Inclusive electron-scattering cross sections have been measured for ^(56)Fe in the quasielastic region at electron energies between 0.9 and 4.3 GeV, at scattering angles of 15° and 85°. Longitudinal and transverse response functions at a q of 1.14 GeV/c have been extracted using a Rosenbluth separation. The experimental Coulomb sum has been obtained with aid of an extrapolation. The longitudinal response function, after correction for Coulomb distortion, is lower than quasifree-scattering-model predictions at the quasielastic peak and on the high-ω side

Beck, D.

Boyd, G.

Chen, J. P.

Filippone, B. W.

Jourdan, J.

McKeown, R. D.

Milner, R. G.

Potterveld, D. H.

Walker, R. C.

Woodward, C.

English

Caltech Authors - Main

VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
Longitudinal and Transverse Response Functions in 56Fe(e,e') 
at Momentum Transfer near 1 Ge VIc 
J.P. Chen, (I) Z. E. Meziani, <2> D. Beck, <3> G. Boyd, (3) L. M. Chinitz, (I) D. B. Day, (I) L. C. Dennis, <4> G. 
Dodge, <2> B. W. Filippone, <3> K. L. Giovanetti, (I) J. Jourdan, <3> K. W. Kemper, <4> T. Koh, <2> W. 
Lorenzon, <6 > J. S. McCarthy, (I) R. D. McKeown, <3> R. G. Milner, (3) R. C. Minehart, (I) J. 
Morgenstern, <s> J. Mougey, (?) D. H. Potterveld, (3) 0. A. Rondon-Aramayo, (I) R. M. Sealock, (I) L. C. 
Smith, (I) S. T. Thornton, (I) R. C. Walker, (3) and C. Woodward (3) 
O>commonwealth Center for Nuclear and High Energy Physics and Physics Department, 
University of Virginia, Charlottesville, Virginia 22901 
(l) Physics Department, Stanford University, Stanford, California 94305 
o> Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California 91125 
(4 )Physics Department, Florida State University, Tallahassee, Florida 32306 
ts>service de Physique Nuc/eaire-Haute Energie, Centre d'Etudes Nucleaires de Saclay, F-91191 Gif-sur-Yvette CEDEX, France 
(6 ) lnstitut fiir Physik, Universitiit Basel, CH -4056 Basel, Switzerland 
(?)Continuous Electron Beam Accelerator Facility, 12070 Jefferson Avenue, Newport News, Virginia 23606 
(Received 8 October 1990) 
Inclusive electron-scattering cross sections have been measured for 56Fe in the quasielastic region at 
electron energies between 0.9 and 4.3 GeV, at scattering angles of 15° and 85°. Longitudinal and trans-
verse response functions at a q of 1.14 GeV/c have been extracted using a Rosenbluth separation. The 
experimental Coulomb sum has been obtained with the aid of an extrapolation. The longitudinal 
response function, after correction for Coulomb distortion, is lower than quasifree-scattering-model pre-
dictions at the quasielastic peak and on the high-w side. 
PACS numbers: 25.30.Fj 
Quasielastic scattering is one of the main reaction 
mechanisms in inclusive high-energy electron scattering 
from nuclei, corresponding to electrons scattering elasti-
cally from a moving bound nucleon. The gross features 
of the quasielastic cross sections have been successfully 
described by single-particle models, such as shell models 1 
and Fermi-gas models. 2 With increasing momentum 
transfers, excitation of nucleon resonances becomes in-
creasingly important in the quasielastic region. In the 
last decade, the separate response functions for longitu-
dinal and transverse virtual photons have been extracted 
for various nuclei at three-momentum transfers q up to 
550 MeV/c, 3- 7 with the expectation that the resonances 
make very little contribution to the longitudinal part 
of the cross section. The results for 12C, 4 4°Ca and 
48Ca, 5•6 and for 56Fe, s-? show a surprising common 
feature: The experimental longitudinal response func-
tions RL are lower than the Fermi-gas calculations by up 
to 50%. This "quenching" of RL diminishes when q in-
creases but is still as much as 30% at q =550 MeV/c, the 
highest momentum transfer for which data have been 
available. The experimental Coulomb sum, which is the 
integral of RL over the energy loss, is also suppressed. 
The quenching of RL was also observed in (e ,e'p) experi-
ments. 8 This has been one of the most puzzling out-
standing problems in nuclear physics. 
The problem has motivated considerable theoretical 
effort. One approach invokes an effective increase in the 
radius of the nucleon in the nuclear medium (swollen nu-
cleon), resulting from manifestations of the quark sub-
