Measurements of shadowing in low |Q2| electroproduction on nuclei

Nuclear Physics B151 (1979) 367-388 © North-Holland Publishing Company

M E A S U R E M E N T S OF SHADOWING IN LOW IQ 21 ELECTROPRODUCTION ON NUCLEI J. BAILEY *, D.R. BOTTERILL **, H.E. MONTGOMERY *** and P.R. NORTON ** Science Research Council, Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, UK A. DEL GUERRA, A. GIAZOTTO, M.A. GIORGI and A. STEFANINI Istituto di Fisiea dell'Universita, Piazza Torricelli 2, 1-561 O0 Pisa, Italy Istituto Nazionale de Fisica Nucleare, Sezione di Pisa, via Livornese, S. Piero a Grado, L56010 Pisa, Italy G. MATONE Laboratori Nazionali di Frascati, L00044 Frascati, Rome, Italy Received 7 November 1978

Measurements of inelastic electron scattering have been made in the range 2.2 < v < 3.8 GeV and 0.1 < 1021 < 0.3 (GeV/c) 2, on a selection of nuclei ranging from hydrogen and deuterium to uranium, by measuring the scattered electron only. Detailed calculations have been made of the contribution of radiative tails to the measured yield. The results show a small 'shadowing' consistent with other electroproduction experiments, and also with photoproductionexperiments in this v range, but the shadowing decreases rapidly as IQ21 increases.

1. Introduction The fact that the photon contains a hadronic component is exhibited b y the phenomenon of 'shadowing' in photoproduction on nuclei. The photo-absorption cross section on a nucleus is less than the sum of the cross sections on the individual nucleons [ 1 - 4 ] . Quantitative calculations of the shadowing have been made using the vector dominance model, either in its simple [5] or more generalised [6] form. The common prediction of such models is that shadowing should also be observed in electroproduction cross sections at low values of the m o m e n t u m transfer 02. * Present address: IKO, Postbus 4395, Amsterdam 1006, The Netherlands. ** Present address: Rutherford Laboratory, Chilton, Didcot, Oxon OX11 OQX. *** Present address: CERN, CH-1211 Geneva 23, Switzerland. 367

368

£ Bailey et al. / Shadowing in low [Q2I electroproduction

Experiments so far performed using single-arm electron scattering at IQ21 ~> 0.25 (GeV/c) 2 have shown no convincing evidence for shadowing [7,8], while one experiment on electron scattering at IQ 21 --- 0.1 (GeV/c) [9] has seen a small but significant shadowing. It is therefore of interest to look for shadowing at small IQ2 I.

2. Experimental method The experiment was performed in a 5 GeV extracted electron beam from the Daresbury synchrotron NINA. Electrons scattered from various targets were detected in a small-aperture magnetic spectrometer comprising six scintillation counter hodoscopes. Electrons were identified in~a threshold CO s (~erenkov counter and a lead.lucite shower counter. The spectrometer has been described in detail in an earlier publication [ 10]. For this experiment the momentum acceptance was +4% and the angular acceptance ~0.6 msr. The beam intensity was measured by a secondary emission monitor (SEM) downstream of the target. The following nuclei were used as targets: (a) Hydrogen and deuterium: the same target vessel was used as in previous experiments with this apparatus. It was 100 mm long, 38 mm in diamater, with hemispherical end-caps of thin stainless steel. The vacuum vessel had thin stainless steel windows at the entrance and exit; (b) The elements Be, C, A1, Co, Nb, Ta, U: the liquid hydrogen/deuterium target was raised above the beam line and the heavy targets inserted. They were clamped in foil holders on the rim of a rotatable wheel. Any chosen target could be moved remotely into the beam. The targets were contained within a helium enclosure to reduce the background counting rate.

Table 1 Details of the targets used Element

Z

N

Target thickness (g " cm-2)

Rad. length (target) (X 10-2)

Rad. length (total) (x 10 -2)

H D Be C A1 Co Nb Ta U

1 1 4 6 13 27 41 73 92

0 1 5 6 14 32 52 108 146

0.700 1.620 1.038 0.800 0.342 0.199 0.136 0.090 0.099

1.11 1.28 1.59 1.87 1.42 1.46 1.37 1.32 1.66

2.59 2.77 2.53 2.81 2.36 2.40 2.31 2.26 2.60

Z Bailey et al. /Shadowing in low IQ21electroproduction

369

Details of the targets may be found in table 1 (target thicknesses are quoted in g • cm-2). The radiation lengths quoted are (a) for the target material itself and (b) the total radiator in the line (including the target). The solid target thicknesses were chosen to be approximately equal in radiation lengths to that of deuterium. In consequence, the counting rate and background-to-signal ratio were worse for the targets of highest atomic weight. Data were taken mainly at two spectrometer angle settings 6 ° and 8 ° , and five m o m e n t u m settings between 1.2 GeV/c and 2.8 GeV/c. We also present data at 8.5 °, 2.25 GeV/c, taken during the earlier stages of the experiment, preliminary results of which have already been reported [11 ]. The relevant kinematic values for these settings are shown in table 2. Our conventional notation is: E, incident electron energy; E', scattered electron energy; 0e, electron scattering angle; q2 = _Q2 = - 2 E E ' ( 1 - cos 0e), 4-momentum transfer to nuclei; v = E - E', electron energy loss; 1¢, effective mass of hadronic final state given by I4/5 = M s + 2 My - Q2, where M is the mass of the target: we shall always calculate W and other derived variables assuming the target particle is a proton; ~ ' = 1 + W2/Q 2 scaling variable; x' = 1/co'. Backgrounds were measured: (a) for hydrogen and deuterium by remotely moving an empty d u m m y target cell into the beam: (b) for the solid targets by measuring the rate from an empty foil holder. They were typically "~14% for hydrogen, 9% for deuterium, and ranged from 2--4% for beryllium, through 8 - 1 5 % for cobalt, to 16-25% for tantalum and uranium. A further source of background was of electrons Table 2 K'mematic settings E

