Measuring the surface thickness of the weak charge density of nuclei
MMeasuring the surface thickness of the weak charge density of nuclei
Brendan Reed, Z. Jaffe, C. J. Horowitz, and C. Sfienti Center for the Exploration of Energy and Matter and Department of Physics,Indiana University, Bloomington, IN 47405, USA Institut f¨ur Kernphysik, Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany (Dated: September 16, 2020)The present PREX-II and CREX experiments are measuring the rms radius of the weak chargedensity of
Pb and Ca. We discuss the feasibility of a new parity violating electron scatteringexperiment to measure the surface thickness of the weak charge density of a heavy nucleus. OncePREX-II and CREX have constrained weak radii, an additional parity violating measurement ata momentum transfer near 0.76 fm − for Pb or 1.28 fm − for Ca can determine the surfacethickness.
I. INTRODUCTION
Where are the protons located in an atomic nucleus?Historically, charge densities from elastic electron scat-tering have provided accurate and model independentinformation [1]. These densities are, quite literally, ourcurrent picture of the nucleus and they have had an enor-mous impact. They have helped reveal sizes, surfacethicknesses, shell structure, and saturation density of nu-clei.An equally important but much more challenging ques-tion is where are the neutrons located in an atomic nu-cleus? Very fundamental nuclear structure informationcould be extracted if we also had access to accurate neu-tron densities. For example, knowing both the protonand the neutron densities would provide constraints onthe isovector channel of the nuclear effective interaction,which is essential for the structure of very neutron richexotic nuclei. Because of nuclear saturation, we expectthe average interior baryon density of
Pb to be veryclosely related to the saturation density of nuclear matter ρ . Since we already know the charge and proton densi-ties with high precision, determining the interior neutrondensity of Pb would allow new insight into nuclear sat-uration and the exact value of ρ [2].However, compared to charge densities, our presentknowledge of neutron densities is relatively poor andmodel dependent. Often neutron densities are deter-mined with strongly interacting probes [3] such as an-tiprotons [4, 5], proton elastic scattering [6], heavy ioncollisions [8], pion elastic scattering [9], and coherentpion photo-production [10]. Here one typically measuresthe convolution of the neutron density with an effectivestrong interaction range for the probe. Uncertaintiesin this range, from complexities of the strong interac-tions, can introduce significant systematic errors in theextracted neutron densities (see Ref. [11] for a recentreview of neutron skin measurements).It is also possible to measure neutron densities, orequivalently weak charge densities, with electro-weak in-teractions, by using neutrino-nucleus coherent scattering[12–15] or parity violating electron scattering [16, 17].Since the weak charge of a neutron is much larger than that of a proton, the weak charge density of a nucleus isvery closely related to its neutron density. Compared tostrongly interacting probes, parity violation provides aclean and model-independent way to determine the weakcharge density and it is likely affected by much smallerstrong interaction uncertainties. In the last few decades,great theoretical [16, 18–23] and experimental [17, 24]efforts have been made to improve parity violating elec-tron scattering experiments. At Jefferson laboratory, theradius of the weak charge density of Pb was first mea-sured in the PREX experiment [17, 26], and has beenrecently measured with higher accuracy in the PREX-IIexperiment [27], while the weak radius of Ca is beingmeasured in the CREX experiment [28].It is a slight misnomer to say PREX-II and CREXare directly measuring weak radii. Strictly speaking theradius is defined by the derivative of a weak form fac-tor (see below) in the limit of the momentum transfergoing to zero. However for practical reasons, PREX-IIand CREX do not measure at zero momentum trans-fer but at small finite momentum transfers. ThereforePREX-II and CREX depend on not just the weak ra-dius but also, to some degree, on the surface thicknessof the weak density. In