Systematic study of Coulomb distortion effects in exclusive (e,e'p) reactions

45 Pages·0.38 MB·English

by  V. Van der Sluys

SSF96-09-02
Systematic study of Coulomb distortion effects in exclusive (e,e p)
′
reactions
V. Van der Sluys, K. Heyde, J. Ryckebusch and M. Waroquier
7 Department of Subatomic and Radiation Sciences
9
9 University of Gent
1

n Proeftuinstraat 86
a
J
B-9000 Gent, Belgium
1
3
(February 9, 2008)

1
v
3
0
0
Abstract
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0
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A technique to deal with Coulomb electron distortions in the analysis of
/
h
t (e,e′p) reactions is presented. Thereby, no approximations are made. The
-
l
c
u suggested technique relies on a partial-wave expansion of the electron wave
n
v: functions and a multipole decomposition of the electron and nuclear current
i
X
in momentum space. In that way, we succeed in keeping the computational
r
a
timeswithinreasonablelimits. Thistheoreticalframeworkisusedtocalculate
the quasielastic (e,e′p) reduced cross sections for proton knockout from the
valence shells in 16O, 40Ca, 90Zr and 208Pb. The final-state interaction of the
ejected proton with the residual nucleus is treated within an optical potential
model. The role of electron distortion on the extracted spectroscopic factors
is discussed.
21.10.Jx, 21.60.Jz, 24.10.Eq, 25.30.Fj
Typeset using REVTEX
1
I. INTRODUCTION
For a long time it has been recognized that the exclusive (e,e′N) reaction in the quasielas-
tic (QE) region is a powerful tool for studying the single-particle motion inside the nucleus,
and is a testing ground for the different available nuclear models. One of the principal in-
terests in the exclusive (e,e′N) reaction is to extract the nucleon spectral function P(p~,E)
from the cross section. This spectral function can be interpreted as the joint probability to
remove a nucleon with momentum p~ from the target nucleus and to find the residual system
at an excitation energy E. Related to these spectral functions, spectroscopic factors and
occupation numbers are often studied. They are a measure for the validity of the indepen-
dent particle model (IPM). The spectroscopic factor S (E) gives the probability to reach
nljm
the single-particle state specified by the quantumnumbers nljm in the residual nucleus at
an excitation energy E. The occupation number N gives the number of nucleons in the
nljm
single-particle state nljm in the target nucleus and involves an integration of the spectro-
scopic factors over the complete excitation energy range [1]. In the IPM the states above
(under) the Fermi level are completely empty (filled) and the total hole (particle) strength
is situated at a fixed single-particle energy. The deviation from full (no) occupancy for the
orbits below (above) the Fermi level is a measure for correlations neglected in this mean-field
approach.
The occupation probabilities in even-even nuclei have been calculated within several the-
oretical frameworks. Most models go beyond the mean-field approach and partially account
for short- and/or long-range nucleon-nucleon correlations [1–6]. Occupation probabilities
for the single-particle states which considerably deviate from the IPM value were obtained.
Moreover, itisdemonstratedthatthesingle-particleholestrengthisfragmentedover abroad
range of energy. In particular, occupation numbers for the proton 3s1/2 orbit in 208Pb have
been calculated varying from 1.42 [3] to 1.66 [7] pointing towards a strong depletion of this
hole state in the ground state of 208Pb. From an experimental point of view, the CERES
method [8] was developed in an attempt to obtain absolute occupation numbers from experi-
2
mental data. The model uses only relative spectroscopic factors and allows to account, in an
approximate way, for the strengths at high missing energies, not accessible for experiment.
With this method, the 3s1/2 occupation number in 208Pb is found to be 1.57(10).
Although the advantages of the quasielastic (e,e′N) process to study spectroscopic fac-
tors are widely recognized, the extraction of these factors from experiment is still not free of
