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Meson-exchange current effects in the magnetic electroexcitation of 48CA
LE P YSICS
Nuclear Physics A537 (1992) 585-605 North-Holland
ESON-EXCHANGE CURRENT EFFECTS IN T E MAGNETIC ELECTROEXCITATION OF "C J.E. AMARO and A.M. LALLENA Departamento de Fisica Moderna, Universidad de Granada, E-18071 Granada, Spain Received 27 February 1991 (Revised 7 June 1991) Abstract: The effects of the meson-exchange currents in the electroexcitation of magnetic states in 4 'Ca below 14 MeV ofexcitation energy and for momentum transfers up to 3.5 fin -, have been calculated. In particular, their dependence with the momentum transferred to the target nucleus as well as with the excitation energy and the multipolarity of the nuclear states involved is analyzed in detail. Results for both pure IpI h and RPA excitations are shown and compared. Two magnetic transition with multipolarity 2 - have been found to be appreciably i.ifluenced by the exchange currents of pionic range.
1 . Introduction As is well known, meson-exchange currents (MEQ produce non-negligible effects contributing to electromagnetic properties of nuclei such as e.g. magnetic moments and (e, e') magnetic form factors. In particular, these effects are crucial in processes involving light nuclei even at low-q transfer. In these nuclei one can bring theory and experiment into agreement only after the consideration of MEC. Typical examples of this situation are the deuteron electrodisintegration at threshold') and the elastic magnetic form factor of 3 He [ref. 2) ] . On the other hand, a major difficulty appears in medium and heavy nuclei . In these nuclei wave functions are described within some uncertainties and thus MEC effects cannot be unambiguously identified . This has restricted the investigation concerning nuclei with A>4 to a few cases in which the nuclear states involved are supposed to have simple enough wave functions . Among them we mention here the magnetic (e, e') form factors for transitions to the high-spin excited states in '1-7) closed-shell nuclei 3,4) and for elastic and inelastic scattering in both medium and heavY 3 ) nuclei with closed-shell :E1 nucleon. In all these works it has been found that the magnitude of the (e, e') cross section is increased by the MEC of pionic range. However, the effect cannot be clearly isolated by comparing with the available experimental data as in the case of light nuclei we cited above. This is so mainly because of two facts. First, one must include in the calculations additional nuclear structure effects, such as ground-state and tensor correlations, which strongly interfere with the MEC ones and which are difficult to manage in these nuclei . Second, though the enhancement is not negligible, the inclusion of MEC only 0375-9474/92/$03 .50 @ 1992 - Elsevier Science Publishers B.V. All rights reserved
386
J.E Amaro, A.M. Latlena / Meson-exchange currents
produces a smooth modification in the shape of the form factors calculated in the impulse approximation and this modification is within the experimental error bars in most cases. espite the fact that the obtained results do not show clear evidence of MEC in medium and heavy nuclei, the simplicity of the (e, e') experiment makes it the ideal tool to proceed with a systematic analysis of MEC effects from both theoretical and experimental points of view. In this paper we shall investigate these effects by making a statistical analysis of them on the full magnetic excitation spectrum of the nucleus. In particular, we will study how these effects depend on the relevant magnitudes of the process, namely the excitation energy and multipolarity of the excited states involved and the momentum transferred to the target nucleus. To do that we have evaluated the magnitude of the contribution- of MEC to the form factors corresponding to the electroexcitation of magnetic states in 48 Ca. The reason for choosing this nucleus is twofold. First, it is a doubly closed-shell nucleus, which permits the application of the RPA formalism when calculating the excited states . Second, tizz neutron excess implies, at least in principle, a large decoupling of proton and neutron degrees of freedom mainiy in the low-lying excitation energy region. As a consequence, one can hope for nuclear RPA wave functions with only a few 1p1h components which means a reduction in the nuclear problem uncertainties. In this situation we expect to get reliable results. In sect. 2 we give details of these calculations. Sect. 3 is devoted to the discussion of the results . In the last section we present our conclusions. 2.
escription of the calculations
As we mentioned above we are interested in the form factor for the electroexcitation of magnetic nuclear states . In plane-wave Born approximation, which is the framework in which we will perform the calculations, this form factor is given b Y 8) IF
r (q )12
71 Z2
(-
Q'2 +tan 210 2 2q
01
O'M
where q is the recoil factor, Z is the nuclear charge, a is the differential cross-section, o-m is the Mott cross-section, Q, = ((o, q) is the four momentum transferred from the electron to the nucleus and 0 is the scattering angle. As for any doubly closed-shell nuclei, the form factor for 48Ca can be written as ,/4-7r
(q)110'), FT(q) = (J'jjTjmag z
(2)
where 10') and JJ') are the ground and excited states, respectively, and Tmag im (q) is the magnetic multipole operator which is given by mag
Tjm
(q) =
1
d rji (qr)
- J(r),
J.E Amaro, A.M. Lallena / Meson-exchange currents
587
with jj (qr) a spherical Bessel function, YjmL a vector spherical harmonics and J(F) the nuclear current. In our calculation we have assumed that this current contains both one- (J(O') and two-body (J(" )) terms i(r) = J(O)(r) +P .) (r)
(4)
Likewise the form factor can be separated into two terms as follows FT(q) = F(O)(q) + Pe')(q) T T The first one is the form factor in the so-called impulse approximation and it is associated to the individual nucleons. The second one is produced by the presence of meson-exchange pieces in the nuclear current. In this work we have only considered the pionic (J( ")) and the 7r-seagull (X'sr') ) currents, which are expected to give the most important contributions to the form factor in the energy and q-transfer regions of interest to us. Thus, additional terms such as A-isobar and p-meson-exchange currents have not been taken because they are expected to give appreciable contributions for higher q-values and energies . The energi--.s and wave functions ofthe magnetic states o f 48 Ca have been obtained by performing a RPA calculation in the framework of the Landau-Migdal theory of finite Fermi systems. In this work the nuclear mean field is described by means of a (phenomenological) spherical Woods-Saxon potential I df (r, RLS, V(r)=Vo f(rRO,aO)---(1-s)M2C2 2 r dr A
1
with
h2
I
aLS)
+ Vc (r) ,
where m is the nucleon mass and Vc(r) is the Coulomb potential generated by a uniform charge distribution of radius Rc . The parameters of the potential that we have used are shown in table 1 . These parameters have been adjusted to fit the experimentally known energies near the Fermi surface (see column 2 in table 2) and the r.m.s. charge radius of "Ca, rc = 3 .48 fin [ref. 9)] . The Woods-Saxon well has been diagonalized in a truncated harmonic-oscillator basis, thus producing an automatic discretization of the continuum. In table 2 we also show (column 3) the results obtained for the orbits close to the Fermi level after diagonalizing the TABLE I
Parameters of the Woods-Saxon potential R a, Ro 1fi) (fffl) (MeV) protons neutrons
-59.50 -50.00
4.36 4.36
(fffl)
0.53 0.53
À
(fin)
RLs
al_s (fM)
c (fin)
26.0 28.0
4 .36 4.36
0.53 0.53
4.5428
J. E
388
inaro, A.M. Lallena / lt-leson-e.-
Single-particle energies for the orbits near the Fermi level. The results obtained by diagonalizing the Woods-Saxon potential are compared with the experimental values considered to adjust the corresponding parameters NeLtrons
Protons 111j
E (MeV)
&, (MeV)
181j
E (MeV)
E,,,, (MeV)
Id~,,:? 2s, /2 1 f7/2 2p~,,:! I f,;,:! 2p, /2
-15,557 -15.344 -9.406 -3,718 -1 .361 -1 .130
-1&167 -15,807 -9.628 %5" -5.821 -6.113
2s, /:! Id3/2 1f7/2 2P3/:! 2p,/2 1~53/2 199/2
-15 .698 -15 .619 -9.550 -4.919 -2.733 -2.216 1 .276
-12.545 -12.525 -9.945 -5.141 -3.119 -1.555 -1.131
potential. In the actual calculations the experimental values have been kept. For the remaining orbits not shown in table 2 we have taken the single-particle energies given by the diagonalization of the potential . The obtained r.m.s. charge radius is r, = 3.33 fin, which adequately describes the experimental value. he RPA calculation has been performed using a configuration space composed by 70 single-particle states, which corresponds to proton and neutron orbits lying at energies below -25 MeV. The residual interaction we have considered is the one developed by the Kilichfor Stony Brook group 'o) the study of magnetic properties ofnuclei. This interaction consists of a long-range part based on the 7r- and p-exchange potentials and a short-range piece which is simulated by a phenomenological zero-range Landauigdal interaction. In momentum space the form of the interaction is ' o")
K,Q) = Co(go
where
-
a 2 + gocr I
.
a 2 T I . T2)+ V,,(q) + J 27r
n(
2
m 7r
a 1 - qcr- 2
q
2
q
+m,,
)
T
1
VP (q) ,
7
2
and ) = 4ri
ere
C = h27T2/ mk 0
2 f~ 1 M2 0 (3
Xq- &'X 2 2 q +M P
F is the inverse of the density of states
lp 1 .
lp
2
at the Fermi surface and
f,, andf, are, respectively, the pion-nucleon and the rho-nucleon coupling constants . Finally, the dimensionless parameters go and go have been fixed to reproduce the energies of the two first excited states . The particular values we have used are given in table 3.
