Theoretical study on nuclear structure by the multiple Coulomb scattering and magnetic scattering of relativistic electrons

0.92 MB·

by  Jian Liu

Theoretical study on nuclear structure by the
multiple Coulomb scattering and magnetic scattering
of relativistic electrons
6
1
0
2

n
a Jian Liu a, Xin Zhangb Chang Xub Zhongzhou Renb
∗
J

6
2
aCollegeofScience, ChinaUniversityofPetroleum(EastChina),Qingdao266580, China
]
h
-t bDepartmentofPhysics, NanjingUniversity, Nanjing210093, China
l
c
u
n
[

1
v Abstract
7
3
9
6 Electron scattering is an effective method to study the nuclear structure. For the odd-A
0
. nuclei with proton holes in the outmost orbits, we investigate the contributions of pro-
1
0
6 ton holes to the nuclear quadrupole moments Q and magnetic moments µ by the multi-
1
: ple Coulomb scattering and magnetic scattering. The deformed nuclear charge densities
v
i
X
are constructed by the relativistic mean-field (RMF) models. Comparing the theoretical
r
a
Coulomb and magnetic form factors with the experimental data, the nuclear quadrupole
moments Q and nuclear magnetic moments µ are investigated. From the electron scatter-
ing, the wave functions of the proton holes of odd-A nuclei can be tested, which can also
reflectthevalidityofthenuclear structuremodel.
Keywords: Electronscattering, nuclear longitudinal formfactors, nuclear magneticform
factors
PACS:21.10.Gv,24.80.+y,25.30.Bf
PreprintsubmittedtoNuclearPhysicsA 27January2016
1 Introduction
Electronscatteringat highenergiesis oneofthemostpowerfultoolstoinvestigate
the nuclear structure, because the electron-nucleus interaction is well understood
[1,2,3,4,5,6]. According to the contributions from the Coulomb and magnetic in-
teractions, electron scattering can be divided into the Coulomb scattering (or lon-
gitudinal scattering) and magnetic scattering [7]. In the past decades, the nuclear
charge densities of many stable have been determined accurately by the electron
Coulombscattering[8,9].RecentlywiththedevelopmentofRadioactiveIonBeam
(RIB) facilities, some exotic nuclei far away from the β stability line with short
half-lives can be produced [10,11]. It is valuable for us to explore the charge den-
sitydistributionsofexoticnucleiwiththeelectromagneticprobes.Forthispurpose,
the new generation electron scattering facilities are under construction at RIKEN
[12,13,14] and GSI [15,16]. It is expected that in the near future elastic electron
scatteringoff exoticnucleican beachieved[17,18].
Besides the nuclear Coulomb scattering, the electron magnetic scattering is an-
other fundamental method to explore the nuclear magnetic properties and valence
nucleon orbits [19]. Different from the Coulomb scattering where all the charged
nucleons contributeequally, themagneticelectron scattering is largely dueto con-
tributions of the unpair nucleon in the outmost orbit. The electron magnetic scat-
tering providesamodel-independentmethodto determinethenuclear shellenergy
level and wave functions of the valence nucleon. Therefore the electron magnetic
scattering experiments have been widely used to investigate the valence structure
ofthenuclei[20,21,22,23,24,25,26].
Corresponding author
∗
Emailaddress:[email protected].(JianLiu).
2
With the rapid development of the experiments, great efforts have also been de-
voted to the theoretical studies of electron scattering off nuclei in the last decade
[27,28,29,30,31,32,33,34,35,36,37,38,39].Theplane-waveBornapproximation(PWBA)
is a simple method to solve the electron scattering where the effects of nuclear
Coulomb field on the scattering electrons are neglected. The phase-shift analy-
sis method is an effect way to study the electron scattering which includes the
Coulomb distortion effects by the exact phase-shift analysis of Dirac equations.
This calculation method is also referred to as the distorted-wave Born approxima-
tion (DWBA) [40,41]. The phase-shift analysis method is an accurate method to
calculate the cross sections of scattering electrons and has been applied in many
theoreticalstudies[32,33,34,35,36,37,38].
