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Electron scattering
Electron Scattering INGO SICK Department of Physics, University of Basel, Basel, Switzerland
ELECTRON
SCATTERING
EXPERIMENTS
The first part of these lectures deals w i t h t h e purely experimental aspects of electron scattering. The m o t i v a t i o n for doing these experiments, the results obtained, and their interpretation in terms of physics will be described in the following sections. The experimental equipment at todays accelerators really starts right at the end of the accelerator. These machines produce beams of a quality barely adequate for the experiments planned. A careful preparation of the beam by the beam switchyard therefore is needed. With energy-loss spectrometers, the now-standard setup used, the beam switchyard that carries the beam from the accelerator to the target actually becomes an integral part of the spectrometer used to detect and energy-analyze the scattered electrons. The higher-energy electron accelerators used at present for nuclear structure studies produce beams of energy B~700 MeV (Bates), 500 MeV (NIKHEF) and 700 MeV (Saclay). The energy resolution of these beams is typically AE/E=2.10-3, the duty cycle of order 1%. The typical setup that has developed over the past ten years incorporates a beam switchyard that distributes the beam to different experimental halls; it simultanously serves to energy-disperse the beam such that on target different beam energies are correlated with different locations. A rotator moves the energy dispersion, which is produced in the horizontal plane, to the vertical one. The electrons scattered by the target are analyzed in the vertical plane by a magnetic spectrometer that measures the energy and (explicitly or implicitly) the geometric location of the scattering vertex on target. Having m e a s u r e d both scattered and initial electron energy, the energy loss of the electron, the quantity of interest for nuclear physics, can be deduced with an accuracy far superior to the AE of the beam. This type of setup has a number of features tageous for high-energy electron scattering
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that make it very advanexperiments. These ex-
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Ingo Sick
p e r i m e n t s suffer from two m a j o r problems: the low c o u n t i n g rates, and not very good e n e r g y resolution. The former is due to the w e a k ness of the e l e c t r o m a g n e t i c c o u p l i n g (which will be an a d v a n t a g e w h e n i n t e r p r e t i n g the s c a t t e r i n g data). The latter is due to the low m a s s of the electron, w h i c h r e q u i r e s high energy, i.e. very small AE/E, in order to achieve short w a v e length and good spatial resolution. The e n e r g y loss s p e c t r o m e t e r t e c h n i q u e s a l l o w to use the full b e a m intensity, w h i c h helps to i n c r e a s e countrates. The energy r e s o l u t i o n obt a i n a b l e is a factor 10-20 b e t t e r than the one of the i n c i d e n t b e a m ; w i t h an o v e r a l l r e s o l u t i o n of AE/E~I0 -4 achievable, e x p e r i m e n t s w i t h an e n e r g y r e s o l u t i o n of <50 K e V become p o s s i b l e d e s p i t e the large incident energies. Fig. 1 shows the layout of a typical switchyard, the one of the ALS m a c h i n e at S a c l a y (Leconte, 1980). In order to start w i t h a b e a m of w e l l d e f i n e d optical properties, the b e a m is focussed at the end of the a c c e l e r a t o r to a v e r t i c a l line. A c o l l i m a t o r of 0.5 mm width, u s e a b l e o n l y at low a v e r a g e b e a m i n t e n s i t y (low r e p e t i t i o n rate), allows to check the fraction of the b e a m that is w i t h i n the d e s i r e d spot. The m a g n e t labelled B4 d e f l e c t s the beam by ~45 °, and p r o v i d e s the m a i n e n e r g y dispersion. Its m a g n e t o - o p t i c a l properties, a d j u s t e d by the i n c l i n a t i o n of e n t r a n c e and exit p o l e t i p angles, are such that a m o n o - e n e r g e t i c b e a m is focussed to a v e r t i c a l line at the l o c a t i o n of the e n e r g y slits. The slit e n e r g y - d e g r a d e s those e l e c t r o n s that fall o u t s i d e the e n e r g y band of AE/E=I0- ~ desired; the v e r t i c a l ext e n s i o n of the b e a m at this l o c a t i o n serves to p r o t e c t the slits a g a i n s t d a m a g e from the high intensity, t y p i c a l l y 100~A, beam. The q u a d r u p o l e lens QD serves to further i n c r e a s e the energy d i s p e r s i o n of the b e a m by about a factor of 2; QD can be tuned such as to obtain at the target the d i s p e r s i o n r e q u i r e d by the spectrometer. The m a g n e t B5 d e f l e c t s the b e a m by a n o t h e r ~45o; the m a i n f u n c t i o n of B5 is to d e f l e c t away those e l e c t r o n s that have been e n e r g y - d e g r a d e d by the slits, such that they can be r e m o v e d by collimators. D o w n s t r e a m of B5, a set of 5 q u a d r u p o l e m a g n e t s Q R I + Q R 5 rotates the e n e r g y d i s p e r s i o n of the b e a m to the v e r t i c a l plane. As c o m p a r e d to the usual o r i e n t a t i o n of q u a d r u p o l e lenses, a r r a n g e d such that x- and y-coordinates are not coupled, these q u a d r u p o l e s are r o t a t e d by 45 ° . By an a p p r o p r i a t e choice of q u a d r u p o l e s t r e n g t h and distance, the e n t i r e r o t a t o r o b t a i n s o p t i c a l p r o p e r t i e s such that x and y of the b e a m are exchanged, w i t h o u t any further f o c u s s i n g action. The pair of q u a d r u p o l e s QFI, QF2, focusses the b e a m onto the target. For a m o n o c h o r m a t i c i n c i d e n t beam, a h o r i z o n t a l line image of a few m m w i d t h is produced. For a b e a m of finite e n e r g y w i d t h AE/E a v e r t i c a l d i s p e r s i o n of ~imm/10 -4 is produced. The d e c o u p l i n g of v a r i o u s parameters, d i s p e r s i o n (QD), r o t a t i o n (QR), m o n o c h r o m a t i c image (QF), helps to e ~ f i c i e n t l y adjust the b e a m p r o p e r t i e s desired. The s p e c t r o m e t e r images a point on the target to a point in the focal plane d e p e n d i n g on e l e c t r o n energy. For e n e r g y - l o s s s p e c t r o m e ters, an i m p o r t a n t q u a n t i t y is the d i s p e r s i o n D=AX/AE, w h e r e AX is the t a r g e t c o o r d i n a t e in the d e f l e c t i o n p l a n e and AE the e n e r g y change for fixed focal plane coordinate. For a g i v e n s p e c t r o m e t e r d i s p e r s i o n D, the s w i t c h y a r d d i s p e r s i o n is chosen to be -D. Hereby all electrons, i n d e p e n d e n t of their initial energy, are focussed
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Electron Scattering
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onto the same point in the focal plane if they have lost the same amount of energy in the target. The position along the focal plane thus no longer corresponds to scattered energy, as it does in conventional systems working with pointlike spotsize on target, but to energy loss. The spectrometers used always deflect in the vertical plane; the horizontal deflection mode used in most lightqon spectrometers is not advantageous for electron scattering. Vertical deflection allows the largest range of scattering angles. This feature is of great importance when one tries to separate charge and magnetic form factors by exploiting their dominance at small and large scattering angles, respectively. In addition, vertical deflection decouples the measurement of energy from the one of scattering angle. This scattering angle must be m e a s u r e d with a precision superior to the one determined by the spectrometer solid angle defining slits if a degradation of the m i s s i n g - e n e r g y resolution due to variation of recoil nucleus energy is to be avoided. In most spectrometers the transverse extension of the beam on target is mapped to a fairly narrow focal plane. This helps to decrease the size of the detector and reduces background. In these cases, the information on the scattering angle is contained in the angle between e l e c t r o n - t r a j e c t o r y and spectrometer median plane. A number of different types of spectrometers reflecting the progressing art of design are presently in use at the different facilities.
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Here, we show o n l y one, the c l a s s i c a l ~180 ° s p e c t r o m e t e r as p r e s e n t l y used, e.g. in Saclay. In this type of spectrometer, fig. 2, the e l e c trons are d e f l e c t e d by an i n h o m g e n e o u s m a g n e t i c field, w i t h a field index chosen to p r o v i d e f o c u s s i n g in both the d e f l e c t i o n plane and p e r p e n d i c u l a r to it. W i t h such a spectrometer, one can a c h i e v e an energy r e s o l u t i o n of ~1.5"10 -4 over the full solid angle of 4.5 msr, and 10 -4 over m u c h of it. The b a f f l e s (fig. 2) stop stray e l e c t r o n s p r o d u c e d by p r i m a r y e l e c t r o n s o u t s i d e the 10% energy a c c e p t a n c e of the spectrometer. The large d e f l e c t i o n angle, the g o o d s h i e l d i n g of the d e f l e c t i o n v o l u m e by the m a g n e t yoke t o g e t h e r w i t h a thick s h i e l d i n g (0.8m iron) of the focal plane p r o v i d e s e x c e l l e n t background rejection; cross s e c t i o n s less than i0-38cm2 can be measured. M o r e recent s p e c t r o m e t e r designs, as the one at Bates (Bertozzi,1979) or N I K H E F (de Vries, 1978) feature several d e f l e c t i n g m a g n e t s w i t h h o m o g e n e o u s fields and curved e n t r a n c e / e x i t pole faces. T h e s e pole face c u r v a t u r e s give better c o n t r o l over s p e c t r o m e t e r aberrations, and allow, as in the case of the N I K H E F QDD, to a c h i e v e a rather flat focal plane. The q u a d r u p o l e - 2 dipole (QDD) a r r a n g e m e n t p r e s e n t l y rep r e s e n t s the o p t i m u m c o m b i n a t i o n for good r e s o l u t i o n , l a r g e solid angle and small size (cost). The q u a d r u p o l e creates a cross over of e l e c t r o n t r a j e c t o r i e s w i t h i n the DD magnets, thus d i m i n i s h i n g the g a p s i z e for given solid angle. A m u l t i p o l e e l e m e n t in b e t w e e n the two d i p o l e s can be used to c o r r e c t h i g h e r order aberrations. This N I K H E F QDD p r e s e n t l y achieves < i ~ -4 r e s o l u t i o n for 5 m s r solid angle.
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Electron Scattering
The next generation spectrometer probably will feature one additional element : a position sensitive thin detector placed after the first dipole, at a place where an intermediate image of the target is formed (fig. 3). In such an arrangement (Zeidmann, 1982) the "first" 3QD spectrometer serves to measure the target coordinates, while the "second" 3D spectrometer serves to measure the electron energy. Decoupling these two quantities gives new degrees of freedom, and promises excellent resolution, a few times 10 -5 , in combination with large, 20msr, solid angle. The focal plane detector has two principal functions : to identify the scattered particles as an electron, and to record with high precision the geometry of its track. This has to be done in a very short time, like ius. The large dynamical range of electron scattering cross sections (up to 12 decades have.been measured, see next section), together with the poor duty cycle of todays accelerators can lead, at low momentum transfer, to very high instantaneous counting rates. In order to identify the scattered electron, the most constraining capability is the one to reject pions. This can be achieved by using Aerogel- or Gas-Cerenkov counters; due to their low index of refraction, these counters do not respond to pions of less than ~i GeV/c momentum. The Gas-Cerenkov counter, ~30cm deep and containing Freon at 1 atmosphere pressure used at Saclay, gives a 100% rejection of pions. To suppress noise, a fast coincidence (20ns) with two rows of plastic scintillator detectors is required. In order to avoid problems due to multiple scattering, these detectors are located after the wire chambers used to measure the track geometry.
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The l o c a l i z a t i o n d e t e c t o r that has found g e n e r a l a c c e p t a n c e in electron s p e c t r o m e t e r s is the m u l t i w i r e d r i f t chamber. This type of detector is thin e n o u g h to allow m u l t i p l e m e a s u r e m e n t s of the same track. It gives e x c e l l e n t spatial r e s o l u t i o n (~0,2mm) w i t h a ressonably small n u m b e r of active wires (typically one per 5 mm). The type of d r i f t c h a m b e r used is d i c t a t e d by the fact that for the s p e c t r o m e t e r s u s e d the angle b e t w e e n focal plane and track is far from 90°~ t y p i c a l l y it is c l o s e r to 30-50 ° . In this case the v e r t i cal drift c h a m b e r (fig. 4) d e v e l o p e d l a r g e l y at Bates (Bertozzi,1977), r e p r e s e n t s a very e c o n o m i c a l m e a n s to a c c u r a t e l y locate the track. The c h a m b e r contains, in r e g u l a r alternation, active (thin) and passive (thick) wires, a r r a n g e d such that b e t w e e n w i r e and h i g h - v o l t a g e plane cells of rather c o n s t a n t e l e c t r i c field are created. Secondary e l e c t r o n s p r o d u c e d in 3 or m o r e cells drift to the sense wires, w h e r e the signal is amplified; the arrival time is a m e a s u r e of the d i s t a n c e b e t w e e n track and sense wire. W i t h 3 or m o r e cells triggered, one c o o r d i n a t e of the track can be r e c o n s t r u c t e d i n d e p e n d e n t of a b s o l u t e time m e a s u r e m e n t s . (The i n c l i n a t i o n of the track is obtained as well, but w i t h a p r e c i s i o n g e n e r a l l y not sufficient). Since such a m e a s u r e m e n t only r e q u i r e s the signal from the 3 wires g i v i n g the s h o r t e s t times, every third w i r e can be c o n n e c t e d to the same time m e a s u r i n g device. As m e n t i o n e d above, b o t h a m e a s u r e m e n t in the d i s p e r s i v e d i r e c t i o n and the one o r t h o g o n a l to it are needed. Such m e a s u r e m e n t s can be d o n e s u c c e s s i v e l y , since the m u l t i p l e s c a t t e r i n g due to M W P C w i n d o w s is low e n o u g h to not d e t e r i o r a t e the track information. Two chambers, one in the focal plane, one ~20cm behind, w i t h wires o r t h o g o n a l to the s p e c t r o m e t e r m i d p l a n e can be used to d e t e r m i n e e l e c t r o n e n e r g y and the c o m p o n e n t of the s c a t t e r i n g angle out of the h o r i z o n t a l plane. Two f u r t h e r c h a m b e r s w i t h wires o r i e n t e d at 45 ° r e l a t i v e to the first ones, can be used to m e a s u r e d i s t a n c e and angle r e l a t i v e to the spect r o m e t e r midplane; this allows to r e c o n s t r u c t the s c a t t e r i n g angle. Given the m u l t i t u d e of chambers, the r e a d o u t system can b e c o m e q u i t e complex. N e v e r t h e l e s s , fast r e a d o u t (in ~s) is d e s i r e d to keep deadtime at low m o m e n t u m t r a n s f e r small. The perhaps m o s t c o m p l e x s y s t e m is the one p r e s e n t l y being i n s t a l l e d at Saclay. The focal plane is e l e c t r o n i c a l l y s e g m e n t e d into 4 i n d i v i d u a l sectors in order to r e d u c e pileup. In every sector, and for every one of the 4 c h a m b e r s ~ 3 timet o - d i g i t a l c o n v e r t e r s (TDC) r e g i s t e r the times of w i r e s i-l, i, i+l. The i n f o r m a t i o n on w i r e n u m b e r i is r e g i s t e r e d by p a t t e r n units which, like the T D C ' s , a r e read out u p o n a t r i g g e r f u r n i s h e d by the fast coi n c i d e n c e b e t w e e n C e r e n k o v and P l a s t i c counters. This d e t e c t o r is c a p a b l e to y i e l d b o t h the g o o d spatial r e s o l u t i o n (0,2mm) r e q u i r e d for g o o d e n e r g y r e s o l u t i o n and the h i g h count rate c a p a b i l i t y required for (lower resolution) c o i n c i d e n c e e x p e r i m e n t s of the type (e,e'p). Once the track of the s c a t t e r e d e l e c t r o n is c o m p l e t e l y m e a s u r e d in the focal plane, the e n e r g y r e s o l u t i o n a c h i e v a b l e a c t u a l l y b e c o m e s b e t t e r than w h a t is i m p l i e d by the s p e c t r o m e t e r o p t i c a l properties. By s o f t w a r e corrections, m a n y of the s p e c t r o m e t e r a b e r r a t i o n s can be removed.
Electron Scattering
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The t a r g e ~ u s e d for e l e c t r o n s c a t t e r i n g e x p e r i m e n t s have t h i c k n e s s e s of t y p i c a l l y t 1 0 0 m g / c m 2. For such targets the c o n t r i b u t i o n of energy straggling, r o u g h l y t.0.4 M e V / g c m -2, is of the o r d e r of o t h e r cont r i b u t i o n s to the o v e r a l l resolution. This s t r a g g l i n g is the dominant c o n t r i b u t i o n of the target as long as the t a r g e t angle can be c h o s e n to b i s e c t the d i r e c t i o n of i n c o m i n g and s c a t t e r e d electron. If the target, at v e r y large s c a t t e r i n g angle, m u s t be used in ref l e c t i o n geometry, the m u c h l a r g e r c o n t r i b u t i o n of e n e r g y loss, ~t-4 M e V / g c m -2, imposes a m u c h s m a l l e r t a r g e t thickness. A m a j o r p r o b l e m w i t h targets results from the h i g h b e a m i n t e n s i t i e s n e e d e d to c o m p e n s a t e for the low cross sections. With, say, 50~A e l e c t r o n b e a m intensity, m a n y t a r g e t s will m e l t rapidly. Although the use of the d i s p e r s e d b e a m t e c h n i q u e leads to a s p r e a d i n g out of the b e a m over ~icm, this often is not sufficient. M o v i n g the target h o r i z o n t a l l y , in order to spread the heat over a larger area, helps. The b e s t w a y to r e m o v e heat, h o w e v e r ', is to cool the t a r g e t w i t h a jet of H 2 - g a s of ~i T o r r pressure. This H 2 is p r o d u c e d u s i n g a c l o s e d c i r c u i t s y s t e m w i t h Roots pumps and can be p o i n t e d at the t a r g e t u s i n g nozzles. Due to the low density, this H 2 does not int e r f e r e w i t h the experiment, but it r e q u i r e s thin w i n d o w s (capable of w i t h s t a n d i n g a few T o r r pressure) to s e p a r a t e b e a m l i n e and spect r o m e t e r v a c u u m from the s c a t t e r i n g chamber. In o r d e r to o b t a i n a c c u r a t e cross sections, w i t h s y s t e m a t i c a l e r r o r s of the order of 1%, the target t h i c k n e s s and h o m o g e n e i t y m u s t be known to at least this accuracy. R e l a t i v e target t h i c k n e s s p r o f i l e s are best o b t a i n e d using x - r a y absorption; w i t h a y - s o u r c e of e n e r g y a p p r o p r i a t e l y c h o s e n to y i e l d a factor of 2~i0 absorption, the density p r o f i l e can be m e a s u r e d using a small Ge d e t e c t o r . The absolute density, a v e r a g e d over the area of the target hit by the beam, is o b t a i n a b l e from the c o m b i n a t i o n of one a b s o l u t e m e a s u r e m e n t and the r e l a t i v e d e n s i t y d i s t r i b u t i o n . This one a b s o l u t e r e f e r e n c e p o i n t can be o b t a i n e d by m e a s u r i n g w i t h a s u i t a b l e t a r g e t (often of n a t u ral i s o t o p i c composition) weight, area and r e l a t i v e d e n s i t y p r o f i l e of the entire target. For the o v e r a l l a c c u r a c y of the e x p e r i m e n t a l cross s e c t i o n s to be determined, a n u m b e r of a d d i t i o n a l v a r i a b l e s m u s t be k n o w n w i t h good p r e c i s i o n : e l e c t r o n b e a m energy, i n t e g r a t e d e l e c t r o n b e a m charge, s p e c t r o m e t e r solid angle and focal plane d e t e c t o r e f f i c i e n c y . The i n c i d e n t e l e c t r o n e n e r g y in g e n e r a l is o b t a i n e d from an a c c u r a t e m e a s u r e m e n t of the s c a t t e r e d e l e c t r o n energy. The field m a p s of high. r e s o l u t i o n s p e c t r o m e t e r s o f t e n are w e l l k n o w n from the m e a s u r e m e n t s p e r f o r m e d to v e r i f y the e n e r g y resolution. A field m e a s u r e m e n t at one point, p e r f o r m e d u s i n g a n u c l e a r m a g n e t i c r e s o n a n c e probe, then is s u f f i c i e n t to d e t e r m i n e the field integral, hence, e l e c t r o n e n e r gy, to a few parts in 104 . The e l e c t r o n b e a m i n t e n s i t y can be i n t e g r a t e d using t o r o i d p u l s e t r a n s f o r m e r s or a F a r a d a y cup. The latter is m o s t s u i t a b l e for absolute c h a r g e d e t e r m i n a t i o n s , and y i e l d s an a c c u r a c y of t y p i c a l l y a f r a c t i o n of one percent. For t a r g e t s of a p p r e c i a b l e thickness, m u l tiple s c a t t e r i n g c a u s e s a few p e r c e n t of the e l e c t r o n s to m i s s the PPNP-N*
418
Ingo Sick
F a r a d a y cup, u n l e s s b e a m r e f o c u s s i n g q u a d r u p o l e s d o w n s t r e a m of the t a r g e t are employed. T h e m i s s i n g f r a c t i o n can be d e t e r m i n e d by m e a suring ratios of F a r a d a y cup and t o r o i d i n t e g r a t e d c h a r g e s w i t h and w i t h o u t target. The s p e c t r o m e t e r solid a n g l e in g e n e r a l is k n o w n w i t h an a c c u r a c y of a f r a c t i o n of one p e r c e n t fro m the g e o m e t r i c a l p r o p e r t i e s of the sol i d - a n g l e d e f i n i n g slits. S o m e care is n e e d e d if the solid angle is not d e f i n e d by the slit in front of the first m a g n e t i c e l e m e n t of the s p e c t r o m e t e r , b u t r a t h e r by some i l l - d e f i n e d p i e c e of the v a c u u m chamber; this s i t u a t i o n u s u a l l y o c c u r s w h e n the b i g g e s t p o s s i b l e solid angle is used. U n d e r these c i r c u m s t a n c e s , the p r e c i s i o n is lower, since the s o l i d angle d e p e n d s on b e a m size, e l e c t r o n e n e r g y and alike. T h e q u a n t i t y m o s t d i f f i c u l t to m e a s u r e u s u a l l y is the focal p l a n e detector efficiency. It o f t e n is d e t e r m i n e d via a m e a s u r e m e n t of an a c c u r a t e l y k n o w n r e f e r e n c e cross section; the 12C and H 2 e l a s t i c cross s e c t i o n s g e n e r a l l y are used for this purpose. An i n d e p e n d e n t d e t e r m i n a t i o n of the a b s o l u t e e f f i c i e n c y is p o s s i b l e if the d e t e c t o r has e n o u g h e l e m e n t s to o v e r d e t e r m i n e e l e c t r o n i d e n t i f i c a t i o n and track m e a s u r e m e n t . In this case the r e d u n d a n c y of the d e t e c t i o n system can be e x p l o i t e d to d e t e r m i n e the e f f i c i e n c y of e v e r y i n d i v i d u a l element. W i t h t o d a y ' s c o m b i n a t i o n s of several ~[WPC, P l a s t i c and C e r e n c o v counters, this r e d u n d a n c y is a v a i l a b l e at low m o m e n t u m transfer, w h e r e b a c k g r o u n d r e j e c t i o n is of no concern. This proced u r e can be e x p e c t e d to p r o v i d e in the future the s t a n d a r d m e t h o d to d e t e r m i n e a b s o l u t e e f f i c i e n c i e s w i t h ~1% accuracy. An alternative p r o c e d u r e , using a s p e c i a l s p e c t r o m e t e r w i t h a p a r t i c u l a r l y simple d e t e c t o r , has b e e n d e v e l o p e d at Mainz (Reuter, 1982). For e x p e r i m e n tal p a r a m e t e r s d e t e r m i n e d w i t h the a c c u r a c i e s q u o t e d above, cross s e c t i o n s can be m e a s u r e d w i t h s y s t e m a t i c a l u n c e r t a i n t i e s of one to s e v e r a l percent. The m o s t p r e c i s e e x p e r i m e n t s , a i m i n g at the d e t e r m i n a t i o n of a b s o l u t e r e f e r e n c e cross sections, have r e a c h e d 0.5% sys t e m a t i c a l e r r o r (Cardman, 1980).