E'

0e

Q2

v

(GeV)

(GeV)

(degrees)

(GeV/c) 2

(GeV)

5.0

1.2 1.6 2.0 2.4 2.8 1.2 1.6 2.0 2.4 2.8 2.25

6 6 6 6 6 8 8 8 8 8 8.5

0.066 0.088 0.110 0.131 0.153 0.117 0.156 0.195 0.234 0.272 0.247

3.8 3.4 3.0 2.6 2.2 3.8 3.4 3.0 2.6 2.2 2.75

x'

0.008 0.012 0.017 0.023 0.031 0.015 0.021 0.030 0.041 0.054 0.041

J. Bailey et al. / Shadowing in low IQ21 electroproduction

370

Table 3 Background-subtracted event yields (not renorrnalised to a c o m m o n target density) E'

0 (degrees)

H

D

Be

1.2 1.6 2.0 2.4 2.8

6 6 6 6 6

8.782 9.055 9.971 11.757 15.562

± 0.177 ± 0.143 -+ 0.152 ± 0.182 ± 0.213

13.069 14.386 17.367 21.675 29.314

± 0.185 ± 0.238 ± 0.203 ± 0.227 ± 0.266

12.120 10.104 10.990 12.598 17.081

± 0,192 ± 0,239 -+ 0,252 -+ 0,249 ± 0,298

12.881 10.010 9.037 10.693 13.707

± 0.309 ± 0.210 ± 0.216 ± 0.227 ± 0.242

1.2 1.6 2.0 2.4 2.8

8 8 8 8 8

2.923 3.168 3.767 4.584 5.890

± 0.072 ± 0.065 ± 0.066 ± 0.081 -+ 0.099

4,817 5.825 6,975 8,863 11,427

+- 0.072 ± 0.077 ± 0.102 ± 0.122 ± 0.153

3.570 3.695 4.263 5.578 7.710

-+ 0,070 5 0.057 ± 0,091 ± 0.077 ± 0.127

3.294 3.008 3.300 4.214 6.043

-+ 0.073 ± 0.079 +- 0.075 -+ 0.074 ± 0.114

2.25

8.5

3.417 +- 0.099

6,445 ± 0.074

C

4.112 +- 0.077

30-

eD ',,C

.Q

•

A.~

I

Nb

20-

Q.

g "~ 10-~

.

>-

o

o

1

~ E' (GeV)

Fig. 1. Electroproduction yields as a function of E ' for various elements for E = 5 GeV and Oe = 6 °.

J. Bailey et al. /Shadowing in low [Q21 electroproduction

A1

Co

Nb

7.057 4.306 3.879 4.350 5.902

-+0.123 ± 0.105 ± 0,078 ± 0,104 ± 0.104

4.788 2.707 2.365 2.798 3.401

± 0.119 ± 0.073 ± 0.060 ± 0.076 -+ 0.078

3.574 2.077 1.827 2.027 2.445

-+ 0.093 -+ 0.064 +- 0.051 -+ 0.066 -+ 0.067

1.595 1.278 1.415 1.787 2.500

± 0.036 ± 0.025 ± 0,022 ± 0.047 ± 0.052

0.969 0.745 0.845 1.049 1.495

± 0.027 ± 0.021 ± 0.020 ± 0.027 ± 0.035

0.723 0.570 0.575 0.784 1.041

± 0.022 ± 0.021 ± 0.018 ± 0.023 ± 0.027

1.358 ± 0.043

0.744 ± 0.014

0.535 -+ 0.012

Ta

U

2.801 ± 0.070

3.625 +- 0.080

1.406 ± 0.047 1.392 -+ 0.062 1.656 -+0.055

1.520 -+ 0.048 1.652 ± 0.063 1.930 -+ 0.058

0.403 -+ 0.017

0.473 -+ 0.017

0.511 -+ 0.020 0.718 +- 0.022

0.573 -+ 0.021 0.758 -+ 0.022

0.346 -+ 0.010

0.400 ± 0.012

371

originating f r o m n ° Dalitz decay. This c o n t r i b u t i o n was d e t e r m i n e d by measuring the e ÷ yield (by reversing the polarity o f the spectrometer magnets). It varied f r o m less than 1% at high scattered electron energy to ~ 5 % at the lowest scattered energy and was a p p r o x i m a t e l y the same for all elements. Some typical background subtracted yields are shown in fig. 1, and the c o m p l e t e set in table 3. T h e y have n o t been renormalised to a c o m m o n target density. The increase in yields at low E ' arises f r o m contributions to the observed cross section f r o m radiative tails o f elastic peaks, w h i c h at lowest E ' are larger than the inelastic cross section. These c o n t r i b u t i o n s arise f r o m : (a) c o h e r e n t electron-nucleus scattering; (b) quasi-elastic electron-nucleon scattering; (c) o t h e r collective nuclear effects such as c o h e r e n t nuclear level excitation and scattering f r o m clusters within the nucleus. These tails had to be calculated and subtracted f r o m the measured data. Only c o h e r e n t nuclear elastic scattering and quasi-elastic scattering f r o m nucleons were considered. All our estimates o f o t h e r processes indicate that t h e y are negligible,

