..n ...«, 5‘1 . , ‘ www.mrnxnfi \ 4‘ a? , 4 . 3Q 5 A . . fir . . . . .. LT. .r h 23.1., .1. . .n 3.3.: 2-143: Lnumfluuun ! L :1... , viii Tran” nix} 4.16% 1.. J. ‘z..\ ‘ afl 3.1! :::.11.u...p 1..TA.,?M- (2.2) This choice of using secondary quantization is not required; it is just more conve- nient for our purposes. There have been many discussions in the literature of how to represent many-fermion wave functions, often as Slater determinants. Two such works are: [35,36], and an in-depth treatment can be found in [37]. To evaluate eq.(2.2), one must decide whether the wave function is made of fermions or bosons. Then, one must make eq.(2.2) normal ordered by using either commutation (bosons) or anti-commutation relations (fermions). Since we will be working exclusively with fermions in this work, the rest of the formalism will be developed for fermions. The anti-commutation relations for fermions are: {an aj} = aiaj + aja, = 0, (2.3) {0:101} = 01 {aI,a,-} = 0.5, 10 where 6,, is the overlap of the two orbitals: 03' = (ca-I451)- (2-4) In the simple case of orthogonal orbitals, then 6,,- = 6,], the Kronecker symbol. By moving all creation operators to the left (or all destruction operators to the right), one completes all possible Wick contractions [38]. The fact that creation operators anti-commute with themselves (eq.(2.3)), means that switching any two particles in eq.(2.1) induces an overall minus sign in the wave function, which satisfies the Pauli Principle. 2.2 Non-orthogonal Orbitals In the present work, we will need the orbitals to be non-orthogonal. To demonstrate how these work, we first need a reference set of complete orthogonal orbitals: 012 = (1|?) = 612, ZI1><1| = 1, {01,03} = 612, II?) = Z<1|fi>|1>= 26mm, 651 = 6b, (film = 1. (2.5) The numbers refer to orthogonal orbitals, whereas Greek labels indicate non—orthogonal orbitals, which in the last line of the above equation we indicated are normalized. Now we have the non-orthogonal orbitals in terms of orthogonal ones. It is then shown how 11 the general overlap expression, eq.(2.3), is obtained. We define creation and destruc- tion operators of the non-orthogonal orbitals: a; = 2015a] aa = 2100101 , (2.6) 1 to obtain the anti-commutation relation: {am 0],} = Zealowml, a3} = 200101,, = 00,3. (2.7) 12 1 Thus, we obtained the result shown in eq.(2.4). 2.3 Examples To demonstrate the machinery of the formalism, we will work with a two-body wave function \pafl : NGLGEIO), (28) with the normalization (wafilwafi) = (Olaaafiagailol = 0001666 _ “afigfla: (2-9) The normalization, written in terms of overlaps in the right-hand side of eq.(2.9), looks like a determinant, which in fact it is. By completing all possible Wick contractions, we generate a determinant of a matrix of overlaps. Generalizing to many-body wave functions: Ida ..... A)=N1'[ai|0) N=[det(g,,)]-1/2 12 ((1)111 ..... vAl¢ul ----- IJA> = [det (“vii/j) det (“uiujH—l/z det (“I/u) ' (2-10) It has just been shown how the normalization generalizes to a many-body wave function. Before going further, we would like to include an example on how one can pick the quantum numbers for the created particles. In this two-body wave function example, it is desired to have a pair of particles in a certain spin state ISM). Now our wave function looks like ISA/I) = NS 2: ClS/glolJ/202alalaga2lo)’ (2-11) 0102 where the subscripts 1 and 2 refer to the spatial wave functions, ¢1 and $2, whatever they may be, and C311?!“ J2m2 is the notation used for Clebsch—Gordan coefficients. The overlap of two such functions is (S'M'ISM)=N§,NS Z CSfo/ZflCIS/AgoLI/2a2(Ola1810232aga2alal]0)' (2-12) 01023132 Here, the matrix element is equal to (Olalslaizsfllgazajallm = 5013160232 — 60132602316126211 (2-13) as always, the single particle overlap is 012 = jwf(r)i/)2(r)d3r = 031. (2.14) To complete the overlap, we must sum up over the spin projections and use the following symmetry property of the Clebsch-Gordan coefficients: 13 CSM _ (_)sl+82—SC1$M (2.15) slm1,s2m2 _ s2m2,slm1' Using this property and completing the sum yields (S'M’lSM) = 653.61.,MIINS|2[1+(—)5|012|2] (2.16) 1 N 2 . 2.17 '3' 1+(-)Sl012|2 ( ’ One immediately notices the possibility that the denominator of the normalization vanishes for S = l and 012 = 1. This is because this state is forbidden by Fermi statis- tics. The triplet spin state means parallel spins (and, to be complete, the symmetric combination of paired spins), and 612 = 1 means the two spatial wave functions are identical, most likely caused by them being at the same point in space. These two things happening simultaneously we know is forbidden by Fermi statistics. This will be further developed after we discuss expectation values. Now that we have a normalized wave function, we can compute expectation values. Here, we go through the computation of the expectation value of a one-body Operator, such as kinetic energy (which is diagonal in spin, which makes the example simpler). The form of our spin-independent one-body operator is: 0(1) = [Zal(r)0(l)aa(r)d3r, (2.18) where 0 (sans caret) is the “operational” part of the operator (e.g., for the :1: position operator, this would be 2:), whereas 0(1) is the full second-quantized form of the operator. The the single-particle operators correspond to the localized states so that: (0|a0(r)a]all0) = 6001 /6(r — x)i/)1(x)d3:c = 60012p1(r). (2.19) 14 For our two-body system, eq.(2.11), we have (SM|0<1>|SM)= |Ns|2 Z /( (0|a131agsga;0(l)aaa;o2a]al[0)d3r (2.20) 001023132 After one computes all sums and Wick contractions, one obtains four terms: (SMIO“)ISM) = INS)? /();(.)o<1>.),(r) + ¢;(r)0w2(r) +(—)Sl61'2¢:(r)0“>w2(r) + 0.225(r)0“)2/21(r)1}d3r- (2.21) If Omhad been the kinetic energy operator, the result would have been the kinetic energy of the single particles 1 and 2, plus (or minus) the kinetic energy of the overlap between the two orbitals. We now turn our attention to two—body operators. The form of a spatial two-body operator is :éf/Zaah' W 0“’(I‘ r’)aa(r ’a) arrfld3 d3r.’ (2.22) We find expectation values in the same way as before: we sandwich the operator by the wave function and perform all possible contractions. In this example, there are actually fewer possibilities with the two-body operator, though this is not usually the case in larger systems. There are actually four terms, but symmetry and the factor 11,- from eq.(2.22) allows us to write the expectation value as (SMIO‘2’|3M> = lel’ / lw5(r’)wi(r)0"’¢1(r)¢2((r’)) +(—)Sw;(r'wxriomwrw.(r’>1d3rd3r'. (2.23) The first term is the direct term, and the second the exchange term. We will now go through another example to demonstrate how the singularity in the denominator of the normalization (eq.(2.16)) is removed. Consider a two particle 15 system where the particles are identical and have parallel spins (for simplicity, we will work in 1-dimension). The wave function of the second particle, $2 is the same as the first particle, only displaced by some distance d (i.e., 1&2 = ¢1(:r — d)). The normalization is eq.(2.16) with S = 1. We take a closer look at the overlap, 012, 012 = /¢f(x)tp1(:1: — d)d:r = 012(d). (2.24) The overlap is a function of the distance d. Expanding the overlap in a Maclaurin series about d = 0, we obtain N49,;-D 912(1) = 012(0)+0’,2(0)d+6’1’2(0) +..., (2.25) = 1+ 61’,(0)§+ .... (2.26) In moving from the first line to the second, we used the fact that 612 is at a maximum at d = 0, which allows us to eliminate the term proportional to its first derivative. The normalization is now, to lowest order, 1 N 2: . ' 3' 0126:2(0) (2.27) We now would like to find the expectation value of the kinetic energy, T. We know from the general formula, eq.(2.21), that there are four terms. The first two are the diagonal matrix elements of the two particles, and are equal, since the kinetic energy does not depend on translation of the origin: / «(Ham-mm = / Wm — driwx — d)dx = To (228) The cross terms are also equal to one another: f ¢I($)T¢1(a= — d>dx = f we — arr/mm = t0(d)- (2.29) 16 The result of the cross terms is called to(d) because at d = 0, to = To. As with the overlap, we then expand t0(d): t0(d) = t0(0)+t3(0)d+t3(0)§ (2.30) = To+t3(0)§. (2.31) Once again, we have used the fact that t0(d) has a maximum at d = 0 to remove the first derivative term. Plugging in the known terms into eq.(2.21), we have =2n—2axmmw) (T) 1—agm2 asm Plugging in the eXpansionS 0f 912(d) and t0(d), we have £22m Simplifying, we find that any dependence on d disappears, and we obtain a finite result: (Taoism) + rad» 2/2 = WWJW ra+ifl1 ash (T) Z 2 672(0) Here we see that the kinetic energy is increased when one moves the two particles together. This is expected, as one of the two particles must be promoted to a higher energy level in order to satisfy the Pauli Principle. Thus, the possibility of having zero in the denominator of a normalization is not a problem of the theory. In fact, it will appear again many times throughout this work; it is just how the theory accounts for Fermi statistics. We now turn to another general part of the formalism, the method of projection into good states of angular momentum. 17 2.4 Angular momentum projection Figure 2.1: Here is a diagram of a two-particle wave function that has been rotated by an angle 19. In order to project into a state of good angular momentum, one must find the overlap of these two wave functions and use a projection operator to pick the desired angular momentum quantum numbers. As our wave function is currently written, it is not in a state of good angular momentum. Here, we will outline the method used to project the wave function into states of good angular momentum. The basic approach is to always calculate overlaps between a wave function and another wave function which has been rotated by some angle and then using a projection technique to pick out the desired quantum numbers (illustrated in Figure 2.1). The general procedure is in many texts, for example [39]. The general formula for axially symmetric systems is: de K09) (wlexp (229.11, ) H|7,[))dc0329 (\I’JMIHIWJM)= , deK (19) («2| exp (wa ) |¢)d 00319 (2.35) where 6"”? denotes a rotation about the y-axis, and d‘IQK is an element of the reduced rotation matrix that picks out the particular state of interest. It has total spin J and projection M = K, K being the projection of angular momentum along the body- 18 fixed axis. In this convention, the body-symmetry axis is taken to be the z-axis. These matrix elements of finite rotation can be found in any book on angular momentum such as [40]. In the case of “He, the formula simplifies. In the model presented in detail in the next chapter, all spins are coupled to singlets, and thus the only effect of the rotation operator is to rotate the single-particle spatial wave functions, and the projection operator essentially picks an orbital momentum which is then equal to the total angular momentum, and selects the parity of the state [7r = (—)‘]. We select the projection of angular momentum equal to zero, which simplifies the rotation matrix elements, dgow) = PJ(COS 29), where PJ(COS 19) is the J th Legendre polynomial. As will be presented in section 3.2, “Li is more complicated. Lithium-6 has a ground state with J = 1, which in our model comes from the spin of the deuteron. There are now two contributions to the total spin, one from orbital angular and the other from spin, so there is no simplification of eq.(2.35) in the case of “Li. Even though we often work with non-orthogonal orbitals, wave functions of dif- ferent angular momentum remain orthogonal. We demonstrate this in the following derivation, considering two “He wave functions with different projections of angular momentum. (swam): / D Mm )an / DM,0(§R’)\II(§ft’r)d$?’), (2.36) where Dim is the Wigner rotation matrix element, and if? is the rotation operator. We can rewrite eq.(2.36) into =/ 0,; (a)D,,’.,,( 32’)(\II(§ftr)|\II(§ft’r))d§R’d§R. (2.37) Now we move the rotation operators around in the matrix element by multiplying 19 through by the inverse rotation, Ji‘l: (‘I’Gfirll‘l’flfi'rll = (‘I’(l‘)|‘1’(§frl§f3’r))- (2-38) Let 3?” = 32-191”, and therefore .5R’ = 3133”. Returning to the full expression, we have / [19,; 133;,“ (mn')(ql(r)|xp(§ir'r ))d§Rd§R". (2.39) We rewrite the second D-function Dfl'ofgwf”) = 2 07.130522), (2.40) it and substitute this result into the full expression, and then integrate over 3%, which generates the required result: 8 2 A I I I = 2L: 16mm. f>nto(w )déR’. (2.41) Now it is clear that wave functions of differing angular momenta are orthogonal, and that the projection process comes down to the integration over one rotation (which for axially symmetric systems further simplifies to one angle). More details about the projection process will be shown in chapter 4 with specific single-particle wave functions. This concludes the chapter on the formalism used in the present work. In the next chapter, we apply the formalism to the two nuclei of interest in a general way. We sketch out the framework of the calculations, starting with the wave functions and going through the calculations of general operators. This sets up the next chapter, where we delve into detailed calculations with a specific choice of single-particle basis functions. 20 Chapter 3 Skeletons of the Six-particle Systems 3.1 Helium-6 3.1.1 Alpha-dineutron configuration We apply the formalism now to “He. We will first work with the alpha-dineutron configuration, which is a two—center model (see Figure 3.1). Since this particular configuration has two centers, there will be some broad sim- ilarities with the two-particle example of the previous section. The wave function is (1:) = Na;aipa{a[,aga12|0), (3.1) where the subscript p indicates protons, and the remaining particles are neutrons. The minus signs indicate spin projection, and particles with subscript 1 are the neutrons in the alpha particle, while those with subscript 2 are the dineutron. Next, the wave 21 Figure 3.1: Helium-6 as an alpha particle plus dineutron. The dineutron is constructed as a spin singlet, and the parameter d describes the distance between the two centers. function is normalized: 2 (\II|\II) = (0|a_2a2a_1a1a_papa],a:pa]a:1a;ai2|0) = 6?, (006,1 — 0(2) , (3.2) 912 is defined as in (2.14). 00, and 0,; are also overlaps, but with particles at the same center, which in many cases is one. We keep the overlap notation, however, because for many of the calculations it is not equal to one, due to the process of projection into states of good angular momentum. The normalization has now been obtained 1 N 2 = . 3.3 ] l “(22 (“used “ “f2l2 ( ) The initial 0?, factor in the denominator comes from the protons, while the remaining expression comes from the neutrons. The neutron part looks like (2.16) with S = 1. The triplet spin occurs because the particles at the different centers may have parallel spins (e.g., there is a spin-up neutron in the alpha and in the dineutron), but they are singlets with respect to their own centers, and the triplets are coupled together 22 to make an overall spin of zero for the “He nucleus. Another interesting feature of the normalization, is that it could be zero if 0%, = 600,). This would be the case, for example, if the two centers came together, and the single-particle wave functions of the external neutrons were not radically different from those in the alpha particle. Then, as the two centers coincide, the overlaps become the same, and the denominator of the normalization becomes zero(see the discussion in the example following eq.(2.23)). This is because when the two centers are in the same place, four identical spin-1/2 fermions are at the same spatial location, which is forbidden by the Pauli principle. This is a very important feature of the theory, and will be discussed further during the calculation of observables. One-body operators Now that the wave function is properly normalized, we can calculate expectation val- ues. We will first go through a one-body operator, of which there are many examples (kinetic energy, mean-square radius, quadrupole moment, etc). The matrix element ‘ we must evaluate is: (0(1)) 2 fZ(0]a_2aga_1a1a_papa,]aaa],a:pa]atlagaig|0)d37‘ (3.4) 0' As with the 2-particle—example, there are four terms: terms centered at the alpha, terms centered at the dineutron, and two kinds of overlap terms (which for most operators are identical, but we will keep them separate here). The terms are: (0(1))p = (a|0(1)la)200(0a9d ‘— 9i2)2: (00))" = 26a (293.03 + 01‘. - 303.6302) , (3.5) 23 where the first two lines in eq.(3.5) are the proton and neutron parts, respectively, of the contribution from the alpha particle. The matrix element in the beginning of each term is the normal expectation value of the operator: (alOmla) = / 272330127433). (3.6) The shorthand used throughout this work for these matrix elements is Ia) denotes a wave function centered at the alpha (which assumes that all four particles have the same spatial wave functions), and Id) denotes a wave function centered at the dineutron. The collection of overlaps that follows the matrix element are the remaining parts of the wave function that are not involved in the normalization. Thus, the matrix element from the protons in the alpha particle will be followed by overlaps involving neutrons. Continuing with the rest of the terms, (0(1)>d = (d|0"’|d>203. (9.294 - 0i2) , (3-7) (0(1))... = (a|0<1>|d)20§ (9?2 — 912900)) , (3.8) (0%. = 26: (0i. — 0.26.9.1). (3.9) These terms are then summed together and multiplied by the normalization. As mentioned before, when the centers come together, the norm develops a singularity in the denominator. If one looks at the one-body terms (eq.(3.5) and following), if the centers coincide, then the overlaps become equal, and each term is zero. Thus, the singularity in the denominator will be removed by the zero in the numerator, generating physical results. More concrete evidence of this will be shown in the next chapter. 24 Two-body operators We now turn our attention to two—body operators. As with the one—body operators, we sandwich the operator by the wave function, and perform all Wick contractions to find its expectation value. What will be shown here is the sum of all terms for a given geometric configuration, i.e., unlike in (3.5), only the last line will appear, and there will not be separate entries for proton and neutron contributions. This is not to say that two-body operators cannot distinguish protons and neutrons, but for brevity, only the total contribution from each geometrical matrix element will be given. For operators that are sensitive to isospin, the form of their expectation values will be given in the section that discusses that individual operator. Before listing all the terms, please note the following convention for the order of integration variables in the individual matrix elements: <12|0|34> = (l/Jl(X1)¢2(X2)IOW3(X2)¢4(3(1))- (3-10) 25 Now, the list of the terms resulting from the calculation of the expectation value of a two-body operator: (0(2))0 = (aa|0(2)|aa) (60:93 + 0112 — 66¥2000d) , (3.11) (0(2))..232 = (daIO‘Z’lad) (8039.1-602012), (3-12) (090......) = (aa|0<2>|ad)2(20§,0.. 4030120)), (3.13) (0(2)),mm = (da|0(2)|aa)2(26]’290—30§6126d), (3.14) (0(2)>ddda = —(dd|0(2)|da)202012, (3-15) (0(2))addd = —(ad|0(2)|dd)263012, (3-16) (0(2))..de = (aa|0<2>|dd)0§0§2, (3.17) (0(2))dada = (dd|0(2)|aa)9§9i2, (3-18) (0(2))..dda = (da|0(2>|da)2(203,9f, 430,), (3.19) (0(2)), = (dd|0(2>|dd)0;;. (3.20) These are summed together and multiplied by the norm, just as in the case for one- body operators. Once again, a zero divided by zero situation resolves itself amicably, which will be seen in more detail in the next chapter. This completes the necessary formal calculations for the a—dineutron configuration of “He. All operators here were assumed to be spin-singlet operators. Due to the choice of wave function for “He, no operators that affect spin (e.g., L - S) have non-zero expectation values. 3.1.2 Cigar configuration The other extreme in picturing an alpha particle plus two additional valence particles is a particle-alpha-particle chain, colloquially referred to as the cigar configuration (pictured in Figure 3.2). The cigar configuration has a higher degree of symmetry than the alpha dineutron configuration (D00), vs. C00,, in Schoenflies point group 26 Figure 3.2: Helium-6 pictured in the cigar configuration, which is a neutron-alpha- neutron chain. The external neutrons are constructed as a spin singlet, and the pa- rameter d describes the distance between the central alpha and the external neutrons (they are equidistant from the alpha). theory parlance), which will have important consequences that shall be seen in the results chapter. The wave function in the cigar configuration is m—m p11) = Nagaipa1a1,Z(—)l/2-mai ai |0). (3.21) The main difference between this wave function and eq.(3.1) is the sum over the spin projections of the external neutrons. Formally, this should have been done in the previous configuration, but since the neutrons in that wave function are located at the same spatial point, the only thing accomplished by summing over the spin projections is the quadrupling of the number of terms. Here, with the neutrons at different locations, the sum introduces important correlations that preserve the higher 27 symmetry of the cigar configuration. The normalization is (MW) 2 N2Z(0|a_mramra_1ala_papa],al_pa]alla] a] [0) m —m mm’ = N226: [63, (63, + 63,) + 26‘}2 — 20f200 (0,, + 61)] , (3.22) 1 N2 = . .2 263162.06. + 63.) + 26:2 — 266.6. (6.. + 6.)] (3 3) As before, the normalization is a collection of terms involving various overlaps. 00, remains the overlap inside the alpha particle, 0,, is the overlap of an external neutron with itself, 012 is the overlap between the alpha particle and an external neutron, and 01 is defined below: 6. = (is) = / 61(r)¢—(r)d3r- (3.24) 03+)- is a wave function centered at the right(+) or left(-) side of the alpha particle, so this term is an exchange term introduced by the sum over spin projections in eq.(3.21). Once again, if all of the particles are brought to the same point where all overlaps become equal, the denominator will go to zero, because there will be four s-wave neutrons at the same point (if the alpha and dineutron wave functions are identical). The calculation of matrix elements proceeds in a similar way to the previous configuration. Here we list the terms for one-body and two-body operators. First, the one-body terms: (00>). = (a|0(1)|a)20a (26: (6,“, + 6:) + 26:;2 - 30:30,, (6.. + 61)) (3.25) (0(1)). = (:l:|0(1)|:l:)20a (60.6., 46%,) (3.26) (0(1)):t = (il0(l)]:F>20a (“019$ - 9f2) (3-27) (0(1)>ai = (aIO‘l’IiWfi (29f2 - 612190: (9n + 91)) (3-28) (0(1))10 = (:1:|0<1)|a)26§ (20.?2 — 6126.. (6., + 61)) . (3.29) 28 It should be noted that in all of the terms above, :1: can be flipped to :1: in all places in the matrix element without changing the result. If all overlaps become equal, each term becomes equal to zero. Here is the list of terms involving a general, spin-singlet, two—body operator in the cigar configuration of “He: (0”), = (aa|0(2)]aa) (60: (03, + 0:) + 20‘]2 — 60?,00, (0,, + 035)) , (3.30) (090,212 _—. (ia|0(2>|ai)26§ (460.6,, — 36%,) , (3.31) (00002,, = (ia|0<2>|a;)26§ (4000,, — 36%,) , (3.32) (0(2))000, = (aa]0(2)|a:l:)20a012 (46f, — 36.. (0,, + 61)) , (3.33) (0(2))4033 = (:l:07]0(2)]aa)20a012 (46f, — 36,, (0,, + 61)) , (3.34) (Oman)... = —293012, (3-37) (0(2))..44; = —(a i l0‘2’l 4= 62292012, (338) (0(2))...2. = (aal0‘2’l 1 0263.63, (3.39) (0(2))1104 = (i a: l0‘2’laa>29§0¥2, (3-40) (0(2))0110 = (a :1: |0<2>|ai)26§ (26f2 — 6,6,.) , (3.41) (0(2))..440. = (a i l0‘2’la¥)29§ (2012 - 0091) , (3-42) <0‘2’>+2_2 = <+ — IO‘Z’I — +>6;‘., (3.43) (0(2)>+——+ = H“ — '09)] + “>93.- (3-44) As before, when all overlaps are equal, the terms sum to zero. Also, if one takes the limit where the cigar configuration becomes the alpha-dineutron (i.e., +,-—+ d and 0,,,0i —2 0d), the previous terms become the list of terms for the alpha-dineutron configuration. 29 With these matrix elements, one can calculate many properties of the cigar con- figuration of “He. We will now move on to the next section which will discuss the interference of these two configurations. 3. 1.3 Interference term The overall composition of “He is a mixture of the two previously mentioned non- orthogonal configurations. That is, [‘1’) = Cllwll + 02002): (345) where 161 is the alpha-dineutron wave function, 2122 is the cigar configuration wave function, and c1 and Cg are weighting coefiicients. There are many such systems in nature, systems which have a potential with multiple minima, thus allowing a mixture of configurations. One of the simplest example is the ammonia molecule, NH3. Its trigonal pyramidal inverts, something which can be measured in the microwave region ( [41—43]). Helium-6 is a more complex mixture, and part of our goal is to determine c1 and c2. In order to do this, we must solve the following eigenvalue problem: (wlllel) (26101062) (c1) = E 1 Whit/12) (Cl). (3.46) (Ipleli/Jll f¢2lHl¢2l C2 (262021) 1 02 The normalization is c? + c3 + 2c“:2 («60162) = 1. (3.47) In eq.(3.46), 1:1 is the Hamiltonian operator, E is the energy which will also be de- termined by solving this equation, (161]1/22) is the overlap of the two configurations, and (z/JIIH [102) is the off-diagonal matrix element of the Hamiltonian between the two configurations. We need to determine two of these quantities: the overlap of the 30 configurations and the Hamiltonian between the two configurations. First, we need to represent both systems in the same coordinate system. This is not trivial, as the center-of-mass of the two systems is not at the same point. Figure 3.3 illustrates the two configurations. The center-of-mass of the cigar system is in the alpha particle, whereas in the alpha-dineutron it is between the alpha and dineutron. In calculations such as the mean-square radius, the location of the center-of-mass is very important, and we must make sure that each configuration is properly referenced from the center-of-mass. ”m Figure 3.3: The two configurations of Helium-6 pictured together. The alpha- dineutron components are filled with diagonal stripes, the cigar configuration com- ponents are open circles and are labeled in italic script. In the figure, the distance between the alpha and the dineutron is the same as the distance across the entire cigar configuration. Now that we are oriented, we will compute the overlap of the two configurations: (101062) = N1N2(Ola—zaza—iaia_papa],,a]_,,,a[,a’_1,0101|0). (3.48) The primes on the labels of the creation operators indicate that these are particles located at a different location, while the plus and minus labels retain the meaning 31 from before ((3.24)). It is not necessary to sum over spin projections here, as in (3.21), as the neutrons in the bra are at the same spatial location, so the sum over the projections in the ket does not introduce any new information. After one contracts all the operators in the above equation, one obtains: V5 2"[9 00d+0d— + 0d000+60-_ 93,963 (96+9a- + 6d—00+)] \/6g( 6 ,6, — 6, )62, [62, (62 + 62) + 26,, — 2612.6, (6, + 6,)] (Walla: (3.49) The overlaps in the numerator of eq.(3.49)are new overlaps between the two configu- rations, while the ones in the denominator are the overlaps from the normalizations of the individual configurations, eq.