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Charge distributions of nuclei of the 1p shell
ANNALS
OF
PHYSICS:
Charge
8, 119-171 (1959)
Distributions
of Nuclei
of the lp
Shell*?
ULRICH MEYER-BERKHOUT~ High-Energy
Physics
Laboratory,
Stanford
University,
Stanford,
California,
AND KENNETH W. FORD§ AND ALEX E. S. GREEN// University
of California,
Los
Alamos
Scienfi$c
Laboratory,
Los
Alamos, New Mexico
New data have been obt,ained at energies of 160 to 420 Mev for the scattering of electrons by Beg, Bra, B”, N14, and O16. A detailed analysis of these elastic scattering data and of other available data for p-shell nuclei is carried through, using various methods and a wide variety of assumed functional forms of the nuclear charge distribution. It is possible to fix rather accurately the spherically symmetric part of the charge distributions of Li6, C’s, Nl4, and 016, and wit.h less accuracy, those of Be9 and B”. Cent,ral densities are uncertain in all cases. Quadrupole scattering appears to be important in N14, in Blr, and possibly in Beg, and a crude estimate of t,he quedrupole moment of N’4 is inferred from the data. The quadrupole scattering depends sensitively on the source of the quadrupole moment, as well as on its value. Regularities within the /p-shell are discussed, and connection made with the properties of heavier nuclei. Be9 and all heavier nuclei have a common value of peak particle density corresponding to a mean particle spacing of (1.13 f 0.01) X lo-r3 cm. Only Li6 fails to reach this density of saturated nuclear matter. * The part of this work carried out at Stanford was supported by the joint program of the Office of Naval Research and the U. S. Atomic Energy Commission, and by the U. S. Air Force, through the Office of Scientific Research of the Air Research and Development Command. t The part of t.his work carried out at Los Afamos was supported by the U. S. Atomic Energy Commission. $ Present address: Erstes Physikalisches Institut der Universitat, Heidelberg, Germany. $ Present address: Physics Department, Brandeis University, Waltham 54, Massachusetts, A part of t.his work was carried out at Brandeis with the support of the National Science Foundation. 11Present address: Physics Section, Convair, San Diego, California. 119
120
MEYER-BERKHOUT
TABLE NEW
EXPERIMENTS
Energy M-1
Element
I
ON NUCLEI
OF THE lp
SHELL Angles (degrees)
nBeg
160 300
33-100 33-74
8’0
160
34-130
gBll
160 300 420
33-130 34-74 34-70
TN’4
160 420
33-100 33-73
,016
160
33-100
I. UNITS
All quantities in this paper with the dimensions of length (e.g., T) will be written in units of 10P3 cm. All quantities of momentum transfer or inverse length (Q) will be written in units of 1013cm-‘. The nuclear charge density (p) will have units of (number of protons)/( lo-l3 cm)3, and the nuclear mass density units of (number of nucleons)/( lo-l3 cm)“. Quadrupole moments will have units of 1O-26cm2, but following previous practice, cross sections will be expressed in units of cm’/steradian. II.
EXPERIMENTS
The experiments whose results are discussed in this paper’ have been performed with the Stanford high-energy linear accelerator. The elements, energies, and angles for which new data have been obtained are summarized in Table I. A. APPARATUS The scattering apparatus and the method of taking data were similar to those reported in earlier papers (1, 9). Somerecent improvements, however, have not been mentioned before.’ The magnetic field of the energy deflecting magnet in the bunker was measured by a proton resonance signal. This made it possibleto reduce energy fluctuations of the incident electron beam during one run to a 1 The experiments reported here have been carried out by the first-named author, M.-B. The analysis of the data has been carried out jointly by all three authors. 2 The authors wish to thank Dr. F. Bumiller, Dr. J. Oeser, and Mr. S. E. Sobottka, were responsible for these improvements of the apparatus.
U. who
NUCLEAR
CHARGE
DISTRIBUTIONS
121
negligible amount. In addition the energy of the electrons could be reproduced accurately from run to run. Two ionization chambers, each of which was split into two subsections and positioned behind the target just in front of the Faraday cup made it possible to stabilize the horizontal and vertical beam spot position on the target. This was accomplished by feeding error signals taken from the two sections of each ionization chamber to Helmholtz deflection coils located approximately 25 feet in front of the scattering chamber along the beam pipe. A rotating coil was used to check the magnetic field in the 36-in. double focusing spectrometer at one particular point. The electron current was measured as usual by a Faraday cup and integrated by a vibrating reed electrometer. B.
RESOLUTION
The beam was analyzed magnetically so that the energy band was approximately 0.2 % wide for the new measurements on Beg, Bn, and N14 at 300 and 420 Mev. The energy slit of the 36-in. magnetic spectrometer was set to 0.2 %. This resulted in an overall resolution of about 0.35 %, depending slightly on the size and shape of the beam spot on the target. This was sufficient to resolve clearly elastic scattering from inelastic scattering events in which the lowest levels in Be’, B”, and h-l4 at 2.43, 2.14, and 2.31 Mev, respectively, were excited. No Beg measurements were performed at 420 Mev since the E2 transition from the ground state to the first excited state at 2.43 Mev turned out to be very strong at the larger scattering angles and therefore was not well resolved from groundstate scattering at this energy. No scattering from a level at about 1.8 Mev (3) could be observed in Be’. This, however, is not surprising since the 1.8-Mev level is known to be very broad. The lowest levels in B” and N14 at 2.14 and 2.31 Mev, probably reached by Ml transitions from the ground state, were barely excited in these experiments so that it was hard to detect them at all at most angles. Accordingly there were no serious resolution problems for B” and N14. The 160-Mev data on Be’, N14, and 016 were taken with an overall resolution of roughly 0.75 %. The overall angular resolution in these experiments was estimated to be about f1.2’, as is described in Ref. L. This should be sufficiently low not to smear out possible diffraction structure in the angular distribution curve of elastically scattered electrons. The electrons were detected and counted in the usual manner with a liquid fluorocarbon cerenkov counter (n = 1.276). C.
INELASTIC
SCATTERING
Estima.ted ratios of cross sections for elastic scattering to inelastic scattering from the first excited levels are given in Table II for some of the p-shell nuclei at some typical angles and energies. If no errors are given, the numbers should be considered as crude estimates only. The first excited level in B” at 0.72 Mev would of course not have been resolved at 420 Mev from ground-state scattering
TABLE
II
INELASTIC CROSS SECTION MEASUREMENTS Level (Mev)
Angle (Degrees)
(r (inelast.)
Lia a
426
2.19
40
0.6
f
25%
Li7 B
426
4.61
40
0.6
f
25%
Be9
160 300
2.43”, 2.43d
B”
160 300
420
CP
187 420
N’4
160 420
d
2.14 2.14
2.14
60 39 42 45 48 51 54 58 62 66 70
3609
b
-O.lP 0.3 0.4
f f -0.5 0.63 f 0.7 f 1.05 f 1.5 f 2.0 f 2.0 f 1.8 f
33-100 34-45 50-54 58 70 74
SO.1’ SO.06 so.1 so.13 $0.25 516
50
NO.25
58
-0.5
37 43 46 58
so.15 10.25 50.1, 50.2,
46
-1.1
0 257, 2570 20% 20% 107, 20%
207, soy, 20% probably
Ml
excit
4.46 Mev and 5.03 Mev levels unresolved 4.46 Mev and 5.03 Mev levels unresolved
4.46 Mev and 5.03 Mev levels unresolved
see Refs. 6 and 6 see Ref. 2 2.31 2.31
33-100 34-61
40.15 $0.2 -
0”
Remarks
c (elast.)
6.06) 6.14 6.91 i 7.12)
35 45 55 70 122
Ml excit. Ml excit. unresolved, excited
unresolved
f 15$?& unresolved -25 unresolved 2 & 25% unresolved
for
but strongly 0 2 40”
NUCLEAR
CHARGE
DISTRIBUTIONS
123
with our present equipment. Therefore B” measurements were restricted to 160 Mev. Comparison measurements between B” and Bn performed at 160 Mev between 33” and 130” with 0.5 % resolution revealed essentially no differences between the angular distribution curves for these two nuclei. There was some weak indication of a slightly steeper drop of the cross section as a function of angle for Bn, but this was definitely an effect not outside statistics. To the extent that such a conclusion can be drawn from measurements at rather low momentum transfers it must therefore be assumed that the radial distribution of charge is rather similar for these two isotopes. Modifications in the angular distribmion curve due to the rather different quadrupole deformations of these two stable boron isotopes are too small to be measurable at this energy except near 130” for the strongly deformed B” nucleus. D.
TARGETS
Be9 targets, known to be of high purity, 0.150-inch and O.lOO-inch thick were used. In case of the boron measurements metallic boron powder enriched to 99 % (96 %) with B” ( B1’) served as target material. The targets were diskshaped, about 2 inches in diameter and 0.75 g/cm2 thick. Two 0.0005-inch thick aluminum foil end windows, one on each side, sealed with O-rings membrane-like onto the aluminum target frame had enough tension and strength to hold the boron powder firmly in the target. Only a slight curvature of the end windows resulted but this was irrelevant owing to the fact that the beam spot was automatically kept in a fixed position on the target during each run.3 From the target dimensions one computes an Al*‘:B atomic ratio of about 1: 270, which is too 3 Absolutely flat boron targets boron powder mixed with a small very high pressure. Heating the led to quite stable and flat boron contamination and therefore was
of sufficient strength amount of hydrocarbon boron cake afterwards targets. The method finally abandoned.
were made by compressing metallic binding agent in a steel box under for several hours to about 150°C however introduced a slight carbon
Footnotes to Table II 8 G. Bltrleson, private communication. s The 0.477.Mev level in Lir was not resolved from ground state scattering. 0 At 160 Mev the cross section for inelastic scattering from the 2.43.Mev level in Be9 strongly increases with increasing angle. No scattering could be detected from a broad level at 1.8 Mev. d The broad level at 1.8 Mev cannot be resolved from the 2.43.Mev level but the energy of the inelastic peak clearly favors scattering from the 2.43.Mev level. e Whenever a - sign appears in front of the cross section ratio the value represents a crude estimate only. f Whenever the 5 sign appears the actual cross section ratio may be much smaller than the upper limit quoted in the table. g At 420 Mev and at 480 Mev, scattering from the first four excited levels between 6.06 and 7.12 Mev is considerably stronger than elastic scattering for 19 > 35”.
124
MEYER-BERKHOUT
small to give rise to any appreciable background. This was proved also by taking data with an empty but otherwise identical dummy target. Chemical contaminations of the target material can falsify the angular distribution curve of elastically scattered electrons. In particular the possibility of filling in a diffraction minimum by scattering events from contaminating elements has to be considered provided such events are unresolvable from the scattering process under study. According to the manufacturers specifications, the Bn (B”) powder used as target material was chemically at least 97 % (98 %) pure. Spectroscopic analysis confirmed that metallic impurities amounted to less than 1%. This is hardly enough to affect the angular distribution curve appreciably, even in the region where according to some nuclear models a diffraction dip should occur. Carbon, nitrogen, and oxygen which often happen to be contaminating elements in metallic boron of course escape detection by spectroscopic analysis. At the momentum resolutions employed in these experiments, nitrogen- and oxygen-scattering would have been resolved at most energies and scattering angles from boron elastic scattering and therefore could not distort the boron elastic scattering measurements. Simple kinematic considerations show for example that an electron scattered elastically at a typical angle and energy of our experiment has a momentum approximately half way between the momentum of an electron scattered elastically from 0’” and one which has excited one of the four first levels in 016 between 6.06 and 7.12 Mev. The N14 elastic scattering peak should also, if of sufficient intensity, have been resolved, at least partially, from the B” elastic scattering peak, but was not observed, whereas the cross section for scattering of the first excited level at 2.31 Mev in Xl4 is too small to be of any importance at all scattering angles and energies investigated so far. Therefore, no attempt was made to determine the oxygen and nitrogen content of the boron target material quantitatively. Carbon impurities, however, could be much more serious. Chemical analysis4 led to the result that the carbon content amounted to about 1.1% in the B” 4 As a check, the Cl* content was also determined by bombarding simultaneously a (CH2), target of known thickness and the boron target with 24-Mev electrons from the Stanford Mark II linear accelerator. After bombardment the resulting Cl’ 20.3.min b+ activity was measured by counting the 0.511 Mev annihilation radiation from both targets with two NaI scintillation spectrometers. Care was taken that target geometry, window width, etc. were approximately the same. The number of annihilation quanta counted in a fixed time interval then immediately gave the ratio of 0 nuclei and therefore Cl* nuclei per square centimeter in both targets provided geometry, absorption, efficiency, etc. were the same. Assuming this to be the case, the Cl2 content was found to be in fair agreement with the result from chemical analysis. An annihilation radiation with approximately two minutes half-life originating from the boron targets can be assigned to 0’6 produced by the 016 (7, n) 016 reaction. For reasons discussed above, however, no attempt was made to determine the 016 content by a similar method. No 10 min activity from N’3 was detectable.
NUCLEAR
CHARGE DISTRIBUTIONS
125
sample and to about 0.63 % in the B” sample. Comparison of cross sections for C” with those to be expected for boron showsthat this is too small a contamination to falsify the boron angular distribution curves of elastically scattered electrons. All measurements on K14 were made with a liquid nitrogen target of 99.9 % purity. To avoid oxygen condensation and exchange the target reservoir was filled in a nitrogen atmosphere. After filling, target and reservoir were sealed off with an toil valve. The N14target itself was about 2.5 inches in diameter and 1 g/cm” thick. The end-windows were made of O.OOl-inch nickel-plated stainless steel (leading to an Fe: K atomic ratio of about 1: 100). A slight curvature of the target end-windows was irrelevant for reasons already discussed above. The stainless steel windows introduced a small background due to electrons scattered from iron. nuclei. Therefore in the beginning of each run a few points were taken with an empty target in order to allow background subtraction. At most angles and energies the observed scattering from the unfilled target was completely negligible. In particular it turned out to be impossible to ascribe the “missing” dip in the N14420-Mev angular distribution curve to iron scattering. For CL6the same water target was used as the one described in Ref. 2. E. CROSS-SECTION
MEASUREMEXTS
Absolute cross sections were determined at one angle at each energy by comparison with free proton scattering, as is described in Ref. 2. The results of these absolute cross-sectionmeasurements are listed in Table III. It seemslikely, however, on the basis of peak height comparison between the results of different runs, that the cross sections at 160 Mev and 80” for B” and Xl4 listed in Table III are too small by a factor of two because of an erroneous reading of the current integrator scaleduring that particular run. Unfortunately, the measurement has not yet been repeated. The procedure of taking data is described in Ref. 2. Most of the angular distribution curves presented in this paper are based on measured areas under the elastic scattering peak. Whenever of sufficient importance the bremsstrahlungstraggling and the Schwinger-Suura correction were applied to the data. Usually these angular distribution curves agreed very well with those derived from peak heights. Whenever discrepancies occurred a weighted average was taken. A few points, especially in the region of large momentum transfers are based solely on peak heights. The overall accuracy of the relative cross sections is believed to be about lo-15 % for most points except those at the very largest angles. Estimated errors of the absolute cross sections are given in Table III. Even if the statistical error was smaller than 10% an error of &lo% is assigned to the relative cross section since reproducibility usually was limited to about that order of magnitude, This probably is due to drifts in various parts of the ex-
126
MEYER-BERKHOUT
TABLE SUMMARY Element
3Li6
Energy WV)
Lab. angle (degrees)
OF ABSOLUTE Absolute cross section (10-80 cm2jsterad)
III CROSS Smoothing factor%‘/fl
SECTIONS Overall
renormalizationb “THIJ
F
H
A
187 426
40
0.24
4Be9
160 300
80 54
1.04 0.34
f f
3070 25%
1.09 1.00
1.25
-1.4
1.1oe
gBll
160 300 420
80 54 43
1.36 0.65 0.26
f f f
40%‘0 30% 30%
1.82’ 1.00 1.18
0.87
0.81g
,C’2
187 420
70 40
0.0028h 0.41 f
1.00 1.16
1.13
1.11
4oy,j
,N’4
160 420
80 40
1.50 0.45
f f
30%’ 2570
1.99’ 1.00
1.21
-
80’6
160 240 360 420
80 60 55 40
2.4 1.1 0.009 0.105
f f f f
3Ooj, 5070; 40%; 20%;
1.12 1.08
1.04
1.20
none f 25%”
1.08
1.10
0.95d
1.00
1.08’
-1.2k
1.08’
8 The smoothing factor is the number by which the cross section at each energy is multiplied to produce a smooth q-F(q) versus q curve. It is arbitrarily taken as 1 for some energy. ,J means the experimental cross section; U’ is the smoothed cross section. b The overall renormalization is the factor by which the smoothed cross section is multiplied to give the theoretical cross section. F means Fourier method, H means Hermite method, and A means a typical one of the simple analytic charge density functions. Footnotes give the charge distributions used. 0 Burleson and Hofstadter, Ref. 7. d Family II, r1 = 1.75, n = 2.25. e Hollow exponential (Z e-x), (+)I’* = 2.75. f The absolute measurements for Bu and N14 at 160 Mev appear to be too small by about a factor of two, as discussed in Section II. g Family II, r1 = 1.95, n = 2.75. h Fregeau and Hofstadter, Ref. 5; Fregeau, Ref. 6. i Oscillator, (Y = $-$, rP = rs = 1.637. j Ehrenberg et al., Ref. R. k Oscillator, 01 = s, rP = rs = 1.667. 1 Oscillator, 01 = 2, rp = ra = 1.764.
NUCLEAR
CHARGE
127
DISTRIBUTIONS
perimental equipment. A tremendous improvement of the data could probably be achieved by repeating these measurements with the new multi-channel electron detection device recently put into operation by H. W. Kendall. Multichannel detection of the electrons in the image plane of the 36-in. magnetic spectrometer not only yields more precise data but in addition offers the possibility OF taking dat,a much faster and therefore of extending the measurements into the very interesting region of still larger momentum transfers. ICI.
ANALYSIS
OF DATA SYMMETRIC
IN
TERMS CHARGE
OF STATIC nENSITIES
SPHERICALLY
Rather complete elastic scattering data for electrons of 160 Mev to 120 Mev are now available for six p-shell nuclei: 3Li6, 4Beg, gB1l, ,@, 7N14, and gO16. (In addition, B1’ was studied at 160 Mev.) For the present analysis we made use of, in addition to the new data reported above, some data for Li6, Cl’, and 016 which have been previously reported and analyzed (2, 4-8). In this section the data are analyzed in terms of a static spherically symmetric charge distribution. The kinematic effects of nuclear recoil are included, but corrections due to quadrupole moment, magnetic moment, and dispersion effect are ignored. These corrections are discussed separately in Section IV, but only the quadrupole effect is examined in any detail. The approach of this section is probably valid for C” and 016 (for which quadrupole moment and magnetic moment effects vanish exactly’), and is probably also valid for Li6, for which the scattering data do not extend as far as the first diffraction minimum. On the other hand, this approach is almost certainly not valid for B” and N14, and probably not valid for Be’. For each of these three elements, a qualitative inspection of the data indicates the filling in of the first diffraction minimum and strongly suggests significant contributions to the scattering from sources other than a static spherically symmetric charge distribution. Nevertheless, an analysis first of the scattering from all six nuclei in terms of a static sp.herical charge seems worthwhile. One would like to know, for example, whether any reasonable spherical charge distribution can account for the observed scattering from N14, or whether the data demand the inclusion of a quadrupole effect. For this element, neither the value of the quadrupole moment nor the yuadrupole form factor is well known. The interpretation of the data will therefore ultimately rest heavily on a theory of the quadrupole effect. The situation is 5 The incoherent quadrupole effect, as defined in Section IV, vanishes for spinless nuclei, but deformation in a spinless nucleus could still make itself felt by altering the average radial distribution of charge. The scattering is still correctly analyzed in terms of a spherical charge distribution. This coherent quadrupole effect is important for isotope shifts and might be detected in electron scattering from two even-even isotopes which have significantly ditl’erent deformations (e.g., Srn’So and SmlS2).