3. Computation of the radiative tails To calculate the elastic and quasi-elastic internal bremsstrahlung tails we used the well-known o n e - p h o t o n exchange 'exact cross section' (formula ( A . 2 4 ) o f

372

J. Bailey et al. /Shadowing in low IQ21electroproduction

Tsai [ 12]). The 'single-arm' electron scattering cross section is given by d2o

1 = / d(cos Ok) [GI(E,E', 0e, 0k,MT) WI(Q 2) dEPd~2e -1 + G2(E , E', Oe, 0k, Mr) W2(Q2)] .

(1)

d[2e is the solid angle of the electron. The functions G 1 and G2 include the vertex and virtual photon propagator corrections and the multiplicative terms accounting for multi-photon emission [8]. 0k is the angle between the real (emitted) photon and the virtual photon, M T is the target mass, and I¢ 1 and I¢2 are the elastic structure functions of the target (this notation follows that of ref. [8]). 3.1. Coherent scattering from the nucleus M T was set equal to the mass of the nucleus. W1 was set to zero, and W2 equal to Z 2 [F(Q2)] 2, where F(Q 2) is the nuclear elastic electromagnetic form factor. Fig. 2 shows an example of the integrand as a function of cos 0k. The three peaks are conventionally known as q, s and p. The solid line represents W2 = Z 20.e., F(Q 2) = 1) and the dashed line the true value of W2 = Z 2 [F(Q2)] 2. The effect of the form factor is to suppress the s-peak by two orders of magnitude but to leave the q-peak almost unchanged. The integral is dominated by the region of very low x/Q 2, and it is therefore critical that an accurate representation of the form factgr_ is used in this region which for this experiment, independent of target and kinematics, is x/Q 2 < 1.5 fm -1. To this end, fits were made to existing data on low-energy electron-nucleus elastic scattering. In general the standard parametrisations of nuclear form factors were considered inadequate. Details of the fits are presented in the appendix. 3.2. Hydrogen The same basic formula was used, with M r set to the proton mass. For the form factor, a modified dipole was used, as done by Stein et al. [8]. 3.3. Quasi-elastic scattering The quasi-elastic scattering from nucleons bound in the nucleus differs from that for free nucleons in three respects: (i) the nucleons are not at rest in the lab frame; (ii) the nucleons are bound; (iii) the Pauli principle can restrict scattering into new states which are already occupied. This causes a suppression of quasi-elastic scattering at small values of Q2. These effects have to be allowed for by assuming a particular model of the nucleus, and thus give model-dependent systematic uncertainties in the calculation

J. Bailey et al. / Shadowing in low IQ21electroproduction

373

s

-27

+

lO

-28

10

10-2~

10-30

+ii

10-31 '

10 u}

%

-32

10-3~

o

I I

10- 3 4 '

"o o) C "o

t I I I

10- 3 5 '

I I

lo-~

I I

10- 3 7

10- 3 8 . 10- 39' I I t

lo-~.

o.b6 o;o7 0'.;0 0;26

13.80

2.60

~Q--2fm-1

10 - 8

10- 7

10-6

10- 5

10 -4

10 -3

10-2

10-1

1 - COS Ok

Fig. 2. The radiative elastic electron-uranium cross section as a function of the angle of the radiated p h o t o n for E = 5 GeV, 0 e = 6 ° and E' = 1.6 GeV. The solid line represents F(Q 2) = 1, and the dashed line the true nuclear form factor.

of the quasi-elastic tails. The various nuclei were treated as follows: Deuterium: the suppression was calculated in the model of Bernab6u [I 3], which gives the simple result that the suppression of the quasi-elastic scattering is compensated by the coherent nuclear scattering from deuterium. This differs from the suppression factor (1 - F~)(Q2)) (Fb = deuteron form factor) used by some other authors [8].

374

J. Bailey et al. / Shadowing in low IQ2Lelectroproduction

Heavier nuclei: the nuclei were treated as a Fermi gas using the model of Moniz [14]. The nucleons were given a Fermi momentum kF which we obtained from the experimental study of quasi-elastic scattering by Whitney et al. [15]. Following Moniz, the nucleon mass was replaced by an effective mass which varied, depending on kinematics, between 0.6 GeV and the free nucleon mass. Modifications to the quasi-elastic tails due merely to the Fermi momentum and binding energy of the nucleon were in general small since they resulted in an effective shift in the

C

Be

1.0-

1.0-

S

s

0.208 GeV/c

0.5

011

012 013 Q2e~ (GeV/c)2

0.5-

014

/~1

GeV/c

// 011

A~

012 013 Q2e~.(GeV/c)2

0'.4

Co, Nb

1.0"

1.0 ¸

S

S

0.5- ~ K F

0'.~

=0.238 GeV/c

012 013 Q2e£(GeV/c)2

O.5-

0~4

011

012 013 Q2e~ (GeV/c)2

Ta, U 1.0 Fermi gas

S

m----

0.5. ~ K F

0.1 i

Bernabeu

--0.265 GeV/c

o'2 013 Q2e.~(GeV/c)2

014

Fig.3.Suppression factors for quasi-elastic scattering.