this paper we quantify this de-pendence and explore how the surface thickness can bedetermined by measuring the parity violating asymmetryat a second, somewhat higher, momentum transfer. Ev-idently such an experiment would provide the next stepafter PREX-II and CREX. The ultimate goal for parityviolation experiment on heavy nuclei will be, as we haveshown in previous work [29] the determination of the en-tire weak density distribution ρ W ( r ) by measuring parityviolation at several different momentum transfers.Our formalism for describing parity violating electronscattering and how this depends on properties of the weakdensity including the surface thickness is presented inSec. II. In Sec. III we present results for the sensitivityof the PREX-II and CREX experiments to the surfacethickness. Next, in Sec. IV we explore the feasibility of anew experiment to measure the surface thickness of theweak density of Pb or Ca. We conclude in Sec. V. a r X i v : . [ nu c l - t h ] S e p II. FORMALISM
The parity violating asymmetry for longitudinally po-larized electrons scattering from a spin zero nucleus, A pv ,is the key observable which is very sensitive to the weakcharge distribution. The close relationship between A pv and the weak charge density ρ W ( r ) can be readily seenin the Born approximation, A pv ≡ dσ/d Ω R − dσ/d Ω L dσ/d Ω R + dσ/d Ω L ≈ − G F q πα √ Q W F W ( q ) ZF ch ( q ) . (1)Here dσ/d Ω R ( dσ/d Ω L ) is the cross section for positive(negative) helicity electrons, G F is the Fermi constant, q the momentum transfer, α the fine structure constant,and F W ( q ) and F ch ( q ) are the weak and charge formfactors respectively, F W ( q ) = 1 Q W (cid:90) d rj ( qr ) ρ W ( r ) (2) F ch ( q ) = 1 Z (cid:90) d rj ( qr ) ρ ch ( r ) . (3)These are normalized so that F W ( q →
0) = F ch ( q →
0) = 1. The charge density is ρ ch ( r ) and Z = (cid:82) d rρ ch ( r )is the total charge. Finally, the weak charge density ρ W ( r ) and the total weak charge Q W = (cid:82) d rρ W ( r ) arediscussed below.The elastic cross-section in the plane wave Born ap-proximation is, dσd Ω = Z α cos ( θ )4 E sin ( θ ) (cid:12)(cid:12) F ch ( q ) (cid:12)(cid:12) , (4)with θ the scattering angle. However, for a heavy nucleus,Coulomb-distortion effects must be included. These cor-rect both Eqs. 1 and 4 and can be included exactly bynumerically solving the Dirac equation for an electronmoving in the coulomb and axial vector weak potentials[18]. Figure 1 shows the cross section and Fig. 2 theparity violating asymmetry A pv for 855 MeV electronsscattering from Pb or Ca.Parity violating experiments directly depend on theweak density ρ W ( r ). However, theoretical calculationsoften determine ρ W ( r ) by folding single nucleon weakform factors with point proton ρ p ( r ) and neutron ρ n ( r )densities. For completeness, we review this procedurehere.If one neglects spin-orbit currents that are discussedin Ref. [31], and other meson exchange currents [32] onecan write, ρ W ( r ) = (cid:90) d r (cid:48) (cid:8) G Zn ( | r − r (cid:48) | ) ρ n ( r )+ 4 G Zp ( | r − r (cid:48) | ) ρ p ( r ) (cid:9) . (5)Here G Zn ( r ) and G Zp ( r ) are the Fourier transforms of theneutron and proton single nucleon weak form factors [33],4 G Zn ( r ) = Q n G pE ( r ) + Q p G nE ( r ) − G sE ( r ) , (6) θ (deg.)10 -6 -5 -4 -3 -2 -1 d σ / d Ω ( m b / s r) Pb Ca FIG. 1: Differential cross section including Coulomb distor-tions for 855 MeV electrons elastically scattered from
Pb(solid black line) and Ca (dashed red line) versus scatteringangle. θ (deg.)010 -5 A pv Pb Ca FIG. 2: Parity violating asymmetry including Coulombdistortions for 855 MeV electrons elastically scattered from
Pb (solid black line) and Ca (dashed red line) versusscattering angle. Symmetrized Fermi weak charge densitiesare used (see text). G Zp ( r ) = Q p G pE ( r ) + Q n G nE ( r ) − G sE ( r ) , (7)where G pE ( r ) and G nE ( r ) are Fourier transforms of theproton and neutron electric form factors. Finally G sE ( r )describes strange quark contributions to the nucleon elec-tric form factor [34–37]. This is measured to be small sowe assume G sE ( r ) ≈ (cid:82) d r G Zn ( r ) = Q n , and (cid:82) d r G Zp ( r ) = Q p . The weak charge of theneutron is Q n = −
1, while the weak charge of the protonis Q p ≈ .