During the studies of electron scattering, the nuclear form factors can be decom-
posed into several multipoles according to the selection rules, such as C , C and
0 2
M . For the Coulomb scattering off spherical nuclei, only the C contributions
1 0
need to be taken into consideration. For the deformed nuclei, the high multipoles
C (λ 2) can reflect the contributions of nuclear deformation. In the last decade,
λ
≥
some researches focused on the Coulomb scattering off spherical nuclei where the
C contributions are precisely calculated under the DWBA method [29,32,33,37].
0
Asweknow,mostofthenucleiarefoundtobedeformedintheirgroundstatesboth
theoretically and experimentally [42,43,44], and the high multipoles C (λ 2)
λ
≥
need bo be included during the studies. In Ref. [45], the dependence of the cross
section on the parameter ξ of DWBA calculations of C multipole were investi-
2
gatedfor27Al.FortheC andC contributions,thereareaslodetailedcomparative
0 2
analysisonthePWBA and DWBA calculationsinRefs. [46,47,48,49].
On the basis of studies done before, in this paper the Coulomb and magnetic mul-
tipolesaresystematicallyinvestigatedfortheodd-Anucleiwithprotonholesinthe
3
outermost orbits. The nuclei 14N, 27Al and 39K are chosen as the candidate nuclei.
Their nuclear quadrupoleand magnetic momentsare assumed as the contributions
of the proton holes in the outermost orbits. The density distributions of deformed
nuclei areconstructed bytherelativisticmean-field (RMF)model.After providing
the deformed density distribution, the nuclear longitudinal form factors F (q) are
L
derived under the PWBA method. Then we take into account the Coulomb distor-
tion corrections. By comparing the theoretical F (q) with the experimental data,
L
the nuclear quadrupole moments Q can be extracted from the electron scattering
experiments.
Besidesthenuclearlongitudinalscattering,thenuclearmagneticformfactorsF (q)
M
are also investigated in this paper. By the longitudinal scattering, we can analyze
the contributions of proton hole to the total charge density distributions.However,
bythemagneticscattering,wecanfurthertestthewavefunctionsandenergylevels
of the proton hole. The wave functions of the proton hole are calculated under the
theoretical framework of RMF model. Then we investigate the nuclear magnetic
moments µ and compare them with the experimental data. Besides, the nuclear
magneticformfactors F (q)arefurtherstudiedwherethemagneticmultipoles M
M L
are derived under the PWBA method. The results are also compared with the ex-
perimental data. Combining the Coulomb scattering and magnetic scattering, the
nuclear quadrupole moments Q and magnetic moments µ can be extracted, which
teststhevalidityofthenuclearstructuremodel.
The paper is organized as follows. In Sec. 2.1, the deformed density distribution
model is constructed and in Sec. 2.2, the theoretical framework of Coulomb and
magnetic scattering is presented. In Sec. 3, the nuclear longitudinal form factors
F (q)2arecalculatedandthenuclearquadrupolemomentsQareextracted.InSec.
L
| |
4, we study the nuclear magnetic moments µ and nuclear magnetic form factors
4
F (q)2. Finally,asummaryisgiveninSec. 5.
M
| |
2 Formalism
In this section, the main formulas of electron-nucleus scattering are outlined. The
calculation scheme is divided into two parts. First, the nuclear density distribution
is constructed where the nuclear deformation and magnetic moment are assumed
as the contributions of proton hole in the outermost orbit. Then according to the
deformed density model, the nuclear Coulomb form factors F (q)2 and magnetic
L
| |
form factors F (q)2 are investigated.
M
| |
2.1 Deformed densitydistributionmodel
In order to study the nuclear longitudinal form factors, the nuclear proton density
distribution is constructed as the superposition of a spherical part and deformed
part:
ρ(r)=ρ0(r)+ρd(r) (1)
p p
where ρ0(r) is spherical symmetric and ρd(r) describes the nuclear deformation.
p
When the nuclear deformation is not big, ρ0(r) is considered as the main part of
p