014

J. Bailey et al. / Shadowing in low IQ21electroproduction

375

scattered energy spectrum of the electron, and far from the quasi-elastic peak the yield does not depend strongly on E'. These effects were small enough (a few per cent) to neglect. Far more significant was the suppression of the low-Q2 quasielastic scattering by the Pauli principle. The Moniz model gave values for the quasielastic tails which, when compared with the free-nucleon tails, allowed the calculation of a suppressionifactor to be applied to the free-nucleon tails at each value of Q2. These suppression factors are shown for each nucleus in fig. 3. The Fermi gas model is expected to give a reasonable description of the heavier nuclei, but has also been used with considerable success for lighter nuclei, in particular by Stanfield et al. [16] for carbon. However, an alternative approach can be made using shell-model wave functions. Bernab6u [ 13] has calculated the shellmodel suppression factor for carbon, which is shown in fig. 3 compared with the Fermi gas model suppression factors. It should be noted that since the dominant contribution to the quasi-elastic tail comes from the s-peak, the Q21 on this graph is the Q2 of the true electron-nucleon scattering, which is considerably smaller than our experimental Q2 because of radiation emitted by the electrons. The two models give broad agreement at high Q2, but disagree at low Q2, where the suppression is largest. Heimlich et al. [ 17], using both their own data and that of Stanfield et al. [16] have extracted the suppression factor experimentally for carbon. The results would appear to favour the Fermi gas model, although it is clear that where the suppression is large there is considerable uncertainty in the calculated quasi-elastic tails.

4. Results The event yields are shown in table 4 in the arbitrary units of events per SEM count normalised to the equivalent thickness of hydrogen,ltogether with the statistical counting and background-subtraction errors. We also show the computed contributions for coherent elastic and quasi-elastic tails. The total systematic error is evaluated as the linear sum of the following contributions. (i) An uncertainty of 2% in the experimental event yield due to uncertainty in the density of the targets. (ii) 5% of the computed coherent elastic tail. This is regarded as a reasonable uncertainty associated with the calculation of tails and form factors, although at our lowest Q2 setting, where this tail dominates, there are indications that this error may be underestimated. (iii) 2% of the computed quasi-elastic tail. This error is made smaller than the corresponding coherent ltail since in the calculation of ratios of cross sections between iheavy nuclei and deuterium it largely cancels out. (iv) An uncertainty of 15% in the calculation of the suppression of the quasielastic tail, i.e., 15% of the suppressed portion of the tail. This serves to increase the systematic error at low Q2 where the suppression may be least reliable.

Table 4 Event yields per nucleon, with calculated coherent elastic and quasi-elastic tails and statistical and systematic errors E'

0 (degrees)