The RPA calculation produces 36 excited states with multipolarities ranging from J' = 1 + to J' = 8 - with the exception of the 7 + which is not present in the energy
a e') MEC the magnetic each see Y being maximum will this experiment and each the we allows information calculated q being spectrum states us the of q-value present Then are index point "'(q) collaborators to (QRI) the even energies, Iinclusion is of F(')(q)l form us interested the trust the these influence are we = to (a) we is or singularities RPA value of for of iHowever, in factor avoid have reasonably in shifted odd, concerning stands shall and excited spins ref the the ph the of magnetic amplitudes, Amaro, where in Aj(ph)e(lp+lh+J+I) computed of respectively ") 13) MEC shifting the form design reliability impulse In ofthe (that and for to MEC (first (MeV (second inherent states principle the A(0) at factor qrnax F(0) (solid well residual parities the of is, states typical some aits in -column) or fin') the up approximation +Aq procedure the Lallena reaches and excitation the of zeros described column) (ex), problems due line) to Besides it interaction Landau-Migdal q-values form e-factor our of form of is +Aq)12 G(a) effect )12 This /by to aevident the 14 48Ca and produces Meson-exchange aresults _= 3MEC factor factor some maximum 0to MeV) (x) IF(o)(q -,/(2a the of different As of magnitude the )12 by used only e (third which Thus measure are the MEC we amplitude that by the (p, (dashed isThus Aq 12 in)12 + In zero-range in anucleonic corresponding we can given this 1) p') will smooth one 0g(" fig calculated As the column) levels dr is and define After currents work asee experiment the as a r2jj obviously eIpositive could peaks b~ line) 12consequence, we 6(a) Aj shown will much identified (qr) the influence 4,14) variation the densities show =evaluate together of = by be Xj* inclusion spectrum, quadratic value Cof the in 1p1h hide or modifying used the + performed the (x) fig (form Iin of containing calculated of according the state the 1YYj*, observed ,with 2both the by MEC of erelative factor, which As effects factor means factor MEC us form the the by we in X to
J.E
.M.
589 TABLE
Parameters piece C,,
90 386 .04
.7
.7
.
region energy excitations
.
(e, Fujita excited
. .
permits For i
FeT
e jpjh
Z
where to
(6)
.
. e(q)
T
IFT(q IF(o)(q T
.
.
for can factor maxima
.
T .
.
which increment e(qmax)
this
G(a) phj
JO
phJ
the
where
ac
.
and all At
e(q) of
.
.a,,
determine
IFT(qmax
F(O'(qmax) T IF(o)(qmax) T
T
. .
.
.
IE
S40
aro,
Lm llena /
currents
5+
12
l +--------
_ 28
(0
l + --------'
2--------0 _______-
-
26 4----------
0--------------0
(6 ,7
6-
(3+ + (6 ,6 ) `
l+ 2
(` 2 _ ` 6 _ ) (4 /5+,2
-
2
-
-
2
Me)
-
BL P. /k .
Fig. 1 . Excitation spectrum obtained in the RPA calculation considered in this work.
-~..
~. . -..
~I
J.E Arnaro, A.M. Lallena / Meson-exchange currents
qeff
-1
M
591
1
Fig. 2. Typical example of the effect of the MEC in the magnetic (e, e') form factor. The solid (dashed) curve corresponds to the calculation performed with (without) the consideration of MEC.
an increase of the form factor due to MEC, whereas if e < 0 the form factor will be reduced. We will be also interested in the relative effect of the MEC, irrespective of its sign. In that case the absolute value lei will be considered. Due to its statistical nature, it is expected that the conclusions obtained by means of this procedure will be applicable to other nuclei . 3.
esults
The first important point observed in our results concerns with the mixing of 1p1h configurations appearing in the RPA wave functions. In this respect it has been found that some RPA states with multipolarities J' = 1', 2- and 4 - are the most affected by the consideration of these non-dominant configurations. On the other hand, those with J' = 3 +, 5 + , 6 - and 8 - show wave functions which can be supposed to be built with only a small number of 1p1h components. As an example we give in table 4 the Aj amplitudes corresponding to the 1p1h configurations of the 5 + and 2 - states obtained at 5 .35 and 6.00 MeV, respectively, in the calculation. Only those components with amplitudes Aj :- 0. 1 are shown. As we can see, the wave function of the 5 + state is dominated by the P( 2P3/2 If7/2 _' ) configuration while that of the 2- state shows a considerable mixing of 1p1h states. This fact is more apparent in fig. 3 where the form factors corresponding to both states are 9
I E~
392
maro, A.M. Lallena /
eson-exchange currents
TAME 4 Amplitudes Aj for the lplh components of be RPA mve functions of the 5' and 2- states obtained at 135 and 6.00 MeV, respectively . Only those values larger than 0.1 have been tabulated Iplh con6gumuon rdpvy Idi /12 ) rYp 3/ 2 , 2s ï112 ) -zr(2p3~ :!, Id,~' ) Id
5' ; 5.35 MeV
2 ; 6.00 MeV 0.171 0,159 0.188 -0.114 -1501
IpIh configuration t,(2p,/2,
5+, 5.35 MeV
ld3/'2)
P(2P3/ :!, 2sl/'2) v(2P3/2, Id .;/. ) v( 2 P3/2, IfÙ12)
1.061
2 ; 6.00 MeV 1211 1182 1552
shown. In the case of the 5' (see fig . 3a) the inclusion of the small components (solid curve) only produce a smooth modification of the form factor obtained by considering only the dominant component (dashed curve). On the other hand, the situation for the 2 - state is quite different, as we can see in fig. 3b. Therein the three calculations correspond to the form factor obtained by considering the full RPA wave function (solid curve), a wave function including all the 1p1h components given in table 4 (dashed line) and, finally, a wave function with those components of amplitude Aj :-::- 0.5 only (dotted curve). As we can see the corresponding form factor appears to be strongly affected by the inclusion of the small components of the wave function . [Further details concerning this particular aspect can be found in ref. 14).] is fact has led us to calculate, in the same way as described above, the QRI for all the 1p1h configurations, we find up to 14 MeV. These configurations are given in table 5. In these cases the form factor can be evaluated by summing in eq. (6) only over one ph configuration and putting Aj (ph) = 1 . It is expected that the com?arison of the results obtained in both the 1plh and RPA calculations will inform us about the differences one can observe when the wave functions are very simple and it wiH be helpful to a better understanding of the RPA results. Moreover, the calculation corresponding to the single lplh configurations will allow us to distinguish between protonic and neutronic excitation and to obtain additional information that cannot be extracted with the RPA wave functions. ith this in mind we have evaluated the QRI factor e(q) for all the 1p1h configurations and RPA states below 14 MeV and for the peaks of the corresponding (e, e') form factors found at values of the momentum transfer up to 3.5 fm - ' . In what follows we analyze the most relevant results obtained in this calculation. 3.1 . AVERAGE
EHAVIOR OF QRl
e start by discussing the main characteristics observed in the average behavior of the e factor. In table 6 we show the mean values and standard deviations of QRI evaluated for both I p I h and RPA wave functions (see columns 3 and 6, respectively)-
J.E Amaro, A.M. Lallena / Meson-exchange currents
593
CT LL
10
-3
I
I
48Ca(2 - ;6.OOMeV)
: - /,
I
C~L_
1 10 0.0 -7
3
, ,' % .
1
1 .0
: _F
1 2.0
-
3.0
q eff [W1 1
Fig. 3. (e, e') form factor for the (a) 5' state at 5.35 MeV and (b) 2- state at 6.00 MeV. Solid curves correspond to the calculation performed with the full RPA wave functions . Dashed (dotted) curves have been obtained by including all the lp1h components with Aj -- 0.1 (Aj -_ 0.5) in the wave function . MEC effects are not considered here.
M Amaro, A.M. Lallena / Me.çan-e.xchange currenis
594
TABLE 5
lp1h exdta6ons Ar which be QRI factor has been calculated in this work lp1h configuration
Energy (MeV)
Multipolarities
IpIh configuration
Energy (MeV)
=Of,~ Z , 2sj,'2 )
1 d~~A!)
63 6.54 9,26 9.62 9,919 10.05 10.35
4 2- ,422222- ,4-
v(2p jj2 , lf~1'2 ) 02p, 12 , If 02) 02N/2v Idp':!) v(2p,-Aj2 ,2s 1/'2 ) '.. (I fS/2' I f 412) "( 1 99/2' 1 fK12) P(2p,/ :!, WOO
4.80 6.83 7.38 7.40 8.39 8.81 9.41
.r(2p :,~ 2 , 1 d 02)
13.87
2-,4-
2s,Q) v(Igq,, 2 , Id j/1:! )
r. (2p,,
:! , 2s jjz ) Tr(2p,j/z , WOO -" ( I ~,,, 2 , 2s j,~2 ) -(2p, :! , ld~~?:!)