128
MEYER-BERBHOUT
similar for Beg, while for B” only the quadrupole form factor is very uncertain. Similarly, although magnetic moments are well known, magnetic form factors are not. We therefore begin by ignoring altogether corrections due to nuclear moments and nuclear dispersion. Moreover, as will become clear in the discussion of the data below, even where quadrupole and/or magnetic effects are significant in the neighborhood of the diffraction minimum, a reasonably good picture of the spherically symmetric part of the charge distribution can still be obtained by omitting from the analysis the data at momentum transfers near the diffraction minimum. This procedure is unambiguous for N14,is lessclear cut for B’l, and is quite uncertain for Be’. I[It may be worthwhile to summarize at this point the previous experimental information on the spectroscopic quadrupole moment of these three nuclei. There is very little experimental information on the quadrupole moment of Be’. The value of 2 (see Section I) used in Section IV is based on an observation (9) of the quadrupole splitting of the magnetic resonance line in a single crystal of Be3AlzSi601s. The quadrupole moment derived from this observation must be regarded as highly uncertain. The quadrupole moment of B” is much better known. The hyperfine structure splitting of the 2p ‘P3,z state in free Bn has been measured with very high precision (10) using the atomic beam resonance method. From the measured hyperfine structure splitting and the known magnetic moment of the B” nucleus, a rather reliable value of Q(B”) = 3.55 can be derived, which value we adopt in Section IV. For B1’ one obtains Q(B1’) = 7.4. Although the N14 quadrupole coupling constant, eQ ~3~V/dz~, is known with good precision for several molecules containing N14, only a very crude value of the N14 quadrupole moment can be derived from such measurements. This is due to the well-known fact that the molecular electron wave functions are not well enough known to yield a reliable value of d2V/&z2. This fact is reflected in the values of the quadrupole moment derived by different authors from essentially the samemeasurements. Townes and Dailey (11) give Q( N14) = 2, while Bassompierre (12) gives a value of 0.7. For the calculations in Section IV we adopt the Townes-Dailey value, Q = 2, which may easily be wrong by a factor of two or even more.) A. METHODS
OF ANALYZING
1. Phase Shift
THE DATA
Analysis
Even for the light elements, a phase shift analysis is required for accurate predictions of the scattering near a diffraction minimum. The Born approximation is nevertheless very useful and of course simpler by far than the phase shift method. Most of our analysis has been based on the Born approximation, but a
NUCLEAR
CHARGE
129
DISTRIBUTIONS
number of phase shift calculations have been carried out where they seemed necessary, or desirable to check the Born approximation. The phase shift method requires a high-speed computer. We have fortunately had available two phase shift analysis programs, one for the Univac computer at Livermore’ and one for the IBM-704 computers at Los Alamos. The Livermore program was formulated by Yennie et al., and has been reported by them (14) .* Some details of the Los Alamos formulation have been included in the recent paper dealing with recoil effects (15), and results for heavier elements will be reported later (16). The Los Alamos program includes recoil effects, and a calculation of the mean square deviation of a theoretical curve from the data, which the Livermore program does not. On the other hand, the Los Alamos program is restricted to a class of charge distributions for which the potential can be expressed explicitly as a function of radius, while the Livermore program does not have this restriction. The following families of charge distributions have been studied with the phase shift analysis. We here omit normalization constants. (a) The Fermi two-parameter function: P-
(1 + exp [CT - r~>lal]-‘,
(1)
in which, roughly, r1 determines the size and a the surface thickness of the nucleus. (The conventional parameters, half-density radius, c, and surface thickness, t, are functions of both r1 and a.) (b) Family II: 1 _ 35 e--n(l-Z) ) P -
1,
!’ /2
-nk-1)
e
7
x
(2) 1
where x: = T/Q , and n is a surface thickness parameter. The name “Family II” stems from an earlier reference (17) where various properties of this function are discussed. The Family II function and the Fermi two-parameter function are very similar if t,he half-density radius, c, and the 90 %-IO % surface thickness, t, are adjusted to have the same values, provided t < G, but the Family II function is mathematically much simpler to work with. (c) Family III: This three-parameter function, which takes its name from 6 We are indebted to Miss Pat Curly for her help with the operation of the computer at Livermore Laboratory. 7 The Llos Alamos phase shift analysis program was originally developed by Hill, Freeman and Ford, and some results were given by Ford and Hill (IS). The program was recoded and improved by J. Wills and B. Hill. These workers have carried out most of the calculations referred to in the present paper, by the other methods discussed below as well as by the phase shift analysis method. * We are indebted to D. G. Ravenhall for making available to us this code. the
130
MEYER-BERKHOUT
Ref. 17, is equal to the family II function multiplied by sinh SX/SX, with s < n. It can exhibit central depression of charge density. (d) Harmonic oscillator function: P-
exp( -r’/r,“)
+ 4r,/r,)3(r/r,>2
exp( -r2/rp”),
(3)
where 01 = ( 56) (2 - 2). This would be the charge density of two 1s protons moving in an oscillator potential with radial parameter rs , plus (2 - 2) lp protons moving in an oscillator potential with radial parameter T, . Thus r8 > rP would correspond roughly to the charge density of 2 protons in a potential more nearly square than the oscillator potential. The further possibility of regarding (Y as a phenomenologically adjustable parameter has been considered in the present work only for Be’. 2. Born
Approhxztion
The theory of the Born approximation for the elastic scattering of electrons has been summarized by Hofstadter (A). Most of our analysis has been based on the Born approximation because of its simplicity and because it makes possible the simultaneous analysis of data at several energies. Figure 1 demonstrates the degree of validity of the Born approximation for the lightest and heaviest elements considered here, Li” and 016, up to the highest momentum transfers for which data exist. On the low side of the first diffraction minimum, the Born approximation and exact phase shift calculations are in very close agreement for 016 and for lighter elements, provided the radial constant for the Born approximation is chosen to be a few tenths of one Fercent larger than for the phase shift analysis (corresponding to the slight increase of electron momentum in the vicinity of the nucleus). At the minimum and for some distance beyond it, the Born approximation is inaccurate. However, since one knows in what way the Born approximation fails, and knows that it)s error is not sensitive to the particular choice of charge distribution function, it can be usefully applied to all of the elements studied here. In particular, it can be used to discard unacceptable charge distributions. The “Fourier method” and the “Hermite method” discussedbelow make use of the Born approximation. In addition, we have used the more standard approach of assuming some reasonably simple functional form for the charge distribution and with it calculating the predicted scattering to compare with experiment. In the Born approximation, as well as in the phase shift method, we have used the Family II function, Eq. (2)) and the harmonic oscillator function, Eq. (3). In addition, Born approximation calculations have been made with the following charge distribution functions. (a) The Slater function: p N Csf-ns~--k8r + CprnP+‘,
(4)
NUCLEAR
CHARGE
DISTRIBUTIONS
131
with C, and C, being normalized to yield the number of s-protons for the volume integral of the first term and the number of p-protons for the second term. The two terms can then be readily interpreted in terms of simple approximate s- and p-wave radial functions, but in practice this Slater function did not prove very successful in fitting the data. (b) The one-parameter functions: These are the group of simple functions listed by Hofstsdter (18). All of the above functions were tried over a wide range of their adjustable parameters for all of the data, and surprisingly few gave satisfactory fits at all energies, Most of the one-parameter functions, for example, gave no satisfactory fit to ang element. In this connection, it is significant that the lower energy data proved almost equally as valuable as the higher energy data in restricting the set of acceptable charge distributions. The data at several energies taken together provided a much greater restriction on the charge distribution than did the 420-Mev data alone. (Of course 420-Mev data at more forward angles would have been equally valuable, but since it was not obtainable at the time these experiments were performed, lower energy data was needed to supply the knowledge about the scattering at low momentum transfers.) Kinematic recoil corrections were included in the Born approximation calculation,s. 3. Fourier
Method
Since the Born approximation is rather accurate for the light nuclei, one might consider the possibility of working backward directly from the experimentally determined form factor, F(q), to the charge distribution, p(r). The quantity rp(r) is just the Fourier sine transform of @‘(cl) :
v(r)=
(27w’~
s,-qF(q)
sin pr dq,
where we choosethe normalizations F( 0) = 1, and 4?rJ pr2dr = 2. The straightforward determination of p from (5) is of course not possible because F( 4) is known only over a limited range of 4. Nevertheless one might ask whether the unknown parts of F(n) are sufficiently unimportant that the charge distribution can be approximately inferred directly from the data (presented as a form factor). In this subsection and the next, we discuss two methods of analysis which are closer in spirit to this approach than to the usual trial and error approach. These methods do not presuppose a simple analytical charge distribution, and in the limit that F(q) is known everywhere they could be made equivalent to carrying out the IFourier transformation (5). We note that in (5) q is the momentum transfer in the center-of-momentum frame, and / F(q) 1 = (~/a~)“~, where ~0 is the point-nucleus cross section in Born approximation, and v is the measured cross section translated to the center-
132
MEYER-BERKHOUT 1O-2g
-
I
I
I
I
I
I
Li6 426 MEV
1o-34 30
40
50 SCATTERING
FIG.
60
ANGLE
la
FIG. 1. Comparison of Born approximation and phase shift analysis. (a) Scattering of 426.Mev electrons by Lie. Both curves are calculated for Family II charge distribution with n = 2.25, T, = 1.75. Most of the discrepancy can be removed by choosing in the Born approximation a slightly larger wave number, or equivalently a larger nuclear radius constant. The difference is, however, less than the uncertainty in the determination of rt , and therefore not taken into account in the subsequent analysis. (b) Scattering of 420.Mev electrons by O16. Both curves are calculated for the oscillator charge distribution with a = 2, r, = pP = 1.754. If the radial constant for the Born approximation is chosen to be 0.3% larger, the small discrepancies below and well above the diffraction minimum are removed. (All of the curves in this figure were calculated without recoil corrections.)
NUCLEAR
CHARGE
133
DISTRIBUTIONS
016420
MEV
8f3 SCATTERING FIG.
ANGLE
IN
DEGREES
lb
of-momentum frame. In the extreme relativistic limit, (co2 J.;O/sin” $50))
go = (Ze”/2E)”
(6)
where the angle 0 and the energy E are in the center-of-momentum frame: E tan
= 0 =
(1 (1
-
6)&b, +
E set
(7) h,)
tan
fhab
,
134
MEYER-BERKHOUT
expressions correct to first order in h = (electron energy)/( nuclear mass). See also Eq. (48) in Ref. 4. (In the actual calculations, more accurate expressions were used. ) The inverse form to (5) is qF( q) = 4~2~’ lm rp( r) sin qr dr , We find it convenient to plot this quantity, $‘(a), rather than F( 4) itself, as a function of 4. Observe that for small q, @(q) = 4. This fact permits a reliable extrapolation of the experimental qF(q) downward to q = 0. (The lowest order deviation of qF(q) from a straight line measures (1’). The lower energy data is therefore valuable in helping to fix the rms radius.) At large q, only weak conditions of physical reasonablenesslimit the behavior of qF(q)-e.g., that p(r) should be everywhere positive, or that p(r) E 0 beyond some distance b. We take advantage of the second assumption, assuming p = 0 for r > b, and write rp(r) as a Fourier series, rp( r) = c C, sin( n?rr/b) ,
r 5 b.
9L=l
(9)
From (8) and (9) it follows at once that the coefficients C, are given by, C, = (ZPd)qnF(qn),
(10)
q,, = m/b.
(11)
where
Accordingly, once a cutoff radius is assumed, one may determine the Fourier components of rp( r) simply by reading values of qF( n) from a graph of experimental data at the qn values given by ( 11)-provided no qn value lies too close to a diffraction minimum. We have applied this method for cutoff radii b = 5 and b = 6. As will be evident in later figures, the charge densities inferred are not very sensitive to the choice of b (except near r = 0, where all methods fail to determine the charge density with any accuracy). The number of Fourier coefficients which can be read from the data are four in most cases.Higher coefficients are arbitrarily set equal to zero. In terms of the Fourier coefficients, the normalization conditions becomes 2 = 4b2c ( - 1) “+lCn/n. Because of the decline of C, with n and because of the n in the denominator, it is reasonable to assume that the unknown higher coefficients contribute very little to the normalization sum. With this assumption, (12) provides a convenient way to normalize the experimental data, whose absolute value is not
NUCLEAR
CHARGE
DISTRIBUTIONS
135
well known. We scale the whole qF(q) curve up or down by whatever factor is required to satisfy ( 12). This factor has in each case been well within the experimental absolute error, and is nearly the same for b = 5 and b = 6. This procedure is required for self consistency but may have extra value in fixing the absolute cross sections to better accuracy than the present measurements. The mean square radius for the Fourier charge distribution is given by (T’) = b2 -
(24b4/7r22)
c
( -1)““Cn/n3.
The terms in this sum diminish rapidly, and the mean square radius is determined mainly by the first coefficient. This coefficient falls near the maximum of the qF(q) curve and is therefore determined by low energy or small angle data. To illustrate the application of the Fourier series method, we show in Fig. 2 the experimental values of pF(q) for 0”. The vertical lines show the values of q,&and q,F(p,) based upon the assumed cutoff radius b = 5. The importance of the low-energy data should become immediately clear from this diagram. The S60- and .i20-Mev data fall at too large 4 values and hence fail to fix C1 . On the other hand, the 240- and 160-Mev data proscribe the value of C1 quite sharply and hence determine the major component of the charge distribution. The experimental study was extended to 160 Mev in the light of this observation. The components of rp(r) are shown in Pig. :3 along with the Fourier sum based upon the assumed cutoff radius b = 5. Also shown on this diagram is the charge distribution inferred by the Fourier series method using b = 6 as well as the distributison obtained on the basis of the harmonic oscillator model. The agreement between the t,hree curves is gratifying. The experimental data for 0’” does not go to sufficiently high energies to determine the fifth Fourier coefficient when b = 5. The uncertainty in this coefficient reflects itself in a corresponding uncertainty as to the shape of the charge distribution. Reasonable extrapolations suggest that C, = 0 f 0.005. This corresponds to an uncertainty of about 1% in the normalization condition (12). However, the uncert’ainty in the fifth component has a rather appreciable effect upon the central charge density. At the center each component contributes /h(O)
= c, m/b.
(14)
The factor n in the numerator amplifies the influence of the higher components upon the central density. The uncertainty in central density associated with the uncertainty in the fifth component in the case of 016 is indicated on Fig. 17. This is the most favorable of the six elements, however, with relatively small latitude in the choice of the fifth component. In other cases the central density is much more poorly determined. While the uncertainty as to the magnitude of qF(q) beyond the range of measurement reflects itself in a small uncertainty in the inferred distribution
136
MEYER-BERKHOUT
v n
A l
160 240 360 420
MEV MEV MEV MEV
DATA DATA OATA DATA
0.2
0.1 l-
L 0
2
q FIG. 2. Form factor for 016, showing experimental points with statistical errors only. The theoretical curves are for the oscillator model and for the Fourier model with b = 6. The oscillator curve has the parameters of the “pure oscillator”-i.e., rs = rP ,01 = Z-from Table IV. The modified oscillator listed in Table IV (from Ref. 2) gives a better fit to the data. The Fourier form factor with b = 5 is scarcely distinguishable from that for b = 6, and is therefore also an excellent fit. The vertical lines designate the values of q,, used in the Fourier method with b = 5. The guessed limits on qF(q) for the fifth Fourier coefficient
are also shown. functions, the uncertainty as to the sign of the Fourier coefficients reflects itself in possible gross changes in the charge distribution function. In the case of 016 the success of the harmonic oscillator model for which the qF(q) function goes negative serves as a guide to infer that the actual qF(g) functions follow the same pattern. If, however, we were to make the assumption that qF(g) merely dips to near zero and then rises again, i.e., the third and fourth Fourier coef-
NUCLEAR
0.14
I
CHARGE
I
137
DISTRIBUTIONS
I
I
I
I
0.12
rp(r)
0.10
I-
0.00
I-
0.06
,-
0.04
0.02
0
(
-0.02
-0.04
- 0.06
0
I
I
I
I
I
I
I
e
3
4
5
6
7
r FIG. 3. T p(r) for 016. Fs and Fe designate Fourier method b = 6. S. H. 0. designates the oscillator model. H designates shown are the individual Fourier components for 6 = 5.
with the
cutoff radii b = 5 and Hermite method. Also
ficients are taken positive, then these Fourier components would contribute a peak at the center of the nucleus rather than a hollow. The inferred distributions would then have a very considerably different character from the ones shown in Figs. 3 and 17. The assumption that F(q) is everywhere positive in the experimentally covered
138
MEYER-BERKHOUT
region is of course not very sensible for Cl2 and 0’” which exhibit obvious diffraction minima. But B” and N14 show barely discernible minima and one must inquire more seriously about the sign of F (4) at the largest measured momentum transfers in these cases. Interpolation and extrapolation from Cl2 and 016 suggest the negative sign. The negative sign is also strongly suggested by the fact that if F(p) is assumed to remain positive, the peak charge density inferred is nearly twice the largest charge density deduced for any other nucleus, light or heavy. Figure 4 compares theoretical qF(q) curves with the N14 data for the two assumptions that F(p) remains positive and that F(q) goes negative. Both 0.5
1
7N’4 . .’
420 MEV ABSOLUTE MEV
0.4
I
2
DATA, NORMALIZE0 MEASUREMENT DATA,
TO
X I.41 b X 1.99)
3
9 FIG. 4. Form factor for Nr4, comparing theoretical curves with experimental points. Fb(-) labels the approximate fit obtained with the Fourier method, b = 5, assuming a change in sign of F(q). Fr,(+) labels the attempted fit obtained with the same method but assuming F(q) > 0 over the whole experimental range. The R’S(-) curve is not a very good fit and the FCC+) curve is a poor fit. Also shown is the best fit with the oscillator model.
NUCLEAR
CHARGE
139
DISTRIBUTIONS
theoreticad curves of course pass through the experimental pnF(qn) points, but one must look between these points as well to see if the Fourier analysis is providing everywhere a good fit to the data. (See also Fig. 7, for Bn, where the fourier method gives a better fit.) We conclude for N14 that a diffraction minimum should be present but is being filled in by other effects. The spherically symmetric part of the charge distribution can nevertheless be determined reasonably accurately if it is assumed that F(q) becomes negative at large 4. Hermite Method Another series expansion technique has been tried which does not require a sharp radial cutoff in the charge density. Assume that v-p(r) may be expanded in terms of N functions fi(r) according to
4.
rp(?“) = ci=o Cifi(T1.
(15)
Choose any N values of Q where data exist and read the corresponding qjF(qj) from a graph of the experimental data. Then qjF(qj)
= C MiXi
values
7
(16)
sin qjr dr.
(17)
where Mii = 4~25
I* fi(r)
By solving the N simultaneous equations (16) with the N2 known coefficients (17), one may find the N coefficients Ci . The normalization condition requires
(18) and, as in the Fourier method, may be used to normalize the absolute values of the data. (The renormalizations required in the two methods were found to be nearly the same.) The success of this expansion technique will depend upon the choice of functions. The approximate validity of the harmonic oscillator model for light nuclei suggests that the fi might be chosen to be simple polynomials multiplied by a gaussian weighting factor. We find a particular generalization of the Hermite polynomials convenient. A generalized Hermite polynomial is (19) H,(z,v) with
= (-l)“e”l‘“(d/d~)“e-““‘,
v >, 0 and n = 0, 1, 2, . . . . A useful integral
(19) for determining
radial mo-
140
MEYER-BERKHOUT
ments is m e-“r2xmHn(3,~) dx = s-w
0, for m < n, and for m - n = 2~ + 1 ’ m! (?r/~)~‘~/p!(4~)!, for m - 72= 2~
(20)
where p is any positive integer or zero. The usual Hermite polynomial (arising, for example, in the one-dimensional oscillator problem) is obtained by setting Y = 1. Instead we choose the set of functions with Y = 34: f;(r)
= 2(27r)-3’2a2H2i+1(ar,f~)e-1’2a2r2.
(21)
These lead to the matrix coefficients (20) M;j = (a/2)(
-l)i(qj/a)2i+1
exp[->$(qi/a)“].
(22)
Equation (20) implies that only the first function contributes to the normalization, and the constant in the definition (21) has been so chosen that Co = Z. Only the first two functions contribute to the mean square radius, which is given by (r2) = 3(Co + 2C1)/a2Z.
(23)
To apply this expansion technique, one must first guessthe value of a. For this guessthe simultaneous equations (16) are solved using the M;j given by (22). If the first coefficient Codoes not equal Z, a new CYis chosen and the process repeated.g In several cases, the normalization could not be achieved with any value of CLThen the absolute value of the experimental data was renormalized (always within the experimental uncertainty) until a properly normalized solution was obtained. The charge distributions inferred by this Hermite method for N 5 4 are shown in some of the following figures. B. ANALYSIS
OF DATA
The experimental data reduced to center-of-momentum form factors are shown for 016 in Fig. 2, for N14in Fig. 4, and for the other four elements in Figs. 5-8. Also shown in these figures are sample theoretical form factors. The data at different energies for the same element are adjusted as shown in Table III in order to cause the qF(p) values to lie along a single smooth curve as a function of p. The analysis of the data is discussedbelow separately for each element. The consequencesof the sign ambiguity of F(q) are also discussed.Finally, regularities noted within the p-shell are summarized. Table IV gives the charge distribution functions which provide good fits to the data. To be acceptable, the charge distribution function was required to pro9 An iterative method using an IBM 704 code by Mr. Benny Hill was used to solve the simultaneous equations and to converge upon the best value of LY. Hand calculation is also feasible, however, if the qi values are chosen to be equally spaced.
NUCLEAR
I 0
CHARGE
I I
141
DISTRIBUTIONS
l
187
MEV
OATA
A
426
MEV
DATA
1 3
4
FIG. 5. Form factor for Li6, experimental points and some theoretical curves. In this figure and in Figs. 2, 4, 6, 7, and 8, statistical errors only are shown, the data being renormalized as indicated in Table III. For good agreement with each theoretical curve, some further overall renormalization may be required, so that a good fit may not at first glance appear very good in these figures. The illustrated curves all provide good fits, except in Fig. 4. The following notation is used: Fb , Fourier method with cutoff radius b, Eq. (9); H, Hermite method, Eqs. (15) and (21); superscript (+), F(q) assumed to remain positive; superscript (-) or no superscript, F(g) assumed to change sign; S. H. O., oscillator model, Eq. (3) ; Slater, Eq. (4) ; II, Family II, Eq. (2) ; Fermi, Eq. (1) ; 5 and 6, following the numbering in ‘Ref. 18, are the forms 5eeZ and (1 + z)ez, respectively. Form factor and momentum transfer are in the center-of-momentum system.
vide a good relative fit to the data at all energies, and to predict an absolute cross section within experimental error at each energy.” For C” and 016 at all 10 As indicated in Section II, the absolute cross-section measurements at 160 Mev are believed to contain an error of a factor of 2. The theoretical
for Bii and Nr4 absolute cross
142
MEYER-BERKHOUT
IqF(q)l
FIG.
6. Form
factor
for
BeQ. See caption
of Fig.
5
energies and angles (phase shift analysis where necessary), and for the other elements below the first diffraction minimum, a good relative fit was required to have a weighted mean square deviation of theory from experiment less than 1.5 times the statistical mean square deviation [definitions and formulas given by Ford and Hill (IS)]. The Fermi two-parameter and some shell model phase shift analysis fits were judged by eye. For the elements which indicate some filling in of the diffraction minimum (Be, B, and N), no objective criterion for a good fit could be employed, and theoretical and experimental curves were compared by eye, the angular region of supposed filling in being overlooked. The Fourier fits to N14 were not very good (see Fig. 4) but are included in Table IV for illustrasections were not required times these measurements).
to agree
with
these
two
measurements
(but
do agree
with
0.5
-. r
0.5
NUCLEAR
CHARGE
143
DISTRIBUTIONS
56” . A m
160 300 420
MEV MEV MEV
DATA DATA DATA
IqW\
0.2 6
0.1
0
: K4 FIG.
I 7.
2
Form factor for B”. See caption of Fig. 5
tive purposes. This makes the results more unreliable, but still permits a reasonably good definition of the charge density. As already emphasized by other authors (2, 4-r)) the absolute cross sections, even though not very accurate, also provide a valuable constraint on the charge density. The still quite appreciable uncertainty in the rms radii (see Li6 discussion below) arises from uncertainty in the absolute cross sections at low q-value. The charge distribution coefficients obtained by the Fourier and Hermite methods are given in Tables V and VI. 3Lia. Because of the validity of the Born approximation, the quantity rp(r) seems to be mathematically the most meaningful thing to present, and we therefore present a few curves of this type. However, physically, one is more interested in p(r) itself, and we present curves of p versus r for all elements.
144
MEYER-BERKHOUT
0.5
I
I
I
I
p
. 187 Q 420
MEV MEV
DATA DATA
0.2
0.
( FIG.
8.
