some of this work at the Canadian Undergraduate Physics Conference. I would like to thank my parents, without whose love and support I could not have gotten this far. Finally, I owe an espe- cially heartfelt thank you to Sonia Bacca, my supervisor, who has taught me and helped me enormously over the last year. ix Chapter 1 Introduction The interaction of nucleons with the electromagnetic field (photons or electrons) is well understood and enables a clear separation of the analysis of the probe from that of the nuclear dynamics, which is still not well understood. In particular, ab- initio methods are very valuable for understanding nuclear dynamics. An ab-initio calculation is “from first principles” in that it describes the quantum mechanical problem as nucleons interacting via a nuclear Hamiltonian. In cases where the Schrödinger equation is solved without introducing approximations, we say such calculations are exact. In this work we perform exact, ab-initio calculations and the terms are used interchangeably1. Exact calculations also allow us to test the role that three-body forces play in the determination of electromagnetic observables. Understanding what observ- ables are sensitive to many-body forces is important for constraining self-consistent models of many-body forces such as Chiral Perturbation Theory [4]. Due to the obvious computational constraints, exact calculations have been re- stricted to the light nuclei with mass numbers about A≤ 12−16 for bound states. In the case of the electromagnetic breakup observables, which depend on contin- uum properties of the nucleus, i.e. properties of unbound states, then typically we are restricted to about A≤ 7. Often, the full continuum response must be calculated in these situations, for which some powerful tools including the Lorentz Integral Transform technique have been developed. 1Note that in the literature this may not generally be the case. 1 In this work, I have studied the photonuclear sum rules of light nuclei including 4He and 6He. Photonuclear sum rules are related to the moments of the photonu- clear cross section which can be compared with experiment. I have used semi- realistic potentials in my calculations, i.e. potentials that do not include high l parts of the interaction, to simplify the calculations. This may lay the groundwork for more sophisticated future calculations. Of particular interest is the polarizability sum rule, which allows a direct cal- culation of the electric polarizability of the nucleus. When a nucleus is placed in an electric field, the field displaces the nucleons and creates a dipole moment in the nucleus centre of mass. In the weak field limit, the induced dipole is directly proportional to the external field: Dz = αEEz. (1.1) Where αE is the polarizability of the nucleus. Of the several nuclear systems we can study, “halo” nuclei like 6He are of particular interest. The name “halo” is given to nuclei with anomalously large matter radius attributed to loosely bound outer neutrons. Scattering experiments suggest that these nuclei have radii much larger than other isotopes of the same element and are often comparable to nuclei with much larger mass numbers. For example, while the radius of 4He is measured to be about 1.7 fm [5], the radius of 6He is found to be about 3.7 fm [6]. Figure 1.1 plot the relative size of light nuclei. For 6He, the neutron halo makes the electric polarizability markedly interesting because we would expect it to be much larger than other helium isotopes, especially the extremely compact 4He. The diffuse neutron halo also makes it particularly difficult to achieve convergent results with 6He as many basis states are needed to capture the long tail. In Table 1.1, I have presented previous ab-initio calculations of nuclear polar- izability of hydrogen and helium isotopes. Note the lack of values for 6He. I will be presenting the first computations of this observable for 6He. There is a single experimental measurement of the polarizability of 6He, and the estimated error on this value is large due to the method used to extract it from the data [7]. The electric polarizability also has importance in high-precision atomic spec- troscopy. Here, accurate measurements of the energy difference between atomic 2 Figure 1.1: A nuclear chart showing the relative sizes of light. Highlighted are 4He, 6He, and some other halo nuclei. Nucleus αE(fm3) Reference 2H 0.6328(17) [8] 0.6314(19) [9] 3H 0.139(2) [10] 0.139 [11] 3He 0.149(5) [10] 0.145 [12] 4He 0.0683(8)(14) [10] 0.0655(4) [1] 6He Table 1.1: Ab inito calculations of electric dipole polarizability of various nu- clei. Note there is no value at all for 6He. 3 energy levels (also called the transition frequency when referring to the resonant laser frequency) are becoming precise enough to see effects of nuclear structure [13]. In general, transition frequencies between S states in single-electron atoms and ions are determined by a initial value derived from a relativistic hydrogen-atom model, to which corrections are applied resulting from Quantum Electrodynam- ics (QED) and nuclear effects. The QED effects are much more important than nuclear effects in light nuclei, but recent progress in this area ensures the highest- order terms have already been calculated and remaining terms are estimated to have smaller effect than nuclear corrections [14]. The biggest nuclear corrections are as- sociated with the non-zero charge radius of the nucleus, and due to the fact that it is not a rigid body. The latter effect is dominated by the electric polarizability, which is effectively a measure of how much the electron deforms the nucleus and “drags” the deformation as it orbits (in analogy to the way the moon creates the tides). Recent measurements of hydrogen-like atoms and ions can be used as mea- surements of the charge radius by fitting the charge-radius correction to the experi- mental data while using calculated values for the QED and polarization corrections [10], and it has been shown that these kinds of measurements agree with the results from direct measurements of proton radius using electron scattering [15, 16]. Although at the current stage [17], electric polarizability plays only a marginal role in comparison to other contributions to atomic observables in the case of six- body nuclei, future improvements of experimental precision will require reliable values for this observable. While the electric polarizability is perhaps the most interesting observable in terms of its implications to other research, the other sum rules I have studied in- clude the bremsstrahlungs sum rule (BSR) and the Thomas-Reiche-Kuhn (TRK) sum rule, which have mostly theoretical interest. The former is relatable in several ways to the geometric structure of the nucleus. This may be useful in understand- ing the neutron halo in 6He and potentially other halo nuclei. The later is related to the so-called TRK enhancement factor, which is used in the study of nuclear potential models by providing a way to quantify the role of exchange effects in the interaction (for more details, see A.1). 4 Chapter 2 Theoretical Framework In this chapter we discuss the photonuclear cross section as a way to gain infor- mation about nuclear observables coupled to the electromagnetic field, and the use of sum rules to relate photonuclear moments to tractable computable expressions. Before introducing the more generic case, we consider a concrete example with a classical analogue; namely, the electric polarizability of a nucleus, which can be thought of as the effect of a perturbation by a small external electric field. 