PRAMANA
— journal of
c Indian Academy of Sciences
physics
Vol. 57, Nos 2 & 3
Aug. & Sept. 2001
pp. 545–556
Success and limits of the relativistic mean field
description of nuclear properties
Y K GAMBHIR
Department of Physics, Indian Institute of Technology, Powai, Mumbai 400 076, India
Abstract. The talk presents the current status and the perspectives of the relativistic mean field
(RMF) description of various nuclear properties. Some remarkable successful applications of RMF
to several different class of nuclear properties are first sketched in a short list. Three selective applications of RMF to the:
Loosely bound nuclei
Anti-proton (p) annihilation
Spin observables (asymmetry parameter) in the parity-violating experiments
with different motivations, are discussed in detail. The talk ends with a partial list of possible future
directions.
Keywords. Relativistic mean field theory; binding energies; radii; nuclear skin-halo; anti-proton
annihilation; parity-violating asymmetry parameters.
PACS No. 21.60.-n
1. Introduction
The astonishing success of the Dirac phenomenology in accounting the spin observables
(asymmetry parameter and spin rotation function) in the intermediate energy polarized
proton nucleus scattering, led spurt in activity in the applications of the relativistic mean
field (RMF) theory to a variety of nuclear properties. Since then numerous successful
applications of RMF have been reported. It is neither advisable nor practical to discuss
these in this short presentation. Therefore, in this talk three applications of RMF, relevant
to the current activity and the present scenario in nuclear physics, have been selected for
detailed discussion. Before proceeding further, first a short list of successful applications
of RMF is presented below:
Intermediate energy p-A scattering :
Successful description of spin observables
(potential shape changes : Wine bottle bottom shape)
Ground state (g.s.) properties :
545
Y K Gambhir
Relativistic Hartree and relativistic Hartree Bogoliubov (RHB)
Spherical, deformed and super-deformed nuclei.
Results:
B.E. : < 0.5 %
Radii : Few percent
Shape co-existence
Large deformations in g.s.
Isotopic shifts
Cranked RMF:
Production of superheavy elements (SHE)
Super deformed bands
Identical bands
Prediction of shell gaps for different combinations of (N; Z ).
Time Dep. & Rel. RPA : Consistent
Temperature dep. RMF:
Vanishing of shell effects
Phase transitions
Equation of state (EQS)
It is appropriate at this point to present relevant essentials of RMF formulation. The RMF
still works at the level of nucleons and mesons. It starts with a Lagrangian describing the
Dirac spinor nucleons interacting only via the electromagnetic (em) and meson fields. The
mesons considered are the scalar sigma ( ), vector omega (! ) and iso-vector vector rho
(). The variation principle yields the equations of motion. In the mean field approximation, replacing the fields by their expectation values, one ends up with a set of coupled
RMF equations: The Dirac equation with potential terms involving meson and em fields
describing the nucleon dynamics and a set of Klein-Gordon type equations with sources
involving nucleonic currents and densities, for mesons and the photon.
The pairing correlations which are important for open shell nuclei can be taken into
account in a very phenomenological way using occupation numbers of the BCS type based
on experimental pairing gap deduced from odd-even mass differences (see e.g. [1]). This
simple procedure (constant gap approximation) works well in the valley of beta-stability,
where experimental masses are known. However, for the proper treatment of the pairing
correlations and to describe correctly the scattering of Cooper pairs into the continuum in
a self-consistent way one needs to extend the present relativistic mean-field theory to a
continuum relativistic Hartree–Bogoliubov theory. Such relativistic Hartree–Bogoliubov
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Success and limits of the relativistic mean field
(RHB) equations have been derived by Kucharek and Ring [2] starting from the RMF
Lagrangian. The RHB equations read as:
U
h + V k =
h Ek
U
V
k
;
(1)
where is the Lagrange multiplier (Fermi energy), E k is the quasiparticle (qp) energy and
the coefficients Uk and Vk are four-dimensional Dirac quasiparticle spinors normalized as
Z
(Uk+Uk0 + Vk+Vk0 ) d3 r =
Ækk0 :
(2)
The mean field single nucleon Dirac Hamiltonian h is
= p + g! !0 + g00 + e(1
h
3 )=2 A0
+ (M + g ):
(3)
Here, , ! 0 , 00 and A0 (Lorentz time-like components) are the meson and em fields
which are to be determined self-consistently from the Klein–Gordon equations:
+ m2 = g s
+ m2! !0 = g! B :
g2 2
g3 3 ;
(4)
(5)
The scalar density s and the baryon density B ,
s
=
X
k
Vk Vk ;
B
=
X
k
Vk+ Vk ;
(6)
with similar equations for the 00 and A0 fields having corresponding source (densities)
terms. The sum over k runs only over all the particle states in the no-sea approximation.