ambiguities. For example, depending on the model used in the analysis of the 208Pb(e,e′p)
reaction, the spectroscopic factor for the transition to the groundstate in 207Tl ( 3s1/2
hole) varies from 0.40 [9] to 0.71 [10]. A reliable determination of spectroscopic factors
requires an accurate knowledge of the (e,e′N) reaction mechanism (photoabsorption mecha-
nism, final-state interaction (FSI) of the ejected nucleon with the residual nucleus) and the
exact treatment of the Coulomb distortion of the scattered electrons, especially for heavy
nuclei.
In this paper we present results from systematic calculations of (e,e′p) cross sections for
a number of even-even target nuclei and various kinematical conditions and confront them
with data taken at NIKHEF. The extracted spectroscopic factors are compared with the
corresponding values deduced within other theoretical approaches [10–13]. Much attention
is paid to the effect of electron distortion on the calculated cross section. It is pointed out
that, especially for scattering off heavy nuclei, an exact treatment of these effects is highly
needed in order to reproduce the shape of the measured cross sections and, consequently, to
obtain reliable spectroscopic factors.
This paper is organized as follows. In section II the theoretical formalism for the (e,e′N)
reaction is outlined. The derivation of the cross section is divided in two subsections treat-
ing the electron and the nuclear aspect of the (e,e′N) reaction. The technical details are
dealt with in appendix A. The numerical details of the adopted approach are discussed in
section III. The formalism is applied to electro-induced one-proton knockout reactions from
a number of medium-heavy target nuclei in section IV. Finally, some conclusions are drawn
in section V.
3
II. FORMALISM
A. Cross section
~
In this paper we describe the process in which an electron with four-momentum k(ǫ,k)
and spin polarization m is scattered from a target nucleus at rest with a rest mass M .
sk A
The detected electron is characterized by its four-momentum k′(ǫ′,~k′) and spin polarization
ms′. The energy transfer to the nucleus ω = ǫ ǫ′ is supposed to be sufficient to eject a
k −
nucleon N (proton or neutron) with four-momentum p (E ,~p ) and spin projection m
N N N sN
out of the target nucleus leaving the residual nucleus with four-momentum p (E ,p~ ). The
B B B
differential cross section and the Feynman amplitude m for this process are related as
fi
follows:
d4σ 1
= ǫ′2 p~ E m 2 δ(ω S E E E +M +M ) . (1)
dǫ′dΩ dΩ dE (2π)5 | N| N | fi| − N − x − N − B N B
e N N i,f
X
Throughout this paper we adopt naturaland unrationalized Gaussian (α = e2) units. In this
relation S stands for the separation energy of a nucleon out of the target nucleus and E
N x
denotes the excitation energy of the residual nucleus. The rest masses of the ejected nucleon
and the residual nucleus are given by M and M . The angles Ω (θ ,φ ) and Ω (θ ,φ )
N B e e e N N N
specify thescattered electron and ejected nucleon with respect tothe chosen reference frame.
At thispoint this reference frameis not further specified. The sum implies a summation
i,f
P
over all final states (electron and nuclear) and anaverage over the initial states (electron and
nuclear). We only have to sum over these final states which satisfy the energy conservation
relation.
In the Born approximation the transition amplitude m can be written in terms of
fi
matrixelements of the electron Jµ and nuclear Jµ charge-current four-vector in momentum
el nucl
space in the following way:
1 1
m = d~q < f J ( ~q ) i >< f Jµ (~q ) i > . (2)
fi −2π2 ω2 ~q 2 +iη e| el,µ − | e n| nucl | n
Z −| | Xµ
4
The initial and final electron states are denoted by i > and f >. The target nucleus and
e e
| |
final nuclear state consisting of a residual nucleus and an ejected nucleon are represented by
i > and f >.
n n
| |
The Feynman amplitude m can further be rewritten as follows
fi
1 1
m = d~q < f ρ ( ~q ) i > < f ρ (~q ) i >
fi 2π2  ~q 2 e| el − | e n| nucl | n
Z | |
1 
+ ( 1)λq < f J ( ~q ) i >< f J (~q ) i > . (3)
ω2 ~q 2 +iη  − e| el,λq − | e n| nucl,−λq | n 
−| | λqX=±1 
 
