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Nuclear photon scattering by 12C
Nuclear Physics North-Holland
A506 (1990) 307-331
NUCLEAR
K.P. SCHELHAAS’,
Max-Planck-lnstitut
PHOTON
J.M. HENNEBERG,
SCATTERING
BY ‘*C*
N. WIELOCH-LAUFENBERG*, B. ZIEGLER
fGr Chemie, Kernphysikalische M. SCHUMACHER
II. ~~.~~~~~s~~~s ~nst~tui der ~niuersit~t
Arbeitsgruppe,
U. ZURM~HL3
and
D 6500 Mainz, Fed. Rep. Germany
and F. WOLF4
~~tt~n~en, D 3400 Giittingen,
Fed. Rep. Germany
Received 26 June 1989 (Revised 4 September 1989) Abstract: Cross sections
for photon scattering by the “C nucleus have been measured in the energy range from 15 to 140 MeV. Quasimono~hromatic tagged photons allowed the separation of elastic and inelastic processes. An additional measurement with continuous bremsstrahlung improved the accuracy of the elastic part of the cross section considerably. In analyzing the bremsstrahlung runs, the inelastic components were taken from the tagged photon results and from literature. The cross sections were analyzed in terms of giant resonances, proton-neutron substructures and subnuclear excitations. Form factors for charge and exchange current were applied to take into account the finite nucleon and exchange-current distribution in *‘C. For the E2 resonance, a coupled-channel model was used, which is applicable to an isolated resonance supe~mposed on a continuum. The average sum of the static electric and magnetic polarizabilities of bound nucleons was found to be in agreement with the value of the free proton.
E
NUCLEAR REACTlONS deduced El, E2 resonance
‘*C( -y, y) (y, y’), E = 15-140 MeV; measured o(E, @). “C parameters, radius parameters, polarizability of bound nucleons.
1. Introduction In previous papers l-6) elastic photon scattering was treated by theory and experiment, and nuclear structure quantities were deduced. In order to attain an as complete picture as possible, it is essential to recognize the need of cross sections in a wide energy range and for all angles. There are regions that are specifically sensitive to one or the other measurable physical quantity, with rather small influence by others, as shown for the form factors in the case of *08Pb [ref. “)I. The conclusions drawn Pb can also be applied to ‘*C, for which the only main difference is, that the for *OS * This work was supported by the Deutsche Forschungsgemeinschaft, ’ Present address: Lurgi AG, 6000 Frankfurt/M., Fed. Rep. Germany. ’ Present 3 Present 4 Present
address: address: address:
037%9474/9O/SO3.50 (Noah-Holland)
Gei GmbH, Elisabethenstr. 44 i/2, 6100 Darmstadt, Bosch AG, 3200 Hildesheim, Fed. Rep. Germany. Siemens AG, 8000 Miinchen 83, Fed. Rep. Germany. @ Elsevier
Science
Publishers
B.V.
SFB 201. Fed. Rep. Germany.
K.P. Schelhaas et al. / Nuclear photon scattering
308
r.m.s. radius
is 2.2 times smaller.
60” are increasingly zero-degree dispersion
obscured
scattering relations
cross
section
and the optical
were taken from literature
Since the forward-going
by electromagnetic
was calculated theorem.
scattering
shower processes from total
cross
These total absorption
7-9). The final data set though,
should
angles
below
in the target, the
present
sections
via
cross sections a consistent
picture of scattering and absorption of photons below meson threshold, in which the real part of the scattering amplitude is partly given by the r-meson production cross section via the dispersion integral. The scattering cross sections were obtained for 60”, 90”, 120”, and 150”, using both tagged photons and bremsstrahlung. The energy range between 15 and 140 MeV is limited by electromagnetic background at the low part and by the onset of 7r” production at the high part. For the end energy of 143 MeV chosen for the bremsstrahlung measurement, it was checked experimentally that the 7~’ decay photon contribution, kinematically allowed between 48 MeV and 95 MeV, did not contribute in amplitude more than 5% of the scattered photons. The analysis follows the same steps as in the case of *“Pb in ref. 5). In the following sections, only the differences of the experimental setups and of the analyses are described in detail. These differences comprise mainly three points: (i) The source of quasimonochromatic photons was changed from positron annihilation-in-flight to tagged photons. The new experimental setup is described in sect. 2.1. (ii) In the analysis of the bremsstrahlung data, a contribution from inelastic scattering has to be taken into account 1*276).A short abstract of the related theory, and the experimental results are given in sect. 3.1. (iii) The two existing previous measurements did not agree on the E2 strength, found distributed around now comprise sufficiently
32 MeV [ref. ‘j)], or above 50 MeV [ref. ‘,‘)I. The results large regions in energy and angle, and allow a more
definite answer to the E2 question. The data analysis is based on a special case of coupled channels theory which is described in very basic terms in sect. 3.2. In sect. 3.3, the final results are given, in an attempt to combine all published and new cross sections
in a consistent
picture.
2. Experiment 2.1. TAGGED
PHOTONS
The general outline of the scattering setup is given in fig. 1. Electrons from the electron accelerator MAMI A with 143 MeV energy and 50 KeV energy spread hit the bremsstrahlung target in a small spot of about 1 mm diameter. The long-time stability in angle and position was better than 0.2 mrad and 0.5 mm, respectively. The electrons were bent downwards and backwards by an angle of 150” inside the gap of the main deflecting magnet. 44 plastic scintillators were placed in the focal plane inside the magnetic field. They define the simultaneously measured photon
309
K.P. Schelhaas et al. / Nuclear photon scattering
Lead
H
Nickel
/ LO.O1 mm Al -Tarset
\
184 MeV electrons Precision
collimators
lm Fig. 1. Top view of the tagging setup. The electron beam from the MAMI-accelerator enters from the left and is bent down by the tagging dipole. 44 electron detectors are placed along the focal plane in the magnet gap. The collimated photon beam hits the scattering target which is viewed by four wellshielded Nal detectors.
energies,
and the energy
widths
of the tagged
quasimonochromatic
parts
of the
bremsstrahlung spectrum. Table 1 summarizes all relevant data on the tagging magnet system, and the photon detectors. In order to obtain similar counting rates in all tagging
channels,
at the low-energy
part of the electron
distribution
several
neigh-
bouring counters have been combined into one channel. The midpoint energies and the energy width of the resulting 29 channels are given in table 2. The measuring routine consisted of 2 steps. First, the photon flux was measured with one of the four NaI spectrometers placed at 0” directly into the photon beam, with greatly reduced electron current. For each coincident event of the photon detector with one or more electron counters, the pulse height of the photon spectrometer, the time differences between the photon and electron signals, and the pattern to identify the detectors involved were recorded. In the off-line analysis, pulse-height spectra of photons in coincidence with each electron channel were generated which represent the NaI responses to photons with energy and energy width determined by the position and width of the given electron counter, or group of counters respectively. The tagging efficiency, Q, defined as the ratio of coincidences to singles electron rate, was determined for each tagging channel. The
310
K.P. Schelhaas et al. / Nuclear photon scattering
TABLE 1 Parameters of the tagging system and of the scattering geometry Electron energy, E, Bremsstrahlung target: material thickness Distance bremsstrahhmg target to quadrupole Quadrupole: effective length aperture field gradient Distance quad~pole-dipole Dipole: entrance rotation angle gap field strength Momentum dispersion Momentum acceptance Tagged photon energy, k
184 MeV aluminium 9iLm 50 mm 82 mm 20 mm -4.8 T/m 200 mm 11° 60 mm 1.3 T 2.0 MeV/c cm 40 to 170 MeV/c 14 to 144 MeV
Electron angle acceptance
>5E 7
Focal plane Scattering target: density thickness NaI solid angles 60”, 90”, 120” and 150° NaI dimensions 60” and 90” 120” and 150”
Cl curved, inside dipole field 2.05 g/cm3 50 mm 11.9, 18.6, 11.9 and 4.15 msr 10” 0 x lo” 10” 0 x 14”
k m,,c*
numerical values, between 0.8 and 0.9, agree very well with a ~l~ulation ‘O)based on the Bethe-Heitler formula I’). An example for a NaI response is shown in fig. 2. Second, the NaI spectrometers were placed in shielded boxes as shown in fig. 1, looking at the ‘*C scattering target. For this geometry, the same parameters as in step one were registered simultaneously for the 4 NaI spectrometers and the 29 tagging channels. A typical time-difference spectrum is given in fig. 3, that served TABLE 2 Midpoint energies k and energy widths Ak of the 29 tagging channels, in MeV
#
k
Ak
1 2 3 3 5 4 7 8 9 10
24.5 26.5 28.5 30.3 32.4 34.3 36.3 38.7 51.3 53.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.9 1.9
1
*
k
Ak
11 12 13 14 15 16 17 18 19 20
55.3 57.3 59.3 61.3 63.3 65.2 67.2 69.2 71.1 83.5
1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 2.0
1
#
k
Ak
21 22 23 24 25 26 27 28 29
86.5 90.5 94.6 98.6 102.6 117.5 122.7 129.2 135.7
3.8 3.8 3.7 3.8 4.0 3.8 5.5 5.4 3.4
K.P. Schelhaas et al. / Nuclear photon scattering
Channel
Number
(0.8
311
MeV/Channei)
Fig. 2. Response of one of the 10” 0 x lo” NaI spectrometers to photons of 98.6 MeV with 3.8 MeV energy spread. The full curve is a 4-parameter fit “) which was used to determine the number of scattered photons.
as basis
for the substraction
of accidental
counts,
as shown
in fig. 4. The true-
coincidence spectrum finally, fig. 5, displays as main features the peaks of elastic and inelastic scattering and a coincident electromagnetic background, extending from the elastic peak to smaller energies. From the number of registered scattered photons, N,, and the number of electron counts, N,, and combining the two steps of the measuring routine, one has the resulting
cross section
The number
of target atoms per cm’, NT, and the solid angles seen by the NaI crystals, AC!, are given in table 1. The shape and magnitude of the electromagnetic background in fig. 5 was calculated as described in sect. 2.2.2 of ref. ‘). A comparison with the measured pulse height distribution is presented in fig. 6. The measured points agree very well with the shape and absolute magnitude of the calculation. The origin of the background is well described by two step processes: (i) Wide-angle pair production with successive bremsstrahlung, and (ii) Compton electrons and pair particles in forward direction, followed by wide-angle bremsstrahlung. The background rate without scattering
312
K.P. Schelhaas et al. / Nuclear photon scattering 120
I
I
I
1 I
II -100
E=34.3MeV
1
8= 150’
1
20 80
90
100
110
TDC Channel
Numbers
120
130
(0.4
w/Channel)
I 150
140
Fig. 3. Time-difference spectrum of photon and electron pulses within a window of 25 ns. Two regions For the accidentals 7,, for the accidental rate, TV, and the (true+accidental) rate, r,+=, are indicated. the pulse height distribution of all 29 tagging channels are identical in shape. They were added, multiplied from the spectrum of TV+. Time runs from right by the ratio of 7t+a to the sum of all 7,) and substracted to left. Neutron pulses appear well separated, about 10 ns after the photon peak.