structure (partial deconfinement). 9 The increased nu-
cleon size modifies the nucleon form factors and reduces 
RL. Since the modification of the form factors increases 
with increasing q, so does the reduction of RL. Other 
calculations attempt to improve agreement with the data 
by including final-state interactions, relativistic effects, 
two- and many-body correlations, 10 and off-shell ef-
fects. 11 Recently, relativistic a-w-p models within the 
framework of quantum hadrodynamics using the ran-
dom-phase approximation have been used to calculate 
RL, Rr, and the Coulomb sum. 12•13 Although these 
models reduce the discrepancy between theory and ex-
periment, they generally fail to reproduce both response 
functions. So far, no single model has succeeded in 
describing both the longitudinal and the transverse 
response functions over the full range of the available 
data. Since the various models have different depen-
dences on momentum transfers, measurements at high 
momentum transfer can be expected to constrain the 
models significantly. 
In this Letter, we present measurements of RL and Rr 
for 56Fe(e,e') at a momentum transfer near I GeV /c. 
The experiment (NE9) was performed at the Stanford 
Linear Accelerator Center (SLAC) with the NP AS 
(Nuclear Physics at SLAC) facility. Electron beams 
with energies ranging from 0.9 to 4.3 GeV were scat-
tered from a 6%-radiation-length natural-iron target. 
Scattered electrons were detected at scattering angles of 
15° for incident energies of 2.7, 3.3, 3.6, 3.9, and 4.3 
GeV, and 85° for 0.9, 1.1, and 1.25 GeV, using the 8-
GeV /c spectrometer with its associated detection system. 
The momenta of the scattered electrons range from 2 to 
@ 1991 The American Physical Society 1283 
VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
4 GeV for 15° data, and from 0.3 to 0.6 GeV for 85° 
data, covering the quasielastic peak region. The detec-
tion system consists of ten planes of multiwire propor-
tional chambers, a threshold gas Cerenkov counter, a 
five-layer total-absorption lead-glass shower counter, and 
three planes of plastic-scintillation counters. It has been 
described in detail elsewhere. 14 
Statistical errors in the measured cross sections are 
about 1.4% for 20-MeV-wide bins. Extensive efforts 
were made to minimize systematic errors, with the main 
uncertainties being the determination of the spectrom-
eter acceptance and background at low momenta (P 
< 600 MeV /c), the absolute normalization, and the ra-
diative corrections. 
The 8-Ge VIc spectrometer had been studied extensive-
ly at high momenta (P > I Ge V /c), 15 but little was 
known about its response at low momenta. The momen-
tum dependence of the acceptance was studied using a 
Monte Carlo simulation. Some test data were taken to 
check the acceptance at high momentum transfer and to 
check the Monte Carlo model of the spectrometer. The 
acceptance at low momenta was determined using the 
Monte Carlo model, with an estimated uncertainty of 
1.5%. The main background at low momenta arises 
from scattering by the magnet pole tips. Some test data 
were obtained with effectively all of this background el-
iminated by inserting a collimator between the two di-
pole magnets. By comparing these measurements with 
the measurements without the collimator, we determined 
that at 85° the background ranged from -2% at the 
quasielastic peak to -5% on the high-energy-loss side. 
After correction, we estimated that this background con-
tributed an uncertainty of about 1% in the cross sections 
at 85°. 
Elastic-scattering cross sections from hydrogen were 
measured and compared to a parametrization using a 
best fit to the measured proton form factors, 16 providing 
an absolute normalization of the data. The radiative 
corrections were made using the formulas of Mo and 
Tsai. 17 The radiative tails from elastic scattering were 
calculated and found to be negligible for all our data. 
The procedure of Stein et a/. 18 was used for continuum 
radiative corrections, which were up to 30% for the 15° 
data but were less than 3% for the 85° data. A sum-
mary of the systematic errors in the cross sections at the 
quasielastic peak and their propagation into the deter-
mination of RL and RT is given in Table I (also see Ref. 