If
-0 f, :! ,
` ) Id Q
10.78
2 - ,4 - ,6 -
01 fs/2 , Idj! )
10.97
v( I
11 .39 11 .41
Multipolarities 3', 5' 3' 221', 3', 5' 2-,4-,6-,822-,423', 5' 5+
10.99
2s 1 /2)
12.92
i,(3s,/ :!, If7__/12)
4-
The first row corresponds to the results obtained by including all the maxima of 10-4 and interest to us. In the second and third rows those peaks with jF(O)(q)j< T P" ) ( q) I < 10- respectively, have not been included . Columns 2 and 5 give the T
number of peaks considered in each case. As a first result we point out that the values of j :E cr found for 1p1h and RPA states are close. Another interesting aspect is that both the QRl mean value and the standard deviation reduce with respect to the other calculations is only those peaks with IF"(q)j :_~- 10' are considered (see T rc ;~; 3). This fiau. iiaeans that on average the bigger the one-body form factor is, the smaller the relative increment due to MEC becomes. Finally, we can state that the mean effect of MEC in the (e, e') form factor is to produce an enhancement of the one-body part by -20%, though a reduction is found in some cases. The similarity observed between lplh and RPA results does not appear in fig. 4. Therein we show the bar diagrams corresponding to the distribution of the peaks with respect to the QR1 value, for both lp1h and RPA wave functions. Here only those maxi'ma with I F(O :- 10-4 have been included . Thus these results correspond T '(q) 1 :to the data in row 2 of table 6. The reason for imposing such a restriction lies in TABLE 6
Mean influence of MEC in 1p1h and RPA excitations lplh
q ~~ -- 3.5 fm -- ' 10-4 1 F'O'l T 10-3 I F'O'l T
RM
W of man
e±a
IOU
# of man
e±a
id ± 0'
87 76 57
0.28 :t 0.70 0.21 :E 0.34 0.15 :1--0.19
0.41 ±: 0.63 0.29 --h 0.27 0.20--EO.14
124 110 70
0.26 :E 0.85 0.211:0.38 0.18 :EO.21
0.39±0.80 0.30±0.32 0.22-±0.17
J.E Amaro, A.M. Lallena / Meson-exchange currents 25,
595
1
20 15 10
r-4-1 20 15 10
0
1
H I I I I I I I I
-100 -80 -60 40 -20
0
e F%1
20
40
[_T_~ [E] 60 80 100
Fig. 4. Distribution of the values of QRl for the peaks of the form factor satisfying the conditions I F(rO) I >. 10-4 for (a) lp1h and (b) RPA excited states . q,,.., -_ 3.5 fin` and 48 the fact that smaller maxima are out of the range experimentally measured in Ca. Besides the consideration of these small peaks makes the standard deviation to increase considerably in both lplh and RPA cases (see row I in table 6) and the results would be distorted because of this fact . As we can see in fig. 4a, in the lp1h case there is a certain symmetry around the interval [ 10%, 20%] which shows a frequency which is notably greater than that of the remaining intervals. The situation for the RPA results (see fig. 4b) is rather different . Now the distribution is peaked in the interval [20%,30%], that is a 10% shift with respect to the lp1h result, but the symmetry around the maximum has disappeared . However, both effects wake the J value to be very similar to that of lplh . Additional information about the MEC influence in the (e, e') form factor can be obtained from the absolute value of QRI, jej . This magnitude will give us the strength of the MEC effect independently of the fact that it can be either an increase or a reduction of the one-body form factor. In columns 4 and 7 of table 6 we show the corresponding mean values and the standard deviations . As we can see the results for lp1h and RPA excited states are again very similar and the same conclusions as for the e factor can be drawn. Now we find that the average influence of MEC is around 30% .
IE Ainam A.R LaWna / Meson-exchange currents
s we mentioned above, the calculation performed for the 1p1h states permits the analysis of the results for protons and neutrons separately . In order to proceed with this study we have evaluated the same magnitudes as in table 6 for such Ip1h excitations . The obtained results are shown in table 7. There we can see that the ean value j is bigger for protons than for neutrons, which is more apparent when re peaks are included in the statistics . Moreover, the data dispersion is larger for neutronic than for protonic states. This asymmetry between both kinds of excitation almost disappears when we turn to the absolute value of the corresponding e factor. In this case the mean value Ar proton and neutrons practically coincides, tbough the standard deviation still shows some differences. he main conclusion we can draw after the analysis of the values of the QRI that we have done is the Mmila6ty between the obtained results for 1p1h and RPA excitations. This means that the KAng of 1p1h configurations present in the RPA wave functions induces almost negligible effects in the contribution of the MEC to the (e, e) form Actor, at least in what refers to the mean values of the e factor. Taking this fact into account we go deeper in the behavior of the QRI and discuss its dependence with the relevant magnitudes of the process. 3.2,
EPENDENCE OF QRI WITH q, E AND J -
ow we are interested in analyzing how the effect of MEC depends on the three fdndamental magnitudes involved in the electroexcitation of the nucleus, that is the energy and spin of the excited state and the momentum transferred from the electron to the nucleus. That is, we want to study the behavior of the e factor with each of these variables. To do that we have determined the dependence of the average value of jej by considering all the yaks appearing at each value of the variable of interest in each case. The results have been plotted in figs. 4-6. Therein the corresponding behavior of the absolue value of QRI with the momentum transfer (fig. 5) and the spin (fig. 6) and energy (fig. 7) of the excited states are shown for both (a) 1p1h and (b) RPA states. The error bars give the corresponding standard deviations . e first aspect that must be pointed out is the similarity found between the results obtained for the simple 1p1h and the RPA excitations. This fact shows again that the Nxing of 1plh configurations involved in the RPA wave functions does TABLE 7
Mean influence of MEC in protonic and neutronic 1p1h excitations Protons # of max . q,,,, -- 3.5
fin' 1 POI , 10-4 T 10-3 10"I a T
34 31 24
132±00 0.28 ±0 .31 T16±0A1
Neutrons
1,jI ± 0,
# of max.
O±o,
1,jI ± 0,
OAO ±156 0.30±0 .29 118±108
B
125±036 0.17±0.35 114±123
OA2 ± OA8 0.28 ±0.27 121±117
45 33
J.E. Amaro, A.M. Lattena / Meson-exchange currents
597
200 jej 150
100
50
0
1
0
0.
40
1 . ill I 116
0
0
.6f
i
1W
100
4)
50~ 00
0
0.
1.
2. q [fm -' l
3.
4.
Fig . 5. Behavior of jej with the momentum transer q for (a) 1plh and (b) RPA excited states. The peaks included satisfy the same conditions as in fig . 4.
not greatly interfere with MEC effects . Thus these effects appear to be rather independent of the particular combination of 1p1h configurations in the RPA wave functions . In what refers to the dependence with q, it can be seen in fig. 5 that the lel values are clearly below -50% up to q - 2.0-2.5 fm- '. Above this value lei grows and reaches, in both 1p1h and RPA states, values near 100% at q - 3.5 fm- ' . We can state that on average it is necessary to go to high momentum transfer in order to find a big effect due to MEC. In any case there are some particular cases in which such effects are considerably large even at q < 2 fin' . We will analyze such cases in the next subsection .
598
AE Ainam AM LaWna / Meson-exchange currents 1004-
1
- I
I
I
I
I
I
I
(a)
75
5 40
I I
40
I
I
I
(b)
1
50
25
-6
0
I
1
I
2
I
3
I
4
I
i
5
I
6
I
7
I
8
I
9
FQ. 6. Same as in hg. 5 but W the dpendenne of jel with the spin of the exciad states .
The behavior of the absolute value of QRI with the nuclear spin J shows some interesting characteristics. In the case of Iplh states (see fig. 6a) the corresponding mean value appears to be constant (with values slightly above 25%) and the dispersion is similar in all the cases with the exception of the peaks with J = 5 and 8 . The shua6on is different for the RPA states . As we can see in fig. 6b the mean values show small variations between the different J. Thus for J = 2 and 6, only the Jej factor is over the 25%, while the average MEC influence for J = 1, 3,4 is clearly reduced with respect to that ofthe I pI h states . On the other hand, standard deviations suggest that the absolute value of the QRI must be considered as independent of J in practice .
J.E. Amaro, A.M. Lallena 200, lel [%]
1
Meson-exchange currents 1
9
10 .
12 .
599
(a) 150~
100
50
0.
lei [%]
(b) 150~
100
50 .-
0 .1 4.
6.
14 .
E [MeV]
Fig . 7 . Same as in fig . 6 but for the dependence of jej with the energy of the excited states.
