Nuclear spin features relevant to ab initio nucleon-nucleus elastic scattering
R. B. Baker, M. Burrows, Ch. Elster, K. D. Launey, P. Maris, G. Popa, S. P. Weppner
aa r X i v : . [ nu c l - t h ] F e b Nuclear spin features relevant to ab initio nucleon-nucleus elastic scattering
R. B. Baker, M. Burrows,
Ch. Elster, K. D. Launey, P. Maris, G. Popa, and S. P. Weppner Institute of Nuclear and Particle Physics, and Department ofPhysics and Astronomy, Ohio University, Athens, OH 45701, USA Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Natural Sciences, Eckerd College, St. Petersburg, FL 33711, USA (Dated: February 1, 2021)
Background:
Effective interactions for elastic nucleon-nucleus scattering from first principles require the use ofthe same nucleon-nucleon interaction in the structure and reaction calculations, as well as a consistent treatmentof the relevant operators at each order.
Purpose:
Previous work using these interactions has shown good agreement with available data. Here, we studythe physical relevance of one of these operators, which involves the spin of the struck nucleon, and examine theinterpretation of this quantity in a nuclear structure context.
Methods:
Using the framework of the spectator expansion and the underlying framework of the no-core shellmodel, we calculate and examine spin-projected, one-body momentum distributions required for effective nucleon-nucleus interactions in J = 0 nuclear states. Results:
The calculated spin-projected, one-body momentum distributions for He, He, and He display char-acteristic behavior based on the occupation of protons and neutrons in single particle levels, with more nucleonsof one type yielding momentum distributions with larger values. Additionally, we find this quantity is stronglycorrelated to the magnetic moment of the 2 + excited state in the ground state rotational band for each nucleusconsidered. Conclusions:
We find that spin-projected, one-body momentum distributions can probe the spin content of a J = 0 wave function. This feature may allow future ab initio nucleon-nucleus scattering studies to inform spinproperties of the underlying nucleon-nucleon interactions. The observed correlation to the magnetic moment ofexcited states illustrates a previously unknown connection between reaction observables such as the analyzingpower and structure observables like the magnetic moment. I. INTRODUCTION
The study of atomic nuclei is dependent on nuclear reactions to extract both reaction- and structure-related ob-servables. From a theoretical perspective, one way to study these nuclear reactions is by reducing the many-bodyproblem to a few-body problem and isolating the relevant degrees of freedom [1]. This few-body problem can thenbe solved with the use of effective interactions, which are often called optical potentials. While different techniqueshave been implemented to construct these effective interactions from first principles, e.g. Refs. [2, 3], here we focuson the use of the ab initio no-core shell model (NCSM) [4–6] and symmetry-adapted no-core shell model (SA-NCSM)[7, 8] to provide the relevant structure inputs. Specifically, we combine one-body densities for the target calculatedfrom these methods with scattering approaches formulated to use structure and reaction input on an equal footingin a systematic expansion. For elastic scattering of protons and neutrons from nuclei, a microscopic optical potentialcan be derived with a Watson expansion of the multiple scattering series [9–14]. This spectator expansion allows forthe use of the same nucleon-nucleon (
N N ) interaction when calculating the one-body densities which must be foldedwith the
N N scattering amplitudes. By using realistic
N N and three-nucleon (3 N ) interactions derived from chiraleffective field theory [15–19], we can implement this procedure in a fully ab initio way, provided we include all relevantterms in the spectator expansion at each order.Recent work [3] was able to construct and implement effective nucleon-nucleus interactions that include the spinof the struck target nucleon consistently at leading order. The results of using those effective interactions to studya nucleon elastically scattering off selected nuclei in J = 0 states yielded only small changes in some of the spinobservables when compared to previous work where the spin of the struck target nucleon was neglected. However,the pattern in where those