14 for more detail). 
The longitudinal and transverse response functions 
were obtained using the Rosenbluth formula with the 
plane-wave Born approximation (PWBA): 
~- {[~)2 dndw -aM q2 RL(q,w) 
+ [ ~ [ 7) +tan 2 [ ~) ]RT(q,w)}, (1) 
1284 
TABLE I. Uncertainties at the quasielastic peak (in%). 
t'l.u/u t'l.u/u 
Source (I 5o) (85°) t'l.RdRL t'l.RdRT 
Acceptance 0.2 1.5 10.6 2.3 
Pole-tip scattering 0.1 1.0 7.0 1.5 
Radiative corrections 1.0 1.0 9.9 1.6 
Normalization 7.3 2.5 
Other systematic 7.8 1.5 
Total systematic 19.3 4.3 
Statistical 1.4 1.3 13.4 2.1 
where aM is the Mott cross section, w is the energy loss, 
and Q 2 =- q;J =q 2 - w2 is the four-momentum transfer 
squared. The effect of Coulomb distortion of the in-
cident and the scattered electrons was approximated by 
using an effective-momentum-transfer approach, 2 
where V c =- 3Za/2R is the mean value of the electro-
static potential of the nucleus, with R = ( t ) 112(r 2) 112 
and (r 2) 112 being the nuclear rms radius. This correc-
tion, to first order, takes the place of a more complicated 
distorted-wave Born-approximation 19 (DWBA) calcula-
tion. A DWBA calculation 20 agrees with the effective-
momentum-transfer approach to better than 1% in RL 
and RT at this momentum transfer for 56Fe. 
Figure 1 shows the results of RL and RT for 56Fe at 
q elf= 1.14 Ge VI c. Only statistical errors are shown in 
the plots. The cross-section data were smoothed while 
performing interpolations using a spline fit. The experi-
mental results are compared with three theoretical calcu-
lations. The dashed curve is the relativistic Fermi-gas 
model of Van Orden. 21 The average separation energy 
and the Fermi momentum are taken to be 36 MeV and 
260 MeV /c, respectively. 22 The solid curve is a PWBA 
quasifree-scattering calculation, which uses a realistic 
momentum distribution obtained from a previous SLAC 
experiment. 23 Both calculations are in agreement with 
the experimental data for RL on the low-energy-loss side 
of the quasielastic peak, but overestimate RL from the 
top of the peak to the high-energy-loss side. At this high 
momentum transfer RT has large contributions from 
processes other than quasielastic scattering, such as 
meson exchange and !:;. resonance excitation, and these 
contributions increase with increasing energy loss. Since 
the calculations shown include only the quasielastic pro-
cess, the calculated RT values are lower than the data as 
expected, except at the lowest energy transfers. The dot-
ted curve is a calculation by Ji, 12 which uses the local-
density random-phase approximation (RP A) to a relativ-
istic a-w-p model including the vacuum-polarization 
effect. 
Figure 2 shows separated response functions with and 
without using the effective-momentum-transfer correc-
VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
0.005 
0.004 
0.003 
~ 
I 0.002 > Ill 
e 0.001 
.... 
~ 0.000 
300 400 500 600 700 800 
(a) 
0.05 
(I) 
0.04 Ill 
0.03 
~ 
i 0.02 > Ill 
e 0.01 
.. 
~ 
0.00 
300 400 500 600 700 800 
OJ (MeV) 
(b) 
FIG. I. (a) Longitudinal and (b) transverse response func-
tions for 56Fe at an effective three-momentum transfer qerr 
= 1.14 GeV /c (circles), where the use of the effective momen-
tum transfer is an approximate method to include the Cou-
lomb-distortion effect. Only statistical errors are shown. The 
dashed curves are from a Fermi-gas-model calculation by Van 
Orden (Ref. 21), and solid curves are from a PWBA quasi-
free-scattering calculation with a realistic momentum distribu-
tion. The dotted curves are from an RP A calculation by Ji 
(Ref. 12). 
tion. The Coulomb distortion is nearly 20% for RL at 
the top of the peak and is only a few percent for RT. 
To obtain the experimental Coulomb sum an integra-
tion of RL is performed at constant three-momentum 
transfer, after dividing out the nucleon charge form fac-
tor with a relativistic correction: 24 
f. Wmu RL (q, (J)) C(q)= w.t dm Z[GE(Q2)]2' (3) 
where met means that m starts just above the elastic 
peak, and Wmax is the maximum value of the energy loss 
for which RL is not zero. The effective nucleon charge 
form factor is 
[GE(Q2)] 2 = {[G~(Q2)] 2+ (N/Z)[G~ (Q 2)] 2} 
I +Q 2/4M 2 X N 
I+Q 2/2M~ ' (4) 
where MN is the nucleon mass. Z and N are the num-
