Pauli blocking and final-state interaction in electron-nucleus quasielastic scattering

0.19 MB·English

by  Lon-chang

Pauli blocking and final-state interaction in
electron-nucleus quasielastic scattering
9
0 Lon-chang (L.C.) Liu†
0
2 Theoretical Division, Group T-16, Mail Stop B243

n
a Los Alamos National Laboratory, Los Alamos, NM 87545 USA
J

2
2

Abstract
]
h
t The nucleon final-state interaction in inclusive electron-nucleus quasielastic scattering is
-
l
c
studied. Based on the unitarity equation satisfied by the scattering-wave operators, a door-
u
n
[ way model is developed to take into account the final-state interaction including the Pauli

1 blocking of nucleon knockout. The model uses only experimental form factors as the input
v
1
and can be readily applied to light- and medium-mass nuclei. Pauli blocking effects in these
6
5
latter nuclei are illustrated with the case of the Coulomb interaction. Significant effects are
3
.
1 noted for beam energies below ∼ 350 MeV and for low momentum transfers.
0
9
0
: † e-mail address: [email protected]
v
i
X
Keywords: Nuclear response function, Pauli blocking.
r
a
PACS: 25.30.Dh, 25.30.Fj
1
1 Introduction
The dominant contribution to electron-nucleus reactions at energies below the pion
production threshold comes from quasielastic electron-nucleus scattering[1], in which a tar-
get nucleon is knocked out to the continuum by the incoming electron. While exclusive
quasielastic experiments can provide detailed nuclear structure information of the struck
nucleon, inclusive experiments allow us to study various general properties of the reaction
dynamics [2]-[5]. As the final-state interaction (FSI) between the knocked-out nucleon and
the residual nucleus can affect the calculated spectra[5], it must be properly evaluated. For
exclusive experiments, optical potentials are often used to calculate the FSI [6]–[8]. Because
these nonhermitian potentials differ from the potential that binds the nucleon in the nucleus,
they generate nucleon scattering wavefunctions that are not orthogonal to the bound-state
wavefunction of the nucleon. This nonorthogonality leads to overestimated (spurious) contri-
bution to nucleon knockout cross sections as the momentum transfer ~q → 0. Many methods
were proposed to restore the orthogonality[9]-[13]. For inclusive experiments, nuclear final
states are not measured. Hence, in principle, a real-valued potential is to be used for FSI
calculations. If one solves simultaneously the bound-state and scattering problems with a
same real-valued potential, then the above-mentioned orthogonality difficulty will not occur.
However, very often, particularly in the case of nonrelativistic treatment of FSI in inclusive
experiments, one uses phenomenological energy-dependent potentials[14]-[16]. These poten-
tialsdiffer fromthepotentialthatbinds thenucleon. Inthis respect, thelack oforthogonality
exists in practice and it is of interest to improve the implementation of the required orthog-
onality in inclusive calculations. In this work, we develop a new approach to FSI in inclusive
quasielastic scattering, which does not need an explicit use of potentials while implements
the needed orthogonality at all FSI energies on a same footing.
2
Because the distortion of the electron waves in the initial and finalstates can be taken
into account by the DWBA method and is of no relevance for the discussion presented in
this work, we will, therefore, use plane waves for the electrons so as to show more clearly
the effects of blocking spurious knockouts in the new approach. The theory is developed in
Section 2 and its application is given in Section 3. Discussion and conclusions are presented
in Section 4.
2 Electron quasielastic scattering from a nucleus
The one-photon exchange, one-nucleon knockout amplitude, A, is illustrated in Fig.1
where the four-momenta of the on-shell particles (external lines of the diagram) are denoted
by p = (E ,~p ) with i = (0,1,2,C,A). The four-momentum of the photon is q = p −
i i i 0
p ≡ (ω,~q). With the Bjorken-Drell convention [17] for the metric, single-particle state
2
normalization, and reaction cross section, the quasielastic scattering differential cross section
equals to
d2σ (2π)4
= δ3(p~ +~p −~p −~p −~p )δ(E +E −E −E −E )
dΩ dE v 0 A 1 2 C 0 A 1 2 C
2 2 Z in spins
X
m M 1 |p~ |E dp~ dp~
e A |A|2 2 2 1 C ,(1)
E0EA ! 2(2JA +1) (2π)3(E2/me)(2π)3(E1/MN)(2π)3(EC/MC)
where v = E E / (p ·p )2 −p2p2 is the relative velocity in the initial channel, J is the
in 0 A 0 A 0 A A
q
spin of the target nucleus, and the summation is over the spin projections of the external
particles.