(3.23) and eq.(3.3). The new overlaps are (primes always refer to cigar wave function coordinates): 6,, = (ala’), (3.50) 6,, = (d|+), (3.51) 6,_ = (d|—), (3.52) 6,, = (.63), (3.53) 03+ = (04+), (3-54) 6,_ = (04—). (3.55) With these overlaps, we can calculate the overlap of the two configurations (eq.(3.49)). They also appear in the terms resulting from the calculation of expectation values, which is shown next. In order to calculate the expectation value of the Hamiltonian, we need the expec- tation values of one-body and two—body operators. The procedure is exactly the same 32 as was shown before, so now we list only the results. First, the one-body operators: (00))“, = (0400002 X (020,011+6d— + 2Oaaga+6a—0¢21a — 3030,6110: (0d+6cr— + 0d—00+))(13'56) (0291. = 0§, (05,103+ — 0605+), (00)), = (d|0<1>|a’)6§, (20,+0,_0,, — 6,, (0,+0,_ + 0.1-0,,» , (0(1)),+ = (a|0(1)|+)0,2m0,, (6,_6,, — 6,,6,_), (0(1)>a— = (0]0(1)]_)6c2moda (00+0da _ 90,004+). (3.57) (3.53) (3.59) (3.60) ( 3.61) As before, these terms are summed together, then divided by a normalization factor, which in this case is the denominator of eq.(3.49). N ow, we proceed with the two-body 33 operators: (00)») (0(2) >a2d+ (0(2)>02d~ (0(2)>aaa+ (0(2))aaa— (0(2))6665 (0(2))d+d— (0(2))d+da (0(2))(1—(10 (0(2))d+a— (0(2))d_.+ (0(2) )a-l-da (0(2)>a-da (0906+..— (0(2))6666 (aalO‘z) |a'a') [60:06d+0d— + 6121a0a+60— _ 30000110: (0d+60— + Oct—001+” 1 (3'62) (660(2)) + 5')63’m (46,,6,_ —— 36,,6,_), (630(2)) — a’)0,2m (40,,0,+ — 30,,0,+), (aa]0(2)la'+) (26,,6,_63, - 363,6,,6,_) , (aa|0(2)|a’——) (26,,6,,_63,, — 30§00d00d+) , (3.63) (3.64) (3.65) (3.66) (da’]0(2)]alai>6cra Hebert—00+ _ 300:0 (602+0d— + 00—0d+)] (367) (66110“) + —>02..., —(dd|0<2>|a’+)6§,6,_, -0§.9a+, —(da|0(2)| '" +>63a0daa —(da|0(2)| + _>6306d0n (da|0(2)| + (1)63, (26,,6,_ —— 6,,6,_) , (35.06)) + 696?... (26.1.6... -— 6....6...) , (aaIO‘Z’I + —>6§.63.., (dd|0<2>|a’a')6§,6,,6,_. (3.68) (3.69) (3.70) (3.71) (3.72) (3.73) (3.74) (3.75) (3.76) As always, these terms are collected and summed, and divided by the normalization. With these operators, we can complete the calculation of the expectation value of the Hamiltonian, and thus we can find the minimum energy with eq.(3.46). With the value of E, we can find an expression for CI in terms of c2: 6 : _H12 — E<¢IWJ2)C 1 H11 _E 27 (3.77) where H12 = (1/11|H|1/22). This value for CI is then substituted into eq.(3.47). It should 34 be noted that when dealing with off-diagonal matrix elements such as (161]1122), the overall sign is arbitrary, so in eq.(3.46), (tbllH [162) and (161]1/12) have an arbitrary sign (though once one is chosen for one of these, it determines the sign of the other). How the choice of sign is made is discussed in the next chapter. Once c1 and c2 have been determined, the expectation value of any other operator (an observable not in the Hamiltonian) can be computed: (O) = Cffi/JIIOWI) + C§<¢2IOWJ2> + 0162(WIIOW) +(1112l0li/21ll- (3-78) We now have completed the discussion of the basic formalism behind the “He calculations. The calculation with a specific choice of single-particle wave function will be discussed in the following chapter. We now move on to the discussion of the other nucleus of interest, “Li. 3.2 Lithium-6 The other main subject of this work, “Li, composes 7.5% of natural lithium [44]. As mentioned before, it is well studied experimentally. We study it here because it is the beta-decay product of “He and is also a difficult test of structure theories. Before going into the two configurations of “Li, we will discuss the projection into good states of angular momentum “for “Li, as it is the same for both configurations. As mentioned in section 2.4, “Li has a ground state spin equal to one, which, in our model, comes from the deuteron spin. In this case, we must use the full general formula, eq.(2.35). When we rotate the “Li wave function, we have not only the spatial rotation, as in “He, but also the rotation of the spin part of the wave function. In the case of “Li, this means rotating a spin one object. The d-matrices for spin-one objects can be found in [40]. We can rewrite eq.(2.35) to show the effect of the spin-rotation 35 explicitly: 1 61.09361 exp (266;) Hl6>dt.(6)dcos6 (\II’MlHl‘I’JM) = . , f dfdt.(6)dcos6 (3.79) where (b is the spatial wave function, and the (1,1,), is the factor that comes from rotating the spin wave function. Since this projection does not have a definite parity, unlike in “He, we also must project into the desired parity of the state (in “Li’s case, positive). The parity projection Operator is: liP’ 2 I6; J") = Iv; 6). (3.80) where P’ is the parity operator which inverts the coordinates of \II through the origin. The parity can be positive or negative, whichever is desired determines which sign is chosen in eq.(3.80). Now we have the method of choosing specific J"r states in “Li, and can move on to the discussion of the specific configurations. 3.2.1 Alpha-deuteron configuration The spatial picture of this system is the same as in Figure 3.1. The wave function is |\II) = Najaipajaifljdaldlm, (3.81) where the designations for the alpha particle are the same as in eq.(3.3), (1;, creates the proton in the deuteron, and aid creates the neutron in the external deuteron. The deuteron, and hence “Li, has spin=1. For convenience, we take the spins of the external proton and neutron parallel and in the “up” projection. The normalization expression for “Li turns out to be the same as in “He, (eq.(3.3)). In fact, the expressions for all the spin— and charge-independent operators are identical to the case of helium, and will not be repeated here. Operators that are sensitive to charge will see a difference, 36 but for spin- and isospin—independent operators, the results in all configurations of “Li match the corresponding case in “He. For operators sensitive to isospin, those results will be given in the next chapter in the details about particular operators. We turn now to the case of spin-dependent operators in the alpha-deuteron configuration. Unfortunately, spin-dependent operators are difficult to treat generally, so we will work with the two specific ones of interest for our calculations, the first one being the spin-orbit operator, L - S. The spin-orbit Operator is 1 L - S = L,S, + L,S,, + L,S, = -2- (LS. + L_S+) + L,S,, (3.82) where in the term on the far right we rewrote the expression to be in terms of the spherical generators of the rotation group. Li and Si are raising and lowering operators for orbital angular momentum and spin angular momentum. They raise (or lower) the projection of the relevant angular momentum on the chosen quantization axis. Li has a spatial definition and will be dealt with in the next chapter. All that needs to be mentioned now is that for all of our calculations, (L_) = — (L+), therefore all expressions in this chapter will be shown in terms of (L1,). This is because the rotation that generates the angular momentum is about the y-axis, thus only L, is non-vanishing. The Byzantine inner workings of spin and rotation Before going into the machinery of the operator, we need to look at the effects of rotation on the spin part of the wave function. Since this operator affects spin, the effect of the angular momentum projection process needs to be taken into account. Protons and neutrons are spin-1 / 2 particles, and thus obey the following rotation law: . 1 1 , 62.1566 = 263336))? >. (3.83) 37 The dxinW) matrix is fairly simple: 19 - 19 cos — —— srn — d¥,2m(19) = 2 2 . (3.84) sin 322 cos % For example, if we rotate a pair of particles with parallel spins pointing “up” (i.e., in the spin state |S M ) = |11)), we get the following expression: - 19 19 19 19 ERaLbHO) = (cos 50.], + sin 50:) (cos 2])“ + sin 50:) [0), (3.85) where the + and - denote the spin projection of the particle. Collecting and simpli- fying, we obtain: sin 19 19 19 =(coszia151+sin2§a:5:+ 2 (3151+a161))|0>. (3.86) Using the following relation, I + —) ) + |00)) (3.87) 1 = $010 1 —+=—10—00, 3.88 | > fit) > I >> < > we can rewrite the result in terms Of spinors: . 19 sin19 19 11 = 2—11 —— '2— — . . §R| ) cos 2| )+ fl |10)+31n 2|1 1) (389) Now that we have these relations, we can move on to the detailed effects of the operator. The L - S operator appears in a two-body potential, 30 we examine how the op— erator operates on pairs of particles. We are looking at matrix elements of the kind: (v.5) = $66; SMIV(r)(L—S+ + 146916); 6'46), (360) 38 Table 3.1: This table shows the results of the spin raising and lowering operator acting between any combination of spin-1 states in terms of the rotation angle, 19, and the potential and orbital angular momentum operators, VLi. I11) |10) |1 — 1) (11] — sin 19(VL+) — cos 19(VL+) sin 19(VL+) (10] cos 19(VL+) — sin 19(VL+) — cos 19(VL+) (1 — 1] sin 19(VL+) cos 19(VL+) —sin19(VL+) where a, 6, 7 and 6 are spatial wave functions, V(r) is some spatial form-factor of the potential, and S, S’, M, andM’ are the spin quantum numbers of the pair of particles. Focusing now on the Si operators, they have the following effect: (1M|S,|3M') = \/2 — M’(M’ i 1)6S,,6M,M,,1. (3.91) We must go through all the matrix elements, couple the bra and ket to good states of spin, then select the non-vanishing terms. By doing derivations like that began in eq.(3.85) and using the definition of the operator in eq.(3.91), we can construct a table containing all combinations for the action of the Si operator (Table 3.1). Spin-orbit continued With the matrix elements of 5+ and S_, we now need to multiply them by the correct combinations of overlaps to obtain the complete matrix element. For example, if we are talking about the complete matrix element involving the spin-up proton in the alpha particle and the proton in the deuteron, we need to simplify the following expression: (mania—1ala—péipéiailaldlm, (3.92) where the tilde (ii) denotes that operator has been rotated with respect to the un- marked operators. The operator matrix element in the beginning of eq.(3.92) is taken care of by using Table 3.1, but we must contend with the rest of the expression, keeping in mind the effects of rotating the operators. In Table 3.2, we collect all the 39 non-zero combinations for overlaps. There are certain overlaps that are zero, these are of the form (0|a,a_,a;aj,|0), (3.93) where the a, operators refer to a particle created or destroyed in the alpha particle. These particles have to be either all protons or all neutrons. Any term with the overlap (3.93) in it is zero, due to the orthogonality of spins. Two particles of the same type in the alpha particle are in a singlet spin state (the annihilation operators in this example), but the created particles have parallel spins, and are thus clearly in the triplet spin state, therefore, the overlap vanishes. This is also true for the transpose of (3.93). After one combines all the correct matrix elements from Table 3.1 with the ap- propriate overlap from Table 3.2 and sums together all terms from a certain spatial geometry, we are left with only two terms: (145),“. = —2 sin 19(VL+),2,2 (2636, — 636%,) , (3.94) (VLS>adda = 28111 79adda (620,, — 026:2) . (3.95) These are the two terms which contribute to the spin-orbit interaction for a spin-orbit interaction that does not depend on isospin (heretofore tacitly assumed). They both vanish in the non-rotated picture (19 = 0). Tensor operator The other spin-dependent operator we will examine is the spherical tensor operator. The spherical tensor operator, S12, is generally written as 3 312:;(01'rlfa’2'rl-01'02, (3-96) 40 Table 3.2: The overlaps for the “Li wave function involving rotated operators. The entire list of overlaps is quite extensive. In the interest of brevity, we list only half of the overlaps here, because if one changes every proton into a neutron and vice-versa in each line, the overlap expression is the same. To further reduce the size of the table, each overlap has a certain symmetry with respect to its transpose. It either changes sign or does not. Those that are the same as their transpose are followed by a superscript “+”, and those that change sign are followed by a superscript “-”. Overlap Value (Olanda_1ala_p&]_ p&[&] 16,, Hd|0) 032 g (03 0,, - 030%,) (Olanda-1alapa:pa]a: la I,‘d|0) :—“'g“ (030,, — 030%,) (Olanda_ 1010160306110] land]0>+ 032 122 (030,, " 631692) (Olanda_1a_papa],q]pctr]Tlald|0)+ 0:“)? 09:0,, + sin2 129- 020%, (Olandala_papa],a_p a _1a,,d|0) —S”.,; (030,, — 030%,) (0|a,,da1a_,,a.,,al(1.]_p,,a]a,,d|0)J"+ osczfig (03 0,, — 030%,) (Glandapda— 10'1a’la—lapda’ndlo)+ 032 2 (931031 " “a“fz“n)2 (0|andapda_la_pa]p ii] 1a],,,a,, d|0)+ (c0322‘90a0n +23in213-0f,) (0]a,,da,,da_1a_,,(3.],a’f la)r a n‘dIO) “”2‘“ (c0322 ‘90,, 0,, + sin2 -0f,) (0,0,, — 0%,) (Olandapda—la—pdT—pajapdandlo)- 51319 (cos2 —0,, 0,0,, + Sin2 30%,) (0,0,, — 0%,) (Olandapda_1a_p&],&]&],dé],d|0)+ “——‘“4 ‘9 (230 0,, —02,) (0|a,,a,,a,a,,at61,61,6Ld|0)- —S'—'g" cos“ — '29,,(6 0,, —6f,)2 (Olandapdalap&:p&]c'ipdc~znd|0)‘ ——“—2-"”’ —cos2 g—(0a0n 0%,) 2 (0|a,,da,,,)a.1a,,,a;',a]a,’,’,dii],d|0)Jr os4 '9 2(0: 0,, —02 ,) (Glandapda_ 1a1a;a]a’_ lc1,,,,|0)+ —cos2 - 32(020n 012 — 0 c,01,) (0]a,,da,,,)a_10.1a]_]:,a]at landIO) 3’3” (2030,012— 00,03, (Olandapda_1a_pa],a:pfz: la ,,+d|0) — (cos2 $030,012 + sin2 $000?» (Olandapda_1a_,,a],a]_,,a]a,,d|0) j“? (020,012 — 0,09,) (Olandapdala_pa],&l_pfi]_ ,aLdIO)“ 3m“ (020,, 012 — 0 00:13,) (Olandapdalawaéqigakr‘t ald|0)+ —co2s2 — 3(020n 012 — 0 0,01,) (Ola- 1a1a_papa],a_pa_,r&], d+]0) —02,012 (0|andapda- 1a_,,a!a]_ pala’ ]—d+1IO> 929% (Ola- lala_,,a,,a,[,a)r pa]a_ ] 1+|0) 0,4, where 0,- : 23,-, and r = r1 — r;. The tensor force is the interaction between particles’ spin and relative motion. If one chooses the coordinate system carefully (i.e., placing the z-axis along r), one can rewrite the tensor operator as 41 As with the spin-orbit operator, the radial dependence of the force must be added in by hand, lA/tensor = l/t (r) 812- (3-98) A table similar to Table 3.1 can be constructed for the tensor operator. This table is Table 3.3. The correct combination is picked out for the given term out of this table, then combined with the proper overlap from Table 3.2. A careful glance at Table 3.3 may reveal something slightly unsettling. Matrix elements that appear to be complex conjugates of each other, (11|5'129R|10) and (10|512§R|11), differ in the table by a factor of two. The source of this difference is twofold. First, the tensor operator in the form of (3.97) acts differently on the different K states of spin-one systems (projection along the symmetry axis in the body-fixed frame): Sm1in mlin, (3%) aflm)==—qmy (3mm Second, and more important, is the location of the quantization axis. In the bra, the quantization axis is a lab frame axis which we have chosen to lie along the symmetry axis of the nucleus. In the ket, we have an axis that rotates with the nucleus, and thus can have any of the three projections in the laboratory frame. As we can see, however, in eq.(3.99), the tensor operator acts differently among the various spin projections in the body-fixed frame, thus when bringing the body-fixed axis back in line with the lab axis, the order of rotation and operation by the tensor operator is important. That is, the tensor operator, in the form of eq.(3.97) does not commute with the rotation operator. Thus, the matrix elements are different, as they are not truly Hermitian conjugates of each other. Returning back to the tensor operator in 6Li, once again, for a given spatial geom- etry of the matrix element, there are many terms. These are summed over to give the 42 Table 3.3: Tabulated here are the results from the tensor Operator, 8'12, operates between any pair of triplet wave functions. All results are proportional to the radial dependence of the tensor force, Vt. The angle that appears in the chart is the rotation angle involved in the angular momentum projection process. Ill) |10) |1— 1) (11| 2cos2g -—\/23in19 2sin2-‘22 (10| —2\/2$in0 —4 c0319 2/25in0 (1— 1| 2sin2 1’ «231w 2cos2 2 '2 total result for the given spatial form-factor integral. These results are listed below: (Vt), = (aall/tlaa)0‘,’, cos2 3, (3.101) (V,)azdz = (ale,|da)20§0§ cos2 :22, (3.102) (Vt), = (ddlmddw: c0321; (3.103) (V,)aaad = — (aathlad)20f,0a 0032 g, (3.104) (14),... = —26?26. 1,3 (3.105) (l/t)ddda = — (ddll/tlda)263612 C082 2, (3.106) (V,)addd = — (aletldd)20f,012 cos2 3, (3.107) (14)),de = (aam|dd)626f cos2 2, (3.108) a 2 2 (14)),10010 = (dletlaa)0,2,02, cos2 322’ (3.109) (V,))C,dda = ad V, ad 20202 0052 19 3.110) a 12 '5. As usual, these are summed together and divided by the normalization. It is interest- ing to note that there is a contribution from the alpha particle (eq.(3.101)). Helium-6 also contains an alpha particle, but the tensor operator vanishes in its case. This is an effect of the overall spin structure of the 6—body wave function. In eq.(3.101), the alpha particle tensor matrix element is proportional to 0%,; in 6He, this term comes from the term between the two protons, but the tensor interaction vanishes for spin singlets (like the protons in the alpha particle), and therefore the terms in the alpha 43 particle vanish in the case of 6He. It should also be noted, that for a pure alpha particle with no external particles, the tensor operator vanishes in our model. This concludes the section on the alpha-deuteron configuration of 6Li. We now move on to the cigar configuration. 3.2.2 Cigar configuration The cigar configuration of 6Li is where the spatial extent of the deuteron is much larger, such that there is a particle on each side of the alpha particle, just as in the diagram for the cigar configuration of 6He (Figure 3.2), only one of the external particles is a proton, and one a neutron. The wave function is |‘II) = NaIaipaIail (—)1/2‘TaI+aI_,_|0), (3.111) where + and - indicate the right or left spatial position. As in the alpha-deuteron case, both external particles are created in the “up” spin-projection, but the sum is over isospin. The deuteron is an isosinglet, and we must sum over the projections of isospin in order to preserve the proper quantum numbers. This means, we still have the same cross terms that were in the cigar configuration of 6He. In the alpha- deuteron configuration, the normalization and all spin-independent operators had identical results in 6Li as in 6He, which is also the case for the cigar configuration. Therefore, we proceed with showing the results of the spin-orbit and tensor operators in the cigar configuration. The results for the spin-orbit operator follow below. There are additional overlaps to those shown in Table 3.2, but these are easily guessed at. Any term involving the cross term between the right and left side particles substitutes a 91: for 0,, which would 44 appear in Table 3.2. Now, the list of terms of the spin-orbit operator: (VLS>+2—2 (VLS)+——+ (VLS>02i2 (VLS)a2+— (VLs>a++a (VLS>a+—a (viS)++—a (ViS)+--a — sin 6(VL,)+2_26:, — sin 6(VL+)+__+6‘;, —2 sin 19(VL+)02i2 (20:9,, — 936$) 1 —2 sin 19(VL+),,2+_ (263,61 — 03,022) , 2 sin 19(VL+>a++a (901911 — 2601922) 1 2sin 19(VL+),,+_,, (00035 —- 2901922) 1 4 sin 0(VL+)++_G0?,0O,, 4 Sin 7.9 (VI/.1. > +_ -06?200 . 45 (3.112) (3.113) (3.114) (3.115) (3.116) (3.117) (3.113) ( 3.119) We now list the results for the tensor operator in the cigar configuration: (V00. 0212 (my, (V1)+2—2 (V1)+——+ (Vt)aaai (W103, i2¥a ai¥2 (Wm... (V0011; (V0044, (Vt)ai~=Fa (V0043— (Vt>+a—a 29 (aa|V,|aa)0‘,‘, cos2 5, 19 (a :t |V,| :1: (1)20202, cos2 2’ (a i 1141 1 0121391, c082 g (+ -IV1I— +)04 cos2 12, a 2 19 (+ — IV1|+ —)0: cos2 5, 0 _ (gall/,|a:l:)20a0f, cos2 5, — (iaIthaa) 200,02, cos2 2, —(;1 :F Imai)26§,6,,cos2 g, 19 —(a :F [14' 3F 221:)262912 COS2 E, 19 —262612 COS2 a, 19 —(a i: |V,| :1: i)262012COS2 2’ (a :t |V,|a:l:)20,2,02, cos2 3, (a $ |V}|a:l:)20i02, cos2 322’ (aathl + —)0C2,02, cos2 %, '0 (+ — IV,|aa)0,2,02, cos2 2' (3.120) (3.121) (3.122) (3.123) (3.124) (3.125) (3.126) (3.127) (3.128) (3.129) (3.130) (3.131) (3.132) (3.133) (3.134) As before, these are summed together and divided by the normalization to yield a complete expectation value. This concludes the section on the cigar configuration. Next, we move to the inter- ference term of 6Li. 46 3.2.3 Interference term As was the case in 6He, the overall wave function of 6Li is a combination of the alpha-deuteron configuration and the cigar configuration. The same procedure was followed for 6Li as in 6He (beginning with eq.(3.46)). The one and two-body operators described in the section on the interference term of 6He are the same in 6Li. As in the previous two sections, we only need to describe the results for the spin-orbit and tensor operators. The overlaps for the interference term are not the same as in the alpha-deuteron case which is listed in Table 3.2. It is not necessary, however, to construct a new table. Table 3.2 gives the angular dependence, and one can translate the alpha-deuteron overlaps into interference term overlaps with their definitions in eq.(3.50—3.55). Now, without further ado, the results of the spin-orbit Operator in the interference term: (VLS>a2d+ = ‘(VL+>a2d+ (29:896— — 93101960191110 Sin 191 (3-135) (Vlea2d— = —(VL+),,2d_ (292,896+ — Qingdagw) sin 191 (3-136) (VLS)daa-+- = daa+ (93894— — 29310194019010 sin '91 (3-137) (VL3)da,,_ = (VL+)daa_ (63,6,1+ — 20§a0da0a+) sin 6, (3.138) (VLS).,+.,_ = (VL+)a+d_0§a0da sin 6, (3.139) (VLS)a—d+ = 6Z..cos2§, (3.144) (V,)aaa+ = —(aa|V,|a’+)0,2,a0a+0aa cos2 %, (3.145) (14)aaa_ = —(nn|v,|n'—)62 (imam, cos2 9, (3.146) d“ 2 (V,)d_a+ = —(da|V,| + —)0f,a0da cos2 g, (3.147) (V,)d+da = —(dd|V,l + a’)0ga0a_ cos2 3, (3.148) (14),, = —(dd|V,| — a')0ga0a+cos2 .123, (3.149) dada = (ddll/tla’a’)0c2m0a+0a_ COS2 g, (3.151) 19 (V,)a+da = (adll/tlo/+)02 0da0a_ cos2 —, (3.152) 00 2 Vt a—da = ad 14 a'— 02 9,1000% COS2 22. (3.153 (10 2 With the spin-orbit and tensor interaction, one can then complete a calculation of the expectation value of the Hamiltonian for the interference term. One can then minimize 6Li in the same way as was done for 6He. Here now ends the chapter on the methods and formalism used in this study. The formalism was introduced through some simple examples, and then we applied the formalism to obtain some formal results in a very general sense for 6He and 6Li. The next chapter will go into further detail with specific choices for single-particle wave functions, and inter-particle interactions. 48 Chapter 4 Gaussian Approximation In the previous chapters, the general formalism was described. The formalism was then applied to the two nuclei of interest, 6He and 6Li. The many expressions for expectation values were left in terms of matrix elements of an operator of a certain type. In this chapter, a specific single-particle basis will be selected, and these matrix elements for all Operators will be derived. After the calculation of the matrix elements, numerical results will be given. 4. 1 Helium-6 The single particle wave function chosen is the Gaussian wave function. This is a function of the form f (cc) = Ae‘“(x““°)2. This wave function is the ground state wave function of the quantum harmonic oscillator, and thus is a suitable wave function for any system around a potential minimum. Also, Gaussians can be integrated an- alytically, which greatly simplifies the calculations. Our specific Gaussians are also real, which also reduces the number of terms needed to be calculated, as the forward and reverse matrix elements are nearly always the same. The asymptotic behavior of the Gaussian is not correct, as it falls off too fast. The true asymptotics should be exponential. It will be shown later that for many observables, this is not critical, 49 however. The Gaussian approximation in the alpha-dineutron configuration will now be discussed. 4.1.1 Alpha-dineutron configuration The alpha-dineutron configuration is pictured in Figure 3.1. The single-particle wave functions are: w... (r) = (33” exp (— M) = (aw- (r — d/3)2) , (4.1) (r + 2d/3)2) , (4.2) role NI? where 1110, refers to a constituent of the alpha particle and has parameters u, the oscillator length, and d, which describes the distance separating the two clusters; 45,; refers to a particle in the dineutron, with the same parameter (1 as in 1,00,, and w for its oscillator length. The coefficient of d is chosen so that the origin of the coordinate system is at the classical center-of-mass of the system. By looking again at the alpha-particle wave function, we can illustrate another nice property of the Gaussian: (pa (r) = N exp (J21 (r — d/3)2) = Ne-"(r2+d2/9)/2 :1,- (ur - d)" , (4.3) n. n which shows that every partial wave is wrapped up inside each Gaussian displaced with respect to the center-of-mass. We can select d to lie along the z-axis, which makes our wave functions (eqs[4.1,4.2]) look like we, (1:,y, z) = (gr/4 exp (—§ (.732 + y2 + :52 + d2 - 2zd/3)) (4.4) (1),) (33,31, 2) 2 (gr/4 exp (—; (x2 + y2 + 22 + d2 + 4zd/3)) . (4.5) We do not, however, work very often in the body-fixed frame. Instead, we find 50 the overlaps and matrix elements between a wave function and another wave function that has been rotated with respect to the first wave function, as it was outlined in section 2.4 in the previous chapter. We will rotate the wave functions around the y-axis using the matrix 00819 0 —sin19 a}: 0 1 0 . (4.6) sin 19 0 cos 19 We now write the rotated wave functions (denoted by the tilde): lb?) = (gym exp (—§ (:1:2 + y2 + 22 + d2 — 2d (:1: sint? + zcos 19) /3)) (4.7) Id) = (gr/4 exp (:3 (:1:2 + y2 + 22 + d2 + 4d (:1: sin19 + zcos 19) /3)). (4.8) With the rotated wave functions, all overlaps and matrix elements can be calculated. We will begin with the overlaps and normalization, and then proceed with the matrix elements. The overlaps are 0., a (aIci) = exp (312—2 (cos19 — 1)) (4.9) 6,) E (dld) = exp (260d2 (0031? — 1)) (4.10) 9.2 2 (old) = (dlé) = (2min) exp (mm—”@2— (5 + 4.203(9)) . (4.11) u+w (u+w) By taking the body-fixed frame limit (00819 = 1), one can see that the overlaps make sense. The alpha and dineutron overlaps become one, as the Gaussian wave functions are normalized, and the overlap between the two centers remains, but becomes one if d is zero and the oscillator lengths are equal for each cluster. The only reason the overlaps within the alpha and the dineutron are not one is because we rotate about the center-of-mass which does not coincide with either center. The symbol 012 is used for the overlap between the two centers (centers 1 and 2) first because it is equal to 51 its transpose, which then gives us more notational options when we come to different overlaps in later sections. The normalization (eq.(3.3)) is reproduced here for easy comparison with the overlaps: 1 N2 : . 93 (90.4% - 9%)2 When d goes to zero, all the overlaps tend towards one, which causes the denominator of the normalization to vanish. As mentioned before, this is because of the Pauli prin- ciple, and when d is zero, four s-wave neutrons are at the same point in space, which is forbidden by Fermi statistics. Note that this is exactly the case when the oscillator lengths are equal. If the oscillator lengths are different, the wavefunctions of the alpha particle and external neutrons are no longer completely identical. The denominator would still be very small (being one minus the ratio of twice the geometric mean of the two oscillator lengths divided by their sum), but not identically zero. The first expectation value shown here is the one-body particle density. For the purpose of presentation, this was done in the body-fixed frame with the wave functions in eq.