120
.. . .. . . ..I..
. . . . .._...................I....
100 80
‘;
0
,,tme + occidental
-
-
~_________________________________________________-_____________________________________~
20 0 l....
I.. . . . Channel
Number
(0.8
MeV/channel)
Fig. 4. Pulse-height distribution for 98.6 MeV tagged photons. Upper part: Pulses falling into time difference of the pulses in 7,+, and 7.. The coincident regions. 7t+a and 7, Lower part: Normalized background has been calculated and checked experimentally (see fig. 6). Above 70 MeV, the inelastic process leading to the first excited state of 12C cannot be separated from the elastic one.
K..P.
et ak 1 Nuclear
40 f
60
50 T;i;
30
9”
20
6
10
C
Sehelhaas
-..-,.-.‘r-‘.-,.‘.‘~.~‘....-.~~‘.‘~-~.~
-
v)
100 80
v
60
E = 24.5
+
B 0 a t; g
photon
E = 26.5
313
scattering
MeV; 8=150”
Meti B =90’
40 20 0
I....l....)....l....l....l.-..l..-.I.*..1 30
40
Channel
50
Number
(0.4
60
70
MeV/~hannei)
Fig. 5. Pulse-height distribution of true coincidences at 24.5 and 26.5 MeV tagging energy. The full line is a fit of the measured response of the NaI spectrometer to the elastic and inelastic scattering peak, separated by 4.4 MeV. Also shown dashed is the electromagnetic coincident background.
i0
Fig. 6. Comparison
of measured and calculated electromagnetic backgrounds photon energies 55 and 112 MeV, each for two angles.
for the monochromatic
314
K.P. Schelhaas et al. / Nuclear photon scattering 3
TABLE
The 3 cascaded
carbon
scattering
experiment; Scattering
angle
targets
density
for the bremsstrahlung
2.03 g/cm3
Angle to primary beam (deg)
(deg) 60 90,120 150
90 45 90
Thickness (mm) 3.75 7.03 16.3
target was negligible above 8 MeV. Neutron induced counts could be eliminated by a proper choice of the time window.
2.2. BREMSSTRAHLUNG
MEASUREMENT
The experimental setup, and also the measuring routine was exactly the same as described in ref. 5, for the photon scattering experiment by 208Pb.The total running time was roughly two days with the MAMI accelerator at 143 MeV electron energy, and 50 ~_LAaverage electron current. The three cascaded scattering targets were machined from high purity graphite with density and dimensions as given in table 3. The electromagnetic background was calculated by integrating the “monochromatic” spectra of fig. 6 over the bremsspectrum photon number distribution. The
60
Photon
Ener
Fig. 7. The electromagnetic background for the 4 angles of the bremsstrahlung experiment, in units of an effective scattering cross section. It is calculated absolutely and found in agreement with the measured spectra of fig. 6.
315
K.P. Schelhaas et al. / Nuclear photon scattering
40 cm polyethylene
absorber,
which
was inserted
in between
scattering
the detectors, is shown in fig. 2 of ref. ‘). It stops all secondary from the scattering target. Part of their energy is transformed photons,
that can partly reach the NaI detector.
Monte Carlo techniques.
The background
target
and
electrons originating into bremsstrahlung
These two steps were simulated
spectra,
calculated
absolutely
by
and checked
against the clearly separable “monochromatic” spectra of fig. 6, are given for the conditions of the simultaneous measurement at four angles in fig. 7. Due to the selection of different target thicknesses for the four scattering angles, and due to an optimal choice of scattering geometry and shielding, this background produced in the giant resonance region, even at 60” scattering angle, not more than 50% of the registered
counts. 3. Results
Fig. 8 provides an overview on the size of the cross sections contributing counting rates of the photon detectors. The largest counting rate, at photon around 0.5 MeV, is produced by atomic Compton effect with roughly Similar in intensity are neutron capture y-rays, and the following p- and up to 7 MeV in the NaI crystals. These two effects are fed by all photon in the primary spectrum, and in fact limit the photon absolute cross section experiment.
intensity
to the energies 1 mb/sr. y-decay energies
for this high-resolution
0 20
Photon Et%gy Fig. 8. Nuclear
resonance
fluorescence
50
40
(MeV)
from discrete (15.1 MeV, (right-hand scale).
left-hand
scale)
and
GDR
states
316
K.P. Schelhaas et al. / Nuclear photon scattering
The largest direct nuclear effect from the target is resonance fluorescence from the 15.1 MeV level of r*C, with an integrated differential cross section of about 100 p.b - MeV/sr. This resonance-fluorescence radiation is -20 times more intense than the giant resonance scattering. Taking into account the bremsstrahlung spectrum shape, the ratio of low-energy Compton plus neutron capture y-rays to scattered photons within 1 MeV at giant resonance energies is roughly 106. In fig. 9 all measured scattering cross sections are plotted, together with the zero-degree scattering cross section, calculated from the total absorption measurement. These data, together with the elastic and inelastic cross sections from the tagged photon measurement, are analyzed in the following 3 subsections. Anticipating the final result, the full lines in fig. 9 show the consistent picture obtained with the data set of table 4.
3.1. INELASTIC
SCAlTERING
Inelastic scattering to the first excited state of “C has been measured using the tagged-photon technique ‘). These data are supplemented by some new cross sections obtained with the tagged-photon setup of sect. 2.1. All these cross sections are
Fig. 9. An overview on the simultaneously measured bremsstrahlung results, together with the tagged photon results, and with the zero-degree scattering cross sections, calculated from the total absorption cross section.
31-i
K. P. Schelhaas et al. / Nuclear photon scattering
plotted
in fig. 10. With
distribution
the help
of the well known,
almost
13), data from other angles were transformed
the energy
dependence
from this rather
of these
small
inelastic
data set. Instead,
the data, will be used to determine
cross sections a theoretical
the ratio of inelastic
isotropic,
angular
to 8 = 135”. It is clear that cannot
be taken
description,
directly
consistent
to elastic cross sections
function of energy, with only a single parameter taken from experiment. Following Hayward 13), the scattering cross section for a spin-zero nucleus be written
with as can
with g,=b(l+cos2
Av(E)=~C
g,-&(13+cos2
e))
e)
+ ED&&,
(fll~ll~~(~ll~llO~~,~ k
0.6
I .
I
’
I
’
I.
I.
I.
I
’
!
$.o.s-
:
2 *
0.4 -
C
40
80
Photon
Er%gy
100
(MeV)
Fig. 10. A collection of all inelastic photon scattering cross sections feeding the 4.4 MeV, 2+ level of “C, in the photon range between 15 and 140 MeV, and transformed to 135” scattering angle; squares from ref. ‘), circles present work. The full curve is deduced from elastic cross sections excluding the quasideuteron states 13). Dashed is the elastic cross section multiplied by pinpI = 0.4. The difference between full and dashed lines is due to the interference of only the elastic process with the seagull term. The curve labelled QD includes quasideuteron states as intermediate states for inelastic scattering.
318
K.P. Schelhaas et al. / Nuclear photon scattering
with 1
1
Ek-E-firk+E,+E-l-$jrk The index k specifies all possible
intermediate
I .
states and v is the angular
momentum
transferred to the nucleus, which, in the case of a spin-zero nucleus, is also the spin of the final state. v = 0 describes elastic scattering, v = 2 the inelastic scattering to the 2+ levels, which is found to be the largest contribution 13,r4) to the inelastic scattering cross section. If we assume a nearly constant ratio pine, for the square of the inelastic to the elastic reduced matrix elements with v = 0 and v = 2 for all states k, making up the giant resonance, one can obtain the only unknown, pine,, from the measured inelastic cross sections of fig. 10, with the condition that the sum of elastic and inelastic cross sections fits the bremsstrahlung data. Before these steps will be followed, and be justified by comparison with the experiment, the term containing the Thomson amplitude D = -Z2e2/AMc2, which interferes only with the elastic cross section, has to be replaced 16) by the seagull amplitude S = -Ze’/ Mc2, which at E + 0 combines with the low-energy limit of FOk to give the correct Thomson amplitude. The inelastic amplitude vanishes at energies large compared to Ek since there is no seagull amplitude to interfere with. Therefore the inelastic cross sections ratios rapidly fall off from the value pine, in the giant resonance region where S is small in comparison, approaching zero at higher energies, as shown in fig. 10. This behavior demonstrates the more basic nature of the seagull term, compared to the Thomson amplitude, for a composite nucleus. The ratio 0.4 of the square of the reduced matrix elements for inelastic to elastic scattering was obtained from a fit by eye of the calculated curve to the experimental data in fig. 10. This value 0.4 has been used to calculate scattered photon spectra. Just below the giant dipole, around 18 MeV photon energy, there is a small energy region where inelastically scattered photons dominate the photon spectrum. Fig. 11 shows this part of the spectrum for 0 = 150”. The average measured photon spectrum compares well with the distribution expected for the inelastic component which was determined
from fig. 10. The assumption
giant dipole
is not fully consistent
of a constant
with the measurement.
Pine, over the peak of the However,
the trend is the
same as seen in fig. 10. For an exact determination of pi”,,(k) in the GDR region, more accurate data are required. For the final description of the bremsstrahlung scattering data, a second inelastic component, feeding the lowest T = 1,2+ state at was found 16.1 MeV was found necessary. The value Pin,=,=0.33 for this transition to fit the data best. At 16.1 MeV, a small peak was observed, shown in fig. 12, which indeed contains the right number of photons. However, since this level also decays through the 4.4 MeV level, and also by (Y-emission, the origin of this small peak cannot be attributed to the decay of this level, fed by inelastic scattering. Other open questions are the role of the exchange currents and the quasi-deuteron intermediate states in feeding inelastic channels. In the present analysis the
1%P. Schelhaaset nl. / Nuclear photon scattering
319
eiostic+ inelastic-.....