Form
factor for CP. See caption of Fig. 5
Figure 9 shows rp(r) deduced by the Hermite and Fourier methods for Li6. The three curves are in remarkably close agreement. In particular, the results of the Fourier method are obviously insensitive to the choice of cutoff distance b, within reasonable limits. The Hermite method also implies a very rapid cutoff because of its gaussian factor. Note in Table III that these three curves yield rms radii of 2.5 to 2.65, while the other acceptable charge distributions have rms radii around 2.8, owing to the contributions of their longer tails. It seems reasonable that the higher figure is more nearly correct, since such a light nucleus
NUCLEAR
CHARGE
DISTRIBUTIONS
145
might be expected to have a rather diffuse surface. (For 016, where the oscillator model is successful, this kind of discrepancy among the rms radii does not exist.) However, it should be emphasized that the rms radius is not well determined. Its value can be made definite only by a rather accurate measurement of the absolute cross section for a momentum transfer around Q = 0.8. (Observe in Table III that the required renormalizations in Li6 are significantly different for the different methods of analysis.) Figure 10 shows most of the acceptable charge densities for Li6 (see also Ref. 7). This figure emphasizes the fact that the central density of charge is completely uncertain,, Beyond r M 1, however, the charge density is rather well defined. Also one rnay conclude that the maximum charge density in Li6 is pmax = 0.064 f 0.006. 4Beg. Figure 11 shows most of the acceptable charge densities in Be’, showing again clea,rly the lack of knowledge of charge density for r 5 1. For this particular element, the Fourier methods for cutoff radii b = 5 and b = 6 differed by the greatest amount, and only b = 5 gave an acceptable fit. This comes about because even for b = 6, one can read from the data only three Fourier coefficients, which is insufficient for a cutoff radius of 6. One may say, for b = 6, that the Fourier method is trying, with inadequate means at its disposal, to produce a smooth but very asymmetric curve in the interval 0 5 r 5 6. Since the oscillator charge distribution is successful for the heavier elements in this group, we remark that attempts were made to fit the Beg data with the charge density function (3)) holding r8 = r, and varying a! and rs , as well as fixing Q!and adjusting rs and rP . No combination of parameters gave a successful fit. The theoretical curves fell more rapidly than the experimental data at the largest momentum transfers. After the quadrupole scattering is included (Section IV) the oscillator charge distribution can no longer be ruled out. From Fig. 11 we conclude that the maximum charge density in Beg is pmax = 0.086 f 0.006. sB’l. The B” data presented in Fig. 7 suggest that quadrupole and/or magnetic scattering are coming in at large momentum transfers to distort the @‘(q) curve. One of the theoretical curves shown in Fig. 7 accordingly is presented without any effort to fit the data at large y. Three “acceptable” charge densities based on the same approach are shown in Fig. 12. They provide good fits to the data except for y 2 1.8. They agree closely with each other for r 2 2, and they imply for B” a maximum charge density pmax + 0.086 f 0.005. (The Fermi function listed in Table IV fits the data only for 4 5 1.7). The results of assuming instead that all of the scattering up to the largest angles observed comes from a static spherically symmetric charge distribution are shown in Fig. 13. None of the assumed simple functional forms of p(r) could provide a fit to all the data but the Fourier and Hermite methods could be forced to fit
146
MEYER-BERKHOUT
TABLE
IV
ACCEPTABLE CHARGE DENSITY FUNCTIONS FOR ~-SHELL NUCLEI" (r2)1/2p
Parametersb
Element ~~~
Familyb
3Li6 c
II xecx d Fourier Fourier Hermite Fermi Oscillatorc
b b (Y r1 a
= = = = =
5, 3 terms 6, 4 terms 0.732, 4 terms 1.20, a = 0.70 55, rs = 2.65, rp = 1.07
2.84 2.80 2.49 2.53 2.65 2.75 2.82
xecx Slater Fourier Fermig Oscillatore
n, b r1 a
= = S =
4, np = 6, k, = 2.80, 5, 3 terms 0.90, a ES 0.79 35, r8 = rp = 1.609
2.75 2.68 2.51 2.999 2.26~
rBe9f
aC’2
,N14 ‘. g
SO’6 In
~1 = 1.75,
n = 2.25
IIK IIa Fermig (1 + x)epx g Fourier(+) i Fourier(+) i Hermite r Hermiteg Oscillatorg
b b (Y 01 (Y
= = = = =
5, 4 terms (better fit than 6, 4 terms 0.92, 4 terms 0.92, 4 terms 1, Ts = rp = 1.55g
II Oscillator Oscillatork Fermi Fourier Fourier Hermite
t-1 = a = 01 = 1-1 = b = b = (Y =
2.31, n = 3.5 x, rp = rs = 1.637; 1.12, rp = rs = 1.71 (better 2.24, a = 0.50 5, 4 terms 6, 5 terms 0.92, 5 terms
II Fermi Oscillator Fourier(+) Fourier Hermite
TI TI (Y b b (Y
= E = = = =
2.50, n = 5.0 2.30, a c% 0.50 54, r,, = rs = 1.667 5, 4 terms (not a good fit) 5, 4 terms (not an excellent 1.08, 4 terms
a a b b 01
= = = = =
2, rp = rs = 1.764” 1.60, rp = ~8 = 1.82 (better 5, 4 terms 6,5terms 0.82, 5 terms
Oscillator Oscillator0 Fourier Fourier Hermite
i
TI = 1.85, n = 2.50 ~1 = 1.95, n = 2.75 7-1 E 2.00, a G 0.55
k, = 2.30
(better
fit than
line
fit
line
2.769 2.709 2.569 2.508 2.34’ 2.35’ 2.36’ 2.418 2.258
above)
below)
than
line
above)
2.45 2.57 2.46 2.57 2.43 2.40
fit)
fit than
2.72 2.41i 2.50 2.54 2.41 2.36 2.47
line
above)
2.65m 2.70 2.66 2.62 2.64
NUCLEAR
CHARGE
DISTRIBUTIONS
147
the data fairly well (see Fig. 7). These methods of analysis are consistent with each other and give charge distributions very different from those shown in Fig. 12. There are, however, several reasons for not believing these results. (1) The maximum charge density, pinax = 0.15 f 0.03, is much larger than found in any other nucleus. (2) All three curves have negative charge density at large r. This is a very small effect, and possibly not significant. (3) The curves have an “unusual” shape. It seems unlikely that any reasonably shaped average potential would lead to radial wave functions for the protons to give these charge densities. This idea has not been put to a yuantitiative test. (4) Theory. Quadrupole scattering from Bn ought to be significant, and these curves assume it is not present. &“. Most of the acceptable charge density functions for C’* are shown in Fig. 14. This nucleus and 01” appear to have the best defined charge densities. Even here, however, it is clear that the central density is not well determined, and it cannot even be stated whether there is any central depression of charge as the shell model predicts. Also, as indicated in Table IV, the rms radius is not well determined. The rapid-cutoff models (oscillator, Hermite, and Fourier) have rms radii of 2.36 to 2.50, while the long-tailed models (Fermi and Family II) have rms radii of 2.54 and 2.72. The latter provide slightly less satisfactory fits Footnotes
to Table
IV
a Most functions and methods listed in this paper were tried for each element. Exceptions are given in footnotes below. b Charge density families and notation for parameters are defined as follows: Fermi, Eq. (1) ; II, Eq. (2) ; Oscillator, Eq. (3) ; Slater, Eq. (4) ; F ourier, Eq. (9) ; Hermite, Eqs. (15) and (21). The one-parameter families indicated explicitly are completely characterized by the rms radius. c Family III not tried for Lie. d Same result also obtained in Ref. 7. * Result quoted from Ref. 7. r Family III not tried for Beg. Hermite method indicated data required renormalizing, but Hermite analysis not completed. g Good fit only when sufficient quadrupole scattering included. ‘1 Family III not tried for B”. Fourier method with assumed diffraction minimum not tried. 1 Superscript (+) means that it has been arbitrarily assumed that F(q) remains positive, an effort being made to force a fit to the data ignoring the probable presence of a filled-in diffraction minimum. j These numbers are in agreement with those of Ref. 2, where for the oscillator model with 01 = 96, the best parameters were found to be rP = r, = 1.65, (r2)i1* = 2.42. k Result quoted from Ref. 2. 1 Family III not tried for Ni4. m Families II and III not tried for 016. n Same result obtained in Ref. 2. 0 Result quoted from Ref. 2. p Radii in this table are actual charge radii and not the radii of proton distributions (see Ref. 2).
148
MEYER-BEREHOUT
TABLE FOURIER
COEFFICIENTS
Element
b
OF CHARGE 100 Ci
V DENSITIES
FOR ~-SHELL
100 CP
100 CB
3Li6
5
4.002
2.48,
0.726
nBe9 gB’i
6 5 5(+)
3.117 5.312 6.945 5.356 8.142 6.42 9.40 8.99 9.904 8.066
2.666 3.388 4.30s 4.509 4.338 5.43 4.57 4.34 3.344 5.484
1.154 1.14s 1.14 1.67 -4.044 1.145 -1.08 1.02 -1.12 0.21,
6(+, 5 6 5 5(f) 5 6
,CP ,N’4 80’6
NUCLEIC
100 Ca
100 c5 -
0.336 0.70 0.735 -0.665 -0.682 -0.65 0.65 -0.549 -0.893
-
-0.30s
8 The coefficients C, are defined by Eqs. (9) and (lo), and the data have been renormaliaed as indicated in Table III in order to satisfy the normalization condition (12). The units of C, are 10Z6 cmm2. The superscript (+) means that the signs of the higher coefficients have been taken to be positive, although a negative sign is more likely correct.
HERMITE Element 3Li6 gBil ,C’2 TN’4 ,016
(Y 0.73 0.92(+) 0.92 0.92 1.08 0.82
COEFFICIENTS
CO 3.0 5.0 5.0 6.0 7.0 8.0
& The coefficients Ci are defined renormalized as indicated in Table 2. The Cc are dimensionless. The IV and V.
TABLE VI FOR SOME Cl 0.201 1.23 1.59 1.66 4.33 2.10
~-SHELL cz
0.0340 -0.0132 0.292 -0.055, 0.665 -0.0392
NUCLEI” c1 -5.0 x -0.0248 0.0376 -0.0276 0.0271 -0.015,
CC lo-4
-0.0017 -0.0009
by Eqs. (15), (16), (17), and (21). The data have been III in order to satisfy the normalization condition C, = superscript (+) has the same significance as in Tables
than the rapid-cutoff models,ll but cannot be ruled out at this time. From Fig. 14 we conclude that the maximum charge density for Cl2 is pmsx = 0.080 f 0.005. 7N14.Shown in Fig. 15 are three acceptable functions ~-p(r) for N14 (still ignor11 The Mev data this model above the from the p-values.
Fermi function giving a nearly perfect fit to the 420-Mev C’s data also fits the 187. up to angles of about 100”. At still larger angles the angular distribution curve for drops too fast and Fregeau’s last experimental points lie up to a factor of two theoretical curve. The only points in the 420-Mev data which deviate somewhat theoretical angular distribution curve in the same direction occur at just these
NUCLEAR
CHARGE
149
DISTRIBUTIONS
0.07
0.06
0.05
0.04
rp(r1 0.03
0.02
0.01
I
0
2
3
4
5
6
7
in caption
of Fig.
r FIG.
9. r p(r)
for
Li6.
Notation
for
this
and
succeeding
figures
5
ing quadrupole effects) for which the form factor passes through zero, and one function for which the form factor is assumed to remain positive. The form factors corresponding to the curves labelled Fi+’ and F:-’ are shown in Fig. 4. The F’+’ curve fits the data less well, but it is interesting to see the difference in the charge distributions based on these two assumptions. The charge density itself is shown in Fig. 16 for each of these four models. The three “minus” curves are very similar to each other and differ markedly from the “plus” curve-in the same qualitative way as for B” (Figs. 12 and 13). There is strong reason to believe that the “minus” curves are correct, and the “plus” curves incorrect, based on the fits to the data, Fig. 4; on analogy with neighboring nuclei Cl2 and O16; and on the theory of the quadrupole effect, discussed below. Figure 16 implies for N14 pmaa = 0.087 f 0.004. 8016. Several acceptable charge density functions for 016 are shown in Fig. 17. The density for r 2 2 is determined to the highest accuracy in this case. Observe in Table VI that for 016 the Hermite coefficients Ci for i 2 2 are very small. If
150
MEYER-BERKHOUT
r
1 P
I ’
II F
0.06
t g(r)
0.04
0.03
Q02
0.0 I
L
0
I
2
3
4
5
6
7
r FIG.
10. Charge
density
p(r)
for
Li6.
For
notation,
see caption
of Fig.
5
they are ignored relative to COand C1 , then the charge density from the Hermite method is exactly eyuivalent to that from the oscillator model with suitably adjusted parameters. The leading two terms in the Hermite function shown in Fig. 17 correspond to an oscillator with (Y = 2.48 and r, = rs = 1.73. For 016, fits with the Family II function were not attempted, but attempted fits with the similar Fermi two-parameter function were not successful. The parameters which gave a fit at all but the largest observed momentum transfers were ~1 = 2.57, a = 0.505. The theoretical cross section at 420 Mev in this case fell below the experimental points in the angular region 65 to 75 degrees by up to a factor of two, and predicted a second diffraction minimum between 80 and 85 degrees. (The oscillator model predicts no second minimum.) From Fig. 17 the maximum charge density in 016 is determined to be pmax = 0.079 f 0.003.
NUCLEAR 0.N
CHARGE
I
I
151
DISTRIBUTIONS
I
I
I
I
0.0:
O.OI
0.0;
o.oz
p(r)
O.Ot
0.04
0.03
0.02
0.0 I
0
I I
I
2
4
3
I
I
5
6
7
r FIG.
11. Charge
density
p(r)
for
Be 9. For
notation,
see caption
of Fig.
5
Regularities in the p-Shell Parameters which have become standard for describing the charge distributions of heavy elements are the half-density radius, c, and the 90 %-lo % surface thickness, t. These are not well defined for the light elements under study. But there is a convenient way to define substitutes for these quantities based on the near constancy of the peak density of nuclear matter (21) .
152
MEYER-BERKHOUT
0.14
0.08 p(r) 0.06
0.04
0.02
0 r
12. Charge density p(r) for B1l, assuming F(q) goes negative, and overlooking data at large momentum transfer (see Fig. 7). For notation, see caption of Fig. 5. FIG.
the
Hofstadter (4) observed that the central densities of charge of nuclei decrease gradually from A = 40 to A = 209. If one assumesthe samespatial distribution for neutrons as for protons, then p(M) = (A/Z)p(Z), where p(M) is the matter density and p(Z) the density of charge. Then it appears that the peak (or central) density of matter, p,,(M), is . constant for all medium and heavy nuclei, with a value of about 0.166 nucleons/( lo-l3 cm)“. Central densities of charge and of matter for seven elements from Ca to Bi are shown in Fig. 18. The indicated errors represent the difference in densities found in the Stanford analysis with the Fermi function and in the Los Alamos analysis (16) of the same data with the Family II function. The true uncertainty is perhaps somewhat greater. Also shown in Fig. 18 are the peak densities of charge inferred in the present analysis of light elements, and the corresponding peak matter densities if the neutrons have the same spatial distribution as the protons. The peak matter densities inferred in this way for Be and B are slightly high, suggesting the pos-
NUCLEAR
CHARGE
153
DISTRIBUTIONS
0.16
p(r)
0.06
0.06
0.04
0.02
0
-0.02
I
2
3
4
5
6
7
r FIG. 13. Charge density P(T) for B”, assuming F(q) remains positive data in the analysis (see Fig. 7). The superscript (+) is to be understood bols in this figure. For notation, see caption of Fig. 5.
and including all on all of the sym-
sibility that the neutrons are in fact slightly more extended. The peak density of Li is low, indicating that finally at A = 6 the concept of nuclear matter has lost its usefulness, and the nuclear surface has penetrated to the center of the nucleus. It is a remarkable fact, however, that all the way from A = 9 to A = 209 there appears to be no significant change in the peak density of nuclear matter.
154
MEYER-BERKHOUT
0.14 I-
0.12
0.06
c
p(r) /
0.06
0.04
0.02 S.H.O.,H,FJ
0 FIG.
14.
I I
1 2
3
4
5
I 6
Charge density p(r) for C12.For notation, see caption of Fig. 5
Presumably then the correct density which should result from a theory of bulk nuclear matter is 0.166 A 0.005. This corresponds to a mean particle spacing of 1.13 * 0.01. For the light nuclei, we can therefore define a sensible half-density radius as the radius where the charge density has half of the average peak value. We take 0.040 as this half density, and call the corresponding radius ~(0.04). It is determined for these six light elements to about 2 %. (Contrast the definition given in Ref. 7, which is much more model dependent. Our definition avoids any reliance on the very uncertain region in the center of the nucleus.) This half-density radius divided by A”3 is plotted in Fig. 19, together with the half-density radii for heavier elements. There is an evident regularity-a steep ascent from Li to C and a more gradual increase throughout the periodic table toward the asymptotic value 1.13 for infinitely heavy nuclei. The errors in Fig. 19 are taken to be 2 % for the p-shell nuclei, 1.5 % for Ca, V, and Co, and 1% for the heavy elements. The crude average value 1.07 for c/A”~ which has been used before (4) is indicated in the figure.
NUCLEAR
CHARGE
DISTRIBUTIONS
r/9(r) 0.06
I
0
2
3
4
5
6
7
I FIG.
15. T p(r)
The mean square radii,
for
N14. For
notation,
see caption
(r2), or correspondingly
of Fig.
the equivalent
5
uniform
radii,
R = [5(~~)/3]~'~,are also of interest. R/All3 is plotted in Fig. 20. It diminishes regularly” from about 1.9 for Li to 1.18 for Pb (and should also approach 1.13 in the limit of infinitely heavy nuclei). The crude average value 1.20 for R/A"3 which is often used is shown in the figure. Errors based on the present analysis are shown for the p-shell nuclei. It is not known very well how accurate are the rms radii for medium and heavy nuclei. Therefore no errors are indicated for the other points. The substantial errors in the rms radii for the light nuclei can be reduced only when more accurate absolute cross-section measurements at momentum transfer around Q z 0.8 become available. The data shown in Figs. 18-20 are summarized for the p-shell nuclei in Table VII. The corresponding numbers, averaged over Au, Pb, and Bi, are also shown for comparison. Table VII includes surface thickness for the light elements. Regarding 0.080 as the “standard” peak charge density, we define the surface thick12 The regular stadter (4).
trend
in
mean
square
radii
has
been
pointed
out
previously
by Hof-
156
MEYER-BERKHOUT 0.20
I
I
I
I
I
I
I
I
4
5
6
0.18
0.18
p(r)
0.10
0.08
0.08
S.H.O.
0.04
0.02 F(-) H S H 0 5”“’
0
I I
I
I
2
3
r FIG.
16. Charge
density
p(r)
for
Nl4.
For
notation,
see caption
of Fig.
5
nessas the distance from p = 0.072 to p = 0.008. (For Li, the inner edge of the surface is then at T = 0.) For Be through 0, the surface thickness is constant, within the error, and equal to 1.9 f- 0.2. This is somewhat smaller than for heavier nuclei. In view of the arbitrary way of defining surface thickness, this may be without significance. On the other hand, these light nuclei have large proton separation energies, and, correspondingly, the proton wave functions fall more rapidly to zero outside the nuclear potential than would the wave functions of
NUCLEAR
CHARGE
157
DISTRIBUTIONS
0.06
p(r) 0.06
0.04
0.0 2
0
2
3
4 5 6 7 r FIG. 17. Charge density p(r) for 0 16.The approximate uncertainty in the central density arising from uncertainty in the fifth Fourier coefficient is shown. For notation, see caption of Fig. 5. I
protons at the top of the Fermi sea in most heavier elements. (A possible close connection between proton separation energy and surface thickness is indicated
by the fact that Bizogseemsto show a significantly greater surface thickness than does Pbzo8.) IV.
CORRECTIONS
TO THE
STATIC
SPHERICALLY
SYMMETRIC
NUCLEUS
Among possible correction effects, me have examined quantitatively only the effect of a deformed charge distribution. Brief discussionsof other effects are also
given below. A. NUCLEAR
CURRENTS
AND MAGNETIZATION
The effect of the proton’s magnetic moment on the scattering of electrons by protons
has been considered
in detail
(4) and affords
a basis for estimating
the
magnetic contribution to scattering from heavier elements. The ratio of magnetic
CENTRAL
FIG. 18. Peak the
data,
and
densities of charge are not theoretical.
DENSITIES
and of matter
in nuclei.
The
HALF-DENSITY
FIG. 19. Half-density oretical. used for
radii The dashed horizontal heavy nuclei.
divided by Al/a. See discussion line shows the rough average
158
curves
show
the trend
of
RADII
in text. The curve is not thevalue of 1.07 which has been
2.0
EQUIVALENT
LI
- UNIFORM
RADII
Be
R A’/3
B
1.5
.O 6
--
---
Ca
Q N
0
v
------
co 0
In Sb
----
Bi
--
Au pb Au ; i\ 1.0, 1.0,
, 3
I 2
I
, 4
dashed nuclei.
20. Equivalent horizontal line
uniform shows
radii divided by AI/z. the rough average value
TABLE SUMMARIZED
Pmnx Element
3Li6 rBeg SB” 02 TN'4 80’6 Au-Pb-Bi
(protons/10W9 cm3) f 0.006 f 0.006 i 0.005 zt 0.005 f 0.004 f. 0.003
0.79 0.84 0.94 1.01 1.045 1.03
0.067
f
1.11
0.002
theoretical. used for
The heavy
IN THE D-SHELLS
R/AI’”
b ,l$>;;,
(lo-‘3 cm)
0.064 0.086 0.086 0.080 0.087 0.079
The curve is not of 1.20 sometimes
VII
PARAMETERS
~W.WlA”‘3
:
i/3
A FIG.