Nucleonic
Wave
Functions
In this section, I w i l l d i s c u s s the a p p l i c a t i o n of e l e c t r o n , s c a t t e ring to the i n v e s t i g a t i o n of n u c l e o n i c w a v e f u n c t i o n s in nuclei. To do so, I p i c k one p a r t i c u l a r e x a m p l e that allows to p r e s e n t the physics in a t r a n s p a r e n t w a y : the use of e l a s t i c e l e c t r o n s c a t t e r i n g for the d e t e r m i n a t i o n of n u c l e a r c h a r g e d e n s i t i e s . The m o t i v a t i o n for s t u d y i n g e l a s t i c e l e c t r o n s c a t t e r i n g r e s u l t s from our d e s i r e to m e a s u r e the shape of nuclei. As we w i l l see below, e l e c t r o n s c a t t e r i n g p r o v i d e s us w i t h a tool that c o m e s q u i t e c l o s e to the " e l e c t r o n m i c r o s c o p e " we are u s e d to r e l a t e to the i n v e s t i g a tion of s p a t i a l p r o p e r t i e s of v e r y small objects. For the p a r t i c u l a r case d i s c u s s e d below, the c h a r g e d i s t r i b u t i o n of a n u c l e u s can be measured. T h i s c h a r g e d i s t r i b u t i o n p r o v i d e s us w i t h the r a d i a l (and s o m e t i m e s azimuthal) d i s t r i b u t i o n of p r o t o n s in nuclei. This quantity is of g r e a t i n t e r e s t for the q u a n t i t a t i v e u n d e r s t a n d i n g of n u c l e o n wave functions.
Electron Scattering
419
E l e c t r o n s as a p r o b e are p a r t i c u l a r l y s u i t a b l e for a c c u r a t e d e t e r m i n a t i o n s of c h a r g e d e n s i t i e s . The i n t e r a c t i o n of an e l e c t r o n w i t h the n u c l e u s is known, and weak. C o n s e q u e n t l y , the o n l y u n k n o w n s occ u r r i n g are the n u c l e a r p r o p e r t i e s , and t h e y can be e x t r a c t e d f r o m e x p e r i m e n t in a q u a n t i t a t i v e w a y since the w e a k n e s s of the e l e c t r o m a g n e t i c i n t e r a c t i o n a l l o w s for a (basically) e x a c t d e s c r i p t i o n of the s c a t t e r i n g process. In o r d e r to d i s c u s s e l a s t i c s c a t t e r i n g , I w i l l c o n c e n t r a t e on one p a r t i c u l a r nucleus, the one of tin, w h i c h has a m a g i c p r o t o n n u m b e r Z=50. P r o t o n t r a n s f e r r e a c t i o n s i n d i c a t e that Z=50 is a v e r y g o o d shell closure; n e i t h e r h o l e s in the n o r m a l l y o c c u p i e d s h e l l s (~ig9/2) nor p a r t i c l e s e x c i t e d to the n o r m a l l y e m p t y shells, h a v e b e e n o b s e r ved. For such a n u c l e u s , the m o s t s o p h i s t i c a t e d c a l c u l a t i o n s of g r o u n d state w a v e f u n c t i o n s , the o n e s o b t a i n e d from H a r t r e e - F o c k theory, are a p p l i c a b l e , a n d a l l o w a q u a n t i t a t i v e c o m p a r i s o n to e x p e riment. In addition, the tin i s o t o p e s p r o v i d e a large v a r i a t i o n of n e u t r o n n u m b e r (A=I12+124 are stable); the s t u d y of isotopic d i f f e r e n ces then a l l o w s to s t u d y w h e t h e r the p r e s e n c e of an o p e n n e u t r o n shell a p p r e c i a b l y i n f l u e n c e s the p r o t o n c o n f i g u r a t i o n and density. S o m e i0 y e a r s ago, we p e r f o r m e d at S t a n f o r d an e x p e r i m e n t (Ficenec, 1972) on all the s t a b l e tin i s o t o p e s ; the d a t a r e s u l t i n g from this e x p e r i m e n t cover the r a n g e of m o m e n t u m t r a n s f e r up to 2 . 7 f m -I. In o r d e r t o e x t e n d the r e g i o n of m o m e n t u m t r a n s f e r , we c o m p l e t e d 2 y e a r s ago an e x p e r i m e n t (Cavedon, 1982) at Saclav, w h e r e we r e a c h e d for ll6sn and 124Sn, q = 3 . 5 f m -I. The d a t a r e s u l t i n g from these two e x p e r i m e n t s are shown in fig. 5. The c r o s s s e c t i o n s n o w c o v e r a r a n g e of m o r e than 12 decades, and the lowest p o i n t m e a s u r e d has a cross s e c t i o n of i 0 - 3 7 c m 2. T y p i c a l s y s t e m a t i c u n c e r t a i n t i e s are 3% and 1% for c r o s s s e c t i o n and cross s e c t i o n ratios; s t a t i s t i c a l e r r o r s v a r y b e t w e e n <1% at low q, to 100% at the h i g h e s t q.
V(~) i0~ [0°
|0-I IO-2 i0-3
Fig.5
i0-4 lO-S
iO-6 [0-7 i0-s i0 -9
i0-*o
i0-n
l
l
2
3
Q(FM "I)
E l a s t i c cross s e c t i o n s f o r 1 2 4 S n (in mb/sr).
420
Ingo Sick
In order to extract the charge d e n s i t y from the data, I will, for the moment, assume the v a l i d i t y of the Plane W a v e Bron A p p r o x i m a t i o n (PWBN). This w o u l d r e q u i r e that Z.~<< i , w h i c h is not the case. But we will see that the p r o c e d u r e can be e a s i l y m o d i f i e d to account for this a p p r o x i m a t i o n . In PWBA, the cross section for e l a s t i c a l l y a s p i n - z e r o nucleus can be d e r i v e d easily: da d--~ (6,E)
= -da (6,E) deMott
Here aM_tt is the M o t t cross a p o i n t ~ i k e o b j e c t of charge Z2.e2
cos26/2
d~Mott
4E 2
sin46/2
1 = ~ f p(r)
sin(qr) qr
for
(i)
describing
the scattering
from
(2)
E is the e l e c t r o n e n e r g y and 6 the s c a t t e r i n g angle. c o n t a i n i n g the n u c l e a r physics, the form factor F(q) F o u r i e r t r a n s f o r m of the n u c l e a r charge d e n s i t y p(r) F(q)
an e l e c t r o n
F2 (q)
section Z
da
scattering
The q u a n t i t y is given as the
4n r2dr
(3)
The v a r i a b l e of i m p o r t a n c e for e l e c t r o n scattering, the m o m e n t u m t r a n s f e r q g i v e n by the e l e c t r o n to the nucleus, is q = 2.E.sin6/2
(4)
(Center of mass effects, d e p e n d i n g on terms ted above, but trivial to include).
of order
l/A,
are n e g l e c -
E q u a t i o n (3) shows that, for CO charge scattering, the p h y s i c s inform a t i o n d e p e n d s on q only, and not on E,6 separately. This v a r i a b l e plays a m o s t i m p o r t a n t role for the i n v e s t i g a t i o n of the nucleus. Ac cording to eq. (3) the n u c l e a r charge d e n s i t y is sampled w i t h a sin(qr)/qr-function. For a g i v e n m a x i m a l m o m e n t u m t r a n s f e r q m a x the c h a r g e d e n s i t y is sampled w i t h a r e s o l u t i o n no finer than 1.5/qmax, this latter q u a n t i t y being, a p p r o x i m a t e l y , the F W H M of one of the peaks of sin(qr)/qr. The q u a n t i t y 1 . 5 / q m a x thus d i r e c t l y plays the role of the r e s o l v i n g power of our e l e c t r o n microscope. If we w a n t to see the d e t a i l e d features of the nucleus, we m u s t p u s h the e x p e r i m e n t to l a r ~ q m a x . E q u a t i o n (3) s u g g e s t s an easy w a y to e x t r a c t experiment. I n v e r t i n g the F o u r i e r t r a n s f o r m 0 (r) = Z _
4~
I n t e g r a t i n g over p(r). The first t e c h n i q u e (Sick, 3He and 4He.
/ F(q)
sin (qr) qr
q2dq
the charge yields
density
from
(5)
the e x p e r i m e n t a l l y d e t e r m i n e d form factors yields a p p l i c a t i o n of this Direct F o u r i e r T r a n s f o r m (DFT) 1978) c o n c e r n e d the d e n s i t y of very light nuclei,
Electron Scattering
421
The a p p l i c a t i o n of eq. (5) m e e t s two d i f f i c u l t i e s . First, P W B A is not a good a p p r o x i m a t i o n for h e a v y nuclei. Second, the m a x i m u m mom e n t u m t r a n s f e r of e x p e r i m e n t does not reach i n f i n i t y as r e q u i r e d by the upper i n t e g r a t i o n limit. Let us d i s c u s s the s o l u t i o n for those p r o b l e m s in turn. W h i l e eq. (I) is not v a l i d for large Z, one can e a s i l y d e t e r m i n e exp e r i m e n t a l form factors F e x _ such that eqs. 3, 5 are a p p l i c a b l e w i t h out loss of accuracy. In o r d e r to t r a n s l a t e cross s e c t l o n s into P W B A form factors, we can d e t e r m i n e a m o d e l charge d i s t r i b u t i o n that fits the e x p e r i m e n t a l cross sections. A c h a r g e d i s t r i b u t i o n d e p e n d i n g on a n u m b e r of p a r a m e t e r s is used to n u m e r i c a l l y solve the D i r a c equation that d e s c r i b e s w i t h o u t a p p r o x i m a t i o n (basically) the e l e c t r o n in the e l e c t r o s t a t i c field of the nucleus. The p a r a m e t e r s of this m o d e l d e n s i t y are v a r i e d until the cross s e c t i o n s c o m p u t e d agree w i t h the e x p e r i m e n t a l ones. P u t t i n g (Sick, 1974) F~xp(q)
= ~exp °mod
• F~o d
(q)
(6)
w h e r e q u a n t i t i e s h a v i n g the index "mod" are c o m p u t e d density, y i e l d s e x p e r i m e n t a l form factors that have of C o u l o m b d i s t o r t i o n removed. G i v e n the fact that tion d e p e n d s on v e r y global p r o p e r t i e s of p(r) o n l y tually) , the r e m o v a l of it via a m o d e l d e n s i t y does to Fex p any d e p e n d e n c e on the m o d e l density. (This by using m a n y d i f f e r e n t p(r) to e v a l u a t e eq. (6)).
using the m o d e l all the e f f e c t s the C o u l o m distor(mainly on Z, acnot i n t r o d u c e inhas b e e n v e r i f i e d
The Fex p for 124Sn r e s u l t i n g from the a p p l i c a t i o n of eq. (6) are shown in fig. 6. The m a i n d i f f e r e n c e to fig. 5 shows up in the dif-
F2(Q) iO-2
°°.,° iO-3
4 [O-'t
•
t%
Fig.6
i0 -s
*l i0 -6
i0 -7
%
I
i0 -a iO-9
i0 -Io 10- n
J
3
tl"1 Q(~*)
Experimental form tors for 124Sn.
fac-
422
Ingo Sick
fraction Fexp-
minima
of d~/d~,
which
turn
into
zeros
(and s i g n - c h a n g e s ) i n
The second d i f f i c u l t y m e n t i o n e d above c a n n o t be taken care of by a s u i t a b l e t r e a t m e n t of data only. The lack of i n f o r m a t i o n due to finite qmax can be m i n i m i z e d by p u s h i n g e x p e r i m e n t to the largest q, i.e. lowest cross section possible. However, this l i m i t a t i o n is no reason to resort to p r o c e d u r e s that d e t e r m i n e p(r) by i n t r o d u c i n g e x p l i c i t or implicit model assumptions. The d e n s i t y at any radius can be d e t e r m i n e d by s t u d y i n g eq. ( 5 ) p(r,qmax )
Z = 4-~
qmax [ F(q) o
sin(qr) qr
2 • q.dq
(7)
as a function of the upper i n t e g r a t i o n limit. The c o n v e r g e n c e of ~(r,qmax) towards the a s y m p t o t i c value p(r,~) = p(r) gives a d i r e c t estimate of the lack of k n o w l e d g e due to finite qmax" The way
result of eq. (7) can be d i s c u s s e d for r = O, where P(O,qma x) =
in a p a r t i c u l a r l y
transparent
qmax I F(q)q2dq o
(8)
For 124Sn, p(O,qmax) is shown in fig. 7. At every d i f f r a c t i o n m i n i m u m of do/d~, F(q) changes sign. Hence, p(O,qmax) r e p r e s e n t s a damped o s c i l l a t o r y function that c o n v e r g e s towards p(O). The d i f f e r e n c e b e t w e e n the last m e a s u r e d m a x i m u m - and m i n i m u m - v a l u e of p(O,qmax) can be taken as an honest, m o d e l i n d e p e n d e n t e s t i m a t e for the u n c e r t a i n t y p(O) is d e t e r m i n e d with. As is shown in fig. 7, p(O) has c o n v e r g e d to better than 2% to its a s y m p t o t i c value at q = 3.5 fm -I. This e x p l a i n s w h y e x p e r i m e n t has been p u s h e d to such a high m o m e n t u m transfer. To d e t e r m i n e p to
O(O,Q) ,
IO0
Fig.7
/ .000
0
" 1.0
" 2.0
' 3,0
' = Q(FY1-1)
C o n v e r g e n c e of p(O,qm~x) as function o~ qmax"
Electron Scattering
423
be£ter than 1% it typically takes a value of F2(qmax) of i0 -I0, independent of mass number; only then the contribution over the not-measured part of F 2 is small enough. The typical (ee)-experiment, which does not go beyond qmax~2fm -I, does not determine p(O) to better than 20%. For radii r>O, the sin(qr)/qr-term adds additional sign-changes of the integrand (eq. (7)). Consequently, the convergence will be more rapid. This DFT method, first applied to heavy nuclei in the interpretation of the 40Ca data (Sick, 1979), I consider to be the optimal method to extract the physics from the data. No model dependence whatsoever is introduced, the relation between data and density is most transparent, and systematical or statistical errors of the data can trivially be taken into account by error propagation. The results obtained for the charge density of 124Sn is shown in fig. 8. The density shows the familiar Fermi-distribution-like shape, with a small oscillatory component superimposed on small radii. In fig. 8 the experimental density is compared to 3 different HartreeFock predictions (Negele 1970, Campi 1972, Decharg~ 1980). These curves have been obtained by starting from an effective density dependent nucleon-nucleon interaction VNN determined either phen0menologically, or derived from the NN-interaction known from NN-scattering. In these HF-calculations the Schr6dinger equation for a nucleon moving in an average potential determined by VNN and the wave functions of the other (A-l) nucleons is solved selfconsistently. The major approximation made in these calculations is the assumption that the nucleons occupy the lowest shell-model orbits, and the cursory treatment of short-range NN correlations via the density-dependence of the effective force.
0,08
Fig. 8
0
2
4
6
R(FM)
Charge density of 124Sn, compared to DDHF predictions.
424
Ingo Sick
The c o m p a r i s o n b e t w e e n e x p e r i m e n t and c a l c u l a t i o n shows features typical for a number of other semimagic nuclei b e t w e e n 160 and 208pb. The global d e n s i t y is quite w e l l reproduced; in p a r t i c u l a r the surface region, and the n u c l e a r rms-radius, is w e l l explained. In the n u c l e a r interior, on the other hand, the t h e o r e t i c a l p r e d i c t i o n s show m u c h m o r e o s c i l l a t o r y s t r u c t u r e than given by experiment. The o s c i l l a t o r y s t r u c t u r e is due to the p r e s e n c e of s h e l l - e f f e c t s . The d e n s i t y is a sum of n u c l e o n radial w a v e functions squared, and these R2(r) e x h i b i t zeros and m a x i m a a c c o r d i n g to p r i n c i p a l q u a n t u m n u m b e r and a n g u l a r m o m e n t u m of the shell. In the total density, the s t r u c t u r e of these R2(r) still shows up as the o s c i l l a t o r y c o m p o n e n t of p(r). In order to e x p l a i n the excess of s t r u c t u r e of the DDHF densities, we can come back to the a p p r o x i m a t i o n s m e n t i o n e d above. In a real nucleus, w h i c h d e v i a t e s from the ideal " c l o s e d - s h e l l " picture, some n u c l e o n s will be e x c i t e d to h i g h e r lying shells, via 2 p - 2 h - e x c i t a tions, for instance. These a d m i x t u r e s of d i f f e r e n t c o n f i g u r a t i o n s can d i l u t e the shell e f f e c t s in p(r). S t r o n g s h o r t - r a n g e correlations b e t w e e n nucleons, a c o n s e q u e n c e of the r e p u l s i v e core of VNN, disfayours a p p r e c i a b l e e x c u r s i o n s of p(r) away from a c o n s t a n t m a x i m u m density; q u a n t i t a t i v e c a l c u l a t i o n s (Strayer, 1973) c o n f i r m that this does lead to a r e d u c t i o n in the amount of shell s t r u c t u r e in p(r). In o r d e r to d i s t i n g u i s h e x p e r i m e n t a l l y b e t w e e n S n - i s o t o p e s are p a r t i c u l a r l y favourable. When 124Sn, the a d d i t i o n of 8 n e u t r o n s is e x p e c t e d e f f e c t on the c o n f i g u r a t i o n admixtures, since the shells of interest for p r o t o n e x c i t a t i o n s A study of the 1 2 4 s n - l 1 6 s n i s o t o p i c d i f f e r e n c e s o f t n e s s of the proton c o n f i g u r a t i o n could be d i f f e r e n c e b e t w e e n HF and experiment.
these two points, the going from ll6sn to to have a c o n s i d e r a b l e these n e u t r o n s o c c u p y above the Fermi level. can reveal w h e t h e r a r e s p o n s i b l e for the
/24
1.5
l.o
{,
~
--
0.5 I
0,25
Fig.9
1.25
Cross
section
2 25
ratios
3 25 Q(FMi)
124sn/l16sn.