changes occurred suggests that a closer investigation is required. Specifically, the “closedshell” nuclei He and O exhibited no changes in their elastic scattering results and the N = Z nucleus C showedonly minimal changes. In contrast, the deviations in the analyzing power A y and spin-rotation parameter Q forproton elastic scattering on He and He were much larger and it is a goal of this paper to examine the cause of thesedeviations in greater detail.Progress in ab initio nuclear structure, both in terms of the development of more specialized realistic interactions[20–23] and numerical improvements in the many-body methods [24–27], have illustrated the large extent to whichfirst principles calculations can describe nuclear states. In particular, the properties of excited states are increasinglywell described from first principles, including collectivity in light to medium-mass nuclei [7, 8] and the emergenceof rotational bands [28–33]. Indeed, ab initio calculations have shown that only a few equilibrium shapes dominatewithin low-lying states, and that members of a rotational band have the same shape(s) and, in addition, exhibit verysimilar spin content [8, 29]. This corroborates earlier studies, starting with the pioneering work of Refs. [34, 35] andincluding large shell-model calculations [36–38]. That the nature of rotational bands can provide insights into orbitalangular momentum and spin components of nuclear wave functions is also shown in Refs. [39, 40]. This is of particularimportance here because the nuclear spin properties are often probed by a nuclear observable such as the magneticmoment, which is zero for J = 0 states. In this paper, we exploit the similarity of the spin content within membersof a rotational band, and by calculating magnetic moments for the first excited 2 + states, one can probe correlationsbetween the spin features of the target detected by its magnetic moment and those detected by the spin-projectionmomentum distribution. Closely correlated results would imply that one can readily use the spin-projected momentumdistribution in J = 0 states to inform spin features in these states, or that measured magnetic moments can informspin properties of effective interactions.In this work, we seek to expand on the formalism presented in our previous work [3] and provide more physicalinsight into the effects of explicitly including the spin of the struck target nucleon in the effective interaction. In Sec. IIwe discuss the relevant derivations for the leading-order effective interaction, with a focus on the spin-dependent termsthat arise. In Sec. III we show results for these spin-dependent terms in the He isotope chain ( He, He, and He)and discuss their physical interpretations. Furthermore, we note a correlation between these spin-dependent termsand a more traditional spin-related observable, the magnetic moment. We discuss our conclusions in Sec. IV.
II. THEORETICAL FRAMEWORK
Calculating elastic nucleon-nucleus scattering observables in an ab initio fashion requires the interaction betweenthe projectile and the target nucleus. In a recent work [3] this effective interaction was derived and calculated inleading order of the spectator expansion of multiple scattering theory for elastic scattering of protons (neutrons) offa 0 + ground state in selected nuclei. Here explicit care is taken so that the N N interaction is treated on the samefooting in the structure as well as the reaction part of the calculation.For completeness we start with the explicit expression for the effective leading-order interaction describing thescattering of a proton from a nucleus in a 0 + ground state, b U p ( q , K NA , ǫ ) = X α =n , p Z d K η ( q , K , K NA ) A p α (cid:18) q , (cid:18) A + 1 A K NA − K (cid:19) ; ǫ (cid:19) ρ K s =0 α (cid:0) P ′ , P (cid:1) + i ( σ (0) · ˆ n ) X α =n , p Z d K η ( q , K , K NA ) C p α (cid:18) q , (cid:18) A + 1 A K NA − K (cid:19) ; ǫ (cid:19) ρ K s =0 α (cid:0) P ′ , P (cid:1) + i X α =n , p Z d K η ( q , K , K NA ) C p α (cid:18) q , (cid:18) A + 1 A K NA − K (cid:19) ; ǫ (cid:19) S n,α (cid:0) P ′ , P (cid:1) cos β + i ( σ (0) · ˆ n ) X α =n , p Z d K η ( q , K , K NA ) ( − i ) M p α (cid:18) q , (cid:18) A + 1 A K NA − K (cid:19) ; ǫ (cid:19) S n,α (cid:0) P ′ , P (cid:1) cos β. (1)The term η ( q , K , K NA ) is the Møller factor [41] describing the transformation from the N N frame to the