I 5
(lo-l3
2.70 2.60 2.50 2.50 2.45 2.65
z!z f f z!z f f -
0.15 0.20 0.20 0.15 0.05 0.05
0 cm)
td (lo-l3
cm)
1.92 1.62 1.45 1.41 1.31 1.36
f f * f f zk
0.11 0.12 0.12 0.08 0.03 0.03
2.80 2.0 1.7 2.1 1.7 1.95
f f f f f f
0.0% 0.1 0.2 0.3 0.1 0.1
1.18
f
0.02
2.4
f
0.3
8 Note that the radii in this table are radii of charge distributions and not of proton distributions. b ~(0.04) is the radius where p = 0.040 and is determined in each case to within 2%. c R is the “equivalent uniform radius,” R = [N (r2)]““. d The “surface thickness” t is here defined as the distance from the radius for which p = 0.072 to the radius for which p = 0.008. e For Li6, the inner radius for evaluating the surface thickness is taken to be zero. 159
6
160
MEYER-BERKHOUT
scattering to charge scattering at a given momentum transfer and angle is roughly proportional to ~1’ 1Fv j”/Z” ( F, 1’) where p is the magnetic moment, 2 the nuclear charge, and F,, and F, are form factors associated with the distributions of magnetic moment and of charge (2%‘). (We here overlook the fact that different form factors may be associated with spin moment and orbital moment.) Now p2/Z2, which is 7.8 (nuclear magnetons)’ for the proton, is considerably smaller for the other elements considered here: 0.075 for Li6, 0.085 for Be’, 0.29 for B”, and 0.0033 for N14. At 420 Mev and 60”, an interesting region in these experiments, the magnetic scattering from the proton is about one-half of the charge scattering. Thus magnetic moment scattering is negligible for all lp-shell nuclei investigated so far if the magnetic form factor F,, does not exceed the charge form factor F, . However, the monopole charge scattering possesses at least one diffraction minimum (on any reasonable model), and at this minimum the magnetic form factor may in fact be large compared to the monopole charge form factor.13 Magnetic scattering may therefore be a significant contributor to the filling in of the diffraction minimum, and needs to be examined more carefully. Since the magnetic moment distribution is probably more peaked toward the edge of the nucleus than is the charge distribution, the magnetic form factor F,, should have its first minimum at a smaller momentum transfer than the first minimum of F, . In order to obtain what should be an upper limit to the magnetic scattering effect, we suppose that the secondary maximum of Fp falls at the first minimum of the monopole part of F, (the quadrupole effects will be discussed separately below). A d-function for the magnetic moment distribution,
with F, = sin(qr,,,r)/qrsUrf , has just this property provided the surface radius rsurf is suitably chosen. For reasons of simplicity, monopole charge scattering cross sections were computed by phase shift analysis for the pure oscillator charge distribution with the only free parameter rP = r8 = a0 adjusted to fit the experimental data at small momentum transfers where other contributions to the scattering are small. By comparing this cross section at its minimum with the corresponding magnetic moment scattering cross section estimated in Born approximation [with reUrf( N14) z 2.61 on the basis of the Rosenbluth equation one arrives at the conclusion that even for such an extreme model magnetic moment scattering cannot account for the observed filling-in of the diffraction minimum for N14. For B”, however, which has a rather large magnetic moment and where in addition the smaller value of 2 leads to a considerably deeper minimum in 13 The calculated
form factor at a diffraction by a phase shift analysis
minimum is to be understood and does not vanish.
as (a/u~J~‘~,
where
CT is
NUCLEAR
CHARGE
DISTRIBUTIONS
161
the predicted monopole charge scattering cross section, such a model [with rsurf(B1’) E 2.251 results in a large filling-in of the monopole charge scattering diffraction minimum, sufficient to account for the experimental observation. Less extreme models predict much less, but still appreciable magnetic scattering for B”. For example, an exponential magnetic moment distribution was considered with an rms radius chosen to be 2.22, the same as that of the oscillator charge distribution which fits the low-momentum-transfer data. This model also predicts a substantial filling-in of the monopole diffraction minimum. Similar estimates of the magnetic scattering from Be’ [with r,,,,,(Be”) E 2.11 led to the result that magnetic scattering could lead to an appreciable filling-in of the diffraction minimum (for this rather extreme model). At 300 Mev, the observed deviations from the harmonic-oscillator-model predictions (see Fig. 23) may arise partially from magnetic scattering. A more realistic model (magnetic moment distribution proportional to squared Ip wave function) predicts magnetic scattering in the interesting region about ten times smaller than does the extreme delta function model. In that case magnetic scattering would be of possible importance only in B”. For Li” results were similar as for Be’, but it should be a particularly interesting case when measurements are extended to larger momentum transfers. Because of the low 2, the Born approximation is nearly valid; and the quadrupole moment is extremely small. These two facts mean that the charge scattering should fall to a very low value at the first diffraction minimum. The small magnetic scattering may, therefore, be measurable by the extent to which it fills in the diffraction minimum. For the other elements, for reasons outlined above, it seems unhkely that anything definite can be learned about magnetic scattering unless the measurements are extended to considerably larger momentum transfers. The formal expressions from which the contributions of nuclear current and magnetization to elastic electron scattering can be evaluated for any definite nuclear model have been given by Schiff (93). We have not attempted quantitative evaluation of these expressions for the present analysis. Our crude analysis indicates, however, that magnetic moment scattering is probably negligible for N14 and small for Li6 and Beg at all angles and energies investigated so far. But for Beg and more especially for B” it may perhaps be comparable to monopole plus quadrupole scattering (see subsection D below), at the largest momentum transfers observed so far. Nuclear currents and magnetization can give rise to electric quadrupole and higher multipole effects, as well as to magnetic dipole effects. The electric quadrupole effect from this source, which flips the spin of the scattered electron ($3)) will be ignored in this work. The nuclear charge gives rise only to electric multipole effects. The quadrupole scattering from nuclear charge alone does not affect
162
MEYER-BERKHOUT
the electron spin, and to distinguish it we will use the expression no-spin-flip quadrupole scattering. It should be noted that the current- and magnetization-produced scattering can in principle be separated experimentally from the charge-produced scattering, provided dispersion effects (subsection C) are negligible. In Born approximation the charge scattering is equal to the point nucleus crosssection multiplied by a function of the momentum transfer 4 only. But the current-magnetization scattering contains additional explicit dependence upon the scattering angleI (9/t) (or equivalently upon the energy). If performed with good precision and extended over a sufficiently large angular interval, measurements on the angular dependence of the elastic cross section at two different energies but covering identical ranges of momentum transfer should make it possible to disentangle the contributions of charge scattering and of current-magnetization scattering [as has been done for the proton (4)]. The present data for B” at 300 and 420 Mev, although overlapping in the region between Q= 1.3 and q = 2.0, are hardly precise enough to permit a decision about this point. It may, therefore, be dhout significance that these data seem to favor pure charge scattering. However, the 82” point taken at 300 Mev is slightly high (Fig. 7) and, if not erroneous, may indicate some current-magnetization scattering. B. RECOIL
EFFECT
The dynamic effect of recoil has been shown to be negligible for zero spin nuclei (15). No theory of the dynamic effect has been given for nuclei of nonzero spin. The recoil current, Zev, (nuclear charge X velocity of recoiling nucleus), is of the same order-for momentum transfers of several hundred Mev/c-as the internal nuclear current. Therefore, the recoil effect might be expected to be of the same order as the magnetic effect. The kinematic effects of recoil have been included in the present analysis. The Schwartz-Gartenhaus correction (25, %‘6)15 has been omitted in considering the oscillator charge distribution. C. NUCLEAR
DISPERSION
Theoretical estimates indicate that the contribution of virtual excitation of nuclear excited states to elastic electron scattering is small. A recent theoretical treatment has been given by Schiff (27)) and calculations have been carried out by Downs (88). It has also been argued, erroneously, that the experiments support this conclusion because the same static nuclear charge distribution gives a 14 For no-spin-flip scattering, tan* (‘$9) (&/da) is a function of q only. scattering, the expression tan2 (J&Y) (da/&)/[1 + 2 tan* (‘$0) ] is a function I5 The omission of this correction means that the shell-model interpretation cillator charge distribution is somewhat in error, but it does not represent principle in the analysis.
For spin-flipof q only. of the osany error in
NUCLEAR
CHARGE
DISTRIBUTIONS
163
good fit to experimental results for the same nucleus at several different energies. The statement is true, but no conclusions about dispersion corrections can be drawn from it. For the light nuclei, one may rephrase the statement in this way: Experimental form factors, F( 4)) deduced at several different energies, lie along a single smooth curve when plotted as a function of g. There are, in fact, substantial regions of overlap in 4 of the data at different energies, but the agreement rests on suitably renormalizing the absolute cross-section data at each energy. The required renormalization is within the experimental error of the absolute measurements, which is Sor these experiments rather large. We have examined the predicted angle-dependence and q-dependence of the dispersion corrections, using some simplified approximate formulas of Downs (28)) which should be good enough for this purpose, and we conclude that quite large dispersion corrections could be present-indeed larger than is reasonable-and still go undetected because of the uncertainties in the absolute cross-section measurements. The experimental detection of the dispersion corrections will be possible only when much more accurate absolute cross-section measurements are made over the same interval of momentum transfer q at two or more different energies. Even then, the separation of dispersion effects and current-magnetization scattering will be difficult. On the other hand, the existence of the pronounced diffraction minima in the scattering from C” and 016, with apparently no significant filling-in as compared to the phase shift analysis predictions, does indicate the smallness of dispersion corrections. There is no apparent reason why the dispersion scattering should have a minimum at or very near the minimum of the static charge scattering. We conclude that the static charge distributions deduced in this analysis must be regarded as effective charge distributions which may include some nonstatic effects of nuclear polarizability, but that these effects are very likely unimportant. D.
QUADRUPOLE
MOMENT
In Born approximation the effect of a nonspherical charge distribution on electron scattering may be calculated very simply.16 It depends, as one might expect, on the spatial distribution of the source of the quadrupole moment, reducing to a simple dependence on the value of the static quadrupole moment only at low momentum transfer. We follow the analysis and notation of Schiff (25’). Define the ground state charge density p as the expectation value of 2 &( r - rl) in the M = 1 substate, and write p(r) = PO(~) + pz(r>YzO(Q + . -- . (24) 16 In this quadrupole “no-spin-flip”.
section only the charge effect is considered, i.e., the monopole plus no-spin-flip scattering, as defined in subsection A. We will generally omit the adjective
164
MEYER-BERKHOUT
Since no net charge is associated with pz , etc., the spherically still obeys the normalization condition, 4?r p&y s
dr = 2.
The quadrupole part of the charge density, quadrupole moment by
symmetric
part
(25)
p2 , is related to the spectroscopic
Q = 2(4~/5)“‘Spl(T)r~
dr.
(26)
(Note that pz (and thus Q) already include a projection factor, and vanish for I = Oort5.) For unpolarized electrons and nuclei, the scattering cross section is given in Born approximation by IJ = uo[ 1Fo I2 + 1 F2
17,
(27)
where go is the point nucleus scattering cross section (in Born approximation), F. is the usual form factor for a spherical charge distribution,
and F2 , the quadrupole
form factor, is given by
Fz(q) = z-‘(4rri’r,)“‘Sj,(4r)pi(r)r2 In these expressionsjo projection factor,
dr.
andjz are spherical Bessel functions, P, = I(21
-
1)/(1
+ 1)(21+
3).
(29)
and PI is a quadrupole (30)
It should be noticed that since the quadrupole form factor, Fz , is a function of q only, Eq. (27) implies that one can always find a spherically symmetric charge distribution which will give the same elastic scattering as the general charge distribution (24). Of course this pseudo-charge-distribution may be unreasonable in various ways. Most seriously it might not be normalized to the correct total charge; or it might be negative at some values of r; or it might show unreasonable oscillations. (The total charge condition was not a very stringent requirement in Section III because of the rather low accuracy of the absolute cross-section measurements.) At small q, Fezis given to leading order by F2 z (1/30)(5/P$‘“(Q/Z)q2.
(31)
NUCLEAR
CHARGE
165
DISTRIBUTIONS
At the large q-values of interest here, (31) is not valid and F, is model dependent. AS the simplest model, consider a deformed uniform charge distribution, with quadrupole charge density given approximately by
pz = 4~(5/47r)""(Q/R")S(r
- R),
(32)
where Q is the spectroscopic quadrupole moment, Then F$ = 4~(5/PI)““(Q/ZR”)j,(qR).
(33)
This model is too idealized to be particularly interesting. For comparisons with experiment we consider instead the following three models. (a) Undeformed p-shell. If the quadrupole moment arisesfrom protons moving in the p-shell of an undeformed potential, the radial dependence of pz is the same as that of the p-shell part of PO. For the harmonic oscillator potential the quadrupole charge density is p2(r)
= (4/37r)5-“*(
Q/ao7)r2
exp( -r2/aoZ),
(34)
*a0* exp ( - ,1/&‘ao2),
(35)
and the quadrupole form-factor is
Fz(q) = (1/30)(5/f%)"2(QlZao2)~
where we have set r, = a0 [see Eq. (3)]. (b) Deformed Oscillator Potential. The charge density in the isotropic oscillator (rp = rs .= a,) is given by pa(r)
+ 3a)][l
= [2Z/p3’*ad(2
+ a(r2/ao2)]
exp( -~*/a:),
(36)
where (Y = (2 - 2)/3. We imagine a small deformation of the potential such that the equipotential surfaces are given by r/( 1 + /~I’zo) = constant. To lowest order the quadrupole charge density may then be found by replacing r by r/( 1 + pYZo) in (36) and expanding to first order in p. The result is p2 = -@rdpO/dr, or pz(r)
= (4Qr2/3~[(5/2)a
+
1]5”“u0’][1 - a +
LYT*/UO~]
exp(-rr2/ao2).
(37)
The quadrupole moment is related to the deformation by
Q = %(5/a) The quadrupole form factor is given by
Fz(q) = ~~o(5/PI)"*(Q/Zao2)q2a02 (1 - [a/2(2 + 5n)lq*ao2} exp(-$&ao2).
(39)
166
MEYER-BERKHOUT
It should be observed that the qusdrupole contribution to the scattering [Eq. (27)] is of second order in p. The deformation gives rise also to a monopole contribution of second order in p, arising from the spherically symmetric part of the second order term in the expansion of p(r). This term represents the average smearing out of the surface due to deformation and could be present also in zerospin nuclei. (It is the term giving rise to isotope shift anomalies.) Because of our phenomenological approach, it does not seem worthwhile to take explicit account of the deformation-induced monopole contribution. (c) Shell Collective Model. The model of Ferrell and Visscher ($9) provides a theoretical justification for the classical deformed potential model discussed above, and leads to identical predictions for the quadrupole scattering. They define “collective” (quadrupole) states by a procedure similar to that described under Eq. (36). A small deformation is introduced into the shell model wave function, and the collective wave function is defined as the derivative of this function with respect to the deformation parameter at zero deformation. The collective state defined in this way consists of that particular combination of excited shell model states with maximum or near-maximum quadrupole matrix elements with the original state. The ground state is then taken to be a shell model state (full s-shell plus partially filled p-shell) plus a small admixture of the associated collective quadrupole state, plus other small admixtures which are irrelevant for calculating quadrupole effects. The charge density then becomes the diagonal matrix element of Z&(r - rl) for the original shell model state plus a cross term linear in the mixture amplitude, plus second order terms which are ignored. The zero order term gives PO + pia)Yzo , the spherically symmetric part of the charge distribution plus the quadrupole part arising from the p-shell configuration. The first order term gives #Yzo , the collective quadrupole part. In the oscillator model, pib) is the same as for model (b), Eq. (37)) and the quadrupole form factor is that given by Eq. (39).” It should be noted that the shell model function from which one starts can be an arbitrarily complicated mixture of states within the p-shell, provided they all have the same radial dependence. The charge densities (36) and (37) are dictated only by the radial functions, and all complexities of the angular functions are absorbed in the number Q. (d) Deformed p-shell. As a model intended mainly as a further test of the sensitivity of predictions to assumptions about the source of the quadrupole moment, we examine the case that the s-shell remains spherical while the p-shell is deformed. The results for this model can be obtained most simply by letting (Y --, cu in Eqs. (37) and (39). The model is not very realistic because parity17 Aside from normalization, the coefficient 56 (a) (~CX + correctly as l,&.
2)-l
the result (37) is given in Ref. (29). For oxygen, should be >iz, but is given as sic. For carbon
however, it is given
NUCLEAR
-
CHARGE
=HARMONIC SCATTERING
o,b
( PHASE ELASTIC SCATTERING
SCATTERING
167
DISTRIBUTIONS
WELL a,=
ELASTIC MONOPOLE 1.625x IO-l3 CM. a = 5/3
SHIFT NO
ANALYSIS SPIN FLIP
( BORN
ANGLE
1 QUADRUPOLE
APPROXIMATION
IN
1
DEGREES
21. Monopole and quadrupole scattering from Nl”. The monopole curve is based on the oscillator model and the three quadrupole curves are based on models (a), (b), and (c) in the text. normalized to a spectroscopic quadrupole moment of 2.0. FIG.
excitation of the 1s shell probably does not require much more energy than parit’y-conserving excitation of the lp shell for the p-shell nuclei. We expect that some combination of models (a) and (b) should be realistic for the nuclei under study. Then t’he factors Q in (35) and (39) should be Q. and Qb, the parts of Q arising from the p-shell and from excitation out of the p-shell; arkd the quadrupole form factors should be added before squaring, i.e., conserving
v-
1F,, I* + / Fia’ + F;“’ 12.
(40) Comparison with Experiment Detailed calculations with the mixed model represented by (40) have been carried out by Fallieros and Pal (SO)‘* using the basic ideas of Ref. 2.9. We thereI8 We are indebted to these workers for supplying us with their results in advance of publication., and for illuminating discussions about their work.
168
MEYER-BERKHOUT
fore present only the predictions of the three limiting models discussed above, for the elements N14, Bn, and Be’. These predictions are shown in Figs. 21-23 for each of these models (a), (b) , and (c) . Since quadrupole moments are not accurately known, the best available experimental value is chosen for each element (see discussion in Section III), and each curve plotted as though that entire quadrupole moment arose from one source: spherical p-shell, deformed nucleus, or deformed p-shell. Because the squares of form-factors add, cross sections are plotted rather than form factors. On each graph is shown also an illustrative prediction of the scattering from the spherically symmetric part of the charge in the oscillator model, with the single radial parameter r8 = rP = a~ chosen to fit the experimental data for low momentum transfers. Figure 21 shows that the filling-in of the diffraction minimum in N14 can be ac10-Z’ -=
HARMONIC ’
lO+l
SiATTEhlNG
a,b,c,
WELL
a
( PHASE
SHIFT ~0
-*a
I
-
I
lO-30
I\1
spec. = 3.55
ANALYSIS SPIN
( BORN
I
I
x IO-= I
I
I
I
70'
:; 80"
90'
MONOPOLE
,:m’
ELASTIC SCATTERING
7
ELASTIC
I+,=
1
FLIP
OUADRUPOLE
APPROXIMATION
CM? Ihl
1
aO= I 55 x Iti3 CM {I I I
t-t-l
IIlYlIIIII
z
+ z p s VI
IO
-3,
lo-3; 30° 400
50"
60'
SCATTERING
FIG. 22. Monopole sumed spectroscopic
' ANGLE
30° IN
and quadrupole scattering from quadrupole moment is 3.55.
40°
50°
60°
70'
DEGREES
B”.
See caption
of Fig.
21. The
as-
NUCLEAR
CHARGE
169
DISTRIBUTIONS
counted for by a spherical p-shell quadrupole moment of about 1.8 X 1O-26 cm”. But if the source of the quadrupole moment is deformation, a moment of 5.6 X 1O-26 cm” would be required to produce the same filling in. Sample combinations whichcouldfitthe420-Mevdataare (l&O), (1.5,1), (1.2,2), (1.0,2.6), (0,5.6), where the first number of each pair is the quadrupole moment from the p-shell, and the second number is the additional quadrupole moment from collective admixtures or deformation (the total quadrupole moment being just the sum of these two numbers). For the first contribution Visscher and Ferrell (31) give a theoreticad value of about 1.1, while the maximum possible value ($1) is about 1.8-a value which it is highly unlikely should be realized. Within the somewhat limited framework of these models, we therefore conclude tentatively that the -
=HARMONIC
SCATTERING
WELL
ELASTIC
( BORN
MONOPOLE
APPROXIMATION
30' SCATTERING
FIG. 23. Monopole sumed spectroscopic
ANGLE
and quadrupole scattering quadrupole moment is 2.0.
from
IN
40'
1
50°
60'
70°
80'
90° 100'
DEGREES
Be9. See caption
of Fig.
21. The
as-
170
MEYER-BERKHOUT
quadrupole moment of N14 is greater than 2, and possibly as great as 4, with at least half of the moment being contributed by collective effects, i.e., excitation out of the s- and p-shells. It should be borne in mind, however, that other incoherent effects may be contributing appreciably to the scattering, particularly spin-flip scattering from nuclear currents. If these are found to be important, the quadrupole moment inferred from experiment will have to be revised downward. Figure 22 shows the quadrupole scattering in B” for an adopted value of Q = 3.55. A quadrupole moment of this order of magnitude can evidently account for the large angle data, but it seems impossible at present to conclude anything quantitative about the value of Q. As with X14, the quadrupole scattering in the interesting region depends markedly on the source of the moment. In addition there is with B’l considerably more uncertainty about what to take for the monopole scattering, and the position of the diffraction minimum is less well determined. Furthermore, magnetic scattering is probably important in B”. Similar curves for Be’, with an adopted Q of 2.0, are shown in Fig. 23. No definite conclusions can be drawn, particularly since no 420-Mev data were obtained for this element, and the position of the diffraction minimum is uncertain. ACKNOWLEDGMENTS We wish to thank Professor R. Hofstadter for his continued interest in this work. One of us (U. M.-B.) is indebted to Professor Hofstadter for offering him the opportunity of a most pleasant participation in the work of the Stanford electron scattering group during the past two years. He also wishes to express his appreciation to Professor Hofstadter for many encouraging discussions during t,he execution of these experiments. We wish to express our thanks to the members of the operating crew of the Stanford accelerator, and to all members of the electron scattering group for their continued help. Mr. C. N. Davey was of great technical assistance in setting up these experiments. We are indebted to Dr. David L. Hill for many contributions to the early phases of the analysis. Mr. J. G. Wills and Mr. B. J. Hill have carried out much of the numerical analysis reported here, and have been very helpful throughout the course of this work. We thank Professor L. I. Schiff and Professor R. A. Ferrell for profitable discussions of quadrupole scattering. Mr. A. Pettinger was of great assistance in coding and running the quadrupole calculations on the Stanford IBM Computer. We wish to thank the authorities of the University of California Radiation Laboratory at Livermore and the Los Alamos Scientific Laboratory for the use of computing machines, particularly Dr. S. Fernbach at Livermore, and Dr. C. Mark at Los Alamos. RECEIVED:
February
25, 1959 REFERENCES
1. E. E. CHAMBERS AND R. HOFSTADTER, Phys. Rev. 103, 1454 (1956). 2. H. F. EHRENBERG, R. HOFSTADTER, U. MEYER-BERKHOUT, D. G. RAVENHALL, SOBOTTBA, Phys. Rev. 113, 666 (1959).
AND S. E.
NUCLEAR
CHARGE
DISTRIBUTIONS
171
3. C. II. MOAK, A. GALONSKY, R. L. TRAUGHBER, AND C. M. JONES, Phys. Rev. 110,1369 (1958). 4. R. HOFSTADTER, in “Annual Review of Nuclear Science.” Vol. 7, p. 231. Annual Reviews, Stanford, 1957. 5. J. FREGEAU AND R. HOFSTADTER, Phys. Rev. 99, 1503 (1955). 6. J. FRECCEAU, Phys. Rev. 104, 225 (1956). 7. R. HOFSTADTER AND G. BURLESON, Phys. Rev. 112, 1282 (1958). 8. J. STREIB, Phys. Rev. 100, 1797 (1955) and private communication. 9. J. HAT’I’ON, B. V. ROLLIN, AND E. F. W. SEYMOUR, Phys. Rev. 33, 672 (1951). 10. G. WESSEL, Phys. Rev. 92, 158 (1953). il. C. H. TOWNES AND B. P. DAILEY, J. Chem. Phys. 17, 782 (1949). 12. A. BASSOMPIERRE, Compt. Rend. 240, 285 (1955). IS. K. FORI) AND D. HILL, in “Annual Review of Nuclear Science,” Vol. 5, p, 25. Annual Reviews, Stanford, 1955. 14. D. R. YENNIE, D. G. RAVENHALL, AND R. N. WILSON, Phys. Rev. 96, 500 (1954). 16. FOLDY, K. W. FORD, AND YENNIE, Phys. Rev. 113, 1147 (1959). 16. K. W. FORD, B. HILL, D. HILL, AND J.G. WILLS (to be published). 17. D. L. HILL AND K. W. FORD, Phys. Rev. 94, 1617 and 1630 (1954). 18. R. HOFSTADTER, Revs. Modern Phys. 28, 214 (1956). 19. W. GR~BNER AND N. HOFREITER, “Integraltafel,” Vol. II, p. 177. Springer-Verlag, Berlin, 1950. 20. ERD&LY[, MAGNUS, OBERHETTINGER, AND TRICOMI, “Table of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1953. 21. L. R. B. ELTON, Revs, Modern Phys. 30, 557 (1958). 22. B. HAHPJ, D. G. RAVENHALL, AND R. HOFSTADTER, Phys. Rev. 101, 1131 (1956). 23. L. I. SCIIIFF, Phys. Rev. 96, 765 (1954). 24. L. J. TASSIE, Nuovo cimento Ser. 10, 6, 1497 (1957). 25. C. SCHWARTZ, private communication. 26. L. J. TASSIE AND F. C. BARKER, Phys. Rev. 111, 940 (1958). 27. L. I. SCIUFF, Nuovo cirnento 6, 1223 (1957). 28. B. W. DOWNS, Phys. Rev. 101, 820 (1956). 29. R. FERRELL AND W. VISSCHER, Phys. Rev. 104, 475 (1956). 20. S. FALLIEROS AND M. Ii. PAL (to be published). $1. W. VISS~HER AND R. FERRELL, Phys. Rev. 107, 781 (1957).