ρ (r)and ρd(r) isreferred to asthedeformed correction toρ0(r).
p p p
For the nuclei with a proton hole in a closed shell, the nuclear deformation is as-
sumed as the contributions of the proton hole. The relativistic mean-field (RMF)
modelisusedtocalculatethedeformedprotondensitydistribution.Theadvantage
of RMF model is that the spin degrees of freedom of nucleons are treated micro-
scopically and the spin-orbit splitting is given automatically, since it is essentially
5
a relativistic effect. In RMF model, the spherical symmetric part ρ0(r) can be cal-
p
culated by[50]:
A
1
ρ0(r) = ( t)(G (r)2 + F (r)2), (2)
p 2 − i | i | | i |
i=1
X
where G and F are the upper and lower components of the Dirac spinors for the
i i
occupiedstates.Forthedeformedpartρd(r),thecontributionoftheprotonholecan
p
bedescribed as [45]:
1 1
ρd(r) = β (G (r)2 + F (r)2)( Y (θ,ϕ)2), (3)
p dr2 | j | | j | 4π −| lm |
where l and j are the orbital and total angular momentum of the proton hole, re-
spectively.The total proton number Z is not changed because ρd(r)d3r = 0. The
p
parameterβ isintroducedto adjustthenuclearquadrupolemoRment:
d
1
Q = ρ (r)(3z2 r2)dτ. (4)
p
e −
Z
V
Thesphericalharmonics Y (θ,ϕ)2 in Eq.(3)can befurtherexpanded:
lm
| |
1
(2l+1)2 2
Y (θ,ϕ)2 = ( 1)m lm,l mL0 l0,l0L0 Y , (5)
| lm | − 4π(2L+1) h − | ih | i L0
L ( )
X
where l m ,l m LM istheClebsch-Gordan (C. G.)coefficient. SubstitutingEqs.
1 1 2 2
h | i
(2), (3) and (5) into Eq. (1), the deformed proton density distribution has the fol-
lowingform:
2l
ρ (r) = ρ0(r)+ ρL(r)Y , (6)
p p p L0
L=2
X
where
1
(2l+1)2 2
ρL(r) = ( 1)mβ lm,l mL0 l0,l0L0 (G (r)2 + F (r)2).
p − d 4π(2L+1) h − | ih | i | j | | j |
( )
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(7)
For the proton hole with the orbital angular momentum l, the spherical harmonics
expandedvalueLinEq.(7)cantakealltheevennumbersfrom2to2laccordingto
the properties of C. G. coefficients. With theEqs. (2) and (7), the deformed proton
densitydistributionin Eq.(6)can becalculated.
2.2 Nuclear electromagneticform factors
Duringthescatteringprocess,theelectronscatteringcrosssectioncanbecalculated
by theformula:
dσ dσ
= η F(q)2, (8)
dΩ dΩ · ·| |
!M
withtheMottcross section
dσ Z2α2Cos2θ
= 2, (9)
dΩ 4E2Sin4θ
!M i 2
and therecoil factor η = [1+ 2ESin2(θ/2)]2. The form factor F(q)2 can be further
M | |
developedas:
θ
F(q)2 = F (q)2 +(1+2tan2 )F (q)2. (10)
L M
| | | | 2 | |
F (q)2 is the longitudinal term (or Coulomb term) and F (q)2 is the magnetic
L M
| | | |
term. The longitudinal and magnetic form factor can be decomposed into several
multipolesaccordingto theselection rules:
∞ ∞
F (q)2 = F (q)2, and F (q)2 = F (q)2. (11)
L Cλ M Mλ
| | | | | | | |
λ=0,even λ=1,odd
X X
7
2.2.1 Longitudinalformfactor
Thelongitudinalterm F (q)inEq.(10)canbeobtainedbyfoldingtheprotonform
L
factor F (q) withtheelectricform factorofasingleprotonGp(q):
p E
F (q)2 = Gp(q)2 F (q)2, (12)
| L | E ·| p |
wherethedetailform ofGp(q)can befound inRefs. [51] and[52].
E
UnderthePWBA method,theprotonform factor F (q)can becalculated by:
p
1
F (q) = d3rρ (r)eiqr, (13)
p p ·
Z
Z
where ρ (r) is the deformed proton density of Eq. (6). The exponential function
p
eiqr inEq. (13)can beexpandedas asumofthesphericalharmonics:
·
eiq·r = 4π(2L′ +1)·iL′ · jL′(qr)·YL′0(θ,ϕ), (14)
L
X′ p
where j (qr) is the spherical Bessel function. Substituting Eqs. (6) and (14) into
L
Eq. (13) and taking into account the orthogonality and normalization of the spher-
ical harmonics, the proton form factor can be obtained by squaring F (q)2 and
p
| |
averagingitovertheorbitalangularmomentumprojectionmfrom ltol.Through
−
thederivationunderthePWBA method,theprotonformfactorcanalsobedivided
intotwoparts:
FPW(q)2 = FPW(q)2 + FPW(q)2. (15)
| p | | C0 | | Cλ |
FPW(q)2 in Eq. (15) can be seen as the contribution of the spherical part ρ0(r) on
| C0 | p
nuclearlongitudinalformfactor:
2
1
FPW(q)2 = Gp(q)2 d3rρ0(r)j (qr) , (16)
| C0 | E · Z p 0
Z !
8
and FPW(q)2 can beseen as thecontributionofthedeformed part ρd(r):
| Cλ | p
2
l 2l
1
FPW(q)2 = Gp(q)2 lm,l mL0 l0,l0L0 I (q) , (17)
| Cλ | E · 2l+1 h − | ih | i L
 