0 30
15
Fig. 11. The measured photon spectrum for elastic and inelastic processes, together with calculations for inelastic processes (full line) elastic processes (dashed line), and elastic-plus-inelastic processes, as measured. Inelastic processes can clearly be identified around 18 MeV, the same parameter pin._,= 0.4, as deduced from fig. 10, has been put into the calculation. In agreement with the data shown in fig. 10, an energy-dependent increasing p,,,,(k) in the GDR region seems to be indicated by the data which, however, should be improved in accuracy to obtain this energy dependence explicitly.
assumption is made that exchange contributions to the giant resonances do, and quasideuteron states do not feed inelastic channels. The influence of the quasideuteron states - assumed to follow also Pinel=0.4 - is shown in fig. 10. 3.2. THE
E2 GIANT
ISOVECTOR
RESONANCE
AND
COUPLED-CHANNEL
THEORY
It proved useful, in the case of “*Pb [ref. 5)], to look for El/E2 interferences I’) as indication for the position and strength of E2 resonances on top of the ubiquitous El strength. For the present experiment, the ratio da(150”)/d~(60”) is the most sensitive quantity. This ratio is shown in fig. 13. Candidates for a possible “E2 pattern” appear mainly at 32 MeV and possibly at 28 MeV. The remaining deviation from the pure-dipole value, 1.4, is due to the form factor multiplying the seagull term. Including an E2 resonance at 32 MeV describes the observed ratio indeed very well, as shown with the full line in fig. 13. However, the absorption cross section for this E2 line, when added to the continuum absorption cross section at 32 MeV clearly contradicts the total absorption measurement ‘). This absorption cross section is shown in fig. 14, labelled “E2 added”. The following consideration is an attempt to reconcile the two experimental results on scattering and absorption cross sections.
320
K.P. Schelhaas et al. / Nuclear photon scattering 80
13
14
Photon Fig. 12. The scattered-photon
pulse-height
15
Energy
16
17
(MeV)
distribution at the energy of the lowest T = 1,2+ state in “C, at 16.1 MeV.
Photon
Energy
(MeV)
Fig. 13. Cross-section ratio do(150”)/du(60”); the full line is a calculation with two E2 resonances with 0.33 and 0.75 isovector sums for the lines at 26 and 32.3 MeV respectively; dashed is the calculation with El only.
321
K.l? Schelhaas et al. / Nuclear photon scattering
. ’
121.
z
-
. I . . . . 1 ’
. ’
.l
E2 added
&Jo._S ‘;
a-
$ z: F 0
8-
s ‘Z f?
4.
:
2.
;
.
.:’
q = 0.5
$ 0
..*.‘*...‘*...
40
25 Pho&
Energy &)
at 32.2 MeV produces the E2 Fig. 14. Total absorption cross section’) for “C. The narrow resonance pattern of fig. 13. Its addition to the absorption cross section is shown as full line, labelled “E2 added”, that is clearly at variance with the experimental data. The meaning of the curve “q = 0.5” is explained in the text. It refers to one of the line profiles, shown in fig. 15.
The basic theory for a resonance, on top of and coupled to a continuum, developed by Fano lx) for the case of autoionizing states in atomic continua, by Weidenmiiller 19) in nuclear physics. The theory is reviewed in ref. *‘). The main
results
are reproduced
was and
here. The
situation is characterized by a narrow resonance cp at an excitation energy Ek which is above the particle emission threshold. The continuum, unperturbed by the resonance is given by I,!J~ and the transition matrix element to the continuum “just wave function. outside resonance” is ( lCIE) T) ?P,), with PO denoting the ground-state The coupling between by the matrix element
continuum
state and resonance
By virtue
cp is changed to @: m V,,&,(E - El)-’ @=cp+P I cl
state at energy E is described
of this interaction,
dE’ ,
which is the wave function
for the discrete state including an admixture of continuum The true wave function pE states due to its interaction V,, with the continuum. from the complete system can be expanded in terms of cp and lCIEin the form pa = acp + I where interactions with other ratio R of the true resonance
bEsGIEdE’ ,
continua or far off resonances transition to the unperturbed
are ignored. For the (just below or above
322
resonance)
K.P. Schelhaas et al. / Nuclear photon scattering
continuum
transition,
Fano obtained
R =W&WJ)2 (~EPlW2
=- (9+s)2 (I+e2)
’
with
The quantity
is the ratio of the cross section for an excitation of the modified discrete state to that for excitation into a band of continuum states of width lY The quantity q is called the profile index. It determines the shape of the absorption pattern, superimposed on the flat energy distribution given by the continuum alone. Line profiles for several values of q are reproduced 20) in fig. 15. The case of the E2 state at 32 MeV in i2C may be similar to the situation described above. There is a single particle continuum with a narrow E2 resonance superim-
_8
-6
-4
Relative
-2
0
energy
2 parameter
4
6
e
Fig. 15. Line profiles for a narrow resonance at E = 0, interacting with a continuum. q = 0 is the case of vanishing coupling of the resonance directly to the ground state.
323
K.P. Schelhaas et al. / Nuclear photon scattering
posed, which can be described
by a 2-particle
excitation.
The actual resonance
may
be characterized by a sizeable coupling V,, to the single particle continuum, leading to a width r of about 1 MeV, and a smaller coupling to the ground state of “C. The magnitude
of q therefore
can be rather
small,
and the line profile,
shown
in
fig. 15, resembles a dip in the absorption cross section, rather than a peak. Dip-like cross sections indeed are observed in atomic photoionization cross sections *I), in the far-ultraviolet energy region. The width of such a dip can be interpreted similar to the width of an absorption peak; it is given by V,, that describes the almost exclusive damping via the continuum state. A phase factor to a matrix element since the scattering amplitude contains
for elastic scattering products of matrix
cannot be observed elements with their
complex conjugates, as can be seen in the expressions with v = 0, reproduced sect. 3.1. The scattering angular distribution is the same for a small positive
in or
negative addition of an imaginary part to a larger scattering amplitude. This special case of elastic scattering has not been treated, to our knowledge, in the literature. The few remarks may be sufficient justification to try a representation of the data by choosing a suitable q-value. This is done in fig. 14 where it is shown by the curve labelled “q = 0.5”, that the theory indeed fits the data well. The continuum multiplying
states primarily are El in character. However, the quasideuteron part, and the retardation
the need for a formfactor factor applied to the El
resonances, emphasizes the presence of continuously distributed E2 (and higher multipole) strength, that, by application of these factors as in the present analysis, is taken into account implicitly. In this sense, one should take the results shown in fig. 14 as a further proof of the importance of finite nuclear size effects. It appears possible within the present cross section uncertainties that also part of the broad El resonance at 28 MeV consists actually of quasideuteron strength. Furthermore, and more importantly, the existence of this E2 continuum was directly proved by Hayward et al. “) in an earlier photon scattering experiment. In their analysis the quasideuteron cross sections were not given a form factor. Therefore, the E2 component appears explicitly with a constant strength of 1.4 mb at the photon energies between 26 and 36 MeV. This value is in excellent agreement with the 1.5 mb required in the dip region for the explanation of the effect of the E2 resonance
as dip in the total cross section.
In summary, the narrow E2 resonance, seen in the angular distribution at 32 MeV, essentially does not increase the total cross section. Instead, it cuts a window into the E2 continuum overlapping this energy region.
3.3. CONSISTENT
DESCRIPTION
OF PHOTON
SCATTERING
AND
PHOTON
ABSORPTION
The analysis of the elastic scattering data with the inclusion of the total absorption cross section was performed exactly the same way as described for the lead experiment, adding the inelastically scattered photon spectra to the elastic ones before
324
K.P. Schelhaas et al. / Nuclear photon scattering
comparing with experiment. The inelastic cross sections of sect. 3.1 and the E2 results of sect. 3.2 were used. Starting with the total absorption cross section, the measured cross sections 7*9)were parametrized with a small number of lorentzians, with
thresholds
energies
and
retardation
reflect the varying
with thresholds
factors.
contributions
The
different
values
for the threshold
from ( y, p), ( y, n) and (‘y, np) reactions,
at 16.0, 18.7 and 27.4 MeV respectively.
The multipolarities
were
assumed to be El and E2 for the nuclear part, and Ml for the delta-resonance. Form factors were applied to the seagull terms and to the lorentzian supposed to describe the quasideuteron region. The form factors have less influence on the cross sections than in the case of “*Pb, a consistent picture however is impossible without their inclusion. Examples of calculations with modified and without form factors are given at the end of this section, in fig. 20. For the proton distribution a 3-parameter Fermi distribution was used with parameters from literature 22). For the exchange-current form factor the result found for 208Pb [ref. ‘)I was included by reducing the half-density radius by a factor 0.6, compared to the proton form factor. The real parts of the different types of amplitudes having different angular distributions (El, E2, Ml and quasideuteron) were calculated by separate dispersion integrals up to 800 MeV. In a trial and error procedure, all parameters were varied until the best representation of the measured sum of elastic and inelastic photon spectra was found. The total cross section for this final result deviates only slightly from the measured total cross section ‘). Both cross sections are shown in fig. 16. The differences are within the systematic errors of the absorption measurement which allow a smoothly varying deviation of kO.2 mb. This difference is not the size and direction, found by Fuller 23), in his attempt to reconcile total absorption cross sections and the sum of particle-emission cross sections. The A-resonance region is shown in fig. 17, together with all available total hadronic cross sections. The 60” scattering data justify deducing a number for the sum of the static electric, (Y,and magnetic, p, polarizabilities of an average nucleon in “C. Using the dispersion integral, the quantity ((Y+p) was found to be (11.9* 0.7) x lop4 fm3 for the energy range from 50 to 140 MeV, shown in fig. 18. This value is slightly smaller than the experimental value of an averaged equal number of free neutrons and protons, but it is within the combined errors bars. The series of figs. 19-21 shows the detailed comparison of the experimental results with the data set which
gives the best overall consistency with scattering, absorption, and the con‘). Some calculated curves showing the clusions drawn from the “‘Pb measurement effects of form factors at 140” scattering angle are included. These figures are explained in the captions thereof. Table 4 summarizes all parametrized results. They are not results of a rigorous least-squares fit, and may therefore reflect to a small extent the way of approaching the trial and error procedure. The present values are very well consistent with former data 1*6), taken at different angles in the giant resonance region. However, at energies above 30 MeV and mainly at forward angles, an apparent discrepancy cannot be explained. In contrast to the procedure followed
325
K.P. Schelhaas et al. / Nuclear photon scattering 25
0
Photon Energy (MeV) Fig. 16. The measured inelastic bremsstrahlung
in this
total absorption cross sections ‘), and the curve representing scattering data. For the sake of clarity many experimental energies have been combined into a small number of points.
best the elastic and points at different
paper,
one would have to alter the total absorption cross section ‘) 1*2,24),and more than allowed by statistics and systematical errors, in considerably order to include these low-value cross sections above 30 MeV. This remaining discrepancy may be reconsidered in the light of the coupled-channel picture of sect. 3.2. Indeed, continuum
scattering
allowing would data.
a coupling
change
Using
of El resonances
the absorption
the results
with part of the underlying
cross section
which is required
of ref. ‘), it may be that the inclusion
El
to fit the of such a
coupled channel picture reduces the discrepancy seen in fig. 21. A final conclusion cannot be drawn, however, before the theory connecting absorption and scattering, which is valid in this special case, has been developed adequately, including also inelastic processes. In fig. 22 finally, the different parts of the photon absorption cross section are plotted, in order to show their relationship as function of energy.