1.2

6

1.6

2.0

2.4

2.8

1.2

6

6

6

6

8

Yield• nucleon

Coherent tail

Quasi tail

Subtracted yield

Statistical error

Systematic error

H D Be C A1 Co Nb Ta U

8.782 5.663 8.170 11.263 14.430 16.820 18.410 21.760 25.500

5.688 1.902 6.409 9.656 12.775 17.599 14.778 19.675 25.448

0.789 1.106 1.252 1.038 0.910 0.865 0.774 0.791

3.094 2.972 0.655 0.355 0.617 -1.689 2.767 1.311 -0.739

0.177 0.080 0.130 0.270 0.250 0.420 0.480 0.540 0.560

0.318 0.177 0.717 0.992 1.192 1.477 1.355 1.650 2.019

H D Be C A1 Co Nb

9.055 6.235 6.811 8.752 8.806 9.515 10.700

4.679 1.167 3.534 4.871 4.907 5.694 5.870

1.015 1.235 1.378 1.149 1.013 0.949

4.376 4.053 2.042 2.503 2.750 2.808 3.881

0.143 0.103 0.160 0.184 0.215 0.370 0.330

0.298 0.174 0.477 0.622 0.610 0.667 0.687

H D Be C A1 Co Nb Ta U

9.971 7.527 7.408 7.902 7.936 8.313 9.410 10.920 10.700

4.194 0.759 1.865 2.332 1.997 2.783 3.150 3.891 4.757

1.223 1.370 1.522 1.274 1.145 1.098 0.973 1.002

5.777 5.545 4.173 4.048 4.665 4.385 5.162 6.056 4.941

0.152 0.088 0.170 0.190 0.160 0.210 0.260 0.360 0.340

0.304 0.194 0.366 0.423 0.396 0.453 0.487 0.543 0.585

H D Be C A1 Co Nb Ta U

11.757 9.394 8.492 9.350 8.896 9.835 10.440 10.810 11.620

4.045 0.508 0.932 1.045 0.930 1.550 1.550 1.855 2.240

1.441 1.530 1.700 1.427 1.287 1.240 1.110 1.123

7.712 7.445 6.030 6.605 6.539 6.998 7.650 7.845 8.257

0.180 0.100 0.170 0.200 0.210 0.270 0.340 0.480 0.440

0.336 0.229 0.312 0.358 0.345 0.399 0.407 0.417 0.454

H D Be C A1 Co Nb Ta U

15.562 12.700 11.510 11.980 12.070 11.950 12.600 12.860 13.580

4.147 0.353 0.448 0.455 0.494 0.795 0.696 0.815 1.009

1.708 1.786 1.983 1.680 1.505 1.435 1.296 1.311

11.415 10.639 9.276 9.542 9.896 9.650 10.469 10.749 11.260

0.213 0.110 0.200 0.210 0.210 0.270 0.340 0.430 0.410

0.415 0.297 0.332 0.363 0.367 0.388 0.391 0.396 0.421

H D Be

2.923 2.087 2.406

1.648 0.433 1.296

0.351 0.428

1.275 1.303 0.682

0.072 0.031 0.047

0.100 0.060 0.170

Table 4 (continued) E'

1.6

2.0

2.4

2.8

2.25

0 (degrees)

8

8

8

8

8.5

Yield/ nucleon

Coherent tail

Quasi tail

Subtracted yield

Statistical error

Systematie error

C A1 Co Nb

2.880 3.262 3.406 3.724

1.785 1.803 2.146 2.237

0.477 0.400 0.358 0.341

0.618 1.059 0.902 1.146

0.064 0.074 0.100 0.110

0.217 0.221 0.243 0.251

H D Be C A1 Co Nb Ta U

3.168 2.525 2.490 2.630 2.613 2.618 2.936 3.131 3.327

1.320 0.227 0.521 0.643 0.592 0.914 0.960 1.163 1.410

0.397 0.438 0.488 0.420 0.381 0.363 0.329 0.336

1.848 1.901 1.531 1.499 1.601 1.323 1.613 1.639 1.581

0.065 0.033 0.038 0.062 0.050 0.070 0.110 0.130 0.120

0.096 0.064 0.111 0.129 0.124 0.142 0.149 0.160 0.177

H D Be C A1 Co Nb

3.767 3.022 2.874 2.885 2.893 2.970 2.960

1.137 0.125 0.205 0.230 0.262 0.435 0.392

0.426 0.458 0.509 0.436 0.397 0.378

2.630 2.471 2.211 2.146 2.195 2.138 2.190

0.066 0.044 0.060 0.066 0.045 0.070 0.090

0.104 0.072 0.091 0.098 0.100 0.113 0.110

H

D Be C AI Co Nb Ta U

4.584 3.885 3.760 3.685 3.654 3.688 4.038 3.970 4.031

1.042 0.071 0.082 0.095 0.124 0.190 0.155 0.170 0.217

0.457 0.476 0.531 0.464 0.425 0.409 0.373 0.379

3.542 3.357 3.202 3.059 3.066 3.073 3.474 3.427 3.435

0.081 0.053 0.052 0.065 0.096 0.095 0.118 0.155 0.148

0.118 0.089 0.096 0.099 0.101 0.107 0.111 0.109 0.113

U D Be C AI Co Nb Ta U

5.890 4.953 5.197 5.284 5.112 5.256 5.362 5.578 5.333

1.017 0.041 0.036 0.045 0.053 0.074 0.051 0.054 0.074

0.498 0.511 0.550 0.498 0.460 0.441 0.405 0.416

4.873 4.414 4.650 4.689 4.561 4.722 4.870 5.119 4.843

0.099 0.066 0.086 0.100 0.106 0.123 0.139 0.171 0.155

0.143 0.110 0.121 0.124 0.121 0.127 0.128 0.131 0.128

H

3.417 2.785 2.772

0.820 0.058 0.070

0.360 0.369

2.597 2.367 2.333

0.099 0.032 0.052

0.089 0.064 0.071

2.777 2.617 2.755 2.687 2.812

0.108 0.164 0.132 0.146 0.190

0.361 0.332 0.315 0.228 0.299

2.308 2.121 2.308 2.313 2.323

0.087 0.050 0.062 0.074 0.084

0.076 0.077 0.077 0.073 0.081

D Be C A1 Co Nb Ta U

J. Bailey et al. / Shadowing in low IQ21electroproduction

378

4.1. Deuterium-hydrogen ratio The ratio oD/o H is shown in table 5 and as a function of Q2 in fig. 4. (The cross sections from now on are always calculated per nucleon.) We also show electroproduction data of Eickmeyer et al. [18], and photoproduction data of Armstrong et al. [19], Meyer et al. [20] and Michalowski et al. [4], all for 2 < v < 4 GeV. The consistency of all these data is excellent and the transition to photoproduction is smooth. This is contrary to the conclusions of Stein et al. [8], who claimed a suppression of deuterium cross sections at low E'.

4.2. Heavier targets Using the hydrogen and deuterium data we can compute for each element Aeff/a , defined as

AOA Aeef/A = 2NOD - ( N - Z )

op '

(cross sections per nucleon).