2.1 Electric Polarizability The perturbed Hamiltonian resulting from applying an electric field with magni- tude Eext pointed along the z direction is H = H0−EextDz, where Dz is the z com- ponent of the induced dipole moment. Due to parity symmetry of the ground state, the first order correction 〈0| (-Eext Dz)|0〉 is zero. The second order correction is E20 =− 1 2 αE2ext, where α corresponds to the classical definition of polarizability; namely, if |0〉 is the ground state of the full Hamiltonian H we have: αE = lim Eext→0 〈0|Dz |0〉 Eext . 5 By using the standard perturbation theory arguments, we can thus show αE = α ∑n | 〈n|Dz |0〉 |2 En−E0 . (2.1) Where the sum ranges over all possible final states |n〉, including continuum states. By convention, the fine structure constant α is included, which absorbs various electromagnetic units from the dipole operator. In this work, αE will always refer to the electric polarizability while α will always refer to the fine structure constant. In the following sections we will show that equation (2.1) can be calculated as a sum rule for the photonuclear cross section and that, in a similar way, other sum rules, including the bremmstrahlungs and Thomas-Reiche-Kuhn sum rule, will lead to analagous formulas related to other electromagnetic observables with theoretical interest. 2.2 Photoabsorption Cross Sections The photoabsorption cross section can be defined in terms of a response function R(ω), where ω is the photon energy: σ γ(ω) = GωR(ω), where G is a constant which we define as G = 4pi2α . The response function encodes the effect of a small perturbation by a photon to the system. For the most general effect of photoabsorption, the response function is written in terms of the electromagnetic current four-vector Jµ = (ρ, j). R(ω) =∑ n | 〈n|Jµ(q) |0〉 |2δ (En− (E0+ω)), (2.2) where q is the momentum transfer (for photoabsorption q = ω). As before, the sum over final states must also include an integral over continuum states. We perform a multipole decomposition on Jµ , first by expanding ρ in terms of 6 Coulomb multipoles: ρ(q) = 4pi∑ lm ilYlm(q̂)Clm(q), Clm(q) = i−l 4pi ∫ dq′ρ(q′)Ylm(q̂′). Where Ylm are the spherical harmonics. An analogous expansion of j can be per- formed in terms of the vector spherical harmonics1: j(q) = 4pi∑ Jlm YJl1m (q̂)J m Jl(q), JmJl(q) = 1 4pi ∫ dq′j(q′)YJl1m (q̂′). In the low energy limit, the current multipoles JmJl can be expressed in terms of the Coulomb multipoles, and of the Coulomb multipoles the dipole is the most dominant. 2 It is thus possible to relate the full response (equation (2.2)), which can in principle be measured, to the unretarded dipole response: RE1UR(ω) =∑ n | 〈n|Dz(ω) |0〉 |2δ (En− (E0+ω)). (2.3) For this reason we can consider the unretarded dipole (E1UR) cross section: σE1URγ (ω) = GωR E1UR(ω), where Dz refers to the unretarded dipole operator Dz = ∑Ai=1 riτ z i /2. Unretarded refers to the neglect of relativistic effects. From now on when referring to the photonuclear cross section, I shall be referring to the E1UR approximation. 1If unfamiliar with the vector spherical harmonics, one can find definitions and lists of properties in [18] 2This is called the Siegert theorem, a full derivation can be found in [19]. 7 2.2.1 Photonuclear Moments and Sum Rules We define the moments of the photonuclear cross section, labeled by k, as mk = ∫ ∞ ωth dωωkσE1URγ (ω), where ωth is the smallest frequency where photoabsorption is allowed (due to en- ergy conservation). Applying the definitions from the previous section we have: mk = G∑ n | 〈n|Dz |0〉 |2(En−E0)k+1. (2.4) Various ways of evaluating the sum over final states will result in expressions gen- erally referred to as sum rules. In particular, we will consider three sum rules: the Thomas-Reiche-Kuhn (TRK), bremsstrahlungs (BSR), and polarizability sum rule (PSR) given respectively by ΣTRK = m0 = G 2 〈0| [D, [H,D]] |0〉 , ΣBSR = m−1 = G 〈0|D†zDz |0〉 , ΣPSR = m−2 = G∑ n | 〈n|Dz |0〉 |2 En−E0 . Derivations of the first two equations can be found in the Appendix section A.1.1 and A.2.1. In the next section we will interpret the above sum rules in terms of other nuclear observables. 2.3 Sum Rules and Nuclear Observables To handle the sum over final states in the expressions for the photonuclear sum rules (2.4), one takes advantage of the closure relation: ∑ n |n〉〈n|= 1. By performing this analysis, we can connect the sum rules to various nuclear ob- servables. We will briefly go over some of the results here; complete derivations 8 are provided in Appendix A. 2.3.1 Polarizability Sum Rule We now refer back to the discussion in section 2.1 and see that evidently ΣPSR = 2pi2αE , so that calculation of the sum rule amounts directly to calculation of the electric polarizability. In order to avoid the sum over final states, we can rewrite the sum rule (as shown in A.3.1) ΣPSR =−G 〈0| D † zDz E0−H |0〉 . (2.5) In later chapters, we will see how to compute the right hand side of (2.5) by using the Lanczos algorithm. 2.3.2 Thomas-Reiche-Kuhn Sum Rule The TRK sum rule is defined by ΣTRK = G 2 〈0| [D, [H,D]] |0〉 . (2.6) We can split the Hamiltonian in (2.6) into a kinetic and potential energy part: ΣTRK = G 2 (〈0| [Dz, [T,Dz]] |0〉+ 〈0| [Dz, [V,Dz]] |0〉) . The kinetic energy part can be explicitly calculated (A.1.2): G 2 〈0| [Dz, [T,Dz]] |0〉= G NZ2mA , so that we can write ΣTRK = G NZ 2mA ( 1+κTRK ) , 9 where κTRK = 2mA NZ 〈0| [Dz, [V,Dz]] |0〉 . The TRK sum rule thus allows for a computation of κTRK , known as the TRK enhancement factor, which is a measure of how much the potential does not com- mute with the dipole operator. If the potential has derivatives or isospin-dependent terms, then it will not commute with Dz. For a purely central potential (i.e. V = V(r)), κTRK is zero. The TRK enhancement factor is of purely theoretical interest and is a useful comparison of different potential models. 2.3.3 Bremstrahlungs Sum Rule The BSR sum rule is defined as ΣBSR = G 〈0|D†zDz |0〉 . There are a number of interesting ways to relate the BSR sum rule to geometric properties of the nucleus. In particular, we can show (A.2.2): ΣBSR = G ( NZ A )2 〈 R2PN 〉 , (2.7) where RPN is the distance between the proton and neutron centres of mass. We define 〈 r2p 〉 to be the mean square proton radius with respect to the nuclear centre of mass, and 〈 r2pp 〉 to be the mean square distance between protons. We can show that (see A.2.3): ΣBSR = G ( Z2 〈 r2p 〉− Z(Z−1) 2 〈 r2pp 〉) . An identical derivation can be done for neutrons (A.2.4): ΣBSR = G ( N2 〈 r2n 〉− N(N−1) 2 〈 r2nn 〉) . (2.8) Also, if we let 〈 r2pn 〉 be the mean square distance between protons and neutrons we 10 can show (A.2.5): ΣBSR = G NZ 2 (〈 r2pn 〉−〈r2p〉−〈r2n〉) . (2.9) Finally, if 〈 r′2p 〉 is the mean square distance of protons from the proton centre of mass, we have (A.2.6): ΣBSR = G ( Z2 〈 r2p 〉−Z2 〈r′2p 〉) . (2.10) 11 Chapter 3 Computational Techniques As evidenced by the formulas in the previous chapter, calculating photonuclear sum rules requires computing the nuclear ground state, as well as applying the dipole operator. In order to do this we need a concrete representation of the ground state, which we compute by expanding in a particular basis and solving the Schrödinger equation. In practice, we use a truncated model space to compute our observables. By expanding the model space and repeating the calculations we can examine the convergence behaviour. The quality of convergence