4. Conclusion Elastic and inelastic photon scattering cross sections over a wide energy range and from 60” to 150” scattering angle, as well as total photon absorption cross sections, form the basis for a consistent description of photon scattering and photon absorption. Several different aspects of nuclear structure physics are involved, besides the general reliance on dispersion relations.
K.P. Schelhaas et al. / Nuclear photon scattering
326 6
I....,....,....,....,....,....
.
..I
‘*c
Y
,,Extrapolation
--._““____<
0
...*‘**..‘..**‘.... 100
of nuclear
section
r...‘tr-.-*-’ -__--_-___ - - - - _ ________.__ -_--._
----_---_---_____________
150
cross
,:’
200
250
300
Photon
350
energy
400
450
0 !501
(MeV)
Fig. 17. The same cross section as fig. 16 for the A-resonance region, that enters the dispersion integral for the real parts of the scattering amplitude; squares from ref. *), circles from ref. 9).
0
1.
I
.I
.I
.I
.I
.I
50
.I.
I
.
150
Photon
eAi?gy
(MeV)
Fig. 18. The 60” scattering cross section, together with the theoretical description, including different nucleonic cross sections. The labelled numbers are sums of average electric and magnetic polarizabilities of the 12 constituents of “C, given in units of 10m4 fm3, and calculated using the minus-second-moment sum rule.
0
20
18
22
24
26
26
30
16
20
Fig. 19. Elastic-plus-inelastic scattering cross giant dipole region. The curves are obtained consistency of scattering and absorption cross appear at the energies of
40
60
80
100
22
24
26
26
30
3: 2
Photon Energy (MeV]
Photon Energy (MeV]
sections from the bremsstrahlung measurement, in the with the data set which results from the best overall sections. The inelastic cross sections are shown as they the inelastically scattered photons.
120
Photon Energy (MeVj
40
60
80
100
120
140
Photon Energy (MeV]
Fig. 20. The same cross section as fig. 19 for the energy region above 30 MeV, that is simultaneously covered in the bremsstrahlung experiment. The dashed curve at 150’ is calculated with F,, = 1 and the dotted curve is calculated with an increased nuclear radius for F=(q), by 10%. The point given as an open square, at 120” scattering angle and at 140 MeV photon energy, is the lowest-energy point of ref. 26).
0
20
30
40
Photon
50
Energy
60
20
(MeVf
30
40
Photon
50
Energy
60
70
(MeV]
Fig. 21. Comparison of the experimental results for elastic scattering with the calculation using the same data set (table 4) as for figs. 16, 17, 19 and 20; open circles from ref. r), squares from ref. “), and closed circles are present resultsz4). The Illinois and NBS data, not taken at the same scattering angles, are transformed using the dipole, (1 + cos* 0), angular distribution.
20
40
Photon Fig. 22. The partition (right), contributions.
60
80
Energy
300 MN
120
Photon
Energy
(MeV}
of the total absorption cross section into the different nuclear (left), and nucleonic The El lorentzian, labelled #5 in table 4, is shown dashed in the left-hand part.
329
K.P. Schelhaas et al. / Nuclear photon scattering
TABLE 4 Data set for the overall
description
of the measured
cross sections
(a) Lorentzians
El
E2
1.5 0.6 1.5 1.3 22.0 100
18.7 18.7 18.7 18.7 17.0 27.4
main El cross section AE = 7 MeV; quasideuteron
26.0 32.3
0.5 1.3
18.0 18.0
0.33 isovector 0.75 isovector
0.142 0.029 0.129 0.057 1.082 0.547
1 2
0.008 0.027
limit
TRK-units (mb . MeV)
140 800 800 800
1.73 1.41 1.96 11.36
Type
see ref. *‘); deduced
0,
W+ 8rr+ 7Y rr”
(d) Form factor ref. 22)
parameters
Type F,=
F,
J=,X (e) Average
sums sums
integral
(MeV)
(c) rr-production,
Ml
22.0 22.9 23.6 25.8 28.0 100
1 2 3 4 5 6
Upper
El
&h
Es
(b) Total absorption
Remarks
l-
TRK
Type *
sum of electric
(mb)
Remarks
all cross sections nuclear only, without quasideuteron all nuclear; no mesonic all cross sections
from total cross sections Effective
8*9)
threshold
Remarks
(MeV)
0.13 0.177
m, m-+35
direct reaction
0.068 0.068 0.164
m, m-+35 mm+45
resonance reaction: W,,, = 1270 MeV r = 240 MeV
of a Fermi
c (fm) 2.35 1.50
and magnetic
3-parameter
2
charge-density
(fm)
0.52 0.52 nucleonic
distribution,
with w = -0.149,
see
Remark from electron
scattering
polarizability
TRK is the dipole sum unit: 60 NZ/A (mb . MeV); I& is the resonance energy (MeV); r is the resonance width (MeV); Ethr is the particle emission threshold (MeV). For the definition of the p-production parameters see ref. ‘s).
330
K.P. Schelhaas
et al. / Nuclear photon scattering
The total cross section, found consistent with the present scattering results, and integrated to meson threshold, resulted in 1.73 classical sums, which is in good agreement whelming found guided
with the direct
measurement
part is El in character.
at 32.3 MeV, where the “dip” by a coupled
continuum integrated
interaction. absorption
channels
‘) of 1.67 classical
An E2 strength
in the absorption
consideration,
This E2 strength cross section.
is taken
comprises
TRK sums. The over-
of 0.75 isovector cross section as evidence
E2 sums was at this energy,
for a resonance-
not more than 1.6% of the total
Form factors have been found necessary as in the case of “‘Pb, and were scaled according to the r.m.s. radii. A better determination of the exchange form factor, compared to the lead case, requires better data around 120 MeV at backward angles. Similarly, an improvement of the value for the sum of the electric and magnetic static polarizabilities requires better data at similar energies at 60” or smaller angles. Inelastic processes have been identified that are consistent with transitions to the first excited state at 4.4 MeV and, possibly, to the lowest T = 1,2+ state at 16.1 MeV. The ratio Pinei of the square of the reduced matrix elements for inelastic to elastic scattering was found to be 0.4 and roughly constant over the giant resonance region (20-40 MeV). The role of the quasideuteron intermediate states, or more generally the role of exchange currents, remains open for future investigation, as well as the energy dependence of Pine, (E) in the giant resonance region. In view of these results on a number of nuclear structure related quantities, and not the least in view of the modest beam time required on a modern C.W. electron accelerator (2 days), the type of experiment, described here, appears very promising for future investigations.
We thank
the crew of the MAMI
accelerator,
especially
K.H. Kaiser,
for their
excellent job; K.H. Musshoff deserves special thanks for his untiring skill setting Discussions with E. Hayward, H.A. up the most complicated experiments. Weidenmiiller and J. Ahrens are gratefully acknowledged.
References 1) D.H. Wright, P.T. Debevec, L.J. Morford and A.M. Nathan, Phys. Rev. C32 (1985) 1174 2) D.H. Wright, Thesis, University of Illinois at Urbana Champaign, 1983 3) B. Ziegler, New vistas in electro-nuclear physics, ed. E.L. Tomusiak, H.S. Caplan and E.T. Dressier (Plenum, New York, 1986) p. 293 4) H. Arenhiivel, New vistas in electro-nuclear physics, ed. E.L. Tomusiak, H.S. Caplan and E.T. Dressier (Plenum, New York, 1986) p. 251 5) K.P. Schelhaas, J.M. Henneberg, M. Sanzone-Arenhiivel, N. Wieloch-Laufenberg, U. Zurmiihl, B. Ziegler, M. Schumacher and F. Wolf, Nucl. Phys. A489 (1988) 189 6) E. Hayward and R.G. Leicht, Ann, AC. Sci. 40 (1980) 99; W.R. Dodge, E. Hayward, R.G. Leicht, M. McCord and R. Starr, Phys. Rev. C28 (1983) 8 7) J. Ahrens, H. Borchert, K.H. Crock, H.B. Eppler, H. Gimm, H. Gundrum, M. Kroning, P. Riehn, G. Sita Ram, A. Ziegler, Nucl. Phys. A251 (1975) 479
K.P. Schelhaas et al. / Nuclear photon scattering 8) J. Arends, 9)
10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)
J. Eyrink,
A. Hegerath,
K.G. Hilger,
B. Mecking,
G. Niildeke
331 and H. Rost, Phys. Lett.