OF
PHYSICS:
Charge
8, 119-171 (1959)
Distributions
of Nuclei
of the lp
Shell*?
ULRICH MEYER-BERKHOUT~ High-Energy
Physics
Laboratory,
Stanford
University,
Stanford,
California,
AND KENNETH W. FORD§ AND ALEX E. S. GREEN// University
of California,
Los
Alamos
Scienfi$c
Laboratory,
Los
Alamos, New Mexico
New data have been obt,ained at energies of 160 to 420 Mev for the scattering of electrons by Beg, Bra, B”, N14, and O16. A detailed analysis of these elastic scattering data and of other available data for p-shell nuclei is carried through, using various methods and a wide variety of assumed functional forms of the nuclear charge distribution. It is possible to fix rather accurately the spherically symmetric part of the charge distributions of Li6, C’s, Nl4, and 016, and wit.h less accuracy, those of Be9 and B”. Cent,ral densities are uncertain in all cases. Quadrupole scattering appears to be important in N14, in Blr, and possibly in Beg, and a crude estimate of t,he quedrupole moment of N’4 is inferred from the data. The quadrupole scattering depends sensitively on the source of the quadrupole moment, as well as on its value. Regularities within the /p-shell are discussed, and connection made with the properties of heavier nuclei. Be9 and all heavier nuclei have a common value of peak particle density corresponding to a mean particle spacing of (1.13 f 0.01) X lo-r3 cm. Only Li6 fails to reach this density of saturated nuclear matter. * The part of this work carried out at Stanford was supported by the joint program of the Office of Naval Research and the U. S. Atomic Energy Commission, and by the U. S. Air Force, through the Office of Scientific Research of the Air Research and Development Command. t The part of t.his work carried out at Los Afamos was supported by the U. S. Atomic Energy Commission. $ Present address: Erstes Physikalisches Institut der Universitat, Heidelberg, Germany. $ Present address: Physics Department, Brandeis University, Waltham 54, Massachusetts, A part of t.his work was carried out at Brandeis with the support of the National Science Foundation. 11Present address: Physics Section, Convair, San Diego, California. 119
120
MEYER-BERKHOUT
TABLE NEW
EXPERIMENTS
Energy M-1
Element
I
ON NUCLEI
OF THE lp
SHELL Angles (degrees)
nBeg
160 300
33-100 33-74
8’0
160
34-130
gBll
160 300 420
33-130 34-74 34-70
TN’4
160 420
33-100 33-73
,016
160
33-100
I. UNITS
All quantities in this paper with the dimensions of length (e.g., T) will be written in units of 10P3 cm. All quantities of momentum transfer or inverse length (Q) will be written in units of 1013cm-‘. The nuclear charge density (p) will have units of (number of protons)/( lo-l3 cm)3, and the nuclear mass density units of (number of nucleons)/( lo-l3 cm)“. Quadrupole moments will have units of 1O-26cm2, but following previous practice, cross sections will be expressed in units of cm’/steradian. II.
EXPERIMENTS
The experiments whose results are discussed in this paper’ have been performed with the Stanford high-energy linear accelerator. The elements, energies, and angles for which new data have been obtained are summarized in Table I. A. APPARATUS The scattering apparatus and the method of taking data were similar to those reported in earlier papers (1, 9). Somerecent improvements, however, have not been mentioned before.’ The magnetic field of the energy deflecting magnet in the bunker was measured by a proton resonance signal. This made it possibleto reduce energy fluctuations of the incident electron beam during one run to a 1 The experiments reported here have been carried out by the first-named author, M.-B. The analysis of the data has been carried out jointly by all three authors. 2 The authors wish to thank Dr. F. Bumiller, Dr. J. Oeser, and Mr. S. E. Sobottka, were responsible for these improvements of the apparatus.
U. who
NUCLEAR
CHARGE
DISTRIBUTIONS
121
negligible amount. In addition the energy of the electrons could be reproduced accurately from run to run. Two ionization chambers, each of which was split into two subsections and positioned behind the target just in front of the Faraday cup made it possible to stabilize the horizontal and vertical beam spot position on the target. This was accomplished by feeding error signals taken from the two sections of each ionization chamber to Helmholtz deflection coils located approximately 25 feet in front of the scattering chamber along the beam pipe. A rotating coil was used to check the magnetic field in the 36-in. double focusing spectrometer at one particular point. The electron current was measured as usual by a Faraday cup and integrated by a vibrating reed electrometer. B.
RESOLUTION
The beam was analyzed magnetically so that the energy band was approximately 0.2 % wide for the new measurements on Beg, Bn, and N14 at 300 and 420 Mev. The energy slit of the 36-in. magnetic spectrometer was set to 0.2 %. This resulted in an overall resolution of about 0.35 %, depending slightly on the size and shape of the beam spot on the target. This was sufficient to resolve clearly elastic scattering from inelastic scattering events in which the lowest levels in Be’, B”, and h-l4 at 2.43, 2.14, and 2.31 Mev, respectively, were excited. No Beg measurements were performed at 420 Mev since the E2 transition from the ground state to the first excited state at 2.43 Mev turned out to be very strong at the larger scattering angles and therefore was not well resolved from groundstate scattering at this energy. No scattering from a level at about 1.8 Mev (3) could be observed in Be’. This, however, is not surprising since the 1.8-Mev level is known to be very broad. The lowest levels in B” and N14 at 2.14 and 2.31 Mev, probably reached by Ml transitions from the ground state, were barely excited in these experiments so that it was hard to detect them at all at most angles. Accordingly there were no serious resolution problems for B” and N14. The 160-Mev data on Be’, N14, and 016 were taken with an overall resolution of roughly 0.75 %. The overall angular resolution in these experiments was estimated to be about f1.2’, as is described in Ref. L. This should be sufficiently low not to smear out possible diffraction structure in the angular distribution curve of elastically scattered electrons. The electrons were detected and counted in the usual manner with a liquid fluorocarbon cerenkov counter (n = 1.276). C.
INELASTIC
SCATTERING
Estima.ted ratios of cross sections for elastic scattering to inelastic scattering from the first excited levels are given in Table II for some of the p-shell nuclei at some typical angles and energies. If no errors are given, the numbers should be considered as crude estimates only. The first excited level in B” at 0.72 Mev would of course not have been resolved at 420 Mev from ground-state scattering
TABLE
II
INELASTIC CROSS SECTION MEASUREMENTS Level (Mev)
Angle (Degrees)
(r (inelast.)
Lia a
426
2.19
40
0.6
f
25%
Li7 B
426
4.61
40
0.6
f
25%
Be9
160 300
2.43”, 2.43d
B”
160 300
420
CP
187 420
N’4
160 420
d
2.14 2.14
2.14
60 39 42 45 48 51 54 58 62 66 70
3609
b
-O.lP 0.3 0.4
f f -0.5 0.63 f 0.7 f 1.05 f 1.5 f 2.0 f 2.0 f 1.8 f
33-100 34-45 50-54 58 70 74
SO.1’ SO.06 so.1 so.13 $0.25 516
50
NO.25
58
-0.5
37 43 46 58
so.15 10.25 50.1, 50.2,
46
-1.1
0 257, 2570 20% 20% 107, 20%
207, soy, 20% probably
Ml
excit
4.46 Mev and 5.03 Mev levels unresolved 4.46 Mev and 5.03 Mev levels unresolved
4.46 Mev and 5.03 Mev levels unresolved
see Refs. 6 and 6 see Ref. 2 2.31 2.31
33-100 34-61
40.15 $0.2 -
0”
Remarks
c (elast.)
6.06) 6.14 6.91 i 7.12)
35 45 55 70 122
Ml excit. Ml excit. unresolved, excited
unresolved
f 15$?& unresolved -25 unresolved 2 & 25% unresolved
for
but strongly 0 2 40”
NUCLEAR
CHARGE
DISTRIBUTIONS
123
with our present equipment. Therefore B” measurements were restricted to 160 Mev. Comparison measurements between B” and Bn performed at 160 Mev between 33” and 130” with 0.5 % resolution revealed essentially no differences between the angular distribution curves for these two nuclei. There was some weak indication of a slightly steeper drop of the cross section as a function of angle for Bn, but this was definitely an effect not outside statistics. To the extent that such a conclusion can be drawn from measurements at rather low momentum transfers it must therefore be assumed that the radial distribution of charge is rather similar for these two isotopes. Modifications in the angular distribmion curve due to the rather different quadrupole deformations of these two stable boron isotopes are too small to be measurable at this energy except near 130” for the strongly deformed B” nucleus. D.
TARGETS
Be9 targets, known to be of high purity, 0.150-inch and O.lOO-inch thick were used. In case of the boron measurements metallic boron powder enriched to 99 % (96 %) with B” ( B1’) served as target material. The targets were diskshaped, about 2 inches in diameter and 0.75 g/cm2 thick. Two 0.0005-inch thick aluminum foil end windows, one on each side, sealed with O-rings membrane-like onto the aluminum target frame had enough tension and strength to hold the boron powder firmly in the target. Only a slight curvature of the end windows resulted but this was irrelevant owing to the fact that the beam spot was automatically kept in a fixed position on the target during each run.3 From the target dimensions one computes an Al*‘:B atomic ratio of about 1: 270, which is too 3 Absolutely flat boron targets boron powder mixed with a small very high pressure. Heating the led to quite stable and flat boron contamination and therefore was
of sufficient strength amount of hydrocarbon boron cake afterwards targets. The method finally abandoned.
were made by compressing metallic binding agent in a steel box under for several hours to about 150°C however introduced a slight carbon
Footnotes to Table II 8 G. Bltrleson, private communication. s The 0.477.Mev level in Lir was not resolved from ground state scattering. 0 At 160 Mev the cross section for inelastic scattering from the 2.43.Mev level in Be9 strongly increases with increasing angle. No scattering could be detected from a broad level at 1.8 Mev. d The broad level at 1.8 Mev cannot be resolved from the 2.43.Mev level but the energy of the inelastic peak clearly favors scattering from the 2.43.Mev level. e Whenever a - sign appears in front of the cross section ratio the value represents a crude estimate only. f Whenever the 5 sign appears the actual cross section ratio may be much smaller than the upper limit quoted in the table. g At 420 Mev and at 480 Mev, scattering from the first four excited levels between 6.06 and 7.12 Mev is considerably stronger than elastic scattering for 19 > 35”.
124
MEYER-BERKHOUT
small to give rise to any appreciable background. This was proved also by taking data with an empty but otherwise identical dummy target. Chemical contaminations of the target material can falsify the angular distribution curve of elastically scattered electrons. In particular the possibility of filling in a diffraction minimum by scattering events from contaminating elements has to be considered provided such events are unresolvable from the scattering process under study. According to the manufacturers specifications, the Bn (B”) powder used as target material was chemically at least 97 % (98 %) pure. Spectroscopic analysis confirmed that metallic impurities amounted to less than 1%. This is hardly enough to affect the angular distribution curve appreciably, even in the region where according to some nuclear models a diffraction dip should occur. Carbon, nitrogen, and oxygen which often happen to be contaminating elements in metallic boron of course escape detection by spectroscopic analysis. At the momentum resolutions employed in these experiments, nitrogen- and oxygen-scattering would have been resolved at most energies and scattering angles from boron elastic scattering and therefore could not distort the boron elastic scattering measurements. Simple kinematic considerations show for example that an electron scattered elastically at a typical angle and energy of our experiment has a momentum approximately half way between the momentum of an electron scattered elastically from 0’” and one which has excited one of the four first levels in 016 between 6.06 and 7.12 Mev. The N14 elastic scattering peak should also, if of sufficient intensity, have been resolved, at least partially, from the B” elastic scattering peak, but was not observed, whereas the cross section for scattering of the first excited level at 2.31 Mev in Xl4 is too small to be of any importance at all scattering angles and energies investigated so far. Therefore, no attempt was made to determine the oxygen and nitrogen content of the boron target material quantitatively. Carbon impurities, however, could be much more serious. Chemical analysis4 led to the result that the carbon content amounted to about 1.1% in the B” 4 As a check, the Cl* content was also determined by bombarding simultaneously a (CH2), target of known thickness and the boron target with 24-Mev electrons from the Stanford Mark II linear accelerator. After bombardment the resulting Cl’ 20.3.min b+ activity was measured by counting the 0.511 Mev annihilation radiation from both targets with two NaI scintillation spectrometers. Care was taken that target geometry, window width, etc. were approximately the same. The number of annihilation quanta counted in a fixed time interval then immediately gave the ratio of 0 nuclei and therefore Cl* nuclei per square centimeter in both targets provided geometry, absorption, efficiency, etc. were the same. Assuming this to be the case, the Cl2 content was found to be in fair agreement with the result from chemical analysis. An annihilation radiation with approximately two minutes half-life originating from the boron targets can be assigned to 0’6 produced by the 016 (7, n) 016 reaction. For reasons discussed above, however, no attempt was made to determine the 016 content by a similar method. No 10 min activity from N’3 was detectable.
NUCLEAR
CHARGE DISTRIBUTIONS
125
sample and to about 0.63 % in the B” sample. Comparison of cross sections for C” with those to be expected for boron showsthat this is too small a contamination to falsify the boron angular distribution curves of elastically scattered electrons. All measurements on K14 were made with a liquid nitrogen target of 99.9 % purity. To avoid oxygen condensation and exchange the target reservoir was filled in a nitrogen atmosphere. After filling, target and reservoir were sealed off with an toil valve. The N14target itself was about 2.5 inches in diameter and 1 g/cm” thick. The end-windows were made of O.OOl-inch nickel-plated stainless steel (leading to an Fe: K atomic ratio of about 1: 100). A slight curvature of the target end-windows was irrelevant for reasons already discussed above. The stainless steel windows introduced a small background due to electrons scattered from iron. nuclei. Therefore in the beginning of each run a few points were taken with an empty target in order to allow background subtraction. At most angles and energies the observed scattering from the unfilled target was completely negligible. In particular it turned out to be impossible to ascribe the “missing” dip in the N14420-Mev angular distribution curve to iron scattering. For CL6the same water target was used as the one described in Ref. 2. E. CROSS-SECTION
MEASUREMEXTS
Absolute cross sections were determined at one angle at each energy by comparison with free proton scattering, as is described in Ref. 2. The results of these absolute cross-sectionmeasurements are listed in Table III. It seemslikely, however, on the basis of peak height comparison between the results of different runs, that the cross sections at 160 Mev and 80” for B” and Xl4 listed in Table III are too small by a factor of two because of an erroneous reading of the current integrator scaleduring that particular run. Unfortunately, the measurement has not yet been repeated. The procedure of taking data is described in Ref. 2. Most of the angular distribution curves presented in this paper are based on measured areas under the elastic scattering peak. Whenever of sufficient importance the bremsstrahlungstraggling and the Schwinger-Suura correction were applied to the data. Usually these angular distribution curves agreed very well with those derived from peak heights. Whenever discrepancies occurred a weighted average was taken. A few points, especially in the region of large momentum transfers are based solely on peak heights. The overall accuracy of the relative cross sections is believed to be about lo-15 % for most points except those at the very largest angles. Estimated errors of the absolute cross sections are given in Table III. Even if the statistical error was smaller than 10% an error of &lo% is assigned to the relative cross section since reproducibility usually was limited to about that order of magnitude, This probably is due to drifts in various parts of the ex-
126
MEYER-BERKHOUT
TABLE SUMMARY Element
3Li6
Energy WV)
Lab. angle (degrees)
OF ABSOLUTE Absolute cross section (10-80 cm2jsterad)
III CROSS Smoothing factor%‘/fl
SECTIONS Overall
renormalizationb “THIJ
F
H
A
187 426
40
0.24
4Be9
160 300
80 54
1.04 0.34
f f
3070 25%
1.09 1.00
1.25
-1.4
1.1oe
gBll
160 300 420
80 54 43
1.36 0.65 0.26
f f f
40%‘0 30% 30%
1.82’ 1.00 1.18
0.87
0.81g
,C’2
187 420
70 40
0.0028h 0.41 f
1.00 1.16
1.13
1.11
4oy,j
,N’4
160 420
80 40
1.50 0.45
f f
30%’ 2570
1.99’ 1.00
1.21
-
80’6
160 240 360 420
80 60 55 40
2.4 1.1 0.009 0.105
f f f f
3Ooj, 5070; 40%; 20%;
1.12 1.08
1.04
1.20
none f 25%”
1.08
1.10
0.95d
1.00
1.08’
-1.2k
1.08’
8 The smoothing factor is the number by which the cross section at each energy is multiplied to produce a smooth q-F(q) versus q curve. It is arbitrarily taken as 1 for some energy. ,J means the experimental cross section; U’ is the smoothed cross section. b The overall renormalization is the factor by which the smoothed cross section is multiplied to give the theoretical cross section. F means Fourier method, H means Hermite method, and A means a typical one of the simple analytic charge density functions. Footnotes give the charge distributions used. 0 Burleson and Hofstadter, Ref. 7. d Family II, r1 = 1.75, n = 2.25. e Hollow exponential (Z e-x), (+)I’* = 2.75. f The absolute measurements for Bu and N14 at 160 Mev appear to be too small by about a factor of two, as discussed in Section II. g Family II, r1 = 1.95, n = 2.75. h Fregeau and Hofstadter, Ref. 5; Fregeau, Ref. 6. i Oscillator, (Y = $-$, rP = rs = 1.637. j Ehrenberg et al., Ref. R. k Oscillator, 01 = s, rP = rs = 1.667. 1 Oscillator, 01 = 2, rp = ra = 1.764.
NUCLEAR
CHARGE
127
DISTRIBUTIONS
perimental equipment. A tremendous improvement of the data could probably be achieved by repeating these measurements with the new multi-channel electron detection device recently put into operation by H. W. Kendall. Multichannel detection of the electrons in the image plane of the 36-in. magnetic spectrometer not only yields more precise data but in addition offers the possibility OF taking dat,a much faster and therefore of extending the measurements into the very interesting region of still larger momentum transfers. ICI.
ANALYSIS
OF DATA SYMMETRIC
IN
TERMS CHARGE
OF STATIC nENSITIES
SPHERICALLY
Rather complete elastic scattering data for electrons of 160 Mev to 120 Mev are now available for six p-shell nuclei: 3Li6, 4Beg, gB1l, ,@, 7N14, and gO16. (In addition, B1’ was studied at 160 Mev.) For the present analysis we made use of, in addition to the new data reported above, some data for Li6, Cl’, and 016 which have been previously reported and analyzed (2, 4-8). In this section the data are analyzed in terms of a static spherically symmetric charge distribution. The kinematic effects of nuclear recoil are included, but corrections due to quadrupole moment, magnetic moment, and dispersion effect are ignored. These corrections are discussed separately in Section IV, but only the quadrupole effect is examined in any detail. The approach of this section is probably valid for C” and 016 (for which quadrupole moment and magnetic moment effects vanish exactly’), and is probably also valid for Li6, for which the scattering data do not extend as far as the first diffraction minimum. On the other hand, this approach is almost certainly not valid for B” and N14, and probably not valid for Be’. For each of these three elements, a qualitative inspection of the data indicates the filling in of the first diffraction minimum and strongly suggests significant contributions to the scattering from sources other than a static spherically symmetric charge distribution. Nevertheless, an analysis first of the scattering from all six nuclei in terms of a static sp.herical charge seems worthwhile. One would like to know, for example, whether any reasonable spherical charge distribution can account for the observed scattering from N14, or whether the data demand the inclusion of a quadrupole effect. For this element, neither the value of the quadrupole moment nor the yuadrupole form factor is well known. The interpretation of the data will therefore ultimately rest heavily on a theory of the quadrupole effect. The situation is 5 The incoherent quadrupole effect, as defined in Section IV, vanishes for spinless nuclei, but deformation in a spinless nucleus could still make itself felt by altering the average radial distribution of charge. The scattering is still correctly analyzed in terms of a spherical charge distribution. This coherent quadrupole effect is important for isotope shifts and might be detected in electron scattering from two even-even isotopes which have significantly ditl’erent deformations (e.g., Srn’So and SmlS2).