05, to lowest order. Including radiative cor-rections [40, 41] one has, Q n =-0.9878, and Q p =0.0721.Finally, the total weak charge of a nucleus, Q W = (cid:90) d rρ W ( r ) = N Q n + ZQ p . (8) Nucleus c (fm) a (fm) Q W Ca 3.99595 0.51540 -26.2164
Pb 6.81507 0.61395 -118.551TABLE I: Fermi function fits of the radius c and surfacethickness a parameters, see Eq. 9, to weak charge densitiespredicted by the FSU Gold relativistic mean field interaction[30]. Also listed is the total weak charge Q W , see Eq. 8. To explore sensitivity to the surface thickness, wemodel ρ W ( r ) with a two parameter Fermi function ρ W ( r, c, a ) [42, 43], ρ W ( r, c, a ) = ρ Sinh( c/a )Cosh( r/a ) + Cosh( c/a ) . (9)Here, the parameter c describes the size of the nucleuswhile a describes the surface thickness (see Table I). Thenormalization constant ρ is ρ = 3 Q W πc ( c + π a ) , (10)so that (cid:82) d rρ W ( r, c, a ) = Q W . The r and r momentsof Eq. 9 are [42, 43], (cid:104) r (cid:105) = 35 c + 75 π a , (11) (cid:104) r (cid:105) = 37 c + 187 c π a + 317 π a . (12)For the FSU Gold relativistic mean field interaction [30],or other density functional, we calculate the r and r moments and invert the above Eqs. to determine fit pa-rameters c and a . The results for Pb are listed in TableI and shown in Fig. 3. The Fermi function provides agood fit to ρ W ( r ) and averages over the small interiorshell oscillations.The Fermi function fit for Ca is shown in Fig. 4: inthis case the fit is less good because the interior shell os-cillations have larger amplitudes. Nevertheless, the Fermifunction still provides a good qualitative description of ρ W ( r ) and we expect the interior shell oscillations to onlybe important at larger momentum transfers. Thereforethe Fermi function is adequate for our purposes of pro-viding a simple representation of the surface thickness.Of course, our choice of Eq. 9 introduces some model de-pendence into our analysis. However, we expect this tobe small and other representations of the weak densitysuch as using a Helm form [38] should give very simi-lar results. We explore in Sec. III the sensitivities of thePREX-II and CREX experiments to values of the surfaceparameter a . ρ (f m - ) ρ W (FSU Gold) ρ ch Symmetric Fermi fit
FIG. 3: Minus the weak charge density - ρ W ( r ) of Pbversus radius r . The solid black line is from the FSU Goldrelativistic mean field model [30] while the the long dashedblue line shows a Fermi function fit (Eq. 9). The short dashedred line shows the experimental charge density ρ ch ( r ) fromRef. [1]. ρ (f m - ) FIG. 4: As Fig. 3 except for Ca.