B98 (1981) 423 G. Audit, N. de Botton, J.L. Fame, M.L. Ghedira, J. Martin, E. Mazzucato, C. Schuhl, G. Tamas, E. Vincenz and P. Argan, Proc. XI Europhysics Divisional Conf. on nuclear physics with electromagnetic probes, ed. G. Thomas, Paris, 1985 (North-Holland, Amsterdam, 1985) p. 196 K.H. Krause, Thesis, University of Mainz, 1989 H.W. Koch and J.W. Motz, Rev. Mod. Phys. 31 (1959) 920 C. Wesselborg, University of Koln, private communication E. Hayward, National Bureau of Standards Monograph 118, NBS Washington, 1970 R. Moreh, Proc. Int. School of intermediate energy nuclear physics, ed. R. Berg&e, S. Costa and C. Schaerf (World Scientific Singapore, 1982), p. 1 A.M. Nathan, Phys. Rev. C38 (1988) 92 R. Silbar, C. Werntz and H. Uberall, Nucl. Phys. A107 (1968) 655; R. Silber and H. Uberall, Nucl. Phys. A109 (1968) 146 B. Ziegler, Proc. intermediate energy nuclear physics with monochromatic and polarized photons, ed. G. Matone and S. Stipcich, Frascati, 1980 U. Fano, Nuovo Cim. 12 (1935) 156; Phys. Rev. 124 (1961) 1866 H.A. Weidenmiiller, Nucl. Phys. 75 (1966) 189 H.S.W. Massey, E.H.S. Burhop and H.B. Gilbody, Electronic and ionic impact phenomena, vol. 1 (Oxford Clarendon Press 1969), p. 594 J.A.R. Samson, Phys. Rev. 132 (1963) 2122 C.W. de Jager, H. de Vries and C. de Vries, At. Data Nucl. Data Tables 14 (1974) 479 E.G. Fuller, Phys. Reports 127 (1985) 185 N. Wieloch-Laufenberg, Thesis, University of Mainz, 1988 B. Ziegler, Lecture Notes in Physics 108, ed. H. Arenhiivel and D. Drechsel (Springer, Berlin 1979) p. 148 E. Hayward and B. Ziegler, Nucl. Phys. A414 (1984) 333
A506 (1990) 307-331
NUCLEAR
K.P. SCHELHAAS’,
Max-Planck-lnstitut
PHOTON
J.M. HENNEBERG,
SCATTERING
BY ‘*C*
N. WIELOCH-LAUFENBERG*, B. ZIEGLER
fGr Chemie, Kernphysikalische M. SCHUMACHER
II. ~~.~~~~~s~~~s ~nst~tui der ~niuersit~t
Arbeitsgruppe,
U. ZURM~HL3
and
D 6500 Mainz, Fed. Rep. Germany
and F. WOLF4
~~tt~n~en, D 3400 Giittingen,
Fed. Rep. Germany
Received 26 June 1989 (Revised 4 September 1989) Abstract: Cross sections
for photon scattering by the “C nucleus have been measured in the energy range from 15 to 140 MeV. Quasimono~hromatic tagged photons allowed the separation of elastic and inelastic processes. An additional measurement with continuous bremsstrahlung improved the accuracy of the elastic part of the cross section considerably. In analyzing the bremsstrahlung runs, the inelastic components were taken from the tagged photon results and from literature. The cross sections were analyzed in terms of giant resonances, proton-neutron substructures and subnuclear excitations. Form factors for charge and exchange current were applied to take into account the finite nucleon and exchange-current distribution in *‘C. For the E2 resonance, a coupled-channel model was used, which is applicable to an isolated resonance supe~mposed on a continuum. The average sum of the static electric and magnetic polarizabilities of bound nucleons was found to be in agreement with the value of the free proton.
E
NUCLEAR REACTlONS deduced El, E2 resonance
‘*C( -y, y) (y, y’), E = 15-140 MeV; measured o(E, @). “C parameters, radius parameters, polarizability of bound nucleons.
1. Introduction In previous papers l-6) elastic photon scattering was treated by theory and experiment, and nuclear structure quantities were deduced. In order to attain an as complete picture as possible, it is essential to recognize the need of cross sections in a wide energy range and for all angles. There are regions that are specifically sensitive to one or the other measurable physical quantity, with rather small influence by others, as shown for the form factors in the case of *08Pb [ref. “)I. The conclusions drawn Pb can also be applied to ‘*C, for which the only main difference is, that the for *OS * This work was supported by the Deutsche Forschungsgemeinschaft, ’ Present address: Lurgi AG, 6000 Frankfurt/M., Fed. Rep. Germany. ’ Present 3 Present 4 Present
address: address: address:
037%9474/9O/SO3.50 (Noah-Holland)
Gei GmbH, Elisabethenstr. 44 i/2, 6100 Darmstadt, Bosch AG, 3200 Hildesheim, Fed. Rep. Germany. Siemens AG, 8000 Miinchen 83, Fed. Rep. Germany. @ Elsevier
Science
Publishers
B.V.
SFB 201. Fed. Rep. Germany.
K.P. Schelhaas et al. / Nuclear photon scattering
308
r.m.s. radius
is 2.2 times smaller.
60” are increasingly zero-degree dispersion
obscured
scattering relations
cross
section
and the optical
were taken from literature
Since the forward-going
by electromagnetic
was calculated theorem.
scattering
shower processes from total
cross
These total absorption
7-9). The final data set though,
should
angles
below
in the target, the
present
sections
via
cross sections a consistent
picture of scattering and absorption of photons below meson threshold, in which the real part of the scattering amplitude is partly given by the r-meson production cross section via the dispersion integral. The scattering cross sections were obtained for 60”, 90”, 120”, and 150”, using both tagged photons and bremsstrahlung. The energy range between 15 and 140 MeV is limited by electromagnetic background at the low part and by the onset of 7r” production at the high part. For the end energy of 143 MeV chosen for the bremsstrahlung measurement, it was checked experimentally that the 7~’ decay photon contribution, kinematically allowed between 48 MeV and 95 MeV, did not contribute in amplitude more than 5% of the scattered photons. The analysis follows the same steps as in the case of *“Pb in ref. 5). In the following sections, only the differences of the experimental setups and of the analyses are described in detail. These differences comprise mainly three points: (i) The source of quasimonochromatic photons was changed from positron annihilation-in-flight to tagged photons. The new experimental setup is described in sect. 2.1. (ii) In the analysis of the bremsstrahlung data, a contribution from inelastic scattering has to be taken into account 1*276).A short abstract of the related theory, and the experimental results are given in sect. 3.1. (iii) The two existing previous measurements did not agree on the E2 strength, found distributed around now comprise sufficiently
32 MeV [ref. ‘j)], or above 50 MeV [ref. ‘,‘)I. The results large regions in energy and angle, and allow a more
definite answer to the E2 question. The data analysis is based on a special case of coupled channels theory which is described in very basic terms in sect. 3.2. In sect. 3.3, the final results are given, in an attempt to combine all published and new cross sections
in a consistent
picture.
2. Experiment 2.1. TAGGED
PHOTONS
The general outline of the scattering setup is given in fig. 1. Electrons from the electron accelerator MAMI A with 143 MeV energy and 50 KeV energy spread hit the bremsstrahlung target in a small spot of about 1 mm diameter. The long-time stability in angle and position was better than 0.2 mrad and 0.5 mm, respectively. The electrons were bent downwards and backwards by an angle of 150” inside the gap of the main deflecting magnet. 44 plastic scintillators were placed in the focal plane inside the magnetic field. They define the simultaneously measured photon
309
K.P. Schelhaas et al. / Nuclear photon scattering
Lead
H
Nickel
/ LO.O1 mm Al -Tarset
\
184 MeV electrons Precision
collimators
lm Fig. 1. Top view of the tagging setup. The electron beam from the MAMI-accelerator enters from the left and is bent down by the tagging dipole. 44 electron detectors are placed along the focal plane in the magnet gap. The collimated photon beam hits the scattering target which is viewed by four wellshielded Nal detectors.
energies,
and the energy
widths
of the tagged
quasimonochromatic
parts
of the
bremsstrahlung spectrum. Table 1 summarizes all relevant data on the tagging magnet system, and the photon detectors. In order to obtain similar counting rates in all tagging
channels,
at the low-energy
part of the electron
distribution
several
neigh-
bouring counters have been combined into one channel. The midpoint energies and the energy width of the resulting 29 channels are given in table 2. The measuring routine consisted of 2 steps. First, the photon flux was measured with one of the four NaI spectrometers placed at 0” directly into the photon beam, with greatly reduced electron current. For each coincident event of the photon detector with one or more electron counters, the pulse height of the photon spectrometer, the time differences between the photon and electron signals, and the pattern to identify the detectors involved were recorded. In the off-line analysis, pulse-height spectra of photons in coincidence with each electron channel were generated which represent the NaI responses to photons with energy and energy width determined by the position and width of the given electron counter, or group of counters respectively. The tagging efficiency, Q, defined as the ratio of coincidences to singles electron rate, was determined for each tagging channel. The
310
K.P. Schelhaas et al. / Nuclear photon scattering
TABLE 1 Parameters of the tagging system and of the scattering geometry Electron energy, E, Bremsstrahlung target: material thickness Distance bremsstrahhmg target to quadrupole Quadrupole: effective length aperture field gradient Distance quad~pole-dipole Dipole: entrance rotation angle gap field strength Momentum dispersion Momentum acceptance Tagged photon energy, k
184 MeV aluminium 9iLm 50 mm 82 mm 20 mm -4.8 T/m 200 mm 11° 60 mm 1.3 T 2.0 MeV/c cm 40 to 170 MeV/c 14 to 144 MeV
Electron angle acceptance
>5E 7
Focal plane Scattering target: density thickness NaI solid angles 60”, 90”, 120” and 150° NaI dimensions 60” and 90” 120” and 150”
Cl curved, inside dipole field 2.05 g/cm3 50 mm 11.9, 18.6, 11.9 and 4.15 msr 10” 0 x lo” 10” 0 x 14”
k m,,c*
numerical values, between 0.8 and 0.9, agree very well with a ~l~ulation ‘O)based on the Bethe-Heitler formula I’). An example for a NaI response is shown in fig. 2. Second, the NaI spectrometers were placed in shielded boxes as shown in fig. 1, looking at the ‘*C scattering target. For this geometry, the same parameters as in step one were registered simultaneously for the 4 NaI spectrometers and the 29 tagging channels. A typical time-difference spectrum is given in fig. 3, that served TABLE 2 Midpoint energies k and energy widths Ak of the 29 tagging channels, in MeV
#
k
Ak
1 2 3 3 5 4 7 8 9 10
24.5 26.5 28.5 30.3 32.4 34.3 36.3 38.7 51.3 53.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.9 1.9
1
*
k
Ak
11 12 13 14 15 16 17 18 19 20
55.3 57.3 59.3 61.3 63.3 65.2 67.2 69.2 71.1 83.5
1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 2.0
1
#
k
Ak
21 22 23 24 25 26 27 28 29
86.5 90.5 94.6 98.6 102.6 117.5 122.7 129.2 135.7
3.8 3.8 3.7 3.8 4.0 3.8 5.5 5.4 3.4
K.P. Schelhaas et al. / Nuclear photon scattering
Channel
Number
(0.8
311
MeV/Channei)
Fig. 2. Response of one of the 10” 0 x lo” NaI spectrometers to photons of 98.6 MeV with 3.8 MeV energy spread. The full curve is a 4-parameter fit “) which was used to determine the number of scattered photons.