The pairing potential in eq. (1) is given by
X
pp :
ab = 12 Vabcd
cd
cd
(7)
=
It is obtained from the pairing tensor U V T and the one-meson exchange interacpp in the pp-channel. This V pp is not yet available in the RMF (see ref. [2]).
tion Vabcd
abcd
Guided by the success of non-relativistic Hartree–Fock–Bogoliubov (HFB) investigations
with phenomenological zero or finite range Gogny type interaction in the pp-channel and
pp in the RMF, one adopts this phein the absence of the field theoretic derivation of V abcd
nomenological approach in solving the RHB equations. One therefore uses, finite range
Gogny-D1S interaction [3]:
X
2
Vpp ;
Wi Mi Px Bi P Hi P (j~r1 ~r2 j=i )
(8)
i=1;2
(1 2) =
[ +
+
]e
;
( = 1 2)
where i ; Wi ; Mi ; Bi ; Hi i
; are parameters. The finite range interaction (D1S)
( [3]) automatically guarantees proper cutoff in momentum space.
The RHB eq. (1) for the spherically symmetric case now reduces to a set of four coupled
differential equations for the RHB Dirac spinors U r and V r . These are to be solved self
consistently either in coordinate space itself or by employing the oscillator basis expansion
method. The explicit calculations require the following input information:
()
()
Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
547
Y K Gambhir
Parameters appearing in the Lagrangian
Vpp
We use the most successful Lagrangian parameter set NL3:
Masses (MeV):
mN
= 938, m = 504:89, m! = 780, m = 760
and coupling constants:
g = 10:444, g! = 12:945, g = 4:383, and g2 =
6:6099 fm
1 , g3
= 13:783.
The RHB calculations yields the following results (output):
The single particle energies " i , nucleon spinors i , the fields ; ! 0 ; 00 ; A0 , the occupancies Vi Ui , total binding energy, the deformations, root mean square (rms) radii,
currents and moments (densities), .
The RHB with the Lagrangian parameter set NL3 along with finite range Gogny interaction D1S, which we use here, has been very successful (e.g. see [4]) in describing several
nuclear properties for example the ground state properties of spherical, deformed and superdeformed nuclei spread over the entire periodic table also of light nuclei near the drip
line. We shall discuss here only the application of RMF to loosely bound nuclei.
( )
2. Loosely bound nuclei
One of the major current activity both experimental and theoretical, in nuclear structure is
devoted to the study of loosely bound nuclei. We have selected the chain of Na-isotopes
as a representative example. It involves both proton and neutron rich nuclei. The study of
its iso-spin dependence imposes a stringent test on the iso-vector part of effective nuclear
interaction.