III. SENSITIVITY OF PREX-II AND CREX TOTHE SURFACE THICKNESS
In this section we calculate how A pv depends on thesurface thickness parameter a for the kinematics of thePREX-II [27] and CREX [28] experiments. The PREX-IIexperiment aims to measure the radius R W of the weakcharge distribution of Pb, R W = 1 Q W (cid:90) d r r ρ W ( r ) = 35 c + 75 π a . (13)Here the second relation follows from our Fermi functionin Eq. 9. To calculate the sensitivity to changes in thesurface thickness a we also change c in such a way that R W in Eq. 13 remains constant. We define the sensitivityto the surface thickness (cid:15) a as the log derivative of theasymmetry w.r.t. the log of the surface thickness, (cid:15) a = d ln A pv d ln a = aA pv dA pv da . (14)We approximate this as (cid:15) a ≈ A pv ∆ A where ∆ A is thedifference in A pv calculated with a increased by 1% (atconstant R W ) and the original A pv . This is shown in Fig.5 for Pb at a beam energy of 950 MeV. We see that (cid:15) a is very small at angles < ◦ because at forward anglesone is most sensitive to R W and this has been kept fixed.We also define the sensitivity to R W as the log deriva-tive of A pv w.r.t. the log of R W , (cid:15) R = d ln A pv d ln R W = R W A pv dA pv dR W . (15)We approximate this as (cid:15) R ≈ A pv ∆ A where ∆ A is thedifference in A pv calculated with both c and a increasedby 1% and the original A pv . This is also shown in Fig.5. Both (cid:15) a and (cid:15) R are seen to change sign near the firstdiffraction minimum in the cross section, see Fig. 1.At the PREX-II average kinematics 950 MeV and scat-tering angle θ ≈ . ◦ we find (cid:15) a ≈ . A pv is measured to about 2.5% then one will be sensitiveto a to ± . / .
091 or ± a is known to betterthan 27%, the uncertainty in a will not give a large er-ror in the extraction of R W . We conclude, PREX-II isnot very sensitive to the surface thickness a . This is inagreement with previous work, see for example [39]. θ (deg.)-4-3-2-101 d l n A pv / d l n a Pb sensitivities
950 MeV
FIG. 5: Log derivative of the parity violating asymmetryw.r.t. the log of the surface thickness parameter a (solid blackline) or w.r.t the log of the weak radius R W (dashed red line)for Pb at 950 MeV versus scattering angle.
We repeat these calculations for CREX, see Fig. 6. Atthe CREX average kinematics 2220 MeV and θ ≈ . ◦ we find a much large (cid:15) a ≈ . Ca has a larger surface to volume ratio than θ (deg.)-15-10-505 d l n A P V / d l n a Ca sensitivities
FIG. 6: As Fig. 5 except for Ca at 2220 MeV.
Pb and because the CREX kinematics are closer tothe diffraction minimum. If A pv is measured to ≈ a to 5% / . ≈ a is known to significantly better than 7% (whichmay be unlikely), its uncertainty will be important in theextraction of R W from A pv . We conclude that CREX issensitive to the surface thickness and one must carefullyaddress this sensitivity in any extraction of R W . IV. NEW EXPERIMENT TO MEASURE THESURFACE THICKNESS
In this section we evaluate the statistical error and fig-ure of merit (FOM) for a new parity violating electronscattering experiment to determine the surface thicknessof the weak charge density of either
Pb or Ca. Thesurface thickness of ρ W ( r ) can differ from the known sur-face thickness of ρ ch ( r ) and is expected to be sensitiveto poorly constrained isovector gradient terms in energyfunctionals. One way to constrain these gradient termsis to perform microscopic calculations of pure neutrondrops in artificial external potentials, using two and threeneutron forces. Then one can fit the resulting energiesand neutron density distributions with an energy func-tional by adjusting the isovector gradient terms. It maybe possible to test these theoretically constrained isovec-tor gradient terms by measuring the surface behavior of ρ W ( r ).For reference, we collect in Table II surface thick-ness parameters a from Fermi function fits to the weakcharge density as predicted by a small selection of non-relativistic and relativistic mean field models. The re-sults of these models for Pb have an average value of a ≈ .