The spherical components of the electron and nuclear current operators are taken with re-
spect to therotatingreference frame(x ,y ,z ) (Fig.1 (a)). In thisway thethirdcomponent
q q q
of the current operator is directly related to the charge operator through the charge-current
conservation relation.
B. The leptonic part
In this section we elaborate on the electron matrixelement < f J ( ~q ) i > in the
e el,µ e
| − |
expression for the Feynman amplitude. The relativistic electron charge-current operator in
coordinate space reads
J0(~r) = e Ψˆe†(~r) Ψˆe(~r) ,
 el − (4)
 J~el(~r) = e Ψˆe†(~r) α~ Ψˆe(~r) ,
−
with Ψˆe(~r) the electron field operator in coordinate space. The initial and final electron
wave functions are defined according to
< ~r i >= Ψe(~r) , (5)
| e ~k
< ~r f >= Ψe (~r) , (6)
| e ~k′
and stand for four-dimensional Dirac spinors. They are solutions of the stationary electron
Dirac equation:
(α~.( i~ )+βm +V) Ψe (~r) = ǫ Ψe (~r) , (7)
− ∇ e ~kmsk ~kmsk
5
where m is the rest mass of the electron and V is the scattering potential. The additional
e
quantumnumber m uniquely determines the electron wave function.
sk
Dealing with high-energetic electrons the electron mass can be neglected with respect
to its total energy and the Dirac equation can be written down in the ultrarelativistic limit
~
(ǫ = k ). In the Dirac-Pauli representation for the α~ and β matrices and in the absence of
| |
an external potential V the solutions of equation (7) are given by :
Ψe (~r) = u (~k,m ) ei~k.~r
~kmsk e sk
= 1  χ1m/s2k(Ωk)  ei~k.~r . (8)
√2 ~σ.~kχ1/2 (Ω )
 |~k| msk k 
 
The spinors χ1/2 (Ω ) can be expressed in terms of the Pauli spinors and the matrixelements
msk k
1
of the Wigner -matrix, i.e.,
2
D
1
χ1/2 (Ω ) = χ1/2(σ) 2 (ϕ ,θ ,0) . (9)
msk k ms Dmsmsk k k
Xms
~
The angles Ω = (θ ,ϕ ) specify the momentum k with respect to the chosen reference
k k k
1
frame (x,y,z). The Wigner (R ) matrix represents the rotation of the reference frame
2 k
D
(x,y,z) over the Euler angles R = (ϕ ,θ ,0) in the basis spanned by the eigenvectors of
k k k
the operators Sˆ2 and Sˆ .
z
Assuming a central potential V = V(r), the electron wave functions are evaluated by
a phase shift analysis based on a partial-wave expansion. Indeed, the Dirac Hamiltonian
(Hˆ = α~.~k+V(r)) commutes with the angular momentum operators Jˆ2 and Jˆ and with the
z
operator Kˆ = β ~σ.L~ + 1 but not with the orbital momentum operator Lˆ2. As such, we
{ }
derived a complete set of operators with common eigenfunctions represented by Ψ˜ǫ (~r):
κjm
Hˆ Ψ˜ǫ (~r) = ǫ Ψ˜ǫ (~r) ,
κjm κjm

 JJˆˆz2ΨΨ˜˜ǫκǫκjjmm((~r~r)) == mj(jΨ˜+ǫκj1m)(~rΨ˜)ǫκ,jm(~r) , (10)
We can construct the partiaKˆl wΨ˜aǫκvjems(~rΨ˜)ǫ =(~r−)κasΨ˜fǫκojlmlo(w~r)s:.
κjm
6
Gǫlj(r) jm (Ω ,σ)
Ψ˜ǫ (~r) = Ψǫ (~r) =  r Yl1/2 r  (11)
κjm ljm i Flǫj(r) jm (Ω ,σ)
 r Yl1/2 r 
 