128
MEYER-BERBHOUT
similar for Beg, while for B” only the quadrupole form factor is very uncertain. Similarly, although magnetic moments are well known, magnetic form factors are not. We therefore begin by ignoring altogether corrections due to nuclear moments and nuclear dispersion. Moreover, as will become clear in the discussion of the data below, even where quadrupole and/or magnetic effects are significant in the neighborhood of the diffraction minimum, a reasonably good picture of the spherically symmetric part of the charge distribution can still be obtained by omitting from the analysis the data at momentum transfers near the diffraction minimum. This procedure is unambiguous for N14,is lessclear cut for B’l, and is quite uncertain for Be’. I[It may be worthwhile to summarize at this point the previous experimental information on the spectroscopic quadrupole moment of these three nuclei. There is very little experimental information on the quadrupole moment of Be’. The value of 2 (see Section I) used in Section IV is based on an observation (9) of the quadrupole splitting of the magnetic resonance line in a single crystal of Be3AlzSi601s. The quadrupole moment derived from this observation must be regarded as highly uncertain. The quadrupole moment of B” is much better known. The hyperfine structure splitting of the 2p ‘P3,z state in free Bn has been measured with very high precision (10) using the atomic beam resonance method. From the measured hyperfine structure splitting and the known magnetic moment of the B” nucleus, a rather reliable value of Q(B”) = 3.55 can be derived, which value we adopt in Section IV. For B1’ one obtains Q(B1’) = 7.4. Although the N14 quadrupole coupling constant, eQ ~3~V/dz~, is known with good precision for several molecules containing N14, only a very crude value of the N14 quadrupole moment can be derived from such measurements. This is due to the well-known fact that the molecular electron wave functions are not well enough known to yield a reliable value of d2V/&z2. This fact is reflected in the values of the quadrupole moment derived by different authors from essentially the samemeasurements. Townes and Dailey (11) give Q( N14) = 2, while Bassompierre (12) gives a value of 0.7. For the calculations in Section IV we adopt the Townes-Dailey value, Q = 2, which may easily be wrong by a factor of two or even more.) A. METHODS
OF ANALYZING
1. Phase Shift
THE DATA
Analysis
Even for the light elements, a phase shift analysis is required for accurate predictions of the scattering near a diffraction minimum. The Born approximation is nevertheless very useful and of course simpler by far than the phase shift method. Most of our analysis has been based on the Born approximation, but a
NUCLEAR
CHARGE
129
DISTRIBUTIONS
number of phase shift calculations have been carried out where they seemed necessary, or desirable to check the Born approximation. The phase shift method requires a high-speed computer. We have fortunately had available two phase shift analysis programs, one for the Univac computer at Livermore’ and one for the IBM-704 computers at Los Alamos. The Livermore program was formulated by Yennie et al., and has been reported by them (14) .* Some details of the Los Alamos formulation have been included in the recent paper dealing with recoil effects (15), and results for heavier elements will be reported later (16). The Los Alamos program includes recoil effects, and a calculation of the mean square deviation of a theoretical curve from the data, which the Livermore program does not. On the other hand, the Los Alamos program is restricted to a class of charge distributions for which the potential can be expressed explicitly as a function of radius, while the Livermore program does not have this restriction. The following families of charge distributions have been studied with the phase shift analysis. We here omit normalization constants. (a) The Fermi two-parameter function: P-
(1 + exp [CT - r~>lal]-‘,
(1)
in which, roughly, r1 determines the size and a the surface thickness of the nucleus. (The conventional parameters, half-density radius, c, and surface thickness, t, are functions of both r1 and a.) (b) Family II: 1 _ 35 e--n(l-Z) ) P -
1,
!’ /2
-nk-1)
e
7
x
(2) 1
where x: = T/Q , and n is a surface thickness parameter. The name “Family II” stems from an earlier reference (17) where various properties of this function are discussed. The Family II function and the Fermi two-parameter function are very similar if t,he half-density radius, c, and the 90 %-IO % surface thickness, t, are adjusted to have the same values, provided t < G, but the Family II function is mathematically much simpler to work with. (c) Family III: This three-parameter function, which takes its name from 6 We are indebted to Miss Pat Curly for her help with the operation of the computer at Livermore Laboratory. 7 The Llos Alamos phase shift analysis program was originally developed by Hill, Freeman and Ford, and some results were given by Ford and Hill (IS). The program was recoded and improved by J. Wills and B. Hill. These workers have carried out most of the calculations referred to in the present paper, by the other methods discussed below as well as by the phase shift analysis method. * We are indebted to D. G. Ravenhall for making available to us this code. the
130
MEYER-BERKHOUT
Ref. 17, is equal to the family II function multiplied by sinh SX/SX, with s < n. It can exhibit central depression of charge density. (d) Harmonic oscillator function: P-
exp( -r’/r,“)
+ 4r,/r,)3(r/r,>2
exp( -r2/rp”),
(3)
where 01 = ( 56) (2 - 2). This would be the charge density of two 1s protons moving in an oscillator potential with radial parameter rs , plus (2 - 2) lp protons moving in an oscillator potential with radial parameter T, . Thus r8 > rP would correspond roughly to the charge density of 2 protons in a potential more nearly square than the oscillator potential. The further possibility of regarding (Y as a phenomenologically adjustable parameter has been considered in the present work only for Be’. 2. Born
Approhxztion
The theory of the Born approximation for the elastic scattering of electrons has been summarized by Hofstadter (A). Most of our analysis has been based on the Born approximation because of its simplicity and because it makes possible the simultaneous analysis of data at several energies. Figure 1 demonstrates the degree of validity of the Born approximation for the lightest and heaviest elements considered here, Li” and 016, up to the highest momentum transfers for which data exist. On the low side of the first diffraction minimum, the Born approximation and exact phase shift calculations are in very close agreement for 016 and for lighter elements, provided the radial constant for the Born approximation is chosen to be a few tenths of one Fercent larger than for the phase shift analysis (corresponding to the slight increase of electron momentum in the vicinity of the nucleus). At the minimum and for some distance beyond it, the Born approximation is inaccurate. However, since one knows in what way the Born approximation fails, and knows that it)s error is not sensitive to the particular choice of charge distribution function, it can be usefully applied to all of the elements studied here. In particular, it can be used to discard unacceptable charge distributions. The “Fourier method” and the “Hermite method” discussedbelow make use of the Born approximation. In addition, we have used the more standard approach of assuming some reasonably simple functional form for the charge distribution and with it calculating the predicted scattering to compare with experiment. In the Born approximation, as well as in the phase shift method, we have used the Family II function, Eq. (2)) and the harmonic oscillator function, Eq. (3). In addition, Born approximation calculations have been made with the following charge distribution functions. (a) The Slater function: p N Csf-ns~--k8r + CprnP+‘,
(4)
NUCLEAR
CHARGE
DISTRIBUTIONS
131
with C, and C, being normalized to yield the number of s-protons for the volume integral of the first term and the number of p-protons for the second term. The two terms can then be readily interpreted in terms of simple approximate s- and p-wave radial functions, but in practice this Slater function did not prove very successful in fitting the data. (b) The one-parameter functions: These are the group of simple functions listed by Hofstsdter (18). All of the above functions were tried over a wide range of their adjustable parameters for all of the data, and surprisingly few gave satisfactory fits at all energies, Most of the one-parameter functions, for example, gave no satisfactory fit to ang element. In this connection, it is significant that the lower energy data proved almost equally as valuable as the higher energy data in restricting the set of acceptable charge distributions. The data at several energies taken together provided a much greater restriction on the charge distribution than did the 420-Mev data alone. (Of course 420-Mev data at more forward angles would have been equally valuable, but since it was not obtainable at the time these experiments were performed, lower energy data was needed to supply the knowledge about the scattering at low momentum transfers.) Kinematic recoil corrections were included in the Born approximation calculation,s. 3. Fourier
Method
Since the Born approximation is rather accurate for the light nuclei, one might consider the possibility of working backward directly from the experimentally determined form factor, F(q), to the charge distribution, p(r). The quantity rp(r) is just the Fourier sine transform of @‘(cl) :
v(r)=
(27w’~
s,-qF(q)
sin pr dq,
where we choosethe normalizations F( 0) = 1, and 4?rJ pr2dr = 2. The straightforward determination of p from (5) is of course not possible because F( 4) is known only over a limited range of 4. Nevertheless one might ask whether the unknown parts of F(n) are sufficiently unimportant that the charge distribution can be approximately inferred directly from the data (presented as a form factor). In this subsection and the next, we discuss two methods of analysis which are closer in spirit to this approach than to the usual trial and error approach. These methods do not presuppose a simple analytical charge distribution, and in the limit that F(q) is known everywhere they could be made equivalent to carrying out the IFourier transformation (5). We note that in (5) q is the momentum transfer in the center-of-momentum frame, and / F(q) 1 = (~/a~)“~, where ~0 is the point-nucleus cross section in Born approximation, and v is the measured cross section translated to the center-
132
MEYER-BERKHOUT 1O-2g
-
I
I
I
I
I
I
Li6 426 MEV
1o-34 30
40
50 SCATTERING
FIG.
60
ANGLE
la
FIG. 1. Comparison of Born approximation and phase shift analysis. (a) Scattering of 426.Mev electrons by Lie. Both curves are calculated for Family II charge distribution with n = 2.25, T, = 1.75. Most of the discrepancy can be removed by choosing in the Born approximation a slightly larger wave number, or equivalently a larger nuclear radius constant. The difference is, however, less than the uncertainty in the determination of rt , and therefore not taken into account in the subsequent analysis. (b) Scattering of 420.Mev electrons by O16. Both curves are calculated for the oscillator charge distribution with a = 2, r, = pP = 1.754. If the radial constant for the Born approximation is chosen to be 0.3% larger, the small discrepancies below and well above the diffraction minimum are removed. (All of the curves in this figure were calculated without recoil corrections.)
NUCLEAR
CHARGE
133
DISTRIBUTIONS
016420
MEV
8f3 SCATTERING FIG.
ANGLE
IN
DEGREES
lb
of-momentum frame. In the extreme relativistic limit, (co2 J.;O/sin” $50))
go = (Ze”/2E)”
(6)
where the angle 0 and the energy E are in the center-of-momentum frame: E tan
= 0 =
(1 (1
-
6)&b, +
E set
(7) h,)
tan
fhab
,
134
MEYER-BERKHOUT
expressions correct to first order in h = (electron energy)/( nuclear mass). See also Eq. (48) in Ref. 4. (In the actual calculations, more accurate expressions were used. ) The inverse form to (5) is qF( q) = 4~2~’ lm rp( r) sin qr dr , We find it convenient to plot this quantity, $‘(a), rather than F( 4) itself, as a function of 4. Observe that for small q, @(q) = 4. This fact permits a reliable extrapolation of the experimental qF(q) downward to q = 0. (The lowest order deviation of qF(q) from a straight line measures (1’). The lower energy data is therefore valuable in helping to fix the rms radius.) At large q, only weak conditions of physical reasonablenesslimit the behavior of qF(q)-e.g., that p(r) should be everywhere positive, or that p(r) E 0 beyond some distance b. We take advantage of the second assumption, assuming p = 0 for r > b, and write rp(r) as a Fourier series, rp( r) = c C, sin( n?rr/b) ,
r 5 b.
9L=l
(9)
From (8) and (9) it follows at once that the coefficients C, are given by, C, = (ZPd)qnF(qn),
(10)
q,, = m/b.
(11)
where
Accordingly, once a cutoff radius is assumed, one may determine the Fourier components of rp( r) simply by reading values of qF( n) from a graph of experimental data at the qn values given by ( 11)-provided no qn value lies too close to a diffraction minimum. We have applied this method for cutoff radii b = 5 and b = 6. As will be evident in later figures, the charge densities inferred are not very sensitive to the choice of b (except near r = 0, where all methods fail to determine the charge density with any accuracy). The number of Fourier coefficients which can be read from the data are four in most cases.Higher coefficients are arbitrarily set equal to zero. In terms of the Fourier coefficients, the normalization conditions becomes 2 = 4b2c ( - 1) “+lCn/n. Because of the decline of C, with n and because of the n in the denominator, it is reasonable to assume that the unknown higher coefficients contribute very little to the normalization sum. With this assumption, (12) provides a convenient way to normalize the experimental data, whose absolute value is not
NUCLEAR
CHARGE
DISTRIBUTIONS
135
well known. We scale the whole qF(q) curve up or down by whatever factor is required to satisfy ( 12). This factor has in each case been well within the experimental absolute error, and is nearly the same for b = 5 and b = 6. This procedure is required for self consistency but may have extra value in fixing the absolute cross sections to better accuracy than the present measurements. The mean square radius for the Fourier charge distribution is given by (T’) = b2 -
(24b4/7r22)
c
( -1)““Cn/n3.
The terms in this sum diminish rapidly, and the mean square radius is determined mainly by the first coefficient. This coefficient falls near the maximum of the qF(q) curve and is therefore determined by low energy or small angle data. To illustrate the application of the Fourier series method, we show in Fig. 2 the experimental values of pF(q) for 0”. The vertical lines show the values of q,&and q,F(p,) based upon the assumed cutoff radius b = 5. The importance of the low-energy data should become immediately clear from this diagram. The S60- and .i20-Mev data fall at too large 4 values and hence fail to fix C1 . On the other hand, the 240- and 160-Mev data proscribe the value of C1 quite sharply and hence determine the major component of the charge distribution. The experimental study was extended to 160 Mev in the light of this observation. The components of rp(r) are shown in Pig. :3 along with the Fourier sum based upon the assumed cutoff radius b = 5. Also shown on this diagram is the charge distribution inferred by the Fourier series method using b = 6 as well as the distributison obtained on the basis of the harmonic oscillator model. The agreement between the t,hree curves is gratifying. The experimental data for 0’” does not go to sufficiently high energies to determine the fifth Fourier coefficient when b = 5. The uncertainty in this coefficient reflects itself in a corresponding uncertainty as to the shape of the charge distribution. Reasonable extrapolations suggest that C, = 0 f 0.005. This corresponds to an uncertainty of about 1% in the normalization condition (12). However, the uncert’ainty in the fifth component has a rather appreciable effect upon the central charge density. At the center each component contributes /h(O)
= c, m/b.
(14)
The factor n in the numerator amplifies the influence of the higher components upon the central density. The uncertainty in central density associated with the uncertainty in the fifth component in the case of 016 is indicated on Fig. 17. This is the most favorable of the six elements, however, with relatively small latitude in the choice of the fifth component. In other cases the central density is much more poorly determined. While the uncertainty as to the magnitude of qF(q) beyond the range of measurement reflects itself in a small uncertainty in the inferred distribution
136
MEYER-BERKHOUT
v n
A l
160 240 360 420
MEV MEV MEV MEV
DATA DATA OATA DATA
0.2
0.1 l-
L 0
2
q FIG. 2. Form factor for 016, showing experimental points with statistical errors only. The theoretical curves are for the oscillator model and for the Fourier model with b = 6. The oscillator curve has the parameters of the “pure oscillator”-i.e., rs = rP ,01 = Z-from Table IV. The modified oscillator listed in Table IV (from Ref. 2) gives a better fit to the data. The Fourier form factor with b = 5 is scarcely distinguishable from that for b = 6, and is therefore also an excellent fit. The vertical lines designate the values of q,, used in the Fourier method with b = 5. The guessed limits on qF(q) for the fifth Fourier coefficient
are also shown. functions, the uncertainty as to the sign of the Fourier coefficients reflects itself in possible gross changes in the charge distribution function. In the case of 016 the success of the harmonic oscillator model for which the qF(q) function goes negative serves as a guide to infer that the actual qF(g) functions follow the same pattern. If, however, we were to make the assumption that qF(g) merely dips to near zero and then rises again, i.e., the third and fourth Fourier coef-
NUCLEAR
0.14
I
CHARGE
I
137
DISTRIBUTIONS
I
I
I
I
0.12
rp(r)
0.10
I-
0.00
I-
0.06
,-
0.04
0.02
0
(
-0.02
-0.04
- 0.06
0
I
I
I
I
I
I
I
e
3
4
5
6
7
r FIG. 3. T p(r) for 016. Fs and Fe designate Fourier method b = 6. S. H. 0. designates the oscillator model. H designates shown are the individual Fourier components for 6 = 5.
with the
cutoff radii b = 5 and Hermite method. Also
ficients are taken positive, then these Fourier components would contribute a peak at the center of the nucleus rather than a hollow. The inferred distributions would then have a very considerably different character from the ones shown in Figs. 3 and 17. The assumption that F(q) is everywhere positive in the experimentally covered
138
MEYER-BERKHOUT
region is of course not very sensible for Cl2 and 0’” which exhibit obvious diffraction minima. But B” and N14 show barely discernible minima and one must inquire more seriously about the sign of F (4) at the largest measured momentum transfers in these cases. Interpolation and extrapolation from Cl2 and 016 suggest the negative sign. The negative sign is also strongly suggested by the fact that if F(p) is assumed to remain positive, the peak charge density inferred is nearly twice the largest charge density deduced for any other nucleus, light or heavy. Figure 4 compares theoretical qF(q) curves with the N14 data for the two assumptions that F(p) remains positive and that F(q) goes negative. Both 0.5
1
7N’4 . .’
420 MEV ABSOLUTE MEV
0.4
I
2
DATA, NORMALIZE0 MEASUREMENT DATA,
TO
X I.41 b X 1.99)
3
9 FIG. 4. Form factor for Nr4, comparing theoretical curves with experimental points. Fb(-) labels the approximate fit obtained with the Fourier method, b = 5, assuming a change in sign of F(q). Fr,(+) labels the attempted fit obtained with the same method but assuming F(q) > 0 over the whole experimental range. The R’S(-) curve is not a very good fit and the FCC+) curve is a poor fit. Also shown is the best fit with the oscillator model.
NUCLEAR
CHARGE
139
DISTRIBUTIONS
theoreticad curves of course pass through the experimental pnF(qn) points, but one must look between these points as well to see if the Fourier analysis is providing everywhere a good fit to the data. (See also Fig. 7, for Bn, where the fourier method gives a better fit.) We conclude for N14 that a diffraction minimum should be present but is being filled in by other effects. The spherically symmetric part of the charge distribution can nevertheless be determined reasonably accurately if it is assumed that F(q) becomes negative at large 4. Hermite Method Another series expansion technique has been tried which does not require a sharp radial cutoff in the charge density. Assume that v-p(r) may be expanded in terms of N functions fi(r) according to
4.
rp(?“) = ci=o Cifi(T1.
(15)
Choose any N values of Q where data exist and read the corresponding qjF(qj) from a graph of the experimental data. Then qjF(qj)
= C MiXi
values
7
(16)
sin qjr dr.
(17)
where Mii = 4~25
I* fi(r)
By solving the N simultaneous equations (16) with the N2 known coefficients (17), one may find the N coefficients Ci . The normalization condition requires
(18) and, as in the Fourier method, may be used to normalize the absolute values of the data. (The renormalizations required in the two methods were found to be nearly the same.) The success of this expansion technique will depend upon the choice of functions. The approximate validity of the harmonic oscillator model for light nuclei suggests that the fi might be chosen to be simple polynomials multiplied by a gaussian weighting factor. We find a particular generalization of the Hermite polynomials convenient. A generalized Hermite polynomial is (19) H,(z,v) with
= (-l)“e”l‘“(d/d~)“e-““‘,
v >, 0 and n = 0, 1, 2, . . . . A useful integral
(19) for determining
radial mo-
140
MEYER-BERKHOUT
ments is m e-“r2xmHn(3,~) dx = s-w
0, for m < n, and for m - n = 2~ + 1 ’ m! (?r/~)~‘~/p!(4~)!, for m - 72= 2~
(20)
where p is any positive integer or zero. The usual Hermite polynomial (arising, for example, in the one-dimensional oscillator problem) is obtained by setting Y = 1. Instead we choose the set of functions with Y = 34: f;(r)
= 2(27r)-3’2a2H2i+1(ar,f~)e-1’2a2r2.
(21)
These lead to the matrix coefficients (20) M;j = (a/2)(
-l)i(qj/a)2i+1
exp[->$(qi/a)“].
(22)
Equation (20) implies that only the first function contributes to the normalization, and the constant in the definition (21) has been so chosen that Co = Z. Only the first two functions contribute to the mean square radius, which is given by (r2) = 3(Co + 2C1)/a2Z.
(23)
To apply this expansion technique, one must first guessthe value of a. For this guessthe simultaneous equations (16) are solved using the M;j given by (22). If the first coefficient Codoes not equal Z, a new CYis chosen and the process repeated.g In several cases, the normalization could not be achieved with any value of CLThen the absolute value of the experimental data was renormalized (always within the experimental uncertainty) until a properly normalized solution was obtained. The charge distributions inferred by this Hermite method for N 5 4 are shown in some of the following figures. B. ANALYSIS
OF DATA
The experimental data reduced to center-of-momentum form factors are shown for 016 in Fig. 2, for N14in Fig. 4, and for the other four elements in Figs. 5-8. Also shown in these figures are sample theoretical form factors. The data at different energies for the same element are adjusted as shown in Table III in order to cause the qF(p) values to lie along a single smooth curve as a function of p. The analysis of the data is discussedbelow separately for each element. The consequencesof the sign ambiguity of F(q) are also discussed.Finally, regularities noted within the p-shell are summarized. Table IV gives the charge distribution functions which provide good fits to the data. To be acceptable, the charge distribution function was required to pro9 An iterative method using an IBM 704 code by Mr. Benny Hill was used to solve the simultaneous equations and to converge upon the best value of LY. Hand calculation is also feasible, however, if the qi values are chosen to be equally spaced.
NUCLEAR
I 0
CHARGE
I I
141
DISTRIBUTIONS
l
187
MEV
OATA
A
426
MEV
DATA
1 3
4
FIG. 5. Form factor for Li6, experimental points and some theoretical curves. In this figure and in Figs. 2, 4, 6, 7, and 8, statistical errors only are shown, the data being renormalized as indicated in Table III. For good agreement with each theoretical curve, some further overall renormalization may be required, so that a good fit may not at first glance appear very good in these figures. The illustrated curves all provide good fits, except in Fig. 4. The following notation is used: Fb , Fourier method with cutoff radius b, Eq. (9); H, Hermite method, Eqs. (15) and (21); superscript (+), F(q) assumed to remain positive; superscript (-) or no superscript, F(g) assumed to change sign; S. H. O., oscillator model, Eq. (3) ; Slater, Eq. (4) ; II, Family II, Eq. (2) ; Fermi, Eq. (1) ; 5 and 6, following the numbering in ‘Ref. 18, are the forms 5eeZ and (1 + z)ez, respectively. Form factor and momentum transfer are in the center-of-momentum system.
vide a good relative fit to the data at all energies, and to predict an absolute cross section within experimental error at each energy.” For C” and 016 at all 10 As indicated in Section II, the absolute cross-section measurements at 160 Mev are believed to contain an error of a factor of 2. The theoretical
for Bii and Nr4 absolute cross
142
MEYER-BERKHOUT
IqF(q)l
FIG.
6. Form
factor
for
BeQ. See caption
of Fig.
5
energies and angles (phase shift analysis where necessary), and for the other elements below the first diffraction minimum, a good relative fit was required to have a weighted mean square deviation of theory from experiment less than 1.5 times the statistical mean square deviation [definitions and formulas given by Ford and Hill (IS)]. The Fermi two-parameter and some shell model phase shift analysis fits were judged by eye. For the elements which indicate some filling in of the diffraction minimum (Be, B, and N), no objective criterion for a good fit could be employed, and theoretical and experimental curves were compared by eye, the angular region of supposed filling in being overlooked. The Fourier fits to N14 were not very good (see Fig. 4) but are included in Table IV for illustrasections were not required times these measurements).
to agree
with
these
two
measurements
(but
do agree
with
0.5
-. r
0.5
NUCLEAR
CHARGE
143
DISTRIBUTIONS
56” . A m
160 300 420
MEV MEV MEV
DATA DATA DATA
IqW\
0.2 6
0.1
0
: K4 FIG.
I 7.
2
Form factor for B”. See caption of Fig. 5
tive purposes. This makes the results more unreliable, but still permits a reasonably good definition of the charge density. As already emphasized by other authors (2, 4-r)) the absolute cross sections, even though not very accurate, also provide a valuable constraint on the charge density. The still quite appreciable uncertainty in the rms radii (see Li6 discussion below) arises from uncertainty in the absolute cross sections at low q-value. The charge distribution coefficients obtained by the Fourier and Hermite methods are given in Tables V and VI. 3Lia. Because of the validity of the Born approximation, the quantity rp(r) seems to be mathematically the most meaningful thing to present, and we therefore present a few curves of this type. However, physically, one is more interested in p(r) itself, and we present curves of p versus r for all elements.
144
MEYER-BERKHOUT
0.5
I
I
I
I
p
. 187 Q 420
MEV MEV
DATA DATA
0.2
0.
( FIG.
8.