60 fm with a variance of ± .
02 fm, a ±
3% range.For Ca the models give a ≈ . ± .
025 fm, a ± a in a new parityviolating experiment to have a model independent deter-mination. Model a [ Pb] (fm) a [ Ca] (fm)SIII 0.5792 0.5053SLY4 0.6040 0.5247SV-min 0.6056 0.5386TOV-min 0.6111 0.5435UNEDF0 0.6155 0.5458IUFSU 0.6079 0.5298FSU Garnet 0.6106 0.5264NL3 0.6096 0.5235TABLE II: Surface thickness parameter a of Fermi functionfits to the weak charge densities of Pb and Ca, see Eq. 9. θ (deg.)-5-4-3-2-101 d l n A pv / d l n a Pb sensitivities
855 MeV
FIG. 7: As Fig. 5 except for
Pb at 855 MeV.
We now examine the optimal kinematics for an ex-periment. The total number of electrons N tot that arescattered into a solid angle ∆Ω in a measurement time T is, N tot = IT ρ tar dσd
Ω ∆Ω . (16)Here I is the beam current and ρ tar is the density ofthe target in atoms per cm . For simplicity, we neglectradiative corrections. The statistical error in the deter-mination of the surface thickness a is ∆ a ,∆ aa = (cid:16) N tot A pv P (cid:15) a (cid:17) − , (17)where P is the beam polarization. This depends on thefigure of merit ( F OM a ) that we define as, F OM a = dσd Ω A pv (cid:15) a . (18)One can adjust the scattering angle (or momentum trans-fer) to maximize F OM a . This in turn will minimize the θ (deg.)-14-12-10-8-6-4-2024 d l n A P V / d l n a Ca sensitivities
855 MeV
FIG. 8: As Fig. 5 except for Ca at 855 MeV. run time necessary to achieve a given statistical error inthe determination of a . Likewise, the statistical error inthe determination of the weak radius R W is,∆ R W R W = (cid:16) N tot A pv P (cid:15) R (cid:17) − . (19)This is closely related to the figure of merit for the de-termination of R W that we define as, F OM R = dσd Ω A pv (cid:15) R . (20)The statistical error in the determination of R W scaleswith ( F OM R ) − / . θ (deg.)10 -13 -12 -11 -10 -9 -8 F O M ( m b / s r) FIG. 9: Figures of merit
F OM a and F OM R versus scatteringangle θ to measure the weak surface thickness (solid blackcurve) or radius (dashed red curve) of Pb at 855 MeV.
In Figs. 7 and 8 we plot sensitivities (cid:15) a and (cid:15) R for Pband Ca and we show the figure of merits
F OM a and F OM R in Figs. 9 and 10. At a laboratory energy of 855
10 15 20 25 θ (deg.)10 -13 -12 -11 -10 F O M ( m b / s r) FIG. 10: Figures of merit
F OM a and F OM R versus scatter-ing angle θ to measure the weak surface thickness (solid blackcurve) or radius (dashed red curve) of Ca at 855 MeV.
MeV, the maximum in
F OM R for Pb occurs near ascattering angle of 5 degrees. This is indicated by a redarrow in Fig. 9 and corresponds to a momentum transfer q = 0 .
38 fm − , see Table III. A maximum in F OM a for Pb occurs near 10 degrees or q = 0 .
76 fm − . A parityexperiment near this momentum transfer will be sensitiveto the surface thickness a . Note there is also a localmaximum in F OM a near 6.5 degrees (or q = 0 .
49 fm − )in Fig. 9. However a measurement of A pv at this q maybe linearly dependent with the PREX-II measurement atonly slightly smaller q ≈ .
38 fm − . This could make itdifficult to separately determine both R W and a . Parameter Pb Ca R W − − a − − TABLE III: Near optimal momentum transfer q that giveslarge figures of merit, for measuring the weak radius R W orsurface thickness a for the nuclei Pb and Ca.