Finally, no clear conclusions can be drawn concerning the dependence of lei with the excitation energy. This is so because of the relative small number of data we have. The RPA results (see fig. 7b) show a certain oscillatory behavior with two bumps at 6-7 MeV and 11-12 MeV, respectively . The same cannot be stated for the 1p1h states because of the lack of data in the region around 12 MeV (see fig. 7a). 3 .3 . OUTSTANDING STATES
In order to finish the discussion of our results concerning the MEC effects we will now analyze the excited state having peaks in which such effects are considerably
600
AE Amaro, AR Lallena / Meson-exchange currents TABLE 8
Outstanding lplh states in "Ca. Those levels with peaks in which lel -_ 341/o are shown lpIh configuration 14 If.5/2, lf7/12) v0pj ,2 , Id .V 3/2 )
E, (MeV)
(fm-1)
IF") T 'l (X IO-2)
e M
1 .3 2.2
0.24 0.20
36.b 54.9
4-
0.6 1 .5 1 .9
0.14 0.32 0.43
-51.4 61 .0 56 .5
10.990
2-
1.2
0.28
47 .0
13.656
3'
2.1
0.11
45 .4
8.390 9.406 10 .970
v0f,, 2, 2s t/12)
K Ig z , 2s, /1:! )
j 7r 1+ 22-
q
remarkable, thus allowing for an easier identification when comparing with the experiment. In what follows we restrict our attention to those cases in which the corresponding form-factor peaks satisfy IF'T0'1::- 10". As we can see in table 6 the mean variation intervals of lei are [6%,34%], in the case of the lpIh states, and [5%,39%] in that olf the RPA ones. Those states with form-factor maxima in which the absolute value of QRI is above these intervals are shown in tables 8 and 9 for 1p1h and RPA states, respectively . Some interesting aspects deserve a comment. First, it is important to note that the multipolarities appearing in both tables are the lowest. This fact is rather striking. In effect, as we said above the attempts to identify MEC tracks in medium and heavy nuclei have been restricted till now to nuclear states with simple enough wave functions . In this sense lower multipolarities were i-iot taken into account because they generally involve a large mixing of 1p1h configurations. However, our results indicate that, first, lower multipolarities present the larger effects due to MEC and second, the 1p1h and RPA states show a similar TABLE 9
Outstanding RPA states in 4"Ca. Those levels with peaks in which lei -_ 38% are shown E, (MeV) 7.27 8.99 9.88 10.16 10.92 11 .40 11 .81 12 .33
IF")i T
e
0.13 0.16 0.16 0.20 0.69 0.49 0.34 0.16
82 .4 47 .6 83 .0 38 .0 41 .8 38 .4 79.5 70.7
( X 10-2) 2221+ 4224-
1.7 2.2 2.1 1 .2 2.0 1.4 1.0 1.3
J.E. Amaro, A. M. Lailena / Méson-exchange currenis
60 1
behavior. These two facts inform us again about the large decoupling between MEC effects and the configuration mixing appearing in the nuclear wave function. As a conclusion, we can state that magnetic transitions to low-J levels seem to be the best places to look for the effects of MEC. A second important aspect is that all the e-values given in these two tables are ` )2- ph configuration at 10.97 MeV shown positive, with exception of the IV( I f5/2 9 1 d 3/2 in table 8. This last case is specially interesting because the MEC contribution produces a reduction of the one-body form factor in the first peak and an enhancement in the second one, the effect being bigger than a 50% in absolute value in both cases . Apart from this particular point, it can be seen by comparing the results included in both tables that the QRI reach higher values for the RPA states than for the I pIh configurations. This is the only effect that can be ascribed to the mixing of 1p1h configurations, which seems to produe a coherent sum of the MEC effects coiresponding - to the different 1p1h components involved in the nuclear wave function . The results obtained for the pure 1p1h states show interesting characteristics that must be discussed. We note that all the configurations shown in table 8 are of neutronic type. This points out again the above mentioned asymmetry between protons and neutrons concerning the MEC contributions . In order to go deeper into this situation we have analyzed the form factors at the maxima for those 1p1h configurations which simultaneously appear for both protons and neutrons . The obtained results are shown in table 10. We have found that, in general, MEC contributions are of the same order in both kind of configurations. In several cases, TABLE 10
Comparison of the maxima of the form factors corresponding to the I pI h configurations simultaneously appearing for protons and neutrons in our calculations lp1h configuration
Fr
(2p,/2, ld3/2)
2-
(If,/2, ld3/'2)
2-
q (fm -1)
I F'OT l ( X 10-2)
I Fr" I I ( X 10-3)
q (fm -1)
IFT'O II ( X 10-2)
1 .1 2.2
0.285 0.380
0.073 0.536
1 .1 2.2
1 .4 2.7 1 .9
1 .060 0.256 1 .120
0.193 0.201 0.142 0.324 0.032 0.429
0.070 0.162
0.161 0.220 0.466 0.469
I f5/ 2, 2s ,/'2)
2-
1.2 2.3
0.780 0.070 0.776 0 .428 0.193
(2P3/2, ld3/'2)
2-
QP3/2,2s~/' 112 )
2-
0.6 1 .7 0.7 2.1
0.605 0.543
4-
Neutrons
Protons
0.605 0.184
0.6 1.5 2.7 1 .9 1 .2 2.3 0.6 1 .6 0.7 2.0
0.282 0.119 0.075 0.164 0.388 0.341
IF""I T
( X 10-3) 0.119 0.502 0.428 1 .090 0.216 1 .080 0.600 0.149 0.132 0.180 0.180 0.443
602
IL Amam A.AL Ladna / Meson .e.-
some of them those of table 8, the neutronic one-body form factor is roughly half that ofthe corresponding protonic state, mainly due to the presence ofthe convection term in the last one. In these circumstances it is obvious that the e factor is appreciably larger for neutrons . In the remaining cases the one-body form factor is of the same order in both kind of configurations, because in these states the convection current gives a contribution opposite to that of the spin-magnetization term. III these cases the QRI is similar for the protonic and the neutronic Ip1h excitation . Concerning the RPA states we must point out that the obtained results are promising in what refers to the identification of the MEC effects in medium nuclei . o this respect three of the states quoted in table 9 are particularly interesting. These are the two 2 - levels at 7.27 and 9.88 MeV and the I' at 10.16 MeV. The last one has been extensively studied in relation to the role of non-standard nuclear effects Q-isobir, Wq mv) in nuclei. Despite the 38% increment due to MEC in the second scattering maximum, no definitive indication of them can be unambiguously established from the comparison with the experimental data [see refs. 15,16)] . The two 2 - states show considerably large increments due to the exchange currents . he one at 9.88 MeV should be compared with the 2 - state obser-.1ed at 10.0 MeV, which is the only level with this multipolarity in that energy region (see fig. 1). Fig. 8 shows the magnetic form factor evaluated with the corresponding RPA wave function . The solid (dashed) curve has been obtained with (without) the inclusion 10 -4
1 48
Ca(2- ;9.88
eV)
10 , 5
N
U
U_
10
A
r_
10IL Io qeV
QW 1 I
Fig. 8. Squared form factor for the theoretical 2 - state at 918 MeV of excitation energy. The calculadon has been peWmed with (solid curvc) and without (dashed curve) the inclusion of MEC.
J.E Amaro, A.M. Latlena / Meson-exchange currents l o-4
1 48
9
U-
603
1
Ca(2';6 .89MeV)
10
10
10
-8 L 0 .0
Fig. 9. Squared form factor for the 2 - state at 6.89 MeV of excitation energy. The calculation has been performed with (solid curve) and without (dashed curve) the inclusion of MEC . The wave function of the RPA state obtained at 7.27 MeV in our calculation has been used. Experimental data have been taken from ref. 12 ) .
of MEC, the effect of which is apparent . Unfortunately, no experimental data for the (e, e') cross section of this state ") are available. The situation is different in what refers to the second 2 - state. As we can see in fig. I there are two levels between 6 .5 and 7 MeV of excitation energy that have been ascribed as 2- states in the experiment 12 .13 ) . Though not enough data could be obtained for the (e, e') form factor corresponding to state at 6.69 MeV, empirical information is available for the level at 6 .89 MeV [ref. 12)]. In fig . 9 we compare the calculated form factor with (solid curve) and without (dashed curve) the inclusion of MEC with the experimental data which have been taken from ref. 12 ) . As we can see the impulse approximation results underestimate these data by a factor -2, and only after the consideration of the exchange effects one can obtain a rather good description of them . This can then be considered, to the best of our knowledge, as one of the first signatures of MEC effects in medium nuclei . 4. Conclusions
The quadratic relative increment for the peaks of the magnetic (e, e') form factor has been calculated for lplh and RPA states in "Ca.
604
J.E Amaro, A.
Lallena /
'eson-e.ychange currents
irst, we have found that QRI is approximately distributed around the interval
I Yo, 30%]. The mean absolute inHuence of MEC in those peaks satisfying T'"I > 10- ' is found to be of 22% with a mean dispersion of 17% for the RP st tes. Similar results have been obtained for the 1p1h states . The inclusion of s Her peaks produces an enhancement of both the mean value and the standard 0 eviatio I what refers to the behavior of MEC effects with the relevant magnitudes inv Ived in the (e, e') process, our results show the necessity for a larger number of ata. In these circumstances the dependence of the absolute QRI with the the nuclear states involved in the transition, its multipolarity excitation energy and the momentum transferred to the nucleus cannot be determined in a fully clear anner. In any case we have found that, on average, jej grows with q above -23-15 &-% shows a certain oscillatory behavior with E and appears to be more r less independent of the multipolarity J. inally, we have found that those states with low spin are the ones in which MEC e ects can become important enough to be apparent when comparing with the experiment. In some RPA cases the QRI factor reaches values of -80% . It is also important to note that the mixing of I p I h configurations does not appear to strongly interfere with MEC effects, given rise only to a coherent contribution ofthe different 1p1h cornponents included in each particular wave function . We have found that the form factor of the state observed at 6.89 MeV can be explain only after the consideration of the MEC in the calculation, thus showing one of the first clear identifications of MEC tracks in medium and heavy nuclei.
a
his work has been supported in part by the DGICYT (Spain) under contract 870969 and by the Junta de Andalucia (Spain) . eferences 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
B. Hois, Prog. Part . Nucl. Phys. 13 (1985) 117 1. Sick, Prog. Part. Nucl. Phys. 13 (1985) 149 T. Suzuki and H . Hyuga, Nucl . Phys. A402 (1983) 491 A.M . Lallena, PhD Thesis, Universidad de Granada (1984) ; S. Krewald, A.M. Lallena and J.S. Dehesa, Nucl. Phys. A448 (1986) 685 A. Arima, Y. Horikawa, H . Hyuga and T. Suzuki, Phys. Lett. 40 (1978) 1001 J .F. Mathiot, PhD Thesis, Universit6 P. et M . Curie,, Paris (1981) ; J.F. Mathiot and B . Desplanques, Phys. Lett. 13101 (1981) 141 P.G. Blunden and B . Castel, Nul. Phys. A445 (1985) 742 ; P.G. Blunden, Phys. Lett. B164 (1985) 258 T. deForest and J .D. Walecka, Adv . Phys. 15 (1966) 1; H. Oberal, Electron scattering from complex nuclei, (Academic Press, New York, 1971) Table of isotopes, Ed. C.M. Lederer and V.S. Shirley, (Wiley, New York, 1978) 1 Speth, V. Klemt, J. Wambach and G.E. Brown, Nucl. Phys. A343 (1980) 382 G. Co' and A .M. Lallena, Nucl. Phys. 510 (1990) 139 J.E. Wise et al., Phys. Rev. C31 (1985) 1699
I E Amaro, AM Lattena / Meson-exchange currents 13) Y. Fujita ef aL, Phys. Rev. C37 (1988) 45 14) J.E. Amaro, Thesis, Universidad de Granada, Granada (1990) ; J.E. Amaro and A.M. Lallena, to be published 15) W. Steffen ef at., Nucl. Phys . A404 (1983) 413 16) W. Weise, Nucl. Phys. A396 (1983) 373c; A. Hiiruig, M. Kohno and W. Weise, Nud. Phys . A420 (1984) 399; A.M. Lallena and J.S. Dehesa, Phys. Lett . B176 (1986) 9; K. Takayanagi, K. Shimizu and A. Arima, Nucl. Phys. (1988) 313; J.E. Amaro and A.M. Lallena, Phys. Lett . B261 (1991) 229
Nuclear Physics A537 (1992) 585-605 North-Holland
ESON-EXCHANGE CURRENT EFFECTS IN T E MAGNETIC ELECTROEXCITATION OF "C J.E. AMARO and A.M. LALLENA Departamento de Fisica Moderna, Universidad de Granada, E-18071 Granada, Spain Received 27 February 1991 (Revised 7 June 1991) Abstract: The effects of the meson-exchange currents in the electroexcitation of magnetic states in 4 'Ca below 14 MeV ofexcitation energy and for momentum transfers up to 3.5 fin -, have been calculated. In particular, their dependence with the momentum transferred to the target nucleus as well as with the excitation energy and the multipolarity of the nuclear states involved is analyzed in detail. Results for both pure IpI h and RPA excitations are shown and compared. Two magnetic transition with multipolarity 2 - have been found to be appreciably i.ifluenced by the exchange currents of pionic range.