Form
factor for CP. See caption of Fig. 5
Figure 9 shows rp(r) deduced by the Hermite and Fourier methods for Li6. The three curves are in remarkably close agreement. In particular, the results of the Fourier method are obviously insensitive to the choice of cutoff distance b, within reasonable limits. The Hermite method also implies a very rapid cutoff because of its gaussian factor. Note in Table III that these three curves yield rms radii of 2.5 to 2.65, while the other acceptable charge distributions have rms radii around 2.8, owing to the contributions of their longer tails. It seems reasonable that the higher figure is more nearly correct, since such a light nucleus
NUCLEAR
CHARGE
DISTRIBUTIONS
145
might be expected to have a rather diffuse surface. (For 016, where the oscillator model is successful, this kind of discrepancy among the rms radii does not exist.) However, it should be emphasized that the rms radius is not well determined. Its value can be made definite only by a rather accurate measurement of the absolute cross section for a momentum transfer around Q = 0.8. (Observe in Table III that the required renormalizations in Li6 are significantly different for the different methods of analysis.) Figure 10 shows most of the acceptable charge densities for Li6 (see also Ref. 7). This figure emphasizes the fact that the central density of charge is completely uncertain,, Beyond r M 1, however, the charge density is rather well defined. Also one rnay conclude that the maximum charge density in Li6 is pmax = 0.064 f 0.006. 4Beg. Figure 11 shows most of the acceptable charge densities in Be’, showing again clea,rly the lack of knowledge of charge density for r 5 1. For this particular element, the Fourier methods for cutoff radii b = 5 and b = 6 differed by the greatest amount, and only b = 5 gave an acceptable fit. This comes about because even for b = 6, one can read from the data only three Fourier coefficients, which is insufficient for a cutoff radius of 6. One may say, for b = 6, that the Fourier method is trying, with inadequate means at its disposal, to produce a smooth but very asymmetric curve in the interval 0 5 r 5 6. Since the oscillator charge distribution is successful for the heavier elements in this group, we remark that attempts were made to fit the Beg data with the charge density function (3)) holding r8 = r, and varying a! and rs , as well as fixing Q!and adjusting rs and rP . No combination of parameters gave a successful fit. The theoretical curves fell more rapidly than the experimental data at the largest momentum transfers. After the quadrupole scattering is included (Section IV) the oscillator charge distribution can no longer be ruled out. From Fig. 11 we conclude that the maximum charge density in Beg is pmax = 0.086 f 0.006. sB’l. The B” data presented in Fig. 7 suggest that quadrupole and/or magnetic scattering are coming in at large momentum transfers to distort the @‘(q) curve. One of the theoretical curves shown in Fig. 7 accordingly is presented without any effort to fit the data at large y. Three “acceptable” charge densities based on the same approach are shown in Fig. 12. They provide good fits to the data except for y 2 1.8. They agree closely with each other for r 2 2, and they imply for B” a maximum charge density pmax + 0.086 f 0.005. (The Fermi function listed in Table IV fits the data only for 4 5 1.7). The results of assuming instead that all of the scattering up to the largest angles observed comes from a static spherically symmetric charge distribution are shown in Fig. 13. None of the assumed simple functional forms of p(r) could provide a fit to all the data but the Fourier and Hermite methods could be forced to fit
146
MEYER-BERKHOUT
TABLE
IV
ACCEPTABLE CHARGE DENSITY FUNCTIONS FOR ~-SHELL NUCLEI" (r2)1/2p
Parametersb
Element ~~~
Familyb
3Li6 c
II xecx d Fourier Fourier Hermite Fermi Oscillatorc
b b (Y r1 a
= = = = =
5, 3 terms 6, 4 terms 0.732, 4 terms 1.20, a = 0.70 55, rs = 2.65, rp = 1.07
2.84 2.80 2.49 2.53 2.65 2.75 2.82
xecx Slater Fourier Fermig Oscillatore
n, b r1 a
= = S =
4, np = 6, k, = 2.80, 5, 3 terms 0.90, a ES 0.79 35, r8 = rp = 1.609
2.75 2.68 2.51 2.999 2.26~
rBe9f
aC’2
,N14 ‘. g
SO’6 In
~1 = 1.75,
n = 2.25
IIK IIa Fermig (1 + x)epx g Fourier(+) i Fourier(+) i Hermite r Hermiteg Oscillatorg
b b (Y 01 (Y
= = = = =
5, 4 terms (better fit than 6, 4 terms 0.92, 4 terms 0.92, 4 terms 1, Ts = rp = 1.55g
II Oscillator Oscillatork Fermi Fourier Fourier Hermite
t-1 = a = 01 = 1-1 = b = b = (Y =
2.31, n = 3.5 x, rp = rs = 1.637; 1.12, rp = rs = 1.71 (better 2.24, a = 0.50 5, 4 terms 6, 5 terms 0.92, 5 terms
II Fermi Oscillator Fourier(+) Fourier Hermite
TI TI (Y b b (Y
= E = = = =
2.50, n = 5.0 2.30, a c% 0.50 54, r,, = rs = 1.667 5, 4 terms (not a good fit) 5, 4 terms (not an excellent 1.08, 4 terms
a a b b 01
= = = = =
2, rp = rs = 1.764” 1.60, rp = ~8 = 1.82 (better 5, 4 terms 6,5terms 0.82, 5 terms
Oscillator Oscillator0 Fourier Fourier Hermite
i
TI = 1.85, n = 2.50 ~1 = 1.95, n = 2.75 7-1 E 2.00, a G 0.55
k, = 2.30
(better
fit than
line
fit
line
2.769 2.709 2.569 2.508 2.34’ 2.35’ 2.36’ 2.418 2.258
above)
below)
than
line
above)
2.45 2.57 2.46 2.57 2.43 2.40
fit)
fit than
2.72 2.41i 2.50 2.54 2.41 2.36 2.47
line
above)
2.65m 2.70 2.66 2.62 2.64
NUCLEAR
CHARGE
DISTRIBUTIONS
147
the data fairly well (see Fig. 7). These methods of analysis are consistent with each other and give charge distributions very different from those shown in Fig. 12. There are, however, several reasons for not believing these results. (1) The maximum charge density, pinax = 0.15 f 0.03, is much larger than found in any other nucleus. (2) All three curves have negative charge density at large r. This is a very small effect, and possibly not significant. (3) The curves have an “unusual” shape. It seems unlikely that any reasonably shaped average potential would lead to radial wave functions for the protons to give these charge densities. This idea has not been put to a yuantitiative test. (4) Theory. Quadrupole scattering from Bn ought to be significant, and these curves assume it is not present. &“. Most of the acceptable charge density functions for C’* are shown in Fig. 14. This nucleus and 01” appear to have the best defined charge densities. Even here, however, it is clear that the central density is not well determined, and it cannot even be stated whether there is any central depression of charge as the shell model predicts. Also, as indicated in Table IV, the rms radius is not well determined. The rapid-cutoff models (oscillator, Hermite, and Fourier) have rms radii of 2.36 to 2.50, while the long-tailed models (Fermi and Family II) have rms radii of 2.54 and 2.72. The latter provide slightly less satisfactory fits Footnotes
to Table
IV
a Most functions and methods listed in this paper were tried for each element. Exceptions are given in footnotes below. b Charge density families and notation for parameters are defined as follows: Fermi, Eq. (1) ; II, Eq. (2) ; Oscillator, Eq. (3) ; Slater, Eq. (4) ; F ourier, Eq. (9) ; Hermite, Eqs. (15) and (21). The one-parameter families indicated explicitly are completely characterized by the rms radius. c Family III not tried for Lie. d Same result also obtained in Ref. 7. * Result quoted from Ref. 7. r Family III not tried for Beg. Hermite method indicated data required renormalizing, but Hermite analysis not completed. g Good fit only when sufficient quadrupole scattering included. ‘1 Family III not tried for B”. Fourier method with assumed diffraction minimum not tried. 1 Superscript (+) means that it has been arbitrarily assumed that F(q) remains positive, an effort being made to force a fit to the data ignoring the probable presence of a filled-in diffraction minimum. j These numbers are in agreement with those of Ref. 2, where for the oscillator model with 01 = 96, the best parameters were found to be rP = r, = 1.65, (r2)i1* = 2.42. k Result quoted from Ref. 2. 1 Family III not tried for Ni4. m Families II and III not tried for 016. n Same result obtained in Ref. 2. 0 Result quoted from Ref. 2. p Radii in this table are actual charge radii and not the radii of proton distributions (see Ref. 2).
148
MEYER-BEREHOUT
TABLE FOURIER
COEFFICIENTS
Element
b
OF CHARGE 100 Ci
V DENSITIES
FOR ~-SHELL
100 CP
100 CB
3Li6
5
4.002
2.48,
0.726
nBe9 gB’i
6 5 5(+)
3.117 5.312 6.945 5.356 8.142 6.42 9.40 8.99 9.904 8.066
2.666 3.388 4.30s 4.509 4.338 5.43 4.57 4.34 3.344 5.484
1.154 1.14s 1.14 1.67 -4.044 1.145 -1.08 1.02 -1.12 0.21,
6(+, 5 6 5 5(f) 5 6
,CP ,N’4 80’6
NUCLEIC
100 Ca
100 c5 -
0.336 0.70 0.735 -0.665 -0.682 -0.65 0.65 -0.549 -0.893
-
-0.30s
8 The coefficients C, are defined by Eqs. (9) and (lo), and the data have been renormaliaed as indicated in Table III in order to satisfy the normalization condition (12). The units of C, are 10Z6 cmm2. The superscript (+) means that the signs of the higher coefficients have been taken to be positive, although a negative sign is more likely correct.
HERMITE Element 3Li6 gBil ,C’2 TN’4 ,016
(Y 0.73 0.92(+) 0.92 0.92 1.08 0.82
COEFFICIENTS
CO 3.0 5.0 5.0 6.0 7.0 8.0
& The coefficients Ci are defined renormalized as indicated in Table 2. The Cc are dimensionless. The IV and V.
TABLE VI FOR SOME Cl 0.201 1.23 1.59 1.66 4.33 2.10
~-SHELL cz
0.0340 -0.0132 0.292 -0.055, 0.665 -0.0392
NUCLEI” c1 -5.0 x -0.0248 0.0376 -0.0276 0.0271 -0.015,
CC lo-4
-0.0017 -0.0009
by Eqs. (15), (16), (17), and (21). The data have been III in order to satisfy the normalization condition C, = superscript (+) has the same significance as in Tables
than the rapid-cutoff models,ll but cannot be ruled out at this time. From Fig. 14 we conclude that the maximum charge density for Cl2 is pmsx = 0.080 f 0.005. 7N14.Shown in Fig. 15 are three acceptable functions ~-p(r) for N14 (still ignor11 The Mev data this model above the from the p-values.
Fermi function giving a nearly perfect fit to the 420-Mev C’s data also fits the 187. up to angles of about 100”. At still larger angles the angular distribution curve for drops too fast and Fregeau’s last experimental points lie up to a factor of two theoretical curve. The only points in the 420-Mev data which deviate somewhat theoretical angular distribution curve in the same direction occur at just these
NUCLEAR
CHARGE
149
DISTRIBUTIONS
0.07
0.06
0.05
0.04
rp(r1 0.03
0.02
0.01
I
0
2
3
4
5
6
7
in caption
of Fig.
r FIG.
9. r p(r)
for
Li6.
Notation
for
this
and
succeeding
figures
5
ing quadrupole effects) for which the form factor passes through zero, and one function for which the form factor is assumed to remain positive. The form factors corresponding to the curves labelled Fi+’ and F:-’ are shown in Fig. 4. The F’+’ curve fits the data less well, but it is interesting to see the difference in the charge distributions based on these two assumptions. The charge density itself is shown in Fig. 16 for each of these four models. The three “minus” curves are very similar to each other and differ markedly from the “plus” curve-in the same qualitative way as for B” (Figs. 12 and 13). There is strong reason to believe that the “minus” curves are correct, and the “plus” curves incorrect, based on the fits to the data, Fig. 4; on analogy with neighboring nuclei Cl2 and O16; and on the theory of the quadrupole effect, discussed below. Figure 16 implies for N14 pmaa = 0.087 f 0.004. 8016. Several acceptable charge density functions for 016 are shown in Fig. 17. The density for r 2 2 is determined to the highest accuracy in this case. Observe in Table VI that for 016 the Hermite coefficients Ci for i 2 2 are very small. If
150
MEYER-BERKHOUT
r
1 P
I ’
II F
0.06
t g(r)
0.04
0.03
Q02
0.0 I
L
0
I
2
3
4
5
6
7
r FIG.
10. Charge
density
p(r)
for
Li6.
For
notation,
see caption
of Fig.
5
they are ignored relative to COand C1 , then the charge density from the Hermite method is exactly eyuivalent to that from the oscillator model with suitably adjusted parameters. The leading two terms in the Hermite function shown in Fig. 17 correspond to an oscillator with (Y = 2.48 and r, = rs = 1.73. For 016, fits with the Family II function were not attempted, but attempted fits with the similar Fermi two-parameter function were not successful. The parameters which gave a fit at all but the largest observed momentum transfers were ~1 = 2.57, a = 0.505. The theoretical cross section at 420 Mev in this case fell below the experimental points in the angular region 65 to 75 degrees by up to a factor of two, and predicted a second diffraction minimum between 80 and 85 degrees. (The oscillator model predicts no second minimum.) From Fig. 17 the maximum charge density in 016 is determined to be pmax = 0.079 f 0.003.
NUCLEAR 0.N
CHARGE
I
I
151
DISTRIBUTIONS
I
I
I
I
0.0:
O.OI
0.0;
o.oz
p(r)
O.Ot
0.04
0.03
0.02
0.0 I
0
I I
I
2
4
3
I
I
5
6
7
r FIG.
11. Charge
density
p(r)
for
Be 9. For
notation,
see caption
of Fig.
5
Regularities in the p-Shell Parameters which have become standard for describing the charge distributions of heavy elements are the half-density radius, c, and the 90 %-lo % surface thickness, t. These are not well defined for the light elements under study. But there is a convenient way to define substitutes for these quantities based on the near constancy of the peak density of nuclear matter (21) .
152
MEYER-BERKHOUT
0.14
0.08 p(r) 0.06
0.04
0.02
0 r
12. Charge density p(r) for B1l, assuming F(q) goes negative, and overlooking data at large momentum transfer (see Fig. 7). For notation, see caption of Fig. 5. FIG.
the
Hofstadter (4) observed that the central densities of charge of nuclei decrease gradually from A = 40 to A = 209. If one assumesthe samespatial distribution for neutrons as for protons, then p(M) = (A/Z)p(Z), where p(M) is the matter density and p(Z) the density of charge. Then it appears that the peak (or central) density of matter, p,,(M), is . constant for all medium and heavy nuclei, with a value of about 0.166 nucleons/( lo-l3 cm)“. Central densities of charge and of matter for seven elements from Ca to Bi are shown in Fig. 18. The indicated errors represent the difference in densities found in the Stanford analysis with the Fermi function and in the Los Alamos analysis (16) of the same data with the Family II function. The true uncertainty is perhaps somewhat greater. Also shown in Fig. 18 are the peak densities of charge inferred in the present analysis of light elements, and the corresponding peak matter densities if the neutrons have the same spatial distribution as the protons. The peak matter densities inferred in this way for Be and B are slightly high, suggesting the pos-
NUCLEAR
CHARGE
153
DISTRIBUTIONS
0.16
p(r)
0.06
0.06
0.04
0.02
0
-0.02
I
2
3
4
5
6
7
r FIG. 13. Charge density P(T) for B”, assuming F(q) remains positive data in the analysis (see Fig. 7). The superscript (+) is to be understood bols in this figure. For notation, see caption of Fig. 5.
and including all on all of the sym-
sibility that the neutrons are in fact slightly more extended. The peak density of Li is low, indicating that finally at A = 6 the concept of nuclear matter has lost its usefulness, and the nuclear surface has penetrated to the center of the nucleus. It is a remarkable fact, however, that all the way from A = 9 to A = 209 there appears to be no significant change in the peak density of nuclear matter.
154
MEYER-BERKHOUT
0.14 I-
0.12
0.06
c
p(r) /
0.06
0.04
0.02 S.H.O.,H,FJ
0 FIG.
14.
I I
1 2
3
4
5
I 6
Charge density p(r) for C12.For notation, see caption of Fig. 5
Presumably then the correct density which should result from a theory of bulk nuclear matter is 0.166 A 0.005. This corresponds to a mean particle spacing of 1.13 * 0.01. For the light nuclei, we can therefore define a sensible half-density radius as the radius where the charge density has half of the average peak value. We take 0.040 as this half density, and call the corresponding radius ~(0.04). It is determined for these six light elements to about 2 %. (Contrast the definition given in Ref. 7, which is much more model dependent. Our definition avoids any reliance on the very uncertain region in the center of the nucleus.) This half-density radius divided by A”3 is plotted in Fig. 19, together with the half-density radii for heavier elements. There is an evident regularity-a steep ascent from Li to C and a more gradual increase throughout the periodic table toward the asymptotic value 1.13 for infinitely heavy nuclei. The errors in Fig. 19 are taken to be 2 % for the p-shell nuclei, 1.5 % for Ca, V, and Co, and 1% for the heavy elements. The crude average value 1.07 for c/A”~ which has been used before (4) is indicated in the figure.
NUCLEAR
CHARGE
DISTRIBUTIONS
r/9(r) 0.06
I
0
2
3
4
5
6
7
I FIG.
15. T p(r)
The mean square radii,
for
N14. For
notation,
see caption
(r2), or correspondingly
of Fig.
the equivalent
5
uniform
radii,
R = [5(~~)/3]~'~,are also of interest. R/All3 is plotted in Fig. 20. It diminishes regularly” from about 1.9 for Li to 1.18 for Pb (and should also approach 1.13 in the limit of infinitely heavy nuclei). The crude average value 1.20 for R/A"3 which is often used is shown in the figure. Errors based on the present analysis are shown for the p-shell nuclei. It is not known very well how accurate are the rms radii for medium and heavy nuclei. Therefore no errors are indicated for the other points. The substantial errors in the rms radii for the light nuclei can be reduced only when more accurate absolute cross-section measurements at momentum transfer around Q z 0.8 become available. The data shown in Figs. 18-20 are summarized for the p-shell nuclei in Table VII. The corresponding numbers, averaged over Au, Pb, and Bi, are also shown for comparison. Table VII includes surface thickness for the light elements. Regarding 0.080 as the “standard” peak charge density, we define the surface thick12 The regular stadter (4).
trend
in
mean
square
radii
has
been
pointed
out
previously
by Hof-
156
MEYER-BERKHOUT 0.20
I
I
I
I
I
I
I
I
4
5
6
0.18
0.18
p(r)
0.10
0.08
0.08
S.H.O.
0.04
0.02 F(-) H S H 0 5”“’
0
I I
I
I
2
3
r FIG.
16. Charge
density
p(r)
for
Nl4.
For
notation,
see caption
of Fig.
5
nessas the distance from p = 0.072 to p = 0.008. (For Li, the inner edge of the surface is then at T = 0.) For Be through 0, the surface thickness is constant, within the error, and equal to 1.9 f- 0.2. This is somewhat smaller than for heavier nuclei. In view of the arbitrary way of defining surface thickness, this may be without significance. On the other hand, these light nuclei have large proton separation energies, and, correspondingly, the proton wave functions fall more rapidly to zero outside the nuclear potential than would the wave functions of
NUCLEAR
CHARGE
157
DISTRIBUTIONS
0.06
p(r) 0.06
0.04
0.0 2
0
2
3
4 5 6 7 r FIG. 17. Charge density p(r) for 0 16.The approximate uncertainty in the central density arising from uncertainty in the fifth Fourier coefficient is shown. For notation, see caption of Fig. 5. I
protons at the top of the Fermi sea in most heavier elements. (A possible close connection between proton separation energy and surface thickness is indicated
by the fact that Bizogseemsto show a significantly greater surface thickness than does Pbzo8.) IV.
CORRECTIONS
TO THE
STATIC
SPHERICALLY
SYMMETRIC
NUCLEUS
Among possible correction effects, me have examined quantitatively only the effect of a deformed charge distribution. Brief discussionsof other effects are also
given below. A. NUCLEAR
CURRENTS
AND MAGNETIZATION
The effect of the proton’s magnetic moment on the scattering of electrons by protons
has been considered
in detail
(4) and affords
a basis for estimating
the
magnetic contribution to scattering from heavier elements. The ratio of magnetic
CENTRAL
FIG. 18. Peak the
data,
and
densities of charge are not theoretical.
DENSITIES
and of matter
in nuclei.
The
HALF-DENSITY
FIG. 19. Half-density oretical. used for
radii The dashed horizontal heavy nuclei.
divided by Al/a. See discussion line shows the rough average
158
curves
show
the trend
of
RADII
in text. The curve is not thevalue of 1.07 which has been
2.0
EQUIVALENT
LI
- UNIFORM
RADII
Be
R A’/3
B
1.5
.O 6
--
---
Ca
Q N
0
v
------
co 0
In Sb
----
Bi
--
Au pb Au ; i\ 1.0, 1.0,
, 3
I 2
I
, 4
dashed nuclei.
20. Equivalent horizontal line
uniform shows
radii divided by AI/z. the rough average value
TABLE SUMMARIZED
Pmnx Element
3Li6 rBeg SB” 02 TN'4 80’6 Au-Pb-Bi
(protons/10W9 cm3) f 0.006 f 0.006 i 0.005 zt 0.005 f 0.004 f. 0.003
0.79 0.84 0.94 1.01 1.045 1.03
0.067
f
1.11
0.002
theoretical. used for
The heavy
IN THE D-SHELLS
R/AI’”
b ,l$>;;,
(lo-‘3 cm)
0.064 0.086 0.086 0.080 0.087 0.079
The curve is not of 1.20 sometimes
VII
PARAMETERS
~W.WlA”‘3
:
i/3
A FIG.