bers of protons and neutrons in the nucleus, respectively. 
The dipole form factor is used for G~(Q 2 ), the proton 
electric form factor. G~(Q 2 ) =0 is used for the neutron 
electric form factor. The use of other parametrizations 
of the form factors, for instance, Hohler's 25 parametriza-
tion 8.2, will change the Coulomb sum by no more than 
5%. Since the data do not cover the whole peak, extra-
0.005 
0.004 
0.003 
~ 
I 0.002 > Ill 
6 0.001 
.... 
~ 0.000 
300 400 500 600 700 800 
(a) 
0.05 
i 
0.04 !I! 
0.03 f. f. 
i 
,_f. 
0.02 ,_f. > 
"" 
Ill 
6 0.01 
.. 
~ 0.00 
300 400 500 600 700 800 
OJ (MeV) 
(b) 
FIG. 2. (a) Longitudinal and (b) transverse response func-
tions for 56 Fe at q -1.14 GeV /c (crosses) compared with these 
at q.rr=1.14 GeV/c (circles). Only statistical errors are shown. 
The solid curves are the PWBA calculation. 
potations into the unmeasured region were performed. 
The contribution to the Coulomb sum from the unmea-
sured tails at both sides was estimated to be about I 0% 
by using exponential extrapolation. The results, along 
with a comparison to the Fermi-gas calculation, are 
shown in Table II. 
To better compare with the results at low momentum 
transfers, a Coulomb sum defined as a simple integration 
of RL over energy loss along constant three-momentum 
transfer was also computed: 
(5) 
Figure 3 presents the result along with the previous mea-
surements and the Fermi-gas calculation. The Coulomb 
sum continues to be lower than the calculation at this 
high momentum transfer by about 30%, with an uncer-
tainty of about 20%. 
In conclusion, RL and RT at q =1.14 GeV/c were ob-
tained for quasielastic scattering from 56Fe. Except on 
the low-energy-loss side of the peak, RT includes large 
effects from the ~ and higher nucleon resonances. The 
measured longitudinal response RL is lower than the cal-
culation at the top and on the high-m side of the peak. 
TABLE II. Coulomb sum. 
Data Error ( ±) Fermi gas RPA model 
q=1.14GeV/c 0.95 
q.rr-1.14GeV/c 0.76 
0.29 
0.23 
1.10 
1.10 
1.16 
1.16 
1285 
VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
10.0 
5.0 
u 
1.0 
0.5 
0 250 500 750 1000 1250 
q (MeV/c) 
FIG. 3. Coulomb sum for 56Fe. The circle is the result of 
this experiment at qetr=1.14 GeV/c. Total error, including 
both statistical and systematic errors, is shown. The squares 
are the Saclay (Ref. 6) and Bates (Ref. 7) data. The solid 
curve is from the Fermi-gas calculation. 
The experimental Coulomb sum, with a rather large un-
certainty, is lower than the Fermi-gas calculation. The 
RL quenching persists at this high momentum transfer, 
showing no strong q dependence. Within I standard de-
viation, the RL strength missing at this q is no more than 
50%. Since RL is a linear combination of the nucleon 
form factors squared, this result constrains swollen-
nucleon models to have no more than 30% reduction in 
the nucleon form factors at this high momentum 
transfer. 
We acknowledge the support of the SLAC staff. We 
thank Dr. X. Ji and Dr. M. Traini for providing us with 
their calculations. This work was supported by U.S. 
Department of Energy Grants No. DE-FG05-86ER40-
261 and No. DE-FG05-88ER40390 (University of Vir-
ginia) and No. DE-FG03-88ER40439 (Stanford Uni-
versity), National Science Foundation Grants No. 
PHY -88-17296 (Cal tech) and No. PHY -89-10648 
(Florida State University), and the Commonwealth of 
Virginia Center for Nuclear and High Energy Physics. 
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Donnelly, Nucl. Phys. AlSO, 393 (1970). 
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61, 1706 (1988); K. F. von Reden eta/., Phys. Rev. C 41, 1084 
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14J. P. Chen, Ph.D. thesis, University of Virginia, 1990; S. 
Dasu eta/., Phys. Rev. Lett. 61, 1061 (1988); S. Dasu eta/., 
Phys. Rev. Lett. 61,2591 (1988). 
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(unpublished). 
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Y. S. Tsai, SLAC Report No. SLAC-PU8-848, 1971 (unpub-
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38,2799 (1988); M. Traini, Phys. Lett. B 213, I (1988); G. Co 
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Longitudinal and Transverse Response Functions in ^(56)Fe(e,e')