1 . Introduction As is well known, meson-exchange currents (MEQ produce non-negligible effects contributing to electromagnetic properties of nuclei such as e.g. magnetic moments and (e, e') magnetic form factors. In particular, these effects are crucial in processes involving light nuclei even at low-q transfer. In these nuclei one can bring theory and experiment into agreement only after the consideration of MEC. Typical examples of this situation are the deuteron electrodisintegration at threshold') and the elastic magnetic form factor of 3 He [ref. 2) ] . On the other hand, a major difficulty appears in medium and heavy nuclei . In these nuclei wave functions are described within some uncertainties and thus MEC effects cannot be unambiguously identified . This has restricted the investigation concerning nuclei with A>4 to a few cases in which the nuclear states involved are supposed to have simple enough wave functions . Among them we mention here the magnetic (e, e') form factors for transitions to the high-spin excited states in '1-7) closed-shell nuclei 3,4) and for elastic and inelastic scattering in both medium and heavY 3 ) nuclei with closed-shell :E1 nucleon. In all these works it has been found that the magnitude of the (e, e') cross section is increased by the MEC of pionic range. However, the effect cannot be clearly isolated by comparing with the available experimental data as in the case of light nuclei we cited above. This is so mainly because of two facts. First, one must include in the calculations additional nuclear structure effects, such as ground-state and tensor correlations, which strongly interfere with the MEC ones and which are difficult to manage in these nuclei . Second, though the enhancement is not negligible, the inclusion of MEC only 0375-9474/92/$03 .50 @ 1992 - Elsevier Science Publishers B.V. All rights reserved
386
J.E Amaro, A.M. Latlena / Meson-exchange currents
produces a smooth modification in the shape of the form factors calculated in the impulse approximation and this modification is within the experimental error bars in most cases. espite the fact that the obtained results do not show clear evidence of MEC in medium and heavy nuclei, the simplicity of the (e, e') experiment makes it the ideal tool to proceed with a systematic analysis of MEC effects from both theoretical and experimental points of view. In this paper we shall investigate these effects by making a statistical analysis of them on the full magnetic excitation spectrum of the nucleus. In particular, we will study how these effects depend on the relevant magnitudes of the process, namely the excitation energy and multipolarity of the excited states involved and the momentum transferred to the target nucleus. To do that we have evaluated the magnitude of the contribution- of MEC to the form factors corresponding to the electroexcitation of magnetic states in 48 Ca. The reason for choosing this nucleus is twofold. First, it is a doubly closed-shell nucleus, which permits the application of the RPA formalism when calculating the excited states . Second, tizz neutron excess implies, at least in principle, a large decoupling of proton and neutron degrees of freedom mainiy in the low-lying excitation energy region. As a consequence, one can hope for nuclear RPA wave functions with only a few 1p1h components which means a reduction in the nuclear problem uncertainties. In this situation we expect to get reliable results. In sect. 2 we give details of these calculations. Sect. 3 is devoted to the discussion of the results . In the last section we present our conclusions. 2.
escription of the calculations
As we mentioned above we are interested in the form factor for the electroexcitation of magnetic nuclear states . In plane-wave Born approximation, which is the framework in which we will perform the calculations, this form factor is given b Y 8) IF
r (q )12
71 Z2
(-
Q'2 +tan 210 2 2q
01
O'M
where q is the recoil factor, Z is the nuclear charge, a is the differential cross-section, o-m is the Mott cross-section, Q, = ((o, q) is the four momentum transferred from the electron to the nucleus and 0 is the scattering angle. As for any doubly closed-shell nuclei, the form factor for 48Ca can be written as ,/4-7r
(q)110'), FT(q) = (J'jjTjmag z
(2)
where 10') and JJ') are the ground and excited states, respectively, and Tmag im (q) is the magnetic multipole operator which is given by mag
Tjm
(q) =
1
d rji (qr)
- J(r),
J.E Amaro, A.M. Lallena / Meson-exchange currents
587
with jj (qr) a spherical Bessel function, YjmL a vector spherical harmonics and J(F) the nuclear current. In our calculation we have assumed that this current contains both one- (J(O') and two-body (J(" )) terms i(r) = J(O)(r) +P .) (r)
(4)
Likewise the form factor can be separated into two terms as follows FT(q) = F(O)(q) + Pe')(q) T T The first one is the form factor in the so-called impulse approximation and it is associated to the individual nucleons. The second one is produced by the presence of meson-exchange pieces in the nuclear current. In this work we have only considered the pionic (J( ")) and the 7r-seagull (X'sr') ) currents, which are expected to give the most important contributions to the form factor in the energy and q-transfer regions of interest to us. Thus, additional terms such as A-isobar and p-meson-exchange currents have not been taken because they are expected to give appreciable contributions for higher q-values and energies . The energi--.s and wave functions ofthe magnetic states o f 48 Ca have been obtained by performing a RPA calculation in the framework of the Landau-Migdal theory of finite Fermi systems. In this work the nuclear mean field is described by means of a (phenomenological) spherical Woods-Saxon potential I df (r, RLS, V(r)=Vo f(rRO,aO)---(1-s)M2C2 2 r dr A
1
with
h2
I
aLS)
+ Vc (r) ,
where m is the nucleon mass and Vc(r) is the Coulomb potential generated by a uniform charge distribution of radius Rc . The parameters of the potential that we have used are shown in table 1 . These parameters have been adjusted to fit the experimentally known energies near the Fermi surface (see column 2 in table 2) and the r.m.s. charge radius of "Ca, rc = 3 .48 fin [ref. 9)] . The Woods-Saxon well has been diagonalized in a truncated harmonic-oscillator basis, thus producing an automatic discretization of the continuum. In table 2 we also show (column 3) the results obtained for the orbits close to the Fermi level after diagonalizing the TABLE I
Parameters of the Woods-Saxon potential R a, Ro 1fi) (fffl) (MeV) protons neutrons
-59.50 -50.00
4.36 4.36
(fffl)
0.53 0.53
À
(fin)
RLs
al_s (fM)
c (fin)
26.0 28.0
4 .36 4.36
0.53 0.53
4.5428
J. E
388
inaro, A.M. Lallena / lt-leson-e.-
Single-particle energies for the orbits near the Fermi level. The results obtained by diagonalizing the Woods-Saxon potential are compared with the experimental values considered to adjust the corresponding parameters NeLtrons
Protons 111j
E (MeV)
&, (MeV)
181j
E (MeV)
E,,,, (MeV)
Id~,,:? 2s, /2 1 f7/2 2p~,,:! I f,;,:! 2p, /2
-15,557 -15.344 -9.406 -3,718 -1 .361 -1 .130
-1&167 -15,807 -9.628 %5" -5.821 -6.113
2s, /:! Id3/2 1f7/2 2P3/:! 2p,/2 1~53/2 199/2
-15 .698 -15 .619 -9.550 -4.919 -2.733 -2.216 1 .276
-12.545 -12.525 -9.945 -5.141 -3.119 -1.555 -1.131
potential. In the actual calculations the experimental values have been kept. For the remaining orbits not shown in table 2 we have taken the single-particle energies given by the diagonalization of the potential . The obtained r.m.s. charge radius is r, = 3.33 fin, which adequately describes the experimental value. he RPA calculation has been performed using a configuration space composed by 70 single-particle states, which corresponds to proton and neutron orbits lying at energies below -25 MeV. The residual interaction we have considered is the one developed by the Kilichfor Stony Brook group 'o) the study of magnetic properties ofnuclei. This interaction consists of a long-range part based on the 7r- and p-exchange potentials and a short-range piece which is simulated by a phenomenological zero-range Landauigdal interaction. In momentum space the form of the interaction is ' o")
K,Q) = Co(go
where
-
a 2 + gocr I
.
a 2 T I . T2)+ V,,(q) + J 27r
n(
2
m 7r
a 1 - qcr- 2
q
2
q
+m,,
)
T
1
VP (q) ,
7
2
and ) = 4ri
ere
C = h27T2/ mk 0
2 f~ 1 M2 0 (3
Xq- &'X 2 2 q +M P
F is the inverse of the density of states
lp 1 .
lp
2
at the Fermi surface and
f,, andf, are, respectively, the pion-nucleon and the rho-nucleon coupling constants . Finally, the dimensionless parameters go and go have been fixed to reproduce the energies of the two first excited states . The particular values we have used are given in table 3.