These results, together with the statistical and systematic errors discussed above are presented in table 6. We also show here the effect of using the Bernab6u [13] model to calculate the suppression of the quasi-elastic tail for carbon, and also (but as an illustration, since the suppression is specifically for carbon) for beryllium. Fig. 5 shows the data for each element plotted as a function of Q2. Points with systematic errors in Aeff/A >~0.1 have been omitted. We also show a selection of photoproduction data for 2 < v < 4 GeV from Brookes et al. [3], Heynen et al. [1 ], and

Table 5 Deuterium: hydrogen cross section ratio (per nucleon) E' (GeV)

0 (degrees)

OD/OH

Statistical error

Systematic error

1.2 1.6 2.0 2.4 2.8 1.2 1.6 2.0 2.4 2.8 2.25

6

0.961 0.926 0.960 0.965 0.932 1.022 1.029 0.940 0.948 0.906 0.911

0.061 0.038 0.029 0.026 0.020 0.063 0.040 0.029 0.026 0.023 0.037

0.114 0.075 0.061 0.052 0.043 0.093 0.064 0.046 0.040 0.035 0.040

8

8.5

J. Bailey et al. / Shadowing in low IQ21electroproduction

379

1.1-

1.0-

E O D

0.9-

;¢t

E (3-t-

0.8-

O

0.7-

olo

o11

o12

0~3

Q2 (GeV/c) 2 Fig. 4. T h e experimental d e u t e r i u m / h y d r o g e n cross section ratio. The errors s h o w n are statistical only. • This experiment, o ref. [4], a ref. [19], o ref. [18], • ref. [20].

Michalowski et al. [4], and the Q2 = 0.1 electroproduction data of Eickmeyer et al. [9]. Our data show the following features. 0) For 0.1 < Q2 < 0.2 (GeV/c) 2 and elements between Be and Co we observe significant shadowing, not inconsistent with the data of Eickmeyer et al. [9], and not very different from that seen in photoproduction at the same v. (ii) For Q2 > 0.25 (GeV/c) 2 we do not observe any shadowing. (iii) For the heavy elements we see no conclusive evidence for shadowing, although our errors are much larger for these elements. (iv) The use of the Bernab~u model reduces the apparent shadowing for C and Be but does not eliminate it. Most vector dominance models predict more shadowing that is seen experimentally, although more refined models (e.g., Ditsas and Shaw [12]) have reduced the shadowing predicted and give a more reasonable representation of the data. Nevertheless, the dependence of the shadowing on Q2, which we see seems to be too fast to be accommodated by such models. Similar results to ours have been obtained on C and A1 in a DESY experiment [29], while other electroproduction data [7,8] have been at higher Q2 and no conelusive evidence for shadowing has been claimed. However, a muon scattering

£ Bailey et aL/ Shadowing in low IQ21 electroproduetion

380 Table 6

Aeff/A, with statistical and systematic errors Statistical error

Systematic error

Aeff/A

0.221 0.119 0.208 -0.570 0.936 0.445 -0.251

0.044 0.091 0.084 0.143 0.165 0.184 0.190

0.243 0.334 0.402 0.500 0.463 0.561 0.686

0.474 0.399

Be C A1 Co Nb

0.508 0.618 0.681 0.698 0.967

0.042 0.048 0.056 0.094 0.087

0.121 0.156 0.154 0.169 0.178

0.726 0.853

6

Be C A1 Co Nb Ta U

0.756 0.730 0.843 0.794 0.936 1.101 0.900

0.034 0.036 0.032 0.040 0.050 0.069 0.065

0.073 0.080 0.078 0.087 0.096 0.110 0.114

0.891 0.874

2.4

6

Be C A1 Co Nb Ta U

0.813 0.887 0.879 0.943 1.032 1.061 1.118

0.026 0.029 0.031 0.039 0.049 0.067 0.063

0.051 0.055 0.054 0.063 0.066 0.069 0.076

0.863 0.946

2.8

6

Be C AI Co Nb Ta U

0.879 0.897 0.933 0.913 0.893 1.025. 1.076

0.022 0.022 0.022 0.028 0.034 0.043 0.042

0.042 0.042 0.044 0.046 0.049 0.052 0.056

0.895 0.911

1.2

8

Be C A1 Co Nb

0.522 0.474 0.812 0.691 0.877

0.039 0.050 0.060 0.079 0.088

0.133 0.168 0.174 0.189 0.197

0.765 0,741

1.6

8

Be C AI Co Nb Ta U

0.803 0.789 0.841 0.694 0.846 0.858 0.826

0.025 0.035 0.030 0.039 0.060 0.071 0.065

0.066 0.073 0.072 0.079 0.084 0.091 0.099

0.910 0.901

E'

0 (degrees)

Element

1.2

6

Be C A1 Co Nb Ta U

1.6

6

2,0

Aeff/A

(Bernab6u)

J. Bailey et al. / Shadowing in low IQ21electroproduction

381

Table 6 (continued) E'

Statistical error

Systematic error

Aeff/A

0.901 0.868 0.890 0.870 0.893

0.030 0.031 0.025 0.033 0.041

0.048 0.047 0.049 0.054 0.054

0.926 0.894

Be C A1 Co Nb Ta U

0.960 0.911 0.915 0.920 1.042 1.032 1.036

0.023 0.024 0.032 0.033 0.040 0.051 0.049

0.041 0.038 0.039 0.042 0.046 0.047 0.049

0.960 0.916

8

Be C A1 Co Nb Ta U

1.066 1.062 1.037 1.079 1.117 1.184 1.124

0.027 0.028 0.029 0.033 0.037 0.045 0.042

0.041 0.039 0.039 0.042 0.043 0.048 0.047

1.066 1.058

8.5

Be C AI Co Nb Ta U

0.996

0.027

0.043

1.003

0.979 0.904 0.986 0.996 1.004

0.039 0.025 0.031 0.037 0.041

0.042 0.042 0.045 0.046 0.050

0 (degrees)

Element

2.0

8

Be C AI Co Nb

2.4

8

2.8

2.25

Aeff/A

(Bernab~u)

experiment at BNL [22] has seen evidence for shadowing at v ~ 3 GeV, Q2 ~ 0.5 (GeV/c) 2 (x ~
5. Conclusions We have observed shadowing in inelastic electron scattering on light elements which is compatible with photoproduction experiments at low Q2, but no evidence is seen for shadowing at Q2 > 0.25 (GeV/c) 2.