can be used to characterise the theoretical errors of our calculations. For A= 3,4 nuclei, the No Core Shell Model (NCSM) has been used to calcu- late some of the observables we are looking at [10]. The standard NCSM is based on a harmonic oscillator basis, which has a Gaussian tail, and thus it is hard for this method to converge rapidly for nuclei with long tails, like halo nuclei. We are interested in computing observables for A= 6,7 nuclei including the halo nucleus 6He, so we use a basis built on the hyperspherical harmonic basis which has the correct exponential asymptotic behaviour built in. 3.1 Hyperspherical Basis The Hyperspherical formalism is based on a generalization of the standard solu- tion to the two-body problemin which you you move to the center of mass frame and consider a single body with relative coordinates and reduces mass. The hy- 12 perspherical coordinates are built up from the Jacobi coordinates(see section B.1), which remove the centre of mass motion and consider A−1 radial coordinates with 2(A− 1) corresponding polar angles. Using the radial Jacobi coordinates ηi, we construct a hyper-radial coordinate ρ related to the “size” of the system: ρk2 = k ∑ i=1 ηi2 = 1 k k+1 ∑ i< j (ri− r j)2 . and through a recursive process generate A−2 hyper-angles ϕi, which correspond to polar angles in and abstract 3(A−1) dimensional space. By transforming the Laplace operator into hyperspherical coordinates we find ∆= ∂ 2 ∂ρ2 + 3(A−1)−1 ρ ∂ ∂ρ − 1 ρ2 K̂2, where K̂ is an operator that depends on the angles and hyper-angles but not the hyper-radius. One interprets K̂ to be a generalization of the angular momentum operator l̂, and we refer to khat as the grandangular momentum operator. More details are provided in Appendix B. Using these coordinates, we can construct the hyperspherical basis Rν(ρ)HK(Ω), where HK(Ω) is made by the eigenstates of K̂2, called the hyperspherical harmon- ics. The HK(Ω) must be constructed to be antisymmetric since we are working with fermions. 3.2 Evaluation of Matrix elements One of the advantages of using an antisymmetrized hyperspherical basis is the way we can simplify matrix elements. With any totally antisymmetrized states |ψ〉, |ψ ′〉 and a one-body operator O(1) = ∑Ai=1Oi, we can reduce the matrix element to that of an operator that acts on only one particle: 〈ψ|O(1) |ψ〉= A〈ψ|OA |ψ〉 . 13 Similarly, for a two-body operator O(2)=∑Ai< jOi j, we can write the matrix element as 〈ψ|O(2) ∣∣ψ ′〉= A(A−1) 2 〈ψ|OA,A−1 ∣∣ψ ′〉 . In the case of one-body operators, we note that when we use Jacobi coordinates, that last coordinate ηA−1 is proportional to the position of the last particle with respect to the centre of mass of the system: ηA−1 = √ A A−1r ′ A. And in fact the last Jacobi coordinate is also proportional to the hyper-radius ρ: ηA−1 = ρ sinϕA−1. If we have the matrix element of a one body operator O(1), we can write it in terms of an operator that acts only on the last particle, and so in terms of the magnitude, direction, and spin-isospin components of the last particle coordinates: 〈ψ|O(1) |ψ〉= A〈ψ|OA |ψ〉= A〈ψ|O(r′A, r̂′A,σA,τA) ∣∣ψ ′〉 , where now we can replace r′A above with √ A−1 A ρA−1 sin(ϕA−1). The spin-angular and isospin portions of the matrix element can now be calcu- lated analytically using standard angular momentum methods. The hyper-angular and hyper-radial portions of the matrix element are reduced to two one-dimensional integrals since the wavefunction |ψ〉 and |ψ ′〉 are both expanded in terms of the hy- perspherical basis, which are products of hyper-radial and hyper-angular functions. The details of these final integrals are left out and can be found in [20] and [19]. In practice, we have used Gaussian Quadrature to compute these integrals numeri- cally. Now let us consider two-body operators. We can write two-body matrix ele- ments in terms of the last two particles matrix elements only, 〈ψ|O(2) ∣∣ψ ′〉= A(A−1) 2 〈ψ|OA,A−1 ∣∣ψ ′〉 . 14 If our operator only cares about the particle relative position and spin-isospin (for example, a central potential), then A(A−1) 2 〈ψ|OA,A−1 ∣∣ψ ′〉= A(A−1) 2 〈ψ|O(r′A− r′A−1,σA,σA−1,τA,τA−1) ∣∣ψ ′〉 . Then we can use the reverse-choice for our Jacobi coordinates, i.e. we choose the last Jacobi coordinate ηA−1 as pointing between the last two particles, ηA−1 = √ 1 2 (r′A− r′A−1), then ηA−2 points between the third-to-last particle and the centre of mass of the last two, and so on. Then we have: 〈ψ|O(2) ∣∣ψ ′〉= A(A−1) 2 〈ψ|O( √ 2ρA−1 sin(ϕA−1, η̂A−1,σA,σA−1,τA,τA−1) ∣∣ψ ′〉 . Using very similar ideas to those in the previous section, we can write the ma- trix element as a sum of products of hyper-radial/hyper-angular, spin-angular and isospin parts. The later two are again calculable analytically, while the former is again reduced to two one-dimensional integrals which can be computed numeri- cally. 3.3 The Lanczos Algorithm The Lanczos algorithm is used to tridiagonalize matrices, i.e. to convert them to the following form: H = a0 b1 0 0 . . . b1 a1 b2 0 . . . 0 b2 a2 b3 . . . 0 0 b3 a3 . . . ... ... . . . . . . . . . by recursively generating an orthonormal set of vectors {|φi〉}. Tridiagonalization routines are commonly used to simplify hermitian matrices because there are very efficient algorithms for computing eigenvalues and eigen- 15 vectors of tridiagonal matrices. The Lanczos algorithm in particular has the prop- erty that even when applied to a truncated linear operator on an infinite dimensional Hilbert space, the eigenvalues of the resulting finite matrix approach the eigenval- ues of the original operator. This makes it a useful tool for tridiagonalizing the Hamiltonian and computing the ground state energy in a truncated model space. In addition to this useful feature, we have a simple recursive expression for the determinant of a tridagonal matrix: det(H) = a0det[H]{2,...,n}−b21det[H]{3,...,n} (3.1) where [H]2,··· ,n is the matrix made from the 2nd to nth columns and rows of H. This property will be used in section 3.4.1 to derive the closed form expression for the polarizability sum rule. To formulate the algorithm, suppose we start with a matrix H. First, a vector of unit length must be chosen for |φ0〉. In general one can try to choose a starting vector in such a way as to optimize convergence, but in our case we make a par- ticular choice of |φ0〉 for analytic reasons which will be described in the following sections. Once |φ0〉 is selected, the others are calculated recursively by bi+1 |φi+1〉= H |φi〉−ai |φi〉−bi |φi−1〉 . (3.2) where |φ−1〉= 0, 〈 φi|φ j 〉 = δi j, a0 = 〈φ0|H |φ0〉 ,b0 = 0. We can see from (3.2) that 〈 φ j ∣∣H |φi〉= 〈φ j∣∣(bi+1 |φi+1〉+ai |φi〉+bi |φi−1〉) = bi j = i−1 ai j = i bi+1 j = i+1 0 otherwise and thus, in the basis formed by the Lanczos vectors, H is tridiagonal with coeffi- 16 cients ai and bi (called Lanczos coefficients). 