The RPA calculation produces 36 excited states with multipolarities ranging from J' = 1 + to J' = 8 - with the exception of the 7 + which is not present in the energy
a e') MEC the magnetic each see Y being maximum will this experiment and each the we allows information calculated q being spectrum states us the of q-value present Then are index point "'(q) collaborators to (QRI) the even energies, Iinclusion is of F(')(q)l form us interested the trust the these influence are we = to (a) we is or singularities RPA value of for of iHowever, in factor avoid have reasonably in shifted odd, concerning stands shall and excited spins ref the the ph the of magnetic amplitudes, Amaro, where in Aj(ph)e(lp+lh+J+I) computed of respectively ") 13) MEC shifting the form design reliability impulse In ofthe (that and for to MEC (first (MeV (second inherent states principle the A(0) at factor qrnax F(0) (solid well residual parities the of is, states typical some aits in -column) or fin') the up approximation +Aq procedure the Lallena reaches and excitation the of zeros described column) (ex), problems due line) to Besides it interaction Landau-Migdal q-values form e-factor our of form of is +Aq)12 G(a) effect )12 This /by to aevident the 14 48Ca and produces Meson-exchange aresults _= 3MEC factor factor some maximum 0to MeV) (x) IF(o)(q -,/(2a the of different As of magnitude the )12 by used only e (third which Thus measure are the MEC we amplitude that by the (p, (dashed isThus Aq 12 in)12 + In zero-range in anucleonic corresponding we can given this 1) p') will smooth one 0g(" fig calculated As the column) levels dr is and define After currents work asee experiment the as a r2jj obviously eIpositive could peaks b~ line) 12consequence, we 6(a) Aj shown will much identified (qr) the influence 4,14) variation the densities show =evaluate together of = by be Xj* inclusion spectrum, quadratic value Cof the in 1p1h hide or modifying used the + performed the (x) fig (form Iin of containing calculated of according the state the 1YYj*, observed ,with 2both the by MEC of erelative factor, which As effects factor means factor MEC us form the the by we in X to
J.E
.M.
589 TABLE
Parameters piece C,,
90 386 .04
.7
.7
.
region energy excitations
.
(e, Fujita excited
. .
permits For i
FeT
e jpjh
Z
where to
(6)
.
. e(q)
T
IFT(q IF(o)(q T
.
.
for can factor maxima
.
T .
.
which increment e(qmax)
this
G(a) phj
JO
phJ
the
where
ac
.
and all At
e(q) of
.
.a,,
determine
IFT(qmax
F(O'(qmax) T IF(o)(qmax) T
T
. .
.
.
IE
S40
aro,
Lm llena /
currents
5+
12
l +--------
_ 28
(0
l + --------'
2--------0 _______-
-
26 4----------
0--------------0
(6 ,7
6-
(3+ + (6 ,6 ) `
l+ 2
(` 2 _ ` 6 _ ) (4 /5+,2
-
2
-
-
2
Me)
-
BL P. /k .
Fig. 1 . Excitation spectrum obtained in the RPA calculation considered in this work.
-~..
~. . -..
~I
J.E Arnaro, A.M. Lallena / Meson-exchange currents
qeff
-1
M
591
1
Fig. 2. Typical example of the effect of the MEC in the magnetic (e, e') form factor. The solid (dashed) curve corresponds to the calculation performed with (without) the consideration of MEC.
an increase of the form factor due to MEC, whereas if e < 0 the form factor will be reduced. We will be also interested in the relative effect of the MEC, irrespective of its sign. In that case the absolute value lei will be considered. Due to its statistical nature, it is expected that the conclusions obtained by means of this procedure will be applicable to other nuclei . 3.
esults
The first important point observed in our results concerns with the mixing of 1p1h configurations appearing in the RPA wave functions. In this respect it has been found that some RPA states with multipolarities J' = 1', 2- and 4 - are the most affected by the consideration of these non-dominant configurations. On the other hand, those with J' = 3 +, 5 + , 6 - and 8 - show wave functions which can be supposed to be built with only a small number of 1p1h components. As an example we give in table 4 the Aj amplitudes corresponding to the 1p1h configurations of the 5 + and 2 - states obtained at 5 .35 and 6.00 MeV, respectively, in the calculation. Only those components with amplitudes Aj :- 0. 1 are shown. As we can see, the wave function of the 5 + state is dominated by the P( 2P3/2 If7/2 _' ) configuration while that of the 2- state shows a considerable mixing of 1p1h states. This fact is more apparent in fig. 3 where the form factors corresponding to both states are 9
I E~
392
maro, A.M. Lallena /
eson-exchange currents
TAME 4 Amplitudes Aj for the lplh components of be RPA mve functions of the 5' and 2- states obtained at 135 and 6.00 MeV, respectively . Only those values larger than 0.1 have been tabulated Iplh con6gumuon rdpvy Idi /12 ) rYp 3/ 2 , 2s ï112 ) -zr(2p3~ :!, Id,~' ) Id
5' ; 5.35 MeV
2 ; 6.00 MeV 0.171 0,159 0.188 -0.114 -1501
IpIh configuration t,(2p,/2,
5+, 5.35 MeV
ld3/'2)
P(2P3/ :!, 2sl/'2) v(2P3/2, Id .;/. ) v( 2 P3/2, IfÙ12)
1.061
2 ; 6.00 MeV 1211 1182 1552
shown. In the case of the 5' (see fig . 3a) the inclusion of the small components (solid curve) only produce a smooth modification of the form factor obtained by considering only the dominant component (dashed curve). On the other hand, the situation for the 2 - state is quite different, as we can see in fig. 3b. Therein the three calculations correspond to the form factor obtained by considering the full RPA wave function (solid curve), a wave function including all the 1p1h components given in table 4 (dashed line) and, finally, a wave function with those components of amplitude Aj :-::- 0.5 only (dotted curve). As we can see the corresponding form factor appears to be strongly affected by the inclusion of the small components of the wave function . [Further details concerning this particular aspect can be found in ref. 14).] is fact has led us to calculate, in the same way as described above, the QRI for all the 1p1h configurations, we find up to 14 MeV. These configurations are given in table 5. In these cases the form factor can be evaluated by summing in eq. (6) only over one ph configuration and putting Aj (ph) = 1 . It is expected that the com?arison of the results obtained in both the 1plh and RPA calculations will inform us about the differences one can observe when the wave functions are very simple and it wiH be helpful to a better understanding of the RPA results. Moreover, the calculation corresponding to the single lplh configurations will allow us to distinguish between protonic and neutronic excitation and to obtain additional information that cannot be extracted with the RPA wave functions. ith this in mind we have evaluated the QRI factor e(q) for all the 1p1h configurations and RPA states below 14 MeV and for the peaks of the corresponding (e, e') form factors found at values of the momentum transfer up to 3.5 fm - ' . In what follows we analyze the most relevant results obtained in this calculation. 3.1 . AVERAGE
EHAVIOR OF QRl
e start by discussing the main characteristics observed in the average behavior of the e factor. In table 6 we show the mean values and standard deviations of QRI evaluated for both I p I h and RPA wave functions (see columns 3 and 6, respectively)-
J.E Amaro, A.M. Lallena / Meson-exchange currents
593
CT LL
10
-3
I
I
48Ca(2 - ;6.OOMeV)
: - /,
I
C~L_
1 10 0.0 -7
3
, ,' % .
1
1 .0
: _F
1 2.0
-
3.0
q eff [W1 1
Fig. 3. (e, e') form factor for the (a) 5' state at 5.35 MeV and (b) 2- state at 6.00 MeV. Solid curves correspond to the calculation performed with the full RPA wave functions . Dashed (dotted) curves have been obtained by including all the lp1h components with Aj -- 0.1 (Aj -_ 0.5) in the wave function . MEC effects are not considered here.
M Amaro, A.M. Lallena / Me.çan-e.xchange currenis
594
TABLE 5
lp1h exdta6ons Ar which be QRI factor has been calculated in this work lp1h configuration
Energy (MeV)
Multipolarities
IpIh configuration
Energy (MeV)
=Of,~ Z , 2sj,'2 )
1 d~~A!)
63 6.54 9,26 9.62 9,919 10.05 10.35
4 2- ,422222- ,4-
v(2p jj2 , lf~1'2 ) 02p, 12 , If 02) 02N/2v Idp':!) v(2p,-Aj2 ,2s 1/'2 ) '.. (I fS/2' I f 412) "( 1 99/2' 1 fK12) P(2p,/ :!, WOO
4.80 6.83 7.38 7.40 8.39 8.81 9.41
.r(2p :,~ 2 , 1 d 02)
13.87
2-,4-
2s,Q) v(Igq,, 2 , Id j/1:! )
r. (2p,,
:! , 2s jjz ) Tr(2p,j/z , WOO -" ( I ~,,, 2 , 2s j,~2 ) -(2p, :! , ld~~?:!)