As in any Feynman diagram, the intermediate particles are off-mass-shell particles.
This is the case with the intermediate photon, the intermediate nucleon, j, and the corre-
sponding residual nucleus, denoted C(j). However, it is useful to put the intermediate heavy
nucleus, C(j), on its mass shell and to retain only the positive-energy spinors of the nucleon
j. This covariant approximation enables one to use the bound-state nuclear wavefunctions
given by traditional nuclear structure theories in which the negative-energy component of
thewavefunction is notconsidered.[18] Because thedifference amongvariousnuclear masses
M is ≪ M , it is also useful to define M as anaverage ofM and substitute the former
C(j) N C C(j)
for the latter. One thus has
3
p0 p2
q
A= p01 p1
pj (cid:10)((cid:0))y
pA pC
p0C
Fig. 1. Amplitude A for quasielastic scattering. The dashed, wavy, solid, and multiple solid lines
represent, respectively, the electrons, the photon, the nucleon, and the nuclei. Ω(−)† is the wave
operator for the nucleon final-state interaction. A summation over the target nucleon label j is
understood.
ee f(q2) dp~
A= p u(p~ ,s )γ u(p~ ,s ) j
q2 ! 2 2 ν 0 0 JXjµjJC(Xj)sC(j) sXjs′1Z (2π)3(Ej/MN)(EC′ /MC)(E1′/MN)
1 1 1 1
× hp~ s ; p~ J s |Ω(−)†|p~ ′ s′; p~ ′J s i hp~ ′ s′|Jν(0)|p~ s i
12 1 C C C j 12 1 C C(j) C(j) 12 1 j2 j
hp~ 1s ; p~ ′ J s |Γ|p J s i
× j2 j C(j) C(j) C(j) A A A . (2)
" p0j −Ej +iǫ #
In Eq.(2) the abbreviated notations E′ ≡ E (p~ ′),E ≡ E (p~ ) and E′ ≡ E (p~ ′) are
C C C j N j 1 N 1
used. The square of the four-momentum transfer is q2 = ω2 − |~q|2. The four-momentum
conservation at each interaction vertex gives ~p ′= ~p +~q, p′0 = p0 +ω , ~p = ~p −p~ ′, and
1 j 1 j j A C
p0 = E (p~ )−E′ (p~ ′). The J ,µ are the total angular momentum and its third component
j A A C C j j
of the j-th target proton , and = Z being the total number of the target protons. The
Jjµj
e and e denote, respectively, tPhe electron and proton charges, and the u(u) and U(U) the
p
corresponding spinors. The f(q2) is the γpp form factor and
1 1 1 1
hp~ ′ s′|Jν(0)|p~ s i = U(p~ ′,s′)Jν(q)U(p~ ,s ) = d~xhp~ ′ s′|eiq~·~xJν(~x)|p~ s i (3)
12 1 j2 j 1 1 j j 1 2 1 j2 j
Z
where J = (J0,J~) is the electromagnetic current operator. For single-nucleon processes one
canrepresent thetarget nucleus asanactive nucleon iandacorresponding spectator residual
nucleus C(i), i.e.,
|p~ J s i= F(J J ;J ) C(J µ , J s |J s )|p~ ;J µ i|(p~ −p~ );J s i ;
A A A i C(i) A i i C(i) C(i) A A i i i A i C(i) C(i)
JiXJC(i) sCX(i)µi
4
1 1
|J µ i= C( s , ℓ m |J µ ) | s i |Φ i . (4)
i i 2 i i i i i 2 i Jiℓimi
mXi,si
Here F(J J ;J ) ≡ [Jν−1(J )J J |} JνJ ] is the coefficient of fractional parentage,
i C(i) A i C(i) i A i A
with ν being the number of protons in the shell having the momentum J . The C’s are
i
the Clebsch-Gordan coefficients. Upon using the bound-state equation G ΓΦ = Φ (with
0 bd bd
G = (p0 − E + iǫ)−1 and Φ = |J µ i), one obtains the covariant single-particle nuclear
0 j j bd j j
wavefunction given by
hp~ 1s ; ~p ′ J s |Γ|p~ j s i
Φ (~λ )≡ j2 j C(j) C(j) C(j) A A A
{j} j p0 −E +iǫ
j j
1
~
=F(J J ;J )C(J µ , J s |J s ) C( s , ℓ m |J µ ) Φ (λ ) . (5)
i C(i) A j j C(j) C(j) A A 2 j j j j j Jjℓjmj j
Here, {j} stands for the ensemble of quantum numbers J ,µ ,J ,s ,s ,ℓ ,m . Further-
j j j j j
C(j) C(j)
more,~λ = η~p −~p ′ /A = ~p −~p /A with η = (A−1)/A is the relative momentum between
j j C(j) j A
nucleon j and the corresponding residual nucleus C(j).
In Eqs.(1)-(4), the states | i and h | are covariantly normalized, namely, h~k′,s′|~k,s i =
(E(~k)/M)1/2δ(~k′ −~k′)δ . On the other hand, in nonrelativistic nuclear theories the states,
s′s
which we denote | ii and hh |, have the normalization hh ~k′,s′|~k,s ii= δ(~k′ −~k)δ . Hence,
s′s
|~k i = |~k ii (E(~k)/M)1/2. It follows that Φ is related to its noncovariantly normalized
counterpart, φ, by
1/2
E E E
~ j C(j) A ~
Φ (λ ) = φ (λ ) , (6)
Jjℓjmj j M M M ! Jjℓjmj j
N C A
where φ (~λ ) = R (|~λ |)Yℓj(λˆ). Being dependent on the relative momentum ~λ , φ is
Jjℓjmj j jjℓj j mj j j
a spectral wave function. Its relation to the corresponding shell-model wave function is given
in Ref.[19]. Upon introducing Eqs.(2)-(5) into Eq.(1), one can write Eq.(1) in the following
compact form:
d2σ dσ m2 L Wµν
= M e µν , (7)
dΩ2dE2 dΩ2 ! E0E2cos2(θ2/2)!
where L = 1 [u(p~ ,s )γ u(p~ ,s )u(p~ ,s )γ u(p~ ,s )] and
µν 2 s0s2 0 0 µ 2 2 2 2 ν 0 0
P
(2π)3 |p~ |M
Wµν = δ3(p~ +p~ −p~ −p~ −p~ )δ(E +E −E −E −E ) 2 A|f(q2)|2
v 0 A 1 2 C 0 A 1 2 C E E
Z in 2 A
5
dp~ dp~ M M M 2
j i C C N
{jX},{i}Z (2π)6(Ej/MN)(Ei/MN) EC′(j)EC′′(i) EN(p~j +~q)EN(p~i +~q)!
1 1 1  1  1