J. Bailey et al. / Shadowing in low IQ21 eleetroproduction

382

0

"~'~0.8

0

0:1

0:2 r ~ 1 0.3

Be Q2(GeV/c)2 0.1 0.2 0;3

~

2(GeV/c)21

Q2(GeV//c)2 0

"~0.8

Nb

0

~ ~°i Ta

!(GeV/c)2 0

~I°i

1.0l "~'~0.8

0

1.0~ ~ 0.8-

III°0

Q2(GeV/c)2 I 0.1 O.2 0,.3

Co

Fig. 5. Experimental values of Aeff/A as a function of Q2. o ref. [9], • ref. [1], • ref. [3], vref. [4], ref. [21], o this experiment: the systematic error is indicated by the rectangle and the statistical error by the lines.

We would like to thank Professors A. Ashmore and A. Donnachie for their support and encouragement, Professor L. Lovitch for useful discussions, E. Ashburner, K. Connell and D. Clarke for excellent help during the experiment, A.K. Nandi for sterling work on the radiative tails, the Daresbury Laboratory Engineering and Computing Division for their helpful service and the NINA crew for providing good beam.

J. Bailey et aL /Shadowingin low

IQ21electroproduction

383

1.1

< ¢) <

1.0 0.9" 0.8-

10

50

160

560

160

56o

A

(a)

1.1'

<

1.00.9.

< 0.8'

5b

1o

A (b) Fig. 6. Experimental values of Aeff/A as a function o f A compared with p h o t o p r o d u c t i o n . (a) t~ This experiment Q2 = 0.15 (GeV/c) 2, v = 2.2 GeV, o ref. [3], v = 1.7 - 3.0 GeV, • ref. [4], v = 2.0 GeV. (b) t~ This experiment, Q2 = 0.195 (GeV/c) 2, v = 3.0 GeV, o ref. [3], v = 2 . 7 5 3.95 GeV, • ref. [4], v = 3.27 GeV.

Appendix I n i t i a l l y t h e s t a n d a r d f o r m u l a e f o r n u c l e a r f o r m f a c t o r s w e r e u s e d , i.e., 9BeI2C:

F(q) = 1

fIX2 2k(2 + 3a)

e --x2[4k

(A.1)

F 2 (Q2)

81 °1

I

°! u~

oj -,

o,

F2(Q 2 )

I

I

t

I

ob',

3 I

F2(Q 2 )

J J I -

0

v

uo,zlanpo.tdoJlaala IZOI ato l ul. gu!atopattg / 7a la ,(al!alt T

l, Sf,

05

10

15

EL

0

U.

0

05

10

15

vo

lO-4

10-3

%

0

10-4J

10. 3

10. 2

10. 2

J

10-

10-1 •

(e)

0.5 ' % ~ fm-1

1.0

Fig. 7. Nuclear form factor fits. (a) 12C (ref. [24]), (b) 27A1 (ref. [25]), (e) S9Co (ref. [26]), (d) 93Nb (ref. [27]), (e) 181Ta (ref. [28]), (f) 238U (ref. [28]).

10- 4

I0 3

j" v0

10. 2

10 -q .

(d)

1

\

(f)

1.5

~o t~

386

J. Bailey et al. / S h a d o w i n g in low IQ21 electroproduetion

Table 7 Parameterisation of nuclear form factors Element

qmin (fm-I)

qmax (fm-1)

A1

A2

A3

A4

9Be

0

-0

12C

0 1.33

1.33 **

1.004

1.224

-0.844

27A1

0 1.34 1.59

1.34 1.59

1.560 9.84 X 10 -2

2.303 0.101

0 -1.07 X 10 -2

2.026 -3.57 x 10 -3

s 9Co

0 1.00 1.25 1.42

1.00 1.25 1.42 **

2.406 -9.19 X 10 -2 -0.134

-6.00 0.182 0.313

30.5 -1.98 X 10 -2 -0.167

-3.78 -2.47 x 10 -2 2.64 Xl0 -2

93Nb

0 0.87 1.05 1.26 1.46

0.87 1.05 1.26 1.46 1.68

3.096 3.84 X 10 -3 9.63 X 10 -2 - 3 . 3 0 X 10 -2 -0.267

13.14 0.252 7.63 X 10 -2 0.353 0.312

11.16 -0.499 -0.134 -0.313 -0.117

0 0.556 3.78 x 10 -2 7.39 x 10 -2 1.45 x 10 -2

1.68

oo

0 1.06 1.42. 0

1.06 1.42 ** 1.42 **

5.01 -9.35 X 10 -2

31.67 -2.39 X 10 -2

-47.57 0.158

28.91 -9.31 x 10 -2

lSlTa

2aSU

1.42

5.691

6.845

7.35

0.249

1.15

with a = (Z - 2)/3,

k = 3(2 + 5 a ) / ( 2 ( 2 + 3 a ) ) ,

x = q . l .O7 A U3 ,

(qinfm-1);

27A1 - 238U:

e-(bq)2/6 F ( q ) = 1 + 1q2c2 '

b = 2.4 f m ,

c = 1.07AU3 fm.