If
-0 f, :! ,
` ) Id Q
10.78
2 - ,4 - ,6 -
01 fs/2 , Idj! )
10.97
v( I
11 .39 11 .41
Multipolarities 3', 5' 3' 221', 3', 5' 2-,4-,6-,822-,423', 5' 5+
10.99
2s 1 /2)
12.92
i,(3s,/ :!, If7__/12)
4-
The first row corresponds to the results obtained by including all the maxima of 10-4 and interest to us. In the second and third rows those peaks with jF(O)(q)j< T P" ) ( q) I < 10- respectively, have not been included . Columns 2 and 5 give the T
number of peaks considered in each case. As a first result we point out that the values of j :E cr found for 1p1h and RPA states are close. Another interesting aspect is that both the QRl mean value and the standard deviation reduce with respect to the other calculations is only those peaks with IF"(q)j :_~- 10' are considered (see T rc ;~; 3). This fiau. iiaeans that on average the bigger the one-body form factor is, the smaller the relative increment due to MEC becomes. Finally, we can state that the mean effect of MEC in the (e, e') form factor is to produce an enhancement of the one-body part by -20%, though a reduction is found in some cases. The similarity observed between lplh and RPA results does not appear in fig. 4. Therein we show the bar diagrams corresponding to the distribution of the peaks with respect to the QR1 value, for both lp1h and RPA wave functions. Here only those maxi'ma with I F(O :- 10-4 have been included . Thus these results correspond T '(q) 1 :to the data in row 2 of table 6. The reason for imposing such a restriction lies in TABLE 6
Mean influence of MEC in 1p1h and RPA excitations lplh
q ~~ -- 3.5 fm -- ' 10-4 1 F'O'l T 10-3 I F'O'l T
RM
W of man
e±a
IOU
# of man
e±a
id ± 0'
87 76 57
0.28 :t 0.70 0.21 :E 0.34 0.15 :1--0.19
0.41 ±: 0.63 0.29 --h 0.27 0.20--EO.14
124 110 70
0.26 :E 0.85 0.211:0.38 0.18 :EO.21
0.39±0.80 0.30±0.32 0.22-±0.17
J.E Amaro, A.M. Lallena / Meson-exchange currents 25,
595
1
20 15 10
r-4-1 20 15 10
0
1
H I I I I I I I I
-100 -80 -60 40 -20
0
e F%1
20
40
[_T_~ [E] 60 80 100
Fig. 4. Distribution of the values of QRl for the peaks of the form factor satisfying the conditions I F(rO) I >. 10-4 for (a) lp1h and (b) RPA excited states . q,,.., -_ 3.5 fin` and 48 the fact that smaller maxima are out of the range experimentally measured in Ca. Besides the consideration of these small peaks makes the standard deviation to increase considerably in both lplh and RPA cases (see row I in table 6) and the results would be distorted because of this fact . As we can see in fig. 4a, in the lp1h case there is a certain symmetry around the interval [ 10%, 20%] which shows a frequency which is notably greater than that of the remaining intervals. The situation for the RPA results (see fig. 4b) is rather different . Now the distribution is peaked in the interval [20%,30%], that is a 10% shift with respect to the lp1h result, but the symmetry around the maximum has disappeared . However, both effects wake the J value to be very similar to that of lplh . Additional information about the MEC influence in the (e, e') form factor can be obtained from the absolute value of QRI, jej . This magnitude will give us the strength of the MEC effect independently of the fact that it can be either an increase or a reduction of the one-body form factor. In columns 4 and 7 of table 6 we show the corresponding mean values and the standard deviations . As we can see the results for lp1h and RPA excited states are again very similar and the same conclusions as for the e factor can be drawn. Now we find that the average influence of MEC is around 30% .
IE Ainam A.R LaWna / Meson-exchange currents
s we mentioned above, the calculation performed for the 1p1h states permits the analysis of the results for protons and neutrons separately . In order to proceed with this study we have evaluated the same magnitudes as in table 6 for such Ip1h excitations . The obtained results are shown in table 7. There we can see that the ean value j is bigger for protons than for neutrons, which is more apparent when re peaks are included in the statistics . Moreover, the data dispersion is larger for neutronic than for protonic states. This asymmetry between both kinds of excitation almost disappears when we turn to the absolute value of the corresponding e factor. In this case the mean value Ar proton and neutrons practically coincides, tbough the standard deviation still shows some differences. he main conclusion we can draw after the analysis of the values of the QRI that we have done is the Mmila6ty between the obtained results for 1p1h and RPA excitations. This means that the KAng of 1p1h configurations present in the RPA wave functions induces almost negligible effects in the contribution of the MEC to the (e, e) form Actor, at least in what refers to the mean values of the e factor. Taking this fact into account we go deeper in the behavior of the QRI and discuss its dependence with the relevant magnitudes of the process. 3.2,
EPENDENCE OF QRI WITH q, E AND J -
ow we are interested in analyzing how the effect of MEC depends on the three fdndamental magnitudes involved in the electroexcitation of the nucleus, that is the energy and spin of the excited state and the momentum transferred from the electron to the nucleus. That is, we want to study the behavior of the e factor with each of these variables. To do that we have determined the dependence of the average value of jej by considering all the yaks appearing at each value of the variable of interest in each case. The results have been plotted in figs. 4-6. Therein the corresponding behavior of the absolue value of QRI with the momentum transfer (fig. 5) and the spin (fig. 6) and energy (fig. 7) of the excited states are shown for both (a) 1p1h and (b) RPA states. The error bars give the corresponding standard deviations . e first aspect that must be pointed out is the similarity found between the results obtained for the simple 1p1h and the RPA excitations. This fact shows again that the Nxing of 1plh configurations involved in the RPA wave functions does TABLE 7
Mean influence of MEC in protonic and neutronic 1p1h excitations Protons # of max . q,,,, -- 3.5
fin' 1 POI , 10-4 T 10-3 10"I a T
34 31 24
132±00 0.28 ±0 .31 T16±0A1
Neutrons
1,jI ± 0,
# of max.
O±o,
1,jI ± 0,
OAO ±156 0.30±0 .29 118±108
B
125±036 0.17±0.35 114±123
OA2 ± OA8 0.28 ±0.27 121±117
45 33
J.E. Amaro, A.M. Lattena / Meson-exchange currents
597
200 jej 150
100
50
0
1
0
0.
40
1 . ill I 116
0
0
.6f
i
1W
100
4)
50~ 00
0
0.
1.
2. q [fm -' l
3.
4.
Fig . 5. Behavior of jej with the momentum transer q for (a) 1plh and (b) RPA excited states. The peaks included satisfy the same conditions as in fig . 4.
not greatly interfere with MEC effects . Thus these effects appear to be rather independent of the particular combination of 1p1h configurations in the RPA wave functions . In what refers to the dependence with q, it can be seen in fig. 5 that the lel values are clearly below -50% up to q - 2.0-2.5 fm- '. Above this value lei grows and reaches, in both 1p1h and RPA states, values near 100% at q - 3.5 fm- ' . We can state that on average it is necessary to go to high momentum transfer in order to find a big effect due to MEC. In any case there are some particular cases in which such effects are considerably large even at q < 2 fin' . We will analyze such cases in the next subsection .
598
AE Ainam AM LaWna / Meson-exchange currents 1004-
1
- I
I
I
I
I
I
I
(a)
75
5 40
I I
40
I
I
I
(b)
1
50
25
-6
0
I
1
I
2
I
3
I
4
I
i
5
I
6
I
7
I
8
I
9
FQ. 6. Same as in hg. 5 but W the dpendenne of jel with the spin of the exciad states .
The behavior of the absolute value of QRI with the nuclear spin J shows some interesting characteristics. In the case of Iplh states (see fig. 6a) the corresponding mean value appears to be constant (with values slightly above 25%) and the dispersion is similar in all the cases with the exception of the peaks with J = 5 and 8 . The shua6on is different for the RPA states . As we can see in fig. 6b the mean values show small variations between the different J. Thus for J = 2 and 6, only the Jej factor is over the 25%, while the average MEC influence for J = 1, 3,4 is clearly reduced with respect to that ofthe I pI h states . On the other hand, standard deviations suggest that the absolute value of the QRI must be considered as independent of J in practice .
J.E. Amaro, A.M. Lallena 200, lel [%]
1
Meson-exchange currents 1
9
10 .
12 .
599
(a) 150~
100
50
0.
lei [%]
(b) 150~
100
50 .-
0 .1 4.
6.
14 .
E [MeV]
Fig . 7 . Same as in fig . 6 but for the dependence of jej with the energy of the excited states.