Electron Scattering
425
~,~)(R) 0
!
!
Fig.10
Density difference 124Sn-x16Sn, compared DDHF p r e d i c t i o n s .
to
-.008 ~ 0
2
4
6
R(FM)
The ratios b e t w e e n the e x p e r i m e n t a l cross s e c t i o n of 124Sn and ll6sn is shown in fig. 9, the d e n s i t y d i f f e r e n c e in fig. 10. T h i s figure shows that the a g r e e m e n t b e t w e e n e x p e r i m e n t and the d i f f e r e n t t h e o r e tical c a l c u l a t i o n s is quite good. D e s p i t e an u n u s u a l l y small a m p l i tude, the e x p e r i m e n t a l error band c o n t a i n s m o s t of the c a l c u l a t i o n s ~ d i f f e r e n c e s at small radii are of the o r d e r of less than 1% of p(r). (The strong o s c i l l a t o r y s t r u c t u r e in Ap(r) o b t a i n e d w i t h the G-O force at p r e s e n t is not u n d e r s t o o d ) . F r o m fig. 10, and the study of s p e c t r o s c o p i c factors m e n t i o n e d at the b e g i n n i n g of this section, we c o n c l u d e that it is not the s o f t n e s s of the p r o t o n c o n f i g u r a t i o n , i.e. the n o n - m a g i c n a t u r e of Z=50, that is r e s p o n s i b l e for the excess of s h e l l - m o d e l s t r u c t u r e in HF d e n s i ties. Rather, the lack of a p r o p e r t r e a t m e n t of s h o r t - r a n g e NN corr e l a t i o n in c a l c u l a t i o n s w h e n t a k i n g as a base the i n d e p e n d e n t - p a r ticle shell m o d e l (IPSM), m u s t be held r e s p o n s i b l e . B e f o r e a c c e p t i n g this latter c o n c l u s i o n , we a c t u a l l y s h o u l d v e r i f y that the real p r o b l e m does not lie at a m u c h m o r e f u n d a m e n t a l level. In the h i g h d e n s i t y r e g i o n of a nucleus, the v e r y c o n c e p t of the shell model, w i t h i n d e p e n d e n t p a r t i c l e s m o v i n g in o r b i t s d e t e r m i n e d by an a v e r a g e potential, m i g h t loose its sense. It is far from obvious that nucleons, w h i t h an r m s - r a d i u s of 0.85 fm, in a n u c l e a r med i u m w i t h t y p i c a l NN d i s t a n c e s of 2 fm, could obey the g o v e r n i n g p r i n c i p l e s of the IPSM. A k i n d of n u c l e o n soup, w i t h signs of p a r tial d i s s o l u t i o n of n u c l e o n s into their q u a r k - c o n t e n t , could be a m u c h m o r e p l a u s i b l e scenario, p a r t i c u l a r l y at a time w h e n p e o p l e believe in q u a r k - b a g s of 1 fm radius s p a c e d by 2 fm from the c e n t e r of n e i g h b o u r i n g bags. Does the s h e l l - m o d e l c o n c e p t m a k e sense for the n u c l e a r i n t e r i o r ? E x p e r i m e n t a l v e r i f i c a t i o n s c o n c e r n i n g this q u e s t i o n are v e r y rare. M o s t o b s e r v a b l e s are d e t e r m i n e d using s t r o n g l y i n t e r a c t i n g p r o b e s that are s t r o n g l y a b s o r b e d by nuclei. Experimental observations
426
Ingo Sick
therefore mainly concern up t i l l n o w on I P S M w a v e indirect.
s u r f a c e p r o p e r t i e s ; the i n f o r m a t i o n p r o v i d e d f u n c t i o n s in the n u c l e a r i n t e r i o r is q u i t e
In o r d e r to s t u d y the b e h a v i o u r of p r o t o n w a v e f u n c t i o n s in the n u c l e a r i n t e r i o u r , w e h a v e m a d e an e x p e r i m e n t to d e t e r m i n e t h e d e n s i t y d i f f e r e n c e of P b 206 and T 1 2 0 5 . T h e s e h e a v y n u c l e i d i f f e r by a 3s p r o t o n , a s h e l l t h a t has a v e r y c h a r a c t e r i s t i c r a d i a l w a v e f u n c t i o n : A p r o n o u n c e d m a x i m u m at r = O, w i t h 2 n o d e s and 2 s e c o n d a r y m a x i m a . T h e p e a k at r = 0 in p a r t i c u l a r c a n s e r v e as an e x c e l l e n t t e s t for the r e l e v a n c e of I P S M o r b i t s at h i g h n u c l e a r d e n s i t y . A p r i o r i , it l o o k s q u i t e d i f f i c u l t to e x t r a c t R2(r) w i t h an a c c u r a c y of i n t e r e s t for the p u r p o s e s t a t e d . O n e a d d e d p r o t o n o u t of 82 p r o duces quite a small Ap(r), and this density difference does not get
16
£
Fig.ll
205TI/206pb cross section ratios calculated assuming different conf i g u r a t i o n s for t h e added proton-
Fig.12
E f f e c t of 2 0 5 T I c o r e polarization on cross section ratio.
12
o
==
NucLear
I
O8
polarization
I 1
o
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I 2
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o
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1.0
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Electron Scattering
427
c o n t r i b u t i o n s from the 3s-shell only. The p o l a r i z a t i o n of the T1 core d u e to the a d d e d p r o t o n w i l l c o n t r i b u t e as well. In addition, shells other than 3s w i l l c o n t r i b u t e to ~p(r) g i v e n the fact that the 3s s p e c t r o s c o p i c factor is less than one. W h e n t r y i n g to i d e n t i f y the 3 s - c o n t r i b u t i o n , one s h o u l d r e a l i z e its unique property, an u n u s u a l shape. R2(r) r e s e m b l e s a d a m p e d o s c i l lation s i m i l a r to Jo(2~r/l) w i t h a w a v e length I g i v e n by 2 ~ / l ~ 2 f m -I. If R 2 w o u l d h a v e e x a c t l y the shape of j~, the c o r r e s p o n d i n g change AF of the form factor w o u l d be a 6 - f u n c t i o n o c c u r r i n g at a m o m e n t u m t r a n s f e r q-2 fm -I. The 3 s - c o n t r i b u t i o n t h e r e f o r e can be e x p e c t e d to stand out i n ~ / o , and a l l o w an easy i d e n t i f i c a t i o n and s e p a r a t i o n from other c o n t r i b u t i o n s . To d e m o n s t r a t e this, we show in fig. ii the Pb/TI cross s e c t i o n ratio o b t a i n e d by a s s u m i n g that the added p r o t o n w o u l d go in one of the shells 3s, 2d or ig of i n t e r e s t in c o n n e c t i o n w i t h c o n f i g u r a t i o n admixtures. One can i m m e d i a t e l y r e c o g n i z e the ~ - f u n c t i o n - l i k e spike due to the 3s shell. The e f f e c t of the o t h e r shells on ~ T I / o P b is v e r y d i f f e r e n t in shape, and can e a s i l y be s e p a r a t e d (Frois, 1983). In fig. 12 we show the e f f e c t on the cross s e c t i o n ratio due to the p o l a r i z a t i o n of the T1 core by the added 3s proton. Again, the shape of the two i n g r e d i e n t s to aTl/o Pb is v e r y different, since the core p o l a r i z a t i o n lacks the nodes p r e s e n t in R~s. Core p o l a r i z a t i o n s h o u l d not C o m p l i c a t e s i g n i f i c a n t l y the i n t e r p r e t a t i o n of o T l / o P b in terms of R~s. The e x p e r i m e n t a l r e s u l t for the 2 0 6 p b / 2 0 5 T I cross s e c t i o n r a t i o shown in fig. 13. The h i g h e r q - d a t a that show the 3 s - p e a k c o m e
aTl/apB
TI/PB
1.4
1.2
1,0
Fig.13
!
I
I.
2.
;
Q
E x p e r i m e n t a l 2 0 5 T l / 2 0 6 p b cross ratios. The curve represents prediction calculated assuming
3,
section the D D H F S3s=0.7 .
is from
428
Ingo Sick
Ap(r)
206pB-~1 • 008
Fig.14
Experimental density difference, c o m p a r e d DDHF prediction.
to
.004
I
I
I
I
I
2
4
6
8
I
the e x p e r i m e n t we r e c e n t l y c o m p l e t e d at S a c l a y (Cavedon, 1983), the low-q data come from an e a r l i e r e x p e r i m e n t p e r f o r m e d at Mainz (Euteneuer, 1978). The d e n s i t y d i f f e r e n c e e x t r a c t e d from these data is shown in fig. 14. The 3s-shape, w i t h its 3 m a x i m a and 2 nodes, stands out in a s p e c t a c u l a r way If we c o m p a r e this e x p e r i m e n t a l result to a HF p r e d i c t i o n o b t a i n e d by a s s u m i n g a pure 3s~2 hole c o n f i g u r a t i o n for 2 0 5 T l , t h e peak of Ap has an a m p l i t u d e about 40% too large; this d i f f e r e n c e is to be expected, though, since the s t a n d a r d DDHF c a l c u l a t i o n s are not s u f f i c i e n t for o d d - e v e n nuclei. Shell m o d e l c a l c u l a t i o n s for T1 show that the g r o u n d state c o n f i g u r a t i o n is m o r e complicated. B o t h studies of c o u p l i n g of the 3s proton to c o l l e c t i v e e x c i t a t i o n of 206pb, as w e l l as shell m o d e l c a l c u l a t i o n s for the (3s~2) -I (2d3/2) -2 state show that the 3s-hole s t r e n g t h in 205TI amounts to s = 0.7-0.9. The r e m a i n d e r of the s t r e n g t h is m a i n l y found in the 2d3/2 c o n f i g u r a t i o n . These p r e d i c t i o n s are c o n f i r m e d by the result o f - t r a n s f e r reactions. Proton p i c k u p from 206pb m e a s u r e s only 70% of the Z = O s t r e n g t h found in 208pb. A c c o r d i n g l y , we c o m p a r e in fig. 12 to a DDHF c a l c u l a t i o n (Campi, 1982) p e r f o r m e d w i t h the hole s t r e n g t h c o n s t r a i n e d to S3s = 0.7, S2d = 0.3. The r e s u l t i n g p r e d i c t i o n for the cross s e c t i o n ratio is shown also in fig. ii. The e x p e r i m e n t a l data are a m a z i n g l y w e l l reproduced; small d i f f e r e n c e s w i t h smooth radial s t r u c t u r e (fig. 14) can be a t t r i b u t e d to our s o m e w h a t too crude t r e a t m e n t of core polar i z a t i o n effects. F r o m figs. ii, 14 we can c o n c l u d e that the d e v i a t i o n s in p(r) disc u s s e d above are not linked to a b r e a k d o w n of the s h e l l - m o d e l idea. The c o n c e p t of the i n d e p e n d e n t p a r t i c l e orbit retains an i m p p r e s s i v e d e g r e e of v a l i d i t y even in the h i g h - d e n s i t y r e g i o n of nuclei. N u c leons retain their i n t e g r i t y (to a large extent, anyway) and the shell m o d e l a c c o u n t s quite well for their w a v e f u n c t i o n even t h o u g h the e f f e c t s of s h o r t - r a n g e c o r r e l a t i o n b e t w e e n n u c l e o n s are t r e a t e d
Electron Scattering in too c r u d e
Mesonic
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429
in DDHF.
of f r e e d o m
T h e t o p i c of this last s e c t i o n deals w i t h c o n s t i t u e n t s of the n u c l e u s o t h e r than n u c l e o n s . For a g o o d reason, we d i s c u s s it o n l y a f t e r having p r e s e n t e d a d e t a i l e d (but limited) a c c o u n t of w h a t is k n o w n on n u c l e o n i c d e g r e e s of freedom. The e f f e c t s of n o n - n u c l e o n i c c o n s t i t u e n t s in n u c l e i in g e n e r a l are small, and d i f f i c u l t to i s o l a t e in an u n a m b i g u o u s way. U n l e s s we u n d e r s t a n d w e l l the n u c l e o n i c p r o p e r t i e s , we c a n n o t a t t r i b u t e d i f f e r e n c e s w i t h e x p e r i m e n t to the small, n o n n u c l e o n i c d e g r e e s of freedom. W h y do e l e c t r o n s r e p r e s e n t a g o o d p r o b e for the i n v e s t i g a t i o n of n o n n u c l e o n i c d e g r e e s of f r e e d o m ? For h a d r o n probes, the r e a c t i o n m e c h a n i s m can be q u i t e c o m p l i c a t e d . T h i s is true in p a r t i c u l a r if one w a n t s to s t u d y small w a v e f u n c t i o n c o m p o n e n t s . Multistep reactions i n v o l v i n g large c o m p o n e n t s o f t e n c o v e r up the c o n t r i b u t i o n s of i n t e rest, the ones due to small w a v e f u n c t i o n c o m p o n e n t s and a r e a c t i o n m e c h a n i s m s i m p l e e n o u g h to be i n t e r p r e t e d . For e l e c t r o n s the r e a c tion m e c h a n i s m a p r i o r i stays m o r e simple, s i n c e the w e a k n e s s of the i n t e r a c t i o n r e n d e r s m u l t i s t e p r e a c t i o n s less likely. The p r e s e n c e of m e s o n i c d e g r e e s of f r e e d o m in n u c l e i has b e e n k n o w n for a long time. A n u c l e o n - n u c l e o n i n t e r a c t i o n m e d i a t e d by m e s o n s e x c h a n g e d b e t w e e n n u c l e o n s i m p l i e s the p r e s e n c e of ~,p, etc. The p r o m i n e n c e of the A - r e s o n a n c e in the ~-N i n t e r a c t i o n i n d i c a t e s the i m p o r t a n c e of ~ in nuclei. It has b e e n q u i t e d i f f i c u l t , though, to find u n a m b i g u o u s e x p e r i m e n t a l s i g n a t u r e s of these n o n - n u c l e o n i c constituents. S i n c e the e a r l y w o r k of V i l l a r s (1947) it is c l e a r that m a g n e t i c p r o p e r t i e s of n u c l e i are p a r t i c u l a r l y w e l l s u i t e d for the i n v e s t i g a t i o n of m e s o n e x c h a n g e c u r r e n t s (~.~C). C o n t r a r y to c h a r g e s c a t t e r i n g , where ~C do not c o n t r i b u t e at q = 0 they a f f e c t in first o r d e r m a g n e t i c moments. Here, the c u r r e n t s p r o d u c e d by the e x c h a n g e d p a r t i c les can c o n t r i b u t e a l r e a d y at q = O. This observable therefore allows a d i r e c t a c c e s s to the p r e s e n c e of m e s o n s in nuclei. O v e r the years it has i n c r e a s i n g l y b e c o m e c l e a r that m a g n e t i c f o r m f a c t o r s are p a r t i c u l a r l y p r o m i s i n g at large q; t h e r e r c a n c e l l a t i o n s o c c u r r i n g in the n u c l e o n i c c o n t r i b u t i o n s can e n h a n c e the r e l a t i v e ~ C e f f e c t s by large factors. It a l s o has b e c o m e c l e a r that light n u c l e i o f f e r the b e s t c h a n c e to s t u d y MEC; for t h e s e nuclei, the n u c l e o n i c w a v e f u n c t i o n is w e l l e n o u g h k n o w n to a l l o w for a r e a s o n a b l y r e l i a b l e p r e d i c t i o n of this n u c l e o n i c " b a c k g r o u n d " . A c c o r d i n g l y , w e w i l l d i s c u s s in the f o l l o w i n g m a g n e t i c f o r m f a c t o r s of light n u c l e i at large m o m e n t u m t r a n s f e r only. The f i r s t e x a m p l e w e p r e s e n t c o n c e r n s the d e u t e r o n e l e c t r o d i s i n t e g r a tion at t h r e s h o l d . The q ~ O limit of this r e a c t i o n , n + p ÷ + d + y at v e r y low e n e r g y (Riska and Brown, 1972), t o g e t h e r w i t h the A = 3 m a g n e t i c m o m e n t s , was one of the first u n a m b i g u o u s p r o o f s that m e s o n i c d e g r e e s of f r e e d o m are n e e d e d for the q u a n t i t a t i v e u n d e r s t a n d i n g of n u c l e a r reactions.
430
Ingo Sick
I s.1636
, t,
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=74 Fig.15
'
=~
t~,, ;,,t,,~,~,t~L_ Ee. (Mev---)
S p e c t r u m for d(e,e') at 410 M e V incident energy and 155 ° s c a t t e r i n g angle.
d~
d (e e')n p Enp- 0+3 MeV
Fig.16
\, IA
IA~MEC+A
0
5
10
a ~ If,W~
20
Cross section for d(e,e') integrated over is-peak (Enp=O~MeV). The curves have been c a l c u l a t e d by L e i d e m a n n and Arenh~vel.
Electron Scattering
431
D e u t e r o n e l e c t r o d i s i n t e g r a t i o n is p a r t i c u l a r l y i n t e r e s t i n g in the reg i o n O~2 M e V above b r e a k u p threshold. A t these energies, the n+p system e x h i b i t s a resonance, a n e a r l y - b o u n d s i n g l e t - S state. The transition from the initial d e u t e r o n S- and D - s t a t e o c c u r s via two amplitudes 3S-IS, 3D-IS that interfere. At large q, q~3.5 fm -I, the nucleon-only (Impulse-Approximation, IA) a m p l i t u d e s i n t e r f e r e d e s t r u c t i vely and the IA cross s e c t i o n b e c o m e s v e r y small. At the same time the M E C - p r o c e s s e s at large q favour S-D transitions. N e a r q ~ 3 . 5 f m -I, the ratio of MEC to IA cross s e c t i o n s b e c o m e s i0, and r e p r e s e n t s a clean signal, a fact first p o i n t e d out by H o c k e r t (1973). In o r d e r to p r o v i d e data in this r e g i o n of m o m e n t u m t r a n s f e r we performed a few years ago a d(e,e') e x p e r i m e n t at S a c l a y (Bernheim, 1 9 8 1 ) E l e c t r o n s of e n e r g i e s 280+535 M e V and i n t e n s i t y up to 25~A w e r e scattered from a liquid d e u t e r i u m target. At the s c a t t e r i n g angle of 155 ° chosen, this cross s e c t i o n is d o m i n a t e d e n t i r e l y by the m a g n e t i c int e r a c t i o n i n v o v l i n g a s p i n - f l i p of one of the nucleons. An e x a m p l e for the e n e r g y s p e c t r u m of the s c a t t e r e d e l e c t r o n s for an i n c i d e n t e n e r g y of 410 M e V is shown in fiq. 15. One r e c o ~ n i s e s the e l a s t i c
Y%
Fig.17
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D o u b l e d i f f e r e n t i a l cross s e c t i o n d(e,e') for E n p = l . 5 M e V c a l c u l a t e d for d i f f e r e n t s t r o n g - v e r t e x form f a c t o r s (short-dashed, solid and l o n g - d a s h e d for a n u c l e o n radius of 0, 0.48 fm, 0.7 fm, respectively).