hp~ s |Jµ(0)| (p~ +~q) s′′ih (p~ +~q) s′ |Jν(0)| ~p s i Φ∗ (~λ )Φ (~λ )
2 i2 i i 2 1 j 2 1 j2 j {i} i {j} j
sX′1s′1′
h(p~ +~q)1s′′; p~ ′′J s |Ω(−)I(1,C)Ω(−)†|(p~ +~q)1s′; ~p ′J s i . (8)
i 2 1 C C(i) C(i) i j j 2 1 C C(j) C(j)
The dσ /dΩ is the Mott differential cross section and is given by
M 2
dσ e2e2 E2
M = p 2 cos2(θ /2) . (9)
dΩ (2π)2(q2)2 2
2
In Eq.(8)
1 1 dp~ dp~
I(1,C) ≡ |p~ s ;~p J s ihp~ s ;~p J s | 1 C = 1 , (10)
12 1 C C C 12 1 C C C (2π)6(E /M )(E /M )
s1XJCsC Z 1 N C C
as a result of the completeness of free two-particle states. Consequently,
(−) (−)† (−) (−)†
Ω I(1,C)Ω = Ω Ω δ . (11)
i j i j ij
The appearance of δ is a consequence of one-step reaction process in which the residual
ij
nucleus acts as a spectator. Because the nucleon j and the residual nucleus can form bound
states, the unitary equation of the wave operators is[20]
Ω(−)Ω(−)†δ = ( 1−Γ )δ , (12)
i j ij j ij
with
nmax
(n)
Γ = |n ihn | ≡ Γ . (13)
j {j} {j} j
n=0 n
X X
(n)
Here, Γ denotes the projector to the bound state |n i, with n = 0 denoting the nuclear
j {j}
ground state and n 6= 0 the nucleon-emission-stable (NES) excited nuclear states. In the
(n)
single-step reaction model, |n i = |J i ⊗ |J i. Here, a nucleon j is lifted from its
{j} j C(j)
(n)
ground-state orbital (denoted J ) to an excited orbital ( denoted J ,n 6= 0).
j j
(n) (n) (n)† (n) (m)† (n)
The projectors Γ have the properties Γ = Γ and Γ Γ = Γ δ . These
j j j j j j nm
properties allow us to rewrite Eqs.(12) and (13) as
nmax
Ω(−)Ω(−)†δ = ( 1−Γ )δ = 1− |n ihn | 1 |n ihn | δ . (14)
i j ij j ij {j} {j} {j} {j} ij
!
n=0
X
6
This last equation defines the doorway model of the final-state nucleon-nucleus interaction.
Using Eqs.(11)-(14) for the last line of Eq.(8), one obtains, after some angular-
momentum recoupling algebra, that
dp~ dp~ |p~ |
Wµν = 1 C δ3(p~ +~p −~p −~p −~p )δ(E +E −E −E −E ) 2 |f(q2)|2
(2π)3v 0 A 1 2 C 0 A 1 2 C E
Z in 2
×( Ξµν − Ξµν) , (15)
I II
with
1 1 1 1 1
Ξµν = hh ~p s |Jµ(0)|p~ s iihh ~p s |Jν(0)| ~p s ii |φ (~λ )|2 , (16)
I 2 j2 j 12 1 12 1 j2 j {j} j
Xs1 JjµjXℓjmjsj
and
nmax 1
Ξµν = |φ(n)(~λ)|2
II 2 {j}
nX=0 sX′s′′JjµjXℓjmjsj
dp~ 1 1
× j φ∗ (~λ ) hh ~p s |Jµ(0)| (p~ +~q) s′′ ii φ(n)(~λ +η~q)
" (2π)3 {j} j j2 j j 2 {j} j #
Z
dp~ 1 1
× i φ(n)∗(~λ +η~q) hh (p~ +~q) s′ |Jν(0)| p~ s ii φ (~λ ) , (17)
" (2π)3 {j} i i 2 i2 j {j} i #
Z
where ~λ = η~p − A−1p~ is the relative momentum of the nucleon-residual nucleus system
1 C
~ ~
in the final state. The momentum conservation at the γpp vertex gives λ = λ + η~q. For
j
succinctness of notation, Eqs.(16) and (17) are expressed in terms of noncovariantly normal-
ized nuclear wave functions φ , and noncovariant states hh | and | ii. Consequently, various
{j}
normalization factors, of the form (E/M), are implicit.
Eq.(15) is illustrated in Fig.2. Its physics content is as follows. The Ξ leads to cross
I
sections obtained with using plane waves in the final state. The Ξ gives the cross sections
II
for the struck nucleon to remain bound. The subtraction of Ξ from Ξ corrects the spurious
II I
contribution arising from using plane waves. As we shall see, at ~q=0 the subtraction is total;
in other words, the spurious proton knockout is completely blocked. Using the well-known
7
=
(cid:0)
n (cid:0)n (cid:0)n
Fig. 2. The doorway model for Pauli-blocking corrections. As in Fig.1, the subscript j and a sum-
mation over it on both side of the graphic equation is implied.
Lorentz-invariant parametrization[21],[22] of the response tensor Wµν, one obtains
d2σ dσ (q2)2 q2
= M R (ω,|~q|)+ −tan2(θ /2) R (ω,|~q|) (18)
dΩ2dE2 dΩ2 " |~q|4 ! L 2|~q|2 2 ! T #
in the laboratory frame. Here, R and R are, respectively, the transverse and longitudinal
T L
response functions with R = (~e † ) Wij(~e ) (i,j = 1,2,3) and R = W00 =
T λ=±1 q~,λ i q~,λ j L
Ξ00 −Ξ00. P
I II
3 Effects of Pauli Blocking
To illustrate the blocking of spurious nucleon knockout in the doorway model, let us
consider the Coulomb scattering only. In this latter case, Jµ = (ρˆ,~0). Hence, R =0 and
T
d2σ dσ (q2)2
= M R . (19)
dΩ dE dΩ |~q|4 L
2 2 2
In the second quantization ρˆ(~x) = ψˆ†(~x)ψˆ(~x). Upon using the nonrelativistic two-component
proton field ψˆ(~x) = (2π)−3/2 d~k ei~k·~x a χ , one finds that the two matrix elements of
ξ ~k,ξ ξ
J0 in the first square bracketRs in ΞP00 equal to δ while the two matrix elements of J0 in
I s1sj
8
Ξ00 become, respectively, δ and δ . Consequently,
II s′′sj s′sj
dK~ |p~ |
R = δ3(~q +~p −K~) 2 |f(q2)|2 R(ω,|~q|) (20)
L v A E
Z in 2
with
~
dλ
R(ω,|~q|) = δ(ω +E −E −E )
(2π)3 A 1 C
Z
|φ (λ~ )|2 −|φ (~λ)|2 |F00(~q)|2 − ′ |φ(n)(~λ)|2 |F0n(~q)|2 . (21)
 {j} j {j} {j} {j} {j} 
Xj (cid:16) (cid:17) Xj nX6=0
 