Finally, no clear conclusions can be drawn concerning the dependence of lei with the excitation energy. This is so because of the relative small number of data we have. The RPA results (see fig. 7b) show a certain oscillatory behavior with two bumps at 6-7 MeV and 11-12 MeV, respectively . The same cannot be stated for the 1p1h states because of the lack of data in the region around 12 MeV (see fig. 7a). 3 .3 . OUTSTANDING STATES
In order to finish the discussion of our results concerning the MEC effects we will now analyze the excited state having peaks in which such effects are considerably
600
AE Amaro, AR Lallena / Meson-exchange currents TABLE 8
Outstanding lplh states in "Ca. Those levels with peaks in which lel -_ 341/o are shown lpIh configuration 14 If.5/2, lf7/12) v0pj ,2 , Id .V 3/2 )
E, (MeV)
(fm-1)
IF") T 'l (X IO-2)
e M
1 .3 2.2
0.24 0.20
36.b 54.9
4-
0.6 1 .5 1 .9
0.14 0.32 0.43
-51.4 61 .0 56 .5
10.990
2-
1.2
0.28
47 .0
13.656
3'
2.1
0.11
45 .4
8.390 9.406 10 .970
v0f,, 2, 2s t/12)
K Ig z , 2s, /1:! )
j 7r 1+ 22-
q
remarkable, thus allowing for an easier identification when comparing with the experiment. In what follows we restrict our attention to those cases in which the corresponding form-factor peaks satisfy IF'T0'1::- 10". As we can see in table 6 the mean variation intervals of lei are [6%,34%], in the case of the lpIh states, and [5%,39%] in that olf the RPA ones. Those states with form-factor maxima in which the absolute value of QRI is above these intervals are shown in tables 8 and 9 for 1p1h and RPA states, respectively . Some interesting aspects deserve a comment. First, it is important to note that the multipolarities appearing in both tables are the lowest. This fact is rather striking. In effect, as we said above the attempts to identify MEC tracks in medium and heavy nuclei have been restricted till now to nuclear states with simple enough wave functions . In this sense lower multipolarities were i-iot taken into account because they generally involve a large mixing of 1p1h configurations. However, our results indicate that, first, lower multipolarities present the larger effects due to MEC and second, the 1p1h and RPA states show a similar TABLE 9
Outstanding RPA states in 4"Ca. Those levels with peaks in which lei -_ 38% are shown E, (MeV) 7.27 8.99 9.88 10.16 10.92 11 .40 11 .81 12 .33
IF")i T
e
0.13 0.16 0.16 0.20 0.69 0.49 0.34 0.16
82 .4 47 .6 83 .0 38 .0 41 .8 38 .4 79.5 70.7
( X 10-2) 2221+ 4224-
1.7 2.2 2.1 1 .2 2.0 1.4 1.0 1.3
J.E. Amaro, A. M. Lailena / Méson-exchange currenis
60 1
behavior. These two facts inform us again about the large decoupling between MEC effects and the configuration mixing appearing in the nuclear wave function. As a conclusion, we can state that magnetic transitions to low-J levels seem to be the best places to look for the effects of MEC. A second important aspect is that all the e-values given in these two tables are ` )2- ph configuration at 10.97 MeV shown positive, with exception of the IV( I f5/2 9 1 d 3/2 in table 8. This last case is specially interesting because the MEC contribution produces a reduction of the one-body form factor in the first peak and an enhancement in the second one, the effect being bigger than a 50% in absolute value in both cases . Apart from this particular point, it can be seen by comparing the results included in both tables that the QRI reach higher values for the RPA states than for the I pIh configurations. This is the only effect that can be ascribed to the mixing of 1p1h configurations, which seems to produe a coherent sum of the MEC effects coiresponding - to the different 1p1h components involved in the nuclear wave function . The results obtained for the pure 1p1h states show interesting characteristics that must be discussed. We note that all the configurations shown in table 8 are of neutronic type. This points out again the above mentioned asymmetry between protons and neutrons concerning the MEC contributions . In order to go deeper into this situation we have analyzed the form factors at the maxima for those 1p1h configurations which simultaneously appear for both protons and neutrons . The obtained results are shown in table 10. We have found that, in general, MEC contributions are of the same order in both kind of configurations. In several cases, TABLE 10
Comparison of the maxima of the form factors corresponding to the I pI h configurations simultaneously appearing for protons and neutrons in our calculations lp1h configuration
Fr
(2p,/2, ld3/2)
2-
(If,/2, ld3/'2)
2-
q (fm -1)
I F'OT l ( X 10-2)
I Fr" I I ( X 10-3)
q (fm -1)
IFT'O II ( X 10-2)
1 .1 2.2
0.285 0.380
0.073 0.536
1 .1 2.2
1 .4 2.7 1 .9
1 .060 0.256 1 .120
0.193 0.201 0.142 0.324 0.032 0.429
0.070 0.162
0.161 0.220 0.466 0.469
I f5/ 2, 2s ,/'2)
2-
1.2 2.3
0.780 0.070 0.776 0 .428 0.193
(2P3/2, ld3/'2)
2-
QP3/2,2s~/' 112 )
2-
0.6 1 .7 0.7 2.1
0.605 0.543
4-
Neutrons
Protons
0.605 0.184
0.6 1.5 2.7 1 .9 1 .2 2.3 0.6 1 .6 0.7 2.0
0.282 0.119 0.075 0.164 0.388 0.341
IF""I T
( X 10-3) 0.119 0.502 0.428 1 .090 0.216 1 .080 0.600 0.149 0.132 0.180 0.180 0.443
602
IL Amam A.AL Ladna / Meson .e.-
some of them those of table 8, the neutronic one-body form factor is roughly half that ofthe corresponding protonic state, mainly due to the presence ofthe convection term in the last one. In these circumstances it is obvious that the e factor is appreciably larger for neutrons . In the remaining cases the one-body form factor is of the same order in both kind of configurations, because in these states the convection current gives a contribution opposite to that of the spin-magnetization term. III these cases the QRI is similar for the protonic and the neutronic Ip1h excitation . Concerning the RPA states we must point out that the obtained results are promising in what refers to the identification of the MEC effects in medium nuclei . o this respect three of the states quoted in table 9 are particularly interesting. These are the two 2 - levels at 7.27 and 9.88 MeV and the I' at 10.16 MeV. The last one has been extensively studied in relation to the role of non-standard nuclear effects Q-isobir, Wq mv) in nuclei. Despite the 38% increment due to MEC in the second scattering maximum, no definitive indication of them can be unambiguously established from the comparison with the experimental data [see refs. 15,16)] . The two 2 - states show considerably large increments due to the exchange currents . he one at 9.88 MeV should be compared with the 2 - state obser-.1ed at 10.0 MeV, which is the only level with this multipolarity in that energy region (see fig. 1). Fig. 8 shows the magnetic form factor evaluated with the corresponding RPA wave function . The solid (dashed) curve has been obtained with (without) the inclusion 10 -4
1 48
Ca(2- ;9.88
eV)
10 , 5
N
U
U_
10
A
r_
10IL Io qeV
QW 1 I
Fig. 8. Squared form factor for the theoretical 2 - state at 918 MeV of excitation energy. The calculadon has been peWmed with (solid curvc) and without (dashed curve) the inclusion of MEC.
J.E Amaro, A.M. Latlena / Meson-exchange currents l o-4
1 48
9
U-
603
1
Ca(2';6 .89MeV)
10
10
10
-8 L 0 .0
Fig. 9. Squared form factor for the 2 - state at 6.89 MeV of excitation energy. The calculation has been performed with (solid curve) and without (dashed curve) the inclusion of MEC . The wave function of the RPA state obtained at 7.27 MeV in our calculation has been used. Experimental data have been taken from ref. 12 ) .
of MEC, the effect of which is apparent . Unfortunately, no experimental data for the (e, e') cross section of this state ") are available. The situation is different in what refers to the second 2 - state. As we can see in fig. I there are two levels between 6 .5 and 7 MeV of excitation energy that have been ascribed as 2- states in the experiment 12 .13 ) . Though not enough data could be obtained for the (e, e') form factor corresponding to state at 6.69 MeV, empirical information is available for the level at 6 .89 MeV [ref. 12)]. In fig . 9 we compare the calculated form factor with (solid curve) and without (dashed curve) the inclusion of MEC with the experimental data which have been taken from ref. 12 ) . As we can see the impulse approximation results underestimate these data by a factor -2, and only after the consideration of the exchange effects one can obtain a rather good description of them . This can then be considered, to the best of our knowledge, as one of the first signatures of MEC effects in medium nuclei . 4. Conclusions
The quadratic relative increment for the peaks of the magnetic (e, e') form factor has been calculated for lplh and RPA states in "Ca.
604
J.E Amaro, A.
Lallena /
'eson-e.ychange currents
irst, we have found that QRI is approximately distributed around the interval
I Yo, 30%]. The mean absolute inHuence of MEC in those peaks satisfying T'"I > 10- ' is found to be of 22% with a mean dispersion of 17% for the RP st tes. Similar results have been obtained for the 1p1h states . The inclusion of s Her peaks produces an enhancement of both the mean value and the standard 0 eviatio I what refers to the behavior of MEC effects with the relevant magnitudes inv Ived in the (e, e') process, our results show the necessity for a larger number of ata. In these circumstances the dependence of the absolute QRI with the the nuclear states involved in the transition, its multipolarity excitation energy and the momentum transferred to the nucleus cannot be determined in a fully clear anner. In any case we have found that, on average, jej grows with q above -23-15 &-% shows a certain oscillatory behavior with E and appears to be more r less independent of the multipolarity J. inally, we have found that those states with low spin are the ones in which MEC e ects can become important enough to be apparent when comparing with the experiment. In some RPA cases the QRI factor reaches values of -80% . It is also important to note that the mixing of I p I h configurations does not appear to strongly interfere with MEC effects, given rise only to a coherent contribution ofthe different 1p1h cornponents included in each particular wave function . We have found that the form factor of the state observed at 6.89 MeV can be explain only after the consideration of the MEC in the calculation, thus showing one of the first clear identifications of MEC tracks in medium and heavy nuclei.
a
his work has been supported in part by the DGICYT (Spain) under contract 870969 and by the Junta de Andalucia (Spain) . eferences 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
B. Hois, Prog. Part . Nucl. Phys. 13 (1985) 117 1. Sick, Prog. Part. Nucl. Phys. 13 (1985) 149 T. Suzuki and H . Hyuga, Nucl . Phys. A402 (1983) 491 A.M . Lallena, PhD Thesis, Universidad de Granada (1984) ; S. Krewald, A.M. Lallena and J.S. Dehesa, Nucl. Phys. A448 (1986) 685 A. Arima, Y. Horikawa, H . Hyuga and T. Suzuki, Phys. Lett. 40 (1978) 1001 J .F. Mathiot, PhD Thesis, Universit6 P. et M . Curie,, Paris (1981) ; J.F. Mathiot and B . Desplanques, Phys. Lett. 13101 (1981) 141 P.G. Blunden and B . Castel, Nul. Phys. A445 (1985) 742 ; P.G. Blunden, Phys. Lett. B164 (1985) 258 T. deForest and J .D. Walecka, Adv . Phys. 15 (1966) 1; H. Oberal, Electron scattering from complex nuclei, (Academic Press, New York, 1971) Table of isotopes, Ed. C.M. Lederer and V.S. Shirley, (Wiley, New York, 1978) 1 Speth, V. Klemt, J. Wambach and G.E. Brown, Nucl. Phys. A343 (1980) 382 G. Co' and A .M. Lallena, Nucl. Phys. 510 (1990) 139 J.E. Wise et al., Phys. Rev. C31 (1985) 1699
I E Amaro, AM Lattena / Meson-exchange currents 13) Y. Fujita ef aL, Phys. Rev. C37 (1988) 45 14) J.E. Amaro, Thesis, Universidad de Granada, Granada (1990) ; J.E. Amaro and A.M. Lallena, to be published 15) W. Steffen ef at., Nucl. Phys . A404 (1983) 413 16) W. Weise, Nucl. Phys. A396 (1983) 373c; A. Hiiruig, M. Kohno and W. Weise, Nud. Phys . A420 (1984) 399; A.M. Lallena and J.S. Dehesa, Phys. Lett . B176 (1986) 9; K. Takayanagi, K. Shimizu and A. Arima, Nucl. Phys. (1988) 313; J.E. Amaro and A.M. Lallena, Phys. Lett . B261 (1991) 229