We now scale our results in Figs. 9 and 10 to otherenergies. As long as the energy is above say 400 MeV,Coulomb distortions do not depend strongly on energy.As a result A pv depends primarily on momentum transfer q and only weakly on beam energy E , A pv ( E , q ) ≈ A pv ( E , q ) . (21)Therefore (cid:15) i ( q ) in Fig. 6, as a function of q , is veryclose to (cid:15) i ( q ) in Fig. 8. The differential cross section de-pends strongly on q . However at fixed q , it scales approx-imately as E so that dσd Ω ( E , q ) ≈ E E dσd Ω ( E , q ). There-fore F OM i , at fixed q , grows with increasing energy, F OM i ( E , q ) ≈ E E F OM i ( E , q ) . (22) This is true for both i = R and a . If the solid angle of thedetector ∆Ω is fixed, it can be advantageous to measureat as forward an angle as possible, and at a higher beamenergy, because this will increase the figure of merit.A measurement of A pv for Pb at q = 0 .
76 fm − ,see Table III, is sensitive to a . In general, it will alsobe sensitive to R W . However PREX-II and CREX areconstraining R W for both Pb and Ca. Therefore, itshould be possible to extract a from only a single newmeasurement. This would completely determine a Fermifunction model of the weak charge density.The statistical error in the extraction of a scales with F OM − / a . For Pb, the maximum in
F OM a in Fig. 9near 10 degrees is about 400 times smaller than the max-imum in F OM R . Therefore a can be determined to 10%with only a few times larger integrated luminosity (beamtime) than PREX-II is using to determine R W to approx-imately 1%. For Ca the local maximum in
F OM a inFig. 10 near 17 degrees is only ≈ /
100 the maximumin
F OM R . Therefore a can be determined to 10% us-ing comparable integrated luminosity (or beam time) asa 1% measurement of R W . We conclude that a parityviolating electron scattering experiment to measure a isfeasible.A similar experiment will be possible at the A1 spec-trometer facility of the MAMI accelerator. Accordingto the construction of the spectrometers and their ar-rangement on the pivot surrounding the scattering cham-ber, there are limitations in the accessible angular range.Furthermore, only selected beam energies at MAMI areequipped with a special stabilization system which is es-sential for performing parity-violation experiments. Twoscenarios are currently under investigation, at beam ener-gies of 855 and 570 MeV respectively, to optimise the ex-perimental conditions. Cherenkov detectors specificallydesigned for counting experiments [52] will be placed inthe focal planes of the high resolution spectrometers.This set-up will allow to make use of the high precisiontracking detectors of the spectrometer to align the elasticline of Pb with the Cherenkov detectors by changingthe magnetic field setting. The data acquisition electron-ics will be the one of the former A4 experiment [53]. Witha beam intensity of 20 µ A at a scattering angle of 10.35 a10% measurement of the surface thickness will be possi-ble in about 100 days for a beam energy of 855 MeV. Atthe lower beam energy the precision in the extraction ofthe surface radius will be the same for a scattering angleof 15.2 but the total running time will double.
V. CONCLUSIONS
The present PREX-II and CREX parity violating elec-tron scattering experiments are probing the weak chargedensities of
Pb and Ca. These experiments are pri-marily sensitive to the radius of the weak charge density R W but they are also sensitive to the surface thickness a . In this paper we have explored the feasibility of a newparity violating electron scattering experiment to mea-sure a for a neutron rich nucleus. PREX-II or CREXcombined with an additional parity violating measure-ment at a momentum transfer near 0.76 fm − for Pbor 1.28 fm − for Ca will cleanly determine both R W and a . Determining a both sharpens the determinationof R W from PREX II or CREX and determines the av-erage interior weak charge density and baryon density[2]. In particular, the average interior baryon density of Pb is closely related to the saturation density of nu-clear matter.
Acknowledgements
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