432
Ingo Sick
peak (Ee'-290 MeV), the IS r e s o n a n c e (Ee'~288 MeV) and the i n e l a s t i c continuum. W i t h o u t the p r e s e n c e of ~.~C, the IS peak w o u l d be a factor of i0 lower, i.e. invisible. The cross s e c t i o n i n t e g r a t e d o v e r this l s - p e a k is shown in fig. 16, tog e t h e r w i t h p r e v i o u s r e s u l t s at low q. It is c o m p a r e d to a t h e o r e t i cal p r e d i c t i o n (Arenh~vel, 1983) that i n c l u d e s the IA p r o c e s s only; the strong d i s a g r e e m e n t in the r e g i o n w h e r e the S-D i n t e r f e r e n c e p r o duces a m i n i m u m in the cross s e c t i o n is obvious. W h e n i n c l u d i n g the s t a n d a r d MEC c u r r e n t s (fig. 17a,b), the ones due to p i o n e x c h a n g e and pair creation, the a g r e e m e n t w i t h the d a t a is v a s t l y improved. The a d d i t i o n a l c o n t r i b u t i o n of the A r e s o n a n c e t e r m (fig. 17c) has little effect. The a g r e e m e n t b e t w e e n p r e d i c t i o n and e x p e r i m e n t is q u i t e r e a s o n a b l e , and shows that the m a i n p a r t of the M E C is u n d e r s t o o d . In detail, a n u m b e r of p o i n t s are still open, as s h o w n by the m o s t recent c a l c u l a tions of M a t h i o t (1983). In p a r t i c u l a r , at large t r a n s f e r the form f a c t o r s at the h a d r o n i c v e r t i c e s (fig. 17) play an i m p o r t a n t role. T h e s e form factors are not v e r y well known, and t h e i r i n f l u e n c e is p a r t l y c o m p e n s a t e d by i n c l u s i o n of the e x c h a n g e of the Q-meson. T h e i r e f f e c t is shown in fig. 18, w h e r e the c h a n g e from no s t r o n g - v e r t e x form factors (short dashes) to a v e r t e x form factor c o r r e s p o n d i n g to a n u c l e o n - r a d i u s of 0.7 fm (long dashes) is shown. F i g u r e 18 shows that the s h o r t - r a n g e ~ C are still q u i t e m o d e l d e p e n d e n t . Pushing e x p e r i m e n t to h i g h e r q s h o u l d a l l o w to d i s t i n g u i s h b e t w e e n the various i n g r e d i e n t s ; the s t r u c t u r e in the form f a c t o r b e t w e e n 20 and 30 fm -2 c o u l d be q u i t e revealing. We s h o u l d also p o i n t out that there is still an u n c e r t a i n t y r e l a t e d to the c h o i c e of the e l e c t r o m a g n e t i c form factor to be u s e d in the c a l c u l a t i o n of MEC diagram. (G e v e r s u s FI, see d i s c u s s i o n for 3He below). The n e x t e x a m p l e I w a n t to d i s c u s s c o n c e r n s the d e u t e r o n e l a s t i c m a g netic form factor B(q). H i s t o r i c a l l y , b o t h the c h a r g e and m a g n e t i c form f a c t o r have b e e n t a k e n as test cases, and have b e e n a c o n t i n u i n g i m p e t u s for the study of MEC. A l t h o u g h the ~ C e f f e c t s t u r n e d out to be small at low q (the d e u t e r o n b e i n g an i s o s c a l a r object), the d e u t e r o n form f a c t o r s still are v e r y i n t e r e s t i n g . The s i m p l i c i t y of the b o u n d 2 - n u c l e o n s y s t e m a l l o w s a l t e r n a t i v e t h e o r e t i c a l a p p r o a c h e s that can be u s e d to c h e c k the s t a n d a r d MEC c a l c u l a t i o n s . The m a g n e t i c f o r m f a c t o r B(q) has b e e n m e a s u r e d by m a n y d i f f e r e n t exp e r i m e n t s o v e r the p a s t 25 years. T h e m o s t r e c e n t e x p e r i m e n t that p r o d u c e d the h i g h e r - q d a t a is the one d i s c u s s e d a b o v e in c o n n e c t i o n w i t h d i s i n t e g r a t i o n at threshold. The m a g n e t i c f o r m factors are comp i l e d in fig. 19. The n o n r e l a t i v i s t i c d e u t e r o n w a v e f u n c t i o n can be c a l c u l a t e d e x a c t l y by s o l v i n g the S c h r 6 d i n g e r e q u a t i o n for a g i v e n N N - p o t e n t i a l . The m a g n e t i c form factor is o b t a i n e d by f o l d i n g the p o i n t n u c l e o n form f a c t o r s w i t h the n u c l e o n m a g n e t i c form factors. S i n c e the i s o s c a l a r m a g n e t i c n u c l e o n f o r m f a c t o r is not all that a c c u r a t e l y known, this i n t r o d u c e s an u n c e r t a i n t y (~20% at 4 fm -I) in the c a l c u l a t e d B(q).
Electron Scattering
433
The impulse-approximation result, obtained by Gari (1976) using the RSC nucleon-nucleon-interaction, is shown in fig. 19 as a dashed line. The contributions of non-nucleonic degrees of freedom are mainly connected to the pair diagram, for q<5 fm -~. Additional processes, the ~py-diagram in particular, dominate at larger transfer. Adding to the IA contribution the effects of MEC yield the dash-dot curve, which agrees reasonably well with experiment. The contribution of A-components, not yet included in the dash-dot curve, will give a further small modification of B(q). For the deuteron a different theoretical approach, the relativistic IA, is possible• Solving the Bethe-Salpeter equation yields a relativiStically correct wave function. The corresponding form factor includes both the terms equivalent to nonrelativistic IA and the pair diagram (i.e. antinucleons). It has been shown by Gross (1978)that, to lowest order in q2/M2, the relativistic IA and the nonrelativistic IA plus the pair diagram are formally equivalent.
B(q)
10 .3 .
. .,.% '"i,
16 4 .
•
\\
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\~.
',J
•
165 Fig. 19
l
i
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i
1
2
3
4
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Deuteron magnetic form factor• The curves correspond to IA (dashed), I A + ~ C (dash-dot) and relativistic IA (dotted).
434
Ingo Sick
R e l a t i v i s t i c c a l c u l a t i o n s for the d e u t e r o n h a v e b e e n p e r f o r m e d by A r n o l d (1980) and Z u i l h o f (1980). T h e s e a u t h o r s o b t a i n s i m i l a r results; one of their p r e d i c t i o n s is shown in fig. 19 by the d o t t e d curve. This r e s u l t d i f f e r s s i g n i f i c a n t l y from the s t a n d a r d I A + ~ C c a l c u l a t i o n s (dash-dot), and p r o d u c e s a form f a c t o r s m a l l e r t h a n the n o n r e l a t i v i s t i c IA result. W h e r e does this d i s a g r e e m e n t b e t w e e n two (supposedly) e q u i v a l e n t appr o a c h e s come from ? The n o n r e l a t i v i s t i c c a l c u l a t i o n s c o r r e s p o n d to an e x p a n s i o n in first o r d e r q2/M2, w h i l e the r e l a t i v i s t i c c a l c u l a t i o n s m a k e no such expansion. At q - v a l u e s of the o r d e r of one G e V / c this can m a k e an a p p r e c i a b l e d i f f e r e n c e . In addition, Z u i l h o f and T j o n (1980) p o i n t out that, for p s e u d o - s c a l a r ~N coupling, a n o n - n e g l i g i b l e c o n t r i b u t i o n of 2 z - e x c h a n g e gives a c o n t r i b u t i o n that tends to c a n c e l the p a i r term. This c o m p a r i s o n shows that the c a l c u l a t i o n s of MEC m a y not yet be as w e l l u n d e r c o n t r o l as one m i g h t have c o n c l u ded from fig. 16. H i g h e r o r d e r d i a g r a m s , and the r e l a t i v i s t i c e f f e c t s o f t e n n e g l e c t e d , are to be s t u d i e d in o r d e r to r e s o l v e the d i s c r e p a n cy p o i n t e d out above, and d e t e r m i n e w h e t h e r the r e ~ o n s c i t e d are the real cause. For the d e u t e r o n m a g n e t i c form factor, a s e c o n d q u e s t i o n a c t u a l l y is w i t h o u t a final answer, the one c o n c e r n i n g the e f f e c t of A A - c o m p o n e n t s in the g r o u n d state w a v e function. A c c o r d i n g to Gari (1976), the effect of A A - c o m p o n e n t s is q u i t e small, w h i l e F a b i a n (1974) finds a c h a n g e of a factor of 2 of B(q) n e a r q~4 fm -I. P a r t l y this d i f f e r e n ce is r e l a t e d to the u n c e r t a i n t y in the v e r t e x form factors. It is n o t clear, however, that there is not a m o r e b a s i c d i s c r e p a n c y , linked e.g. to a s s u m p t i o n s on the A m a g n e t i c moment. The t h i r d o b s e r v a b l e that gives d i r e c t access to the role of m e s o n exc h a n g e c u r r e n t s in n u c l e i is the 3 H e m a g n e t i c form factor. It was alr e a d y r e c o g n i z e d by V i l l a r s (1947) that the A=3 m a g n e t i c m o m e n t s rec e i v e an a p p r e c i a b l e c o n t r i b u t i o n of ~0.3 m a g n e t o n s . The i s o v e c t o r m a g n e t i c m o m e n t in p a r t i c u l a r is s e n s i t i v e to ~ C . The c a l c u l a t i o n of H a r p e r (1972) s h o w e d that the T - e x c h a n g e , p a i r and A - d i a g r a m s q u a n t i t a t i v e l y e x p l a i n the d i f f e r e n c e b e t w e e n the m a g n e t i c m o m e n t s o b t a i n e d f r o m e x p e r i m e n t and F a d d e e v c a l c u l a t i o n s . A t m o m e n t u m t r a n s f e r s q>O, the 3He m a g n e t i c form f a c t o r gets e s p e c i a l ly i n t e r e s t i n g . As p o i n t e d o u t by B r a n d e n b u r g (1974) F M c o n t a i n s an S- to D - s t a t e a m p l i t u d e that i n t e r f e r e s w i t h the d o m i n a t i n g S2-term. T h i s i n t e r f e r e n c e leads to a p r o n o u n c e d d i f f r a c t i o n m i n i m u m at q2 = 8 fm-2; in a n a l o g y w i t h d e u t e r o n e l e c t r o d i s i n t e g r a t i o n , w e can e x p e c t to find large e f f e c t s of e x c h a n g e c u r r e n t s in t h a t region. T h e i n i t i a l m a g n e t i c s c a t t e r i n g e x p e r i m e n t s on 3He, w h i c h were d o n e at S t a n f o r d (Collard,1965, M c C a r t h y , 1970), c l e a r l y s h o w e d that the d i f f r a c t i o n m i n i m u m p r e d i c t e d near 8 fm -2 w a s absent. F i g u r e 20 shows the m o r e a c c u r a t e d a t a a v a i l a b l e today, the ones r e s u l t i n g f r o m the e x p e r i m e n t s p e r f o r m e d at B a t e s (Dunn, 1983) and S a c l a y (Cavedon, 1982). A d i f f r a c t i o n m i n i m u m is o b s e r v e d , but not at 8 fm -2, but at q2 ~ 17 fm -2 ! The dashed
curve
in f i g u r e
20 g i v e s
the i m p u l s e
approximation
form
Electron Scattering
435
factor c a l c u l a t e d u s i n g the w a v e f u n c t i o n of Torre (1981). This w a v e f u n c t i o n is c a l c u l a t e d by s o l v i n g the F a d d e e v e q u a t i o n in c o n f i g u r a tion space for the R e i d soft core NN interaction. This prediction s t r o n g l y d i f f e r s from the data, as do o t h e r s o b t a i n e d u s i n g d i f f e r e n t NN i n t e r a c t i o n s and d i f f e r e n t t e c h n i q u e s to c a l c u l a t e the 3-body w a v e function. W h i l e the c h a r g e form factor is s e n s i t i v e to short range p r o p e r t i e s of the w a v e f u n c t i o n that are p o o r l y known, the m a g netic form factor d e p e n d s on (better known) l o n g - r a n g e p r o p e r t i e s only; the short range features, o b s e r v a b l e in p r i n c i p l e at large q, are c o v e r e d up by the S-D t r a n s i t i o n term. G i v e n this situation, we m a y trust the IA p r e d i c t i o n to a c o n s i d e r a b l e extent. Deviations must come from the n o n - n u c l e o n i c d e g r e e s of freedom. In fig. 20 we show a r e c e n t c a l c u l a t i o n of S t r u e v e (1983) w h o solves the F a d d e e v e q u a t i o n in m o m e n t u m space u s i n g the P a r i s n u c l e o n - n u c l e o n potential. To the o n e - b o d y form factor have b e e n a d d e d the p a i r - and m e s o n c u r r e n t s a r i s i n g from ~ and p exchange, and the e f f e c t of strongv e r t e x form factors is included. In this c a l c u l a t i o n the A - c o n t r i b u tion is not yet taken into account. To be c o n s i s t e n t , the A has to be a l l o w e d for b o t h in the MEC p r o c e s s (fig. 18c) and in the 3-body force w h i c h p o l a r i s e s and r e n o r m a l i s e s the p u r e l y n u c l e o n i c c o n f i g u r a tion. A c c o r d i n g to a c o u p l e d c h a n n e l c a l c u l a t i o n (Strueve, 1983), t h e o v e r a l l e f f e c t of F M is small for q<5 fm -I. The curve a) in fig. 20 shows that m e s o n i c e f f e c t s (though c a l c u l a t e d i n c o r r e c t l y using FI, see below) do a c c o u n t for the l a r g e s t p a r t of the d i s c r e p a n c y b e t w e e n IA p r e d i c t i o n and experiment. This c l e a r l y r e p r e s e n t s a success of the ~ C picture, and gives us c o n f i d e n c e that these c o n s t i t u e n t s are u n d e r s t o o d to a r e a s o n a b l e d e g r e e of accuracy. This is e m p h a s i z e d by curve b), w h i c h is o b t a i n e d by Riska (1980) using a very simple m i n d e d n u c l e o n i c w a v e function. This curve d e m o n strates that the 3He m a g n e t i c form factor is q u i t e i n s e n s i t i v e to the n u c l e o n i c w a v e function; q u a l i t a t i v e a g r e e m e n t w i t h the d a t a at q2>5 fm -2 is e s s e n t i a l l y a test of our u n d e r s t a n d i n g of MEC. This success of the m e s o n - e x c h a n g e p i c t u r e s h o u l d not m a k e us b e l i e v e that e v e r y t h i n g is under control. One p r o b l e m was a l r e a d y i n d i c a t e d in c o n n e c t i o n w i t h the d e u t e r o n form factor~ here we e m p h a s i z e a related p o i n t that is v e r y i m p o r t a n t for a q u a n t i t a t i v e d e s c r i p t i o n . W h e n c a l c u l a t i n g the m e s o n e x c h a n g e c o n t r i b u t i o n s , one s h o u l d r e s p e c t c u r r e n t c o n s e r v a t i o n ; this c o n s i d e r a t i o n , after all, was a m a j o r one w h e n i n t r o d u c i n g MEC in the first place. It has ben e m p h a s i z e d by A r e n h 6 v e l and others, that this r e q u i r e s the use of the n u c l e o n elect r o m a g n e t i c form factor G e r a t h e r than F 1 in the c a l c u l a t i o n of the MEC c o n t r i b u t i o n . Not r e s p e c t i n g this rule allows to get b e t t e r agreem e n t w i t h the e x p e r i m e n t a l data (Bornais, 1981). F i g u r e 21 t a k e n from S t r u e v e (1983) d e m o n s t r a t e s the i m p o r t a n c e of using G e. The d i f f e r e n c e b e t w e e n G e and F 1 is of o r d e r q2/M2, i.e. of r e l a t i v i stic order. This e m p h a s i z e s again that in the n e x t step ~ C n e e d to be s t u d i e d in a r e l a t i v i s t i c a l l y m o r e c o r r e c t way. As p o i n t e d out at this school by Riska, one of the r e l a t i v i s t i c terms of the p a i r d i a g r a m for S-D t r a n s i t i o n s gives an e f f e c t of the same size as the d i f ference Ge-F I. For this p a r t i c u l a r term, the n o n r e l a t i v i s t i c e x p r e s -
436
Ingo Sick
10 o 3He (e,e) o
BATES
• SACl.~
10 -Z
....
NUCLEON ONLY NUCLEON+MESON EXCHANGE
x
\
e.
10 -¢ bL.
1 | I,
Ii II II II II
(a) (b)
10 - 8
0
10
20
30
(fro-2) Fig.20
3He elastic form factor calculated by Torre (IA, dashed), Strueve (IA+MEC, solid, a) and Riska (IA+MEC, solid, b).
10 0
FORMFACTOR 'De e
I0-I
=
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o U._
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i
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,
.
.
i
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.
.
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.
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.
.
.
.
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3He magnetic form factor calculated using for ~ C as electromagnetic form factor F 1 (dashed) and Ge(solid).
Electron Scatterin~
437
sion for the pair-current calculated with F 1 would be equivalent to the relativistic one calculated with G e. Clearly, effects of this order coming from other amplitudes and for the other exchange diagram, need to be investigated before the MEC contribution quantitatively can be compared to experiment. From the preceding discussion on the A ~ 3 nuclei it is clear that the role of meson exchange currents is reasonably understood. In those cases where the MEC are large, the pair and meson current diagrams (involving both z and 0) account for the larger part of the difference between experiment and nucleon-only prediction. The A-diagram, on the other hand, is still poorly tested, since we have not yet found an observable dominated by the A-term. (For heavier nuclei, where nuclear structure uncertainties introduce large additional uncertainties, the A-contribution is in no better shape). A next step in the understanding of non-nucleonic degrees of freedom obviously will be the question on the role of quarks in nuclei. This question enters already when discussing the s h o r t e r - range MEC; it becomes the dominating issue once we attempt to understand processes at even higher m o m e n t u m transfer. Some of the recent developments are described in other sections of this volume. A systemtic discussion of the role of quarks in electron-nucleus scattering will be the theme of the '84 Erice School.
REFERENCES Arenh6vel, H., W. Leidemann (1983). Deuteron e l e c t r o d i s i n t e g r a t i o n near threshold at high momentum transfer. Nucl.Phys., A393, 385398. Arnold, R. G., C. E. Carlson, F. Gross (1980). Elastic electron deuteron scattering at high energy. Phys.Rev., C21, 1426-1451. Bernheim, M., E. Jans, J. Mougey, D. Royer, D. Tarnowski, S. Turck, I. Sick, G. P. Capitani, E. DeSanctis, S. Frullani (1981). Electron-induced deuteron disintegration at threshold. Phys.Rev.Lett.,
4_66, 402-405 Brandenburg,R. A., Y. E. Kim, A. Tubis (1974). Magnetic form factor of 3He. Phys.Rev. Lett., 32, 1325-27. Bertozzi, W., M. V. Hynes, C.P. Sargent, C. Creswell, P. C. Dunn, A. Hirsch, M. Leitch, B. Norum, F.B. Rad, T. Sasanuma (1977). Focal plane instrumentation: a very high resolution ~.~.IPC system for inclined tracks. Nucl. Instr. Meth., 14___!, 457-467. Bertozzi, W., M. V. Hynes, C. P. Sargent, W. Turchinez, C. W i l l i a m s o n (1979). High resolution spectrometers for electron scattering. Nucl. Instr. Meth., 162, 211-238. Bornais, R., B. Goulard, E. Hadjimichael (1981). F7 Proc. ICOHEPANS and Nucl. Phys., A347, 143-163. Campi, X., D . W . L . Sprung (1972). Spherical nuclei in the local density approximation. Nucl. Phys., A194, 401-442 and priv.comm. Campi, X. (1982), priv. comm. Cardman, L. S., J. W. Lightbody, S. Penner, S. P. Fivozinsky, X. K. Maruyama, W. P. Trower, S. E. W i l l i a m s o n (1980). The charge distribution of 12C. Phys. Lett., 91B, 203-206.
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Cavedon, J. M., B. Frois, D. Goutte, M. Huet, P. Leconte, J. Martino, X. H. Phan, S. K. Platchkov, S. E. Williamson, W. Boeglin, I. Sick, P. deWitt-Huberts, L. S. Cardman, C. N. Papanicolas (1982). Magnetic form factor of 3He. Phys. Rev. Lett., 49, 986-989. Cavedon, J. M., B. Frois, D. Goutte, M. Huet, P. Leconte, C. N. Papanicolas, X.H. Phan, S. K. Platchkov, S. E. Williamson, W. Boeglin, I. Sick (1982). Is the shell-model concept relevant for the nuclear interior ? Phys. Rev. Lett., 49, 978-981. Cavedon, J. M., J. B. Bellicard, B. Frois, D. Goutte, M. Huet, P. Leconte, X. H. Phan, S. K. Platchkov, I. Sick (1982). The charge density distribution of ll6sn and 124Sn. Phys. Lett. II8B, 311314. Collard, H., R. Hofstadter, E. B. Hughes, A. Johansson, M. R. Yearian, R. B. Da~, R. T. Wagner !1965). Elastic electron scattering from 3H and ~He. Phys. Rev., 138, B57. Decharg~, J., D. Gogny, (1980). Hartree-Fock Bogolyubov calculations with the D1 effective interaction on spherical nuclei. Phys. Rev., C21, 1568-1593, and priv. comm. de Vries, C. (1978). In Few body systems and electromagnetic interactions. Ed. C. Ciofidegli Atti, Springer, Berlin, 181-218. Dunn, P., S. B. Kowalski, F. N. Rad, C. P. Sargent, W. E. Turchinez. R. Goloskie, D. P. Saylor (1983). 3He magnetic form factor~ Phys. Rev., C27 , 71-82. Euteneuer, H., J. Friedrich, N. Vogler (1978) The charge distribution differences of 209Bi, 208,206,206,204pb and 205TI. Nucl. Phys., A298, 452-476. Fabian, W., H. Arenh~vel, H. G. Miller (1974). Meson exchange and isobar admixture contributions to elastic electron-deuteron scattering. Z. Physik, 271, 93-96. Ficenec, J. R., L. A. Fajardo, W. P. Trower, I. Sick (1972). Elastic electron-tin scattering. Phys. Lett., 42B, 213-216. Frois, B., J. M. Cavedon, D. Goutte, M. Huet, P. Leconte, C. N. Papanicolas, X. H. Phan, S. K. Platchkov, S. E. Williamson, W. Boeglin, I. Sick(1983). A precise determination of the 3s proton orbit. Nucl.Phys., A396, 409-418. Gari, M., H. Hyuga, B. Sommer (1976). Resonance AA-degrees of freedom in the deuteron and their influence on the electromagnetic form factors. Phys. Rev., C14, 2196-2210. Gross, F. (1978). In Few body systems and nuclear forces I. Ed. H. Zingel, Springer, Berlin, 46-49. Harper, E. P., Y. E. Kim, A. Tubis, M. Rho (1972). Meson exchange currents and 3H, 3He magnetic moments. Phys. Lett., 40B, 533-536. Hockert, J., D. O. Riska, M. Gari, A. Huffman (1973). Meson exchange currents in deuteron electrodisintegration. Nucl.Phys., A217, 14-28. Leconte, P., J. Mougey, A. Tomaso, P. Barreau, M. Bernheim, A. Bussi~re, L. Cohen, J. C. Comoretto, J. Dupont, S. Frullani, C. Grunberg, J. M. Hisleur, J. LeDevehat, M. Lef~vre, G. Lemarchand, J. Millaud, D. Royer, R. Salvaudon (1980). The electron scattering facility at the Saclay 600 MeV linear accelerator. Nucl. Instr. Meth., 169, 401-412. Mathiot, J. F. (1983). What do we learn from the deuteron electrodisintegration near threshold, and thermal n-p capture ? To be publ.