3.4 Implementation Given the form of the sum rules derived above, we see that there are three funda- mental computational tasks which must be performed to calculate them. The first is calculation of the nuclear ground state |0〉, which entails solving the Schrödinger equation and requires calculating matrix elements of the form HKK′νν ′ = 〈K| 〈ν |H ∣∣K′〉∣∣ν ′〉 , where here we identify |K〉, |ν〉 with the hyper-angular-spin-isospin and hyper- radial parts of the hyperspherical harmonic basis states, respectively. As is the case with most algorithms for solving for eigenvalues and eigenvec- tors of hermitian matrices, the Hamiltonian is first trigiagonalized, for which the Lanczos algorithm is used. The number of matrix elements that need to be calculated can be limited by choosing only those basis states which share quantum numbers with the states of interest. In particular, for the ground state we only compute matrix elements for basis states with total angular momentum, isospin, and isospin projection (J,T,Tz) equal to that of the ground state. The hyperspherical harmonics are all character- ized by a grandangular moment quantum number K. We control the size of our model space by generating all states with the correct J,T,Tz quantum numbers and K less than or equal to some maximum number Kmax. Then, by increasing Kmax, we can assess the truncation error of our calculations. We also have to multiply the Hamiltonian by the Lanczos vectors ∣∣φ i〉 as part of the Lanczos algorithm, which corresponds to calculating φ n+1Kν = ∑ K′ν ′ HKK′νν ′φ nK′ν ′−anφKν −bnφ n−1Kν , where φ nKν = 〈K| 〈ν | |φ n〉. As this computation must be performed for every Lanc- zos vector (the number of which is determined by the number of basis states used), this is the most computationally intensive part for the larger model spaces. This is the algorithm that I focused on parallelizing. 17 Finally, we also have to compute 〈0|D†zDz |0〉. To do this we must expand Dz |0〉 by generating the basis states with the quantum numbers determined by the selection rules for Dz, namely ∆J = 1, ∆T = 1, and ∆Tz = 0. So, for the case of 4He, where in the ground state J = 0,T = 0,Tz = 0, we would need to generate states for Dz |0〉 with quantum numbers J = 1,T = 1,Tz = 0. In general there will be multiple transition channels. In particular for the case of 6He we have for the ground state (J = 0,T = 1,Tz = −1), and so we have two transition channels (J = 1,T = 1,Tz =−1) and (J = 1,T = 2,Tz =−1). The com- putations must be performed for each channel and then the results combined to calculate the sum rules. This is described in more detail in later sections. These main computational tasks are performed by a code written by Barnea and Novoselsky (referred to henceforth as the HH code), which computes com- pletely antisymmetrized hyperspherical basis states and computes Lanczos coeffi- cients [21, 22]. I wrote a code from scratch that combined these raw elements into calculated values for the sum rules. The details of this sum rule computation is presented in the next subsection. 3.4.1 Calculating the Sum Rules Recall equation (2.5): ΣPSR =−G 〈0| D † zDz E0−H |0〉 . We rewrite this as ΣPSR =−G 〈0|D†zDz |0〉〈φ0| 1 E0−H |φ0〉 , where |φ0〉 = Dz|0〉√〈0|D†zDz|0〉 , which is by construction of unit length. Let |φ0〉 be the first Lanczos vector, from which the complete set {|φi〉} are recursively defined. Then define xnm = 〈φn| 1E0−H |φm〉. Clearly |φ0〉= (E0−H) 1E0−H |φ0〉 . 18 which we can write component-wise as ∑ n (E0−Hmn)xn0 = δm0, where δi j is the Kronecker delta function. Using Cramer’s Rule for linear systems, we have x00 = det([E0−H]{2,··· ,n}) det(E0−H) . By definition, H is tridiagonal in the |φi〉 basis, so clearly E0−H is also tridiagonal. Thus we can apply the recursive formula for the determinant of a tridiagonal matrix (3.1), the end result of which (in terms of the Lanczos coefficients ai,bi is x00 = 1 E0−a0− b21 E0−a1− b22 E0−a2− b23 . . . . This continued fraction is very fast to compute given the Lanczos coefficients and converges quickly with the number of Lanczos coefficients (in practice the first 1000 terms is more than sufficient to converge to machine precision). The Lanczos coefficients are calculated by the HH code as part of the Lanczos algorithm, and are then used as an input to my code. Finally, we have a number for the polarizability sum rule: ΣPSR =−G 〈0|D†zDz |0〉x00. In practice, for 6He we have two transition channels which will each produce a set of Lanczos coefficients. The results is the sum over each channel: ΣPSR =−G∑ T 〈0|D†zDz |0〉T xT00. (3.3) 19 For the TRK sum rule we have: ΣTRK = G∑ n 〈0|D†z |n〉〈n|Dz |0〉(En−E0), = G∑ n 〈0|D†zEn |n〉〈n|Dz |0〉−〈0|D†z |n〉〈n|Dz |0〉E0, = G (〈0|D†zHDz |0〉−〈0|D†zDz |0〉E0) , = G 〈0|D†zDz |0〉(〈φ0|H |φ0〉−E0), = G 〈0|D†zDz |0〉(a0−E0). where |φ0〉 is defined as before. Thus we can calculate ΣTRK using 〈0|D†zDz |0〉 and the first Lanzcos coefficient. In the case of multiple transitions channels: ΣTRK = G∑ T 〈0|D†zDz |0〉T ( aT0 −E0 ) . (3.4) The BSR sum rule is given simply by ΣBSR = G 〈0|D†zDz |0〉 , or ΣBSR = G∑ T 〈0|D†zDz |0〉T . (3.5) in the case of multiple channels. Equations (3.3), (3.4), (3.5) therefore allow us to compute the sum rules given the Lanzcos coefficients and the value of 〈0|D†zDz |0〉 produced by the HH code. 20 Chapter 4 Calculating the Full Continuum Response The methods described in the previous chapters used the Lanczos technique to calculate the moments of the photonuclear cross section without resorting to com- puting the full continuum response. This response can be computed, however, using a Lorentz Integral transform method. Since some of these cross sections are available from previous work [3], it will be interesting to check our results from the computation of sum rules with the results from direct integration of the cross section, using the definition of the moments: mk = ∫ ∞ ωth dωωkσE1URγ (ω). The Lorentz Integral Transform of the response function (equation (2.3)) is defined: L (σ) = ∫ dω R(ω) (ω−σR)2+σ2I , where σ = σR+ iσI . 21 Plugging in the form of the response, we have: L (σ) = ∫ dω ∫ dn |〈n|Dz |0〉|2 δ (En−E0−ω) (ω−σR)2+σ2I , = ∫ dω ∫ dn 〈0|D†z |n〉〈n|Dz |0〉δ (En−E0−ω) (ω−σR)2+σ2I , = ∫ dn 〈0|D†z |n〉〈n|Dz |0〉 (En−E0−σR)2+σ2I . Now it is useful to expand (En−E0−σR)2+σ2I = (En−E0−σR+ iσI)(En−E0−σR− iσI), and, using the fact that En |n〉= H |n〉, we can apply the closure relation to derive: L (σ) = 〈0|D†z ( 1 H−E0−σR+ iσI )( 1 H−E0−σR− iσI ) Dz |0〉 , = 1 σI Im [ 〈0|D†z ( H−E0−σR+ iσI H−E0−σR+ iσI )( 1 H−E0−σR− iσI ) Dz |0〉 ] , = 1 σI Im [ 〈0|D†z ( 1 H−E0−σR− iσI ) Dz |0〉 ] , =− 1 σI Im [ 〈0|D†z ( 1 E0+σR+ iσI−H ) Dz |0〉 ] , =− 1 σI Im [ 〈0|D†z ( 1 z−H ) Dz |0〉 ] . where z= E0+σR+ iσI . Now we can apply an argument analagous to section 3.4.1 by setting |φ0〉= Dz |0〉√ 〈0|D†zDz |0〉 , 22 so that we can write L (σ) =− 1 σI 〈0|D†zDz |0〉 Im [ 〈φ0| 1z−H |φ0〉 ] , L (σ) =− 1 σI 〈0|D†zDz |0〉 Im 1 z−a0− b21 z−a1− b22 z−a2− b23 . . . . Where again ai,bi are the Lanczos coefficients. Inverting the transformL (σ) to obtain R(ω) is a numerical task accomplished by fitting the transformed response function to an easily transformable ansatz with many parameters. Thus the full response can be calculated from the same Lanczos coefficients used in the sum rule computations. As mentioned above, some cross sections calculated with this method are avail- able for 6He from [? ]. I have used these cross sections to extract values for the photonuclear moments by direct integration which can be compared with the sum rule calculation. The difference between this method and the direct computation of the sum rules is that one has to go through the inversion procedure, which intro- duces additional numerical errors. 