 
l = j 1/2 if κ = (j +1/2) ,
with  − −
 l = j +1/2 if κ = j +1/2 .
We introduce the common notationl
l = j + 1 l = j 1 ,
 2 ⇒ − 2 (12)
 l = j 1 l = j + 1 .
− 2 ⇒ 2
The spherical spin-orbit eigenspinor jm (Ω ,σ) is defined in the following way
Yl1/2 r
jm (Ω ,σ) = < lm 1/2m jm > Y (Ω )χ1/2(σ) . (13)
Yl1/2 r l s| lml r ms
mXlms
Each partial wave (11) can be easily proved to satisfy the eigenvalue equations (10) under
the condition that the radial electron wave functions Gǫ (r) and Fǫ(r) are solutions of the
lj lj
following second-order differential equations:
d2 Gǫ (r)+ dV(r)/dr d Gǫ (r)
dr2 lj E−V(r) dr lj

 d2 F+ǫ(hr()E+−dVV((rr)/)d)r2d−Fκ(ǫκr(+2r1)) + κr dEV−(rV)/(rd)ri Gǫlj(r) = 0 , (14)
dr2 lj E−V(r) dr lj
For each partial wave+ljh(tEhe−sVec(orn)d)2-o−rdκe(rκr−2d1i)ff−ereκrndEtVi−a(rVl)/(erdq)ruiaFtilǫoj(nr)fo=r G0 ǫ. (r) has to be solved
lj
numerically. For the regular solutions one imposes the following boundary conditions:
limGǫ (r) = 0 ,
lj
r→0
d
lim Gǫ (r) = 0 for l > 0 , (15)
r→0 dr lj
and one obtains the corresponding solution for Fǫ(r) through the relation:
lj
Gǫ (r) = (l l)Fǫ(r) . (16)
lj − lj
The asymptotic behaviour of the radial electron wave functions for Coulomb potential
~
scattering are given by (k k )
≡ | |
7
e,ǫ(tot)
sin(kr lπ/2+δ ηln(2kr))
lim Gǫ (r) = (l l) − lj − , (17)
r→∞ lj − k
e,ǫ(tot)
sin(kr lπ/2+δ ηln(2kr))
lim Fǫ(r) = − lj − . (18)
r→∞ lj − k
e,ǫ(tot)
The phase shift δ reflects the influence of the scattering potential V. It consists of two
lj
parts, i.e., the Coulomb phase shift σe and an additional phase shift δe,ǫ. For a Coulomb
l lj
potential generated by the Z protons in the nucleus, the Coulomb phase shift is defined
according to (η = Ze2):
−
σe = argΓ(l+1+iη) . (19)
l
Due to the fact that the scattering potential V is spin-independent, one can easily verify
that the total phase shift is l-independent, i.e.,
e,ǫ(tot) e,ǫ(tot) e,ǫ(tot)
δ = δ = δ . (20)
j lj lj
e(±)
Finally, the electron wave function Ψ (~r) is expanded in terms of these partial waves
~kmsk
Ψǫ (~r):
ljm
Ψe(±) (~r) = aǫmsk (±)Ψǫ (~r) . (21)
~kmsk ljm ljm ljm
X
The initial and final electron wave functions have to satisfy the outgoing (+) respectively
incoming ( ) boundaryconditions. Knowingtheasymptotic behaviour oftheradialelectron
−
wave functions, the coefficients aǫmsk(±) are fixed by
ljm
alǫjmmsk(±) = Dm12smsk(Rk)√4π2ile±iδje,ǫ(tot)(l−l)Yl∗ml(Ωk) < lml1/2ms|jm > . (22)
mXsml
At this point only the scattering potential V remains to be specified. In general the
central Coulomb scattering potential generated by Z protons is given by
1 r ∞
V(r) = 4πZα ρ(r′)r′2dr′ 4πZα ρ(r′)r′dr′ , (23)
− r −
Z0 Zr
with ρ(r) the nuclear charge density normalized according to 4π ∞ρ(r)r2dr = 1. In the
0
R
forthcoming discussion we have taken this charge density to correspond with a homogeneous
spherical charge distribution of Z protons within the nuclear radius R.
8
By switching off the scattering potential V one can easily verify that the solution (21)
coincides with the free electron wave function (8) since the differential equations (14) reduce
to the differential equations for the spherical Bessel functions. In this way a sensitive testing