(4.4-4.5). We use the general expression for a one-body operator (eq.(3.5-3.9)). In this case, the Operator is just I, the identity matrix, and we integrate over the y-coordinate. The results are shown in Figures 4.1, 4.2, and 4.3. The three figures all show the one—body particle density, but for different values of the parameter (.2. In Figure 4.1, they are equal, and the alpha particle (on the right side of the figure) looks bigger than the dineutron cluster. In Figure 4.2, the oscillator length of the dineutron is set to be larger than the alpha particle, which makes the dineutron more sharply peaked, as it is now more focused in space. The last figure, Figure 4.3, shows a more diffuse dineutron. For all three figures, the distance between the two centers was set to be 2.5 frn. There is nothing significant about this distance, it was chosen in order to keep a clear distinction between the two clusters while keeping them close enough so that their densities still overlap somewhat. 52 5—5 Figure 4.1: The expectation value of the one-body particle density in the intrinsic frame of 6He. It is pictured in the xz plane, with the clusters’ centers along the z— axis. In this figure, the oscillator lengths are equal in both clusters to 0.53 fin“2 and d is set equal to 2.5 fm. The alpha particle is centered in the positive 2 region, and the dineutron in the negative 2 region. This is the only expectation value calculated exclusively in the body-fixed frame. If the body-fixed frame expectation value of any other operator is desired, it can be easily obtained from the projected terms by taking c0819 = 1, and no longer integrating over the angles. We will begin our tour through the expectation values of various operators with those operators found in the Hamiltonian, which is: H: 2T.- —T..,,.+V,-,. (4.12) The first two terms in eq.(4.12) are kinetic energy terms. The first one is the sum of the one-body kinetic energy of the six particles, while the second one removes the energy associated with the motion of the center-of—mass of the particles. The last term is the interaction, the details of which will be covered in the subsections devoted to the potential energy. 53 5-5 Figure 4.2: The expectation value of the one-body particle density of “He with u=0.53 fin“2 and w=0.68 fm‘z. The parameter d is set equal to 2.5 frn. 4.1.2 Kinetic energy The kinetic energy calculation determines the amount of energy due to the motion of the particles present in the system. As just mentioned, there are two parts to this calculation, first the one-body kinetic energy of the six particles in the system, and then a correction to remove spurious motion of the center-of—mass. The one-body kinetic energy operator is: . h T = ‘275V2' (4.13) The kinetic energy is diagonal in spin and isospin. For this and all calculations in this work, the nucleons are treated as having the same mass, which for numerical calcu- lations is set equal to 939 MeV. The general form of the expectation value is found in eq.(3.4-3.9). The matrix elements which are summed together in the previously 54 Figure 4.3: The expectation value of the one-body particle density of 6He with u=0.53 frn‘2 and w=0.41 fm‘2. The parameter d is set equal to 2.5 fm. mentioned terms are listed below: (aITIa) = gen, [3 — W] , (4.14) (leld) = 30.1 [3 — W] , (4.15) 11w uwdz (leIa) = (O'T'd) : 1/ + w012 [3 — W (5 + 43)] . (4.16) For these terms, the prefactor, 52 / 2m, has been suppressed, and :1: E cos 19. Since the kinetic energy operator does not change the overlaps at all, the matrix elements are written in terms of the overlaps as well. Many operators have this property, and we will use this simplification in writing the matrix elements whenever possible. Most matrix elements also depend on cos 19, but the angular form (c0519) will be restored if necessary for clarity, or if it is different from cos 19. Additionally, all these matrix elements will now be calculated in the rotated system, so the tilde that was introduced to indicate a rotated wavefunction will now be omitted. All wave functions in the ket should be assumed to have been rotated with respect to the wave functions in the 55 <1>(MeV) d(frn) Figure 4.4: The results of the one-body kinetic energy calculation in the alpha- dineutron configuration of 6He. Thase results were obtained with u = 0.53 fm"2, the standard value for the alpha particle, and w = V. bra unless otherwise noted. With these matrix elements, the expectation value of the one-body kinetic energy can be completed. The one-body kinetic energy as a function of the inter-cluster distance parameter (1 is shown in Figure 4.4. Curvas are seen for the ground state, J = 0, up to the J = 4 state. At large values of d, the states are in the expected order for rotational states. That is, they go in consecutive numbers of J with spacing J (J + 1). Something different is observed at small values of d. Discussion of these features of Figure 4.4 is delayed because first we need to discuss the center-of—mass correction. We want to remove the energy associated with the motion-of—center of mass of the system: __ P2 _ (Zipz‘92 (Ticm — m — m, (4.17) where P and M are the momentum of the center-of-mass and the total mass of the system, respectively, which are re—written in terms of the single-particle quantities 56 on the right-hand side of the equation. Expanding on that, we get a separation into one-body (recoil effect) and two-body terms: #J' The first term is exactly the same as the one-body kinetic energy, while the sec- ond term is a two-body term. When we combine this result with the pure one-body calculation, we obtain the following result for the kinetic energy: 1 (I) = Z [M — 1>> — 2>1 , (4.19) where T (1) is the one-body kinetic energy and Tm is the two-body kinetic energy. We already gave the results for the one-body kinetic energy. Here are given the results of the momentum operator (ii/1V) in the alpha-dineutron configuration of 6He: (alpla) = 300% [sin 19)“: + (00819 — 1) i] (4.20) (dlpld) = —%9d§ [sin 19)? + (cos19 — 1) 2] (4.21) (alpld) = —612§%3 [2 sin 19;“: + (1 + 2 cos 19) 2] (4.22) uwd , . .. (dlpla) = 012m [srn 19x + (2 + cos 19) z] . (4.23) The last two matrix elements are not equal, despite being transposes of each other. This is because the momentum operator is a vector operator (the previous calculation was for the kinetic energy, a scalar). The magnitude of the matrix elements is not changed, but the angular dependence is different. In the absence of rotation (19 = 0), the matrix elements are equal. The matrix elements for the two-body terms can be found by taking the scalar products of any of these terms (including with themselves), and multiplying by a prefactor —h2 / 2m. These matrix elements accompany the over— lap expressions in eq.(3.11-3.20), which are summed up in order to find the overall 57 (MeV) d(fm) Figure 4.5: The results of the kinetic energy calculation in the alpha-deuteron con- figuration of 6He, with the center-of—mass energy removed. The results were obtained with the oscillator parameters 11 = w = 0.53 fm‘z. expectation value. The corrected kinetic energy shown in Figure 4.5 is qualitatively similar to the figure showing only the one—body contribution (Figure 4.4), with the energies reduced by around 12 MeV. At large distances, where the order of the states is rotational, the ground state energy corresponds to the sum of the kinetic energy of an alpha particle and dineutron. At small distances, the figure becomes more interesting, with degeneracies appearing. The energy increases because two of the neutrons are forced into higher orbits. When dis large, all four neutrons can remain in s-waves and J = K, but this is no longer the case when they come close to each other due to the Pauli principle. The lowest state is where J = 0 and J = 2 are degenerate. This is where both extra neutrons go to the p—shell and couple to J = O (the two p—orbital particles can couple to both the 6 = 0 and the II = 2 states to produce J = 0 at d = 0). The next state at small (1 is where J = 1 and J = 3 come together. These are negative parity states, and the only way this can be achieved at d = 0 is to have one particle 58 go to the p-shell, and another particle go to the next shell, the sd—shell. Finally, the J = 4 state stays high because 8 = 4 cannot couple to lower angular momenta. This is where the two particles are pushed into the sd—shell. The states at small of also appear to be equally-spaced, which is characteristic of the quantum harmonic oscillator. This is indeed the cause of the equally spaced levels, as the basis wave functions for the calculation are Gaussians. Because of these basis functions, at d = 0, the system is a spherical harmonic oscillator, and thus, has equally spaced levels in the kinetic energy. The kinetic energy results highlight an interesting feature of the formalism. The system most likely prefers to be at a d-value different from zero (maximum kinetic energy), but not too large, because the nuclear force is short-ranged. The minimum in energy will likely be between the pure s-wave system at large (1, and the oscillator limit at d = O, which results in a picture of s-waves and higher orbits. This is automatically handled by the formalism, and is an advantage over theories that would just place the external particles in p—waves, assuming that the s—waves are occupied by the neutrons in the alpha particle. This feature is a strong point of the formalism. 4.1.3 Interaction Potentials in nuclear physics constitute a large body of work in their own right. For structure studies, one can broadly divide them into two types: mean field potentials, and nucleon-nucleon potentials. The mean field potential averages out the interaction between the nucleons themselves into a one-body potential well, which the nucleons fill. This is an approximation that becomes more valid as the nucleus becomes larger, and since this work deals with light nuclei, we will use interactions that are of the nucleon-nucleon variety. Volkov interaction The first interaction we chose to use was the Volkov potential [45]. The Volkov po— tentials are a set of eight different phenomenological potentials that were designed to 59 fit features of the alpha particle (binding energy, charge radius). They were chosen here because our systems consist of alpha particles plus a few external particles, and the potentials are Gaussian in form. The framework of the Volkov potentials is V0035) = 2A: U(7”-ij) (1‘ m + mPS) 1 (4-24) i _ V6 _ — v7 -20— _ V8 _ —4o— . h l l l l 1 1m 0 1 2 3 4 5 6 mm Figure 4.6: Shown here are the eight sets of Volkov potentials as a function of the distance between the two nucleons. The plot that goes off of the graph, V4, is quite repulsive and finally intersects the ordinate at 331.6 MeV. operator will be addressed. 110:2 3/2 (aalVlaa) = V, (”012—”) exp [11d2 (2: — 1) /9] , (4.27) 2 3/2 (adIVIda) = V. (~fla—-—J V + w + uwa (11111012 (V + 4w) + (V — 2w)2)(1— 11:) + 181/1.1),,2 exp 18 (u + w + uwa2) u [ ,(4.28) 211a2 1111) 3” (aa|V[ad)+ — Va (m) (1—:r) (ua2(u —3w)+4V—8w)+9w (ua2+3) 18D(z/,w) exp [ 3:12] (4.29) WW) _ V m 3” 04+ — a D(w,1/) 4(1—$)(w2a2-2u+4w)+91/(wa2+3)mfl exp[ 18D(w,1/) w... ,(4.30) 61 where D(:z:, y) = 1012(1- + y) + 32: + y, (4.31) wa2 3/2 (ddIVldd) = Va (L002 + 2) exp [4wd2 (x — 1) /9] , (4.32) 41/112012 3” mud2 (5+4x) (dd|V|aa)+ = Va (m) exp I: 9(V+w) :[ , (4.33) 4 2 3/2 (dalVlda) = Va (fi) _uwA (5+4:1:)+2(1—-:r)(1/+2112)2 2 x exp [ 9(1/ + w) A d , (4.34) where A(p, q) 2 pa:2 + qa2 + 4. (4.35) These matrix elements are inserted into the corresponding expression in eqs.(3.11)- (3.20) in order to determine the expectation value of the potential. As listed above, however, these are only for the part of the potential that is proportional to (1 — m) (see eq.(4.24)). The Majorana exchange operator changes things, slightly. As seen in its definition, eq.(4.25), the Majorana exchange operator switches the spatial locations of a pair of particles, and leaves spin-isospin properties unchanged. One can say then immediately that none of the matrix elements in eq.(4.27)-eq.(4.34) with bra or ket at the same spatial location are affected by the operator. These leaves two of the seven terms, (adIVIda) and (da|V|ad). These two are changed into each A other. Thus, for the term of the potential proportional to mP ,j, the same set of overlap terms are used as before, except that (aleIda) is switched with (dalVlad). In other words, instead of eq.(3.12), we have 0(2) 02,. = (da V da 8039,, — 6920f , (4.36) a a 2 62 and (0(2))..dd... = (adIVIda) (2030f, — 0.3.0.1) (4.37) instead of eq.(3.19). The Majorana exchange operator can also be written as xx 1 Pi]- = —4 [1 + a,- - 03- + 1',- - 1',- +(a',--a'j)('r,--1'j)]. (4.38) This form was used to confirm results obtained with the original formula. A plot of the potential results is shown in Figure 4.7. At large values of d, where all the curves come together, is the sum of the potential energies of an alpha particle and dineutron. As d becomes smaller, the different levels appear. The lowest levels are once again J = 0 and J = 2, though the potential breaks the degeneracy. The potential also breaks the degeneracy of J = 1 and J = 3. Interestingly, J = 4 comes in between J = 1 and J = 3. This is due to the Ma jorana exchange operator’s preference for even waves. When the exchange parameter is set equal to zero (see Figure 4.8), the order of states at (1 equal zero is J = 0,2,1,3,4, but for Figure 4.7, m is 0.6 (the standard setting for the Volkov potentials), which makes J = 4 more attractive compared to the odd waves (however, this may not be the case, see 4.2.2). Another comparison that can be made is the effect of different sets of Volkov parameters. The plot in Figure 4.7 is with V1, which has a fairly hard core (V(r) =+60 MeV at d = 0). In Figure 4.9, the expectation value of V2 is shown. Volkov V2 is a soft-core potential, with a value of +0.5 MeV at d = 0. As one can see in the figure, qualitatively, there is not a great change by changing the parameter set. Quantitatively, the V2 potential expectation value is deeper by three MeV at d = 0. Three MeV can be a lot in these loosely bound nuclei, however, the minimum in binding energy is usually far from d=0, and there is less difference between the potentials the higher one goes in d. 63 _9O 1.; I I I I I I I I I I T l f I I I I I I C / : I — 1:0 j -95 f a F _ 1:1 , >- -l -100 " :1 § [ — 1:2 ~ g 1 A -105 - — 3 —- J=3 : —110 — —— 1:4 1 —-115 " / 4 :1/1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1: 0 2 4 6 8 10 d(fm) Figure 4.7: The results of the expectation value calculation of the Volkov potentials (V1) in the alpha-dineutron configuration of 6He. The results were obtained with both oscillator parameters equal to 0.53 fm—z, and the Majorana exchange operator set to 0.6. 64 _90 J I I f l I I I I I I I l I I I I I I I ]_, _100 ; — 1:0 : -110} — 1:1 1 i; -120 ': — 1:2 1 a : “ A ; a 3 —l30: _ 1:3 : -14o ’— J t — 1:4 : I i -150 j j —l60:- 5 r 1 l l l l 1_ l o 2 4 6 8 10 d(fm) Figure 4.8: The expectation value of the Volkov V1 potential in the alpha-dineutron configuration of 6He as a function of d. In this plot, the Majorana exchange parameter was set equal to zero, but all other parameters are the same as in Figure 4.7. 1.] I l r I I T l 1 l fir U r fl I I 41 Y I l J _ 1. _/ _ —95 P — 1:0 : I 1 —1oo_— _ 1:1 4 g : —— 1:2 1 g —105 — — A : - 3 _ — 1:3 -110: - : — J= : -115 1 ~ p . -120 ’- 1 1 1 l 1 1 1 l 1 1 1 I 1 l 1 l-l 0 2 4 6 8 10 d(fm) Figure 4.9: The expectation value of the Volkov V2 (soft core) potential in the alpha- dineutron configuration of 6He. In this plot, the oscillator parameters are both equal to 0.53 fm‘2, and the Majorana exchange parameter is set equal to 0.6. 65 Minnesota potential After the Volkov potentials, the next interaction used was the Minnesota potential (first appeared in [46]; the parameters used in present work are borrowed from [47]). In contrast to the Volkov potentials, the Minnesota potential was designed to fit the n — p and p — p s-wave scattering parameters (the scattering length and effective range). They are also expressed as Gaussians, which makes computations simpler. The form of the potential is: 14,- : [vp + 1/2 (1 + P5914 + 1/2 (1 — P;)1(,][u/2 + 1/2 (2 —- 11) Pf], (4.39) where VR, Vt and V; are the Gaussian form factors for the repulsive, triplet, and singlet potentials, respectively, Pg; is the spin-exchange operator (exchanges the spins of particles 1' and j, giving +1 in triplet states and -1 in singlet states), Pi; is the coordinate exchange operator (Majorana exchange operator), and u is the exchange parameter which should be close to one. Figure 4.10 shows the Minnesota potential in the singlet and triplet channel. Clearly, the triplet channel is more attractive, which makes sense as this is the deuteron-like channel, and the deuteron is the only bound two-nucleon system. Since the form of the Minnesota potential is Gaussian, no new matrix elements need to be listed here. One merely finds the appropriate geometrical term in the list of matrix elements eq.(4.27)-eq.(4.34), and changes the Volkov parameters to Minnesota parameters. However, since the Minnesota potentials explicitly depend on spin, the list of terms in eq.(3.11)-eq.(3.20) need to be re-written in terms of a singlet part and 66 60 r + 1 1 T . . . . 1+ f. r 1 . . 1 r r v . 1 40 - ~ ~ 1 ~ 1 20 - _ —-—- S=0 E _ a _ 3 0 > .. _ —— S=1 _20 _ -40 .. . 1 . 4 5 r(fm) Figure 4.10: This plot shows the singlet (T21) and triplet (T = 0) channels for the Minnesota potential. The singlet potential has a strong repulsive core, with a value of +109 MeV at r = 0. The triplet channel has a milder repulsion and a much deeper attraction. a triplet part. The singlet contribution is (14)., : (aalmaa)(30363+0;‘2-3600f20d), (4.40) (V3)azd2 = (doll/glad) (36304—033), (4.41) (V slaaad=fV 8)daaa = (aa|V,,|ad) (29f29a"6912191294)1 (4-42) (V s=lddda (Valaddd = —4020121 (4-43) (V )adad=331... = (aallélddflfifim (4-44) (Valadda = (dallélda>29§9fe1 (4.45) (V0.1 = (dles92‘1ldd) (4-46) '67 The triplet part is .. = (aelmea) (30:01— 30.01.91). (4.47) 011.212 = (claimed) (5636.1 — 563.03), (4.48) (thaaad = (thdaaa = (aalelad) (6913260: _ 69391290 1 (449) (l/tladda = (doll/Ada) (29:0f2 " 292.911) - (450) One can see that if one sums the triplet and singlet terms together for each geomet- rical term, the list is the same as for the Volkov potentials. The Majorana exchange operator switches the same terms in the Minnesota potential as it did in the Volkov potentials. Terms of the type 02d? are switched with adda in both the singlet and triplet cases. Figure 4.11 shows the expectation value of the Minnesota potential as a function of d. The plot is similar to the plot of the Volkov potentials, but there are some differences. The order of states is the same as in the m = 0.6 Volkov plot, but in the case of the Minnesota potential, the J = 4 state is around halfway between J = 1 and J = 3, rather than being very close to J = 1 as in the Volkov plot. This could be due to the fact that the exchange term accounts for 50% of the Minnesota potential compared to 60% for the Volkov potential. Another difference is that the magnitude of the potential seems to fall slightly more rapidly in the Minnesota potential than the Volkov potential. The large d limit once again corresponds to the sum of the potential energies of an alpha particle and dineutron. Gogny interaction The other interaction examined was the 1970 version of the Gogny interaction [48]. This interaction has four types of contributions: V(r) : mm + was” + VL3(1~)L- s + mania, (4.51) 68 _l l I l ' l I l I F -85 E // j -91; -: E — 1:1 -95 3 3 E : —— 1:2 7: -100:— j 3 : _ 1:3 : —105 _— j z — J=4 2 —110 j j _115 L _ 1 1 4L 1 1 J l l 1 1 l 0 2 4 6 8 10 d(fm) Figure 4.11: Plotted here is the expectation value of the Minnesota potential in the alpha-dineutron configuration of 6He. The oscillator parameters are both equal to 0.53 fm‘2, and the exchange parameter, u, is set equal to one. where Vw(r) is the Wigner or central interaction, VT('r) is the tensor term, VLS(1') is the spin—orbit term, and VLL is a second-order momentum term that we did not consider. The radial dependence was expressed in terms of Gaussian functions, with separate terms for different spin and isospin combinations. This interaction was chosen in order to fit two-nucleon scattering properties and some properties of the deuteron, as well as many properties of heavy spherical nuclei ranging from 16O to 208Pb. We found, however, that this Gogny interaction did not seem suitable for our systems. Its binding energy for our alpha particle was 13 MeV, less than half of the experimental value. This is not too shocking, since the Gogny interaction is essentially an effective interaction for medium and heavy nuclei. 69 4. 1.4 Total energy While not a new operator per se, the next stop on our tour through 6He is the total energy, which is the sum of the previous two operators. The total energy for the alpha-dineutron configuration calculated with the Volkov V1 potential is shown in Figure 4.12. The large (1 limit is the sum of the total energy of an alpha particle and a dineutron. Bound behavior is seen only for J = 0 and J = 2. Also, the level of the minimum of the J = 2 curve is nearly equal to its asymptotic energy, which makes it more like a resonance. It’s also interesting to note that the minimum in energy occurs around d = 3.5 fm. If one looks back at Figure 4.5, one sees that at this value of d, the particles are still mostly in s—waves, with only a small contribution of p—waves. The large increase in kinetic energy caused by accessing higher angular momenta overcomes the increased attraction gained by moving the nucleons closer to each other, causing the external particles to sit further away from the alpha center in this model. The total energy calculated with the Minnesota potential is pictured in Figure 4.13. Like with the Volkov potential, binding only occurs for J = 0 and J = 2. The minimum for J = 2 is deeper this time, but the overall energy is higher. This is due to the fact that 6He is dominated by singlet spin pairs, which in the Minnesota potentials are much less attractive than triplet pairs. Another similarity with the previous plot is that the minimum in energy is located at a value of d where the external particles are still mostly in s—waves, indicating that also for the Minnesota potentials, avoiding the increase in kinetic energy at small values of d is more advantageous than the increase in binding from the potential. This ends the discussion on the operators that make up the Hamiltonian for 6He. We will now move on to some other calculations performed in the alpha-dineutron configuration of 6He. 70 —- J=0 . — J=1 i 10 r r A L -1 % C — 1:2 g 4 A 0 LL] _ V .. -10; -20: f 4 1 l l 1 l 1 J l 1 0 2 4 6 8 10 d(fm) Figure 4.12: This plot shows the total energy of the alpha-dineutron configuration of 6He as a function of (1 calculated with the Volkov V1 potential. The oscillator parameters are both set equal to 0.53 frn‘2 and the Majorana exchange parameter is equal to 0.6. 71 r-T f l l I l l _‘ 30 — : p .1 ~ — 1:0 [— 20 r '- + —— le « f g 10 1: -— J=2 L a >- r A T ”13 i —— 1:3 0 — J=4 : _10 l L l l l l l l 0 2 4 6 8 10 d(fm) Figure 4.13: This plot displays the total energy of the alpha-dineutron configuration of the 6He calculated with the Minnesota potential. The oscillator parameters are both set equal to 0.53 fm‘2 fm and the exchange parameter u is set equal to one. 4.1.5 Mean square radius The mean square radius is a basic property of a nucleus reflecting its spatial extent. Usually, there are two kinds of radius: the charge radius and matter radius. The charge radius reflects the size of the proton distribution in the nucleus, and is the more easily measured of the two, whereas the matter radius is the total size of the nucleus. In some nuclei, these quantities may be the same, or very nearly the same, but in 6He we expect a significant difference between them, because of its extended neutrons. Since the mean square radius is the size of the system, the location of the ori- gin is very important. The radius must be properly referenced in order to supply a meaningful answer. The single-particle wave functions were all re-written in terms of differences in order to guarantee this. Before going into these details, however, the Operator itself has to be properly defined. We start with the matter radius, which is 72 the radius of all particles in the nucleus, referenced to its center-Of-mass: (13,): A120,. (4.52) where RC", is the coordinate of the center of mass, which is defined as 1 A 1 A Rem = MZmiri = EZI‘i, (4.53) where the far right section is the simplifying case where the mass of all constituent particles is the same. With this definition, we can rewrite eq.(4.52) as A(r,2,,)= :1"? ——Zr,z::rj+ AZrinj. (4.54) In the later two sums, there are terms where 3 = j, which are like the one-body terms in the first sum, therefore we can simplify this expression into two terms, one-body and two-body: A(7',2n)= A 7121"? ——Zri- rj. (4.55) i = (alrzld) = V W612 V2 (21 —ZCM)2+w2 (Zz—ZCM)2+2VW (21 —ZCM) (Z2—ZCM) $0 (u+w)2 12' (4.75) The list of terms is now different, as it depends on whether it involves neutrons or protons. The proton term is: (73).. = (al1‘2la)200, (90,0) — 0(2)? (4.76) 76 The neutron one-body terms are: (r2)a = (alrzlaflflifld (600d — 0%)) , (4.77) (7‘2)d = (dlrzldflflg (00,60) — 6h) , (4.78) (72)” = (OIT2ld)40:012 (0f2 '— 006d) , (4.79) where in the last term we have simplified things slightly because the matrix element is equal to its transpose. The two-body matrix elements are: (21 - ZCM)2 (1 + 33) (aalrl ~r2|aa) = 2 6:, (4.80) 2 (ddlr1.r2|dd) = (Z? Z02) (1”)03, (4.81) (0er1 .rzlda) = (‘7‘ ‘ 20“) (Z2; ZCM) (1 + ”609d, (4.82) uz—Z 2+cuz—Z z—Z 1+:c (aalrl-r2|ad) = [ ( 1 CM) (210/ +033” 2 CM)“ )60012, (4.83) wz—Z, 2+uz—Z z—Z 1+2: (ddlrl -r2|dd) = l < 2 CM) (211143)” 2 CM)“ >004)”, (4.84) 2 _ 2 2 _ 2 _ _ (aalrl -r2|dd) = (1 (z1 ZCM) +w (zz ZCM) +22Vw(21 ZCM)(z2 Z0M)$0f2, (u+w) (4.85) (adll‘l . rzlad) = 2””(z1‘ZCMW2—ZCMH(”2(31-ZCM)2+w2(Z2-ZCM)2)392 . (1/+(.u)2 12 (4.86) We have three lists of terms to present: proton-proton, neutron-neutron, and neutron- proton. First, we list the proton-proton term: (1'1 ' 1'2)a = (aalrl ' rzlaa) (000d — 6¥2)2 . (4.87) 77 The neutron-neutron terms are: (r1 -r3)a = (aalrl -r2|aa)9203, (4.88) (r1 -r2)d = (dd|r1 -r2|dd)0:, (4.89) (r1 ' r2)aaad = —(oza|r1 -r2|ad)4626d612, (4-90) (r1 490242 = (adIrl -r2|da)20§ (2600., — 0%,), (4.91) (r1 -r2)ddda = —(dd|r1 -r2|da)40:012, (4.92) (r1 -r2)adad = (aalrl -r2|dd)26:0¥2, (4.93) (r1 - rfiadda = (adlrl - rzlad)262 (26%;, — 00,64) . (4.94) The neutron-proton terms are: (r1 - r2)a = (aalrl - rglaa)4 (0:03 — £20004) , (4.95) (r1 - rfiaaad = (aa|r1 - rglad)800012 ((9?2 — 60.90)) , (4.96) (1‘1 ' r2)02d2 = (adlrl ' rglda)46: (000d — 0%) . (4.97) All the terms required to calculate the matter radius and charge radius of the alpha-dineutron of 6He have been obtained. We calculate this radius for the set of parameters given by a minimization of the expectation value of the Hamiltonian with respect to the three parameters of the calculation ((1, V, w), as there is not a minimum principle for the radii. After a number is obtained from the formulae above, there is one last adjustment that needs to be made. The number obtained from the formulae in this section is for point-like nucleons. In order to obtain a number that the fact that the individual nucleons have a finite size, we use the following formula for the charge radius(see, for example [49], but without the Darwin-Foldy contribution): (43,) = <43) + <83) + gala). (4.98) 78 where (r3) is the number calculated using the methods presented in this section, (R3) is the proton charge radius which is m =2 0.895 fm [50] and (R2,) is the charge radius of the neutron, which is (R3,) = —0.120 fm2 [51]. For the matter radius, we use the same formula, but neglect the negative neutron contribution. This concludes the section on the various radii calculated for the alpha-dineutron configuration of 6He. We move on to the cigar configuration. 