23 Chapter 5 Results 5.1 Testing the Sum Rule Code To test my sum rule code, I compared results from Gazit et al. [1], which performed these sum rule calculations for 4He using the AV18-UIX potentials. Their result at various sizes of model space are shown in Table 5.1, compared to my results computing the sum rules using the same Lanzcos coefficients. The values agree to within what can be attributed to machine precision. As a next step, I calculated the polarizability sum rule for 4He using the same simple potentials that I would be using for 6He: the Minnesota (MN) [23, 24], Argonne V4’ (AV4’) [25], and the Malfiet-Tjon I-III (MT13) [26]. The x-axis denotes the maximum grandangular momentum of the basis states used for the PSR BSR TRK Kmax [1] This work [1] This work [1] This work 16 0.06434 0.06434 2.406 2.405 146.0 145.9 18 0.06473 0.06473 2.410 2.409 146.2 146.2 Table 5.1: Previous calculations from Gazit et al. [1] of the sum rules using the Lanzcos coefficients are compared with our calculations using the same Lanzcos coefficients to test the validity of the sum rule code. We compared our results for largest model spaces denoted by Kmax. 24 Figure 5.1: Polarizability of 4He computed using different sized model spaces denoted by maximum grandangular momentum Kmax. The ex- perimental points were obtained by Compton scattering with experi- mental error indicated by the shaded regions. calculation. As discussed in the Chapter 3, the maximum grandangular momentum is allowed to vary as a way to assess the error due to truncation of our model space. States with even grandangular momentum K have even parity, so for the ground state we used even Kmax, which is why points appear on the plot in increments of two. In fact we are also generating states up to K = Kmax + 1, since the dipole operator carries odd parity, so the odd Kmax+1 states are needed to expand Dz |0〉. Thus, I will refer to model space sizes as Kmax = X/X+1 for even X , to reflect the states used to expand |0〉/Dz |0〉. The values appear fairly well converged in Kmax. We can estimate the trun- 25 Convergence Error MN 1.0% AV4’ 0.1% MT13 1.1% Table 5.2: Convergence error in the polarizability sum rule for 4He with the simple potential models. The value is estimated as the percentage change from the Kmax = 10 to Kmax = 12 model space size. Theory Experiment 0.06830(8)(14) [10] 0.072(4) [2] 0.0655(4) [1] 0.076(8) [27] 0.0635(4) This work Table 5.3: Theoretical and experimental results for the polarizability of 4He. The two other theoretical values were computed using more realistic two- body potential models and included three-body forces. The experimental values were obtained through Compton scattering. cation error by taking the percentage change in the last two points, which we summarize in Table 5.2. There is clearly a non-negligible potential dependence in the theoretical results shown in Figure 5.1. For this reason, we take as our best value the average of the three results in the largest model space, using their stan- dard deviation as the error. The value is compared with existing theoretical and experimental results in Table 5.3. Note that the result from [10] and [1] include realistic two-body forces and three-body forces, whereas ours have neither. These factors both help the comparison with experiment. The deviation of our calcula- tions from the experimental data are not large, but they are also not unexpected, as the semi-realistic potentials used are only expected to reproduce the bulk of the nuclear dynamics and not the details. 5.2 Parallelization The version of the HH code I started with was a serial code, only able to run on a single processor. This is not a problem for the 4He calculations, but the number of 26 Figure 5.2: A cartoon of how the matrix elements in a hermitian matrix can be allocated efficiently between four processors. The colour of each matrix element corresponds to the processor it is assigned to. hyperspherical harmonics becomes large when A > 4, meaning calculations take much longer for 6He. In the interest of saving computation time, I attempted to parallelize the part of the code which multiplied a Lanczos vector by the Hamil- tonian matrix (recall equation (3.2)), as this is the most significant bottleneck for large model spaces1. The parallel algorithm splits the columns of the Hamiltonian matrix and the corresponding components of the vector among the processors in an efficient way and then allows the results to be collected and summed together for the final an- swer. Since the Hamiltonian is hermitian, only the upper or lower triangular el- ements needed to be considered, which meant the effective size of each column was not constant. The simplest method we used was going through the columns in order while rotating the processor which it would be assigned to. So, for instance, (with four processors) the nth processor would be assigned all columns numbered n modulo 4. See Figure 5.2 for a illustration. The implementation of the parallel algorithm was done using the Message 1Some other parts of the code had already been parallelized. 27 Passing Interface (MPI) and run on the TRIUMF Theory Cluster known as cougar2. An actual speed up of computation time was achieved. The magnitude of the speed up will depend greatly on the computation that is being done, since certain parts of the code are parallelized more efficiently than others and the location of the bottle- neck can depend on the nature of the input. Since the focus of the parallelization was the Hamiltonian multiplication in the Lanczos Algorithm, the parallel code is expected to do better with larger Hilbert spaces. For an intermediate sized space using Kmax = 6/7 (corresponding to a matrix dimension of 104) and the MN poten- tial, I was able to observe a decrease in time from about 170 minutes to just over 17 minutes when running on 12 processors, leading to a speed up factor of 10. 5.3 Sum Rules We have calculated all three of the sum rules for 6He using the three semi-realistic potentials we used for 4He. The results for the TRK, BSR, and Polarizability sum rules are plotted in Figures 5.3, 5.4, and 5.5 respectively. We identify two sources of theoretical error, a convergence error and a model dependent error. The con- vergence error is the error due to the truncation of the model space; we estimate it as the percentage change between the last two model space sizes. We estimate the model dependent error as the standard deviation of the values that the three potential models give for a given sum rule. The TRK and BSR sum rules appear better converged than the PSR. From a theoretical point of view, it is interesting to note that the AV4’ potential has the largest TRK sum rule, meaning it has the largest enhancement factor. The TRK sum rule is least sensitive of the model dependence, with a spread of about 5.82%. The BSR sum rules measures the total dipole strength and happens to be bigger as well for the AV4’. The MT13 is has a more converged value for the BSR sum rule, but it is also smaller than the other two. The model dependence for the BSR sum rule is 10.1%. The PSR presents the weakest convergence pattern. The AV4’ and MN val- ues are still growing, while only the MT13 shows a decent convergence pattern. The PSR has the largest model dependence of 31.1%. In fact, we can note that 2http://cougar.triumf.ca 28 Figure 5.3: TRK sum rule computations for different sizes of model space characterized by max grandangular momentum Kmax. the model dependence in the PSR is much larger than the same sum rule for 4He using the same potentials. This is consistent with the results from [3] in which the photoabsoption cross sections were calculated: Since the PSR depends on the inverse-square energy-weighted integral of the cross section, it is most sensitive to the low-energy portion of the cross section, in particular the location of the thresh- old energy (Recall that the threshold energy is the minimum energy required for photoabsorption). The threshold energy of 4He is about 20 MeV, while the thresh- old for 6He is only about 1 MeV (since the neutron halo is loosely bound), so 6He should be much more sensitive to the threshold energy than 4He, which is where [3] found there to be the largest discrepancy between potentials. The computed values for the model dependent error and the convergence error are listed in Tables 29 Figure 5.4: BSR sum rule computations for different size of model space characterized by max grandangular momentum Kmax. MN AV4’ MT13 TRK 4.34% 2.47% 2.66% BSR 5.27% 8.38% 4.76% PSR 22.0% 28.3% 13.4% Table 5.4: Convergence error estimate for sum rules. Computed as the per- centage change in the value from the second-largest to the largest model space. 