N A frame.The functions A p α , C p α , and M p α represent the N N interaction through Wolfenstein amplitudes [42]. Since theincoming proton can interact with either a proton or a neutron in the nucleus, the index α indicates the neutron (n)and proton (p) contributions, which are calculated separately and then summed up. With respect to the nucleus,the operator i ( σ (0) · ˆ n ) represents the spin-orbit operator in momentum space of the projectile. As such, Eq. (1)exhibits the expected form of an interaction between a spin- projectile and a target nucleus in a J = 0 state [43].The momentum vectors in the problem are given as q = p ′ − p = k ′ − k , K = 12 ( p ′ + p ) , ˆ n = K × q | K × q | K NA = AA + 1 (cid:20) ( k ′ + k ) + 12 ( p ′ + p ) (cid:21) , P = K + A − A q , P ′ = K − A − A q . (2)A sketch of the scattering in the N A frame is given in Fig. 1, which includes the incoming momentum k of the projectile,its outgoing momentum k ′ , the momentum transfer q , and the average momentum K NA . The struck nucleon in thetarget has an intial momentum p and a final momentum p ′ . The two quantities representing the structure of thenucleus are the scalar one-body density ρ K s =0 α (cid:0) P ′ , P (cid:1) and the spin-projected momentum distribution S n,α (cid:0) P ′ , P (cid:1) .Both distributions are nonlocal and translationally invariant. Lastly, the term cos β in Eq. (1) comes from projecting ˆ n from the N N frame to the
N A frame. For further details, see Ref. [3].The scalar one-body density is a well known quantity, while the spin-projected momentum distributions have notbeen studied in detail. In general, a spin-dependent nonlocal density can be defined as [3], ρ K s ( p , p ′ ) = * Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A X i =1 δ ( p i − p ) δ ( p ′ i − p ′ ) X q s σ ( i ) K s q s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Φ + , (3)where σ ( i ) K s q s is the spherical representation of the spin operator of the struck nucleon in the target nucleus. Theoperator structure of the effective interaction [44, 45] given by Eq. (1) requires we calculate the projections of thespin operator on the momentum basis given by the vectors q , K , and ˆ n . Due to parity constraints the projections on q and K are zero. The projection along the ˆ n -direction becomes S n ( p , p ′ ) ≡ ρ K s ( p , p ′ ) · ˆ n = X q s ( − q s ρ K s =1 q s ( p , p ′ ) ˆ n − q s , (4)where ˆ n has been written in terms of its spherical components. This equation can be explicitly evaluated as [3] S n ( q , K ) = X q s ( − − q s r π Y − q s ( ˆ n ) X n r ljn ′ r l ′ j ′ l + l ′ X K l = | l − l ′ | K l X k l = − K l X Kk h K l k l , q s | Kk i h J − M , Kk | J − M i ( − K ( − − l s j + 1)(2 j ′ + 1)(2 s + 1)(2 K l + 1)(2 J + 1) l ′ l K l s s j ′ j K X n q ,n K ,l q ,l K h n K l K , n q l q : K l | n ′ r l ′ , n r l : K l i d =1 R n K l K ( K ) R n q l q ( q ) Y ∗ l q l K K l k l ( b q , ˆ K ) D AλJ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a † n ′ r l ′ j ′ ˜ a n r lj ) ( K ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) AλJ E e A b q . (5)Note that h n K l K , n q l q : K l | n ′ r l ′ , n r l : K l i d =1 is a Talmi-Moshinsky bracket used to transform from the pp ′ frame tothe q K frame and the subscript d = 1 is defined in Ref. [46]. The factor e A b q comes from removing the center-of-mass contributions in the one-body density matrix elements (OBDMEs) of the target. The OBDMEs, of which the D AλJ || ( a † n ′ r l ′ j ′ ˜ a n r lj ) ( K ) || AλJ E term in Eq. (5) is the reduced form, are calculated as the inner product of the creation( a † n r ljm ) and annhilation (˜ a n r ljm = ( − j − m a n r lj − m ) operators for single-particle harmonic oscillator states labeledby their ( n r , l, j, m ) values.To facilitate calculations of the spin-projected momentum distribution, we make two choices: 1) use states with J = 0 and 2) choose the vector ˆ q in the z-direction and ˆ K in the x-z plane. This points ˆ n along the negativey-direction, and Eq. (5) simplifies to S n ( q , K ) = ( − i ) X n r ljn ′ r l ′ j ′ p j + 1( − j + s +1 (cid:26) l ′ l s s j (cid:27)X n q ,n K ,l q ,l K h n K l K , n q l q : 1 | n ′ r l ′ , n r l : 1 i d =1 R n K l K ( K ) R n q l q ( q ) X q s = − , Y ∗ l q l K − q s ( b q , ˆ K ) D Aλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a † n ′ r l ′ j ′ ˜ a n r lj ) (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Aλ E e A b q . (6)For