4.2 Cigar configuration The cigar configuration is pictured in Figure 3.2. The single-particle wave functions are: z/JQ(r) 2 (gr/4 exp (—1/7‘2/2) , (4.99) 440-) = (33” exp [—§ (r =1: (1)2] , (4.100) where $0, is a constituent of the alpha particle with oscillator parameter V, and 45,): is the neutron on the right (+) or left (-) of the alpha particle, with oscillator parameter 4.). The parameter d reflects the distance between the alpha particle and the external neutrons. As before, (1 is taken to lie along the z-axis. Also as in the previous section, we show one—particle density plots of this configuration. In Figure 4.14, we see the density plot with oscillator parameters equal to each other. Figure 4.15 shows the same quantity but with 0) larger than V, and Figure 4.16 shows the one-particle density with w smaller than u. The distance parameter d, was once again set to 2.5 fm. This value was chosen for demonstration purposes only. All other expectation values, of course, are calculated in the rotated system. The 79 Figure 4.14: Shown here is the one-particle density of the cigar configuration. It is pictured in the 2:2 plane. In this picture, the oscillator lengths are both equal to 0.53 fm‘2 and d (the distance between the alpha particle and an external particle) is set to 2.5 fm. The alpha particle is located at the origin, and is flanked by both external neutrons. 5—5 Figure 4.15: The one-particle density of the cigar configuration with unequal oscillator parameters. The alpha oscillator parameter, :1, is set equal to 0.53 fin”, and the external neutron parameter, w, is set equal to 0.68 fm‘z. 80 Figure 4.16: The one—particle density of the cigar configuration with 1/=0.53 frn‘2 and w=0.41 fm‘2. rotated single particle wave functions are: ¢a(:r,y, z) = (gr/4 exp [—u(:52 + y2 + 22)/2] , (4.101) 01 3“ —w 2 - 2 2 ¢i(x,y,z)=(;) exp 7(1: :F2xdsm19+y +2 ¥szcos19+d2) . (4.102) The overlaps of the system are: 0.. = (046.) = 1, (4.103) 0,, = (#1:) = exp [1.1112(4- — 1) /2] , (4.104) 04 = (in?) = exp [—wd2 (:1.- + 1) /2] , (4.105) 012 = (:tlé = (0458) = (121‘:pr exp (mi/+3] , (4.106) where :1: E cos '0. Once again, when d is zero, all the overlaps are equal to one (if u = 0)). Which causes the norm, eq.(3.23) (reproduced below), to have zero in its 81 denominator: 1 29821 [93 (972; + 04) + 20112 — 29212200: (on + 021:”. As discussed before, this is a consequence of trying to put four identical fermions at the same spatial location. We move on to calculating the matrix elements between rotated wave functions, beginning with the kinetic energy. As before, tildes in kets will no longer be indicated, as all wave functions in the ket are understood to be rotated with respect to the wave functions in the bra. 4.2. 1 Kinetic energy We start with expectation values of the operators in the Hamiltonian, eq.(4.12). The first of these Operators deal with the kinetic energy. The first being the sum of the one-body kinetic energies, and the second correcting for the center-of-mass motion. The matrix elements of the one-body kinetic energy operator (eq.(4.13)) are: (alTIa) = 37" (4.107) (illet) = 1.19.. [3+E—‘7m—d2] (4.108) (21:01:12) = wag—W] (4.109) (:1:|T|a)2(a|T|:l:) = ”fwol2(3—V':"wdz) (4.110) As in the previous configuration, the factor 712 / 2m has been suppressed. These matrix elements are plugged into eqs.(3.25)-(3.29) to obtain the one-body contribution to the kinetic energy. The center-of-mass correction to kinetic energy was discussed at length in the section on the alpha-dineutron configuration (following eq.(4.17)). All that remains for us to do here is to enumerate the terms pertaining to the cigar configuration. We 82 list here the matrix elements of the momentum operator: (alpla) = 0, (4.111) (:l:|p|:l:) = i%0ng[sin19x+(cosz9—1)2], (4.112) (44pm = $$0ig[sini9x+(l+cosd)2], (4.113) (ilpla) = ¢%30122, (4.114) (alpli) = iz—(‘lj—ifi—Jolg(sin8x+cos02). (4.115) The scalar product of these terms is taken and then inserted into the correct term in the list of two-body terms found in eqs.(3.30)-(3.44). Since the first matrix element, eq.(4.111), vanishes, many of the terms vanish. In Figure 4.17, we show the total kinetic energy contribution to the Hamiltonian of the cigar configuration. Like in the result for the alpha-dineutron configuration (Figure 4.5), the small (1 limit shows an increase in the kinetic energy (because two neutrons are forced into higher orbits) and degeneracy between the J = 0 and J = 2 levels. The large (1 limit corresponds to the kinetic energy of the alpha particle and two free neutrons. An interesting difference between Figure 4.17 and Figure 4.5 is that in the cigar configuration, no odd waves appear. This is due to the higher symmetry of the cigar configuration. For example, if one considers only the two external neutrons (which is reasonable since the alpha particle has all angular momentum quantum numbers equal to zero), we must obey the rule for two nucleons, that (— 1)L+S+T = — 1, or that L+S +T must be odd. Since the two particles are neutrons, that means T = 1, and by construction, they are in the singlet spin state (S = 0), thus the only way to fulfill the condition is to have L be even. Another check on the calculation, is that at d = O, we have six particles at the same point, and the results should be independent of how they got there. In other words, the values at d = 0 should be the same for d = 0 for the cigar configuration and 83 130: 120: 110" 9 : é) _ 7:100 l— . V . 90_ 80: 70- 0 2 4 6 8 10 d(fm) Figure 4.17: The total kinetic energy in the cigar configuration of 6He is shown here. The contribution from the center-of-mass motion has been removed. This plot was obtained with both oscillator parameters equal to 0.53 fm’2. the alpha-dineutron configuration. Upon examining the figures, one can see that this is true. The J = 0 and J = 2 degenerate level comes in at just under 108 MeV (107.6 MeV to be precise), and the J = 4 level is at 133 MeV in both pictures. One might make the observation, that the J = 1 and J = 3 levels certainly do not correspond in both configurations, but this is not a problem. In order for the cigar configuration to produce J = 1 and J = 3 levels at d = 0, there would have to be a sudden break in the symmetry at that value of (1. Similarly, for the inverse to be the case, those levels would have to disappear for the alpha-dineutron configuration just at d = 0, and this sudden creation or destruction of symmetry is not a part of this theory. 4.2.2 Interaction We turn now to the two-body interactions used in the cigar configuration. The inter- actions used for 6He were discussed at length in section 4.1.3, so here we can move on directly to the matrix elements. As before, if a matrix element is equal to its 84 transpose, it will be listed with a subscript- “+” following the matrix element. Also, some matrix elements come in pairs in which the only difference is the sign of the angle term. We will exploit this fact for brevity, and indicate it by the expression B = A(:r: ——> -:I:). The matrix elements for the Volkov potentials are: V02 3/2 (aalVlaa) —Va (W) , (4.116) 2 3/2 2 _ (aiIVlzlza) =Va :woz exp _w(ua +1)(1 x)+2uwd2 , uwa +u+w 2(uwa2+u+w) (4.117) (aiIVI 42a) =(a:l:|V| ia)(a:—+ —£L‘), (4.118) (aalVlazt)+ =Va (%%f)fl)3/2 [-— (”a22l:(3,)ul:)0d2] , (4.119) <4 41% a). =v. (3377“)3” x exp [_ (012012 + 1111202) (3136:), :) 1101a2 + 31/ + 8wwd2] , (4120) (¥ i M ¥ a>+ =<$ i M i a)+(:v -> —:v), (4-121) where D(a, b) is defined in eq.(4.31). Continuing: (.ua2 3” waz(1-a:)+4 2 (d: IF |V| 2F i) —Va (01022 + 2) exp [— 01012 + 2 wd] , (4.122) (is: M tat) =(:v—* -:v), (4.123) 41/012072 3” 110:2 +4 d: V =Va - . d2 , < ZFI |aa)+ ((11 + (1))(z/a2 + an2 + 4)) exp [ I/a2 +1.1Ja2 +4“) ] (4.124) 85 (ialVl :l: a) =Va 41/111012 3/2 ((11 + w) (11012 +02012 + 4)) 2w(1—x)+u(ua2+wa2+4)l 2 X exp [— (u + w) (ua2 + 112012 + 4) “dd ] ’ (4125) (ialVl 2): a) =(:l:a|V| :1: a)(x -—> -:1:). (4.126) These matrix elements are put alongside the appropriate overlap expression from eq.(3.30)-(3.44) and summed together. This is for the terms pr0portional to 1 — m in the Volkov potential, or to u/ 2 in the Minnesota potential. The Majorana exchange operator shuffles many terms among themselves in the cigar configuration. Eq. (4.117 ) is switched with eq. (4.125) and eq.(4.118) is switched with eq.(4.126) when one inserts matrix elements into the list starting with eq.(3.30). The Majorana exchange oper- ator also switches eq.(4.122) with eq.(4.123) and eq.(4.120) with eq.(4.121), but the overlap terms that accompany thase pairs of matrix elements are the same (eqs.(3.43) with(3.44) and (3.35)-(3.38), respectively), so the Majorana operator has no net effect in these instances. The expectation value of the Volkov potential is shown in Figure 4.18 for V1 and in Figure 4.19 for V2. They are similar to the pictures for the alpha-dineutron con- figuration, save for the absence of odd waves. The J = 4 level is bound much more weakly, and J = 0 and J = 2 are bound much more tightly, with a very small separa- tion between the two of them. The potential energy of the cigar configuration seems to fall off faster than in the alpha-dineutron configuration, because as d increases, not only are the external neutrons moving away from the alpha particle, they are also moving away from each other, which is not the case in the other configuration. The large d limit here contains only the potential energy within the alpha particle, as the two neutrons do not attract each other at large distances. As before, there is not much difference between V1 and V2, except near d = 0. As with the kinetic energy, we can compare the potentials in both configurations at d = 0. The first two levels look to be at the same value, and they are to many 86 '80? — J=O j -90: __ 1:2 - 9 D s 1 . A - 3 10°? — 1:4 . -110 / ’1 1 1 1 l l l l l 0 2 4 6 8 10 d(fm) Figure 4.18: Plotted here is the expectation value as a function of d of the Volkov V1 interaction in the cigar configuration of 6He. In this plot, the oscillator lengths were both set equal to 0.53 fm‘z, and the Majorana exchange parameter was set equal to 0.6. 1r ' 1 1 1 1 1 1 1 1 T 1 1 I 1 1 I 1 r 1 1.1 —80 _ _ ' — 1:0 « -90 _ _ ,4 - —— 1:2 : > O 2 - . 71’ -100 ~ ~ 3 ~ _____ 1:4 ‘ —110 — _ _120 L1 1 1 1 1 1 1 1 1 1 1 1 1 1 , , 1 , , , 1_ 0 2 4 6 8 10 d(fm) Figure 4.19: Displayed here is the expectation value of the Volkov V2 potential for three different levels of the cigar configuration of 6He. These results were produced with both oscillator parameters equal to 0.53 fm‘2 and the Majorana exchange pa- rameter equal to 0.6. 87 digits (-117.9 and -116.78 MeV, respectively, for the V1 expectation value), but the J = 4 levels don’t appear to agree (-104.7 MeV for the alpha—dineutron configuration and -97.8 MeV for the cigar). They agree better with the Majorana part of the interaction turned off (-138.1 MeV for the alpha-dineutron case and -l39.4 for the cigar configuration), but the lack of agreement is puzzling. Many checks were performed, including calculating the Majorana effects with the operator in its alternate form, eq.(4.38), but this did not change the results. This behavior of the highly excited state is not yet fully understood. We now turn to the Minnesota potential. The general form of the potential was written in eq.(4.39). The form factors are all Gaussian: Vk(7‘,-j) = Vk exp (—n,-r,-2j) . (4.127) One can use the matrix elements listed in eq.(4.116)-eq.(4.126) with the substitution n,- = 1/012 and the appropriate magnitude instead of Va. We do, however, have to re-write the general cigar two-body matrix elements (eqs.(3.30)—(3.44)) in terms of singlet and triplet spin terms. The terms listed in eqs.(3.30)-(3.44) can be used for 88 the repulsive term in the Minnesota potential. The singlet terms are: (V106 (V0642 (V0642 (8).... = (Wm... (V0944 = (Waxy (V0444: = 40/064112 (141)640— = (Vs)+a—a aiia ai$a 041).. (V04 There triplet terms are: (V011 = (Waznz = (V0642 = (W)aaan=(Vt)naaa = aiia = ai2Fa = = (44114144 [263, (03, + 01.) + 263‘2 — 26?,60 (0,, + 61)] ,(4.128) _—_ (a :1: |V,| 4: a) (30391. — 031912) 1 = (a :1 MI 4 a) (363.91 — 0:612) 1 = (flair/3014;) [26f260 — 303,612 (6.1 + 64)] 1 = —2(¥ i M! i (1)9302, = (q: :1: |V,,| 2F (1)939121 = (aaIVsl + 4291219121 i IVslai)29¢19121 (01 = (a ¥ |V,,[ad:)200012, (i $ |V11| =1: 209311 < :1: =F|V11I i #63,. (4.129) (4.130) (4.131) (4.132) (4.133) (4.134) (4.135) (4.136) (4.137) (4.138) (aaIthaa) [20: (03, + 6i) — 20(260 (0n + 643)] ,(4.139) (a 4: MI is a) (592.91 — 503.011) . (a :t lel 3F 0) (5931921 “ 59:92) 1 (aall/tai) [6613200 — 363,612 (6., + 61)] , (a i IV1Ia=t> (2001912 - 29.3191») 1 (a q: Ithai) (2001012 — 2620i) . (4.140) (4.141) (4.142) (4.143) (4.144) The exchange operator in eq.(4.39) switches the same terms as in the Volkov poten- tials. The expectation value of the Minnesota potential as a function of d is shown 89 T I f r l l W I j .. =0 ., ~80 _ J : > _ _ 0 E A .1 3 -100 — —— 1:4 - _110 / fl 1 a 1 l 1 1 1 l 1 1 1 l 1 1 m l 1 1 m l 0 2 4 6 8 10 d(fm) Figure 4.20: Plotted here is the expectation value of the Minnesota potential in the cigar configuration of 6He. These results were obtained with the oscillator lengths both equal to 0.53 fm"2, and the exchange parameter, 11, equal to one. in Figure 4.20. The plot is similar to the plot in the alpha-dineutron configuration (Figure 4.11), except that the odd levels are missing. The first two levels, J = 0 and J = 2, are very close together, and J = 4 is much less bound than the first two levels. At (1 = 0, as before we see agreement with the alpha-dineutron configuration in the values of the potential for the first two levels (-116.7 MeV and -117.0 MeV), but the J = 4 levels do not coincide (-100.8 MeV in the alpha-dineutron configuration and -99.3 MeV for the cigar configuration), but they are closer than with the Volkov potential. 4.2.3 Total energy We now turn to the sum of the results of the kinetic energy and potential energy, which is the total energy of the cigar configuration. In Figure 4.21, we show the total energy as a function of d obtained with the Volkov V1 potential. Here we see again bound behavior for only J = 0 and J = 2, but at a much higher energy than for the alpha- 90 301 — 1:0 4 20 — 5 . __ 1:2 - 101 j — 1:4 ‘10;W\ — O 2 4 6 8 10 d(fm) (MeV) Figure 4.21: Shown here is the total energy of the cigar configuration, with the Volkov V1 potential used as the interaction. For this plot, the oscillator parameters were both equal to 0.53 fm“2, and the Majorana exchange parameter is 0.6. dineutron configuration (Figure 4.12). The J = 2 excited state appears to be a bound excited state rather than a resonance, which was the case in the other configuration. The large (1 energy corresponds to the total energy of an alpha particle and two free neutrons, which is why it is different than the large (1 limit of the alpha-dineutron configuration. At small d, the first two levels correspond in both configurations, but J = 4 does not, for reasons noted in the section on the potential. Figure 4.22 shows the total energy calculated with the Volkov V2 interaction. The picture is very similar to the previous plot, except the energies are slightly lower. Figure 4.23 shows the same quantity calculated with the Minnesota potential. This spectrum is similar to the other two. It is perhaps the highest in energy, and the first two states are closer to each other than with the Volkov potentials. It also has the same (1 = 0 behavior as the Volkov potentials: agreement for J = 0,2 and a mismatch for J = 4. This concludes the discussion of the operators that make up the Hamiltonian. We 91 1:0 1:2 . 9 : 0 g _ 9 - V 1:4 10 d(fm) Figure 4.22: Plotted here is the total energy of the cigar configuration, with the Volkov V2 potential used as the interaction. For this plot, the oscillator parameters were both equal to 0.53 fm‘z, and the Majorana exchange parameter is 0.6. (MeV) )- )- )- d(fm) Figure 4.23: This plot shows the total energy spectrum of the cigar configuration obtained with the Minnesota potential. In this plot, the oscillator parameters are both equal to 0.53 fm‘2, and the exchange parameter, u, is equal to one. 92 now move on to the discussion of other calculations in the cigar configuration. 4.2.4 Mean square radius In the previous configuration, we had to re—write the coordinates in order to carefully determine the center-of-mass. In the cigar configuration, this is not necessary. Due to its high symmetry, the single-particle wave functions (eqs.(4.101)-(4.102)) do not have to be re—written. The alpha particle is always at the center-of—mass. The form of the Operators for the matter radius and charge radius remain the same (eqs.(4.55) and (4.60), respectively). We begin, then, with the one-body matrix elements: «4314) = ,3 (4.145) (ilr2li) = gn[%+§3_(12:€2] (4.146) (102(4) : 61[%+-d112:i)] (4.147) (:tlr2|a)=(a|r2|i) : 612(u:w+(T::—)2d2j' (4.148) We must separate the one-body terms into proton and neutron terms. The proton term is: (72),, = (alrzlawa [03, (0,2, + 0:) + 20‘:2 — 26f2fla (0,. + 012)] (4.149) 93 The neutron terms are: (alr2|a)0: [0a (031 +02 4) (4112(4wf; (0,0,, — 8%,) , (ilr2|$)03 (90 94 — 0f2 (alr2|:l:)20c2,012 [20(2— (ilr2la)20f,012 [26?2 )1 9 (0n + 0i” 1 - 0c, (0,, + 61)]. (4.150) (4.151) (4.152) (4.153) (4.154) Those are summed together with the proper coefficients shown eqs.(4.55) and (4.60). The two-body matrix elements are formed from the scalar products of the one- body matrix elements (r1) - (r2). They are: - <¥Irl$> (ilrlfl - <$lrli> (ilrli) - ($lrla> - (alrl=F) (ierI) ° (¥|r|a> (Itlrl?) - (alrlfl (alrli) - (alrl?) (ilrla) - <¥lrla> (alrli) - (ilrla> (alrliF) - (ilrla> —03,(1 + 4112/2, tad2 —6n612m ‘9"9‘220/ + 01) (4.155) (4.156) (4.157) (4.158) (4.159) (4.160) (4.161) (4.162) (4.163) (4.164) There are no terms involving the direct alpha terms ((alrla)), because the alpha is at the origin and this matrix element is zero. Because of this fact, the only two-body 94 terms are neutron-neutron terms. They are: (r1-r2)n = (ilrli) - (1139]?)92‘11 (4-165) (1'1 ' 1‘2): = (ilrlfl ‘ (JEIrIiWL‘v (4-166) (r1 11:2)42350, = —(j:|r|i) - ($lrla)20i012, (4.167) (r1 -r2)i2a¢ = —(:t|r|:1:) - (a[r|2F)20g012, (4.168) (1'1 'r2)i¥¢a = “(330143) ' (Tlrlaflagflm (4-169) (1‘1 'P2)43;a4 = ‘(ilrlfl ' (airlimim (4-170) (r1-r2>a4a4 = - (alrl¥)29§9121 (4-171) (r1 -r2)4a:a = (ilrla)-(¥lrla>29§9121 (4-172) (r1°r2)a4a: = (alrli) -(alrl¥>29§9121 (4-173) (1342),,11, : (C1014) - (4|rla)2 (2193,19?2 -— 020"), (4.174) (r1 - 1‘2)04;a = (alrli) - ($lrla)2 (203,0?2 -— 0:04) . (4.175) These terms are summed together and divided by the normalization in order to obtain the matter or charge radius in the cigar configuration. 4.3 Electromagnetic transitions Electromagnetic processes connect the various states of a nucleus. When an excited state decays to the ground state, or when a nucleus is excited into one of its higher levels, the nucleus emits or absorbs an electromagnetic wave. This process has been studied in great length and all details can be found in any text on nuclear physics, such as [10] or [52]. We will just give a brief summary of the theory here. 95 4.3. 1 Background We are interested in the transition probability, Tfi, between an initial and final state (which is the reciprocal of the lifetime for transition from an excited state to a lower state): 87r )1 + 1 f‘ : 71—A[(2)1+ 1)11]2 k2A+1l—/ 17147009301100?)(WWqulfiI/Jmfl’dfl (4-182) Remember that in 6He, the L-value of the projection is the same as the J-value for the state, since the only angular momentum in that system is orbital angular momentum. We then rewrite the matrix element as: (11111101111141) = (418-1011111110 (4.183) 97 Since qu is a tensor operator, we know that 1 _ 1 k _ 3%, 10qu = Z qul’anq(§Iy 1). (4.184) qll We then insert 9232’” into the matrix element in order to use this relation: (¢|§It’“10kq§f?|§2"1§fw)=ZD§;,,(§R’ ((1-lokq,,|1‘r 11111) (4.185) We then use the definition 317132 E 9}” and rewrite the full equation (RHS of eq.(4.182)) as (411M1(1r)|01.q|1/11M(1t )>=// 2001001! ,;:.(11/)D,L,,0(11r111¥')(11 wIqu11|§f%”7,b)d§R’d§R”. (4.186) By using the same relation as in eq.(2.40), we work on performing the first integration (over 82’): / 01171181011118 101111481111? = II__ 871-2 I I I : (—1)q 12L, + 105,);_qu,31,,_,,. (4.187) We can now write the final general formula for the expectation value of tensor oper- ators in our rotated coordinate system: (911M1(W)l041I2/21M(%)>= 2 L'+12_)Z( 1)"q — (1011:1111— qCL43114_q~ qIIM” x/ DM110(9?”))(1/2|0,,q11|§R”1,b)d§R”. (4.188) We then look at eq.(4.188) for an E2 transition between initial state J = O and 98 final state J = 2. For this transition, eq.(4.188) becomes 8W2 II I (I/JZIW'OIt’N Z eiTEEOI'wOO(%)> :"5—' Z: (— 1)q C&%OC§&II,2_qII qIIMII >< [01402181Zeirmowwdw. (4189) Here, we chose to only do the calculation with Ygo, because of the powerful Wigner- Eckart theorem [53,54], which states: (njmlquln'j'm') = ij (njllTklln'j'), (4-190) kmfnfl where Tk is a tensor operator of rank k, and n stands for all other quantum numbers. Once a calculation has been made for one particular qu, all others in the multiplet can be obtained with Clebsch-Gordan coefficients. This is especially easy in our specific case since all the Clebsch-Gordan coefficients are equal to one. Resuming, we perform the sums over q” and M ” and obtain: 8 2 - A = % / 030(12')10M,(¢|Ze,r31/20|12'¢)d12'. (4.191) We then integrate over the Euler angles a and 7, which leads us to the final result: 327r4 2 A , = —5— [(114 ZeiriYgoléR' ¢)d(cos 19). (4.192) Everything is now in terms of the erzYm matrix elements. The form of the operator is (in the lab frame): er2Y20(a:, y, z) = tel/16% (22:2 — 51:2 — y2) . (4.193) This operator requires that the distances be taken from the center-of-mass of the system. Though the sum is over charge, the neutrons contribute through the recoil of 99 the center-of-mass. Thus, we have the same collection of terms as was derived in the discussion of the charge radius (eq. (4.60)). The expectation value of the operator in eq.(4.193) must be calculated for all pos- sible transitions, that is, alpha-dineutron to alpha-dineutron, cigar to cigar, and both cross terms. We start first with the alpha-dineutron to alpha-dineutron transition. Alpha-dineutron to alpha-dineutron The notation for the parameters is the following: all parameters will have a subscipt indicating the state in which they belong, zero for the ground state, and 2 for the excited state. Greek letters indicate the alpha-dineutron configuration, and Latin letters indicate the cigar configuration. We will also use the notation introduced in the radii section in the alpha-dineutron section for referring to particle locations (eq.(4.61)). For the coordinates themselves, 21 and 22 refer to the ground state, and primed coordinates refer to the excited state. The center-of-mass is also different in the excited state, but we need a common reference point in order to properly perform the calculations. Thus, we need the location of the alpha particle and dineutron, z’1 and 2;, respectively, in terms of the center-of-mass of the ground state, Z, and the distance between the alpha and dineutron, d’. In order to obtain expressions for 8’1 and zé, we use the definition of the center-of-mass, eq.(4.53), and the simple relation d’ = z; — Zi- From these two equations, we obtain the relation for z; and 2.3: d'(1/2 + w2ll2 — ( i2)2l — 2V2(9'12)2 + 3Z(V2 + w2)[1 — (9'12)2l ‘25 Z 3(u2+w2>[1—<612)21 ’ (“94) z; = 241221012)? + (14 + 4211320 - < 12>?) - 251, (4.195) (V2 + Wall? - ( i2)2l — 21/209192 where 0’12 is the overlap between the two centers in the excited state. The definition for zi is recursive, but we keep it this way for the sake of brevity. Before listing the matrix elements, we need to list the overlaps for the transitions. 100 The overlaps are: 2,/z/ 1! 3/2 , (a2la0> =( 0 2) exp [“F(V0,V2,Zl,zl)], where F(a,b,z,-,z') = ab [(2, — Z)2 + (z; — Z)2 —— 2:1:(2, — Z)(z£ — Z)] i 2(a + b) Continuing: 2 1120012 3/2 I (42140) = w “111—1890912141 , 2 2,/V 91 3/2 , =( 0 2) exp {—F(V0,L¢J2,21,22)], 2,/w 1! 3” (a2Id0) = (_0w—2) exp [—F(w0, V2122, Zi)l- 0 (4.196) (4.197) (4.198) (4.199) (4.200) We can proceed now with the matrix elements. All terms should be multiplied 5 by the prefactor e i671 that will not be written in the following expressions. The one—body terms are: <02|€T2Y20|010) = 0aaf(VOal/2yzlaz,1)a where 2 ,_ 2 2_ 2 {_ , ,_ {_ f(a,b,z,-,z£)=a(zi Z) (3:1: 1)+2b (z, Z)+4ab.1:(z, Z)(zz (4.201) Z) (a + b)2 101 (4.202) Resuming: =00a6ddG(z/o, (.00, 112,012, 21, 22,2'1, 2;), (4.208) (agagle’I'ZYgolaod0> =00afladG(1/o, wo, V2, V2, 21 , 22,21, 2'1), (4.209) <02d2|€T2Y20|C¥000> =9009daG(l/0,V0,V2,w2,21,21,21,2;), (4.210) (dgagle'rzYQoldodo) =0dd00dG(wo, (120,122, (1)2, 22,22,121, 25), (4.211) (dgdglerzl’goldoao) =0dd0daG(1/o,w0,w2,w2, z1,z2,z§,z§), (4.212) 102 (dgdglerngdaoao) =642iaG(V07 V0, (4)2, (.112, 21, 21, 2;, 2;), (4.213) (a2a2|67‘2}/20|d0d0) =6§dG(w01w07 V21 V21 221 Z2) 2'17 2,1), (4'214) (a2d2l67‘2Y20l010d0) zodagadG(V0>w01w2aI/21 21, 2212;, 3i), (4-215) (d2d2l8T2Y2oldgdo) =0§dG(wo, wo, (112,022, 22, 22, 2;, 22). (4.216) These matrix elements are inserted into the prOper two-body term from eqs.(4.80)- (4.97). This is then squared and divided by the square of the norms of both states, and this completes the contribution of the alpha-dineutron to alpha-dineutron part of the transition. Cigar to cigar In the cigar to cigar transition, the only contribution is from the recoil of the neutrons. This is because our spherical alpha particle is located at the origin, which means the proton charge contribution to the transition is zero. The only terms that contribute then are the pure neutron one-body terms (eqs.(4.150)-(4.154)) and the neutron- neutron two-body terms (eqs.