30 Figure 5.5: Polarizability sum rule computations for different sizes of model space characterized by max grandangular momentum Kmax. The exper- imental point is from Pachucki and Moro [2] using Coulomb breakup, the shaded region is the experimental error. 5.5 and 5.4 respectively. Note that even though there is non-negligible convergence error in all the sum rules, this error is still dominated by the model dependent error, as was the case for 4He. Different potential models may overbind or underbind the ground states of nu- clei. We would expect that a potential that overbinds a nucleus would also compute a polarizability that is too small. In Table 5.6, I have plotted the computed binding energies using the three different potentials for the Kmax = 12 model space. This can be compared to the binding energy from the National Nuclear Data Center: −29.27. I have included the 4He results as well, which can be compared to its 31 Mean Spread TRK 132.3 5.82% BSR 6.599 10.1% PSR 0.812 31.1% Table 5.5: Estimated model-dependent error. Computed as the standard devi- ation from the mean value among potential models used. BE (MeV) rrms (fm) 6He MN -30.50(3) 2.42(7) AV4’ -32.90(3) 2.38(7) MT13 -31.81(9) 2.25(2) 4He MN -23.95(1) 1.410(1) AV4’ -32.25(36) 1.386(2) MT13 -30.74(17) 1.422(2) Table 5.6: Binding energies and root mean square radii for 6He and 4He using Kmax = 12. The error in the last digit is taken to be the difference between the Kmax = 12 and Kmax = 10 values. measured binding energy −28.30. We see that all our potentials overbind 6He. In fact, our calculated values for the PSR are smaller than the experimental result provided by Pachucki and Moro [2] determined from Coulomb breakup of 6He on lead, 5.4 Photoabsorption Cross Section The 6He photoabsorption cross sections were calculated by Bacca et al. [3] using the Lorentz Integral Transform technique discussed in chapter 4. The cross section is only computed up to some finite energy, but the integrals are required to be taken to infinity to calculate the photonuclear moments. In the case of the PSR, how- ever, since the cross section is inverse-square energy-weighted, the tail contributes little to the value of the integral. In fact, attempting to fit the tail to a reasonable 32 Figure 5.6: The photoabsorption cross section computed by Bacca et al. [3] using the AV4’ potential. The inverse-square energy-weighted cross section is included (scaled to be visible) to show how the lack of data going to infinity does not effect the value of the PSR. The model space size used to compute the cross section was Kmax = 12. ansatz and integrating results in a contribution on the order of 0.01%, which is significantly smaller than both the convergence and model dependent error. Fig- ure 5.6 shows the computed cross section from [3] compared to the inverse-square energy-weighted cross section, which has been scaled to appear in the Figure. We can use the integral of the computed cross section as another test of the sum rule computations. Since the cross section was computed with AV4’ and us- ing Kmax = 12, we can compare the integral of the inverse-square energy-weighted cross section to the sum rule calculation using the same potential and model space 33 Integral Sum Rule 0.946 0.951 Table 5.7: PSR of 6He calculated by integrating the cross section compared to sum rule approach. size. This is summarized in Table 5.7. The only difference between the calcula- tion of the cross section and the integral are the fact that the integral introduces numerical errors due to the inversion procedure which are avoided in the sum rule calculation. Thus, we can attribute a small discrepancy between these two values as due to this numerical error. 5.5 Conclusions In this work, I present the first ab-initio calculation of the polarizability sum rule for 6He. The calculation is performed with simple potentials as a starting point. This lays the groundwork for future, more sophisticated calculations. The truncation of the Hilbert space still plays a large role in our computed results for the 6He sum rules, i.e., the calculations are not fully converged. Evi- dently the dipole operator couples more strongly to high grandangular momenta of 6He than 4He. The values may now be beginning to converge, however, and perhaps K = 14 or K = 16 states will show reasonable convergence. These much larger Hilbert spaces can be achieved using our parallel code, but we were unable to complete the calculations due to lack of time. Certainly a proper truncation of the hyperspherical harmonics negligible symmetries to reduce the number of states will have to be used. 34 Bibliography [1] D. Gazit, N. Barnea, S. Bacca, W. Leidemann, and G. Orlandini. Photonuclear sun rules and the tetrahedral configuration of he-4. Phys. Rev. C, 36:061001, 2006. [2] K. Pachucki and A. M. Moro. Nuclear polarizability of helium isotopes in atomic transitions. Phys. Rev. A, 75(3):032521, 2007. 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Solution of the faddeev equations for the triton problem using local two-particle interactions. Nuclear Physics A, 127(1): 161 – 168, 1969. [27] J. L. Friar. Nuclear polarization corrections in µ−he4 atoms. Phys. Rev. C, 16(4):1540–1548, 1977. 37 Appendix A Complete Derivations of Sum Rule Formulas I have attempted to provide convincing derivations of all of the sum rules formulas that were cited in this work. 38 A.1 Thomas-Reiche-Kuhn Sum Rule A.1.1 ΣTRK = G∑ n |〈n|Dz |0〉|2 (En−E0) , = G∑ n 〈0|Dz |n〉〈n|Dz |0〉(En−E0) , = G∑ n 〈0|Dz |n〉(En−E0)〈n|Dz |0〉 , = G∑ n 1 2 〈0|Dz |n〉(En−E0)〈n|Dz |0〉− 12 〈0|Dz |n〉(E0−En)〈n|Dz |0〉 , = G∑ n 1 2 (〈0|Dz |n〉〈n|(EnDz−DzE0) |0〉−〈0|(E0Dz−DzEn) |n〉〈n|Dz |0〉) , = G∑ n 1 2 (〈0|Dz |n〉〈n|(HDz−DzH) |0〉−〈0|(HDz−DzH) |n〉〈n|Dz |0〉) , = G∑ n 1 2 (〈0|Dz |n〉〈n| [H,Dz] |0〉−〈0| [H,Dz] |n〉〈n|Dz |0〉) , = G 2 (〈0|Dz [H,Dz] |0〉−〈0| [H,Dz]Dz |0〉) , = G 2 〈0| [Dz, [H,Dz]] |0〉 . A.1.2 We can split the Haniltonian into a kinetic and potential energy part H = T +V : G 2 〈0| [Dz, [H,Dz]] |0〉= G2 (〈0| [Dz, [T,Dz]] |0〉+ 〈0| [Dz, [V,Dz]] |0〉) , where T is the kinetic energy and V is the potential. We can further simplify the kinetic energy part by actually calculating the com- mutators. Two possible choices of coordinates in which the dipole is easy to define are the lab frame and the centre-of-mass frame. 