J = 0 states, the coupling coefficients require K s = 1 and K l = 1 in the Talmi-Moshinsky bracket. While we relyon Eq. (6) for the numerical implementation of S n ( q , K ), we can also examine one-dimensional functions dependingonly on the momentum transfer, S n ( q ) ≡ Z d K S n ( q , K ) = Z d KK Z d θ q K sin( θ q K ) Z d φ q K S n ( q, K , θ q K , φ q K ) , (7)and on the average momentum, S n ( K ) ≡ Z d qS n ( q , K ) = Z d qq Z d θ q K sin( θ q K ) Z d φ q K S n ( q, K , θ q K , φ q K ) . (8)Note that θ q K and φ q K are the polar and azimuthal angles between q and K . This definition is similar to defining thecharge form factor, which is widely used to characterize momentum distributions in the nucleus.Lastly, it is worth noting the choice of specific coordinates in Eq. (6) prevents further analytical insights viaintegration, but we can use an alternate derivation of Eq. (4) which yields [47] S n ( q , K ) = − i √ π X n r ljn ′ r l ′ j ′ p (2 j + 1)(2 s + 1)( − j + s +1 (cid:26) l ′ l s s j (cid:27)X n q ,n K ,l q ,l K h n K l K , n q l q : 1 | n ′ r l ′ , n r l : 1 i d =1 R n K l K ( K ) R n q l q ( q ) X w ˆ l q ˆ l K h l q | w i h l K | w i (cid:26) l q w l K (cid:27) ( − ) l K P w (cos( θ q K )) | sin( θ q K ) | D Aλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a † n ′ r l ′ j ′ ˜ a n r lj ) (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Aλ E e A b q , (9)where P w (cos( θ q K )) is a Legendre polynomial and the integer w is determined by the angular momentum coupling.Furthermore, we can expand the sum over w and consider the lowest nonzero term. This corresponds to n r = n ′ r = 0and l = l ′ = 1, which only has one nonzero Talmi-Moshinsky bracket when n q = n K = 0 and l q = l K = 1. TheClebsch-Gordan coefficients in Eq. (9) are nonzero when w = 0 ,
2, yielding an angular dependence of the form S n (cos( θ q K )) ∼ P (cos( θ q K )) − P (cos( θ q K )) | sin( θ q K ) | = 32 sin( θ q K ) . (10)Thus, while the full expression in Eq. (9) is complicated, in lowest order it has a relatively simple angular dependence. III. RESULTS AND DISCUSSION
First, we examine the functions S n ( q ) and S n ( K ), given by Eqs. (7) and (8) respectively, for selected He isotopes.For these comparisons, we use the chiral interaction NNLO opt [20] and a large N max value for each calculation, where N max is the maximum number of harmonic oscillator quanta allowed above the valence shell for a given nucleus. Forboth He and He, their S n ( q ) and S n ( K ) behave similarly, though they differ in magnitude (Fig. 2). In contrast,the curves for He are noticeably different, both in magnitude and sign. Notably, the S n ( q ) extend to a few fm − while the S n ( K ) are largely concentrated below 1 fm − . As K is considered a nonlocal variable, this likely reflectsthe nonlocality of S n ( q , K ) is well confined to within the nucleus. While both can be informative, we focus on S n ( q )due to its dependence on physically relevant momentum transfer, q . To better understand these differences and todevelop expectations for what the S n ( q ) for any given nucleus might look like, we examine Eq. (8) more closely. Asthe only required inputs are the one-body density matrix elements (OBDMEs), we would expect S n ( q ) to have somedependence on the underlying shell structure for a given nucleus.To study this in more detail, we performed toy model calculations in which each given harmonic oscillator state iscompletely filled and those nucleons are frozen. This results in there being zero probability of these nucleons movingto a different harmonic oscillator state, which means the resulting S n ( q ) is independent of the choice of nuclearinteraction. This approach can then provide a basis for interpreting the S n ( q ) of a realisitic NCSM calculation. Assuch, in this toy model S n ( q ) displays characteristic behavior based on n r , l , and j values, as shown in Fig. 3. Fortwo nucleons frozen in the 0 s / shell, we find no contribution to S n ( q ) as expected for l = l ′ = 0 in Eq. (8). For twonucleons frozen in the 0 p / shell, we find a contribution equal in magnitude but opposite in sign as four nucleonsfrozen in the 0 p / shell, despite there being more nucleons. In combination, this means full orbitals with the same n r , l values sum to zero S n ( q ) contributions, which can be seen for all HO shells up to l = 3 in Fig. 3. This patternis always such