Electron Scattering
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McCarthy, J. S., I. Sick, R. R. Whitney, M. R. Yearian (1970). Electromagnetic structure of the 3He nucleus. Phys. Rev. Lett., 2_~5, 884-888. Negele, J. (1970). Structure of finite nuclei in the local density approximation. Phys. Rev., CI, 1260-1321, and priv. comm. Reuter, W., G. Fricke, K. Merle, H. Miska (1982). Nuclear charge distribution and rms-radius of 12C. Phys. Rev., C26, 806-818. Riska, D. O., G. E. Brown (1972). Meson exchange effects in n+p+d+y. Phys. Lett., 38B, 193-195. Riska, D. O. (1980). The magnetic form factor of 3He. Nucl. Phys., A350, 227-252. Sick, I. (1974). Model independent nuclear charge densities from elastic electron scattering. Nucl. Phys., A218, 509-541. Sick, I. (1978). In H. Zingl (Ed.) Few body systems and nuclear forces II, Springer, Berlin, 236-246. Sick, I., J. B. Bellicard, J. M. Cavedon, B. Frois, M. Huet, P. Leconte, X. H. Phan, S. Platchkov (1979). Charge density of 40Ca. Phys. Lett., 88B, 245-248. Sick, I. (1983). Electron scattering and nuclear constituents. Nucl. Phys., A396, 455-465. Strayer, M. R., W. M. Basichis, A. K. Kerman (1973). Correlation effects in nuclear densities. Phzs. Rev., C8, 1269-1274. Strueve, W., C. Hajduk, P. U. Sauer (1983). M e s o n i c exchange current contributions to the 3He magnetic form factor. To be publ. T0rre, J., J. J. Benayou, J. Chauvin (1981). Two pion e x c h a n g e three-body potential in three- nucleon states. Z. Physik, 300, 319-322. Villars, F. (1947). The magnetic exchange moments for 3H and 3He. Helv. Phys. Acta, 20, 476-490. Zeidmann, B. (1982) in A national CW GeV electron microton laboratory Argonne National Laboratory, 249-256. Zuilhof, M. J., J. A. Tjon (1980). Electromagnetic properties of the deuteron and the Bethe-Salpeter equation with one-boson exchange. Phys. Rev., C22, 2369-2382.
ELECTRON
SCATTERING
EXPERIMENTS
The first part of these lectures deals w i t h t h e purely experimental aspects of electron scattering. The m o t i v a t i o n for doing these experiments, the results obtained, and their interpretation in terms of physics will be described in the following sections. The experimental equipment at todays accelerators really starts right at the end of the accelerator. These machines produce beams of a quality barely adequate for the experiments planned. A careful preparation of the beam by the beam switchyard therefore is needed. With energy-loss spectrometers, the now-standard setup used, the beam switchyard that carries the beam from the accelerator to the target actually becomes an integral part of the spectrometer used to detect and energy-analyze the scattered electrons. The higher-energy electron accelerators used at present for nuclear structure studies produce beams of energy B~700 MeV (Bates), 500 MeV (NIKHEF) and 700 MeV (Saclay). The energy resolution of these beams is typically AE/E=2.10-3, the duty cycle of order 1%. The typical setup that has developed over the past ten years incorporates a beam switchyard that distributes the beam to different experimental halls; it simultanously serves to energy-disperse the beam such that on target different beam energies are correlated with different locations. A rotator moves the energy dispersion, which is produced in the horizontal plane, to the vertical one. The electrons scattered by the target are analyzed in the vertical plane by a magnetic spectrometer that measures the energy and (explicitly or implicitly) the geometric location of the scattering vertex on target. Having m e a s u r e d both scattered and initial electron energy, the energy loss of the electron, the quantity of interest for nuclear physics, can be deduced with an accuracy far superior to the AE of the beam. This type of setup has a number of features tageous for high-energy electron scattering
411
that make it very advanexperiments. These ex-
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Ingo Sick
p e r i m e n t s suffer from two m a j o r problems: the low c o u n t i n g rates, and not very good e n e r g y resolution. The former is due to the w e a k ness of the e l e c t r o m a g n e t i c c o u p l i n g (which will be an a d v a n t a g e w h e n i n t e r p r e t i n g the s c a t t e r i n g data). The latter is due to the low m a s s of the electron, w h i c h r e q u i r e s high energy, i.e. very small AE/E, in order to achieve short w a v e length and good spatial resolution. The e n e r g y loss s p e c t r o m e t e r t e c h n i q u e s a l l o w to use the full b e a m intensity, w h i c h helps to i n c r e a s e countrates. The energy r e s o l u t i o n obt a i n a b l e is a factor 10-20 b e t t e r than the one of the i n c i d e n t b e a m ; w i t h an o v e r a l l r e s o l u t i o n of AE/E~I0 -4 achievable, e x p e r i m e n t s w i t h an e n e r g y r e s o l u t i o n of <50 K e V become p o s s i b l e d e s p i t e the large incident energies. Fig. 1 shows the layout of a typical switchyard, the one of the ALS m a c h i n e at S a c l a y (Leconte, 1980). In order to start w i t h a b e a m of w e l l d e f i n e d optical properties, the b e a m is focussed at the end of the a c c e l e r a t o r to a v e r t i c a l line. A c o l l i m a t o r of 0.5 mm width, u s e a b l e o n l y at low a v e r a g e b e a m i n t e n s i t y (low r e p e t i t i o n rate), allows to check the fraction of the b e a m that is w i t h i n the d e s i r e d spot. The m a g n e t labelled B4 d e f l e c t s the beam by ~45 °, and p r o v i d e s the m a i n e n e r g y dispersion. Its m a g n e t o - o p t i c a l properties, a d j u s t e d by the i n c l i n a t i o n of e n t r a n c e and exit p o l e t i p angles, are such that a m o n o - e n e r g e t i c b e a m is focussed to a v e r t i c a l line at the l o c a t i o n of the e n e r g y slits. The slit e n e r g y - d e g r a d e s those e l e c t r o n s that fall o u t s i d e the e n e r g y band of AE/E=I0- ~ desired; the v e r t i c a l ext e n s i o n of the b e a m at this l o c a t i o n serves to p r o t e c t the slits a g a i n s t d a m a g e from the high intensity, t y p i c a l l y 100~A, beam. The q u a d r u p o l e lens QD serves to further i n c r e a s e the energy d i s p e r s i o n of the b e a m by about a factor of 2; QD can be tuned such as to obtain at the target the d i s p e r s i o n r e q u i r e d by the spectrometer. The m a g n e t B5 d e f l e c t s the b e a m by a n o t h e r ~45o; the m a i n f u n c t i o n of B5 is to d e f l e c t away those e l e c t r o n s that have been e n e r g y - d e g r a d e d by the slits, such that they can be r e m o v e d by collimators. D o w n s t r e a m of B5, a set of 5 q u a d r u p o l e m a g n e t s Q R I + Q R 5 rotates the e n e r g y d i s p e r s i o n of the b e a m to the v e r t i c a l plane. As c o m p a r e d to the usual o r i e n t a t i o n of q u a d r u p o l e lenses, a r r a n g e d such that x- and y-coordinates are not coupled, these q u a d r u p o l e s are r o t a t e d by 45 ° . By an a p p r o p r i a t e choice of q u a d r u p o l e s t r e n g t h and distance, the e n t i r e r o t a t o r o b t a i n s o p t i c a l p r o p e r t i e s such that x and y of the b e a m are exchanged, w i t h o u t any further f o c u s s i n g action. The pair of q u a d r u p o l e s QFI, QF2, focusses the b e a m onto the target. For a m o n o c h o r m a t i c i n c i d e n t beam, a h o r i z o n t a l line image of a few m m w i d t h is produced. For a b e a m of finite e n e r g y w i d t h AE/E a v e r t i c a l d i s p e r s i o n of ~imm/10 -4 is produced. The d e c o u p l i n g of v a r i o u s parameters, d i s p e r s i o n (QD), r o t a t i o n (QR), m o n o c h r o m a t i c image (QF), helps to e ~ f i c i e n t l y adjust the b e a m p r o p e r t i e s desired. The s p e c t r o m e t e r images a point on the target to a point in the focal plane d e p e n d i n g on e l e c t r o n energy. For e n e r g y - l o s s s p e c t r o m e ters, an i m p o r t a n t q u a n t i t y is the d i s p e r s i o n D=AX/AE, w h e r e AX is the t a r g e t c o o r d i n a t e in the d e f l e c t i o n p l a n e and AE the e n e r g y change for fixed focal plane coordinate. For a g i v e n s p e c t r o m e t e r d i s p e r s i o n D, the s w i t c h y a r d d i s p e r s i o n is chosen to be -D. Hereby all electrons, i n d e p e n d e n t of their initial energy, are focussed
413
Electron Scattering
jo,o
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Beam switchyard and HEI experimental hall of the Saclay ALS accelerator.
onto the same point in the focal plane if they have lost the same amount of energy in the target. The position along the focal plane thus no longer corresponds to scattered energy, as it does in conventional systems working with pointlike spotsize on target, but to energy loss. The spectrometers used always deflect in the vertical plane; the horizontal deflection mode used in most lightqon spectrometers is not advantageous for electron scattering. Vertical deflection allows the largest range of scattering angles. This feature is of great importance when one tries to separate charge and magnetic form factors by exploiting their dominance at small and large scattering angles, respectively. In addition, vertical deflection decouples the measurement of energy from the one of scattering angle. This scattering angle must be m e a s u r e d with a precision superior to the one determined by the spectrometer solid angle defining slits if a degradation of the m i s s i n g - e n e r g y resolution due to variation of recoil nucleus energy is to be avoided. In most spectrometers the transverse extension of the beam on target is mapped to a fairly narrow focal plane. This helps to decrease the size of the detector and reduces background. In these cases, the information on the scattering angle is contained in the angle between e l e c t r o n - t r a j e c t o r y and spectrometer median plane. A number of different types of spectrometers reflecting the progressing art of design are presently in use at the different facilities.
414
Ingo Sick
Here, we show o n l y one, the c l a s s i c a l ~180 ° s p e c t r o m e t e r as p r e s e n t l y used, e.g. in Saclay. In this type of spectrometer, fig. 2, the e l e c trons are d e f l e c t e d by an i n h o m g e n e o u s m a g n e t i c field, w i t h a field index chosen to p r o v i d e f o c u s s i n g in both the d e f l e c t i o n plane and p e r p e n d i c u l a r to it. W i t h such a spectrometer, one can a c h i e v e an energy r e s o l u t i o n of ~1.5"10 -4 over the full solid angle of 4.5 msr, and 10 -4 over m u c h of it. The b a f f l e s (fig. 2) stop stray e l e c t r o n s p r o d u c e d by p r i m a r y e l e c t r o n s o u t s i d e the 10% energy a c c e p t a n c e of the spectrometer. The large d e f l e c t i o n angle, the g o o d s h i e l d i n g of the d e f l e c t i o n v o l u m e by the m a g n e t yoke t o g e t h e r w i t h a thick s h i e l d i n g (0.8m iron) of the focal plane p r o v i d e s e x c e l l e n t background rejection; cross s e c t i o n s less than i0-38cm2 can be measured. M o r e recent s p e c t r o m e t e r designs, as the one at Bates (Bertozzi,1979) or N I K H E F (de Vries, 1978) feature several d e f l e c t i n g m a g n e t s w i t h h o m o g e n e o u s fields and curved e n t r a n c e / e x i t pole faces. T h e s e pole face c u r v a t u r e s give better c o n t r o l over s p e c t r o m e t e r aberrations, and allow, as in the case of the N I K H E F QDD, to a c h i e v e a rather flat focal plane. The q u a d r u p o l e - 2 dipole (QDD) a r r a n g e m e n t p r e s e n t l y rep r e s e n t s the o p t i m u m c o m b i n a t i o n for good r e s o l u t i o n , l a r g e solid angle and small size (cost). The q u a d r u p o l e creates a cross over of e l e c t r o n t r a j e c t o r i e s w i t h i n the DD magnets, thus d i m i n i s h i n g the g a p s i z e for given solid angle. A m u l t i p o l e e l e m e n t in b e t w e e n the two d i p o l e s can be used to c o r r e c t h i g h e r order aberrations. This N I K H E F QDD p r e s e n t l y achieves < i ~ -4 r e s o l u t i o n for 5 m s r solid angle.
C~L'"*'0ls
--
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The 900 M e V meter,
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H i g h r e s o l u t i o n spectrom e t e r p r o p o s e d by ANL.
415
Electron Scattering
The next generation spectrometer probably will feature one additional element : a position sensitive thin detector placed after the first dipole, at a place where an intermediate image of the target is formed (fig. 3). In such an arrangement (Zeidmann, 1982) the "first" 3QD spectrometer serves to measure the target coordinates, while the "second" 3D spectrometer serves to measure the electron energy. Decoupling these two quantities gives new degrees of freedom, and promises excellent resolution, a few times 10 -5 , in combination with large, 20msr, solid angle. The focal plane detector has two principal functions : to identify the scattered particles as an electron, and to record with high precision the geometry of its track. This has to be done in a very short time, like ius. The large dynamical range of electron scattering cross sections (up to 12 decades have.been measured, see next section), together with the poor duty cycle of todays accelerators can lead, at low momentum transfer, to very high instantaneous counting rates. In order to identify the scattered electron, the most constraining capability is the one to reject pions. This can be achieved by using Aerogel- or Gas-Cerenkov counters; due to their low index of refraction, these counters do not respond to pions of less than ~i GeV/c momentum. The Gas-Cerenkov counter, ~30cm deep and containing Freon at 1 atmosphere pressure used at Saclay, gives a 100% rejection of pions. To suppress noise, a fast coincidence (20ns) with two rows of plastic scintillator detectors is required. In order to avoid problems due to multiple scattering, these detectors are located after the wire chambers used to measure the track geometry.
I---6mm~
DL3
DLI
DL2
DL3
0LI
DLZ
12
/ TYPICAL TRACK
Fig.4
/
Configuration ber.
of a vertical
drift cham-
416
Ingo Sick
The l o c a l i z a t i o n d e t e c t o r that has found g e n e r a l a c c e p t a n c e in electron s p e c t r o m e t e r s is the m u l t i w i r e d r i f t chamber. This type of detector is thin e n o u g h to allow m u l t i p l e m e a s u r e m e n t s of the same track. It gives e x c e l l e n t spatial r e s o l u t i o n (~0,2mm) w i t h a ressonably small n u m b e r of active wires (typically one per 5 mm). The type of d r i f t c h a m b e r used is d i c t a t e d by the fact that for the s p e c t r o m e t e r s u s e d the angle b e t w e e n focal plane and track is far from 90°~ t y p i c a l l y it is c l o s e r to 30-50 ° . In this case the v e r t i cal drift c h a m b e r (fig. 4) d e v e l o p e d l a r g e l y at Bates (Bertozzi,1977), r e p r e s e n t s a very e c o n o m i c a l m e a n s to a c c u r a t e l y locate the track. The c h a m b e r contains, in r e g u l a r alternation, active (thin) and passive (thick) wires, a r r a n g e d such that b e t w e e n w i r e and h i g h - v o l t a g e plane cells of rather c o n s t a n t e l e c t r i c field are created. Secondary e l e c t r o n s p r o d u c e d in 3 or m o r e cells drift to the sense wires, w h e r e the signal is amplified; the arrival time is a m e a s u r e of the d i s t a n c e b e t w e e n track and sense wire. W i t h 3 or m o r e cells triggered, one c o o r d i n a t e of the track can be r e c o n s t r u c t e d i n d e p e n d e n t of a b s o l u t e time m e a s u r e m e n t s . (The i n c l i n a t i o n of the track is obtained as well, but w i t h a p r e c i s i o n g e n e r a l l y not sufficient). Since such a m e a s u r e m e n t only r e q u i r e s the signal from the 3 wires g i v i n g the s h o r t e s t times, every third w i r e can be c o n n e c t e d to the same time m e a s u r i n g device. As m e n t i o n e d above, b o t h a m e a s u r e m e n t in the d i s p e r s i v e d i r e c t i o n and the one o r t h o g o n a l to it are needed. Such m e a s u r e m e n t s can be d o n e s u c c e s s i v e l y , since the m u l t i p l e s c a t t e r i n g due to M W P C w i n d o w s is low e n o u g h to not d e t e r i o r a t e the track information. Two chambers, one in the focal plane, one ~20cm behind, w i t h wires o r t h o g o n a l to the s p e c t r o m e t e r m i d p l a n e can be used to d e t e r m i n e e l e c t r o n e n e r g y and the c o m p o n e n t of the s c a t t e r i n g angle out of the h o r i z o n t a l plane. Two f u r t h e r c h a m b e r s w i t h wires o r i e n t e d at 45 ° r e l a t i v e to the first ones, can be used to m e a s u r e d i s t a n c e and angle r e l a t i v e to the spect r o m e t e r midplane; this allows to r e c o n s t r u c t the s c a t t e r i n g angle. Given the m u l t i t u d e of chambers, the r e a d o u t system can b e c o m e q u i t e complex. N e v e r t h e l e s s , fast r e a d o u t (in ~s) is d e s i r e d to keep deadtime at low m o m e n t u m t r a n s f e r small. The perhaps m o s t c o m p l e x s y s t e m is the one p r e s e n t l y being i n s t a l l e d at Saclay. The focal plane is e l e c t r o n i c a l l y s e g m e n t e d into 4 i n d i v i d u a l sectors in order to r e d u c e pileup. In every sector, and for every one of the 4 c h a m b e r s ~ 3 timet o - d i g i t a l c o n v e r t e r s (TDC) r e g i s t e r the times of w i r e s i-l, i, i+l. The i n f o r m a t i o n on w i r e n u m b e r i is r e g i s t e r e d by p a t t e r n units which, like the T D C ' s , a r e read out u p o n a t r i g g e r f u r n i s h e d by the fast coi n c i d e n c e b e t w e e n C e r e n k o v and P l a s t i c counters. This d e t e c t o r is c a p a b l e to y i e l d b o t h the g o o d spatial r e s o l u t i o n (0,2mm) r e q u i r e d for g o o d e n e r g y r e s o l u t i o n and the h i g h count rate c a p a b i l i t y required for (lower resolution) c o i n c i d e n c e e x p e r i m e n t s of the type (e,e'p). Once the track of the s c a t t e r e d e l e c t r o n is c o m p l e t e l y m e a s u r e d in the focal plane, the e n e r g y r e s o l u t i o n a c h i e v a b l e a c t u a l l y b e c o m e s b e t t e r than w h a t is i m p l i e d by the s p e c t r o m e t e r o p t i c a l properties. By s o f t w a r e corrections, m a n y of the s p e c t r o m e t e r a b e r r a t i o n s can be removed.