at Momentum Transfer near 1 GeV/c

VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
Longitudinal and Transverse Response Functions in 56Fe(e,e') 
at Momentum Transfer near 1 Ge VIc 
J.P. Chen, (I) Z. E. Meziani, <2> D. Beck, <3> G. Boyd, (3) L. M. Chinitz, (I) D. B. Day, (I) L. C. Dennis, <4> G. 
Dodge, <2> B. W. Filippone, <3> K. L. Giovanetti, (I) J. Jourdan, <3> K. W. Kemper, <4> T. Koh, <2> W. 
Lorenzon, <6 > J. S. McCarthy, (I) R. D. McKeown, <3> R. G. Milner, (3) R. C. Minehart, (I) J. 
Morgenstern, <s> J. Mougey, (?) D. H. Potterveld, (3) 0. A. Rondon-Aramayo, (I) R. M. Sealock, (I) L. C. 
Smith, (I) S. T. Thornton, (I) R. C. Walker, (3) and C. Woodward (3) 
O>commonwealth Center for Nuclear and High Energy Physics and Physics Department, 
University of Virginia, Charlottesville, Virginia 22901 
(l) Physics Department, Stanford University, Stanford, California 94305 
o> Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California 91125 
(4 )Physics Department, Florida State University, Tallahassee, Florida 32306 
ts>service de Physique Nuc/eaire-Haute Energie, Centre d'Etudes Nucleaires de Saclay, F-91191 Gif-sur-Yvette CEDEX, France 
(6 ) lnstitut fiir Physik, Universitiit Basel, CH -4056 Basel, Switzerland 
(?)Continuous Electron Beam Accelerator Facility, 12070 Jefferson Avenue, Newport News, Virginia 23606 
(Received 8 October 1990) 
Inclusive electron-scattering cross sections have been measured for 56Fe in the quasielastic region at 
electron energies between 0.9 and 4.3 GeV, at scattering angles of 15° and 85°. Longitudinal and trans-
verse response functions at a q of 1.14 GeV/c have been extracted using a Rosenbluth separation. The 
experimental Coulomb sum has been obtained with the aid of an extrapolation. The longitudinal 
response function, after correction for Coulomb distortion, is lower than quasifree-scattering-model pre-
dictions at the quasielastic peak and on the high-w side. 
PACS numbers: 25.30.Fj 
Quasielastic scattering is one of the main reaction 
mechanisms in inclusive high-energy electron scattering 
from nuclei, corresponding to electrons scattering elasti-
cally from a moving bound nucleon. The gross features 
of the quasielastic cross sections have been successfully 
described by single-particle models, such as shell models 1 
and Fermi-gas models. 2 With increasing momentum 
transfers, excitation of nucleon resonances becomes in-
creasingly important in the quasielastic region. In the 
last decade, the separate response functions for longitu-
dinal and transverse virtual photons have been extracted 
for various nuclei at three-momentum transfers q up to 
550 MeV/c, 3- 7 with the expectation that the resonances 
make very little contribution to the longitudinal part 
of the cross section. The results for 12C, 4 4°Ca and 
48Ca, 5•6 and for 56Fe, s-? show a surprising common 
feature: The experimental longitudinal response func-
tions RL are lower than the Fermi-gas calculations by up 
to 50%. This "quenching" of RL diminishes when q in-
creases but is still as much as 30% at q =550 MeV/c, the 
highest momentum transfer for which data have been 
available. The experimental Coulomb sum, which is the 
integral of RL over the energy loss, is also suppressed. 
The quenching of RL was also observed in (e ,e'p) experi-
ments. 8 This has been one of the most puzzling out-
standing problems in nuclear physics. 
The problem has motivated considerable theoretical 
effort. One approach invokes an effective increase in the 
radius of the nucleon in the nuclear medium (swollen nu-
cleon), resulting from manifestations of the quark sub-
structure (partial deconfinement). 9 The increased nu-
cleon size modifies the nucleon form factors and reduces 
RL. Since the modification of the form factors increases 
with increasing q, so does the reduction of RL. Other 
calculations attempt to improve agreement with the data 
by including final-state interactions, relativistic effects, 