that the j = l + state has an overall positive S n ( q ) and the j = l − state has an overall negative S n ( q ). Furthermore, the equal-in-magnitude behavior can be observed regardless of how many nucleons it takes tocompletely fill the shell, e.g. even for filled 0 f / (6 nucleons) versus 0 f / (8 nucleons), the magnitudes of S n ( q )are still the same. Note that while the curves in Fig. 3 only show so called “diagonal” OBDMEs, i.e. a † α ˜ a β where α = β , these OBDMEs are larger than the off-diagonal OBDMEs where α = β . Further, from Fig. 3 we can seethat different n r values will yield different S n ( q ) curves, though they still maintain the same opposite-in-sign andequal-in-magnitude behavior.Given this information, we can now better interpret the S n ( q ) of a realistic NCSM calculation. Separating theOBDMEs to examine contributions to S n ( q ) for specific shells, we see a strong dominance of p / shells in He and He, as shown in Fig. 4. Since we now know that filled shells have equal magnitude, we would expect similar partially-filled shells to have a similar magnitudes in their associated S n ( q ) curve. For both He and He we can see the p / curve has a magnitude of more than twice that of the p / curve, indicating nucleons prefer to fill the p / shells. Thisinterpretation is supported by the ratio of the occupation probabilities calculated in the NCSM as well. Additionally,since we know the n r value changes the S n ( q ) curve, Fig. 4 also indicates the 0 p shell largely dominates the total S n ( q ) since neither He nor He show the second peak indicative of the 1 p shells.Similarly, if we integrate S n ( q ) with respect to q , S n ≡ Z d qq S n ( q ) , (11)we can see that the p / contributions to the S n for the neutrons slightly more than doubles from He to He, asshown in Table I. In contrast, the p / contributions to the S n for the neutrons barely changes. Note that Table I alsoindicates that the proton contributions to S n barely change from He to He, suggesting that the proton informationis mostly the same. The S n values for He are included in Table I for comparison purposes. Specifically, it should benoted that the proton value for He is more negative than the proton values in He and He despite all three nucleicontaining two protons. This possibly indicates the underlying OBDMEs include proton-neutron pairing effects for N = Z nuclei largely suppressed when N = Z .While the function S n ( q ) clearly inherits information from the underlying shell structure, given its operator structurewe would also expect it to provide more general information about the spin content of a given nucleus or nuclearinteraction. To better facilitate those comparisons, we can examine the behavior of S n in more detail. For Heand He, S n as a function of N max is largely consistent for a variety of different nuclear interactions, as shown inFig. 5 for NNLO opt [20], Daejeon16 [21], and LENPIC-SCS at N2LO (NN potential) [16, 48, 49]. For high N max values, we can start to see some slight deviations as the results approach convergence. For He, we can see cleardifferences in the value of S n depending on the choice of interaction. While Daejeon16 provides a result of almostzero, both NNLO opt and LENPIC-SCS yield more negative values, both of which converge toward different values.This indicates the quantity S n probes a portion of nuclear interactions that remains distinct, even when the bulkobservables, e.g. binding energy and rms radius, would be in better agreement, e.g. [21, 49].While S n is not an observable – it is not derived from a Hermitian operator – we do find that it has a strongcorrelation with a well-defined observable: the magnetic moment. Specifically, when S n is calculated for the 0 +gs ofa nucleus, we find a strong correlation between that value and the magnetic moment of the 2 + excited state in theground state rotational band, e.g. µ = h + | M | + i . Using the impulse approximation to the magnetic moment,given as M = r π µ N A X i =1 ( g ℓi ℓ i + g si s i ) (12)where g ℓ p = 1, g ℓ n = 0, g s p = 5 . g s n = − . + state is a rotational ( L = 2) excitation of the0 +gs . To study these correlations, we employed a technique discussed in Ref. [51] and illustrated in Ref. [52]. Briefly,we treat each calculation of µ and S n (for different values of ¯ h Ω and N max ) as elements of two separate vectors. Thecosine of the angle between