as basis
for the substraction
of accidental
counts,
as shown
in fig. 4. The true-
coincidence spectrum finally, fig. 5, displays as main features the peaks of elastic and inelastic scattering and a coincident electromagnetic background, extending from the elastic peak to smaller energies. From the number of registered scattered photons, N,, and the number of electron counts, N,, and combining the two steps of the measuring routine, one has the resulting
cross section
The number
of target atoms per cm’, NT, and the solid angles seen by the NaI crystals, AC!, are given in table 1. The shape and magnitude of the electromagnetic background in fig. 5 was calculated as described in sect. 2.2.2 of ref. ‘). A comparison with the measured pulse height distribution is presented in fig. 6. The measured points agree very well with the shape and absolute magnitude of the calculation. The origin of the background is well described by two step processes: (i) Wide-angle pair production with successive bremsstrahlung, and (ii) Compton electrons and pair particles in forward direction, followed by wide-angle bremsstrahlung. The background rate without scattering
312
K.P. Schelhaas et al. / Nuclear photon scattering 120
I
I
I
1 I
II -100
E=34.3MeV
1
8= 150’
1
20 80
90
100
110
TDC Channel
Numbers
120
130
(0.4
w/Channel)
I 150
140
Fig. 3. Time-difference spectrum of photon and electron pulses within a window of 25 ns. Two regions For the accidentals 7,, for the accidental rate, TV, and the (true+accidental) rate, r,+=, are indicated. the pulse height distribution of all 29 tagging channels are identical in shape. They were added, multiplied from the spectrum of TV+. Time runs from right by the ratio of 7t+a to the sum of all 7,) and substracted to left. Neutron pulses appear well separated, about 10 ns after the photon peak.
120
.. . .. . . ..I..
. . . . .._...................I....
100 80
‘;
0
,,tme + occidental
-
-
~_________________________________________________-_____________________________________~
20 0 l....
I.. . . . Channel
Number
(0.8
MeV/channel)
Fig. 4. Pulse-height distribution for 98.6 MeV tagged photons. Upper part: Pulses falling into time difference of the pulses in 7,+, and 7.. The coincident regions. 7t+a and 7, Lower part: Normalized background has been calculated and checked experimentally (see fig. 6). Above 70 MeV, the inelastic process leading to the first excited state of 12C cannot be separated from the elastic one.
K..P.
et ak 1 Nuclear
40 f
60
50 T;i;
30
9”
20
6
10
C
Sehelhaas
-..-,.-.‘r-‘.-,.‘.‘~.~‘....-.~~‘.‘~-~.~
-
v)
100 80
v
60
E = 24.5
+
B 0 a t; g
photon
E = 26.5
313
scattering
MeV; 8=150”
Meti B =90’
40 20 0
I....l....)....l....l....l.-..l..-.I.*..1 30
40
Channel
50
Number
(0.4
60
70
MeV/~hannei)
Fig. 5. Pulse-height distribution of true coincidences at 24.5 and 26.5 MeV tagging energy. The full line is a fit of the measured response of the NaI spectrometer to the elastic and inelastic scattering peak, separated by 4.4 MeV. Also shown dashed is the electromagnetic coincident background.
i0
Fig. 6. Comparison
of measured and calculated electromagnetic backgrounds photon energies 55 and 112 MeV, each for two angles.
for the monochromatic
314
K.P. Schelhaas et al. / Nuclear photon scattering 3
TABLE
The 3 cascaded
carbon
scattering
experiment; Scattering
angle
targets
density
for the bremsstrahlung
2.03 g/cm3
Angle to primary beam (deg)
(deg) 60 90,120 150
90 45 90
Thickness (mm) 3.75 7.03 16.3
target was negligible above 8 MeV. Neutron induced counts could be eliminated by a proper choice of the time window.
2.2. BREMSSTRAHLUNG
MEASUREMENT
The experimental setup, and also the measuring routine was exactly the same as described in ref. 5, for the photon scattering experiment by 208Pb.The total running time was roughly two days with the MAMI accelerator at 143 MeV electron energy, and 50 ~_LAaverage electron current. The three cascaded scattering targets were machined from high purity graphite with density and dimensions as given in table 3. The electromagnetic background was calculated by integrating the “monochromatic” spectra of fig. 6 over the bremsspectrum photon number distribution. The
60
Photon
Ener
Fig. 7. The electromagnetic background for the 4 angles of the bremsstrahlung experiment, in units of an effective scattering cross section. It is calculated absolutely and found in agreement with the measured spectra of fig. 6.
315
K.P. Schelhaas et al. / Nuclear photon scattering
40 cm polyethylene
absorber,
which
was inserted
in between
scattering
the detectors, is shown in fig. 2 of ref. ‘). It stops all secondary from the scattering target. Part of their energy is transformed photons,
that can partly reach the NaI detector.
Monte Carlo techniques.
The background
target
and
electrons originating into bremsstrahlung
These two steps were simulated
spectra,
calculated
absolutely
by
and checked
against the clearly separable “monochromatic” spectra of fig. 6, are given for the conditions of the simultaneous measurement at four angles in fig. 7. Due to the selection of different target thicknesses for the four scattering angles, and due to an optimal choice of scattering geometry and shielding, this background produced in the giant resonance region, even at 60” scattering angle, not more than 50% of the registered
counts. 3. Results
Fig. 8 provides an overview on the size of the cross sections contributing counting rates of the photon detectors. The largest counting rate, at photon around 0.5 MeV, is produced by atomic Compton effect with roughly Similar in intensity are neutron capture y-rays, and the following p- and up to 7 MeV in the NaI crystals. These two effects are fed by all photon in the primary spectrum, and in fact limit the photon absolute cross section experiment.
intensity
to the energies 1 mb/sr. y-decay energies
for this high-resolution
0 20
Photon Et%gy Fig. 8. Nuclear
resonance
fluorescence
50
40
(MeV)
from discrete (15.1 MeV, (right-hand scale).
left-hand
scale)
and
GDR
states
316
K.P. Schelhaas et al. / Nuclear photon scattering
The largest direct nuclear effect from the target is resonance fluorescence from the 15.1 MeV level of r*C, with an integrated differential cross section of about 100 p.b - MeV/sr. This resonance-fluorescence radiation is -20 times more intense than the giant resonance scattering. Taking into account the bremsstrahlung spectrum shape, the ratio of low-energy Compton plus neutron capture y-rays to scattered photons within 1 MeV at giant resonance energies is roughly 106. In fig. 9 all measured scattering cross sections are plotted, together with the zero-degree scattering cross section, calculated from the total absorption measurement. These data, together with the elastic and inelastic cross sections from the tagged photon measurement, are analyzed in the following 3 subsections. Anticipating the final result, the full lines in fig. 9 show the consistent picture obtained with the data set of table 4.
3.1. INELASTIC
SCAlTERING
Inelastic scattering to the first excited state of “C has been measured using the tagged-photon technique ‘). These data are supplemented by some new cross sections obtained with the tagged-photon setup of sect. 2.1. All these cross sections are
Fig. 9. An overview on the simultaneously measured bremsstrahlung results, together with the tagged photon results, and with the zero-degree scattering cross sections, calculated from the total absorption cross section.
31-i
K. P. Schelhaas et al. / Nuclear photon scattering
plotted
in fig. 10. With
distribution
the help
of the well known,
almost
13), data from other angles were transformed
the energy
dependence
from this rather
of these
small
inelastic
data set. Instead,
the data, will be used to determine
cross sections a theoretical
the ratio of inelastic
isotropic,
angular
to 8 = 135”. It is clear that cannot
be taken
description,
directly
consistent
to elastic cross sections
function of energy, with only a single parameter taken from experiment. Following Hayward 13), the scattering cross section for a spin-zero nucleus be written
with as can
with g,=b(l+cos2
Av(E)=~C
g,-&(13+cos2
e))
e)
+ ED&&,
(fll~ll~~(~ll~llO~~,~ k
0.6
I .
I
’
I
’
I.
I.
I.
I
’
!
$.o.s-
:
2 *
0.4 -
C
40
80
Photon
Er%gy
100
(MeV)
Fig. 10. A collection of all inelastic photon scattering cross sections feeding the 4.4 MeV, 2+ level of “C, in the photon range between 15 and 140 MeV, and transformed to 135” scattering angle; squares from ref. ‘), circles present work. The full curve is deduced from elastic cross sections excluding the quasideuteron states 13). Dashed is the elastic cross section multiplied by pinpI = 0.4. The difference between full and dashed lines is due to the interference of only the elastic process with the seagull term. The curve labelled QD includes quasideuteron states as intermediate states for inelastic scattering.
318
K.P. Schelhaas et al. / Nuclear photon scattering
with 1
1
Ek-E-firk+E,+E-l-$jrk The index k specifies all possible
intermediate
I .
states and v is the angular
momentum
transferred to the nucleus, which, in the case of a spin-zero nucleus, is also the spin of the final state. v = 0 describes elastic scattering, v = 2 the inelastic scattering to the 2+ levels, which is found to be the largest contribution 13,r4) to the inelastic scattering cross section. If we assume a nearly constant ratio pine, for the square of the inelastic to the elastic reduced matrix elements with v = 0 and v = 2 for all states k, making up the giant resonance, one can obtain the only unknown, pine,, from the measured inelastic cross sections of fig. 10, with the condition that the sum of elastic and inelastic cross sections fits the bremsstrahlung data. Before these steps will be followed, and be justified by comparison with the experiment, the term containing the Thomson amplitude D = -Z2e2/AMc2, which interferes only with the elastic cross section, has to be replaced 16) by the seagull amplitude S = -Ze’/ Mc2, which at E + 0 combines with the low-energy limit of FOk to give the correct Thomson amplitude. The inelastic amplitude vanishes at energies large compared to Ek since there is no seagull amplitude to interfere with. Therefore the inelastic cross sections ratios rapidly fall off from the value pine, in the giant resonance region where S is small in comparison, approaching zero at higher energies, as shown in fig. 10. This behavior demonstrates the more basic nature of the seagull term, compared to the Thomson amplitude, for a composite nucleus. The ratio 0.4 of the square of the reduced matrix elements for inelastic to elastic scattering was obtained from a fit by eye of the calculated curve to the experimental data in fig. 10. This value 0.4 has been used to calculate scattered photon spectra. Just below the giant dipole, around 18 MeV photon energy, there is a small energy region where inelastically scattered photons dominate the photon spectrum. Fig. 11 shows this part of the spectrum for 0 = 150”. The average measured photon spectrum compares well with the distribution expected for the inelastic component which was determined
from fig. 10. The assumption
giant dipole
is not fully consistent
of a constant
with the measurement.