two- and many-body correlations, 10 and off-shell ef-
fects. 11 Recently, relativistic a-w-p models within the 
framework of quantum hadrodynamics using the ran-
dom-phase approximation have been used to calculate 
RL, Rr, and the Coulomb sum. 12•13 Although these 
models reduce the discrepancy between theory and ex-
periment, they generally fail to reproduce both response 
functions. So far, no single model has succeeded in 
describing both the longitudinal and the transverse 
response functions over the full range of the available 
data. Since the various models have different depen-
dences on momentum transfers, measurements at high 
momentum transfer can be expected to constrain the 
models significantly. 
In this Letter, we present measurements of RL and Rr 
for 56Fe(e,e') at a momentum transfer near I GeV /c. 
The experiment (NE9) was performed at the Stanford 
Linear Accelerator Center (SLAC) with the NP AS 
(Nuclear Physics at SLAC) facility. Electron beams 
with energies ranging from 0.9 to 4.3 GeV were scat-
tered from a 6%-radiation-length natural-iron target. 
Scattered electrons were detected at scattering angles of 
15° for incident energies of 2.7, 3.3, 3.6, 3.9, and 4.3 
GeV, and 85° for 0.9, 1.1, and 1.25 GeV, using the 8-
GeV /c spectrometer with its associated detection system. 
The momenta of the scattered electrons range from 2 to 
@ 1991 The American Physical Society 1283 
VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
4 GeV for 15° data, and from 0.3 to 0.6 GeV for 85° 
data, covering the quasielastic peak region. The detec-
tion system consists of ten planes of multiwire propor-
tional chambers, a threshold gas Cerenkov counter, a 
five-layer total-absorption lead-glass shower counter, and 
three planes of plastic-scintillation counters. It has been 
described in detail elsewhere. 14 
Statistical errors in the measured cross sections are 
about 1.4% for 20-MeV-wide bins. Extensive efforts 
were made to minimize systematic errors, with the main 
uncertainties being the determination of the spectrom-
eter acceptance and background at low momenta (P 
< 600 MeV /c), the absolute normalization, and the ra-
diative corrections. 
The 8-Ge VIc spectrometer had been studied extensive-
ly at high momenta (P > I Ge V /c), 15 but little was 
known about its response at low momenta. The momen-
tum dependence of the acceptance was studied using a 
Monte Carlo simulation. Some test data were taken to 
check the acceptance at high momentum transfer and to 
check the Monte Carlo model of the spectrometer. The 
acceptance at low momenta was determined using the 
Monte Carlo model, with an estimated uncertainty of 
1.5%. The main background at low momenta arises 
from scattering by the magnet pole tips. Some test data 
were obtained with effectively all of this background el-
iminated by inserting a collimator between the two di-
pole magnets. By comparing these measurements with 
the measurements without the collimator, we determined 
that at 85° the background ranged from -2% at the 
quasielastic peak to -5% on the high-energy-loss side. 
After correction, we estimated that this background con-
tributed an uncertainty of about 1% in the cross sections 
at 85°. 
Elastic-scattering cross sections from hydrogen were 
measured and compared to a parametrization using a 
best fit to the measured proton form factors, 16 providing 
an absolute normalization of the data. The radiative 
corrections were made using the formulas of Mo and 
Tsai. 17 The radiative tails from elastic scattering were 
calculated and found to be negligible for all our data. 
The procedure of Stein et a/. 18 was used for continuum 
radiative corrections, which were up to 30% for the 15° 
data but were less than 3% for the 85° data. A sum-
mary of the systematic errors in the cross sections at the 
quasielastic peak and their propagation into the deter-
mination of RL and RT is given in Table I (also see Ref. 
14 for more detail). 