these two vectors, found by taking their inner product, tells us how much these quantitiesoverlap and we can quantify this through a correlation coefficient ζ . The sign of ζ refers to positive or negativecorrelation and the values span | ζ | = 0 (no correlation) to | ζ | = 1 (perfect correlation).Notably, in Fig. 6(a), we see a strong positive correlation between the magnetic moment and S n for N < Z ( ζ C µ,S n = 0 . N = Z ( ζ C µ,S n = 0 . N > Z ( ζ He µ,S n = − . + excited state already existsin the wave function of the 0 +gs though it is not accessible to direct measurement. The strength of this correlation isthe important factor, as the sign comes from the underlying µ and S n values, i.e. both µ and S n are positive in C and C, but µ is negative and S n is positive in He. Additionally, if we separate the angular momentum contributionsto the magnetic moment given by Eq. (12), we can see that these correlations are strongly driven by the spin termsin the magnetic moment, labeled as µ s in Fig. 6(b). When comparing Fig. 6(a) and Fig. 6(b), it can be seen thatthe orbital angular momentum terms in the magnetic moment slightly decrease the strength of these correlations,as the correlation coefficients for the latter plot are slightly larger ( ζ C µ s ,S n = 0 . , ζ C µ s ,S n = 0 . , ζ He µ s ,S n = − . µ and S n is driven by the neutrons in He ( ζ He µ neu s ,S neu n = 0 .
992 versus ζ He µ pro s ,S pro n = − . C ( ζ C µ pro s ,S pro n = 0 .
994 versus ζ C µ neu s ,S neu n = − . C ( ζ C µ pro s ,S pro n = 0 .
995 versus ζ C µ neu s ,S neu n = 0 . IV. CONCLUSIONS AND OUTLOOK
We have examined in detail the one-body, spin-projected momentum distribution S n ( q , K ), a new term that appearsat leading order in the spectator expansion when the spin of the struck target nucleon is explicitly included. Inparticular, we have examined its dependence on the momentum transfer q and showed changes in S n ( q ) for a givennucleus and for a given nucleon type in a nucleus. For the He-isotope chain ( He, He, and He) the S n ( q ) for neutronsincrease in magnitude as more neutrons are added, though the proton contributions in He and He remain the sameand differ from those of He.Noting the underlying shell structure inherent in our calculations, we identified interaction-independent character-istics of S n ( q ) based on which harmonic oscillator shells the nucleons occupy. This allowed for the interpretation of S n ( q ) for a realistic nucleus in better detail and showed that changes to S n ( q ) along the isotopic chain are related towhich harmonic oscillator shells the subsequent nucleons are most probable to occupy. Furthermore, by investigatingthe dependence of the integrated quantity S n for different realistic nuclear interactions, we observed this behavioris largely independent of the interactions employed but noted different interactions may see yield slightly differentoverall values for S n .To better understand implications of this for an observable quantity, we compared calculations of S n for a 0 +gs to themagnetic moment of the 2 + state in that ground state rotational band and found a strong correlation, regardless ofthe nucleus. This correlation was driven by the spin components of the magnetic moment and indicates the quantityof interest here, S n , (which is required to perform leading order calculations of effective interactions in a consistentway) probes spin information in a J = 0 wave function that would normally be accessible only through observables inits excited states. This suggests the exciting possibility that future ab initio nucleon-nucleus scattering studies couldbe sensitive to spin properties of the underlying nucleon-nucleon interaction, thereby providing further insight thanpreviously appreciated. ACKNOWLEDGMENTS
This work was performed in part under the auspices of the U. S. Department of Energy under contract Nos.DE-FG02-93ER40756 and DE-SC0018223, and by the U.S. NSF (OIA-1738287 & PHY-1913728). The numericalcomputations benefited from computing resources provided the Louisiana Optical Network Initiative and HPC re-sources provided by LSU, together with resources of the National Energy Research Scientific Computing Center, aDOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy undercontract No. DE-AC02-05CH11231. [1] C. W. Johnson et al. , in
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000 0 .
000 0 .
000 0 .
000 0 .