Electron Scattering
417
The t a r g e ~ u s e d for e l e c t r o n s c a t t e r i n g e x p e r i m e n t s have t h i c k n e s s e s of t y p i c a l l y t 1 0 0 m g / c m 2. For such targets the c o n t r i b u t i o n of energy straggling, r o u g h l y t.0.4 M e V / g c m -2, is of the o r d e r of o t h e r cont r i b u t i o n s to the o v e r a l l resolution. This s t r a g g l i n g is the dominant c o n t r i b u t i o n of the target as long as the t a r g e t angle can be c h o s e n to b i s e c t the d i r e c t i o n of i n c o m i n g and s c a t t e r e d electron. If the target, at v e r y large s c a t t e r i n g angle, m u s t be used in ref l e c t i o n geometry, the m u c h l a r g e r c o n t r i b u t i o n of e n e r g y loss, ~t-4 M e V / g c m -2, imposes a m u c h s m a l l e r t a r g e t thickness. A m a j o r p r o b l e m w i t h targets results from the h i g h b e a m i n t e n s i t i e s n e e d e d to c o m p e n s a t e for the low cross sections. With, say, 50~A e l e c t r o n b e a m intensity, m a n y t a r g e t s will m e l t rapidly. Although the use of the d i s p e r s e d b e a m t e c h n i q u e leads to a s p r e a d i n g out of the b e a m over ~icm, this often is not sufficient. M o v i n g the target h o r i z o n t a l l y , in order to spread the heat over a larger area, helps. The b e s t w a y to r e m o v e heat, h o w e v e r ', is to cool the t a r g e t w i t h a jet of H 2 - g a s of ~i T o r r pressure. This H 2 is p r o d u c e d u s i n g a c l o s e d c i r c u i t s y s t e m w i t h Roots pumps and can be p o i n t e d at the t a r g e t u s i n g nozzles. Due to the low density, this H 2 does not int e r f e r e w i t h the experiment, but it r e q u i r e s thin w i n d o w s (capable of w i t h s t a n d i n g a few T o r r pressure) to s e p a r a t e b e a m l i n e and spect r o m e t e r v a c u u m from the s c a t t e r i n g chamber. In o r d e r to o b t a i n a c c u r a t e cross sections, w i t h s y s t e m a t i c a l e r r o r s of the order of 1%, the target t h i c k n e s s and h o m o g e n e i t y m u s t be known to at least this accuracy. R e l a t i v e target t h i c k n e s s p r o f i l e s are best o b t a i n e d using x - r a y absorption; w i t h a y - s o u r c e of e n e r g y a p p r o p r i a t e l y c h o s e n to y i e l d a factor of 2~i0 absorption, the density p r o f i l e can be m e a s u r e d using a small Ge d e t e c t o r . The absolute density, a v e r a g e d over the area of the target hit by the beam, is o b t a i n a b l e from the c o m b i n a t i o n of one a b s o l u t e m e a s u r e m e n t and the r e l a t i v e d e n s i t y d i s t r i b u t i o n . This one a b s o l u t e r e f e r e n c e p o i n t can be o b t a i n e d by m e a s u r i n g w i t h a s u i t a b l e t a r g e t (often of n a t u ral i s o t o p i c composition) weight, area and r e l a t i v e d e n s i t y p r o f i l e of the entire target. For the o v e r a l l a c c u r a c y of the e x p e r i m e n t a l cross s e c t i o n s to be determined, a n u m b e r of a d d i t i o n a l v a r i a b l e s m u s t be k n o w n w i t h good p r e c i s i o n : e l e c t r o n b e a m energy, i n t e g r a t e d e l e c t r o n b e a m charge, s p e c t r o m e t e r solid angle and focal plane d e t e c t o r e f f i c i e n c y . The i n c i d e n t e l e c t r o n e n e r g y in g e n e r a l is o b t a i n e d from an a c c u r a t e m e a s u r e m e n t of the s c a t t e r e d e l e c t r o n energy. The field m a p s of high. r e s o l u t i o n s p e c t r o m e t e r s o f t e n are w e l l k n o w n from the m e a s u r e m e n t s p e r f o r m e d to v e r i f y the e n e r g y resolution. A field m e a s u r e m e n t at one point, p e r f o r m e d u s i n g a n u c l e a r m a g n e t i c r e s o n a n c e probe, then is s u f f i c i e n t to d e t e r m i n e the field integral, hence, e l e c t r o n e n e r gy, to a few parts in 104 . The e l e c t r o n b e a m i n t e n s i t y can be i n t e g r a t e d using t o r o i d p u l s e t r a n s f o r m e r s or a F a r a d a y cup. The latter is m o s t s u i t a b l e for absolute c h a r g e d e t e r m i n a t i o n s , and y i e l d s an a c c u r a c y of t y p i c a l l y a f r a c t i o n of one percent. For t a r g e t s of a p p r e c i a b l e thickness, m u l tiple s c a t t e r i n g c a u s e s a few p e r c e n t of the e l e c t r o n s to m i s s the PPNP-N*
418
Ingo Sick
F a r a d a y cup, u n l e s s b e a m r e f o c u s s i n g q u a d r u p o l e s d o w n s t r e a m of the t a r g e t are employed. T h e m i s s i n g f r a c t i o n can be d e t e r m i n e d by m e a suring ratios of F a r a d a y cup and t o r o i d i n t e g r a t e d c h a r g e s w i t h and w i t h o u t target. The s p e c t r o m e t e r solid a n g l e in g e n e r a l is k n o w n w i t h an a c c u r a c y of a f r a c t i o n of one p e r c e n t fro m the g e o m e t r i c a l p r o p e r t i e s of the sol i d - a n g l e d e f i n i n g slits. S o m e care is n e e d e d if the solid angle is not d e f i n e d by the slit in front of the first m a g n e t i c e l e m e n t of the s p e c t r o m e t e r , b u t r a t h e r by some i l l - d e f i n e d p i e c e of the v a c u u m chamber; this s i t u a t i o n u s u a l l y o c c u r s w h e n the b i g g e s t p o s s i b l e solid angle is used. U n d e r these c i r c u m s t a n c e s , the p r e c i s i o n is lower, since the s o l i d angle d e p e n d s on b e a m size, e l e c t r o n e n e r g y and alike. T h e q u a n t i t y m o s t d i f f i c u l t to m e a s u r e u s u a l l y is the focal p l a n e detector efficiency. It o f t e n is d e t e r m i n e d via a m e a s u r e m e n t of an a c c u r a t e l y k n o w n r e f e r e n c e cross section; the 12C and H 2 e l a s t i c cross s e c t i o n s g e n e r a l l y are used for this purpose. An i n d e p e n d e n t d e t e r m i n a t i o n of the a b s o l u t e e f f i c i e n c y is p o s s i b l e if the d e t e c t o r has e n o u g h e l e m e n t s to o v e r d e t e r m i n e e l e c t r o n i d e n t i f i c a t i o n and track m e a s u r e m e n t . In this case the r e d u n d a n c y of the d e t e c t i o n system can be e x p l o i t e d to d e t e r m i n e the e f f i c i e n c y of e v e r y i n d i v i d u a l element. W i t h t o d a y ' s c o m b i n a t i o n s of several ~[WPC, P l a s t i c and C e r e n c o v counters, this r e d u n d a n c y is a v a i l a b l e at low m o m e n t u m transfer, w h e r e b a c k g r o u n d r e j e c t i o n is of no concern. This proced u r e can be e x p e c t e d to p r o v i d e in the future the s t a n d a r d m e t h o d to d e t e r m i n e a b s o l u t e e f f i c i e n c i e s w i t h ~1% accuracy. An alternative p r o c e d u r e , using a s p e c i a l s p e c t r o m e t e r w i t h a p a r t i c u l a r l y simple d e t e c t o r , has b e e n d e v e l o p e d at Mainz (Reuter, 1982). For e x p e r i m e n tal p a r a m e t e r s d e t e r m i n e d w i t h the a c c u r a c i e s q u o t e d above, cross s e c t i o n s can be m e a s u r e d w i t h s y s t e m a t i c a l u n c e r t a i n t i e s of one to s e v e r a l percent. The m o s t p r e c i s e e x p e r i m e n t s , a i m i n g at the d e t e r m i n a t i o n of a b s o l u t e r e f e r e n c e cross sections, have r e a c h e d 0.5% sys t e m a t i c a l e r r o r (Cardman, 1980).
Nucleonic
Wave
Functions
In this section, I w i l l d i s c u s s the a p p l i c a t i o n of e l e c t r o n , s c a t t e ring to the i n v e s t i g a t i o n of n u c l e o n i c w a v e f u n c t i o n s in nuclei. To do so, I p i c k one p a r t i c u l a r e x a m p l e that allows to p r e s e n t the physics in a t r a n s p a r e n t w a y : the use of e l a s t i c e l e c t r o n s c a t t e r i n g for the d e t e r m i n a t i o n of n u c l e a r c h a r g e d e n s i t i e s . The m o t i v a t i o n for s t u d y i n g e l a s t i c e l e c t r o n s c a t t e r i n g r e s u l t s from our d e s i r e to m e a s u r e the shape of nuclei. As we w i l l see below, e l e c t r o n s c a t t e r i n g p r o v i d e s us w i t h a tool that c o m e s q u i t e c l o s e to the " e l e c t r o n m i c r o s c o p e " we are u s e d to r e l a t e to the i n v e s t i g a tion of s p a t i a l p r o p e r t i e s of v e r y small objects. For the p a r t i c u l a r case d i s c u s s e d below, the c h a r g e d i s t r i b u t i o n of a n u c l e u s can be measured. T h i s c h a r g e d i s t r i b u t i o n p r o v i d e s us w i t h the r a d i a l (and s o m e t i m e s azimuthal) d i s t r i b u t i o n of p r o t o n s in nuclei. This quantity is of g r e a t i n t e r e s t for the q u a n t i t a t i v e u n d e r s t a n d i n g of n u c l e o n wave functions.
Electron Scattering
419
E l e c t r o n s as a p r o b e are p a r t i c u l a r l y s u i t a b l e for a c c u r a t e d e t e r m i n a t i o n s of c h a r g e d e n s i t i e s . The i n t e r a c t i o n of an e l e c t r o n w i t h the n u c l e u s is known, and weak. C o n s e q u e n t l y , the o n l y u n k n o w n s occ u r r i n g are the n u c l e a r p r o p e r t i e s , and t h e y can be e x t r a c t e d f r o m e x p e r i m e n t in a q u a n t i t a t i v e w a y since the w e a k n e s s of the e l e c t r o m a g n e t i c i n t e r a c t i o n a l l o w s for a (basically) e x a c t d e s c r i p t i o n of the s c a t t e r i n g process. In o r d e r to d i s c u s s e l a s t i c s c a t t e r i n g , I w i l l c o n c e n t r a t e on one p a r t i c u l a r nucleus, the one of tin, w h i c h has a m a g i c p r o t o n n u m b e r Z=50. P r o t o n t r a n s f e r r e a c t i o n s i n d i c a t e that Z=50 is a v e r y g o o d shell closure; n e i t h e r h o l e s in the n o r m a l l y o c c u p i e d s h e l l s (~ig9/2) nor p a r t i c l e s e x c i t e d to the n o r m a l l y e m p t y shells, h a v e b e e n o b s e r ved. For such a n u c l e u s , the m o s t s o p h i s t i c a t e d c a l c u l a t i o n s of g r o u n d state w a v e f u n c t i o n s , the o n e s o b t a i n e d from H a r t r e e - F o c k theory, are a p p l i c a b l e , a n d a l l o w a q u a n t i t a t i v e c o m p a r i s o n to e x p e riment. In addition, the tin i s o t o p e s p r o v i d e a large v a r i a t i o n of n e u t r o n n u m b e r (A=I12+124 are stable); the s t u d y of isotopic d i f f e r e n ces then a l l o w s to s t u d y w h e t h e r the p r e s e n c e of an o p e n n e u t r o n shell a p p r e c i a b l y i n f l u e n c e s the p r o t o n c o n f i g u r a t i o n and density. S o m e i0 y e a r s ago, we p e r f o r m e d at S t a n f o r d an e x p e r i m e n t (Ficenec, 1972) on all the s t a b l e tin i s o t o p e s ; the d a t a r e s u l t i n g from this e x p e r i m e n t cover the r a n g e of m o m e n t u m t r a n s f e r up to 2 . 7 f m -I. In o r d e r t o e x t e n d the r e g i o n of m o m e n t u m t r a n s f e r , we c o m p l e t e d 2 y e a r s ago an e x p e r i m e n t (Cavedon, 1982) at Saclav, w h e r e we r e a c h e d for ll6sn and 124Sn, q = 3 . 5 f m -I. The d a t a r e s u l t i n g from these two e x p e r i m e n t s are shown in fig. 5. The c r o s s s e c t i o n s n o w c o v e r a r a n g e of m o r e than 12 decades, and the lowest p o i n t m e a s u r e d has a cross s e c t i o n of i 0 - 3 7 c m 2. T y p i c a l s y s t e m a t i c u n c e r t a i n t i e s are 3% and 1% for c r o s s s e c t i o n and cross s e c t i o n ratios; s t a t i s t i c a l e r r o r s v a r y b e t w e e n <1% at low q, to 100% at the h i g h e s t q.
V(~) i0~ [0°
|0-I IO-2 i0-3
Fig.5
i0-4 lO-S
iO-6 [0-7 i0-s i0 -9
i0-*o
i0-n
l
l
2
3
Q(FM "I)
E l a s t i c cross s e c t i o n s f o r 1 2 4 S n (in mb/sr).
420
Ingo Sick
In order to extract the charge d e n s i t y from the data, I will, for the moment, assume the v a l i d i t y of the Plane W a v e Bron A p p r o x i m a t i o n (PWBN). This w o u l d r e q u i r e that Z.~<< i , w h i c h is not the case. But we will see that the p r o c e d u r e can be e a s i l y m o d i f i e d to account for this a p p r o x i m a t i o n . In PWBA, the cross section for e l a s t i c a l l y a s p i n - z e r o nucleus can be d e r i v e d easily: da d--~ (6,E)
= -da (6,E) deMott
Here aM_tt is the M o t t cross a p o i n t ~ i k e o b j e c t of charge Z2.e2
cos26/2
d~Mott
4E 2
sin46/2
1 = ~ f p(r)
sin(qr) qr
for
(i)
describing
the scattering
from
(2)
E is the e l e c t r o n e n e r g y and 6 the s c a t t e r i n g angle. c o n t a i n i n g the n u c l e a r physics, the form factor F(q) F o u r i e r t r a n s f o r m of the n u c l e a r charge d e n s i t y p(r) F(q)
an e l e c t r o n
F2 (q)
section Z
da
scattering
The q u a n t i t y is given as the
4n r2dr
(3)
The v a r i a b l e of i m p o r t a n c e for e l e c t r o n scattering, the m o m e n t u m t r a n s f e r q g i v e n by the e l e c t r o n to the nucleus, is q = 2.E.sin6/2
(4)
(Center of mass effects, d e p e n d i n g on terms ted above, but trivial to include).
of order
l/A,
are n e g l e c -
E q u a t i o n (3) shows that, for CO charge scattering, the p h y s i c s inform a t i o n d e p e n d s on q only, and not on E,6 separately. This v a r i a b l e plays a m o s t i m p o r t a n t role for the i n v e s t i g a t i o n of the nucleus. Ac cording to eq. (3) the n u c l e a r charge d e n s i t y is sampled w i t h a sin(qr)/qr-function. For a g i v e n m a x i m a l m o m e n t u m t r a n s f e r q m a x the c h a r g e d e n s i t y is sampled w i t h a r e s o l u t i o n no finer than 1.5/qmax, this latter q u a n t i t y being, a p p r o x i m a t e l y , the F W H M of one of the peaks of sin(qr)/qr. The q u a n t i t y 1 . 5 / q m a x thus d i r e c t l y plays the role of the r e s o l v i n g power of our e l e c t r o n microscope. If we w a n t to see the d e t a i l e d features of the nucleus, we m u s t p u s h the e x p e r i m e n t to l a r ~ q m a x . E q u a t i o n (3) s u g g e s t s an easy w a y to e x t r a c t experiment. I n v e r t i n g the F o u r i e r t r a n s f o r m 0 (r) = Z _
4~
I n t e g r a t i n g over p(r). The first t e c h n i q u e (Sick, 3He and 4He.
/ F(q)
sin (qr) qr
q2dq
the charge yields
density
from
(5)
the e x p e r i m e n t a l l y d e t e r m i n e d form factors yields a p p l i c a t i o n of this Direct F o u r i e r T r a n s f o r m (DFT) 1978) c o n c e r n e d the d e n s i t y of very light nuclei,
Electron Scattering
421
The a p p l i c a t i o n of eq. (5) m e e t s two d i f f i c u l t i e s . First, P W B A is not a good a p p r o x i m a t i o n for h e a v y nuclei. Second, the m a x i m u m mom e n t u m t r a n s f e r of e x p e r i m e n t does not reach i n f i n i t y as r e q u i r e d by the upper i n t e g r a t i o n limit. Let us d i s c u s s the s o l u t i o n for those p r o b l e m s in turn. W h i l e eq. (I) is not v a l i d for large Z, one can e a s i l y d e t e r m i n e exp e r i m e n t a l form factors F e x _ such that eqs. 3, 5 are a p p l i c a b l e w i t h out loss of accuracy. In o r d e r to t r a n s l a t e cross s e c t l o n s into P W B A form factors, we can d e t e r m i n e a m o d e l charge d i s t r i b u t i o n that fits the e x p e r i m e n t a l cross sections. A c h a r g e d i s t r i b u t i o n d e p e n d i n g on a n u m b e r of p a r a m e t e r s is used to n u m e r i c a l l y solve the D i r a c equation that d e s c r i b e s w i t h o u t a p p r o x i m a t i o n (basically) the e l e c t r o n in the e l e c t r o s t a t i c field of the nucleus. The p a r a m e t e r s of this m o d e l d e n s i t y are v a r i e d until the cross s e c t i o n s c o m p u t e d agree w i t h the e x p e r i m e n t a l ones. P u t t i n g (Sick, 1974) F~xp(q)
= ~exp °mod
• F~o d
(q)
(6)
w h e r e q u a n t i t i e s h a v i n g the index "mod" are c o m p u t e d density, y i e l d s e x p e r i m e n t a l form factors that have of C o u l o m b d i s t o r t i o n removed. G i v e n the fact that tion d e p e n d s on v e r y global p r o p e r t i e s of p(r) o n l y tually) , the r e m o v a l of it via a m o d e l d e n s i t y does to Fex p any d e p e n d e n c e on the m o d e l density. (This by using m a n y d i f f e r e n t p(r) to e v a l u a t e eq. (6)).
using the m o d e l all the e f f e c t s the C o u l o m distor(mainly on Z, acnot i n t r o d u c e inhas b e e n v e r i f i e d
The Fex p for 124Sn r e s u l t i n g from the a p p l i c a t i o n of eq. (6) are shown in fig. 6. The m a i n d i f f e r e n c e to fig. 5 shows up in the dif-
F2(Q) iO-2
°°.,° iO-3
4 [O-'t
•
t%
Fig.6
i0 -s
*l i0 -6
i0 -7
%
I
i0 -a iO-9
i0 -Io 10- n
J
3
tl"1 Q(~*)
Experimental form tors for 124Sn.
fac-
422
Ingo Sick
fraction Fexp-
minima
of d~/d~,
which
turn
into
zeros
(and s i g n - c h a n g e s ) i n
The second d i f f i c u l t y m e n t i o n e d above c a n n o t be taken care of by a s u i t a b l e t r e a t m e n t of data only. The lack of i n f o r m a t i o n due to finite qmax can be m i n i m i z e d by p u s h i n g e x p e r i m e n t to the largest q, i.e. lowest cross section possible. However, this l i m i t a t i o n is no reason to resort to p r o c e d u r e s that d e t e r m i n e p(r) by i n t r o d u c i n g e x p l i c i t or implicit model assumptions. The d e n s i t y at any radius can be d e t e r m i n e d by s t u d y i n g eq. ( 5 ) p(r,qmax )
Z = 4-~
qmax [ F(q) o
sin(qr) qr
2 • q.dq
(7)
as a function of the upper i n t e g r a t i o n limit. The c o n v e r g e n c e of ~(r,qmax) towards the a s y m p t o t i c value p(r,~) = p(r) gives a d i r e c t estimate of the lack of k n o w l e d g e due to finite qmax" The way
result of eq. (7) can be d i s c u s s e d for r = O, where P(O,qma x) =
in a p a r t i c u l a r l y
transparent
qmax I F(q)q2dq o
(8)
For 124Sn, p(O,qmax) is shown in fig. 7. At every d i f f r a c t i o n m i n i m u m of do/d~, F(q) changes sign. Hence, p(O,qmax) r e p r e s e n t s a damped o s c i l l a t o r y function that c o n v e r g e s towards p(O). The d i f f e r e n c e b e t w e e n the last m e a s u r e d m a x i m u m - and m i n i m u m - v a l u e of p(O,qmax) can be taken as an honest, m o d e l i n d e p e n d e n t e s t i m a t e for the u n c e r t a i n t y p(O) is d e t e r m i n e d with. As is shown in fig. 7, p(O) has c o n v e r g e d to better than 2% to its a s y m p t o t i c value at q = 3.5 fm -I. This e x p l a i n s w h y e x p e r i m e n t has been p u s h e d to such a high m o m e n t u m transfer. To d e t e r m i n e p to
O(O,Q) ,
IO0
Fig.7
/ .000
0
" 1.0
" 2.0
' 3,0
' = Q(FY1-1)
C o n v e r g e n c e of p(O,qm~x) as function o~ qmax"
Electron Scattering
423
be£ter than 1% it typically takes a value of F2(qmax) of i0 -I0, independent of mass number; only then the contribution over the not-measured part of F 2 is small enough. The typical (ee)-experiment, which does not go beyond qmax~2fm -I, does not determine p(O) to better than 20%. For radii r>O, the sin(qr)/qr-term adds additional sign-changes of the integrand (eq. (7)). Consequently, the convergence will be more rapid. This DFT method, first applied to heavy nuclei in the interpretation of the 40Ca data (Sick, 1979), I consider to be the optimal method to extract the physics from the data. No model dependence whatsoever is introduced, the relation between data and density is most transparent, and systematical or statistical errors of the data can trivially be taken into account by error propagation. The results obtained for the charge density of 124Sn is shown in fig. 8. The density shows the familiar Fermi-distribution-like shape, with a small oscillatory component superimposed on small radii. In fig. 8 the experimental density is compared to 3 different HartreeFock predictions (Negele 1970, Campi 1972, Decharg~ 1980). These curves have been obtained by starting from an effective density dependent nucleon-nucleon interaction VNN determined either phen0menologically, or derived from the NN-interaction known from NN-scattering. In these HF-calculations the Schr6dinger equation for a nucleon moving in an average potential determined by VNN and the wave functions of the other (A-l) nucleons is solved selfconsistently. The major approximation made in these calculations is the assumption that the nucleons occupy the lowest shell-model orbits, and the cursory treatment of short-range NN correlations via the density-dependence of the effective force.