In obtaining Eqs.(20) and (21) we used the relations dp~ dp~ = dK~d~λ (with K~ ≡ ~p + p~ )
1 C 1 C
and
dp~
i φ(n)∗(~λ +η~q) φ (~λ ) = d~r eiq~·~ri ψ(n)∗(~r ) ψ (~r ) = F0n(~q) . (22)
(2π)3 {j} i {j} i i {j} i {j} i {j}
Z Z
The ′ in Eq.(21) indicates that not every target proton is involved in a 0 → n transition.
HencPe, ′ 1 ≡ Z′ ≤ Z. The δ function in Eq.(21) constrains the energy loss ω and makes
j
ω depenPd on ~λ2 and the average proton separation energy B = M +M −M .
1 C A
In Eq.(21), F00(~q) ≡ Fg.s.→g.s.(~q) is the nuclear (ground-state) form factor of the
{j} {j}
j-th proton with the property F00(0) = 1. For n 6= 0, F0n(~q) ≡ Fg.s.→n(~q) are the transition
{j} {j} {j}
form factors, and F0n(0) = 0. Consequently, when ~q → 0, R → 0; i.e., the knockout of a
{j}
target proton is completely blocked at ~q = 0. We have noted that experimental form factors
are not parametrized with respect to an individual proton but rather with respect to the
whole nucleus as a function of |~q|. (Henceforth, |~q| is denoted as q for a succinct notation.)
It is, therefore, appropriate to introduce
1
F00(~q)= F00(q) ≡ F00(q) ,
{j} Z A
1
F0n(~q)= F0n(q) ≡ F0n(q) (n 6= 0) . (23)
{j} Z′ A
The q-dependence of PBC can be obtained by integrating over all energy loss in Eq.(21).
Using the completeness relation
~ ~
dλ dλ
|φ (~λ )|2 = j |φ (~λ )|2 = 1 , (24)
(2π)3 {j} j (2π)3 {j} j
Z Z
9
one obtains
nmax
dω R(ω,q) = Z 1−|F00(q)|2 −β |F0n(q)|2 ≡ ZL(q) . (25)
 