case for our numerical approach is found.
We want to stress that the problem of Coulomb distortion of the initial and final electron
in the electron scattering process is solved to all orders. Earlier work in this field by Boffi
et al. [11] handled the electron distortion in an approximate way through a high-energy
expansion of the electron wave functions combined with an expansion in powers of Zα.
The DWEEPY code [11] used in the analysis of the NIKHEF data adopts this approximate
treatment of electron distortion. To lowest order inZα it was proved that electron distortion
effects could be approximated by an effective momentum approach (EMA). This means that
the plane wave in eq. (8) has to be replaced by
keff
ei~k.~r ei~keff.~r , (24)
−→ k
with
3Zα
~keff = (k + )~e . (25)
k
2R
Clearlythisapproachisveryeasytohandleandworthcomparingwiththecompletedistorted
wave approach so that its degree of accuracy can be estimated.
C. The nuclear part
In a previous paper [14], we have shown that at low values of the missing momentum,
meson-exchange currents (MEC) and long-range effects only slightly affect the calculated
(e,e′p) cross section. As we will restrict ourselves to QE (e,e′p) reactions at low missing
momenta only the one-body part of the nuclear four-current is retained. Hereby we adopt
the operator as dictated in the non-relativistic impulse approximation:
ρ (~r) = eGi (~r,ω)δ(~r ~r ) , (26)
nucl E − i
i=1...A
X
9
eGi (~r,ω)
J~ (~r) = E ~ δ(~r ~r )+δ(~r ~r )~
nucl i i i i
i=X1...A( i2Mi (cid:16)∇ − − ∇ (cid:17)
eGi (~r,ω)
+ M δ(~r ~r )~ ~σ .
i i
2Mi − ∇× )
This nuclear charge-current four-vector refers to A non-interacting point-like nucleons with
mass M . To correct for the finite extent of the nucleons, the Sachs electromagnetic form-
i
factors G and G are introduced.
E M
As for the electron wave functions, the final nuclear wave function is determined through
a phase shift analysis after an expansion in partial waves. The final nuclear state is taken to
be a linear combination of one particle-one hole excitations C;ωJM > out of the A-particle
|
groundstate i >withC h,p . Theholestatehischaracterizedbythequantumnumbers
n
| ≡ { }
n ,l ,j and energy ǫ . The continuum particle state is specified by the quantumnumbers
h h h h
p = (l,j) and the energy ǫ = E M . The isospin nature of the particle-hole state is
p N N
−
denoted by t . The particle-hole state in the coupled scheme is defined according to
q
C;ωJM > = < j m jm JM > ( 1)jh−mh ph−1(ω) > , (27)
h h
| − | − |
mXhm
with the uncoupled particle-hole state defined as follows
ph−1(ω) >= c+(ǫ )c i > , (28)
| p p h| n
and ω = ǫ ǫ . The operators c+ and c denote single-particle creation and annihilation op-
p h
−
erators. The radial wave functions for the bound hole states are solutions of the Schr¨odinger
equation with a Hartree-Fock potential generated with aneffective interaction of the Skyrme
type (SkE2) [15]. The continuum particle states are evaluated within an optical potential
model (OPM) [11]. The physical radial wave functions are regular in the origin and behave
asymptotically (r ) according to
→ ∞
φ (r) r→∞ 2µN sin(kpr−lπ/2−ηln2kpr+δlnj,ǫp(tot)) ǫ > 0 ,
 p −→ πkp r p
 q (29)
r→∞
φ (r) 0 ǫ < 0
h h
 −→
10