The longitudinal and transverse response functions 
were obtained using the Rosenbluth formula with the 
plane-wave Born approximation (PWBA): 
~- {[~)2 dndw -aM q2 RL(q,w) 
+ [ ~ [ 7) +tan 2 [ ~) ]RT(q,w)}, (1) 
1284 
TABLE I. Uncertainties at the quasielastic peak (in%). 
t'l.u/u t'l.u/u 
Source (I 5o) (85°) t'l.RdRL t'l.RdRT 
Acceptance 0.2 1.5 10.6 2.3 
Pole-tip scattering 0.1 1.0 7.0 1.5 
Radiative corrections 1.0 1.0 9.9 1.6 
Normalization 7.3 2.5 
Other systematic 7.8 1.5 
Total systematic 19.3 4.3 
Statistical 1.4 1.3 13.4 2.1 
where aM is the Mott cross section, w is the energy loss, 
and Q 2 =- q;J =q 2 - w2 is the four-momentum transfer 
squared. The effect of Coulomb distortion of the in-
cident and the scattered electrons was approximated by 
using an effective-momentum-transfer approach, 2 
where V c =- 3Za/2R is the mean value of the electro-
static potential of the nucleus, with R = ( t ) 112(r 2) 112 
and (r 2) 112 being the nuclear rms radius. This correc-
tion, to first order, takes the place of a more complicated 
distorted-wave Born-approximation 19 (DWBA) calcula-
tion. A DWBA calculation 20 agrees with the effective-
momentum-transfer approach to better than 1% in RL 
and RT at this momentum transfer for 56Fe. 
Figure 1 shows the results of RL and RT for 56Fe at 
q elf= 1.14 Ge VI c. Only statistical errors are shown in 
the plots. The cross-section data were smoothed while 
performing interpolations using a spline fit. The experi-
mental results are compared with three theoretical calcu-
lations. The dashed curve is the relativistic Fermi-gas 
model of Van Orden. 21 The average separation energy 
and the Fermi momentum are taken to be 36 MeV and 
260 MeV /c, respectively. 22 The solid curve is a PWBA 
quasifree-scattering calculation, which uses a realistic 
momentum distribution obtained from a previous SLAC 
experiment. 23 Both calculations are in agreement with 
the experimental data for RL on the low-energy-loss side 
of the quasielastic peak, but overestimate RL from the 
top of the peak to the high-energy-loss side. At this high 
momentum transfer RT has large contributions from 
processes other than quasielastic scattering, such as 
meson exchange and !:;. resonance excitation, and these 
contributions increase with increasing energy loss. Since 
the calculations shown include only the quasielastic pro-
cess, the calculated RT values are lower than the data as 
expected, except at the lowest energy transfers. The dot-
ted curve is a calculation by Ji, 12 which uses the local-
density random-phase approximation (RP A) to a relativ-
istic a-w-p model including the vacuum-polarization 
effect. 
Figure 2 shows separated response functions with and 
without using the effective-momentum-transfer correc-
VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
0.005 
0.004 
0.003 
~ 
I 0.002 > 
Ill e 0.001 
.... 
~ 
0.000 
300 400 500 600 700 800 
(a) 
0.05 
(I) 
0.04 Ill 
0.03 
~ 
i 0.02 > 
Ill e 0.01 .. 
~ 
0.00 
300 400 500 600 700 800 
OJ (MeV) 
(b) 
FIG. I. (a) Longitudinal and (b) transverse response func-
tions for 56Fe at an effective three-momentum transfer qerr 
= 1.14 GeV /c (circles), where the use of the effective momen-
tum transfer is an approximate method to include the Cou-
lomb-distortion effect. Only statistical errors are shown. The 
dashed curves are from a Fermi-gas-model calculation by Van 
Orden (Ref. 21), and solid curves are from a PWBA quasi-
free-scattering calculation with a realistic momentum distribu-
tion. The dotted curves are from an RP A calculation by Ji 
(Ref. 12). 
tion. The Coulomb distortion is nearly 20% for RL at 
the top of the peak and is only a few percent for RT. 
To obtain the experimental Coulomb sum an integra-
tion of RL is performed at constant three-momentum 
transfer, after dividing out the nucleon charge form fac-
tor with a relativistic correction: 24 
f. Wmu RL (q, (J)) 
C(q)= w.t dm Z[GE(Q2)]2' (3) 