39 Using lab-frame coordinaties PCM = A ∑ i=1 pi, RCM = 1 A A ∑ i=1 ri, [PCM,RCM] =−ih̄, [pi,RCM] = 1 A A ∑ j=1 [pi,r j] = −ih̄ A , [PCM,ri] = A ∑ j=1 [ p j,ri ] =−ih̄, Tint = Tlab−TCM = 1 2m ( A ∑ i=1 p2i − 1 A P2CM ) , Dz = A ∑ i=1 (ri−RCM)qi = A ∑ i=1 riqi−ZRCM. First calculate [T,Dz]. [T,Dz] = 1 2m [ A ∑ i=1 p2i − 1 A P2CM, A ∑ j=1 r jq j−ZRCM ] , = 1 2m ( A ∑ i=1 A ∑ j=1 [ p2i ,r jq j ]− A∑ i=1 [ p2i ,ZRCM ]− A∑ j=1 [ 1 A P2CM,r jq j ] + [ 1 A P2CM,ZRCM ]) , = 1 2m ( A ∑ i=1 A ∑ j=1 (pi [pi,r j]q j+[pi,r j]piq j)− A ∑ i=1 2Z [pi,RCM]pi− A ∑ j=1 2 A [PCM,r j]PCM+ 2Z A [PCM,RCM]PCM ) , = −ih̄ m ( A ∑ i=1 piqi− A ∑ i=1 Z A pi− A ∑ j=1 1 A PCMq j+ Z A PCM ) , = −ih̄ m ( A ∑ i=1 ( qi− ZA ) pi− Z A PCM+ Z A PCM ) , = −ih̄ m A ∑ i=1 ( qi− ZA ) ,pi 40 Now calculate [Dz, [T,Dz]]. [Dz, [T,D]] = −ih̄ m [ A ∑ i=1 riqi−ZRCM, A ∑ j=1 ( q j− ZA ) p j ] , = −ih̄ m ( A ∑ i=1 A ∑ j=1 [ riqi, ( q j− ZA ) p j ] − A ∑ j=1 [ ZRCM, ( q j− ZA ) p j ]) , = −ih̄ m ( A ∑ i=1 A ∑ j=1 qi ( q j− ZA )[ ri,p j ]− A∑ j=1 Z ( q j− ZA )[ RCM,p j ]) , = h̄2 m ( A ∑ i=1 ( q2i −qi Z A ) − A ∑ j=1 Z ( q j− ZA ) 1 A ) , = h̄2 m ( Z− Z 2 A − ( Z2 A − Z 2 A )) , = h̄2 m ( Z(Z+N) A − Z 2 A ) , = h̄2 m ZN A . Then we have: ΣTRK = G 2 〈0| [Dz, [H,Dz]] |0〉 , = G 2 h̄2 m ZN A ( 1+κTRK ) , where κTRK = mA h̄2NZ 〈0| [Dz, [V,Dz]] |0〉 . 41 Using centre-of-mass frame coordinates First we have to derive the commutators. As before we have: PCM = A ∑ i=1 pi, RCM = 1 A A ∑ i=1 ri, [pi,RCM] = 1 A A ∑ j=1 [pi,r j] = −ih̄ A , [PCM,ri] = A ∑ j=1 [ p j,ri ] =−ih̄, [PCM,RCM] =−ih̄, Now we define the centre-of-mass frame coordinates: r′i = ri−RCM, p′i = pi−PCM. Thus the commutators are: [ p′i,r ′ j ] = [ pi− 1 A PCM,r j−RCM ] , = [pi,r j]− [pi,RCM]− 1 A [PCM,r j] 1 A [PCM,RCM] , = δi j(−ih̄)− 1A(−ih̄)− 1 A (−ih̄)+ 1 A (−ih̄), = ( δi j− 1A ) (−ih̄),[ PCM,r′i ] = [PCM,ri−RCM] = (−ih̄)− (−ih̄) = 0,[ p′i,RCM ] = [ pi− 1 A PCM,RCM ] = 1 A (−ih̄)− 1 A (−ih̄) = 0. 42 Now we observe that in these coordinates: Tint = 1 2m A ∑ i=1 p′2i , Dz = A ∑ i=1 qir′i. We precede thus: [T,Dz] = 1 2m A ∑ i=1 A ∑ j=1 [ p′2i ,r ′ jq j ] , = 1 2m A ∑ i=1 A ∑ j=1 ( p′i [ p′i,r ′ j ] q j+ [ p′i,r ′ j ] q j ) ), = −ih̄ m A ∑ i=1 A ∑ j=1 ( δi j− 1A ) p′iq j, = −h̄ m ( A ∑ i=1 p′iqi− A ∑ i=1 A ∑ j=1 1 A p′iq j ) , = −ih̄ m A ∑ i=1 ( qi− ZA ) p′i. 43 Then we can calculate: [Dz, [T,Dz]] = −ih̄ m A ∑ i=1 A ∑ j=1 [ qir′i, ( q j− ZA ) p′i ] , = −ih̄ m A ∑ i=1 A ∑ j=1 qi ( q j− ZA )[ r′i,p′i ] , = h̄2 m A ∑ i=1 A ∑ j=1 qi ( q j− ZA )( δi j− 1A ) , = h̄2 m ( A ∑ i=1 q2i − Z A qi− A ∑ i=1 A ∑ j=1 qi ( q j− ZA ) 1 A ) , = h̄2 m ( Z− Z A − ( Z2 A − Z 2 A )) , = h̄2 m ( Z(Z+N) A − Z 2 A ) , = h̄2 m NZ A . And so, as before: ΣTRK = G 2 〈0| [Dz, [H,Dz]] |0〉 , = G 2 h̄2 m ZN A ( 1+κTRK ) , where κTRK = mA h̄2NZ 〈0| [Dz, [V,Dz]] |0〉 . 44 A.2 Bremstrahlung Sum Rule A.2.1 ΣBSR = G∑ n | 〈n|Dz |0〉 |2, = G∑ n 〈0|Dz |n〉〈n|Dz |0〉 , = G 〈0|Dz∑ n |n〉〈n|Dz |0〉 , = G 〈0|D†zDz |0〉 . A.2.2 We can re-write the dipole operator in terms of RPN , the distance between the proton and neutron centres of mass: Dz = 1 2 A ∑ i=1 τzi r ′ i, = 1 2 ( N+Z A ) A ∑ i=1 τzi r ′ i, = 1 2 NZ A ( 1 Z + 1 N ) A ∑ i=1 τzi r ′ i, = NZ A ( 1 Z A ∑ i=1 τzi r ′ i 2 + 1 N A ∑ i=1 τzi r ′ i 2 ) , = NZ A ( 1 Z A ∑ i=1 (1+ τzi ) 2 r′i− 1N A ∑ i=1 (1− τzi ) 2 r′i ) , 1 = NZ A (RPCM−RNCM) = NZA RPN . 45 Now we can re-wrtie the BSR sum: ΣBSR = G 〈0|D†zDz |0〉 , = G ( NZ A )2 〈 R2PN 〉 . A.2.3 Considering that we can also write the dipole operator as a sum over only the proton coordinates: Dz = ∑Zi=1 r′i, we can derive the following relation: D†zDz = Z ∑ i, j=1 r′i · r′ j, = 1 2 ( r′2i + r ′2 j − ( r′i− r′ j )2) , = 1 2 (2Z) Z ∑ i=1 r′2i − 1 2 Z ∑ i, j=1 ( r′i− r′ j )2 , = Z2 〈 r2p 〉− Z(Z−1) 2 〈 r2pp 〉 , where r2p and r2pp are the mean-square proton radius and the mean square proton- proton distance respectively, defined as: r2p = 1 Z Z ∑ i=1 r′2i , r2pp = 1 Z(Z−1) Z ∑ i, j=1 ( r′i− r′ j )2 . Then the BSR sum rule becomes: ΣBSR = G ( Z2 〈 r2p 〉− Z(Z−1) 2 〈 r2pp 〉) . 1Recal that since r′i are the centre of mass coordinates, their sum over all particles is zero. Thus we can add arbitrary constant terms into the sum over r′i’s 46 A.2.4 Observe that since we’re in the centre of mass, we can also re-write the dipole operator as Dz =−∑Ai=1 (1−qi)r′i =−∑Ni=1 r′i. But then we have that D†zDz = N ∑ i, j=1 r′i · r′ j, which has the same form as the previous derivation only now we are summing over neturons instead of protons. Thus we can immediatley derive: ΣBSR = G ( N2 〈 r2n 〉− N(N−1) 2 〈 r2nn 〉) , where r2n and r2nn are defined similarly as for protons. A.2.5 We can also consider the following form for D†zDz: D†zDz = A ∑ i, j=1 τzi τ z jr ′ i · r′ j 4 . We can split this sum into proton-proton, neutron-neutron, and proton-neutron pairs: D†zDz = 1 4 Z ∑ i, j=1 r′i · r′ j+ 14 N ∑ i, j=1 r′i · r′ j− 142 Z ∑ i=1 N ∑ i=1 r′i · r′ j, where we introduce a factor of 2 in the third term because the original sum would double-count each pair. We see that the first two terms, from previous analysis, are multiples of the dipole square-norm: D†zDz = 1 4 D†zDz+ 1 4 D†zDz− 1 4 2 Z ∑ i=1 N ∑ i=1 r′i · r′ j. 47 Thus, we can regroup terms and proceed: D†zDz =− Z ∑ i=1 N ∑ i=1 r′i · r′ j =−1 2 ( Z ∑ i=1 N ∑ i=1 r′2i + r ′2 j − Z ∑ i=1 N ∑ i=1 ( r′i− r′ j )2) , = 1 2 ( NZr2pn−NZr2p−NZr2n ) , = NZ 2 ( r2pn− r2p− r2n ) , where r2pn is the mean-square proton-neutron distance: 1 NZ Z ∑ i=1 N ∑ i=1 ( r′i− r′ j )2 . So then the BSR sum rule can be written: ΣBSR = G NZ 2 (〈 r2pn 〉−〈r2p〉−〈r2n〉) . A.2.6 We can observe that the dipole magnitude is proportional to the magnitude of the proton centre of mass. D†zDz = Z ∑ i, j=1 r′i · r′ j, = Z2 ( 1 Z2 Z ∑ i, j=1 r′i · r′ j ) , = Z2R2p. 48 Then we can see that: D†zDz = Z ∑ i, j=1 r′i · r′ j, = 2 Z ∑ i=1 r′i Z ∑ i=1 r′ j−Z2Rp, = 2Z Z ∑ i=1 r′iRp−Z Z ∑ i=1 R2p, =−Z Z ∑ i=1 ( r′2i − r′2i −2r′iRp+R2p ) , = Z Z ∑ i=1 ( r′2i − ( r′i−Rp )2) , = Z2r2p−Z2r′2p , where r′2p is the mean square distrance between protons and the proton centre of mass: r′2p = 1 Z Z ∑ i=1 ( r′i−Rp )2 . This gives us another expression for the BSR sum rule: ΣBSR = G ( Z2 〈 r2p 〉−Z2 〈r′2p 〉) . 49 A.3 Polarizability Sum Rule A.3.1 ΣPSR = ∫ ∞ ωth dωG R(ω) ω , = ∫ ∞ ωth dω G ω ∑n |〈n|Dz |0〉|2 δ (En−E0−ω), = G∑ n 〈0|D†z |n〉〈n|Dz |0〉 En−E0 , = G∑ n 〈0|D†z |n〉〈n|Dz |0〉 H−E0 , = G 〈0| D † zDz H−E0 |0〉 , =−G 〈0| D † zDz E0−H |0〉 . 50 Appendix B Hyperspherical Formalism Here I have presented some more details about the hyperspherical formalism. The main ideas are sketched, while the more technical derivations are left out. B.1 Jacobi Coordinates The hyperspherical coordinates are recursively defined in terms of a set of Jacobi coordinates. Jacobi coordinates are an extension of the technique used in the two- body problem with a central potential where the center of mass motion is ignored and the relative position vector between the two masses is imagined as the position vector of a single pseudo-particle. The Jacobi coordinates for a two body system are the magnitude and two angles associated with the vector from the regular two-body problem. In a four-body system, we first pick two particles and their relative position vector as before. Next we take the position vector of the third particle with respect to the centre of mass of the first two, and then the relative position of the fourth particle with respect to the centre of mass of the first three. This is illustrated in Figure B.1 We continue this process up to A particles, thus defining A− 1 lengths and 2(A− 1) angles. These are the Jacobi coordinates of the system. It’s easy to see that for A identical mass particles with positions rk in the lab frame, the kth Jacobi 51 Figure B.1: The (unnormalized) Jacobi coordinates for four particles. The η̃ i vectors point along the direction of the normalized η i given by equation (B.1). coordinate is defined: ηk = √ k k+1 ( rk+1− 1k k ∑ i=1 ri ) k = 1, ...,A−1. (B.1) B.2 Hyperspherical Coordinates and the Laplace Operator Suppose we have a set of A− 1 Jacobi coordinates {ηk} with magnitudes ηk and angular parts η̂k. Then we define the first