Pine, over the peak of the However,
the trend is the
same as seen in fig. 10. For an exact determination of pi”,,(k) in the GDR region, more accurate data are required. For the final description of the bremsstrahlung scattering data, a second inelastic component, feeding the lowest T = 1,2+ state at was found 16.1 MeV was found necessary. The value Pin,=,=0.33 for this transition to fit the data best. At 16.1 MeV, a small peak was observed, shown in fig. 12, which indeed contains the right number of photons. However, since this level also decays through the 4.4 MeV level, and also by (Y-emission, the origin of this small peak cannot be attributed to the decay of this level, fed by inelastic scattering. Other open questions are the role of the exchange currents and the quasi-deuteron intermediate states in feeding inelastic channels. In the present analysis the
1%P. Schelhaaset nl. / Nuclear photon scattering
319
eiostic+ inelastic-.....
0 30
15
Fig. 11. The measured photon spectrum for elastic and inelastic processes, together with calculations for inelastic processes (full line) elastic processes (dashed line), and elastic-plus-inelastic processes, as measured. Inelastic processes can clearly be identified around 18 MeV, the same parameter pin._,= 0.4, as deduced from fig. 10, has been put into the calculation. In agreement with the data shown in fig. 10, an energy-dependent increasing p,,,,(k) in the GDR region seems to be indicated by the data which, however, should be improved in accuracy to obtain this energy dependence explicitly.
assumption is made that exchange contributions to the giant resonances do, and quasideuteron states do not feed inelastic channels. The influence of the quasideuteron states - assumed to follow also Pinel=0.4 - is shown in fig. 10. 3.2. THE
E2 GIANT
ISOVECTOR
RESONANCE
AND
COUPLED-CHANNEL
THEORY
It proved useful, in the case of “*Pb [ref. 5)], to look for El/E2 interferences I’) as indication for the position and strength of E2 resonances on top of the ubiquitous El strength. For the present experiment, the ratio da(150”)/d~(60”) is the most sensitive quantity. This ratio is shown in fig. 13. Candidates for a possible “E2 pattern” appear mainly at 32 MeV and possibly at 28 MeV. The remaining deviation from the pure-dipole value, 1.4, is due to the form factor multiplying the seagull term. Including an E2 resonance at 32 MeV describes the observed ratio indeed very well, as shown with the full line in fig. 13. However, the absorption cross section for this E2 line, when added to the continuum absorption cross section at 32 MeV clearly contradicts the total absorption measurement ‘). This absorption cross section is shown in fig. 14, labelled “E2 added”. The following consideration is an attempt to reconcile the two experimental results on scattering and absorption cross sections.
320
K.P. Schelhaas et al. / Nuclear photon scattering 80
13
14
Photon Fig. 12. The scattered-photon
pulse-height
15
Energy
16
17
(MeV)
distribution at the energy of the lowest T = 1,2+ state in “C, at 16.1 MeV.
Photon
Energy
(MeV)
Fig. 13. Cross-section ratio do(150”)/du(60”); the full line is a calculation with two E2 resonances with 0.33 and 0.75 isovector sums for the lines at 26 and 32.3 MeV respectively; dashed is the calculation with El only.
321
K.l? Schelhaas et al. / Nuclear photon scattering
. ’
121.
z
-
. I . . . . 1 ’
. ’
.l
E2 added
&Jo._S ‘;
a-
$ z: F 0
8-
s ‘Z f?
4.
:
2.
;
.
.:’
q = 0.5
$ 0
..*.‘*...‘*...
40
25 Pho&
Energy &)
at 32.2 MeV produces the E2 Fig. 14. Total absorption cross section’) for “C. The narrow resonance pattern of fig. 13. Its addition to the absorption cross section is shown as full line, labelled “E2 added”, that is clearly at variance with the experimental data. The meaning of the curve “q = 0.5” is explained in the text. It refers to one of the line profiles, shown in fig. 15.
The basic theory for a resonance, on top of and coupled to a continuum, developed by Fano lx) for the case of autoionizing states in atomic continua, by Weidenmiiller 19) in nuclear physics. The theory is reviewed in ref. *‘). The main
results
are reproduced
was and
here. The
situation is characterized by a narrow resonance cp at an excitation energy Ek which is above the particle emission threshold. The continuum, unperturbed by the resonance is given by I,!J~ and the transition matrix element to the continuum “just wave function. outside resonance” is ( lCIE) T) ?P,), with PO denoting the ground-state The coupling between by the matrix element
continuum
state and resonance
By virtue
cp is changed to @: m V,,&,(E - El)-’ @=cp+P I cl
state at energy E is described
of this interaction,
dE’ ,
which is the wave function
for the discrete state including an admixture of continuum The true wave function pE states due to its interaction V,, with the continuum. from the complete system can be expanded in terms of cp and lCIEin the form pa = acp + I where interactions with other ratio R of the true resonance
bEsGIEdE’ ,
continua or far off resonances transition to the unperturbed
are ignored. For the (just below or above
322
resonance)
K.P. Schelhaas et al. / Nuclear photon scattering
continuum
transition,
Fano obtained
R =W&WJ)2 (~EPlW2
=- (9+s)2 (I+e2)
’
with
The quantity
is the ratio of the cross section for an excitation of the modified discrete state to that for excitation into a band of continuum states of width lY The quantity q is called the profile index. It determines the shape of the absorption pattern, superimposed on the flat energy distribution given by the continuum alone. Line profiles for several values of q are reproduced 20) in fig. 15. The case of the E2 state at 32 MeV in i2C may be similar to the situation described above. There is a single particle continuum with a narrow E2 resonance superim-
_8
-6
-4
Relative
-2
0
energy
2 parameter
4
6
e
Fig. 15. Line profiles for a narrow resonance at E = 0, interacting with a continuum. q = 0 is the case of vanishing coupling of the resonance directly to the ground state.
323
K.P. Schelhaas et al. / Nuclear photon scattering
posed, which can be described
by a 2-particle
excitation.
The actual resonance
may
be characterized by a sizeable coupling V,, to the single particle continuum, leading to a width r of about 1 MeV, and a smaller coupling to the ground state of “C. The magnitude
of q therefore
can be rather
small,
and the line profile,
shown
in
fig. 15, resembles a dip in the absorption cross section, rather than a peak. Dip-like cross sections indeed are observed in atomic photoionization cross sections *I), in the far-ultraviolet energy region. The width of such a dip can be interpreted similar to the width of an absorption peak; it is given by V,, that describes the almost exclusive damping via the continuum state. A phase factor to a matrix element since the scattering amplitude contains
for elastic scattering products of matrix
cannot be observed elements with their
complex conjugates, as can be seen in the expressions with v = 0, reproduced sect. 3.1. The scattering angular distribution is the same for a small positive
in or
negative addition of an imaginary part to a larger scattering amplitude. This special case of elastic scattering has not been treated, to our knowledge, in the literature. The few remarks may be sufficient justification to try a representation of the data by choosing a suitable q-value. This is done in fig. 14 where it is shown by the curve labelled “q = 0.5”, that the theory indeed fits the data well. The continuum multiplying
states primarily are El in character. However, the quasideuteron part, and the retardation
the need for a formfactor factor applied to the El
resonances, emphasizes the presence of continuously distributed E2 (and higher multipole) strength, that, by application of these factors as in the present analysis, is taken into account implicitly. In this sense, one should take the results shown in fig. 14 as a further proof of the importance of finite nuclear size effects. It appears possible within the present cross section uncertainties that also part of the broad El resonance at 28 MeV consists actually of quasideuteron strength. Furthermore, and more importantly, the existence of this E2 continuum was directly proved by Hayward et al. “) in an earlier photon scattering experiment. In their analysis the quasideuteron cross sections were not given a form factor. Therefore, the E2 component appears explicitly with a constant strength of 1.4 mb at the photon energies between 26 and 36 MeV. This value is in excellent agreement with the 1.5 mb required in the dip region for the explanation of the effect of the E2 resonance
as dip in the total cross section.
In summary, the narrow E2 resonance, seen in the angular distribution at 32 MeV, essentially does not increase the total cross section. Instead, it cuts a window into the E2 continuum overlapping this energy region.
3.3. CONSISTENT
DESCRIPTION
OF PHOTON
SCATTERING
AND
PHOTON
ABSORPTION
The analysis of the elastic scattering data with the inclusion of the total absorption cross section was performed exactly the same way as described for the lead experiment, adding the inelastically scattered photon spectra to the elastic ones before
324
K.P. Schelhaas et al. / Nuclear photon scattering
comparing with experiment. The inelastic cross sections of sect. 3.1 and the E2 results of sect. 3.2 were used. Starting with the total absorption cross section, the measured cross sections 7*9)were parametrized with a small number of lorentzians, with
thresholds
energies
and
retardation
reflect the varying
with thresholds
factors.
contributions
The
different
values
for the threshold
from ( y, p), ( y, n) and (‘y, np) reactions,
at 16.0, 18.7 and 27.4 MeV respectively.