(4.165)-(4.175)). The proper coefficient for each term is found in the charge radius section, in equation (4.60), except that the entire equation is not divided by the total charge of the nucleus as it is for the charge radius. Proceeding to the actual calculation, we first need the overlaps. The overlaps are: (02'00) E 000 = (2%)”, (4.217) no + 722 2,r— 3” ' d2 — 2 d ‘ (izliol 2 6.... = (w) exp —“’°“’” ° + ‘13 “’d" 2) . (4218) 100 + 102 L 2(UJO + 102) 2‘/’UJ21U0)3/2 F 100102 (d3 + d% + 2ilidodg)1 :l: = 9, = ——-— — , 4.219 < 2|¥0) l (wo ‘1' “(U2 exp L 2(100 + 102) ( ) 2 11127110)”2 _ wowzdg l d: a E 6m = ex —— , 4.220 < 2| 0) (no+w2 p 1 2(no+w2)d ( ) 2 3/2 1' (82(10) _ 0.... = ( WW) exp —Ml . (4.221) wo + 72.2 L 2(100 + 71.2), 103 To remind the reader, the oscillator parameters are written with Latin letters because we are referring to the cigar configuration here (n replaces V, and w replaces 0)). We proceed with the matrix elements. We will only list the non-zero elements (those diagonal in the alpha particle will be zero). First, the one-body matrix elements: 0m. (blah/20m) =m [wgd3(3x2 — 1) + 21113.13 + 4w0w2$d0d2] , (4.222) 0 2 0.. (:lzglerleOIio) =—l— [wgdg(3:z:2 — 1) + 27.03813 — 4w0w2$d0d2] , (4.223) (1.00 + 102)2 2w2d2 i 21/ =0m—2——, 4.224 f 2|” 20|ao> (no + 102)? ( ) 2w d2 2 _ 0 0 (02'67‘ nolio) —6anm. (4.225) The non-vanishing two—body matrix elements are: 62 (+2 —2 |€T2Y20| —0 +0) 2' — m [wfidfisz — 1) + 2211ng ‘l' 4wongd0d2] , (4.226) 62, (i2 432 |€T2Y20| $0 $0) = — W [wgdg(32:2 — 1) + 2w§d§ —- 41002022610412] , (4.227) 1112(12(31132 — 1) + 2w0ngd0d2 :t 21/ :t = — awe," 0 0 .22 < 202l67' 20' 31:0 0) (100 + w2)(wo + n2) (4 8) 2w2d2 + 2wongd0d2 a: 21/ d: = — 0,,..0... 2 2 , 4.229 ( 2 4:2 '6?" who 0) (100 + w2)(n0 + 102) ( ) w2d2(3$2 —- 1) — 2w0ngd0d2 i 21/ a: = — 0,. 00,... 0 0 4.230 ( 2012|€T 20' 0 4:0) 1 (we + w2)(w0 + 72.2) ( l 2 2d2 — 2 (1 (i2 :F2 l€T2Y20l00-"F0) = — grlgna 1112 2 1110111227010 2 (4-231) (wo + w2)(n0 + 1112) , 104 2103013 (+2 "‘2 |€T2}/20|(1000) = — 620W, (4.232) 21)de 3:1:2 — 1 (012012le7‘2Y20l +0 ‘0) = " 6(2)"; 2150(+ n2)2 ) (4233) 2w0w2d0d2$ :l: 21/ i =omoan , 4.2 4 (a2 2 '6?" 20'00 > (w0+n2)(no+w2) ( 3 ) (a. 1:. 16221401an = — owe... ”mm” (4.235) (we + n2)(n0 + 102)- This finishes off the cigar to cigar contribution to the reduced transition rate. Next is the first of the cross terms, the alpha-dineutron to cigar transition. Alpha-dineutron to cigar The alpha-dineutron to cigar transition is the first of the two cross terms in the calculation of the B (E2) We would expect its contribution to be less than the direct terms, but could perhaps still be important. The overlap structure is similar to the one- and two—body terms found in the discussion of the interference term in the previous chapter, but it is not the same. We will start with the overlaps, then the list of overlap terms, and finally the matrix elements. Before starting with the overlaps, we will clarify the notations used in this section. For the oscillator parameters, Greek letters will be the alpha-dineutron parameters (still subscripted for absolute clarity), and Latin letters will refer to the cigar configuration. For the distances, 21 and 22 are the locations of the alpha particle and the dineutron, respectively, in the alpha- dineutron configuration. The center-of-mass will be referred to be Z, and the distance parameter in the cigar configuration will be C’ E (12 + Z, where d2 is the normal alpha- external neutron distance used in previous calculations. The introduction of C’ allows us to keep everything in terms of a difference between a coordinate and the center- of-mass. 105 We then continue with the overlaps for this transition: 3/2 _ 2 (92km) 5 em = (Ll/On?) eXp [_V07l2(21 Z) J , V0 + 72.2 204) + 71.2) 2,/w0n2)3/2 [ w0n2(22 - 2?] d E 0.. = —— — , (012l 0) d (we + 722 exp 2(w0 + n2) 2‘/1/ 111 3/2 , ("l—2'00) E 9+0 : (fit—”'23) exp[_F(V01w27 Zlic )l) where F was defined in eq.(4.197). Continuing: ,—— 3/2 (—2|a0) E 0_(, = (2 11010:) exp[—F(u0,w2, 21,6)“:E —’ “'97), Vo+w ,—— 3/2 <+2ld0> E 0+d : (W) 9Xp[—F(WO,’UJ2,ZQ,CI)], ,-——w0w2 3/2 I (—2|d0) E 0_d = (Lu-1;) exp[—F(w0,w2,z2,C )](x —> —a:). (4.236) (4.237) (4.238) (4.239) (4.240) (4.241) We will now list the overlap expressions. The pattern follows the charge radius calculation, and thus we have five terms as in eq.(4.60). The proton one-body term is: (B(E2)>p= The neutron one-body terms are: (B(E2))a=02a [20006+d6_d - 00d (0,4004 + 0_a0+d)] , (B (E 2=)>+d <+2l87‘2Y20ld0>93 0(9aa9—d — 60416—0!) 9 (B (E 2—)> d— =<— 2l67‘2Y20|do)9:a (9aa9+d — 9ad9+a) , (B(E2)>ad =(a2l67‘2Y20ldch21a [29ad9+a9—a — 90m (9+a9—d + 9-09+d)l , (B(E2))+a =<+2leT2Y20la0)6ia0¢1d(g—agad — 9aa9—d) , (B(E2))_a =(-2|67‘2Y20Ia0)0:090d (0+a00d — 00.016411) . 106 (agler2Ygolao)0aa [0300+d6_d + 03d0+00_a—00a60d(0+00_d + 0_00+d)] . (4.242) (4.243) (4.244) (4.245) (4.246) (4.247) (4.248) This finishes the one-body terms. We start the two—body terms with the proton-proton term: (B(E2)(2))p =(O202IBT2Y20|0000) (4.249) X [0309+d6_d + 03d9+06_a '— 0009(10 (6+00_d + 6—06+d)] . The neutron-neutron terms are: (B(E2)(2))a =((1202IeTQléolaoaowiaBerid, (4.250) (B(E2)(2)>rdld =(+2 —2 l€T2Y20ld0d0>93m (4-251) (B(E2)<2>)md = — (agaglerzl’golaodowga (67.06% + awe”), (4.252) (B(E2)(2))aam = - (+202|er2bolaoao)62000d6_d, (4.253) (B(E2)(2))aa,a = — {—262|er2m0|aoao)9§aaad6+d, (4.254) (B(E2)<2>>amd =(+2a2l€7"2Y20l00d0)9c2m (200.9-.. - 04.0.4), (4255) (B(E2)(2))aa,d =(—2agler2l’20|aodo)0:a (260,0...) - 0,0,6“) , (4.256) (B(E2)(2)).d,a =(+2 "‘2 l67‘2Y20laodolggagada (4-257) (B(E2)(2))ldm =(—2 +2 ler2Y20|aod0)0§a00d, (4.258) (B(E2)<2>),dad =(+252|er2n0|dodo)6gao_a, (4.259) (B(E2)(2)),dad =(—2a2|er2Y20|d0d0)0300+a, (4.260) (B(E2)(2))mza =(+2 —2 lerngolaoao)6300§d, (4.261) (B (192)”) >0:de =(a2a2|er2160|d0d0)6200+a0_a, (4262) (B(E2)(2))mad =(+2agler2Y20|doa0)0(2m (20046-0, — 600,64), (4.263) (B(E2)(2)),aad =(—262|er2Y20|doao)6;-’m (26mm... — acme”). (4.264) In the above list, 7‘ has been substituted for “+” and l for “-” in the term labels. We 107 now list the final group of terms, the proton-neutron two-body terms: (B(E2)(2))p,, =(a252[em/200106.926.m [26000+d0_d - 0..., (0+..6_d + 6-06+d)] , (4.265) (B(E2)(2))aaad =<0202|€T2Y20laod0)20aa [260d6+06—a — 60a (6+a6—d + 0—a6+d)] 7 (B(E2)(2)>aara =(02 +2 |€T2Y20l0000>29aa9ad (00410—01 _ 60:06—41) 7 (B(E2 (2)>aard =(02 +2 |€T2Y20ld00)29:a (6006—41 — Bade—a) a (B(E2)(2))aa,d =(62 +2 lerngoldao)26§a (60,064 — 60,20). (B(E2)(2)>aala 2(02 _2 l€T2Y20l00010>29aa9ad (90d6+a — 60a6+d) a ) (4.266) (4.267) (4.268) (4.269) (4.270) The list of terms is now complete. We already have the definitions of the overlaps, so we now move to the matrix elements. The one-body matrix elements are: 113(z1— Z)2(3:z:2 —- 1) (V0 + R2)2 (azlerngolao) =05... (+2l87‘21/20ld0) =0+df(w0) w?) Z27 C’), where the function f was defined in eq.(4.202). Resuming: (-2|€T2Y20|(10) =6—df(w07 “)2: 22) C’)($ '—) —$)3 (+2l8T2Y20laol =9+af(V0, W2, 21, (I), <—2|€T2Y20l010) =9—af(V0,w2,Zl,C,)($ —" ‘33), 028(22 — Z)2(3£L’2 — 1) (L00 + R2)2 (02 |er2|d0) =60.) 108 (4.271) (4.272) (4.273) (4.274) (4.275) (4.276) The two-body matrix elements are: (agaglerngo|aoao) (+2QQICT21/Qolaoao) where 9(a7bauav7213Z17C) = Resuming: (—2a2l€7”2Y20l00C¥0> (01202 lerngolaodo) ((12 +2 lerzY-zoldoao) ((12 ‘2 |€T2Y20|d00fol (02 +2 '67‘2Y20l00d0) (012 —2 l87‘2 Y20l00d0>= ((12 +2 lerzymldodol <02 —2 l67‘2Y20ld0d0> (azagle’rzl/goldochfl :00“! (+2 -2 '67‘2Y20l01000) (+2 —2 I67‘2Y20ld0d0) (+2 —2 |€T2Y20l00dol {—2 +2 |€T2Y20l00d0> :60“) 60d 2 ”0(21—2Z)2(3$ — 1) :60“) (V0 + n2)2 1 I =6000+ag(u0, V0, w21n2) Zl, ZI,C )’ a d/ 2, in order for the distances to properly correspond. There is a change in notation, in order to keep the configurations straight. All the parameters of the alpha-dineutron configuration will be written in Greek letters-11,0), 6, where 6 is the distance parameter. For the cigar configuration, we will use Latin letters—n,w,d, where n is the alpha particle oscillator parameter, w is the oscillator parameter for the external neutrons, and d is the previously mentioned distance parameter. We list 112 the overlaps: 60,, E (ala') = (if—:3)” exp [ 1 8314271)] , (4.298) 90+ E (04+) = (ifi)3/2 exp :_ uw (9d:;il:1<-5: ;)12d6:v): , (4.299) 60— E (04") = (ifi)3/2 exp "_yw (9d:;E;1c: ;)12d6:v): , (4.300) 6,... a (dla') = (:fiffl exp (—-é%%] , (4.301) Since the wave functions are real, the transpose of the overlaps and all matrix elements is the same, thus we will only write these matrix elements with the alpha-dineutron wave function in the bra and the cigar wave function in the ket. The first quantity we calculated was the overlap of the two configurations: fi<¢1l¢2> _ 63m, [020,061+6d— + 031060+60— _ 60100610: (6d+60— + 0d—00+)] \/<¢1l7l’1>(1/’2|1/12) \/(¢1l¢1)(¢2|¢2) (4.304) This quantity as a function of the distance parameters (left equal to each other) is shown in Figure 4.24. For this plot, all four oscillator parameters are equal, so we would expect the overlap to be one at d,6 = 0. We see in the plot that is in fact minus one. This is because the sign between off-diagonal matrix elements is random. So we then add by hand an extra minus sign to make the overlap positive, which defines the overall sign for the matrix elements (i.e., makes the expectation value of the Hamiltonian negative). When the distance is large, the overlap goes to zero, because at large distances, the two configurations are orthogonal. 113 0.0 l—T W W W I I T I —0.2 - —0.4 — Overlap —0.6 l -—0.8 r l I T I I I —l.0 (1 (fm) Figure 4.24: Plotted here is the overlap of the two configurations. They were plotted under the conditions where they should be identical at the origin, so all four oscillator parameters and both distance parameters are equal to each other. 114 4.4. 1 Kinetic energy We need to find the matrix element of the Hamiltonian between the two configura— tions. The first step is the kinetic energy. As before, this will have a one-body and two-body component. The one-body matrix elements are: (a|T|a’) = 6 617:7)(3— $27)), (4.305) (a|T|+) = 6..., —V—2(:+w)(— ”w(9d:;(:‘:;)12d5$)), (4.306) (61TH = 62,,_ jj—XEM (3— Vw(9d:;(:6:;)12d5“)), (4.307) (d|T|+) = 6d+—-——— 2()w+w )-(3 wwl9d23;(;6:2w)24d6x)), (4.308) (le|—) = 6d+———— 2()w+w w)—(3 ww<9d23;(;6:21$24d6$)), (4.309) (leIa') = gnaw—Zn) w(3— 521—5:6—;). (4.310) Once again, the factor that gives the dimensions of energy, h2/ 2m, has been sup- pressed. These matrix elements are inserted into eqs.(3.56)-(3.61) and divided by the normalization. For the center-of-mass correction, we need the expectation values of the momen- tum Operator. They are: (61pm = 0.0%;2, (4.311) (a|p|+) = GHQ—1:171:75 dsin0x+26+3:003921, (4.312) (a|p|-—) = 6,_Zé’i‘fi?) Ldsinflx+ “COS: 4262] , (4.313) (d|p|+) = 99275117,) ldsinflx+ “COS: _462: , (4.314) (d|p|+) = 6d+fiwj ldsin 1912+ “COS: +462: , (4.315) (dlpla') = —6d,,37(2:4:—5n—)6. (4.316) 115 In these matrix elements, the dimensional factor h has been suppressed. All possible scalar products are taken between these matrix elements, and they are inserted into eqs.(3.62)-(3.7 6). The proper weighting of each term is discussed in section 4.1.2. 4.4.2 Interaction We now move on to the two-body interaction. As mentioned for the two configurations, we used interactions of the Gaussian type. We will list the matrix elements using the Volkov parameters Va and a, which are defined in eq.(4.24). These matrix elements are mostly rather lengthy, and we will try to use intermediate notations in order to make things clearer. We will also make use of the a: —> —2: property of many of the matrix elements that was used before in the cigar configuration. The matrix elements are: (mu/(676') —v 4””0‘2 m e ——’”fl— (4 317) _ a (u+n)A(1/,n) xp 9(1/+n) ’ ' 4ua2(/nw 3” N I = — 4. l (“'V“ H V“ (00.72.4212)) 9"" l 7200,22, u,w)l ’ ( 3 8) (aalVa’—) =(aalVa'+)(:c —> —:I:), (4.319) where A(p, q) is defined in eq.(4.35), and D’(a, b, c, d) = a2(a + b)(c + d) + 2(a. + b + c + d), (4.320) and N =4V62[Va2(n + w) + 2nwa2 + 4n + 47.0] + 910612 [ua2(u + n) + 41/ + 271] (4.321) + 12uw6dxA(1/, n). 116 Resuming: 2 3/2 4730: W) ) (4.322) D’(u, n, n, w) _ 62 (71032 (1m + 40m + 51/02) + 41m + 16607:. + 181/w) 18D’(1/,n, n,w) ’ (da|V|a+) =V, ( 402W )3/2 exp [_ 72 D, ( N ] (4.323) (dalVlo/a’) = ( x exp D’(V,n,w,w) V,n,w,w) (dalVla—) =(da|V|a+)(a: —> —:L‘), (4.324) where N =452 (012 (un(w+w)+4ww(u+n))+2(V+4w)(n+w)+18uw) (4.325) + 9wd2[wa2(1/ + n) + 2(1/ + w + 13)] + 24wd61: (6607201 + n) — V + 202) . Resuming: 40.11002 3/2 _ = . 2 (dd'v' + l V“ ((w + mom) (4 3 6) 1602621300, 10) + 910d? ((4)02 (40 + w) + 462 + 4w) — 4.327 X 9"" l 36(4) + w)A(w,w) l ’ ( ) 402a2 nw 3” N’ , = _ 4. 2 (dlela +) V“ (D’(w,n,w,w)) exp [ 72D’(w,n,w,w)l ’ ( 3 8) (dlela’—) =(dleIo/+)(:1: —+ —:1:), (4.329) where N’ 2160162 [4001201 + w) + 2711002 + 4n + 4111] (4.330) + 9wd2[wa2(n + w) + 2n + 440] —— 24ww6dxA(n, w). 117 Resuming: (dalVl + —) —v “"2 ”w 3/2 ex — N (4 331) _ a D’(z/, w,w, w) p 72D’(V, w,w, w) ’ ' (daIVI -— +) =(daIVI + -—)(:1: -—> —a:), (4.332) where N =462[wa2(uw + ww + uw) + 41/112 + 160221) + 181/w] (4.333) + 91in2 [0720/20 + ww + 110)) + 4w + 8w] + 12wa2d6:c(uw + 202211 + 3146). Resuming: (aleIa'+) —V 4&2 V ”mm m ex — N, (4 334) — a D’(u, w, n, w) p 72D’(u, w, 71,01) ’ ° (ad|V|a’—) =(alela'+)(:1: —> —CL‘), (4.335) where N’ =462[Vwa2(n + w) + 4wna2(1/ + w) + 2(1/ + 4w)(n + 111)] (4.336) + 9wd2[uaz(n + w) + 2(1/ + w + 71.)] + 12w6dx[ua2(n + w) + 21/ — 4w]. Resuming: 41/112012 3” _. zva 4. (“Q'V' + ) () ( 337’ 4uw62A(1/, w) + 910612 (11a2 + 4) (u + w) X exp — 7 36(1/ + w)A(V, 7.12) 40271012 3/2 4607162 ' ’ =Va —— . . (dd'v'aal ((w+n)4(w.n)) pl 9(w+n>l (4338) These matrix elements are then inserted into eqs.(3.62)-(3.76) and divided by the normalization to obtain the potential energy in the interference term. 118 All the operators in the Hamiltonian have now been accounted for and we can then follow the minimization procedure discussed in eq.(3.46) and following. This will determine the minimum energy as well as the weights of each configuration, which will be necessary in calculating observables (see eq.(3.7 8)) 4.4.3 Mean square radius In order to calculate mean radii, we need to also take into account the interfer— ence term, as was just mentioned. In the individual configurations, we needed to rewrite the alpha-dineutron configuration to pay special attention to the center-of- mass, whereas we could leave the cigar configuration as it was first written. For the interference term, we will have to re-label particle locations. The alpha particle in the alpha-dineutron term (configuration one) is located at (in the body-fixed frame) 21, and the dineutron (or deuteron, anticipating the problem for lithium) is placed at 22. The alpha particle in the cigar configuration (configuration two), is placed at Z, which is the center-of-mass of the system. The coordinate C is assigned the value Z + d, and one of the cigar particles is placed there. The other particle is placed at 2Z — C. In practice, we always work with differences (i(( — Z), so it is only the physical distance d that is important. With these labels, the overlaps are (very similar to the ones used for the electro- 119 magnetic transitions): 3/2 2 2M) exp (— ””(z‘ ‘ Z) ) , (4.339) exp[—F(V, w, z1,()], (4.340) exp[—F(u, w, 31, C)](:z: —> —:1:), (4.341) exp[—F(w, w, 22, O], (4.342) ww 3/2 2F) exp[—F(w,w,22,()](:c——+—x), (4.343) 6..., = (2%)”? exp (—“m(z2 _ Z?) , (4.344) 2(w + n) where the F function was defined in eq.(4.197). With these overlaps, we can continue with the one-body matrix elements: (alr2la’)=6m( 3 +5inf—Zl2), (4.345) u+n (1/+n)2 (OITQIH =9a+R(V,w.21,C), (4-345) (alr2|—) =60_R(u,w,21,()(x —+ —a:), (4.347) (dlr2|+) :6d+R(w, w, .22, c), (4.348) (dIT2|-) =9d—R(w.w,22,C)($ -+ —a:), (4349) 2 , _ 3 w2(z2 - Z)2 (d|r la) —9da (w + n + (w + "(2 ) , (4.350) where 2 '—Z2 2 {_ 2 -—Z (_ R(a,b,z,~,z,’-)= 3 +a(z, )+b(z, Z) +2aba:(z, )(2:z Z) a+b (6+6)2 ' (4.351) These one-body matrix elements are inserted into the following terms, the first being 120 the proton-proton term: (73);) = (alr2a,>2gaa [0:a6d+6d— + 03a60+6a— _ gaaoda (60+6d— + 60—0d+)] ' (4352) The neutron one-body terms are: (7‘2la = (0930093.... [296696+93— — 966 (95+93_ + 6.,_0.1+)], (4-353) (7‘2)a+ = (0|T2|+>9.2.9da (05—065 — 9550+). (4-354) (r2)a_ = (alr2|—)6:Bda (001.6%, — 6006’“) , (4.355) (2:2).“ = eiia (6.60..- — 6.61.). (4.356) (73).;— = (le2I—Wiia (966.4“ - 601+0da)2 (4-357) (T2>da : (le2IQI)6:a [260+6a—03a _' 00: (00+6d— + 00—0d+)] ' (4358) These are the terms added together with the proper weighting coefficients in eq.(4.60) in order to obtain the one-body contribution to the charge radius calculation. For the matter radius, eq.(4.55) is used along with the general one-body operator expression for the interference term (eqs.(3.56)-(3.61)). Turning to the two-body section of the aforementioned equations, we begin with the two-body matrix elements: 2 (aawla'a) =63, (”—(Zl—fl) , (4.359) V + n (aalrzla’+) =60090+RI(V, u, w, n, 21, 21, C, at), (4.360) (4.361) where R'(a b u v 2- 2k 2’ x) = ab(z.- - Z)(Zk - Z) + avx(z.- - Z)(z' __ Z). (4.362) (a + v)(b + u) 121 Resuming: (aalri’la’w) = (adlr2la'a') = (adlr2l + a’) = (adIr2| — a’) = (ddl7‘2la'+) = (dle2|0'-) = = (adlrzl — +> = (ddlr2l + —) = (adlr2|a'+) = (adlrzla'—) = (dal|7"2 lo/a’) = = 60090_R’(V, V: w) n) Z1) Z1) (7 —$)1 1140(21- Z) (22 — Z) (u+n)(w+n) 0cm: ado ) gaagd+R’(l/awaw7 n) 217 227 (1:17)) gaagd—aRI(V7w7wan7Z19221C1__x) 0d+0daR'(w, w, w, n, 22, 2:2, C, 2:), 0d_0daR’(w, w, w, n, 22, 22, C, —:z:), 90—0d+ 2 — 2 (w+w)(l/+w){Vw(z2—Z)(Z1—Z)_w (C Z) +w$(C - Z) [V(Zl “ Z) _ “(Z2 _ Z)]}’ 00+6d- _ ‘7 _ — w2 — 2 (w +w)(u+w) {Vw(z2 Z) (“1 Z) (C Z) _wa _ z) [V(z, — Z) — w(22 — Z)]l, 2(22—Zl—w2(C—Z)2 w 6d+0d— (U) + w)2 ) 0d060+R’(w7V1w7n1Z21 31,9, CE), 0,1060_R'(w,l/,w,n,22, Z17<7 —$)) 2 (W (Z2 ‘ Z))2 ado: _— ’ (w+n) V2 (31 — le — “’2 (C — Z)2 0,, 0m + (V + w)2 (4.363) (4.364) (4.365) (4.366) (4.367) (4.368) (4.369) (4.370) (4.371) (4.372) (4.373) (4.374) (4.375) For the matter radius, all we need are these matrix elements, then they can be inserted into the general form for two-body operators in the interference term(eqs.(3.62)- (3.76)), with the coefficients in eq.(4.55). For the charge radius, we need to sepa— rate the terms into proton-proton, neutron-neutron, and proton-neutron. The proton- proton term is: (7‘2)p = (aal7‘2la’a') [93a6d+0d— + 93090.4»90— — 6006,10, (60+0d_ + 60.60”” . (4.376) 122 The neutron-neutron terms are: (r2)a = (00 I 7‘2 | a’a’)0:a6d+ 00)-, (73)“),+ = (da|r2I + 01')9,2m [26000d_ — 90—0710] , (7'2)aad_ = (dalrzl — 036:0, [20,306,118 — 00+6da] , (r2)aaa+ = (aalrzla’+)9§09d06d_, (7‘2)aaa— = (aalT2|0'—)93.898898+, (ram = —6£ia (60.4.61- + 60-6.4), (T2>a+d— = —93894m (7‘2)a_d+ = —(da|r2| — +)6200da, (r2)dad+ = —(dd|7"2| + a’)6g000_, (7‘2ldad— = —(dd|7"2| — 0')9289a+, (T2)d+d— = (ddlr2l + —>0:a7 (7‘2ldaa+ = (da|r2| + ‘1')9318 (296a9a— - 96.0ng , (r2)daa_ = ((101er — 009.218 (29da9a+ - 9869.“), (r2)dada = (ddlr2 |a'a')0§afla+00_, (7‘2)a+a— = (aalfll + “>938”;- (7,2)0 = (aalrzla’a’waa [46aa0d+6d— — 29cm (9a+9d— + 60—6d+)] , (r2)aaa+ = (aalr2la'+)20(m (030,0 _ — flaaddafldJ , (r2)aaa_ = (oxalrzlo/—)20(m (03,100+ — 00,00,109“) , (7‘2ldaaa = (dalrzla'allgaa l46da60+00- '7 20cm (90+0d— + 00-039] , (r2)azd+ = (adl7‘2l + 0020310 (6006d_ — 0,1000-) , (r2)02d_ = (adIr2| — (132920 (OOQQH — $000+) . 123 (4.377) (4.378) (4.379) (4.380) (4.381) (4.382) (4.383) (4.384) (4.385) (4.386) (4.387) (4.388) (4.389) (4.390) (4.391) (4.392) (4.393) (4.394) (4.395) (4.396) (4.397) This completes the discussion of the mean-square radii in the interference term of 6He and the discussion of the Gaussian approximation for 6He. We will then move on to the second nucleus of interest, 6Li. 4.5 Lithium-6 The treatment of 6Li in the Gaussian approximation has many similarities with the previous section on 6He. This will allow us to be briefer, and focus on the differences and additions to the calculations outlined in great detail in the preceding pages. The kinetic energies are identical, save for the different projection process, as is the central and Majorana interactions. There are some additional words to say about the spin-orbit and tensor interactions, and then we will discuss the mean-square radius, as that differs slightly because the charge radius depends on proton and neutron number, which is of course different in the case of 6Li. We then will move on to the additional calculation of the quadrupole momentof the ground state of 6Li. This general sequence of discussion will be carried through both configurations and the interference term, with a discussion of the magnetic moment coming just before the before the interference term, and then finally finishing the chapter with a discussion of both nuclei and all configurations in a section on beta decay. 4.5.1 Alpha-deuteron configuration The general set-up of the alpha-deuteron configuration of 6Li was discussed in sec- tion 3.2. The single—particle wave functions are the same as in the case of 6He. The one-particle densities look the same as in the previous configuration. The kinetic and central potential energy calculations are also the same, save for the angular momen- tum projection, which is different (see eq.(3.79)). There is another difference, however, that effects all calculations, and that is the parity projection. The parity projection operator, eq.(3.80), ensures that we are in a good state 124 of parity. This is necessary for lithium, since it has more contributions to the total angular momentum than just the orbital angular momentum as seen in the case of 6He. All the states in 6Li have positive parity, so we will always use the upper sign of eq.(3.80). Because of this operator, all calculations now have two main terms, one where everything is unchanged, and the other where the parity operator has been applied. The effect of the parity Operator is to change the sign of any term prOportional to the rotation angle. No new matrix elements need to be calculated because of this simple rule, but we should note that the norm now consists of two terms, one just as in the case of 6He, plus an additional term affected by the parity operator: 1+1?),- 2 (q: (6L1) | |)II (6Li)) = [63 (6304 — 9%,)2 + (9?, (609.) — (22)] , (4.398) [\DIb—i where a bar over an overlap means that the sign of the angle has been changed from the normal overlap, e.g.: do = exp {—(x—l-gl—Zuf] . (4.399) By looking at eq.(4.9), one can see the only thing that has been affected by the parity operator is the sign of the angle term. This is the same for all alpha-deuteron 6Li calculations. There are always two terms, and the second (the one where the parity operator acts) has all signs changed from what was shown in the alpha-dineutron section of 6He. 4.5.2 Kinetic energy Since we do not need to show any new calculations here, we can go straight to the plots of the results. Figure 4.25 shows the kinetic energy of the system as a function of d. In this plot, the center-of—mass motion has been removed. There are three levels here, J = 1, 2 and 3. There is very little differentiation between them in the kinetic energy. The large d kinetic energy corresponds to the kinetic energy of a deuteron and the kinetic energy of an alpha particle summed together. At small d, we see all three 125 105 " 100: J=l 5 ~ 1 95: € 9 : : E E i v 85, j J=3 80, 75: 70’ d( fm) Figure 4.25: This plot shows the kinetic energy of the alpha-deuteron configuration of 6Li as a function of d. For this plot, both oscillator parameters were set equal to 0.53 fm‘2. levels come together. At small d, a proton and a neutron are forced into the next shell, the p—shell. All states are of even parity, by projection, and there are only minimal contributions from any orbital momenta higher than 8 = 2, so both the particles can sit comfortably in the p—shell. 4.5.3 Interaction For lithium, the same interactions (Volkov and Minnesota) as in the case of helium were used, but there were more kinds of forces (tensor and spin-orbit in addition to central). The central and Majorana exchange are done exactly in the same manner as in helium, with the only difference coming once again from the projection process. The Minnesota potential, however, has some difference as the spin structure of 6Li is different than in 6He. We must re-write the two-body terms into a singlet part and a 126 triplet part. The singlet part is: (V08 a2d2 01de = = (aaII/slaa)2909d (606d — 91,2) 1 = (ad|1/;|da)30§ (609d — Biz) , = (aamlad)60..012(6¥2 - 00494) ~ (aathlaa) (30:03 + 0112 — 39a0d0f2) a (admldaw: (5986.1 — 39%) , (aa|Vt|ad)0a012 (20f2 — 600,08) , —929¥2- (4.400) (4.401) (4.402) We turn now to the discussions of the new operators in the potential. The tensor interaction was discussed before in section 3.2.1. There is not much more to add here, except that the radial form factor that was used was the Volkov potentials. For the Minnesota potential, the tensor interaction was not used. The spin-orbit potential form is . 1 d VLS = ragga exp (—'yr,-2j) L - S = —27rng exp (—*y'r,.2j) L - S. (4.410) In our calculations, V30 was set equal to 20 MeV, 7 was equal to 0.5 fin”, and To, the nuclear radius, was set equal to 1.2 fm. The inner workings of the L - S were discussed in detail in section 3.2.1. Here, we only need to provide the two non-vanishing matrix 127 elements of the VLS operator: (adleSIda) = (VLUW +’y( (V+ 01) )3/) 212T:J:2+V7+V2:11)] sin19 (4.411) xexp[— (1—$)[’7(V 40W)+V0J(V+4w]+18'yV02 d2] 18 [V01 + 7(V + 02)] 41/0.) 3” de2(l/ + 202) (adleSlad) _ ((u + w)(l/ + 02 + 47)) 3(u + 0)) (u + w + 44y)Si “”9 (4412) xexp[ (1-$)[2’7(V-2w)2—4Vw(l/+w)]+36’YVw-l-9Vw(1/+w)d2] 9(" + w)('/ + w + 47) The factor —271‘3V30 has been suppressed because it is the same in both terms. These two matrix elements are plugged into eqs.(3.94) and (3.95) and divided by the norm to obtain the spin-orbit contribution to the interactions. The results of all the interactions with the Volkov potentials are shown in Figure 4.26 (V1) and Figure 4.27 (V2). There is little difference qualitatively between the two sets of parameters. Quantitatively, the V2 potential is about 3 MeV deeper than the V1 potential. These potentials do not distinguish the three states very much, similar to the kinetic energy results. We would then expect in the total energy plots to see the three levels very close together. The results of the Minnesota potential are showed in Figure 4.28. The Minnesota potential seems to show more differentiation between the states of 6Li compared to the Volkov potentials. The magnitude of the interaction at small d is comparable to the Volkov V1 potential, though at large (1 it is smaller. This is due to the lack of a tensor interaction which is an important part of the overall interaction in the deuteron. 4.5.4 Total energy In this subsection, we show the spectra of the alpha-deuteron configuration of 6Li with the three previously mentioned potentials. First, we examine the spectrum calculated with the Volkov V1 interaction (Figure 4.29). As anticipated, the spread in energies 128 T 1' 5 T T W T W T WW I f 1W 1 T j W W 7 T.‘ -105:— . -— 1:1 -110_— ‘ ’>‘ : _ 1 g —115_— 1‘2 — \A/ .4 > . V . -120:‘ __ 1:3 — _125 / 0 2 4 6 8 10 d(fm) Figure 4.26: The expectation value of the Volkov V1 potential as a function of d in the alpha—deuteron configuration of 6Li. The oscillator parameters were both equal to 0.53 fm‘2, the Majorana exchange parameter was 0.6, and the parameters of the spin-orbit potential were Vso=20 MeV and 720.5 fm‘z. l l -105_— j -110:— H g -115'_- __ 1:2 : g i A E - 3 -120:- - _ ——-— J23 —125_7 j —130 — — l 41 0 8 10 d(fm) Figure 4.27: The expectation value of the Volkov V2 potential as a function of d in the alpha-deuteron configuration of “Li. The oscillator parameters were both equal to 0.53 fm‘z, the Majorana exchange parameter was 0.6, and the parameters of the spin-orbit potential were Vso=20 MeV and 7:0.5 fm‘2. 