I 5
(lo-l3
2.70 2.60 2.50 2.50 2.45 2.65
z!z f f z!z f f -
0.15 0.20 0.20 0.15 0.05 0.05
0 cm)
td (lo-l3
cm)
1.92 1.62 1.45 1.41 1.31 1.36
f f * f f zk
0.11 0.12 0.12 0.08 0.03 0.03
2.80 2.0 1.7 2.1 1.7 1.95
f f f f f f
0.0% 0.1 0.2 0.3 0.1 0.1
1.18
f
0.02
2.4
f
0.3
8 Note that the radii in this table are radii of charge distributions and not of proton distributions. b ~(0.04) is the radius where p = 0.040 and is determined in each case to within 2%. c R is the “equivalent uniform radius,” R = [N (r2)]““. d The “surface thickness” t is here defined as the distance from the radius for which p = 0.072 to the radius for which p = 0.008. e For Li6, the inner radius for evaluating the surface thickness is taken to be zero. 159
6
160
MEYER-BERKHOUT
scattering to charge scattering at a given momentum transfer and angle is roughly proportional to ~1’ 1Fv j”/Z” ( F, 1’) where p is the magnetic moment, 2 the nuclear charge, and F,, and F, are form factors associated with the distributions of magnetic moment and of charge (2%‘). (We here overlook the fact that different form factors may be associated with spin moment and orbital moment.) Now p2/Z2, which is 7.8 (nuclear magnetons)’ for the proton, is considerably smaller for the other elements considered here: 0.075 for Li6, 0.085 for Be’, 0.29 for B”, and 0.0033 for N14. At 420 Mev and 60”, an interesting region in these experiments, the magnetic scattering from the proton is about one-half of the charge scattering. Thus magnetic moment scattering is negligible for all lp-shell nuclei investigated so far if the magnetic form factor F,, does not exceed the charge form factor F, . However, the monopole charge scattering possesses at least one diffraction minimum (on any reasonable model), and at this minimum the magnetic form factor may in fact be large compared to the monopole charge form factor.13 Magnetic scattering may therefore be a significant contributor to the filling in of the diffraction minimum, and needs to be examined more carefully. Since the magnetic moment distribution is probably more peaked toward the edge of the nucleus than is the charge distribution, the magnetic form factor F,, should have its first minimum at a smaller momentum transfer than the first minimum of F, . In order to obtain what should be an upper limit to the magnetic scattering effect, we suppose that the secondary maximum of Fp falls at the first minimum of the monopole part of F, (the quadrupole effects will be discussed separately below). A d-function for the magnetic moment distribution,
with F, = sin(qr,,,r)/qrsUrf , has just this property provided the surface radius rsurf is suitably chosen. For reasons of simplicity, monopole charge scattering cross sections were computed by phase shift analysis for the pure oscillator charge distribution with the only free parameter rP = r8 = a0 adjusted to fit the experimental data at small momentum transfers where other contributions to the scattering are small. By comparing this cross section at its minimum with the corresponding magnetic moment scattering cross section estimated in Born approximation [with reUrf( N14) z 2.61 on the basis of the Rosenbluth equation one arrives at the conclusion that even for such an extreme model magnetic moment scattering cannot account for the observed filling-in of the diffraction minimum for N14. For B”, however, which has a rather large magnetic moment and where in addition the smaller value of 2 leads to a considerably deeper minimum in 13 The calculated
form factor at a diffraction by a phase shift analysis
minimum is to be understood and does not vanish.
as (a/u~J~‘~,
where
CT is
NUCLEAR
CHARGE
DISTRIBUTIONS
161
the predicted monopole charge scattering cross section, such a model [with rsurf(B1’) E 2.251 results in a large filling-in of the monopole charge scattering diffraction minimum, sufficient to account for the experimental observation. Less extreme models predict much less, but still appreciable magnetic scattering for B”. For example, an exponential magnetic moment distribution was considered with an rms radius chosen to be 2.22, the same as that of the oscillator charge distribution which fits the low-momentum-transfer data. This model also predicts a substantial filling-in of the monopole diffraction minimum. Similar estimates of the magnetic scattering from Be’ [with r,,,,,(Be”) E 2.11 led to the result that magnetic scattering could lead to an appreciable filling-in of the diffraction minimum (for this rather extreme model). At 300 Mev, the observed deviations from the harmonic-oscillator-model predictions (see Fig. 23) may arise partially from magnetic scattering. A more realistic model (magnetic moment distribution proportional to squared Ip wave function) predicts magnetic scattering in the interesting region about ten times smaller than does the extreme delta function model. In that case magnetic scattering would be of possible importance only in B”. For Li” results were similar as for Be’, but it should be a particularly interesting case when measurements are extended to larger momentum transfers. Because of the low 2, the Born approximation is nearly valid; and the quadrupole moment is extremely small. These two facts mean that the charge scattering should fall to a very low value at the first diffraction minimum. The small magnetic scattering may, therefore, be measurable by the extent to which it fills in the diffraction minimum. For the other elements, for reasons outlined above, it seems unhkely that anything definite can be learned about magnetic scattering unless the measurements are extended to considerably larger momentum transfers. The formal expressions from which the contributions of nuclear current and magnetization to elastic electron scattering can be evaluated for any definite nuclear model have been given by Schiff (93). We have not attempted quantitative evaluation of these expressions for the present analysis. Our crude analysis indicates, however, that magnetic moment scattering is probably negligible for N14 and small for Li6 and Beg at all angles and energies investigated so far. But for Beg and more especially for B” it may perhaps be comparable to monopole plus quadrupole scattering (see subsection D below), at the largest momentum transfers observed so far. Nuclear currents and magnetization can give rise to electric quadrupole and higher multipole effects, as well as to magnetic dipole effects. The electric quadrupole effect from this source, which flips the spin of the scattered electron ($3)) will be ignored in this work. The nuclear charge gives rise only to electric multipole effects. The quadrupole scattering from nuclear charge alone does not affect
162
MEYER-BERKHOUT
the electron spin, and to distinguish it we will use the expression no-spin-flip quadrupole scattering. It should be noted that the current- and magnetization-produced scattering can in principle be separated experimentally from the charge-produced scattering, provided dispersion effects (subsection C) are negligible. In Born approximation the charge scattering is equal to the point nucleus crosssection multiplied by a function of the momentum transfer 4 only. But the current-magnetization scattering contains additional explicit dependence upon the scattering angleI (9/t) (or equivalently upon the energy). If performed with good precision and extended over a sufficiently large angular interval, measurements on the angular dependence of the elastic cross section at two different energies but covering identical ranges of momentum transfer should make it possible to disentangle the contributions of charge scattering and of current-magnetization scattering [as has been done for the proton (4)]. The present data for B” at 300 and 420 Mev, although overlapping in the region between Q= 1.3 and q = 2.0, are hardly precise enough to permit a decision about this point. It may, therefore, be dhout significance that these data seem to favor pure charge scattering. However, the 82” point taken at 300 Mev is slightly high (Fig. 7) and, if not erroneous, may indicate some current-magnetization scattering. B. RECOIL
EFFECT
The dynamic effect of recoil has been shown to be negligible for zero spin nuclei (15). No theory of the dynamic effect has been given for nuclei of nonzero spin. The recoil current, Zev, (nuclear charge X velocity of recoiling nucleus), is of the same order-for momentum transfers of several hundred Mev/c-as the internal nuclear current. Therefore, the recoil effect might be expected to be of the same order as the magnetic effect. The kinematic effects of recoil have been included in the present analysis. The Schwartz-Gartenhaus correction (25, %‘6)15 has been omitted in considering the oscillator charge distribution. C. NUCLEAR
DISPERSION
Theoretical estimates indicate that the contribution of virtual excitation of nuclear excited states to elastic electron scattering is small. A recent theoretical treatment has been given by Schiff (27)) and calculations have been carried out by Downs (88). It has also been argued, erroneously, that the experiments support this conclusion because the same static nuclear charge distribution gives a 14 For no-spin-flip scattering, tan* (‘$9) (&/da) is a function of q only. scattering, the expression tan2 (J&Y) (da/&)/[1 + 2 tan* (‘$0) ] is a function I5 The omission of this correction means that the shell-model interpretation cillator charge distribution is somewhat in error, but it does not represent principle in the analysis.
For spin-flipof q only. of the osany error in
NUCLEAR
CHARGE
DISTRIBUTIONS
163
good fit to experimental results for the same nucleus at several different energies. The statement is true, but no conclusions about dispersion corrections can be drawn from it. For the light nuclei, one may rephrase the statement in this way: Experimental form factors, F( 4)) deduced at several different energies, lie along a single smooth curve when plotted as a function of g. There are, in fact, substantial regions of overlap in 4 of the data at different energies, but the agreement rests on suitably renormalizing the absolute cross-section data at each energy. The required renormalization is within the experimental error of the absolute measurements, which is Sor these experiments rather large. We have examined the predicted angle-dependence and q-dependence of the dispersion corrections, using some simplified approximate formulas of Downs (28)) which should be good enough for this purpose, and we conclude that quite large dispersion corrections could be present-indeed larger than is reasonable-and still go undetected because of the uncertainties in the absolute cross-section measurements. The experimental detection of the dispersion corrections will be possible only when much more accurate absolute cross-section measurements are made over the same interval of momentum transfer q at two or more different energies. Even then, the separation of dispersion effects and current-magnetization scattering will be difficult. On the other hand, the existence of the pronounced diffraction minima in the scattering from C” and 016, with apparently no significant filling-in as compared to the phase shift analysis predictions, does indicate the smallness of dispersion corrections. There is no apparent reason why the dispersion scattering should have a minimum at or very near the minimum of the static charge scattering. We conclude that the static charge distributions deduced in this analysis must be regarded as effective charge distributions which may include some nonstatic effects of nuclear polarizability, but that these effects are very likely unimportant. D.
QUADRUPOLE
MOMENT
In Born approximation the effect of a nonspherical charge distribution on electron scattering may be calculated very simply.16 It depends, as one might expect, on the spatial distribution of the source of the quadrupole moment, reducing to a simple dependence on the value of the static quadrupole moment only at low momentum transfer. We follow the analysis and notation of Schiff (25’). Define the ground state charge density p as the expectation value of 2 &( r - rl) in the M = 1 substate, and write p(r) = PO(~) + pz(r>YzO(Q + . -- . (24) 16 In this quadrupole “no-spin-flip”.
section only the charge effect is considered, i.e., the monopole plus no-spin-flip scattering, as defined in subsection A. We will generally omit the adjective
164
MEYER-BERKHOUT
Since no net charge is associated with pz , etc., the spherically still obeys the normalization condition, 4?r p&y s
dr = 2.
The quadrupole part of the charge density, quadrupole moment by
symmetric
part
(25)
p2 , is related to the spectroscopic
Q = 2(4~/5)“‘Spl(T)r~
dr.
(26)
(Note that pz (and thus Q) already include a projection factor, and vanish for I = Oort5.) For unpolarized electrons and nuclei, the scattering cross section is given in Born approximation by IJ = uo[ 1Fo I2 + 1 F2
17,
(27)
where go is the point nucleus scattering cross section (in Born approximation), F. is the usual form factor for a spherical charge distribution,
and F2 , the quadrupole
form factor, is given by
Fz(q) = z-‘(4rri’r,)“‘Sj,(4r)pi(r)r2 In these expressionsjo projection factor,
dr.
andjz are spherical Bessel functions, P, = I(21
-
1)/(1
+ 1)(21+
3).
(29)
and PI is a quadrupole (30)
It should be noticed that since the quadrupole form factor, Fz , is a function of q only, Eq. (27) implies that one can always find a spherically symmetric charge distribution which will give the same elastic scattering as the general charge distribution (24). Of course this pseudo-charge-distribution may be unreasonable in various ways. Most seriously it might not be normalized to the correct total charge; or it might be negative at some values of r; or it might show unreasonable oscillations. (The total charge condition was not a very stringent requirement in Section III because of the rather low accuracy of the absolute cross-section measurements.) At small q, Fezis given to leading order by F2 z (1/30)(5/P$‘“(Q/Z)q2.
(31)
NUCLEAR
CHARGE
165
DISTRIBUTIONS
At the large q-values of interest here, (31) is not valid and F, is model dependent. AS the simplest model, consider a deformed uniform charge distribution, with quadrupole charge density given approximately by
pz = 4~(5/47r)""(Q/R")S(r
- R),
(32)
where Q is the spectroscopic quadrupole moment, Then F$ = 4~(5/PI)““(Q/ZR”)j,(qR).
(33)
This model is too idealized to be particularly interesting. For comparisons with experiment we consider instead the following three models. (a) Undeformed p-shell. If the quadrupole moment arisesfrom protons moving in the p-shell of an undeformed potential, the radial dependence of pz is the same as that of the p-shell part of PO. For the harmonic oscillator potential the quadrupole charge density is p2(r)
= (4/37r)5-“*(
Q/ao7)r2
exp( -r2/aoZ),
(34)
*a0* exp ( - ,1/&‘ao2),
(35)
and the quadrupole form-factor is
Fz(q) = (1/30)(5/f%)"2(QlZao2)~
where we have set r, = a0 [see Eq. (3)]. (b) Deformed Oscillator Potential. The charge density in the isotropic oscillator (rp = rs .= a,) is given by pa(r)
+ 3a)][l
= [2Z/p3’*ad(2
+ a(r2/ao2)]
exp( -~*/a:),
(36)
where (Y = (2 - 2)/3. We imagine a small deformation of the potential such that the equipotential surfaces are given by r/( 1 + /~I’zo) = constant. To lowest order the quadrupole charge density may then be found by replacing r by r/( 1 + pYZo) in (36) and expanding to first order in p. The result is p2 = -@rdpO/dr, or pz(r)
= (4Qr2/3~[(5/2)a
+
1]5”“u0’][1 - a +
LYT*/UO~]
exp(-rr2/ao2).
(37)
The quadrupole moment is related to the deformation by
Q = %(5/a) The quadrupole form factor is given by
Fz(q) = ~~o(5/PI)"*(Q/Zao2)q2a02 (1 - [a/2(2 + 5n)lq*ao2} exp(-$&ao2).
(39)
166
MEYER-BERKHOUT
It should be observed that the qusdrupole contribution to the scattering [Eq. (27)] is of second order in p. The deformation gives rise also to a monopole contribution of second order in p, arising from the spherically symmetric part of the second order term in the expansion of p(r). This term represents the average smearing out of the surface due to deformation and could be present also in zerospin nuclei. (It is the term giving rise to isotope shift anomalies.) Because of our phenomenological approach, it does not seem worthwhile to take explicit account of the deformation-induced monopole contribution. (c) Shell Collective Model. The model of Ferrell and Visscher ($9) provides a theoretical justification for the classical deformed potential model discussed above, and leads to identical predictions for the quadrupole scattering. They define “collective” (quadrupole) states by a procedure similar to that described under Eq. (36). A small deformation is introduced into the shell model wave function, and the collective wave function is defined as the derivative of this function with respect to the deformation parameter at zero deformation. The collective state defined in this way consists of that particular combination of excited shell model states with maximum or near-maximum quadrupole matrix elements with the original state. The ground state is then taken to be a shell model state (full s-shell plus partially filled p-shell) plus a small admixture of the associated collective quadrupole state, plus other small admixtures which are irrelevant for calculating quadrupole effects. The charge density then becomes the diagonal matrix element of Z&(r - rl) for the original shell model state plus a cross term linear in the mixture amplitude, plus second order terms which are ignored. The zero order term gives PO + pia)Yzo , the spherically symmetric part of the charge distribution plus the quadrupole part arising from the p-shell configuration. The first order term gives #Yzo , the collective quadrupole part. In the oscillator model, pib) is the same as for model (b), Eq. (37)) and the quadrupole form factor is that given by Eq. (39).” It should be noted that the shell model function from which one starts can be an arbitrarily complicated mixture of states within the p-shell, provided they all have the same radial dependence. The charge densities (36) and (37) are dictated only by the radial functions, and all complexities of the angular functions are absorbed in the number Q. (d) Deformed p-shell. As a model intended mainly as a further test of the sensitivity of predictions to assumptions about the source of the quadrupole moment, we examine the case that the s-shell remains spherical while the p-shell is deformed. The results for this model can be obtained most simply by letting (Y --, cu in Eqs. (37) and (39). The model is not very realistic because parity17 Aside from normalization, the coefficient 56 (a) (~CX + correctly as l,&.
2)-l
the result (37) is given in Ref. (29). For oxygen, should be >iz, but is given as sic. For carbon
however, it is given
NUCLEAR
-
CHARGE
=HARMONIC SCATTERING
o,b
( PHASE ELASTIC SCATTERING
SCATTERING
167
DISTRIBUTIONS
WELL a,=
ELASTIC MONOPOLE 1.625x IO-l3 CM. a = 5/3
SHIFT NO
ANALYSIS SPIN FLIP
( BORN
ANGLE
1 QUADRUPOLE
APPROXIMATION
IN
1
DEGREES
21. Monopole and quadrupole scattering from Nl”. The monopole curve is based on the oscillator model and the three quadrupole curves are based on models (a), (b), and (c) in the text. normalized to a spectroscopic quadrupole moment of 2.0. FIG.
excitation of the 1s shell probably does not require much more energy than parit’y-conserving excitation of the lp shell for the p-shell nuclei. We expect that some combination of models (a) and (b) should be realistic for the nuclei under study. Then t’he factors Q in (35) and (39) should be Q. and Qb, the parts of Q arising from the p-shell and from excitation out of the p-shell; arkd the quadrupole form factors should be added before squaring, i.e., conserving
v-
1F,, I* + / Fia’ + F;“’ 12.
(40) Comparison with Experiment Detailed calculations with the mixed model represented by (40) have been carried out by Fallieros and Pal (SO)‘* using the basic ideas of Ref. 2.9. We thereI8 We are indebted to these workers for supplying us with their results in advance of publication., and for illuminating discussions about their work.
168
MEYER-BERKHOUT
fore present only the predictions of the three limiting models discussed above, for the elements N14, Bn, and Be’. These predictions are shown in Figs. 21-23 for each of these models (a), (b) , and (c) . Since quadrupole moments are not accurately known, the best available experimental value is chosen for each element (see discussion in Section III), and each curve plotted as though that entire quadrupole moment arose from one source: spherical p-shell, deformed nucleus, or deformed p-shell. Because the squares of form-factors add, cross sections are plotted rather than form factors. On each graph is shown also an illustrative prediction of the scattering from the spherically symmetric part of the charge in the oscillator model, with the single radial parameter r8 = rP = a~ chosen to fit the experimental data for low momentum transfers. Figure 21 shows that the filling-in of the diffraction minimum in N14 can be ac10-Z’ -=
HARMONIC ’
lO+l
SiATTEhlNG
a,b,c,
WELL
a
( PHASE
SHIFT ~0
-*a
I
-
I
lO-30
I\1
spec. = 3.55
ANALYSIS SPIN
( BORN
I
I
x IO-= I
I
I
I
70'
:; 80"
90'
MONOPOLE
,:m’
ELASTIC SCATTERING
7
ELASTIC
I+,=
1
FLIP
OUADRUPOLE
APPROXIMATION
CM? Ihl
1
aO= I 55 x Iti3 CM {I I I
t-t-l
IIlYlIIIII
z
+ z p s VI
IO
-3,
lo-3; 30° 400
50"
60'
SCATTERING
FIG. 22. Monopole sumed spectroscopic
' ANGLE
30° IN
and quadrupole scattering from quadrupole moment is 3.55.
40°
50°
60°
70'
DEGREES
B”.
See caption
of Fig.
21. The
as-
NUCLEAR
CHARGE
169
DISTRIBUTIONS
counted for by a spherical p-shell quadrupole moment of about 1.8 X 1O-26 cm”. But if the source of the quadrupole moment is deformation, a moment of 5.6 X 1O-26 cm” would be required to produce the same filling in. Sample combinations whichcouldfitthe420-Mevdataare (l&O), (1.5,1), (1.2,2), (1.0,2.6), (0,5.6), where the first number of each pair is the quadrupole moment from the p-shell, and the second number is the additional quadrupole moment from collective admixtures or deformation (the total quadrupole moment being just the sum of these two numbers). For the first contribution Visscher and Ferrell (31) give a theoreticad value of about 1.1, while the maximum possible value ($1) is about 1.8-a value which it is highly unlikely should be realized. Within the somewhat limited framework of these models, we therefore conclude tentatively that the -
=HARMONIC
SCATTERING
WELL
ELASTIC
( BORN
MONOPOLE
APPROXIMATION
30' SCATTERING
FIG. 23. Monopole sumed spectroscopic
ANGLE
and quadrupole scattering quadrupole moment is 2.0.
from
IN
40'
1
50°
60'
70°
80'
90° 100'
DEGREES
Be9. See caption
of Fig.
21. The
as-
170
MEYER-BERKHOUT
quadrupole moment of N14 is greater than 2, and possibly as great as 4, with at least half of the moment being contributed by collective effects, i.e., excitation out of the s- and p-shells. It should be borne in mind, however, that other incoherent effects may be contributing appreciably to the scattering, particularly spin-flip scattering from nuclear currents. If these are found to be important, the quadrupole moment inferred from experiment will have to be revised downward. Figure 22 shows the quadrupole scattering in B” for an adopted value of Q = 3.55. A quadrupole moment of this order of magnitude can evidently account for the large angle data, but it seems impossible at present to conclude anything quantitative about the value of Q. As with X14, the quadrupole scattering in the interesting region depends markedly on the source of the moment. In addition there is with B’l considerably more uncertainty about what to take for the monopole scattering, and the position of the diffraction minimum is less well determined. Furthermore, magnetic scattering is probably important in B”. Similar curves for Be’, with an adopted Q of 2.0, are shown in Fig. 23. No definite conclusions can be drawn, particularly since no 420-Mev data were obtained for this element, and the position of the diffraction minimum is uncertain. ACKNOWLEDGMENTS We wish to thank Professor R. Hofstadter for his continued interest in this work. One of us (U. M.-B.) is indebted to Professor Hofstadter for offering him the opportunity of a most pleasant participation in the work of the Stanford electron scattering group during the past two years. He also wishes to express his appreciation to Professor Hofstadter for many encouraging discussions during t,he execution of these experiments. We wish to express our thanks to the members of the operating crew of the Stanford accelerator, and to all members of the electron scattering group for their continued help. Mr. C. N. Davey was of great technical assistance in setting up these experiments. We are indebted to Dr. David L. Hill for many contributions to the early phases of the analysis. Mr. J. G. Wills and Mr. B. J. Hill have carried out much of the numerical analysis reported here, and have been very helpful throughout the course of this work. We thank Professor L. I. Schiff and Professor R. A. Ferrell for profitable discussions of quadrupole scattering. Mr. A. Pettinger was of great assistance in coding and running the quadrupole calculations on the Stanford IBM Computer. We wish to thank the authorities of the University of California Radiation Laboratory at Livermore and the Los Alamos Scientific Laboratory for the use of computing machines, particularly Dr. S. Fernbach at Livermore, and Dr. C. Mark at Los Alamos. RECEIVED:
February
25, 1959 REFERENCES
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AND S. E.
NUCLEAR
CHARGE
DISTRIBUTIONS
171
3. C. II. MOAK, A. GALONSKY, R. L. TRAUGHBER, AND C. M. JONES, Phys. Rev. 110,1369 (1958). 4. R. HOFSTADTER, in “Annual Review of Nuclear Science.” Vol. 7, p. 231. Annual Reviews, Stanford, 1957. 5. J. FREGEAU AND R. HOFSTADTER, Phys. Rev. 99, 1503 (1955). 6. J. FRECCEAU, Phys. Rev. 104, 225 (1956). 7. R. HOFSTADTER AND G. BURLESON, Phys. Rev. 112, 1282 (1958). 8. J. STREIB, Phys. Rev. 100, 1797 (1955) and private communication. 9. J. HAT’I’ON, B. V. ROLLIN, AND E. F. W. SEYMOUR, Phys. Rev. 33, 672 (1951). 10. G. WESSEL, Phys. Rev. 92, 158 (1953). il. C. H. TOWNES AND B. P. DAILEY, J. Chem. Phys. 17, 782 (1949). 12. A. BASSOMPIERRE, Compt. Rend. 240, 285 (1955). IS. K. FORI) AND D. HILL, in “Annual Review of Nuclear Science,” Vol. 5, p, 25. Annual Reviews, Stanford, 1955. 14. D. R. YENNIE, D. G. RAVENHALL, AND R. N. WILSON, Phys. Rev. 96, 500 (1954). 16. FOLDY, K. W. FORD, AND YENNIE, Phys. Rev. 113, 1147 (1959). 16. K. W. FORD, B. HILL, D. HILL, AND J.G. WILLS (to be published). 17. D. L. HILL AND K. W. FORD, Phys. Rev. 94, 1617 and 1630 (1954). 18. R. HOFSTADTER, Revs. Modern Phys. 28, 214 (1956). 19. W. GR~BNER AND N. HOFREITER, “Integraltafel,” Vol. II, p. 177. Springer-Verlag, Berlin, 1950. 20. ERD&LY[, MAGNUS, OBERHETTINGER, AND TRICOMI, “Table of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1953. 21. L. R. B. ELTON, Revs, Modern Phys. 30, 557 (1958). 22. B. HAHPJ, D. G. RAVENHALL, AND R. HOFSTADTER, Phys. Rev. 101, 1131 (1956). 23. L. I. SCIIIFF, Phys. Rev. 96, 765 (1954). 24. L. J. TASSIE, Nuovo cimento Ser. 10, 6, 1497 (1957). 25. C. SCHWARTZ, private communication. 26. L. J. TASSIE AND F. C. BARKER, Phys. Rev. 111, 940 (1958). 27. L. I. SCIUFF, Nuovo cirnento 6, 1223 (1957). 28. B. W. DOWNS, Phys. Rev. 101, 820 (1956). 29. R. FERRELL AND W. VISSCHER, Phys. Rev. 104, 475 (1956). 20. S. FALLIEROS AND M. Ii. PAL (to be published). $1. W. VISS~HER AND R. FERRELL, Phys. Rev. 107, 781 (1957).