(a.2)

with

It was f o u n d t h a t (A. 1 ) g a v e a g o o d r e p r e s e n t a t i o n o f elastic s c a t t e r i n g d a t a for 9Be, while ( A . 2 ) was n o t s u f f i c i e n t l y a c c u r a t e for o u r purposes. Fits were m a d e t o available d a t a at low q2 elastic e l e c t r o n s c a t t e r i n g t o o b t a i n F ( q 2 ) . T h e q2 range was divided i n t o regions over w h i c h a simple p a r a m e t r i s a t i o n gave a g o o d fit.

J. Bailey et al. /Shadowing in low IQ21electroproduction

A5

A6

A7

A8

A9

387

Alo

A11

Para-

metrisation (A.1) 0.273

5.22 × 10-2

0 0

0 0

0 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 -0.241 0 0 0

0 0 0 0 0

0 0 0 0 0

0 -6.60 x 10-a -1.05

0 1.78 X 10-2 -0.625

-6.21 × 10-2

0

0 -0.757 0

0

0

(A.3) (A.1)

0.913 -0.213 0 0

(A.3) (A.4) (A.2)

0 0 0

0 0 0

0 0 0

(A.3) (A.4) (A.4) (A.2)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

(A.3) (A.4) (A.4) (A.4) (A.4) (A.2)

0 -3.51 X 10-a

0 0

0 0

0 0

0 0

-0.205

8.13 X 10-3

8.50 x 10-2

0

0

(A.3) (A.4) (A.2) (A.3) (A.2)

(i) The first region 0 ~< q ~< q'. N F ( q ) = 1/[1

+ ~ Anq2"l,

N
,

(A.3)

n=l

A 1 = ~(r2), (ref. [231). q' was the highest value o f q over which a good fit could be obtained. (ii) Further regions q > q' (number of these regions determined by goodness of fits). N F(q) = ~ A n q z ( n - ' ) ,

( N < 11).

(A.4)

n=l

Table 7 and fig. 7 show the results of these fits. 9Be was always very well represented by the simple formula (A.1).

J. Bailey et al. /Shadowing in low IQ21 electroproduction

388 References [1] [2] [3] [4] [5] [6] [7] [8] [9] 10] 11] 12] 13] 14] 15] 16] 17] 18] 19] 20] 21] 22] 23] 24] 25] 261 27] 28] 29]

V. Heynen et al. Phys. Lett. 34B (1971) 651. D.O. Caldwell et al., Phys. Rev. Lett. 23 (1969) 1256; Phys. Rev. D7 (1973) 136. G.R. Brookes et al., Phys. Rev. D8 (1973) 2826. S. Michalowski et al., Phys. Rev. Lett. 39 (1977) 737. S. Brodsky and J. Pumplin, Phys. Rev. 182 (1969) 1794; D. Schildknecht, Nucl. Phys. B66 (1973) 398. P. Ditsas et al., Nucl. Phys. B99 (1975) 85; G. Cocho et al., Nucl. Phys. B78 (1974) 269. W.R. Ditzler et al., Phys. Lett. 57B (1975) 201. S. Stein et al., Phys. Rev. D12 (1975) 1884. J. Eickmeyer et al., Phys. Rev. Lett. 36 (1976) 289. A. Del Guerra et al., Nucl. Phys. B99 (1975) 253. J. Bailey et al., Contribution to 17th Int. Conf. on High-energy physics, London, 1974. Y.S. Tsai, SLAC-PUB-848, unpublished. J. Bernab6u, Nucl. Phys. B49 (1972) 186. E.J. Moniz, Phys. Rev. 184 (1969) 1154. R.R. Whitney et al., Phys. Rev. C9 (1974) 2230. K.C. Stanfield et al., Phys. Rev. C3 (1971) 1448. F.H. Heimlich et al., Nucl. Phys. A231 (1974) 509. J. Eickmeyer et al., Phys. Lett. 63B (1976) 104. T.A. Armstrong et al., Phys. Rev. D5 (1972) 1640; Nucl. Phys. B41 (1972) 445. H. Meyer et al., Phys. Lett. 33B (1970) 189. P. Ditsas and G. Shaw, Nucl. Phys. Bl13 (1976) 246; and private communication. M. May et al., Phys. Rev. Lett. 35 (1975) 407. C.W. De Jager et al., Atomic and Nuclear Data Tables 14 (1974) 479. J.A. Jansen et al., Nucl. Phys. A188 (1972) 337. R.H. Lombard et al., Nucl. Phys. A101 (1967) 601. H. CranneU et al., Phys. Rev. 121 (1961) 283. N.G. Shevchenko et al., Soy. J. Nucl. Phys. 5 (1967) 676. B. Hahn et al., Phys. Rev. 101 (1956) 1131. S. Hartwig et al., DESY preprint DESY 77/55 (1977).