0,08
Fig. 8
0
2
4
6
R(FM)
Charge density of 124Sn, compared to DDHF predictions.
424
Ingo Sick
The c o m p a r i s o n b e t w e e n e x p e r i m e n t and c a l c u l a t i o n shows features typical for a number of other semimagic nuclei b e t w e e n 160 and 208pb. The global d e n s i t y is quite w e l l reproduced; in p a r t i c u l a r the surface region, and the n u c l e a r rms-radius, is w e l l explained. In the n u c l e a r interior, on the other hand, the t h e o r e t i c a l p r e d i c t i o n s show m u c h m o r e o s c i l l a t o r y s t r u c t u r e than given by experiment. The o s c i l l a t o r y s t r u c t u r e is due to the p r e s e n c e of s h e l l - e f f e c t s . The d e n s i t y is a sum of n u c l e o n radial w a v e functions squared, and these R2(r) e x h i b i t zeros and m a x i m a a c c o r d i n g to p r i n c i p a l q u a n t u m n u m b e r and a n g u l a r m o m e n t u m of the shell. In the total density, the s t r u c t u r e of these R2(r) still shows up as the o s c i l l a t o r y c o m p o n e n t of p(r). In order to e x p l a i n the excess of s t r u c t u r e of the DDHF densities, we can come back to the a p p r o x i m a t i o n s m e n t i o n e d above. In a real nucleus, w h i c h d e v i a t e s from the ideal " c l o s e d - s h e l l " picture, some n u c l e o n s will be e x c i t e d to h i g h e r lying shells, via 2 p - 2 h - e x c i t a tions, for instance. These a d m i x t u r e s of d i f f e r e n t c o n f i g u r a t i o n s can d i l u t e the shell e f f e c t s in p(r). S t r o n g s h o r t - r a n g e correlations b e t w e e n nucleons, a c o n s e q u e n c e of the r e p u l s i v e core of VNN, disfayours a p p r e c i a b l e e x c u r s i o n s of p(r) away from a c o n s t a n t m a x i m u m density; q u a n t i t a t i v e c a l c u l a t i o n s (Strayer, 1973) c o n f i r m that this does lead to a r e d u c t i o n in the amount of shell s t r u c t u r e in p(r). In o r d e r to d i s t i n g u i s h e x p e r i m e n t a l l y b e t w e e n S n - i s o t o p e s are p a r t i c u l a r l y favourable. When 124Sn, the a d d i t i o n of 8 n e u t r o n s is e x p e c t e d e f f e c t on the c o n f i g u r a t i o n admixtures, since the shells of interest for p r o t o n e x c i t a t i o n s A study of the 1 2 4 s n - l 1 6 s n i s o t o p i c d i f f e r e n c e s o f t n e s s of the proton c o n f i g u r a t i o n could be d i f f e r e n c e b e t w e e n HF and experiment.
these two points, the going from ll6sn to to have a c o n s i d e r a b l e these n e u t r o n s o c c u p y above the Fermi level. can reveal w h e t h e r a r e s p o n s i b l e for the
/24
1.5
l.o
{,
~
--
0.5 I
0,25
Fig.9
1.25
Cross
section
2 25
ratios
3 25 Q(FMi)
124sn/l16sn.
Electron Scattering
425
~,~)(R) 0
!
!
Fig.10
Density difference 124Sn-x16Sn, compared DDHF p r e d i c t i o n s .
to
-.008 ~ 0
2
4
6
R(FM)
The ratios b e t w e e n the e x p e r i m e n t a l cross s e c t i o n of 124Sn and ll6sn is shown in fig. 9, the d e n s i t y d i f f e r e n c e in fig. 10. T h i s figure shows that the a g r e e m e n t b e t w e e n e x p e r i m e n t and the d i f f e r e n t t h e o r e tical c a l c u l a t i o n s is quite good. D e s p i t e an u n u s u a l l y small a m p l i tude, the e x p e r i m e n t a l error band c o n t a i n s m o s t of the c a l c u l a t i o n s ~ d i f f e r e n c e s at small radii are of the o r d e r of less than 1% of p(r). (The strong o s c i l l a t o r y s t r u c t u r e in Ap(r) o b t a i n e d w i t h the G-O force at p r e s e n t is not u n d e r s t o o d ) . F r o m fig. 10, and the study of s p e c t r o s c o p i c factors m e n t i o n e d at the b e g i n n i n g of this section, we c o n c l u d e that it is not the s o f t n e s s of the p r o t o n c o n f i g u r a t i o n , i.e. the n o n - m a g i c n a t u r e of Z=50, that is r e s p o n s i b l e for the excess of s h e l l - m o d e l s t r u c t u r e in HF d e n s i ties. Rather, the lack of a p r o p e r t r e a t m e n t of s h o r t - r a n g e NN corr e l a t i o n in c a l c u l a t i o n s w h e n t a k i n g as a base the i n d e p e n d e n t - p a r ticle shell m o d e l (IPSM), m u s t be held r e s p o n s i b l e . B e f o r e a c c e p t i n g this latter c o n c l u s i o n , we a c t u a l l y s h o u l d v e r i f y that the real p r o b l e m does not lie at a m u c h m o r e f u n d a m e n t a l level. In the h i g h d e n s i t y r e g i o n of a nucleus, the v e r y c o n c e p t of the shell model, w i t h i n d e p e n d e n t p a r t i c l e s m o v i n g in o r b i t s d e t e r m i n e d by an a v e r a g e potential, m i g h t loose its sense. It is far from obvious that nucleons, w h i t h an r m s - r a d i u s of 0.85 fm, in a n u c l e a r med i u m w i t h t y p i c a l NN d i s t a n c e s of 2 fm, could obey the g o v e r n i n g p r i n c i p l e s of the IPSM. A k i n d of n u c l e o n soup, w i t h signs of p a r tial d i s s o l u t i o n of n u c l e o n s into their q u a r k - c o n t e n t , could be a m u c h m o r e p l a u s i b l e scenario, p a r t i c u l a r l y at a time w h e n p e o p l e believe in q u a r k - b a g s of 1 fm radius s p a c e d by 2 fm from the c e n t e r of n e i g h b o u r i n g bags. Does the s h e l l - m o d e l c o n c e p t m a k e sense for the n u c l e a r i n t e r i o r ? E x p e r i m e n t a l v e r i f i c a t i o n s c o n c e r n i n g this q u e s t i o n are v e r y rare. M o s t o b s e r v a b l e s are d e t e r m i n e d using s t r o n g l y i n t e r a c t i n g p r o b e s that are s t r o n g l y a b s o r b e d by nuclei. Experimental observations
426
Ingo Sick
therefore mainly concern up t i l l n o w on I P S M w a v e indirect.
s u r f a c e p r o p e r t i e s ; the i n f o r m a t i o n p r o v i d e d f u n c t i o n s in the n u c l e a r i n t e r i o r is q u i t e
In o r d e r to s t u d y the b e h a v i o u r of p r o t o n w a v e f u n c t i o n s in the n u c l e a r i n t e r i o u r , w e h a v e m a d e an e x p e r i m e n t to d e t e r m i n e t h e d e n s i t y d i f f e r e n c e of P b 206 and T 1 2 0 5 . T h e s e h e a v y n u c l e i d i f f e r by a 3s p r o t o n , a s h e l l t h a t has a v e r y c h a r a c t e r i s t i c r a d i a l w a v e f u n c t i o n : A p r o n o u n c e d m a x i m u m at r = O, w i t h 2 n o d e s and 2 s e c o n d a r y m a x i m a . T h e p e a k at r = 0 in p a r t i c u l a r c a n s e r v e as an e x c e l l e n t t e s t for the r e l e v a n c e of I P S M o r b i t s at h i g h n u c l e a r d e n s i t y . A p r i o r i , it l o o k s q u i t e d i f f i c u l t to e x t r a c t R2(r) w i t h an a c c u r a c y of i n t e r e s t for the p u r p o s e s t a t e d . O n e a d d e d p r o t o n o u t of 82 p r o duces quite a small Ap(r), and this density difference does not get
16
£
Fig.ll
205TI/206pb cross section ratios calculated assuming different conf i g u r a t i o n s for t h e added proton-
Fig.12
E f f e c t of 2 0 5 T I c o r e polarization on cross section ratio.
12
o
==
NucLear
I
O8
polarization
I 1
o
I
I 2
I
I 3
q ( f r o -1)
# 1.6
--3s ..........
2d
(Z Z
1.2
o
SW
U'I .
1.0
¢J
0-8
I
I 1
I
q (fm -'~)
|
Electron Scattering
427
c o n t r i b u t i o n s from the 3s-shell only. The p o l a r i z a t i o n of the T1 core d u e to the a d d e d p r o t o n w i l l c o n t r i b u t e as well. In addition, shells other than 3s w i l l c o n t r i b u t e to ~p(r) g i v e n the fact that the 3s s p e c t r o s c o p i c factor is less than one. W h e n t r y i n g to i d e n t i f y the 3 s - c o n t r i b u t i o n , one s h o u l d r e a l i z e its unique property, an u n u s u a l shape. R2(r) r e s e m b l e s a d a m p e d o s c i l lation s i m i l a r to Jo(2~r/l) w i t h a w a v e length I g i v e n by 2 ~ / l ~ 2 f m -I. If R 2 w o u l d h a v e e x a c t l y the shape of j~, the c o r r e s p o n d i n g change AF of the form factor w o u l d be a 6 - f u n c t i o n o c c u r r i n g at a m o m e n t u m t r a n s f e r q-2 fm -I. The 3 s - c o n t r i b u t i o n t h e r e f o r e can be e x p e c t e d to stand out i n ~ / o , and a l l o w an easy i d e n t i f i c a t i o n and s e p a r a t i o n from other c o n t r i b u t i o n s . To d e m o n s t r a t e this, we show in fig. ii the Pb/TI cross s e c t i o n ratio o b t a i n e d by a s s u m i n g that the added p r o t o n w o u l d go in one of the shells 3s, 2d or ig of i n t e r e s t in c o n n e c t i o n w i t h c o n f i g u r a t i o n admixtures. One can i m m e d i a t e l y r e c o g n i z e the ~ - f u n c t i o n - l i k e spike due to the 3s shell. The e f f e c t of the o t h e r shells on ~ T I / o P b is v e r y d i f f e r e n t in shape, and can e a s i l y be s e p a r a t e d (Frois, 1983). In fig. 12 we show the e f f e c t on the cross s e c t i o n ratio due to the p o l a r i z a t i o n of the T1 core by the added 3s proton. Again, the shape of the two i n g r e d i e n t s to aTl/o Pb is v e r y different, since the core p o l a r i z a t i o n lacks the nodes p r e s e n t in R~s. Core p o l a r i z a t i o n s h o u l d not C o m p l i c a t e s i g n i f i c a n t l y the i n t e r p r e t a t i o n of o T l / o P b in terms of R~s. The e x p e r i m e n t a l r e s u l t for the 2 0 6 p b / 2 0 5 T I cross s e c t i o n r a t i o shown in fig. 13. The h i g h e r q - d a t a that show the 3 s - p e a k c o m e
aTl/apB
TI/PB
1.4
1.2
1,0
Fig.13
!
I
I.
2.
;
Q
E x p e r i m e n t a l 2 0 5 T l / 2 0 6 p b cross ratios. The curve represents prediction calculated assuming
3,
section the D D H F S3s=0.7 .
is from
428
Ingo Sick
Ap(r)
206pB-~1 • 008
Fig.14
Experimental density difference, c o m p a r e d DDHF prediction.
to
.004
I
I
I
I
I
2
4
6
8
I
the e x p e r i m e n t we r e c e n t l y c o m p l e t e d at S a c l a y (Cavedon, 1983), the low-q data come from an e a r l i e r e x p e r i m e n t p e r f o r m e d at Mainz (Euteneuer, 1978). The d e n s i t y d i f f e r e n c e e x t r a c t e d from these data is shown in fig. 14. The 3s-shape, w i t h its 3 m a x i m a and 2 nodes, stands out in a s p e c t a c u l a r way If we c o m p a r e this e x p e r i m e n t a l result to a HF p r e d i c t i o n o b t a i n e d by a s s u m i n g a pure 3s~2 hole c o n f i g u r a t i o n for 2 0 5 T l , t h e peak of Ap has an a m p l i t u d e about 40% too large; this d i f f e r e n c e is to be expected, though, since the s t a n d a r d DDHF c a l c u l a t i o n s are not s u f f i c i e n t for o d d - e v e n nuclei. Shell m o d e l c a l c u l a t i o n s for T1 show that the g r o u n d state c o n f i g u r a t i o n is m o r e complicated. B o t h studies of c o u p l i n g of the 3s proton to c o l l e c t i v e e x c i t a t i o n of 206pb, as w e l l as shell m o d e l c a l c u l a t i o n s for the (3s~2) -I (2d3/2) -2 state show that the 3s-hole s t r e n g t h in 205TI amounts to s = 0.7-0.9. The r e m a i n d e r of the s t r e n g t h is m a i n l y found in the 2d3/2 c o n f i g u r a t i o n . These p r e d i c t i o n s are c o n f i r m e d by the result o f - t r a n s f e r reactions. Proton p i c k u p from 206pb m e a s u r e s only 70% of the Z = O s t r e n g t h found in 208pb. A c c o r d i n g l y , we c o m p a r e in fig. 12 to a DDHF c a l c u l a t i o n (Campi, 1982) p e r f o r m e d w i t h the hole s t r e n g t h c o n s t r a i n e d to S3s = 0.7, S2d = 0.3. The r e s u l t i n g p r e d i c t i o n for the cross s e c t i o n ratio is shown also in fig. ii. The e x p e r i m e n t a l data are a m a z i n g l y w e l l reproduced; small d i f f e r e n c e s w i t h smooth radial s t r u c t u r e (fig. 14) can be a t t r i b u t e d to our s o m e w h a t too crude t r e a t m e n t of core polar i z a t i o n effects. F r o m figs. ii, 14 we can c o n c l u d e that the d e v i a t i o n s in p(r) disc u s s e d above are not linked to a b r e a k d o w n of the s h e l l - m o d e l idea. The c o n c e p t of the i n d e p e n d e n t p a r t i c l e orbit retains an i m p p r e s s i v e d e g r e e of v a l i d i t y even in the h i g h - d e n s i t y r e g i o n of nuclei. N u c leons retain their i n t e g r i t y (to a large extent, anyway) and the shell m o d e l a c c o u n t s quite well for their w a v e f u n c t i o n even t h o u g h the e f f e c t s of s h o r t - r a n g e c o r r e l a t i o n b e t w e e n n u c l e o n s are t r e a t e d
Electron Scattering in too c r u d e
Mesonic
a way
de@rees
429
in DDHF.
of f r e e d o m
T h e t o p i c of this last s e c t i o n deals w i t h c o n s t i t u e n t s of the n u c l e u s o t h e r than n u c l e o n s . For a g o o d reason, we d i s c u s s it o n l y a f t e r having p r e s e n t e d a d e t a i l e d (but limited) a c c o u n t of w h a t is k n o w n on n u c l e o n i c d e g r e e s of freedom. The e f f e c t s of n o n - n u c l e o n i c c o n s t i t u e n t s in n u c l e i in g e n e r a l are small, and d i f f i c u l t to i s o l a t e in an u n a m b i g u o u s way. U n l e s s we u n d e r s t a n d w e l l the n u c l e o n i c p r o p e r t i e s , we c a n n o t a t t r i b u t e d i f f e r e n c e s w i t h e x p e r i m e n t to the small, n o n n u c l e o n i c d e g r e e s of freedom. W h y do e l e c t r o n s r e p r e s e n t a g o o d p r o b e for the i n v e s t i g a t i o n of n o n n u c l e o n i c d e g r e e s of f r e e d o m ? For h a d r o n probes, the r e a c t i o n m e c h a n i s m can be q u i t e c o m p l i c a t e d . T h i s is true in p a r t i c u l a r if one w a n t s to s t u d y small w a v e f u n c t i o n c o m p o n e n t s . Multistep reactions i n v o l v i n g large c o m p o n e n t s o f t e n c o v e r up the c o n t r i b u t i o n s of i n t e rest, the ones due to small w a v e f u n c t i o n c o m p o n e n t s and a r e a c t i o n m e c h a n i s m s i m p l e e n o u g h to be i n t e r p r e t e d . For e l e c t r o n s the r e a c tion m e c h a n i s m a p r i o r i stays m o r e simple, s i n c e the w e a k n e s s of the i n t e r a c t i o n r e n d e r s m u l t i s t e p r e a c t i o n s less likely. The p r e s e n c e of m e s o n i c d e g r e e s of f r e e d o m in n u c l e i has b e e n k n o w n for a long time. A n u c l e o n - n u c l e o n i n t e r a c t i o n m e d i a t e d by m e s o n s e x c h a n g e d b e t w e e n n u c l e o n s i m p l i e s the p r e s e n c e of ~,p, etc. The p r o m i n e n c e of the A - r e s o n a n c e in the ~-N i n t e r a c t i o n i n d i c a t e s the i m p o r t a n c e of ~ in nuclei. It has b e e n q u i t e d i f f i c u l t , though, to find u n a m b i g u o u s e x p e r i m e n t a l s i g n a t u r e s of these n o n - n u c l e o n i c constituents. S i n c e the e a r l y w o r k of V i l l a r s (1947) it is c l e a r that m a g n e t i c p r o p e r t i e s of n u c l e i are p a r t i c u l a r l y w e l l s u i t e d for the i n v e s t i g a t i o n of m e s o n e x c h a n g e c u r r e n t s (~.~C). C o n t r a r y to c h a r g e s c a t t e r i n g , where ~C do not c o n t r i b u t e at q = 0 they a f f e c t in first o r d e r m a g n e t i c moments. Here, the c u r r e n t s p r o d u c e d by the e x c h a n g e d p a r t i c les can c o n t r i b u t e a l r e a d y at q = O. This observable therefore allows a d i r e c t a c c e s s to the p r e s e n c e of m e s o n s in nuclei. O v e r the years it has i n c r e a s i n g l y b e c o m e c l e a r that m a g n e t i c f o r m f a c t o r s are p a r t i c u l a r l y p r o m i s i n g at large q; t h e r e r c a n c e l l a t i o n s o c c u r r i n g in the n u c l e o n i c c o n t r i b u t i o n s can e n h a n c e the r e l a t i v e ~ C e f f e c t s by large factors. It a l s o has b e c o m e c l e a r that light n u c l e i o f f e r the b e s t c h a n c e to s t u d y MEC; for t h e s e nuclei, the n u c l e o n i c w a v e f u n c t i o n is w e l l e n o u g h k n o w n to a l l o w for a r e a s o n a b l y r e l i a b l e p r e d i c t i o n of this n u c l e o n i c " b a c k g r o u n d " . A c c o r d i n g l y , w e w i l l d i s c u s s in the f o l l o w i n g m a g n e t i c f o r m f a c t o r s of light n u c l e i at large m o m e n t u m t r a n s f e r only. The f i r s t e x a m p l e w e p r e s e n t c o n c e r n s the d e u t e r o n e l e c t r o d i s i n t e g r a tion at t h r e s h o l d . The q ~ O limit of this r e a c t i o n , n + p ÷ + d + y at v e r y low e n e r g y (Riska and Brown, 1972), t o g e t h e r w i t h the A = 3 m a g n e t i c m o m e n t s , was one of the first u n a m b i g u o u s p r o o f s that m e s o n i c d e g r e e s of f r e e d o m are n e e d e d for the q u a n t i t a t i v e u n d e r s t a n d i n g of n u c l e a r reactions.