The chain of sodium isotopes is a rare and perhaps a unique example of isotopic chain
which has been investigated both experimentally and theoretically. The isotopic shift measurements have been reported [5] for this chain of isotopes. The knowledge of the charge
(proton) radius r c
: fm [6] rp2 rc2 : of the most stable sodium isotope, 23 Na,
then yields the charge (proton) radii of the remaining isotopes. Recently, the same chain of
isotopes have been used as secondary beams (projectiles) (950A MeV energy) incident on
12 C target and the measured cross sections analysed [7] in the Glauber model. The Glauber
model requires in addition to the effective nucleon–nucleon cross section , the neutron
and proton densities of the target as well as of the projectile. The target (stable nucleus)
densities are supposed to be known from the earlier work (e.g. electron scattering), is
taken to be 0.8 times [8,9] its free value at this energy. The projectile proton and neutron
densities are assumed to be of two parameter Fermi type:
= 2 94
( =
0 64)
( )
k (r) =
0k
(
r
1 + e Rk )=ak ;
(9)
k stands for neutron (n) or proton (p). k0 is fixed from the norm. We only know
= 3:137 fm, and ap = 0:564 fm for 23Na. We require four parameters R p , ap, Rn,
where
Rp
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Success and limits of the relativistic mean field
an , but we have only two: ( R~p and ) known quantities for each isotope.
Therefore, one
needs to impose further restrictions. The following two options were tried [7]:
A.
B.
Rn (A Na) = r0 N 1=3 and Rp (A Na) = Rp (23 Na) = 3.137 fm.
Thus the densities (’s) change only due to changes in surface diffuseness (except
N 1=3 dependence in R n ).
ap (A Na) = an (A Na) = 0.564 fm. In this case all changes in k are due to changes
in Rk .
The extracted [7] densities and radii are termed as experimental and are labelled by EXPT.
A and EXPT. B respectively. Theoretically also, the mean field calculations have become
available [4,10,11]. We now compare the results (radii, skin thickness, cross sections and
density distributions) of the RHB calculations [4,10] with the corresponding experimental
values in figure 1. Clearly, the overall agreement between the theory (RHB) and the experiment is very satisfactory. The deviations are noticed at the finer level. We conclude this
part by the following statement: That the RHB is successful and the parameter set NL3 has
the right content of the iso-vector part of the interaction.
p A annihilation
3. It has been suggested [12,13] that the anti-proton (p) annihilation in which the energy
transferred to the nucleus is almost negligible can profitably be used to explore the nuclear
periphery. Consider, for example, anti-protons slowed down in the matter to less than 1
KeV, then an anti-protonic atom may be formed by Auger electron emission. This then cascades toward the nuclear surface by emitting anti-protonic X-rays. This cascade terminates
when the anti-proton encounters a nucleon at the nuclear surface and annihilates. This annihilation site is far above the lowest Bohr orbit and corresponds to the orbit(s) having
large principal quantum number(s) n
and highest possible angular momentum.
Further, for such an anti-proton annihilation at distant orbits there is a large probability
(Pmiss ) that all pions created during the annihilation miss the target nucleus A Z; N , as a
result, a cold nucleus with mass one unit less (A
) is produced. Schematically:
( = 8 10)
(
1
%
% X (Z 1; N )
p + A(Z; N ) !
& Y (Z; N 1)
&
(
)
)
+ (pions):
(
1 )
( ))
If the target nucleus A Z; N is selected such that both residues X Z
; N and
Y Z; N
are radioactive, so that their relative yields N p; p and N p; n can be
measured accurately by the radio-chemical method. The peripheral factor f p :
(
1)
( ( )
Im(
ap ) Z
1
Pdh (p)
fexpt =
Im(a ) N 1 P (n)
(10)
n
dh
can then be extracted. Here a n (ap ) is the p n(
p p) scattering amplitude. The factor
p
N (p; n)
N (p; p)
Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
549
Y K Gambhir
Figure 1. (a–d): The calculated RHB rms proton (rp ) and neutron (rn ) radii, the skin
thickness (rn rp ) and interaction cross sections ( at 950A MeV projectile energy
on 12 C target) alongwith the corresponding experimental results (expt. A and expt. B
for rn and skin thickness) respectively. (e, f): RHB proton and neutron densities of
23;27;31
Na.