where met means that m starts just above the elastic 
peak, and Wmax is the maximum value of the energy loss 
for which RL is not zero. The effective nucleon charge 
form factor is 
[GE(Q2)] 2 = {[G~(Q2)] 2+ (N/Z)[G~ (Q 2)] 2} 
I +Q 2/4M 2 X N 
I+Q 2/2M~ ' 
(4) 
where MN is the nucleon mass. Z and N are the num-
bers of protons and neutrons in the nucleus, respectively. 
The dipole form factor is used for G~(Q 2 ), the proton 
electric form factor. G~(Q 2 ) =0 is used for the neutron 
electric form factor. The use of other parametrizations 
of the form factors, for instance, Hohler's 25 parametriza-
tion 8.2, will change the Coulomb sum by no more than 
5%. Since the data do not cover the whole peak, extra-
0.005 
0.004 
0.003 
~ 
I 0.002 > 
Ill 
6 0.001 
.... 
~ 
0.000 
300 400 500 600 700 800 
(a) 
0.05 
i 
0.04 !I! 
0.03 f. f. 
i 
,_f. 
0.02 ,_f. > "" Ill 6 0.01 .. 
~ 
0.00 
300 400 500 600 700 800 
OJ (MeV) 
(b) 
FIG. 2. (a) Longitudinal and (b) transverse response func-
tions for 56 Fe at q -1.14 GeV /c (crosses) compared with these 
at q.rr=1.14 GeV/c (circles). Only statistical errors are shown. 
The solid curves are the PWBA calculation. 
potations into the unmeasured region were performed. 
The contribution to the Coulomb sum from the unmea-
sured tails at both sides was estimated to be about I 0% 
by using exponential extrapolation. The results, along 
with a comparison to the Fermi-gas calculation, are 
shown in Table II. 
To better compare with the results at low momentum 
transfers, a Coulomb sum defined as a simple integration 
of RL over energy loss along constant three-momentum 
transfer was also computed: 
(5) 
Figure 3 presents the result along with the previous mea-
surements and the Fermi-gas calculation. The Coulomb 
sum continues to be lower than the calculation at this 
high momentum transfer by about 30%, with an uncer-
tainty of about 20%. 
In conclusion, RL and RT at q =1.14 GeV/c were ob-
tained for quasielastic scattering from 56Fe. Except on 
the low-energy-loss side of the peak, RT includes large 
effects from the ~ and higher nucleon resonances. The 
measured longitudinal response RL is lower than the cal-
culation at the top and on the high-m side of the peak. 
TABLE II. Coulomb sum. 
Data Error ( ±) Fermi gas RPA model 
q=1.14GeV/c 0.95 
q.rr-1.14GeV/c 0.76 
0.29 
0.23 
1.10 
1.10 
1.16 
1.16 
1285 
VOLUME 66, NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 1991 
10.0 
5.0 
u 
1.0 
0.5 
0 250 500 750 1000 1250 
q (MeV/c) 
FIG. 3. Coulomb sum for 56Fe. The circle is the result of 
this experiment at qetr=1.14 GeV/c. Total error, including 
both statistical and systematic errors, is shown. The squares 
are the Saclay (Ref. 6) and Bates (Ref. 7) data. The solid 
curve is from the Fermi-gas calculation. 
The experimental Coulomb sum, with a rather large un-
certainty, is lower than the Fermi-gas calculation. The 
RL quenching persists at this high momentum transfer, 
showing no strong q dependence. Within I standard de-
viation, the RL strength missing at this q is no more than 
50%. Since RL is a linear combination of the nucleon 
form factors squared, this result constrains swollen-
nucleon models to have no more than 30% reduction in 
the nucleon form factors at this high momentum 
transfer. 
We acknowledge the support of the SLAC staff. We 
thank Dr. X. Ji and Dr. M. Traini for providing us with 
their calculations. This work was supported by U.S. 
Department of Energy Grants No. DE-FG05-86ER40-
261 and No. DE-FG05-88ER40390 (University of Vir-
ginia) and No. DE-FG03-88ER40439 (Stanford Uni-
versity), National Science Foundation Grants No. 
PHY -88-17296 (Cal tech) and No. PHY -89-10648 
(Florida State University), and the Commonwealth of 
Virginia Center for Nuclear and High Energy Physics. 
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Longitudinal and Transverse Response Functions in ^(56)Fe(e,e') at Momentum Transfer near 1 GeV/c

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