hyper-radial coordinate ρ1 = η1 and the kth hyper-radial coordinate is defined such that: ρk2 = k ∑ i=1 ηi2 = 1 k k+1 ∑ i< j (ri− r j)2 . (B.2) 52 Figure B.2: Geometric interpretation of the recursive generation of hyper- spherical coordinates The the kth hyperangle is defined such that: ρk−1 = ρk cos(ϕk), ηk = ρk sin(ϕk). We can visualize this geometrically (as shown in Figure B.2) by placing the pre- vious hyper-radius and the next Jacobi coordinate on a plane at right angles. Then the magnitude of their difference is the next hyper radius, and the angle between the hyper-radii is the next hyper-angle. One can show that, if dη̂1 and dη̂2 are the volume elements of the angular parts of the first two Jacobi coordinates in a three-body problem, then the total volume element in the hyperspherical coordinates for the three-body problem is dV6 = ρ25dρ2 sin2(ϕ2)cos2(ϕ2)dϕ2dη̂1dη̂2 ≡ ρ25dρ2dS5, and then the A body volume element can be defined recursively: dV3(A−1)= ρA−13A−4dρA−1dS3A−4≡ ρA−13A−4dρA−1 sin2(ϕA−1)cos3A−7(ϕA−1)dϕA−1dη̂A−1dS3A−7. Note that the volume element only depends on the last hyper-radial coordinate and all the angles, but not on any lower-order radial coordinates. As we will see, this is a useful property for evaluating matrix elements in hyperspherical coordinates. 53 As we can see from (B.2), the hyper-radial coordinate does not depend on which Jacobi coordinates are used and is symmetric under the permutation of par- ticle positions. The whole A body system can thus be described by the centre of mass motion plus one hyper-radial coordinate ρA, A−2 hyper-angular coordinates ϕ(A−1) = {ϕ2, ...ϕA−1}, and 2(A−1) angular coordinates Ω(A−1) = {η̂1, ..., η̂A−1}. (Note that the angular and hyper-angular coordinates do depend on the choice of Jaboci coordinates). Since it appears in the Schödinger equation, we must derive the Laplace opera- tor in hyperspherical coordinates. Recall that for a two-body system, we can write the Laplace operator as ∆(1) = ∆η1− 1 η12 l̂21 , where ∆η1 = ∂ 2 ∂η12 + 2 η1 ∂ ∂η1 , and l̂1 is the relative angular momentum operator. We can easily see that for the three-body problem expressed with two Jacobi coordinates, the Laplace operator becomes ∆(2) = ∆η1 +∆η2− 1 η12 l̂21 − 1 η22 l̂22 . If we apply the transformation into hyperspherical coordinates directly, we find that we can split the Laplace operator into a term that depends only on the hyper-radial coordinate, and an angular term: ∆(2) = ∆ρ2− 1 ρ22 K̂22 , where ∆ρ2 = ∂ 2 ∂ρ22 + 5 ρ2 ∂ ∂ρ2 , and K̂22 =− ∂ 2 ∂ϕ22 −4cot(2ϕ) ∂ ∂ϕ2 + 1 cos2(ϕ2) l̂21 + 1 sin2(ϕ2) l̂22 . We want to recursively define ∆ρA−1 and 1 ρA−12 K̂2A−1 to be the radial and angular part 54 of ∆(A−1). To do this we first observe that ∆(A−1) ≡ ∆ρA−1− 1 ρA−12 K̂2A−1, (B.3) = ∆ρA−2− 1 ρA−22 K̂2A−2+∆ηA−1− 1 ηA−12 l̂2A−1. (B.4) Then we transform ρA−2 and ηA−1 into ρA−1 and ϕA−1. Equating (B.3) and (B.4) gives: ∆ρA−1 = ∂ 2 ∂ρA−12 + 3(A−1)−1 ρA−1 ∂ ∂ρA−1 , and K̂2A−1 =− ∂ 2 ∂ϕA−12 + 3A−9− (3A−5)cos(2ϕA−1) sin(2ϕA−1) ∂ ∂ϕA−1 + l̂2A−1 sin2ϕA−1 + K̂2A−2 cos2ϕA−1 . Defining L̂A−1 = L̂A−2+ l̂A−1 as the total internal angular momentum of the A-body system, one can show that the operators K̂A−1 = {K̂2A−1, ..., K̂22 , L̂2A−1, ..., L̂22, l̂2A−1, ..., l̂21 , L̂zA−1} all commute with each other and thus a hyperspherical state can be described by the associated eigenvalues. B.3 Hyperspherical Harmonics One can see that the hyperpherical harmonics are an extension of the spherical harmonics in a multidimensional space. In 3-dimensional space, we consider the eigenfunctions of the l̂ operator, known as the spherical harmoics. We can ask what the 3(A− 1) dimensional analogue of the spherical harmonics are, namely, the hyperspherical harmonics. We note that K̂1 = {l̂21 , l̂z}. Thus, by definition, the eigenfunctions of K̂1 are the spherical harmonics. We then ask that the hyper- spherical harmonics of order A−1 be the eigenfunctions of K̂A−1. We can construct such eigenfunctions recursively using the spherical harmon- ics. First consider the base case: K̂2 = {K̂22 , L̂22, L̂z2, l̂21 , l̂22}. We can construct an eigenfunction of {L̂22, L̂z2, l̂21 , l̂22} by standard angular momentum coupling (for any 55 operator Q̂, Q refers to the corresponding quantum number): ΦL2M2;l1l2 ( Ω(2) ) = ∑ m1,m2 CL2M2l1m1l2m2Yl1m1(η̂1)Yl2m2(η̂2) Where Ylm refers to the spherical harmonic with angular momentum l and projec- tion z and CL2M2l1m1l2m2 are the Clebsch-Gordan Coefficients [18]. The eigenfunctions of K̂22 (with eigenvalues K2(K2+4), K2 ≥ l1+ l2) are given by [20]: ψK2;l1l2 (ϕ2) =N (K2, l1, l2)(cos(ϕ2)) l1 (sin(ϕ2))l2 P (l2+ 12 ,l1+ 1 2 ) n2 cos(2ϕ2), where P(l2+ 1 2 ,l1+ 1 2 ) n2 is a Jacobi polynomial, n2 = K2−l1−l2 2 , and N2(K2, l1, l2) = √ (2K2+4)n2!Γ(n2+ l1+ l2+2) Γ(n2+ l2+ 32)Γ(n2+ l1+ 3 2) . Then, since K̂2 all commute, we construct the hyperspherical harmonic YK2(Ω(2),ϕ(2)) = ψK2;l1l2 (ϕ2)ΦL2M2;l1l2 ( Ω(2) ) , where KA−1 are the set {KA−1, ...,K2,LA−1, ...,L2, lA−1, ..., l1,MA−1} of quantum numbers associtated with K̂A−1. We recall that sinceK1 = {l1,m1}, YK1 =Yl1m1 . We can now define a recursive procedure to construct higher order hyperspherical harmonics. Given YKA−2 we construct YKA−1 by first coupling YlA−1mA−1 to YKA−2 : ΦLA1MA−1;KA−2lA−1 ( Ω(A−1) ) = ∑ MA−2,mA−1 CLA−1MA−1lA2mA−2lA−1mA−1 YKA−2(Ω(A−2),ϕA−2)YlA−1mA−1(η̂A−1). Then we construct the eigenfunction of K̂2A−1 with eigenvalue KA−1(KA−1+3A−5) [20] ψKA−1;l1KA−1 (ϕA−1)=N (KA−1, l1,KA−2)(cos(ϕA−1)) KA−2 (sin(ϕA−1))lA−1 P (lA−1+ 12 ,KA−2+ 1 2 ) nA−1 cos(2ϕA−1), 56 where nA−1 = KA−1−KA−2−lA−12 and NA−1(KA−1, l1,KA−2) = √ (2KA−1+3A−5)nA−1!Γ(nA−1+KA−2+ lA−1+ 3A−52 ) Γ(nA−1+ lA−1+ 32)Γ(nA−1+KA−2+ 3A−6 2 ) . And finally, since all of K̂A−1 commute, we can construct the (A− 1)th order hy- perspherical harmonic YKA−1 ( Ω(A−1),ϕ(A−1) ) =ψKA−1;lA−1KA−2 ( ϕ(A−1) ) ΦLA−1MA−1 ;KA−2lA−1 ( Ω(A−1),ϕ(A−2) ) . (B.5) Thus defined, the hyperspherical harmonics form an orthonaromal set of eigen- functions on the hypersphere satisfying〈 YKN |YK ′N 〉 = δKN ,K ′N . B.4 Hyperspherical Basis The hyperspherical harmonics as constructed above have no particular symmetry under particle permutation. One must construct completely antisymmetrized hy- perspherical harmonics when dealing with fermions. The more difficult problem of generating anti-symmetrized hyperspherical harmonics is solved in [21, 22]. With anti-symmetrized harmonics, we can build the basis: |K〉 |ν〉 ≡ Rν(ρ)HK(Ω,s1, · · · ,sA, t1, · · · , tA), labelbasis (B.6) where HK is a hyperspherical harmonic and spin-isospin state which is totally an- tisymmetrized, K is a set of quantum numbers including a grandangular momen- tum, total angular momentum L with projection M, total spin/isospin S/T with projection Sz/Tz, as well as those related to permuational symmetry [20]. The hyper-radial part of the wave function (depending on ρ) needs to be expanded in a hyper-radial basis: ψ(ρ) =∑ ν Rν(ρ). The hyper-radial coordinate is a measure of the size of the system and we expect 57 the ground state wavefunction to fall off exponentially with hyper-radius, so we wish to use an expansion of the form RKν(ρ) = e−aρPν(ρ), where Pν(ρ) is a polynomial. A natural choice turns out to be the Laguerre Poly- nomials, because they satisfy the orthogonality relation ∫ ∞ 0 ραe−ρL(α)n (ρ)L (α) m (ρ)dρ = (n+α)! n! δn,m, and can be defined L(α)n (ρ) = n ∑ i=0 (−1)i ( n+α n− i ) ρ i i! . Then, the wavefunction is expanded Ψ=∑ Kν CKνRν(ρ)HK(Ω,σ1, · · · ,σA,τ1, · · · ,τA) =∑ Kν CKν |ν〉 |K〉 . 58
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