000 0 . p / − . − . − . − . − . − . p / .
150 0 .
149 0 .
154 1 .
271 0 .
124 2 . d / .
169 0 .
169 0 .
074 0 .
070 0 .
040 0 . d / − . − . − . − . − . − . f + − . − . − . − . − . − . − . − . − .
096 0 . − .
104 1 . S n broken down by nucleon type and diagonal lj OBDME values for NNLO opt [20] at ¯ h Ω = 20 MeV.Results for He are at N max = 18, for He at N max = 18, and for He at N max = 14. The neutron values for He and Hecalculated correspond applying Eq. (11) to each S n ( q ) in Fig. 4. Note the row labeled f + corresponds to all remaining diagonalOBDMEs and all off-diagonal ODBMEs. See text for further detail. p FIG. 1. Schematic diagram of the scattering plane, indicating the relevant momenta. Note that the normal vector ˆ n in Eq. (2)is perpendicular to both the pp ′ -plane and the q K -plane. −1 ]0.00.10.20.30.40.50.60.70.8 S n ( q ) (a) He He He0 1 2 3 4 5−0.02−0.010.00 −1 ]01234567 S n ( ) (b) He He He0 1 2 3 4 5−0.3−0.2−0.10.0
FIG. 2. (a) The function S n ( q ) for the neutron distribution in He at N max = 18, He at N max = 18, and He at N max = 14calculated with the NNLO opt chiral interaction [20]. (b) The function S n ( K ) for the same values. The bands in each plotindicate variations in ¯ h Ω (16-24 MeV) at that value of N max . The insets show the He results in better detail. −1 ]−2.0−1.5−1.0−0.50.00.51.01.52.0 S n ( q ) (a) , 1s −1 ]−3−2−10123 S n ( q ) (b) sd1p FIG. 3. The function S n ( q ) for protons or neutrons filling each closed shell configuration at ¯ h Ω = 20 MeV for (a) the s , p , and sd shell and (b) the pf shell. −1 ]−0.20.00.20.40.60.8 S n ( q ) He(a) s p p d d f + 0 1 2 3 4 5q [fm −1 ]−0.20.00.20.40.60.8 S n ( q ) He(b) s p p d d f + FIG. 4. The function S n ( q ) for the neutron distribution in (a) He at N max = 18 and (b) He at N max = 14, both with NNLO opt [20] and ¯ h Ω = 20 MeV. The contributions to the total S n ( q ) curve are broken down by individual diagonal lj OBDME values.See text for further discussion.FIG. 5. Integrated S n values from Eq. (11) for neutrons in He, He, and He as calculated by the NNLO opt [20], Daejeon16 [21],and LENPIC-SCS at N2LO (NN potential) [16, 48, 49] interactions. The band for each interaction corresponds to variationsin ¯ h Ω. −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6S n (normalized)−0.6−0.4−0.20.00.20.40.6 μ ( n o r m a li z e d ) (a) He C C−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6S n (normalized)−0.6−0.4−0.20.00.20.40.6 μ μ ( n o r m a li z e d ) (b) He C C FIG. 6. (a) Correlation plot for the magnetic moment of the 2 + state in the ground state rotational band and the integratedtotal (protons + neutrons) S n value for the 0 +gs in He, C, and C calculated with NNLO opt . (b) Same plot for the spincontributions to the magnetic moment. Points for each nucleus were calculated at ¯ h Ω = 16 , ,
24 MeV with N max = 6 , , He and C and N max = 4 , , C. The solid black line is a perfect positive correlation and the dashed gray line is aperfect negative correlation. −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6S pron (normalized)−0.6−0.4−0.20.00.20.40.6 μ p r o μ ( n o r m a li z e d ) (a) He C C −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6S neun (normalized)−0.6−0.4−0.20.00.20.40.6 μ n e u μ ( n o r m a li z e d ) (b) He C C FIG. 7. (a) Correlation plot for the proton spin contributions to the magnetic moment of the 2 + state in the ground staterotational band and the integrated proton S n value for the 0 +gs in He, C, and C calculated with NNLO opt . (b) Same plot forthe neutron spin contributions and the integrated neutron S n value. Points for each nucleus were calculated at ¯ h Ω = 16 , , N max = 6 , ,
10 for He and C and N max = 4 , ,12