Z n6=0
X
 
The ratio β ≡ Z′/Z depends on nuclear excitation mechanisms. The function L(q) gives the
probability for a struck proton to leave the nucleus. Eq.(25) shows how the doorway and
Fermi gas models differ. In the Fermi gas model, the nucleon density distribution |ψ(p~ )|2, is
j
assumed to be θ(|p~ |−k ) where k is the Fermi momentum. Because of the Pauli principle,
j F F
this box-type momentum-space density distribution blocks ψ(p~ ) → ψ(p~ + ~q) transitions
j j
whenever |p~ +~q| ≤ k . For realistic density distributions, there is no such sharp momentum
j F
cutoff in transitions. Instead, the ψ(p~ ) to ψ(n)(p~ +~q) transition can occur at any given ~q
j j
with the probability |F0n(q)|2. Hence, |F00(q)|2 + β |F0n(q)|2 is the probability that
n6=0
the struck nucleon remains bound. With a minus sPign in front of this last quantity, the
second and third terms in Eq.(25) give the blocking correction to nucleon knockout in a
realistic nucleus. We name this correction the Pauli-blocking correction (PBC) because it is
a consequence of the Pauli exclusion principle.
A comment on Eq.(25) is in order. While form factors F00 have been determined
experimentally for a large number of nuclei, experimental information on transition form
factors F0n (n 6= 0) is much less systematic. However, in nuclei with mass number A ≤ 5
there is no NES excited states. Consequently, only the term |F00|2 is needed in Eq.(25). The
L(q) can, therefore, be calculated exactly for these light nuclei with the use of experimental
form factors.
In Fig.3, the functions L(q) = 1 − |F00(q)|2 for two light nuclei are shown. In both
cases L(q) = 0atq = 0andL(q) → 1whenq > 2.7fm−1.Graphically,thePBCisrepresented
by 1−L(q) which is the vertical distance between the curve and the horizontal line passing
through L(q)=1. Fig.3 shows the PBC is complete (i.e., 100%) at q=0 and how it decreases
with increasing q. . Since there is only one bound state in 3He and 4He (the ground states),
1 − |F00(q)|2 represents an exact calculation of L(q) for these nuclei. In nuclei with mass
number A ≥ 6,there areNES states anditsnumber increases with A.To illustratethe effects
of NES states in 1p-shell nuclei, we show in Fig.4 the function L(q) of 12C, assuming β = 1
in Eq.(25). The PBC effects due to |F00|2 and ( |F00|2 + |F0,2+|2 ) are given, respectively,
10