Im(ap )
Im(an)
()
accounts for the ratio of annihilation probabilities and is taken to be 0.63 [12]. P dh p and
Pdh n are the probability of the deep hole states excitation during the distant annihilation
()
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Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
Success and limits of the relativistic mean field
and are negligible as P miss
1
1
Pdh(p)
Pdh(n)
= 1. It is shown [13] that the last term
is close to unity.
p is the direct measure of the ratio of the neutron
This experimental peripheral factor f expt
and proton density at the annihilation site which is almost independent of Z and is taken
to be R
: fm, R being the half density radius of the target nucleus. Explicitly:
+25
p )
fexpt
Z n
N p
at
R + 0:25 fm:
At this site n > p and both n and p are very small. Therefore, the comparison of the
theoretical and experimental peripheral factor imposes a stringent test on the theory used
for the calculation of neutron and proton densities.
The anti-proton (p)-beam from low energy anti-proton ring (LEAR) facility at CERN
is used to form anti-protonic atom which cascades by emitting X-rays till the anti-proton
annihilation site ( R
: fm) which is almost independent of Z . The peripheral factor
p for several nuclei have been extracted [12]. It is found [14,15] that the mean field
fexpt
theories both non-relativistic (HFB with Skyrme type interaction [16]) and relativistic RHB
fail, in fact both underestimate (see figure 2) 3–4 times, the factor f p . This however, is
not surprising. The mean field theories are designed to give accurate overall (average)
description and therefore, may not yield reliable description at the periphery. One may
need to impose a constraint so that it yields the correct asymptotic behavior. It is observed
that the results improve [15] dramatically when the asymptotic behavior is incorporated
phenomenologically.
To conclude this part it is found that the mean field theories in their present form are
inadequate and therefore should be fine tuned to incorporate correctly the asymptotic behavior.
+25
4. Parity violating (PV)-experiments
It has been suggested [17–19] that the PV-experiments are expected to yield reliable neutron densities (n ) in the nucleus. For example the measured parity violating asymmetry
parameter (Ae ) in the longitudinally polarized e A scattering is shown to be very sensitive to the Fourier transform of n . The asymmetry parameter is defined as the difference
between cross sections of the scattering of the right- and left-handed longitudinally polarized electrons. This difference arises from the interference of one-photon and Z 0 boson
exchange. The experiments to measure this parameter A e are feasible at CEBAF which
provides stable beam with high lumininosity and also at Jefferson lab.
We now sketch the relevant steps (for details see [17]) for the derivation of A e . Consider
the elastic electron scattering from the spin zero nucleus. The relevant potential the electron
sees is the Coulomb (VCoul ) plus the weak. The latter originates from the exchange of the
Z 0 boson. The dominating weak part is the nuclear vector V N -electron axial vector A e
current–current interaction:
(
)
Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
( )
551
Y K Gambhir
Figure 2. The ratio of neutron and proton densities as a function of radial distance.
The experimental value of halo factor is indicated at the corresponding annihilation site,
which is R (half density radius) + 2.5 fm. The horizontal error bars indicate uncertainty
in the radial position of the annihilation site.
VN
552
O
A
G X
Ae = pF
Ci
2 i=1
= 5A (~r)
i i e 5 e
Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
Success and limits of the relativistic mean field
with
A (~r)
( )
= 2GpF2 5w (~r)
(11)
()
p n coupling constant and G F is the weak Fermi constant while
Here, Cp Cn is the e
w denotes the weak charge density:
w (~r) =
Z
d3r0 gp (j~r
~r 0 j)
n (r0 ) +
1 4 sin2 w p (r0 ) :
(12)
The proton electric form factor:
gp (j~r ~r 0 j) where
3
e
8
r ;
(13)
= 4:27 fm 1 and
sin2 w = 0:23
for the Weinberg angle.
In the limit of vanishing electron mass, the electron spinor
= 21 (1 5) e
e defines the helicity states
(15)
which satisfy the Dirac equation:
[ p + V (r)] = E (16)
V (~r) = VCoul (~r) A (~r) :
(17)
with
The parity violating (pv) helicity asymmetry,
d+
Ae = dd
+
d
+
d
d
d :
d
(18)
Here, refers to the electron scattering with the potential V .