129 —95 " —100_ —105 j —110: (MeV) —115: -120: —125: d(fm) Figure 4.28: The expectation value of the Minnesota potential as a function of din the alpha-deuteron configuration of “Li. The oscillator parameters are both equal to 0.53 fm‘z, the exchange parameter 0. is equal to one, and the parameters of the spin-orbit potential were the same as in previous plots. There is no tensor interaction in this plot. of the three levels is small. Perhaps 1 MeV separate the J = 1 ground state from the J = 3 excited state. At values of d smaller than about 2 fm, what separation there was almost disappears. All three of these states are bound since they are lower than their asymptotic energies. The minima also occur around 3.5 fm, where the external particles are still mostly in s-waves. The picture is essentially the same for the Volkov V2 interaction (Figure 4.30), except that the curves are shifted around 2 MeV to more negative energies. Figure 4.31 shows the energy spectrum calculated with the Minnesota potential. This spectrum shows differences from the spectra calculated with the Volkov poten- tials. First of all, the three levels are clearly separate for all values of d. The bound states appear in a much deeper well compared to their asymptotic energies than in the Volkov potential, and they occur at less than 3 fm, making the overall nucleus much smaller. The sequence of states remains the same, and the higher energy can 130 (MeV) I I | I I | b) b) N N N N N O 00 OK A N I w 4:. fir T Y 5 fl I I I V I I f T I I I r I C . .— 1 4 '_ — 1:1 l . —— 1:2 . - —( u 4 .J - 4 : —- J=3 : _ /\ _l * H3 .4 bl L 4 1 1 1 1 I 1 1 1 I 1 1 1 0 2 4 6 8 10 d(fm) Figure 4.29: The spectrum of the alpha-deuteron configuration of “Li calculated with the Volkov V1 interaction. The oscillator parameters are both equal to 0.53 fm‘z, the Majorana exchange parameter is equal to 0.6, and the parameters of the spin-orbit potential are the same as in previous plots. (MeV) I N .1; I N Ch I N 00 I w o I u N I w .1; | 2» Ch Y I I I I W T l T I I I l 141 : l l l l 1 1 A : 0 2 4 6 8 10 d(fm) Figure 4.30: The spectrum of the alpha-deuteron configuration of “Li calculated with the Volkov V2 interaction. The oscillator parameters are both equal to 0.53 fm‘z, the Majorana exchange parameter is equal to 0.6, and the parameters of the spin-orbit potential are the same as in previous plots. 131 (MeV) d(fm) Figure 4.31: The spectrum of the alpha-deuteron configuration of “Li calculated with the Minnesota potential. The oscillator parameters are both equal to 0.53 fm‘z, the exchange parameter, u, is equal to one, and the values of the spin-orbit parameters are unchanged from previous plots. be attributed to the lack of a tensor interaction. This concludes the discussion on the operators of the Hamiltonian of “Li. We will now move on to the calculation of other observables. 4.5.5 Mean square radius The calculation of the mean square radius of the alpha-deuteron configuration pro- ceeds in the same manner as has been discussed in section 4.1.5, except that the angular momentum projection is different. Also, one will have to do the parity pro- jection, which has also been discussed previously (see eq.(3.80) and following plus the last paragraph of section 4.5.1). The distinction between charge and matter radius is lost in the case of “Li, as now there is a proton in the external cluster. One just needs to follow the procedure outlined in the previously mentioned section on the matter radius, and that will give the charge and matter radius for “Li. 132 4.5.6 Electric quadrupole moment The quadrupole moment is an indication of the shape of a nucleus. The quadrupole moment operator is: Q0 = (/ 1217316062) = 222 — {1:2 — y“, (4413) and one sums over all particles for a matter quadrupole moment or just over protons for the electric quadrupole moment. As one can see from the form of the operator, the expectation value in spherical nuclei should be zero. If it is positive, then the nucleons are more concentrated along the symmetry axis of the nucleus and is said to be prolate. If it is negative, then the nucleons accumulate around the equator of the nucleus, and the nucleus is said to be oblate. By convention, the quadrupole moment is tabulated in the state of maximum angular momentum projection, |J,M = J). The does not mean any change for us, since all our calculations for the ground state of “Li are in the I11) state. The operator in eq.(4.413) is in the body-fixed frame. In our laboratory frame, the expectation of the quadrupole moment is: 3K2 — J(J +1) (JJIQJKIJJ) = (J +1)(2J + 3) where K is the projection of angular momentum on the body-fixed axis. Before calculating the matrix elements, we must make sure the quadrupole mo- ment has no contamination from the center-of-mass. This is essentially the same procedure as in the mean-square radius calculation (starting with eq.(4.52)). The result is essentially the same as eq.(4.55): A — 1QI — %Qn, (4.415) <9>= A where Q; is the one-body expectation value and Q11 is the two-body expectation value. For the charge quadrupole moment, we can use this formula and divide the 133 answer by two, since half of the particles are protons. We can now give the matrix elements. Note that we must use the same style of coordinates as in the center-of-mass calculations (see eqs.(4.61)-(4.72)). The one-body matrix elements of the Q0 operator (eq.(4.413)) are: (21— Z)2 (1 + 31:)(1-I— 2:) (a|Q0|a) :60, 4 , (4.416) (49044) =63” ‘ Z) (1:32“ + i”), (4.417) _ 2112(21— Z)2 +0)2 (22 — Z)2 (3:1:2 — 1) + 4Vw$(21— Z)(z2 — Z) (alQoldl 4912 (V+02)2 I (4.418) 2 (2:2 — Z)2 + V2 (31 — Z)2 (3:02 — 1) + 4VLUIII(21— Z)(z2 — Z) 202 (dIQOIa) =912 (”+002 ’ (4.419) where Z is the location of the center-of-mass (replaces ZCM from the previous calcu- lations). We have all the ingredients to calculate the one-body expectation value of the quadrupole moment. We then need the two-body terms: 21— Z)2(1+ 32:)(1+ x) (aaIQolaa) =9§< 4 , (4.420) (adIQoIda) =689.1(z1 — Z) (22 — Z) (1 + 32:)(1+ 2:), (4.421) (aalQolad) =08012(z1- Z) (1 + 2:) [w (222(73):)? — 1) + 21/(Z1— 2)], (4422) WWW) =64012 (z, — Z) (1 + 2:) [u (212(13):? — 1) + 202 (22 — 2)], (4.423) (ddeo‘dm 2901012022 — Z) (1 + .6) [0 (42(13):? — 1) + 20 (2.2 — 2)], (4.424) (adIQoIdd) =6d612(22 -— Z) (1 + 1:) [02 (2:61:15? — 1) + 21/(21— 2)], (4.425) 134 (22 — Z)2 (1 + 3$)(1+ cc) WIQOIdd) :63 4 1 (4.426) __ 2 1101(22 - Z) (21 - Z) (3:1:2 +1)+ 23; [0,2 (Z2 _ Z)2 + V2 (21 _ Z)2] (daIQOIdOl) _012 (V + (0)2 3 (4.427) (ddeolaa) ___-92 ”2 (Z1 - Z )2 (3.462 _ 1) + 202 (.22 — Z)2 + 4Vwa: (.21 — 2) (z2 _ z) 12 (V + 0))2 , (4.428) (aalQoldd) =02 2112(Zl— Z)2 + 012 (Z2 — Z)2 (32:2 _ 1) + 4V0211: (zl — Z) (22 _ Z) 12 (V + 0))2 ' (4.429) These matrix elements are inserted into the proper term from eqs.(3.11)-(3.20). This is all that is needed to calculate the quadrupole moment. This concludes the section on the alpha-deuteron configuration of “Li. We now go to the cigar configuration. 4.6 Cigar configuration The cigar configuration of “Li was introduced already in section 3.2.2, and also has many similarities with the “He cigar configuration (section 4.2). We just need to add here the things that are different for lithium, and show the various energy plots. We should note that the parity projection is not necessary in this configuration. The symmetrization procedure (outlined starting with eq.(3.21)) of the external spins takes care of parity of the state. 4.6.1 Kinetic energy The kinetic energy calculation in the cigar configuration proceeds just as in the case of “He, save for the different angular momentum projection factor. We then examine the plot that shows the total kinetic energy of the system (center-of-mass energy removed), which is shown in Figure 4.32. The picture is remarkably similar to the 135 : ‘ ‘ j ' 1 ' fl 1 f I, 105} E 100} € : —— 1:1 I 95': 5 g E 2 90: i 7 : ——-—-— 1:2 1 I; 85: ‘ , . t i 80: 1 : —— 1:3 . 75} € 70- 0 2 4 6 8 10 d(fm) Figure 4.32: The kinetic energy of the cigar configuration of “Li as a function of d. The spurious center-of-mass motion has been removed. The oscillator parameters were both set equal to 0.53 fm"2. alpha-deuteron result (Figure 4.25). The d = 0 value is 108 MeV in both plots, which is what we expect. There is also very little separation between the three levels in both plots. The cigar plot falls perhaps slightly faster than the alpha-deuteron C386. 4.6.2 Interaction The details of much of the two—body interaction calculation have been covered before, especially in the case of the Volkov potentials. These are calculated in exactly the same way as for the cigar configuration of “He, save for projecting into different states. There are the additional tensor and spin-orbit interactions, however. The tensor interaction uses the Volkov form factors, and its relevant terms are eqs.(3.120)-(3.134). The spin- 136 orbit operator introduces new matrix elements. They are: 3/2 2 W ) ”w ‘12 sin19 (4.430) (a :I: IVle i a) = (W + ,(V +01) 4 [V01 + )(V +w)l xex _(1—$)W(’7+V)+’7Vw 2 p( 2[V02+7(V+02)] d)’ (a It IVle 3}: a) = — (a :1: IVle :I: a)(:c —-> -.’L‘) (4.431) 4V0) )3/2 V012d2 (W27) W2, ( ) x exp (— w(1 — 11:) + 4702f) , w + 2') (— + IVle — +) = ‘- (— + IVLSI + —)(.’E -+ -:13). (4.437) 137 _1 I I v I T T I I I I w w I I Y m I I I 11 -80r : _90; _ , — 1:1 9 ~ - o —100— — g - . A —— = > : J 2 . V —110_— j Z _ = 1 —120 J 3 C 0 2 4 6 8 10 d(fm) Figure 4.33: The potential energy as a function of d in the cigar configuration of “Li, calculated with the Volkov V1 interaction. In this plot, the oscillator parameters are both equal to 0.53 fm'2, the Majorana exchange parameter is equal to 0.6, and the spin-orbit parameters are Vso=20 MeV, and 7:05 fm‘2. In the listed matrix elements above, the prefactor —2'yr3 V30 has been suppressed since it appears in all of them. These matrix elements are then plugged into the terms listed in eqs.(3.112)-(3.119) and divided by the norm. We show plots of the Volkov V1 and V2 results in Figures 4.33 and 4.34. These plots include the tensor and spin-orbit interactions. These plots are qualitatively the same, with the only difference being that the V2 plot is shifted two or three MeV lower in energy. At (1 = 0, the plots correspond with the values found in the alpha-deuteron configuration (Figures 4.26 and 4.27). For the Minnesota potential, we must separate the terms into singlet and triplet 138 .1 r f 1’1 w fifif ' Y ‘ I ' T ' 1* _80)— _. -90} L — 1:1 3 A ’ 4 % ‘10“? i :2, . . A '- —— J22 : 3 ~110I - I 0 -120} —— 1:3 -130: J I: 0 8 10 d(fm) Figure 4.34: The potential energy as a function of d in the cigar configuration of “Li, calculated with the Volkov V2 interaction. In this plot, the oscillator parameters are both equal to 0.53 fm‘2, the Majorana exchange parameter is equal to 0.6, and the spin-orbit parameters are unchanged from the previous plot. terms, which is different in lithium than in helium. The singlet terms are: (VSIG = (l/s)azn2 : A S V D Q D H. II (aallélaawa [300, (63, + 6:) — 36?2 ((9,, + 6H] , (a i |V,| :t (1)29: (200,9, — 6%,) , (a i IVsl #1 0920:: (2600,,m — 0%) , (Gall/slai)660012 [26?2 — 001(611 + 055)] . 139 (4.438) (4.439) (4.440) (4.441) The triplet terms are: (V), = (aathIaa) [30: (6,2, + 0:) + 26112 — 30f26a (0n + 0Q] , (4.442) (Wanna: = (a i thl i (1)93. (5680.. - 3912) , (4443) 00.242 = (a 4 MI :1: 903.0491 - 39%,) , (4.444) (mmi = (mu/46219260012 [26%2 — 300, (0., + 01)] , (4.445) (th9401 = ’(3F i Ith i (1)493912, (4-446) (VtIiIFFa = —(i ¥ IVtI i @493912, (4-447) Oiia = (a 3*: Il/tlail29a (2012 — 0a9n) 7 (4-450) (Vt>ai-3Fa = (a 4: IthaiWa (2012 — 086m), (4-451) (V004..— = (964th + 4493.912- (4-452) Figure 4.35 shows the expectation value of the Minnesota potential as a function of (1. As with the other plots, there is little differentiation of the three levels, though there is more here than in the case of the Volkov interactions. The d=0 limit correspond with the alpha-deuteron case (Figure 4.28), with the lowest state coming in at around -128 MeV. 4.6.3 Total energy The energy spectra of the cigar configuration calculated with the Volkov V1 and V2 potentials are shown in Figures 4.36 and 4.37. Once again, all three levels are very close together, even more so, the first two levels. Since the asymptotic state is an alpha particle and two free nucleons, the well is much deeper (in contrast with the other configuration which asymptotically is an alpha particle and bound deuteron). Finally, they are consistent with the alpha-deuteron configuration at (120, where they 140 I fir I—rfi I I If r f W W I 1., -: —901 ———— 1:1 1 r; I i o )- ... g —100— j 9 I 1:2 : v , - —llO: : I — 1:3 I —120 _ :/ 1 l 1 1 1 l 1 1 1 l 1 1 1 J_ 1 1 1 l:: 0 2 4 6 8 10 d(fm) Figure 4.35: The expectation value of the Minnesota potential as a function of d calculated in the cigar configuration of “Li. For this plot, the oscillator parameters are both equal to 0.53 fm‘2, the exchange parameter, u, is equal to one, and the spin- orbit parameters remain unchanged from previous calculations. There is no tensor interaction in this calculation. both have a value of -21.2 MeV (V1) and -23.3 MeV (V2). The story is similar with the Minnesota potential, which is shown in Figure 4.38. There is a more clear separation of states, though they are still very close together. The minimum still occurs close to 1 fm, and the well is still quite deep. As in the previous case, the value of the energy at d=0 (-20.2 MeV) is the same in both configurations. 4.6.4 Mean square radius The mean square radius calculation in the cigar configuration has been discussed be- fore in the “He cigar configuration section (section 4.2.4). For lithium, the calculation is the same except for the projection process. Also, the charge and matter radius are the same for lithium, so only one calculation is necessary. 141 (MeV) d(fm) Figure 4.36: The energy spectrum of the cigar configuration of “Li calculated with the Volkov V1 potential. The oscillator parameters are both equal to 0.53 fm"2, the Majorana exchange parameter is equal to 0.6, and the spin-orbit parameters were not changed from the previous graphs. 1:1 5‘ _ O E L3 J=2 v 1:3 10 d(fm) Figure 4.37: The energy spectrum of the cigar configuration of “Li calculated with the Volkov V2 potential. The oscillator parameters are both equal to 0.53 fm‘2, the Majorana exchange parameter is equal to 0.6, and the spin-orbit parameters were not changed from the previous graphs. 142 (MeV) d(fm) Figure 4.38: The energy spectrum of the cigar configuration of “Li calculated with the Minnesota potential. The oscillator parameters are both equal to 0.53 fm‘2, the exchange parameter, 0., is equal to one, and all spin-orbit parameters are the same as in the previous plots. There is no tensor interaction in this plot. 4.6.5 Electric quadrupole moment The quadrupole moment operator was defined in eq.(4.414). We also must be con- cerned with the center-of-mass contamination, so we must use eq.(4.415) in order to remove the center-of-mass terms. In that equation, there are one-body and two-body terms. The one—body matrix elements are: (alQola> = 0, (4.453) (iIQOIi) = 9n§(l+3x)(1+$), (4.454) 910.142) = 01%20—34x1—m). (4.455) (.uzd2 2 (iIQOIa) Z 012m(3$ ‘1): (4-456) 2012d2 (QIQOIiI = 912W' (4-457) 143 The alpha term vanishes because the alpha is spherically symmetric and is located at the origin. Similarly, all two-body matrix elements which involve a diagonal alpha term vanish, and will not be included in the list. The non-vanishing two-body matrix elements are: (+ — lQol — +) = 435—12414. 3x)(1+ :c), (4.458) (+ — |Q0| + -—) = 43,?1- 3x)(1— 2:), (4.459) (+ — |Q0|a+) = —6n612%%—)—(32 — 1)(1+ 3:), (4.460) <+alQo| - +> = —0..612wa2w(1 + 2:), (4.461) <+ - lQola—) = —01612wa:(1 — 2:), (4.462) (+alQol + —) = 416.55%};(1: — 1)(34: +1), (4.463) (+alQOI + a) = afzflz, (4.464) (+alQol — a) = 43%;5, (4.465) <+ — IQoIaa> = 412%, (4466) (0701le + —) = —6f2(—V“i%—)§(3x2 — 1). (4.467) The one-body matrix elements are plugged into eqs.(3.25)-(3.29), then divided by the norm. For the two body matrix elements, the relevant equations are eqs.(3.30)-(3.44). This then completes the quadrupole moment calculations.- 4.7 Magnetic dipole moment In addition to electric quadrupole moments, nuclei with non-zero angular momentum have magnetic moments as well, which is a measure of the asymmetry in the “magnetic charge” of the nucleus. These magnetic charges come from the intrinsic magnetic moments of the protons and neutrons, and the orbital motion of the protons (neutrons, 144 being uncharged, do not produce any additional magnetic field with their motion). The magnetic dipole moment Operator is A 12... = W 23194040) + 9403.401, (4.468) where [1N is the nuclear magneton (= eh/2Mpc), g; and g, are the orbital and spin g- factors of the i-th nucleon, and m is the projection of the operator, since the magnetic dipole moment is a vector operator. The g-factors are equal to: 1 for a proton 5.586 for a proton 93 = . (4.469) 0 for a neutron —3.826 for a neutron 9: As was the case in the quadrupole moment, we calculate the expectation value of the magnetic moment in the state of maximum projection. We actually calculate the projection of the magnetic moment along the nuclear spin: 0400141) = 711—100(4990. (4470) For our “Li system, we can make a few simplifications. The alpha particle is spinless, and the orbital g-factor of the neutrons is zero, so we can re—write eq.(4.468) as: Z 1‘40 = #N 93(n)S(n) + 93(p)S(p) + 23(9) , (4-471) 1’ where the first two terms are from the spins of the external proton and neutron, and the sum runs only over protons. We can simplify the operator further by working with the total spin of the external particles, S = s(n) + 8(1)), and the total angular momentum, L. Making this substitution, the operator becomes: fio/un = [(9.01) + 93(9))3 + (93(1)) - 93(n))(S(p) - 8(n)) + L] , (4472) NIH 145 where the factor 1/2 appears before L because “Li is an N = Z nucleus, and we can consider half of the orbital angular momentum to come from the protons. The operator s(p) — s(n) plays no role in our calculation as our system is completely symmetric with respect to protons and neutrons. We then take the scalar product of this operator with the total angular momentum, J: S ' J J + J—‘J—J (4.473) (M'JlJ/HN 937m 2J(J+1) 1 where 95 is (93(71) + gs (p)) / 2 or the sum of the magnetic moments of the free proton and free neutron. Since we are working the state of maximum projection, S = 5,, = 1 and J = Jz = 1. We can simplify the first term by using the relation similar to that used for the spin orbit operator: 3 . J = (S+J_ + S_J+) + 5.1,... (4.474) MID—d The terms with raising and lowering operators vanish in the state of maximum pro- jection, and we are left with Ssz = J2. For the second term in eq.(4.473), we use the relation L = J — S. The resulting expression for the magnetic moment is: / _ J2 +J(J+1)—J2 floflN—gs—J+1 2(J+1) I (4.475) where J2 is just a number, not the operator. This is our final expression, which depends only on quantum numbers. It is independent of the spatial configuration. The numerical result will be given in the next chapter and discussed. 146 4.8 Interference term As with the two separate configurations of “Li, there is a lot of repetition with the helium calculations. In the interest of brevity, we will only add the new material here. Thus, there will be no sections on the kinetic energy or mean-square radius, as they are exactly the same as the helium case save for the angular momentum projection. Projecting onto parity is not necessary in the interference term, as all matrix elements already have a partner which is identical save for an angle-dependent term which has the Opposite sign. We will then skip the kinetic energy, and move straight to the interaction. 4.8. 1 Interaction The Volkov central interaction is unchanged from the “He interference term. The tensor term has the Volkov matrix elements plugged into eqs.(3.141)-(3.153). For the spin-orbit interaction, we do have new matrix elements. The matrix elements of the spin orbit operator (eq.(4.410)) with the prefactor —27r3I/;0 suppressed are: 3/2 4‘/V0mw ) wd6[3VOJ + n(V + 201)] sin19 (4.476) (€1lele + (1’) = ( I DZY(V,n,w,w) 12D,(V,n,0),w) x ex N p 72D£Y(V, 71,0), 111)] ’ (£1lele — a’) = — (adIVLSI + a')(2: —-> —:z:), (4.477) where D’,(a,b,c,d) = (a+ b)(c+ d) + 27(a+ b+c+d), and N :46:2 {Vn(02 + w) + 4ww(V + n) + 8’Y[(l/ + 401)(n + w) + 9V0)]} + 9wd2[(V + n)(w + 27) + 270)] — 24wd62:[01(V + n) + 7(20) — V)]. 147 Resuming, we have: 4(/V02nw 3/2 wd6[3V02 + n(V + 202)] , d V ' = — '19 4.47 (a I lea +> (DI,(V,n,02,w)) 12D;(V,n,02, 0)) sm ( 8) x exp ]— N' ] 72D’7(V, 71,02,112) ’ (adIVlea’—) = — (adIVLsIa'+>($ —* —€L‘), (4-479) where N’ =462 [(V + 402)(nw + 27(n + 112)) + V02(4n + w) + 187V02] + 9wd2[V(n + 02) + 27(n + V + 02)] + 12wd63:[V(02 + n) + 27(V -— 202)]. Continuing: 4w(/V02 3/2 wd6[3V02 + w(V + 202)] DIY(V, 10,02, 02) 6D!,(V, 10,02, 11)) (adIVle + —) = ( Sin19 (4.480) T X 9"" [721210.445] (adIVle — +) = — (07lele + —)(:L‘ -—> —-:l?). (4.481) where T =462[w(Vw + 40211) + 5V02) + 27(2V'w + 80211) + 9V02)] + 9wd2[w(V + 02) + 2V02 + 47(V + 02 + 2112)] — 12wd52:(Vw + 20211) + 3V02). These matrix elements are plugged into eqs.(3.135)-(3.140). We remind the reader that in the interference term, Greek letters refer to parameters Of the alpha-deuteron configuration, while Latin letters are used for the parameters of the cigar configura- tion. We have now addressed the tensor and spin-orbit interactions. What is left is the 148 Minnesota potential in the interference term of “Li. As was mentioned before, the Minnesota potential has spin-dependent forces, and one must separate the terms into singlet and triplet terms, and this separation is not the same in the two studied nuclei. In the interference term, the singlet part is: (Vs)a (V9624. 00.9..- (Vs>aaa+ (Vs>aaa— %63..(56..6._—36..0._), (4489) (V0.44— : (adlvtl-a’)%0§.(50..0.+—30..6.+). (4.490) <8)... = 0§i.6... (4.494) 04)...- = ——9?..9a—. (4.496) 0§.(264.0._—0..64_). (4.499) 04).... = (daml —a’>03.. (26.1.6... —0..6.+), (4.500) 04)...- = (Mil/499393.03... (4.501) 04).... = (ddII/Ela’a’)9§.6a+0._. (4.502) The matrix elements for these terms are the same as the Volkov matrix elements. These are plugged into the terms above, and summed together in order to calculate the expectation value of the Minnesota potential. This completes our discussion Of interactions in the interference term of “Li. Next, we look at the electric quadrupole moment. 150 4.8.2 Electric quadrupole moment In order to complete the calculation of the expectation value of the electric quadru- pole moment, we need the contribution from the interference term. The quadrupole moment has been introduced in the previous sections on the subject, so here we just include the matrix elements of the operator in the interference term. As shown in eq.(4.415), there are one-body and two—body terms. The one-body terms are: , _ 21/2 (Z2 — Z)2 (alQola) —0.. (.2 + n), . (4.503) (QIQOI+) =Qa+f(w, V) C) 21)) (4504) (aIQOI_) =00-f(w1 V1C7 Z2)($ _) —.’L'), (4505) (dIQOI+) =6d+f(w1w1C) 22)) (4'506) (deoH =94—f(w.w.C. 22)($ -> ~93). (4-507) , 2 2 - Z 2 (dIQola) =6... “(05:71)? ) , (4.508) where f was defined in eq.(4.202). The two-body terms are: 2 2 2 1 — Z 2 (aalQola'a') =90“, ”(5: n), ) , (4.509) , _ 2V(21—Z)[02(22—Z)+wa:(C—Z)] ((1de0] + a ) —0aa6d+ (I/ + n)(w + w) , (4.510) , _ 2V(21 —Z)[02(22——Z)—w:r(C—Z)] (adeol — a) —0a09d_ (V + ”)0” + w) , (4.511) 151 211(21— Z) [V(Zl - Z) + wx(( — Z)] (aalQola’+) =05505+ (V + 700/ + w) , (4.512) (aalQola’—) =0006..- MZ‘ _ Z1521)? VZ: It)?“ _ Z )1, (4.513) (adeolo/a’) =60a0... 21173;)32; Z), (4.514) (ddeola’+> =0d+0da 2“)” _ 21312342212?“ - Z )1, (4.515) (ddeola’—) =9d_ada 2W? ’ Z 13:23:11?“ ‘ Z )1, (4.516) ((151420) — +) = (V +63%; w) [2.1.422 — Z)(z1 — Z) (4.517) +2w$(C - Z)[V(21 — Z) — we. - Z)] — 52 (c — Z)? (32? — 1)] , 6d—00H- (u + w)(w + w) —2wa:(C — z) [V(z. —- z) - we. — Z)] — .52 (c — Z)? (3.2 — 1)] . 2.5%... — 2)2 — um — z)2(3..2 — 1) (dalQol + —‘) = [2VLU(22 — Z)(Zl — Z) (4.518) (ddeol + —) 26d+6d— (w + w)2 , (4.519) , _ 2w(22—Z)[V(21 —Z)+wx(C—Z)] (daIQOI + a ) —0dafla+ (w + ”X” + w) , (4.520) I _ 2w(zz—Z)[V(Zl “2) -w$(C—Z)] (dOIQol - 0) —94a9a— 2 (a; + n) (V + w) 2 2 . (4-521) (aalQol + —> =6..o.._2” (Z1 ‘ Z) (11(5); 2’ (3.. ‘ 1), (4.522) (ddeolo/a’) =93, 2“2(Z2 ‘ Z>2 (4.523) (w + 12)2 These terms are summed together using the normal overlaps for the interference term (eqs.(3.56)-(3.61) and eqs.(3.62)-(3.76) for the two-body term), and divided by the norm. This completes the discussion of the interference term of 6Li, and indeed the Gaussian approximation of 6He and 6Li as individual nuclei. We just have one final section on the beta decay of 6He, which incorporates the interplay of both nuclei and configurations. 152 4.9 Beta decay Beta decay is how the 6He nucleus transforms into the stable 6Li nucleus. We will start with some background on the subject, then apply it directly to our system. 4.9. 1 Background The process of nuclear beta decay transforms a nucleus to a neighboring nucleus in its isobaric chain: g‘X —»Zfi, Y + e— + .7. (4.524) The form of beta decay illustrated above is the one relevant for the present work, called beta-minus or negatron decay. For completeness, the other types of beta decay are beta-plus or positron decay ’Z‘X —>sz Y + e+ + v, (4.525) and electron capture: 9X + e‘ —->Zfl Y + V. (4.526) The theory of beta decay was first formulated by Fermi in 1934 [55]. Later, it was found out that nuclear beta decay is just part of a class of reactions described by the so—called weak interaction. The weak interaction is mediated by very massive particles (80-90 times the mass of a nucleon), so at normal nuclear energies, Fermi’s contact formulation is essentially valid. One can infer the selection rules from looking at eqs.(4.524) and (4.525). In both of those equations, the right-hand side, in addition to the daughter nucleus, contains two other particles. These particles are electrons (or positrons) and neutrinos (or antineutrinos). These particles are spin one-half particles, and thus can be emitted with total spin zero or total spin one. Since transitions are much more probable if no orbital angular momentum is carried away by the particles, we have two kinds of “allowed” beta decay. The spin zero case is called Fermi beta 153 decay, and the spin one case Gamov-Teller [56]. These selection rules are summarized here: Fermi Gamov — Teller AJ=0 AJ=0,1 (but not0—>0) 71'in = +1 71'in : +1 AJ is the difference in total angular momentum of the initial and final states, |J f —J 1|, and mm is the product of the parities of the initial and final states. With these selection rules, we have the two kinds of allowed beta decay transitions, and can calculate transition probabilities. In beta decay, experimenters often measure the half-life of the given state or nucleus. The half-life is In 2 t1 2 = , (4.527) / T),- where Tfi is the transition rate between a specific set of initial and final states. Inserting the expression for Tfi, eq.(4.527) simplifies to Ko t = , 4.528 where K0 is a collection of fundamental constants: 2'1r3l‘i7 1n 2 K = —— = 6147 4.529 0 mgc4G%~ 8’ ( ) f0 is a dimensionless phase-space integral involving the lepton kinematics in the Coulomb field of the daughter nucleus (known as the Fermi integral), and BF and BGT are the reduced transition probabilities, analogous to the B(E2) of the electro- 154 magnetic transitions. They are: 2 2 9v 2 9A 2 B = B = 4.530 F 2.4-MW” GT 2J.