m= l L=2,even
where X−  X 
1
I (q) = dr j (qr)β (G (r)2 + F (r)2). (18)
L L d j j
Z | | | |
Z
For proton hole with the angular momentum l, there are only contributions for
L = 0,2,4, ,2l,accordingtothesymmetrypropertyofC.G.coefficientsandthe
···
orthogonalitypropertyofspherical harmonics.
TheCoulombdistortioncorrectionsareverysignificantduringthetheoreticalstud-
ies of electron scattering [41]. Under the theoretical framework of Eq. (15) of
PWBA, the longitudinal term F (q)2 can be divided into two parts F (q)2 and
L C0
| | | |
F (q)2.TheCoulombdistortioncorrectionsneedtobeincludedinthesetwoparts
Cλ
| |
and theEq. (15)can berewrittenas:
FDW(q)2= FDW(q)2 + FDW(q)2 (19)
| L | | C0 | | Cλ |
The DWBA calculation FDW(q)2 of C multiple can be done by many methods,
| C0 | 0
such as the relativistic Eikonal approximation [29] and the phase-shift analysis
method[33,35,36].Inthispaper,wecalculatethe FDW(q)2 bythephase-shiftanal-
| C0 |
ysismethodand thedetailedformulas can beseen inRefs. [33,40].
For C multiples, there are few researches on the DWBA calculations FDW(q)2.
λ | Cλ |
In Refs. [48,49], the authors presented full DWBA calculations for the C multi-
2
pole of the inelastic electron scattering. It can be seen in the Fig. 1 of Ref. [49]
that the deviations between the cross sections σDW and σPW mainly located at the
C2 C2
diffraction minima. However at these positions, the nuclear longitudinal form fac-
9
tors F (q)2 aremainlydeterminedbythecontributionsofC multipole,whichcan
L 0
| |
be seen in the left panels of Figs. 2, 4 and 6 of this paper. Besides, when construct
the deformed density distribution, we assume that for the small deformation, the
spherical part ρ0(r) is the majorpart of ρ (r) and the nonspherical part ρd(r) is the
p p p
high order correction of ρ0(r). Therefore, we calculate the spherical contributions
p
F (q)2 accurately where the Coulomb distortion corrections are taken into ac-
C0
| |
count with the DWBA method. For ρd(r), their contribution F (q)2 is calculated
p | Cλ |
with Eq. (17) under the PWBA method. Finally, the longitudinal form factors of
deformed nucleiisgivenas following:
FDW(q)2 FDW(q)2 + FPW(q)2. (20)
| L | ≈ | C0 | | Cλ |
Combining the DWBA calculations of FDW(q)2 and the formula of Eq. (17), the
| C0 |
nuclearlongitudinalformfactor F (q)2 can beobtained.
L
| |
2.2.2 Magneticformfactor
For odd-A nuclei with the proton-hole configuration in a closed shell, the nuclear
magnetic moments are assumed as the contributions of the proton hole. The wave
functionofprotonholecanbedescribedbytheDiracspinorundertheRMFmodel:
i[G(r)/r]Φ inκm
κm
   | i 
ψ = = . (21)
nκm −[F(r)/r]Φ−κm −|nκmi
Themagneticform factorisdefined as [53,19]:
4πf 2(q)f 2 (q) 2j
F2 (q2) = sn c.m. j Tˆmag j 2 (22)
M 2j+1 h || L || i
L=1,odd
X (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
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