5. Calculations
(
)
First the RHB calculations yield the proton and neutron densities p and n which in
turn are used to calculate the weak density w . This then determines the potentials V .
The solution of partial wave Dirac equation with this V (including Coulomb distortions)
. From this the asymmedetermines the phase shifts which in turn yield cross sections dd
try parameter A e is trivially obtained which can be compared with the calculated Fourier
transform of the neutron density n .
Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
553
Y K Gambhir
6. Results
The RHB calculations are carried out [20] using the NL3 Lagrangian parameter set along
with the D1S Gogny pairing interaction for series of Ni, Sn isotopes. The calculated asymmetry parameter A e and squares of normalized Fourier transforms F 2 q =N 2 of neutron
densities as a function of momentum transfer q , for elastic scattering from 58 66 Ni at
850 MeV [20] are shown in figure 3a. We observe:
()
( ()
)
? The magnitude of A e is small ( 10 5).
? Ae at high q contains the information about the neutron density (Fourier transform
of n ).
? At lower energies (500 MeV) the differences between the neighboring isotopes is
small while significant differences appear at 850 MeV and above at 20 Æ . These
?
?
differences can be related to the differences in the neutron density distributions.
The peaks in (figure 3a) the asymmetry parameter A e and in the squares of normalized Fourier transforms F 2 q =N 2 appear at the same position.
The figure of merit:
( ()
F
)
d
= A2e d
can profitably be used to ascertain the differences in n (neutron radii r n ) of chain
of isotopes.
The experiments to measure this small asymmetry parameter A e are shown [17] to be
feasible at CEBAF and Jefferson Lab. The results are expected in very near future.
7. Atomic PV-experiments
The current and future atomic parity violating (PV) experiments aim to measure the electric dipole amplitude between two electronic states having same parity. This transition is
allowed only via PV-interaction and is forbidden otherwise (see
Ne.g.e figure 3b the level diagram of 133 Cs). The dominant contribution arises due to V N
A . The small suppressed
nuclear spin dependent contribution
due
to
the
electron
vector
nuclear
axial vector current–
N N
current interaction V e
A has the same form as that arising due to the interaction of
the PV nuclear current ~jPV with the electron through virtual photon. Therefore, experimentally these two cannot be separated. The experiment includes the total, the sum of all
the three contributions. The first two can be estimated by using the atomic structure theory
along with the standard model. The PV nuclear part enters through the nuclear anapole
moment ~a which is defined as the second spatial moment of the PV nuclear current ~jPV :
(
~a = ( )
Z
)
r2 ~jPV (r) d~r
which is related to a through
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Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
(19)
Success and limits of the relativistic mean field
Figure 3. (a) Parity-violating asymmetry parameters Al (upper panel) and squares
of normalized Fourier transforms of neutron densities (lower panel), as a function of
the momentum transfer q , for elastic scattering from 58 66 Ni at 850 MeV. (b) Level
diagram of 133 Cs.
~a = a (
~
1)j+l+1=2 22(jj ++ 11) jjj j :
(20)
Using the atomic structure theory along with the standard model and the PV atomic experiment, a can be extracted [21,22]. Using suitable nuclear model (e.g. RMF) the a can
also be estimated. For example the extracted (experimental) value of a is 0.750.39 [23]
for 133 Cs I
= + which can be compared with 0.388 [24] the value obtained using
RMF.
We conclude this part by stating that the current and future PV atomic physics experiments may yield accurate information about the neutron densities which may impose a
stringent test on the nuclear structure theory.
( =72 )
Pramana – J. Phys., Vol. 57, Nos 2 & 3, Aug. & Sept. 2001
555
Y K Gambhir
8. Future directions
Fine tune RMF (RHB)
?
?
?
?
Number-N and angular momentum-J projection
Correct asymptotic behavior
Role of continuum
Field theoretic derivation of V pp
Interaction in low p environment
Treatment of Vpp in the continuum.
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