-+1|MGT" ( ) where gv and 9A are the vector and axial vector coupling constants, respectively, J,- is the total spin of the initial state, and M is the matrix element that contains all the nuclear information. A simpler quantity to work with than the half-life is a quantity that removes the Fermi integral: 10g ft = 10g fotl/g, (4.531) where the logarithm is used because ft is often a very large number. The matrix elements in eq.(4.530) involve the Fermi and Gamov-Teller transition Operators. For BF, we have IMFI2 = |l27 (4532) where T3: = 2 ti, (4.533) I: where ti is the isospin raising or lowering operator and is summed over all nucleons. The Gamov-Teller operator is OCT = ati, (4.534) where a' is the spin vector, and ti is the isospin raising or lowering operator as before. This operator is also a one-body operator, and is summed over all nucleons. We now have all the information we need in order to proceed to our specific problem, the decay of 6He. 155 4.9.2 Helium-6 beta decay The beta decay of 6He proceeds like in eq.(4.524): SHe ——>§Li + e“ + 17. Given that 6He has a ground state of 0+, and the ground state of 6Li is a 1+ state, we have a pure Gamov-Teller decay. A decay to any of the excited states of lithium would be a forbidden decay, as AJ > 1. We then just need to concern ourselves with the Gamov-Teller transition from the previous section. Our goal is to calculate BGT, which for our specific case is: 2 B... = Z <6Li<1mf>l Zafit‘iIGHemo» . (4.535) V,m f k where the sums go over k, the neutrons in 6He, m f, the magnetic sub-states of the final state in 6Li, and V, which is the index of the Pauli spin matrices. We work with the following representation of the Pauli matrices: a' = 0,,)? + 0,,)? + 022 (4.536) 1 l = 5(0++U_)i+‘2—z(0+_0_)y+0227 (4'537) where 0 2 0 0 0+ = a- = , (4.538) O 0 2 0 and 0,, is the normal 2 term of the Pauli matrices. We then have to calculate the overlap of 6Li with a 6He nucleus where a neutron has been changed into a proton. To guide our calculation, we will begin by verifying the Gamov-Teller sum rule for our system. 156 For Gamov—Teller decays, a sum rule exists that states: ZIBS'T’i — 88.4] = 3(N. - 2.), (4.539) f where the sum is over the final states, and the sum rule itself is the difference in the 307* value for a given initial state to either 6+ or H‘ decay. The result depends on the proton and neutron numbers of the initial state. In the case of 6He, the sum rule is equal to six. To see if we can obtain this result, we first look back at our starting equation, eq.(4.535). In principle, because of the sum over spin projections and final m states, we would have nine terms in the sum. However, by re—writing the sigma operator in terms of the raising and lowering operators, we can reduce the number. Since the 0i operators flip the spin of a particle, they can only connect 6He’s ground state with the :l:1 projections of the final state in 6Li, while the 0,, operator connects to the longitudinal projection of the final state. The sum in eq.(4.535) runs over all the neutrons in 6He, but if it acts on a neutron in the alpha particle, the matrix element vanishes. This is because it creates a third proton inside the alpha particle where there are already two s—wave protons, thus a proton cannot be created there without an excitation in orbital angular momentum, which is not permitted for an allowed Gamov-Teller transition, so we can simplify our calculation by considering the external neutrons only. In calculating the sum rule, we begin with the term that corresponds to 0,,, which connects to the m f = 21:1 final state in 6Li: 1 (6Li(1;l: 1)|axt+|6He(00)) = -2-(1 :l: 1,,,-|(a+ + o_)|00H.,). (4.540) Now we work with just the m f = 1 final state. Only the 0+ term contributes, and 157 consider only the external particles. We write in first quantization to be clear: i((n+P+ " P+n+) |0+t+l (n+n_ — n_n+)) = %<(n+P+ — P+n+) | (”+11% — P+n+)> = 1- (4.541) This equals one only because we have assumed, in this ideal case, that all parameters are identical between the two nuclei, and then the overlap can be one. Now, for the m f = —1 final state: 1 1 Z<(n—P- — p+n+) IU—t+| (n+n_ “ n_n+)) = 5((71—13— " pm...) I (p_n_ "‘ n—P—)> = ‘1- (4.542) When summed together, the result for 03 is zero. We now look at the 0,, terms. For 0,,, our starting point is 1 -2—2(1 d: 1Li|(a+ — a_)IOOHe). (4.543) The action of the operators is the same as for 0,,, but there is an extra minus sign, which makes the result equal to 22. We then take the magnitude squared, which is four. We have two thirds of the expected su‘m, with one term left, the oz term. The wave function of the longitudinal state of 6Li looks like this in first quantiza- tion (again, referring only to the external particles): W614i); 10) = (|P+n—) + IP—n+) - ln+P—> - ln-P+>). (4-544) [\DIt—l where p is a proton and n is a neutron, and the :l: refer to spin projections. One can see that the wave function is symmetric with regards to spin exchange, and antisymmetric with respect to the “flavor” exchange of proton and neutron (the state is an isosinglet). We now turn to the action of the oz operator which connects 6He with with m f = 0 state in 6Li. We begin with 1 — = \/§ (4-546) We then square the result, we get two, which then completes the sum rule. Now that we have verified the sum rule, we can look into the overlap of our lithium and helium wavefunctions. We must find the overlap of alpha-dineutron with alpha- deuteron, cigar with cigar, and the cross terms. Each overlap is of the form similar to that which was calculated in the interference term (eq.(3.49)), except it is an overlap of a configuration of 6Li with a configuration of 6He where one of the neutrons has been turned into a proton. We will begin with the alpha-dineutron alpha-deuteron overlap. For clarity, we will call the alpha-two—particle cluster configurations cl, the first configuration, and the cigar configurations Cg. We then want to calculate: 0310, (gaagdd — adagad)2 . \/(¢1(Li)|¢1(Li)>(¢1(H€)I¢1(He)) <¢1(Li)|¢1(He)) = (4-547) The denominator in eq.(4.547) contains the usual normalizations of both systems. We now have some new overlaps to list, and we will need a new convention for the symbols of each configuration and nucleus. Greek letters will refer to parameters in lithium, and Latin letters will refer to parameters in helium. Parameters will also be labeled with subscripts. A subscript one will indicate the first configuration of the particular nucleus, and a subscript two will indicate the cigar configuration. With 159 that in mind, we list the overlaps used in eq.(4.547): 3/2 2 _ 0aa=(ala') = (2V”‘"‘) exp [-”‘n1(61+dl 20116130], (4.548) V1 + 72.1 18(1/1 + n1) 2,/w1w1 3/2 2w1w1(6f+ d? '- 2d161$) 0 = d d’ = —— — 4.549 dd < I > (w1+ wl) exp 9(w1+ wl) ’ ( ) (9 d = (ald’) = 2./V1w1 3” exp F_V1w1(5f+ 4d? + 461mm)“ (4 550) a V1 + wl L 1804 + 101) , . 6.15. = (dla') = _2____w1nl 3/2 exp Lwynl (46% + d? + 461(1117)‘ (4 551) L01 + 711 L 18(0)] + H1) - . These are the overlaps contained in eq.(4.547). We then look at the transition between the two cigar configurations: 92‘... (63", + 93.) + 262 02 02 - 263a0an6m (6..., + 0.) (¢2(Li)|¢2(He)) = . a“ a". "a (4552) \/ (¢2(L1)I¢2(Ll))(¢2(He)l’¢2(He)) The overlap in the above equation are: 2 3/2 a... = = (———V”2"’) . (4.553) 1’2 + 72.2 2,/— 3/2 F 2 _ ‘ 0.... = was) = (#3233) exp 3"”? (‘52 "L d3 252d”) , (4.554) 012 + 1112 ) 8(w2 + “’2) J 2,/w2w2 3/2 - £02102 (5% + d3 + 262d2$)- 6 = :l: ' = — , 4.555 i < FF) (w2+w2) exp _ 8(w2+w2) ( ) 2 1121112 3/2 F Vg'wgdg - 0.... = ' = V —— , 4.5 (a|n> (V2 + 1122) €Xp L 8(1/2 + w2)_ ( 56) I 2./w2n2 3/2 ) (4)272263 - no, 2 = —— —— . 4. 6 (nla) (“)2 + n2) exp L 8(w2 + n2)‘ ( 557) The next transition is from the cigar configuration of 6He to the alpha-deuteron configuration of 6Li. The form of the matrix element is: — figga [0306d+6d_ + 93aOQ+00— - 0000(10 (00+0d_ + 60_0d+)] lLi 2He = W W ” ¢ (4.558) The minus sign makes the overlap equal to positive one when all parameters are equal 160 and the distances are set to zero. The overlaps are: 2‘/V177,2 3/2 11177.26? 00,0, = ' = —— -— , 4.559 (ala) (V1 '1' n2) exp 18(111 + 77.2) ( ) 2 (4)1’UJ2 3” ( (4)1102 (166? + 9d§ — 2461d233)) 9 = d +, = .._ _ , 4.560 .. H > (51W) expL 726M...) (( > 2 3/2 r' ' 9d— 2 (dl—') = ( ,/w1w2) exp _w1w2(165f + 9d§ + 2461d2x) ,(4.561) (4)1 + 77.2 L 72(w1 + 7.02) _ 2,/V1w2 3” ' Vle (45? + 9d§ + 1261012511)1 6a = +’ = — , 4.562 + (0| ) (V1 + (1)2) exp _ 72(1/1 + 102) J ( ) 2 3/2 F' 462 _ d '1 6,_ = (6):) = ( V ”11“?) exp —"‘w2( 1 + 9‘13 126‘ 25”) , (4.563) 111 + W2 _ 72(1/1 + 102) J 2 (4)1712 3/2 1' 2.017226% Gazd’ = V —-—————. 4.564 d < la) ((4)1 + 722) €Xp _ 9(0)] ‘1‘ n2) ( ) Finally, we have the transition from the alpha-dineutron of 6He to the cigar configu- ration of 6Li. The matrix element is: _ @000, [662100441041 + 6200+00—a _ 60016110: (0+a6—d + 6—06+d)l 2Li lHe = W W ” \/ (5: + w.) exp _ 720,? + w.) J .( 56 ) 2,/——..;2w1 3” ' w2w1(96§+16d§ + 2462dlx)' __ = — d’ = — 4. 0 d < I > ((4)2 + (1)1) exp _ 72((4)2 + 701) d ,( 568) , 24?an 3/2 F w2n1(95§+ 4d? +12agdlx)' a = = _ , 4.569 6+ (+IO) ((4)2 ‘1' n1) eXp L 72((4)2 + 72.1) J ( ) 0 : (—Ia’) = 2,/w2n1 3/2 exp (_(4)277.1 (96% + 4d? — 12626111!)1 (4 570) _a (4)2 + 77.1 L 72(w2 + m) J ’ j ' 2‘/w2n1 3/2 (- 2w2n1d¥ no: = I = —_ . 4. 71 0 (”'01) ((4)2 + 72.1) exp L 9(w2 + 71.1) ( 5 ) We now have all the overlaps of all the configurations. The total matrix element 161 depends on these overlaps as well as the coefficients of the individual coefficients in their wave functions: (\II(Li)|a't+|‘IJ(He) = ClLiCIHeg-nll + C2LiC2He9-nz2 + CiLiC2He£mzl + C2Li61He9-n12, (4-572) where 911,-; is shorthand for the overlap matrix elements given above, and the first subscript is the initial configuration and the second subscript is the final configuration. The coefficients a, are determined by the minimization of the expectation values of the Hamiltonians of the individual nuclei. Since the overlap does not depend on the magnetic sub-states of 6Li, the result of eq.(4.572) is squared, then multiplied by six to obtain the final result for BGT. Thus, if the overlap was a perfect one, then we would obtain the sum-rule of six. The result is then plugged into the equation (4.573) where the axial-vector coupling constant 9,; is equal to 1.2695 [57], which gives the log ft value for the beta decay of 6He. This concludes the chapter which shows the results of the Gaussian approximation in the two systems of interest in this work. In the next chapter, we will show the numerical results of the calculations and compare them with experiment and other theories. 162 Chapter 5 Numerical Results In this chapter, we show the numerical results of the calculations outlined in the previous two chapters. They are then discussed and compared with experimental findings and the results of other theories. The results will be presented in several tables. The first to be discussed will be 6He, beginning with the optimal variational parameters and relative weights of the configurations, then energies, and finally other observables. We will then follow with the same for 6Li. 5. 1 Helium-6 5 . 1 . 1 Spatial parameters We begin the discussion of the 6He results with showing the optimal values of the variational parameters for the ground and excited states, which are displayed in Table 5.1. The variational parameters are the two oscillator strengths, u and w, and the alpha-external neutron distance d. They are shown in the table for three potentials: Volkov V1 (the column V1), Volkov V2 (column V2), and the Minnesota potential (column M). The table shows that there is little difference in the minimization between the two sets of Volkov parameters, and indeed the Minnesota parameters are also similar, 163 Table 5.1: This table shows the optimized variational parameters of the two config- urations of 6He for the three different potentials considered: Volkov V1 (V1), Volkov V2 (V2), and Minnesota (M). H Um”) w (fm‘2) d (fm) V1 V2 M V1 V2 M V1 V2 M a-2n 0.51 0.51 0.56 0.40 0.40 0.35 3.71 3.71 3.71 cigar 0.50 0.50 0.50 0.48 0.48 0.45 1.61 1.61 1.51 Oz-2n(2+) 0.50 0.50 0.56 0.30 0.30 0.30 3.01 3.01 2.41 cigar(2+) 0.50 0.50 0.51 0.40 0.40 0.42 1.01 1.01 0.91 even for the excited state. This is not too surprising, as was seen in the plots of the previous chapter, there was not a lot of qualitative difference between the potentials. There is an interesting contrast between the two configurations when it comes to the location of the minimum. In the alpha-dineutron, the alpha-neutron distance is rather large, and if one looks at the kinetic energy plot (Figure 4.5), one can see that the kinetic energy is still close to its asymptotic value, meaning the neutrons are still mostly all in s-waves. In the cigar configuration, however, if one looks at its kinetic energy (Figure 4.17), at the point where the minimum occurs, the kinetic energy is much higher than its asymptotic value, which means that two of the neutrons spend most of their time in the p—shell. The minimum is at a smaller value of d in the cigar configuration because the potential falls off faster in this configuration. This is due to the fact that as (1 increases, not only is the attraction between the external neutron and the alpha particle decreasing, but the attraction from the other neutron decreases even faster, which keeps the neutrons closer to the alpha particle in the cigar configuration. 5.1.2 Energy The energy results and relative weights of the configurations are shown in Table 5.2. One can see when comparing the results obtained with the two Volkov potentials, that there is consistently about a 1.5 MeV difference in the energy calculations, and a 9.5 MeV difference between the alpha-dineutron configuration and the cigar con- 164 figuration. This in turn gives similar results for the relative contributions of each configuration in the overall wave function. In the Minnasota potential, the two con- figurations are much closer in energy. This is because 6He is built up mostly of singlet pairs, which have a much weaker attraction in the Minnesota potential. It is then less favorable to have the two external neutrons close to each other than with the Volkov potentials. Thus, the alpha-dineutron configuration is less dominant in the system described by the Minnesota potentials. In the case of the weights of the configura- tions, the balance of the wave function (since the sum of the two 03’s is not one) is carried by the interference term, which means there is a fairly large overlap between the two spatial configurations (around 32% in the case of the Volkov potentials and 39% in the case of the Minnesota potential). As for the energy of the 2+ excited state, the origin of the large gap is the alpha-dineutron configuration, and specifically the kinetic energy. At the larger values of d, the Volkov potentials to not greatly distin- guish between the two levels, but the kinetic energy does, which contributes the most to the large gap between the ground state and first excited state. In the case of the Minnesota potential, the source is in the interference term. Though the two individual configurations match up well with the observed gap, the interference term is still close to 40% of the wave function, and it has a much larger gap, which then creates the large overall gap seen in the table. For considering the excited state, one often needs a larger amount of input than to describe the ground state. For the excited state, the spin-triplet configuration of the external neutrons may become important. According to a few-body calculation [58], the spin-triplet configuration accounts for 32% of the wavefunction of the 2+ excited state. Also, eventually, triton clustering would also become important, but we don’t expect triton clustering to be important just 2 MeV above the ground state. Other theoretical models have also done well in reproducing the binding energy of 6He. The Fermionic Molecular Dynamics (FMD) approach obtains a -29.1 MeV value when working with the realistic Argonne V18 potential [59]. A similar method, An- 165 Table 5.2: The total energy results for the individual configurations and the minimized rasults, plus the weights of the configurations in the wave function and the excitation energy of the 2+ excited state. The calculated energies of the alpha particle and the alpha particle plus dineutron are provided for comparison. (E) (MeV) 6,2 E(2+) (MeV) V1 V2 M V1 V2 M V1 V2 M a—2n -25.7 -27.2 -19.1 .598 .607 .441 +3.891 +3.770 +1.932 cigar -16.3 -17.7 -14.3 .0864 .0824 .168 +1.665 +1.571 +1.002 overall -272 -28.7 -22.1 N/A +4990 +4912 +4714 exp [31] -293 N/A +1.797 a -28.0 -28.9 -251 N/A N/A a+2n -26.1 -272 —19.7 N/A N/A tisymmetrized Molecular Dynamics (AMD), obtains -28.6 MeV with the Volkov V2 interaction (modified with the addition of Bartlett and Heisenberg exchange terms), the Coulomb interaction and a spin-orbit interaction (which does not vanish in their formulation) [60]. AMD calculations also obtained excellent agreement for the ex- cited state of 6He with a result of +1.86 MeV relative to the ground state. Unsur~ prisingly, the ab initio models, Variational and Green’s Function Monte Carlo (VMC and GFMC) [33,61] and No—Core Shell Model (NCSM) [32] achieve almost perfect agreement with experiment with their energy results. These models have many more parameters, including some kind of three—body interaction in order to achieve agree- ment with experiment. Their results for two-body forces only are -23.8 MeV (Argonne V18) [61] and -26.7 (CD—Bonn 2000), respectively. As for the relative contributions of the two configurations, FMD obtains two minima in their calculations, corresponding to our cigar and dineutron configurations. In their results, the cigar configuration only lies 1.1 MeV above the alpha-dineutron configuration. Thus, they agree that the dominant configuration would be the alpha- dineutron configuration, but not nearly as dominant as in our results. This could be a result of the interaction, as we have seen with a more realistic interaction (the Minnesota potential), the two configurations are much closer in energy compared to the Volkov potential results, so perhaps we would obtain a similar result with a more 166 complicated interaction. Another point of view is expressed by Bertulani and Hussein [62]. From elec- tromagnetic dissociation data, they have extracted a B(E 1) value, and from that determined the opening angle between the neutrons in 6He to be 83°. This is an interesting result, as that angle is almost halfway between our two configurations. However, the opening angle of 83° also leads to a matter radius that is much larger than the experimental matter radius, so the question seems far from resolved. A mean field calculation studying dineutron correlations [63] found both cigar-like and dineutron correlations in 6He, both of which were dominated by spin singlets. They also found more particle density in the dineutron configuration than the cigar con- figuration, but their results were obtained with Hartree—Fock—Bogoliubov model with Quasi-particle Random Phase Approximation, which has questionable validity for a six-particle system. Experiments have also been done in order to try and determine the dominant component of the 6He wave function. An experiment that looked at 6He break-up on a 209Bi target found that the cross section for one neutron transfer was one-fourth that of the two-neutron transfer [64]. Another experiment looked explicitly for one and two-neutron transfer with a reaction on copper, and its preliminary results show the two-neutron cross section is greater by two orders of magnitude [29]. Most experiments use the dineutron model in their analysis to calculate things such as reaction cross sections [65], but this has more to do with the ease of the calculation than a completely accurate structural picture of 6He. In other words, the geometrical picture of 6He is still an open question, but from the current results, the cluster picture appears more adequate. 5.1.3 Radii and other observables Table 5.3 shows our results for the charge radius, matter radius, and B(E2); O+ —> 2+ for 6He, and includes experimental results and the results of other theoretical models. 167 As before, for our results, we divide them into columns for each of the different potentials, and show results for each individual configuration. Configuration results for the B (E2) are not really applicable since the transition links both configurations. Items in the table marked “N/ A” in the rows for other theoretical results mean the author is unaware of results for that particular quantity. When looking at our results for the charge radius, the result obtained in the alpha- dineutron configuration most closely matches the experimentally observed number. The cigar configuration is much smaller, since the charge resides at the center-of- mass in this configuration, and thus should just be the size of the alpha cluster. The alpha is enlarged in our model slightly, which gives the value shown in the table compared to the measured 1.67 fm for the alpha particle. This is not a problem, as the presence of the neutrons could certainly cause the alpha particle to swell when compared to an isolated alpha. An enlarged alpha could better overlap with the somewhat distant external neutrons. The cigar component is a small part of the wave function, but the interference term, which comprises 32% of the wave function in the case of the Volkov potentials contributes an even smaller value, which accounts for the slightly undersized charge radius. Since the interference and cigar configurations account for an even larger part of the wave function calculated with the Minnesota, both the charge and matter radii obtained with these wave functions are small. The matter radius calculated with the Volkov potentials is satisfactory, especially when one considers the wide dispersion of experimental values that have been reported. Values of 2.26 [66], 2.33 [67], and 2.52 [68] have also been published from experiments performed in the late 19803 and early 1990s. The other theories also do well for the radii, especially the ab initio models. It should be noted that the AMD calculations add constraints in order to fit matter radii, so it is no surprise that they can produce a large neutron radius. Our result for the B(E2) is comparable with other theories and experiment, but there is a large spread of results. The experiment is a difficult one because the 2+ state 168 Table 5.3: Results for the charge radius (rch), matter radius (rm), and B(E2) calcu- lated with the three different potentials discussed in this work. Experimental values and the values of other theoretical models are also shown. Experimental uncertainties are shown in parentheses after the quoted value. J73) (fm) (”Z (fm) B(E2)0+—> 2+ (e2fm4) V1 V2 M V1 V2 M V1 V2 M a-2n 2.08 2.08 2.05 2.51 2.52 2.36 N/A N/A N/A cigar 1.75 1.75 1.75 2.14 2.14 2.14 N/A N/A N/A overall 1.99 1.98 1.91 2.40 2.40 2.26 2.891 2.932 2.305 exp 205(1) [70] 248(3) [71] 5.4(7) [64], 3.2(6) [69] AMD N/A 2.37 :60 N/A FMD 2.02 :59 2.42 :59; N/A GFMC 2.05 72 N/A 9.05 [73] NCSM 2.03 :74 N/A 1.056 [32] is above the alpha—n — 71 threshold, and the experiments report varying amounts of model dependence in their results (more so in the older experiment [69]). To add to the theoretical results, we quote a few-body calculation, which gives a B(E2) of around 1.0 62 fm4 at the resonance, then grows as more of the continuum is included [58]. As mentioned, this is a difficult experiment, and it seems more experiments should be done before any consensus will be formed on this process. 5. 1.4 Asymptotics Loosely-bound, few-body systems are unique in many ways, one of which is the asymp- totic behavior of the wave function. Since the external particles, in this case neutrons, exist further away from the tightly bound core than the typical range of the nuclear force (1-2 fin), they are in the classically forbidden region. Particles in the classically forbidden region should have exponential asymptotic behavior. In our formulation so far, all particles have Gaussian asymptotics. We wanted to test to see what kind of impact these asymptotics might have. We reasoned that these asymptotics are first most appropriate for the alpha- dineutron configuration. Our minimum in energy occurs at 3.7 fm, which is far into the classically forbidden region. The cigar configuration is more compact, and has a 169 minimum at 1.6 fm, which is a borderline case. We also decided to check the effect in the body-fixed frame. Adding the exponential tail destroys the analyticity of the calculations, and further including angular momentum projection is extremely taxing computationally, therefore we proceeded with calculations in the body-fixed frame. We did not change the alpha-particle wave functions, since it is a tightly bound system. The dineutron wave functions were changed to: (7)3“ exp (—wr2/2) r < R ¢d(7') = 3/4 (i) exp (sz/ 2) exp (-er) 7' 2 R. where R is the matching radius. The calculations were done in spherical coordinates, with the modified tails of the dineutron particles going away from the alpha particle. For our calculations, R was chosen to be 3 fm, which is the approximate point at which the Volkov interaction becomes negligible. In Table 5.4, we show some results from the body-fixed frame with the alternate asymptotics (Exp Asym) and the reference Gaussian asymptotics (Ref Asym). We show them in this form because as a plot, the two curves would be on top of each other. As one can see from the values in the table, the difference is not tremendous. The new asymptotics create a minimum which is lower by about 180 keV, then it falls off slightly faster, becoming shallower by around 300 keV before converging to essentially the same asymptotic value. The matter radius was also calculated for the value of d which is the minimum in the projected ground state (3.71 frn), and the value was 2.342 fm with the new asymptotics and 2.341 with the old asymptotics. In our calculations, the exponential asymptotics did not appear to make a great difference. They should, however, be perhaps investigated further. Perhaps a more realistic potential, or the addition of a long-ranged interaction such as the Coulomb interaction (not necessary for 6He, but would be desirable for 6Li). Also, upgraded computational techniques would make examining the effect of the asymptotics easier, as with the current methods some of the numerical integrals took 48 hours to com- 170 Table 5.4: A table of values at various values of d of the total energy calculated with exponential asymptotics(Exp Asym) and Gaussian asymptotics (Ref Asym) in the body-fixed frame of 6He. For the exponential asymptotics, the matching radius was set to 3 fm. (E) (MeV) (1 Alt Asym Ref Asym 2.1 —11.2654 -11.5031 2.6 -11.8268 -11.6410 3.1 -11.5632 -11.3159 3.6 -10.6260 -10.6390 5.1 —8.3442 -8.6718 7.1 -8.1492 -8.1432 plete, which makes progress on calculating matrix elements (of several integrals) very sluggish, and additionally projected wave functions could be examined. This concludes the section on the numerical results for 6He. We have shown results with fair to good agreement with experimental data. It is important to note that our model is a simple model with six parameters in two configurations (two oscillator parameters and one distance per configuration). AMD calculations have three pa- rameters per basis state, and approximately 150 basis states are used for a converged calculation. FMD calculations have 7A parameters (where A is the number of nucle- ons) per Slater determinant, and the best results are obtained with a superposition of many Slater determinants. They are able to generate very accurate results with these highly computational methods, but we are able to obtain comparable results with physically clear, and simple input. 