430
Ingo Sick
I s.1636
, t,
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S p e c t r u m for d(e,e') at 410 M e V incident energy and 155 ° s c a t t e r i n g angle.
d~
d (e e')n p Enp- 0+3 MeV
Fig.16
\, IA
IA~MEC+A
0
5
10
a ~ If,W~
20
Cross section for d(e,e') integrated over is-peak (Enp=O~MeV). The curves have been c a l c u l a t e d by L e i d e m a n n and Arenh~vel.
Electron Scattering
431
D e u t e r o n e l e c t r o d i s i n t e g r a t i o n is p a r t i c u l a r l y i n t e r e s t i n g in the reg i o n O~2 M e V above b r e a k u p threshold. A t these energies, the n+p system e x h i b i t s a resonance, a n e a r l y - b o u n d s i n g l e t - S state. The transition from the initial d e u t e r o n S- and D - s t a t e o c c u r s via two amplitudes 3S-IS, 3D-IS that interfere. At large q, q~3.5 fm -I, the nucleon-only (Impulse-Approximation, IA) a m p l i t u d e s i n t e r f e r e d e s t r u c t i vely and the IA cross s e c t i o n b e c o m e s v e r y small. At the same time the M E C - p r o c e s s e s at large q favour S-D transitions. N e a r q ~ 3 . 5 f m -I, the ratio of MEC to IA cross s e c t i o n s b e c o m e s i0, and r e p r e s e n t s a clean signal, a fact first p o i n t e d out by H o c k e r t (1973). In o r d e r to p r o v i d e data in this r e g i o n of m o m e n t u m t r a n s f e r we performed a few years ago a d(e,e') e x p e r i m e n t at S a c l a y (Bernheim, 1 9 8 1 ) E l e c t r o n s of e n e r g i e s 280+535 M e V and i n t e n s i t y up to 25~A w e r e scattered from a liquid d e u t e r i u m target. At the s c a t t e r i n g angle of 155 ° chosen, this cross s e c t i o n is d o m i n a t e d e n t i r e l y by the m a g n e t i c int e r a c t i o n i n v o v l i n g a s p i n - f l i p of one of the nucleons. An e x a m p l e for the e n e r g y s p e c t r u m of the s c a t t e r e d e l e c t r o n s for an i n c i d e n t e n e r g y of 410 M e V is shown in fiq. 15. One r e c o ~ n i s e s the e l a s t i c
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D o u b l e d i f f e r e n t i a l cross s e c t i o n d(e,e') for E n p = l . 5 M e V c a l c u l a t e d for d i f f e r e n t s t r o n g - v e r t e x form f a c t o r s (short-dashed, solid and l o n g - d a s h e d for a n u c l e o n radius of 0, 0.48 fm, 0.7 fm, respectively).
432
Ingo Sick
peak (Ee'-290 MeV), the IS r e s o n a n c e (Ee'~288 MeV) and the i n e l a s t i c continuum. W i t h o u t the p r e s e n c e of ~.~C, the IS peak w o u l d be a factor of i0 lower, i.e. invisible. The cross s e c t i o n i n t e g r a t e d o v e r this l s - p e a k is shown in fig. 16, tog e t h e r w i t h p r e v i o u s r e s u l t s at low q. It is c o m p a r e d to a t h e o r e t i cal p r e d i c t i o n (Arenh~vel, 1983) that i n c l u d e s the IA p r o c e s s only; the strong d i s a g r e e m e n t in the r e g i o n w h e r e the S-D i n t e r f e r e n c e p r o duces a m i n i m u m in the cross s e c t i o n is obvious. W h e n i n c l u d i n g the s t a n d a r d MEC c u r r e n t s (fig. 17a,b), the ones due to p i o n e x c h a n g e and pair creation, the a g r e e m e n t w i t h the d a t a is v a s t l y improved. The a d d i t i o n a l c o n t r i b u t i o n of the A r e s o n a n c e t e r m (fig. 17c) has little effect. The a g r e e m e n t b e t w e e n p r e d i c t i o n and e x p e r i m e n t is q u i t e r e a s o n a b l e , and shows that the m a i n p a r t of the M E C is u n d e r s t o o d . In detail, a n u m b e r of p o i n t s are still open, as s h o w n by the m o s t recent c a l c u l a tions of M a t h i o t (1983). In p a r t i c u l a r , at large t r a n s f e r the form f a c t o r s at the h a d r o n i c v e r t i c e s (fig. 17) play an i m p o r t a n t role. T h e s e form factors are not v e r y well known, and t h e i r i n f l u e n c e is p a r t l y c o m p e n s a t e d by i n c l u s i o n of the e x c h a n g e of the Q-meson. T h e i r e f f e c t is shown in fig. 18, w h e r e the c h a n g e from no s t r o n g - v e r t e x form factors (short dashes) to a v e r t e x form factor c o r r e s p o n d i n g to a n u c l e o n - r a d i u s of 0.7 fm (long dashes) is shown. F i g u r e 18 shows that the s h o r t - r a n g e ~ C are still q u i t e m o d e l d e p e n d e n t . Pushing e x p e r i m e n t to h i g h e r q s h o u l d a l l o w to d i s t i n g u i s h b e t w e e n the various i n g r e d i e n t s ; the s t r u c t u r e in the form f a c t o r b e t w e e n 20 and 30 fm -2 c o u l d be q u i t e revealing. We s h o u l d also p o i n t out that there is still an u n c e r t a i n t y r e l a t e d to the c h o i c e of the e l e c t r o m a g n e t i c form factor to be u s e d in the c a l c u l a t i o n of MEC diagram. (G e v e r s u s FI, see d i s c u s s i o n for 3He below). The n e x t e x a m p l e I w a n t to d i s c u s s c o n c e r n s the d e u t e r o n e l a s t i c m a g netic form factor B(q). H i s t o r i c a l l y , b o t h the c h a r g e and m a g n e t i c form f a c t o r have b e e n t a k e n as test cases, and have b e e n a c o n t i n u i n g i m p e t u s for the study of MEC. A l t h o u g h the ~ C e f f e c t s t u r n e d out to be small at low q (the d e u t e r o n b e i n g an i s o s c a l a r object), the d e u t e r o n form f a c t o r s still are v e r y i n t e r e s t i n g . The s i m p l i c i t y of the b o u n d 2 - n u c l e o n s y s t e m a l l o w s a l t e r n a t i v e t h e o r e t i c a l a p p r o a c h e s that can be u s e d to c h e c k the s t a n d a r d MEC c a l c u l a t i o n s . The m a g n e t i c f o r m f a c t o r B(q) has b e e n m e a s u r e d by m a n y d i f f e r e n t exp e r i m e n t s o v e r the p a s t 25 years. T h e m o s t r e c e n t e x p e r i m e n t that p r o d u c e d the h i g h e r - q d a t a is the one d i s c u s s e d a b o v e in c o n n e c t i o n w i t h d i s i n t e g r a t i o n at threshold. The m a g n e t i c f o r m factors are comp i l e d in fig. 19. The n o n r e l a t i v i s t i c d e u t e r o n w a v e f u n c t i o n can be c a l c u l a t e d e x a c t l y by s o l v i n g the S c h r 6 d i n g e r e q u a t i o n for a g i v e n N N - p o t e n t i a l . The m a g n e t i c form factor is o b t a i n e d by f o l d i n g the p o i n t n u c l e o n form f a c t o r s w i t h the n u c l e o n m a g n e t i c form factors. S i n c e the i s o s c a l a r m a g n e t i c n u c l e o n f o r m f a c t o r is not all that a c c u r a t e l y known, this i n t r o d u c e s an u n c e r t a i n t y (~20% at 4 fm -I) in the c a l c u l a t e d B(q).
Electron Scattering
433
The impulse-approximation result, obtained by Gari (1976) using the RSC nucleon-nucleon-interaction, is shown in fig. 19 as a dashed line. The contributions of non-nucleonic degrees of freedom are mainly connected to the pair diagram, for q<5 fm -~. Additional processes, the ~py-diagram in particular, dominate at larger transfer. Adding to the IA contribution the effects of MEC yield the dash-dot curve, which agrees reasonably well with experiment. The contribution of A-components, not yet included in the dash-dot curve, will give a further small modification of B(q). For the deuteron a different theoretical approach, the relativistic IA, is possible• Solving the Bethe-Salpeter equation yields a relativiStically correct wave function. The corresponding form factor includes both the terms equivalent to nonrelativistic IA and the pair diagram (i.e. antinucleons). It has been shown by Gross (1978)that, to lowest order in q2/M2, the relativistic IA and the nonrelativistic IA plus the pair diagram are formally equivalent.
B(q)
10 .3 .
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•
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434
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R e l a t i v i s t i c c a l c u l a t i o n s for the d e u t e r o n h a v e b e e n p e r f o r m e d by A r n o l d (1980) and Z u i l h o f (1980). T h e s e a u t h o r s o b t a i n s i m i l a r results; one of their p r e d i c t i o n s is shown in fig. 19 by the d o t t e d curve. This r e s u l t d i f f e r s s i g n i f i c a n t l y from the s t a n d a r d I A + ~ C c a l c u l a t i o n s (dash-dot), and p r o d u c e s a form f a c t o r s m a l l e r t h a n the n o n r e l a t i v i s t i c IA result. W h e r e does this d i s a g r e e m e n t b e t w e e n two (supposedly) e q u i v a l e n t appr o a c h e s come from ? The n o n r e l a t i v i s t i c c a l c u l a t i o n s c o r r e s p o n d to an e x p a n s i o n in first o r d e r q2/M2, w h i l e the r e l a t i v i s t i c c a l c u l a t i o n s m a k e no such expansion. At q - v a l u e s of the o r d e r of one G e V / c this can m a k e an a p p r e c i a b l e d i f f e r e n c e . In addition, Z u i l h o f and T j o n (1980) p o i n t out that, for p s e u d o - s c a l a r ~N coupling, a n o n - n e g l i g i b l e c o n t r i b u t i o n of 2 z - e x c h a n g e gives a c o n t r i b u t i o n that tends to c a n c e l the p a i r term. This c o m p a r i s o n shows that the c a l c u l a t i o n s of MEC m a y not yet be as w e l l u n d e r c o n t r o l as one m i g h t have c o n c l u ded from fig. 16. H i g h e r o r d e r d i a g r a m s , and the r e l a t i v i s t i c e f f e c t s o f t e n n e g l e c t e d , are to be s t u d i e d in o r d e r to r e s o l v e the d i s c r e p a n cy p o i n t e d out above, and d e t e r m i n e w h e t h e r the r e ~ o n s c i t e d are the real cause. For the d e u t e r o n m a g n e t i c form factor, a s e c o n d q u e s t i o n a c t u a l l y is w i t h o u t a final answer, the one c o n c e r n i n g the e f f e c t of A A - c o m p o n e n t s in the g r o u n d state w a v e function. A c c o r d i n g to Gari (1976), the effect of A A - c o m p o n e n t s is q u i t e small, w h i l e F a b i a n (1974) finds a c h a n g e of a factor of 2 of B(q) n e a r q~4 fm -I. P a r t l y this d i f f e r e n ce is r e l a t e d to the u n c e r t a i n t y in the v e r t e x form factors. It is n o t clear, however, that there is not a m o r e b a s i c d i s c r e p a n c y , linked e.g. to a s s u m p t i o n s on the A m a g n e t i c moment. The t h i r d o b s e r v a b l e that gives d i r e c t access to the role of m e s o n exc h a n g e c u r r e n t s in n u c l e i is the 3 H e m a g n e t i c form factor. It was alr e a d y r e c o g n i z e d by V i l l a r s (1947) that the A=3 m a g n e t i c m o m e n t s rec e i v e an a p p r e c i a b l e c o n t r i b u t i o n of ~0.3 m a g n e t o n s . The i s o v e c t o r m a g n e t i c m o m e n t in p a r t i c u l a r is s e n s i t i v e to ~ C . The c a l c u l a t i o n of H a r p e r (1972) s h o w e d that the T - e x c h a n g e , p a i r and A - d i a g r a m s q u a n t i t a t i v e l y e x p l a i n the d i f f e r e n c e b e t w e e n the m a g n e t i c m o m e n t s o b t a i n e d f r o m e x p e r i m e n t and F a d d e e v c a l c u l a t i o n s . A t m o m e n t u m t r a n s f e r s q>O, the 3He m a g n e t i c form f a c t o r gets e s p e c i a l ly i n t e r e s t i n g . As p o i n t e d o u t by B r a n d e n b u r g (1974) F M c o n t a i n s an S- to D - s t a t e a m p l i t u d e that i n t e r f e r e s w i t h the d o m i n a t i n g S2-term. T h i s i n t e r f e r e n c e leads to a p r o n o u n c e d d i f f r a c t i o n m i n i m u m at q2 = 8 fm-2; in a n a l o g y w i t h d e u t e r o n e l e c t r o d i s i n t e g r a t i o n , w e can e x p e c t to find large e f f e c t s of e x c h a n g e c u r r e n t s in t h a t region. T h e i n i t i a l m a g n e t i c s c a t t e r i n g e x p e r i m e n t s on 3He, w h i c h were d o n e at S t a n f o r d (Collard,1965, M c C a r t h y , 1970), c l e a r l y s h o w e d that the d i f f r a c t i o n m i n i m u m p r e d i c t e d near 8 fm -2 w a s absent. F i g u r e 20 shows the m o r e a c c u r a t e d a t a a v a i l a b l e today, the ones r e s u l t i n g f r o m the e x p e r i m e n t s p e r f o r m e d at B a t e s (Dunn, 1983) and S a c l a y (Cavedon, 1982). A d i f f r a c t i o n m i n i m u m is o b s e r v e d , but not at 8 fm -2, but at q2 ~ 17 fm -2 ! The dashed
curve
in f i g u r e
20 g i v e s
the i m p u l s e
approximation
form
Electron Scattering
435
factor c a l c u l a t e d u s i n g the w a v e f u n c t i o n of Torre (1981). This w a v e f u n c t i o n is c a l c u l a t e d by s o l v i n g the F a d d e e v e q u a t i o n in c o n f i g u r a tion space for the R e i d soft core NN interaction. This prediction s t r o n g l y d i f f e r s from the data, as do o t h e r s o b t a i n e d u s i n g d i f f e r e n t NN i n t e r a c t i o n s and d i f f e r e n t t e c h n i q u e s to c a l c u l a t e the 3-body w a v e function. W h i l e the c h a r g e form factor is s e n s i t i v e to short range p r o p e r t i e s of the w a v e f u n c t i o n that are p o o r l y known, the m a g netic form factor d e p e n d s on (better known) l o n g - r a n g e p r o p e r t i e s only; the short range features, o b s e r v a b l e in p r i n c i p l e at large q, are c o v e r e d up by the S-D t r a n s i t i o n term. G i v e n this situation, we m a y trust the IA p r e d i c t i o n to a c o n s i d e r a b l e extent. Deviations must come from the n o n - n u c l e o n i c d e g r e e s of freedom. In fig. 20 we show a r e c e n t c a l c u l a t i o n of S t r u e v e (1983) w h o solves the F a d d e e v e q u a t i o n in m o m e n t u m space u s i n g the P a r i s n u c l e o n - n u c l e o n potential. To the o n e - b o d y form factor have b e e n a d d e d the p a i r - and m e s o n c u r r e n t s a r i s i n g from ~ and p exchange, and the e f f e c t of strongv e r t e x form factors is included. In this c a l c u l a t i o n the A - c o n t r i b u tion is not yet taken into account. To be c o n s i s t e n t , the A has to be a l l o w e d for b o t h in the MEC p r o c e s s (fig. 18c) and in the 3-body force w h i c h p o l a r i s e s and r e n o r m a l i s e s the p u r e l y n u c l e o n i c c o n f i g u r a tion. A c c o r d i n g to a c o u p l e d c h a n n e l c a l c u l a t i o n (Strueve, 1983), t h e o v e r a l l e f f e c t of F M is small for q<5 fm -I. The curve a) in fig. 20 shows that m e s o n i c e f f e c t s (though c a l c u l a t e d i n c o r r e c t l y using FI, see below) do a c c o u n t for the l a r g e s t p a r t of the d i s c r e p a n c y b e t w e e n IA p r e d i c t i o n and experiment. This c l e a r l y r e p r e s e n t s a success of the ~ C picture, and gives us c o n f i d e n c e that these c o n s t i t u e n t s are u n d e r s t o o d to a r e a s o n a b l e d e g r e e of accuracy. This is e m p h a s i z e d by curve b), w h i c h is o b t a i n e d by Riska (1980) using a very simple m i n d e d n u c l e o n i c w a v e function. This curve d e m o n strates that the 3He m a g n e t i c form factor is q u i t e i n s e n s i t i v e to the n u c l e o n i c w a v e function; q u a l i t a t i v e a g r e e m e n t w i t h the d a t a at q2>5 fm -2 is e s s e n t i a l l y a test of our u n d e r s t a n d i n g of MEC. This success of the m e s o n - e x c h a n g e p i c t u r e s h o u l d not m a k e us b e l i e v e that e v e r y t h i n g is under control. One p r o b l e m was a l r e a d y i n d i c a t e d in c o n n e c t i o n w i t h the d e u t e r o n form factor~ here we e m p h a s i z e a related p o i n t that is v e r y i m p o r t a n t for a q u a n t i t a t i v e d e s c r i p t i o n . W h e n c a l c u l a t i n g the m e s o n e x c h a n g e c o n t r i b u t i o n s , one s h o u l d r e s p e c t c u r r e n t c o n s e r v a t i o n ; this c o n s i d e r a t i o n , after all, was a m a j o r one w h e n i n t r o d u c i n g MEC in the first place. It has ben e m p h a s i z e d by A r e n h 6 v e l and others, that this r e q u i r e s the use of the n u c l e o n elect r o m a g n e t i c form factor G e r a t h e r than F 1 in the c a l c u l a t i o n of the MEC c o n t r i b u t i o n . Not r e s p e c t i n g this rule allows to get b e t t e r agreem e n t w i t h the e x p e r i m e n t a l data (Bornais, 1981). F i g u r e 21 t a k e n from S t r u e v e (1983) d e m o n s t r a t e s the i m p o r t a n c e of using G e. The d i f f e r e n c e b e t w e e n G e and F 1 is of o r d e r q2/M2, i.e. of r e l a t i v i stic order. This e m p h a s i z e s again that in the n e x t step ~ C n e e d to be s t u d i e d in a r e l a t i v i s t i c a l l y m o r e c o r r e c t way. As p o i n t e d out at this school by Riska, one of the r e l a t i v i s t i c terms of the p a i r d i a g r a m for S-D t r a n s i t i o n s gives an e f f e c t of the same size as the d i f ference Ge-F I. For this p a r t i c u l a r term, the n o n r e l a t i v i s t i c e x p r e s -
436
Ingo Sick
10 o 3He (e,e) o
BATES
• SACl.~
10 -Z
....
NUCLEON ONLY NUCLEON+MESON EXCHANGE
x
\
e.
10 -¢ bL.
1 | I,
Ii II II II II
(a) (b)
10 - 8
0
10
20
30
(fro-2) Fig.20
3He elastic form factor calculated by Torre (IA, dashed), Strueve (IA+MEC, solid, a) and Riska (IA+MEC, solid, b).
10 0
FORMFACTOR 'De e
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i
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,
.
.
i
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.
.
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.
i
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.
.
.
.
40
3He magnetic form factor calculated using for ~ C as electromagnetic form factor F 1 (dashed) and Ge(solid).
Electron Scatterin~
437
sion for the pair-current calculated with F 1 would be equivalent to the relativistic one calculated with G e. Clearly, effects of this order coming from other amplitudes and for the other exchange diagram, need to be investigated before the MEC contribution quantitatively can be compared to experiment. From the preceding discussion on the A ~ 3 nuclei it is clear that the role of meson exchange currents is reasonably understood. In those cases where the MEC are large, the pair and meson current diagrams (involving both z and 0) account for the larger part of the difference between experiment and nucleon-only prediction. The A-diagram, on the other hand, is still poorly tested, since we have not yet found an observable dominated by the A-term. (For heavier nuclei, where nuclear structure uncertainties introduce large additional uncertainties, the A-contribution is in no better shape). A next step in the understanding of non-nucleonic degrees of freedom obviously will be the question on the role of quarks in nuclei. This question enters already when discussing the s h o r t e r - range MEC; it becomes the dominating issue once we attempt to understand processes at even higher m o m e n t u m transfer. Some of the recent developments are described in other sections of this volume. A systemtic discussion of the role of quarks in electron-nucleus scattering will be the theme of the '84 Erice School.
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