The multipolarities
were
assumed to be El and E2 for the nuclear part, and Ml for the delta-resonance. Form factors were applied to the seagull terms and to the lorentzian supposed to describe the quasideuteron region. The form factors have less influence on the cross sections than in the case of “*Pb, a consistent picture however is impossible without their inclusion. Examples of calculations with modified and without form factors are given at the end of this section, in fig. 20. For the proton distribution a 3-parameter Fermi distribution was used with parameters from literature 22). For the exchange-current form factor the result found for 208Pb [ref. ‘)I was included by reducing the half-density radius by a factor 0.6, compared to the proton form factor. The real parts of the different types of amplitudes having different angular distributions (El, E2, Ml and quasideuteron) were calculated by separate dispersion integrals up to 800 MeV. In a trial and error procedure, all parameters were varied until the best representation of the measured sum of elastic and inelastic photon spectra was found. The total cross section for this final result deviates only slightly from the measured total cross section ‘). Both cross sections are shown in fig. 16. The differences are within the systematic errors of the absorption measurement which allow a smoothly varying deviation of kO.2 mb. This difference is not the size and direction, found by Fuller 23), in his attempt to reconcile total absorption cross sections and the sum of particle-emission cross sections. The A-resonance region is shown in fig. 17, together with all available total hadronic cross sections. The 60” scattering data justify deducing a number for the sum of the static electric, (Y,and magnetic, p, polarizabilities of an average nucleon in “C. Using the dispersion integral, the quantity ((Y+p) was found to be (11.9* 0.7) x lop4 fm3 for the energy range from 50 to 140 MeV, shown in fig. 18. This value is slightly smaller than the experimental value of an averaged equal number of free neutrons and protons, but it is within the combined errors bars. The series of figs. 19-21 shows the detailed comparison of the experimental results with the data set which
gives the best overall consistency with scattering, absorption, and the con‘). Some calculated curves showing the clusions drawn from the “‘Pb measurement effects of form factors at 140” scattering angle are included. These figures are explained in the captions thereof. Table 4 summarizes all parametrized results. They are not results of a rigorous least-squares fit, and may therefore reflect to a small extent the way of approaching the trial and error procedure. The present values are very well consistent with former data 1*6), taken at different angles in the giant resonance region. However, at energies above 30 MeV and mainly at forward angles, an apparent discrepancy cannot be explained. In contrast to the procedure followed
325
K.P. Schelhaas et al. / Nuclear photon scattering 25
0
Photon Energy (MeV) Fig. 16. The measured inelastic bremsstrahlung
in this
total absorption cross sections ‘), and the curve representing scattering data. For the sake of clarity many experimental energies have been combined into a small number of points.
best the elastic and points at different
paper,
one would have to alter the total absorption cross section ‘) 1*2,24),and more than allowed by statistics and systematical errors, in considerably order to include these low-value cross sections above 30 MeV. This remaining discrepancy may be reconsidered in the light of the coupled-channel picture of sect. 3.2. Indeed, continuum
scattering
allowing would data.
a coupling
change
Using
of El resonances
the absorption
the results
with part of the underlying
cross section
which is required
of ref. ‘), it may be that the inclusion
El
to fit the of such a
coupled channel picture reduces the discrepancy seen in fig. 21. A final conclusion cannot be drawn, however, before the theory connecting absorption and scattering, which is valid in this special case, has been developed adequately, including also inelastic processes. In fig. 22 finally, the different parts of the photon absorption cross section are plotted, in order to show their relationship as function of energy.
4. Conclusion Elastic and inelastic photon scattering cross sections over a wide energy range and from 60” to 150” scattering angle, as well as total photon absorption cross sections, form the basis for a consistent description of photon scattering and photon absorption. Several different aspects of nuclear structure physics are involved, besides the general reliance on dispersion relations.
K.P. Schelhaas et al. / Nuclear photon scattering
326 6
I....,....,....,....,....,....
.
..I
‘*c
Y
,,Extrapolation
--._““____<
0
...*‘**..‘..**‘.... 100
of nuclear
section
r...‘tr-.-*-’ -__--_-___ - - - - _ ________.__ -_--._
----_---_---_____________
150
cross
,:’
200
250
300
Photon
350
energy
400
450
0 !501
(MeV)
Fig. 17. The same cross section as fig. 16 for the A-resonance region, that enters the dispersion integral for the real parts of the scattering amplitude; squares from ref. *), circles from ref. 9).
0
1.
I
.I
.I
.I
.I
.I
50
.I.
I
.
150
Photon
eAi?gy
(MeV)
Fig. 18. The 60” scattering cross section, together with the theoretical description, including different nucleonic cross sections. The labelled numbers are sums of average electric and magnetic polarizabilities of the 12 constituents of “C, given in units of 10m4 fm3, and calculated using the minus-second-moment sum rule.
0
20
18
22
24
26
26
30
16
20
Fig. 19. Elastic-plus-inelastic scattering cross giant dipole region. The curves are obtained consistency of scattering and absorption cross appear at the energies of
40
60
80
100
22
24
26
26
30
3: 2
Photon Energy (MeV]
Photon Energy (MeV]
sections from the bremsstrahlung measurement, in the with the data set which results from the best overall sections. The inelastic cross sections are shown as they the inelastically scattered photons.
120
Photon Energy (MeVj
40
60
80
100
120
140
Photon Energy (MeV]
Fig. 20. The same cross section as fig. 19 for the energy region above 30 MeV, that is simultaneously covered in the bremsstrahlung experiment. The dashed curve at 150’ is calculated with F,, = 1 and the dotted curve is calculated with an increased nuclear radius for F=(q), by 10%. The point given as an open square, at 120” scattering angle and at 140 MeV photon energy, is the lowest-energy point of ref. 26).
0
20
30
40
Photon
50
Energy
60
20
(MeVf
30
40
Photon
50
Energy
60
70
(MeV]
Fig. 21. Comparison of the experimental results for elastic scattering with the calculation using the same data set (table 4) as for figs. 16, 17, 19 and 20; open circles from ref. r), squares from ref. “), and closed circles are present resultsz4). The Illinois and NBS data, not taken at the same scattering angles, are transformed using the dipole, (1 + cos* 0), angular distribution.
20
40
Photon Fig. 22. The partition (right), contributions.
60
80
Energy
300 MN
120
Photon
Energy
(MeV}
of the total absorption cross section into the different nuclear (left), and nucleonic The El lorentzian, labelled #5 in table 4, is shown dashed in the left-hand part.
329
K.P. Schelhaas et al. / Nuclear photon scattering
TABLE 4 Data set for the overall
description
of the measured
cross sections
(a) Lorentzians
El
E2
1.5 0.6 1.5 1.3 22.0 100
18.7 18.7 18.7 18.7 17.0 27.4
main El cross section AE = 7 MeV; quasideuteron
26.0 32.3
0.5 1.3
18.0 18.0
0.33 isovector 0.75 isovector
0.142 0.029 0.129 0.057 1.082 0.547
1 2
0.008 0.027
limit
TRK-units (mb . MeV)
140 800 800 800
1.73 1.41 1.96 11.36
Type
see ref. *‘); deduced
0,
W+ 8rr+ 7Y rr”
(d) Form factor ref. 22)
parameters
Type F,=
F,
J=,X (e) Average
sums sums
integral
(MeV)
(c) rr-production,
Ml
22.0 22.9 23.6 25.8 28.0 100
1 2 3 4 5 6
Upper
El
&h
Es
(b) Total absorption
Remarks
l-
TRK
Type *
sum of electric
(mb)
Remarks
all cross sections nuclear only, without quasideuteron all nuclear; no mesonic all cross sections
from total cross sections Effective
8*9)
threshold
Remarks
(MeV)
0.13 0.177
m, m-+35
direct reaction
0.068 0.068 0.164
m, m-+35 mm+45
resonance reaction: W,,, = 1270 MeV r = 240 MeV
of a Fermi
c (fm) 2.35 1.50
and magnetic
3-parameter
2
charge-density
(fm)
0.52 0.52 nucleonic
distribution,
with w = -0.149,
see
Remark from electron
scattering
polarizability
TRK is the dipole sum unit: 60 NZ/A (mb . MeV); I& is the resonance energy (MeV); r is the resonance width (MeV); Ethr is the particle emission threshold (MeV). For the definition of the p-production parameters see ref. ‘s).
330
K.P. Schelhaas
et al. / Nuclear photon scattering
The total cross section, found consistent with the present scattering results, and integrated to meson threshold, resulted in 1.73 classical sums, which is in good agreement whelming found guided
with the direct
measurement
part is El in character.
at 32.3 MeV, where the “dip” by a coupled
continuum integrated
interaction. absorption
channels
‘) of 1.67 classical
An E2 strength
in the absorption
consideration,
This E2 strength cross section.
is taken
comprises
TRK sums. The over-
of 0.75 isovector cross section as evidence
E2 sums was at this energy,
for a resonance-
not more than 1.6% of the total
Form factors have been found necessary as in the case of “‘Pb, and were scaled according to the r.m.s. radii. A better determination of the exchange form factor, compared to the lead case, requires better data around 120 MeV at backward angles. Similarly, an improvement of the value for the sum of the electric and magnetic static polarizabilities requires better data at similar energies at 60” or smaller angles. Inelastic processes have been identified that are consistent with transitions to the first excited state at 4.4 MeV and, possibly, to the lowest T = 1,2+ state at 16.1 MeV. The ratio Pinei of the square of the reduced matrix elements for inelastic to elastic scattering was found to be 0.4 and roughly constant over the giant resonance region (20-40 MeV). The role of the quasideuteron intermediate states, or more generally the role of exchange currents, remains open for future investigation, as well as the energy dependence of Pine, (E) in the giant resonance region. In view of these results on a number of nuclear structure related quantities, and not the least in view of the modest beam time required on a modern C.W. electron accelerator (2 days), the type of experiment, described here, appears very promising for future investigations.
We thank
the crew of the MAMI
accelerator,
especially
K.H. Kaiser,
for their
excellent job; K.H. Musshoff deserves special thanks for his untiring skill setting Discussions with E. Hayward, H.A. up the most complicated experiments. Weidenmiiller and J. Ahrens are gratefully acknowledged.
References 1) D.H. Wright, P.T. Debevec, L.J. Morford and A.M. Nathan, Phys. Rev. C32 (1985) 1174 2) D.H. Wright, Thesis, University of Illinois at Urbana Champaign, 1983 3) B. Ziegler, New vistas in electro-nuclear physics, ed. E.L. Tomusiak, H.S. Caplan and E.T. Dressier (Plenum, New York, 1986) p. 293 4) H. Arenhiivel, New vistas in electro-nuclear physics, ed. E.L. Tomusiak, H.S. Caplan and E.T. Dressier (Plenum, New York, 1986) p. 251 5) K.P. Schelhaas, J.M. Henneberg, M. Sanzone-Arenhiivel, N. Wieloch-Laufenberg, U. Zurmiihl, B. Ziegler, M. Schumacher and F. Wolf, Nucl. Phys. A489 (1988) 189 6) E. Hayward and R.G. Leicht, Ann, AC. Sci. 40 (1980) 99; W.R. Dodge, E. Hayward, R.G. Leicht, M. McCord and R. Starr, Phys. Rev. C28 (1983) 8 7) J. Ahrens, H. Borchert, K.H. Crock, H.B. Eppler, H. Gimm, H. Gundrum, M. Kroning, P. Riehn, G. Sita Ram, A. Ziegler, Nucl. Phys. A251 (1975) 479
K.P. Schelhaas et al. / Nuclear photon scattering 8) J. Arends, 9)
10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)
J. Eyrink,
A. Hegerath,
K.G. Hilger,
B. Mecking,
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