5.2 Lithium-6 5.2.1 Spatial parameters As with the section on helium, we begin by showing the minimized variational pa- rameters for 6Li. These can be found in table 5.5. The results for the two Volkov potentials are nearly identical once again, though this time there are some very small 171 Table 5.5: The minimized variational parameters of the ground state 6Li and two excited states. Results are shown for the three potentials discussed throughout this work. V (fm‘2) w (fm‘2) (1 (fm) V1 V2 M V1 V2 M V1 V2 M a—d 0.54 0.54 0.56 0.55 0.55 0.60 3.51 3.50 2.71 Cigar 0.50 0.50 0.52 0.50 0.50 0.52 1.21 1.11 1.01 oz-d(2+) 0.53 0.54 0.56 0.52 0.50 0.58 3.51 3.40 2.71 cigar(2+) 0.50 0.50 0.52 0.50 0.52 0.52 1.11 1.11 1.01 oz-d(3+) 0.53 0.53 0.56 0.52 0.52 0.58 3.40 3.40 2.61 cigar(3+) 0.50 0.50 0.52 0.52 0.52 0.52 1.01 1.01 1.01 differences (and more for the excited states). The Minnesota results are fairly similar to the Volkov ones in the oscillator lengths, but are different in the minimum value of d, especially in the case of the alpha-deuteron configuration. The reason for this will be discussed in the section on energy. We also see that the cigar configurations are consistently more compact than the alpha-deuteron configurations, though their single-particle constituents are more diffuse (a lower oscillator parameter means a more spatially diffuse object). The alpha-deuteron configurations have much lower kinetic energies than the cigar configurations because of their large values of d. Even in the case of the Minnesota potential, the alpha-deuteron configuration is 20 MeV lower in kinetic energy. The more rapidly falling potential once again confines the cigar configuration into a smaller space. 5.2.2 Energy In Table 5.6, the ground state energy and relative weights are shown for 6Li. When looking at the table, we see some similarities with the results for 6He. Once again, the Volkov potentials are much stronger in binding the alpha-deuteron configuration rather than the cigar configuration. The disparity is even greater in the case of 6Li, where the margin is almost 12 MeV where it was 9 in the case of 6He. The disparity is less when calculated with the Minnesota potentials. When compared with exper- 172 Table 5.6: The results for the ground state energies of “Li calculated with the Volkov V1, Volkov V2, and the Minnesota potential. The Volkov calculations were done with a tensor interaction, and all three were performed with a spin-orbit interaction. Results for the alpha particle plus deuteron are given as a guide (row “a + d”). (E) (MeV) 6? V1 V2 M V1 V2 M a-d -35.6 -37.4 -27.6 .700 .722 .741 cigar -24.2 -25.8 -21.3 .0493 .0445 .0279 overall -36.5 -38.2 -27.9 N / A a+d -332 -344 -249 N/A exp [31] -32.0 N/A iment, the Volkov potentials over-bind substantially while the Minnesota potential seems to be underbound. The source of the over-binding the Volkov potentials can be traced to the tensor interaction. The Volkov potential was not designed with a tensor interaction, and thus using it as a radial form factor for a tensor interaction produces too much binding. This also, however, keeps the value of d large for “Li despite being over-bound. The tensor interaction in the free deuteron is stronger than when the deuteron is brought closer to the alpha particle, so the tensor interaction effectively pushes the deuteron further away from the alpha particle. In the case of the cigar configuration, asymptotically there is only a free proton and neutron, so there is no tensor interaction, and thus in the cigar configuration, the tensor pulls the two particles closer to the alpha particle. The spin-orbit interaction, which also does not follow any prescription of any particular potential model, appears to only have a small effect with its current set of parameters, affecting the binding energy by at most 500 keV and pulling the particles to slightly smaller values of the distance parameter. Other than the ab initio models (which reproduce the binding energy of “Li very well), there are not many other theoretical models to compare with. The author is unaware of results for 6Li in either AMD or F MD. There are mid-903 calculations of a Russian group using a method inspired by Resonating Group Method that they call Antisymmetrized Multicluster Dynamic Model with Pauli projection (AMDMP), 173 which uses cluster wave functions, nucleon-nucleon potentials, and an alpha-deuteron potential of their own devising [75]. They use a Pauli projection technique in order to exclude Pauli forbidden states, such as all particle sitting in s-waves, though we have seen under certain conditions this is not necessarily forbidden. Their best result for the binding energy of “Li is -31.5 MeV [76] with the Reid soft core potential and their own alpha-deuteron potential. We can also compare with the results of Wildermuth and Tang, who were mentioned in the introduction as earlier pioneers of cluster models. Their best result for the “Li binding energy is -29.9 MeV [24], calculated with an early version of the Minnesota potential. Our results for the relative weights show that the alpha-deuteron configuration dominates for all three potentials, containing between 70-74% of the wave function. The weight of the cigar configuration is quite small, less than 5%, which means the interference term accounts for around 25% of the wave function. There is very little discussion of the cigar configuration of “Li in the literature. One mention was in a recent three-body calculation by Horiuchi and Suzuki [77]. They calculated two— particle correlation functions, and found that while the cigar-like peaks were of equal height in helium and lithium, the deuteron peak in “Li was twice the height of the dineutron peak in “He. This is qualitatively similar to the results of our calculations. In Table 5.7, we report our results for the excited states of “Li. Here our results do not reflect what is observed in experiment. The excitations are far too low, and the levels occur in the wrong order. Only the Volkov V2 interaction in the cigar configuration yields the observed sequence of states. The other theoretical models correctly predict the order of states, and are able to calculate with fair agreement the excitation energy of the levels. The most likely cause is in the interactions, specifically the tensor and spin-orbit interactions. Our method is variational, but our results are lower than the experimental results. It would be interesting to see the results using a potential with a more realistic tensor and spin-orbit part. 174 Table 5.7: Results for the excited states of “Li, calculated with the usual three poten- tials. We also show the results of the ab initio models and AMDMP. Experimental uncertainties are indicated in parentheses. E(2+) (keV) E(3+) (keV) V1 V2 M V1 V2 M a-d 684 687 132 1257 1267 1557 cigar 18 212 503 316 196 1020 overall 536 738 728 1273 1202 1513 exp [31] 4312(22) 2186(2) NCSM [32] 4610 2841 GFMC [61] 4000 2800 AMDMP [76] 4989 2660 5.2.3 Charge radius and other observables In Table 5.8, we show our results for the charge radius, quadrupole moment and magnetic moment. Our charge radius is smaller than the observed charge radius, and those obtained from other theories (except NCSM). From our minimum parameters, one can see why this occurs. First, the radii are smaller than the “He minimum values, as the push and pull between the tensor interaction and the spin-orbit and central potential slightly favors smaller radii. This is especially so in the case of the Minnesota potential where there is no tensor interaction, and thus the Minnesota results are very small. Also, the oscillator parameters are rather large for lithium, which focuses the particles more, causing the radius to be smaller. Our quadrupole moment results are large and positive. Cluster models seem to always achieve a positive value, as commented on by Wiringa in [33]. Horiuchi claims this is caused by using an alpha cluster with four s-wave particles and a tensor interaction [77]. Our large value comes from the dominance of the alpha-deuteron cluster which has the larger size (the quadrupole moment does scale with d, though not as quickly as T2), and is also sensitive to the method of angular momentum projection. The three-body calculation of Horiuchi [77] also gives a positive result, +0164 6 fm2. The magnetic moment, as mentioned in section 4.7, is independent of configuration and geometry. Our value of .69 my is above the pure Schmidt model [78] result of 0.62 MN, but below the 175 Table 5.8: Shown here are the charge radius, quadrupole moment, and magnetic mo— ment calculated for “Li, plus experimental values and values obtained by other theo- retical models. 7% (fm) Q (e fm?) (4 0m) V1 V2 M V1 V2 M V1 V2 M a-d 2.30 2.30 2.08 1.20 1.20 0.77 +.690 +.690 +.690 cigar 2.03 2.02 1.95 0.53 0.50 0.46 +.690 +.690 +.690 overall 2.26 2.27 2.09 1.02 1.04 0.70 +.690 +.690 +.690 exp 252(3) [79],2.55(4) [80] -0.0818(17) [31] +822 [31] NCSM [32] 2.31 -0042 +847 GFMC [72] 2.53 032 +817 AMDMP [76] 2.53 0.49 +829 experimental value. A contribution that would depend on the configuration and the interaction is one proportional to the spin-orbit interaction, but in our estimates it was too small in our model to have a significant effect. Our model is too simple to incorporate things such as meson-exchange currents which also contribute to the magnetic moment. Perhaps a different spin-orbit interaction would allow us to achieve better agreement with experiment. 5.3 Beta decay In Table 5.9 we show the results of the 6He—>“Li beta decay. Our result for the Volkov potentials agrees very well with experiment. The Minnesota result does not, and one might ask why the Volkov results are so much better than the Minnesota result. The Gamov-Teller decay is essentially an overlap of the two wavefunctions. Thus, what is essential are the relative similarity of the minimum variational parameters and the relative weights. For “He, the Minnesota result shows that the wave function is only 44% alpha-dineutron, compared to almost 75% alpha-deuteron for “Li, whereas the Volkov results are around 10% of each other. Also, the oscillator parameters and especially d values are closer together with the Volkov potentials. The energy results may seem at first glance to be also less than ideal, but within the potential model 176 Table 5.9: Shown here are the results of the beta decay calculation for the decay of “He to “Li along with the experimental results and other theoretical calculations. log ft V1 V2 M This Work 2.90 2.90 3.81 exp [31] 2.91 NCSM [32] 2.86 GFMC [73] 2.92 AMDMP [76] 2.90 used, they are not that bad. The over-binding is not that big when compared with the alpha-deuteron threshold (experimentally 1.47 MeV [31]). Thus, the ground state wave functions are actually decent, and the beta decay result reflects this. This helps to also answer why the Volkov result is so close to experiment, despite other results for “Li that do not agree as well with the experimental results. The other theories also reproduce the experimental result with good agreement. As an example of a general principle, a mean field calculation carried out in [52] gives a log ft = 3.07, which shows that for small nuclei, mean field/shell model type approaches do not give the best results. This concludes the chapter listing the numerical results of the calculation. We proceed then to our conclusions and outlook for the future of these calculations. 177 Chapter 6 Conclusions and outlook 6.1 Summary The purpose of this dissertation was to build a simple model that provides a reason- able description of light, loosely-bound nuclei. This was done by the calculation of many observables and comparing them to experimental data; the instrument of the calculations was the Brink Formalism in secondary quantization. After a background of the history of nuclear theory in general and cluster models in particular in Chapter One, the formalism used throughout the dissertation was intro— duced. Simple examples were worked out which illustrated the use of non-orthogonal orbitals and how the formalism accounts for the Pauli principle. Examples were also performed to illustrate the calculation of one- and two—body operator expectation values. The final ingredient in the formalism was the method of projecting into good states of angular momenta, which was explained in the final section of Chapter Two. In Chapter Three, we introduce the formalism to the six-body systems of inter- est. The calculation of expectation values is outlined in a completely general sense; no choice has been made yet about single-particle wave functions or nucleon-nucleon interaction. The two configurations, alpha-two particle cluster and cigar, are also introduced, and the calculation of one-body and two-body expectation values is out- 178 lined for both configurations. The interference term and minimization routine are also introduced. Lithium-6 calculations are then previewed, and are compared and contrasted with the calculations for “He. Finally, the calculation of spin-dependent operators such as the tensor operator and spin-orbit operator are described. Chapter Four delves into a particular choice of variational single-particle wave function, the Gaussian. The Gaussian wave functions are described, then used to calculate several things in both configurations. For “He, the expectation value of the Hamiltonian as a function of alpha-external neutron distance is calculated, along with the charge and matter radii. The kinetic energy calculation in particular illustrates how the formalism handles the Pauli principle. Interactions are chosen and described, and plots are shown of several expectation values as a function of alpha-neutron dis- tance. After the calculations are shown in the two configurations, electromagnetic transitions are introduced and the transition rate is calculated for the transition from the ground state of “He to its first excited state. Next, the calculations in the inter- ference term are shown for all the previously mentioned operators. The next subject is “Li, and operators that are different from those in “He are discussed. These include the tensor operator, spin-orbit operator, electric quadrupole moment and magnetic dipole moment. Finally, the topic of beta decay is introduced and the calculation of the beta decay of “He to “Li is discussed. The penultimate chapter, Chapter Five, reports the numerical results obtained with our model. Starting with “He, the optimized values of the variational parame- ters are reported, then the results for the ground and excited state energies results, weights of the configurations, charge and matter radii, and the quadruploe transition probabilities between the ground and first excited state. All results were reported for the three potentials used throughout this dissertation: the Volkov V1 and V2 interac- tions, and the Minnesota potential. Next, a calculation with different single-particle wave functions with different asymptotic properties was described and tables of re- sults that compared the new asymptotics with the Gaussian asymptotics were shown. 179 All results were compared with experimental data and the results of other theoretical models. Then the results for “Li were reported for energies, excited state excitation energies, relative weights of the configurations, charge radius, electric quadrupole mo- ment and magnetic dipole moment. Finally the results for the beta decay of “He were reported and discussed. 6.2 Conclusions The goal of this study was to use transparent physical input which was at the same time quantum mechanically rigorous, and try and reproduce the main features of the nuclei of interest, “He and “Li. In this case, the study can be considered to have met its goal. The input is simply the two extreme cases of an alpha plus two particles, with the Pauli principle exactly handled through non-orthogonal orbitals and secondary quantization. Helium-6 is fairly well described with our model, and “Li less so, but the description is still adequate. Improvements can be made, and ideas for improving the model will be discussed in the next section. For “He, we were able to reproduce its halo nature, which is seen in the large differ- ence between its charge and matter radius. Our numerical results are slightly smaller than experimental results for each quantity, but the extended neutron structure is evident. We found that “He is loosely-bound, and mostly in the alpha-dineutron con- figuration. Most other theories have come to similar conclusions. The experimental charge radius supports correlated neutrons in “He, as do preliminary transfer reaction data, but more is forth-coming from these experiments. A larger input is needed to be able to accurately describe excited states. Our results with the Minnesota poten- tial were generally worse than with the Volkov potentials. The Minnesota potential produced smaller radii (although the extended neutron structure is still present), and either much less or too much binding, depending on one’s perspective (the energy of “He against the alpha is much higher than with Volkov, but because of the very 180 strongly repulsive dineutron, the binding compared to the asymptotic state of an al- pha particle and separated dineutron is much larger than with Volkov). We also saw that more realistic exponential asymptotic behavior had little effect on bound state properties. Lithium-6 proved to be more of a challenge. With the Volkov potentials, our system is overbound, thanks to the Volkov form factor attached to the tensor interaction. The size is smaller than experiment, and we once again have problems with the excited states. The quadrupole moment is also quite a bit larger than experiment. The Minnesota results are similar in many respects, save the binding and the size is very small, but this is due to the lack of a tensor interaction. Both the Minnesota and Volkov potentials show a very dominant alpha-deuteron structure. Horn these results, it can be concluded that a better treatment of the spin—orbit and tensor interaction should be used to improve the results for “Li. The wave functions themselves do not seem to be too bad, at least in the case of the Volkov potentials, as the beta decay of “He agrees extremely well with experiment. 6.3 Future work Regarding the future, there are two main thrusts: how we can improve the model for “He and “Li without sacrificing its transparency and simplicty, and on which other systems would be interesting to use a similar approach. For improving the current model, one could think about starting to represent single particles as sums of Gaus- sians, although this starts to increase the complexity of the calculation and make the model more similar to other cluster models. For better treatment of excited states, including the contribution spin-triplet dineutron and cigar states in “He would prob— ably achieve better results. We should also do a calculation of something that is more sensitive to asymptotic behavior to see then the difference exponential tails can make. Such a calculation could be of an asymptotic normalization coefficient, or a reaction 181 calculation such as charge exchange. Another improvement would be to introduce a continuous, smoothly changing f (0), where B is the angle between the two exter- nal particles to describe the system instead of two extreme configurations. Finally, as mentioned in the previous paragraph, a potential that properly includes the spin-orbit and tensor interaction would be very interesting to apply to “Li. One such potential is the Argonne V8’. One possible drawback is that most Argonne potentials require a three-body force in order to achieve good agreement with experiment. Finally, the Coulomb interaction should be included, especially for “Li. It is small (820 keV at the distance of the minimum in energy in “Li), but long range and repulsive, which could also be important for improving our description of “Li. As for new calculations, a reaction involving “He or “He and “Li would be very interesting application of our wave function. Our wave function treats Fermi statistics exactly, which is sometimes not the case in reaction calculations. It would be inter- esting to see the role of asymptotics versus Fermi statistics in such a calculation. For other systems, 8He would be a topical system, as its charge radius has also recently been measured [81]. The preliminary result is that it is smaller than the charge radius of “He. We would model “He as a mixture of a dineutron-alpha-dineutron chain and an alpha-tetraneutron, and see what kind of charge radius we obtain. Another system that could be modeled is 7Li as a proton hole orbiting a two—alpha 8Be system. This is more of a pure theoretical interest, though the Li isotope chain has been the subject of a few recent study [79]. Finally, the chain of Be isotopes has always been good for cluster studies, and applying our model to those isotopes would be natural. In this dissertation, we have developed a simple but fully microscopic formalism for describing clusters in nuclei. It has been shown that though simple, the model can still capture essential physics. The formalism is extremely flexible, and it is sure to be applied to other light nuclear systems in the future. 182 Bibliography [1] J. Dalton. A New System of Chemical Philosophy. Peter Owen Limited, London, 1808. [2] J. J. Thomson. Phil. Mag., 44:293, 1897. [3] E. Rutherford. Phil. Mag., 21:669, 1911. [4] H. Becquerel. Compt. Renal, 122:420, 1896. [5] E. Rutherford and T. Royds. Phil. Mag., 17 :281, 1909. [6] J. Chadwick. Nature, 1292312, 1932. [7] W. Heisenberg. Z. Physik, 77:1, 1932. [8] E. P. Wigner. Phys. Rev, 51:947, 1937. [9] N. Bohr and F. Kalckar. Kgl. Danske Videnskab. Selskab, Mat-fys. Medd., 14:10, 1937. [10] J. M. Blatt and V. F. Weisskopf. Theoretical Nuclear Physics. John Wiley & Sons, New York, 1952. [11] D. R. Tilley, H. R. Weller, and G. M. Hale. Nucl. Phys, A541:1, 1992. [12] E. Borie and G. A. Rinker. Phys. Rev. A, 18:324, 1978. [13] J. A. Wheeler. Phys. Rev., 52:1083, 1937. [14] W. Wefelmeier. Z. Physik, 107:332, 1937. [15] C. F. von Weizsacker. Naturwiss., 26:209,225, 1938. [16] U. Fano. Naturwiss., 25:602, 1937. [17] L. R. Hafstad and E. Teller. Phys. Rev, 54:681, 1938. [18] C. Kittel. Phys. Rev., 62:109, 1942. [19] H. A. Bethe. Phys. Rev., 53:842, 1938. [20] R. G. Sachs. Phys. Rev, 552825, 1939. 183 [21] D. R. Inglis. Phys. Rev., 552329, 1939. [22] J. A. Wheeler. Phys. Rev., 59:16, 1941. [23] M. Born and J. R. Oppenheimer. Ann. Physik, 84:457, 1927. [24] K. Wildermuth and Y. C. Tang. A Unified Theory of the Nucleus. Academic Press, New York, 1977. [25] R. Jastrow. Phys. Rev., 792389, 1950. [26] H. Feldmeier and J. Schnack. Rev. Mod. Phys, 72:655, 2000. [27] Y. Kanada-En’yo, H. Horiuchi, and A. Ono. Phys. Rev. C, 52:628, 1995. [28] T. Bjerge. Nature, 138:400, 1936. [29] A. Navin, 2007. Private communication. [30] A. Galonsky and M. T. McEllistrem. Phys. Rev., 982590, 1955. [31] D. R. Tilley, C. M. Cheves, J. L. Godwin, G. M. Hale, H. M. Hoffman, J. H. Kelley, c. G. Sheu, and H. R. Weller. Nucl. Phys, A708z3, 2002. [32] P. Navratil, J. P. Vary, W. E. Ormand, and B. R. Barrett. Phys. Rev. Lett., 87:172502, 2001. [33] B. Wiringa. Nucl. Phys, A631:70, 1998. [34] D. M. Brink. International School of Physics ”Enrico Fermi”, XXXVI:247, 1966. [35] P. L6wdin. Phys. Rev., 97:1475, 1955. [36] S. Rombouts and K. Heyde. J. Phys. A, 27 23293, 1994. [37] J. M. Ziman. Elements of Advanced Quantum Theory. Cambridge University Press, Cambridge, 1969. [38] G. C. Wick. Phys. Rev., 80:268, 1950. [39] P. Ring and P. Schuck. The Nuclear Many-Body Problem. Springer-Verlag, New York, 1980. [40] D. M. Brink and G. R. Satchler. Angular Momentum. Oxford University Press, Oxford, third edition, 1993. [41] F. Hund. Z. Physik, 43:805, 1927. [42] D. M. Dennison and G. E. Uhlenbeck. Phys. Rev., 412313, 1932. [43] G. Herzberg. Infra Red and Raman Spectra. D. Van Nostrand Co.,Inc., New York, 1945. 184 [44] W. D. Ehmann and D. E. Vance. Radiochemistry and Nuclear Methods of Anal- ysis. John Wiley & Sons, New York, 1991. [45] A. B. Volkov. Nucl. Phys, 74:33, 1965. [46] I. Reichstein and Y. C. Tang. Nucl. Phys, A128z529, 1970. [47] D. R. Thompson, M. Lemere, and Y. C. Tang. Nucl. Phys, A286253, 1977. [48] D. Gogny, P. Fires, and R. De Tourreil. Phys. Lett. B, 322591, 1970. [49] H. Esbensen, K. Hagino, P. Mueller, and H. Sagawa. Phys. Rev. C, 7 62024302, 2007. [50] I. Sick. Phys. Lett. B, 576:62, 2003. [51] S. Kopecky, J. A. Harvey, N. W. Hill, M. Krenn, M. Pernicka, and S. Steiner. Phys. Rev. C, 5622229, 1997. [52] J. Suhonen. From Nucleons to Nucleus. Springer-Verlag, New York, 2007. [53] E. P. Wigner. Z. Physik, 432624, 1927. [54] C. Eckart. Rev. Mod. Phys, 2:305, 1930. [55] E. Fermi. Z. Physik, 882161, 1934. [56] G. A. Gamov and E. Teller. Phys. Rev., 492895, 1936. [57] W. M. Yao et al. J. Phys. C, 33:1, 2006. [58] B. V. Danilin, I. J. Thomson, J. S. Vaagen, and M. V. Zhukov. Nucl. Phys, A6322383, 1998. [59] T. Neff and H. Feldmeier. Nucl. Phys, A7382357, 2004. [60] N. Itagaki, A. Kobayakawa, and S. Aoyama. Phys. Rev. C, 682054302, 2003. [61] S. C. Pieper, R. B. Wiringa, and J. Carlson. Phys. Rev. C, 702054325, 2004. [62] C. A. Bertulani and M. S. Hussein. Phys. Rev. C, 762051602, 2007. [63] K. Hagino and H. Sagawa. Phys. Rev. C, 722044321, 2005. [64] J. J. Kolata et al. Phys. Rev. C, 752031302, 2007. [65] E. A. Benjamim et al. Phys. Lett. B, 647230, 2007. [66] L. Chulkov et al. Europhys. Lett., 8:245, 1989. [67] I. Tanihata et al. Phys. Lett. B, 2062592, 1988. [68] I. Tanihata, D. Hirata, T. Kobayashi, K. Sugimoto S. Shimoura, and H. Toki. Phys. Lett. B, 2892261, 1992. 185 [69] T. Aumann et al. Phys. Rev. C, 5921252, 1999. [70] L.-B. Wang et al. Phys. Rev. Lett., 932142501, 2004. [71] A. Ozawa, T. Suzuki, and I. Tanihata. Nucl. Phys, A693232, 2001. [72] S. C. Pieper, V. R. Pandharipande, R. B. Wiringa, and J. Carlson. Phys. Rev. C, 642014001, 2001. [73] M. Pervin, S. C. Pieper, and R. B. Wiringa, 2007. arXiv:0710.1265v2. [74] E. Caurier and P. Navratil. Phys. Rev. C, 732021302, 2006. [75] G. G. Ryzhikh, R. A. Eramzhyan, V. I. Kukulin, and Yu. M. Tchuvil’sky. Nucl. Phys, A563z247, 1993. [76] V. I. Kukulin, V. N. Pomerantsev, Kh. D. Razikov, V. T. Voronchev, and G. G. Ryzhikh. Nucl. Phys, A5862151, 1995. [77] W. Horiuchi and Y. Suzuki. Phys. Rev. C, 762024311, 2007. [78] T. Schmidt. Z. Physik, 1062358, 1937. [79] R. Sanchez et al. Phys. Rev. Lett., 962033002, 2006. [80] C. W. de Jager, H. de Vries, and C. de Vries. At. Data and Nucl. Data Tables, 142479, 1974. [81] Z.-T. Lu. Bulletin of the American Physical Society, 52.9271, 2007. 186 Illflllllljljlfljjgl[ljlj'lgll