Coulomb displacement energies for 1f-2p shell nuclei

NutdearPhysics

Not to

COULOMB

be

A168 (1971) 151-176;

@ North-~o~jand

Punishing

Co., Amsterdam

reproduced by photoprint or microfilm without written permission from the publisher

DISPLACEMENT

ENERGIES

FOR lf-2p

SHELL

NUCLEI

F. D. BECCHETTI J. N. Williams Laboratory

of Nuclear Physics,

University

of Minnesota

t

and The Niels Bohr Institute,

University

of Copenhagen,

Denr~rk

D. DEHNHARD J. H. Williams Laboratory

of Nuclear Physics,

University

of Minnesota

t

and T. G. DZUBAY tt The Palmer Physical

Laboratory,

Princeton

University,

Princeton,

NJ.,

USA 7

Received 14 October 1970 (Revised 4 December 1970) Abstract: Coulomb

displacement energies for 29 stable isotopes between A = 42 and A = 70 have been measured to an accuracy of &6 keV using the (3He, t) reaction at 24.6 MeV incident 3He energy. Coulomb energy shifts between members of an isotopic sequence have been determined to an accuracy of +I3 keV. Triton spectra were obtained at 18” using a calibrated position-sensitive detector in the focal plane of an Enge split-pole magnetic spectrometer. In most of the spectra a single, strong transition could be identified as the transition to the isobaric analogue of the target nucleus ground state and an unambiguous Coulomb displacement energy was calculated from the measured Q-value. The triton spectra for the residual nuclei 56Co, ssCo and 64Cu showed two strong transitions in the energy region expected for the analogue state. In these nuclei, the t3He, t) cross sections to any single state were lower than those expected from the systematics obtained from neighbouring isotopes. This indicates a fragmentation of the analogue strength. Off-diagonal Coulomb matrix elements in the range 5 to 35 keV can account for the observed energy splittings and (“He, t) cross-section ratios. The experimental Coulomb displacement energies were analysed using various models. The experimental isotopic shifts in Coulomb displacement energies were used together with measured shifts in charge radii to extract isotopic shifts in the radii of the neutron excess distribution. NUCLEAR 50.52.53.

E

54Cr

REACTIONS SJMn

54.56.57.58&

42*43.44.46,48Ca, 59C0,

4SSc, 46.47,48.49*

ss,so.61.a2,64Ni,63,65CU

5OTi,

51V,

64.66.67.682n

(3He, t), E = i4.6 MkV; o(E,, 8 = is”), Q(IAS). Ca-Sc, Sc-Ti, Ti-V,‘V-Cr, Cr-Mn, Mn-Fe, Fe-Co, Co-Ni, Ni-Cu. Zn-Ga, Cu-Zn isotopes deduced Coulomb disp!acement energies. Enriched and natural targets. Magnetic particle spectrometer and positionsensitive detector.

t Work supported in part by the US Atomic Energy Commission, Contracts AT (ll-l)-1265 and AT (30”I)-937. This is report number COO-1265-96. tt Present address: Triangle Universities Nuclear Laboratory and Department of Physics, Duke Unive~ity, Durham, N.C., USA. 151

152

F. D. BECCHETTI

et al.

1. Introduction Coulomb displacement energies have been used extensively in the determination of nuclear sizes and in the study of charge-dependent interactions in nuclei. It has been shown lP3) t h a t such energies are a measure of the overlap of the neutron excess with the total nuclear charge distribution and hence afford a method of measuring the difference in proton and neutron radii. The fine structure exhibited by the Coulomb displacement energies within an isotopic sequence has been used to measure the charge dependence of the pairing interaction, Coulomb spin-orbit effects etc. “). Since these effects are small compared to the dominant pure Coulomb interaction, Coulomb displacement energies, AEc, of high accuracy are required. The most direct method of measuring Coulomb displacement energies is to use the (p, n) reaction to populate the isobaric analogue of the target nucleus, since AEc is simply the negative of the Q-value for such a transition. After Anderson et al. ‘), in their study of (p, n) reactions, discovered isobaric analogue states in heavy nuclei, they used this reaction to measure AEc values for a large number of nuclei “). However, it is difficult to obtain good energy resolution in the detection of fast neutrons. The study of isobaric analogue resonances or reactions such as (p, d), (3He, p), etc., where charged particles are detected, is experimentally more suitable; however, the accuracy of the Coulomb displacement energy is often limited by the errors in the nuclear masses of the target and residual nuclei. Furthermore, these reactions often populate analogues of states in unstable nuclei or analogues of excited states for which charge distributions cannot be measured by high-energy electron scattering or other direct methods. This limits the theoretical analysis. Sherr et al. ‘) have demonstrated the advantage of using the (3He, t) reaction in measuring Coulomb displacement energies. As in the case of (p, n), the (3He, t) reaction was observed to selectively populate analogue states. For this reaction AEc = AM,_,+ AM3u,_t-Q(3He, t). The difference in the Q-value for a pair of target nuclei can be used to obtain the isotopic shift in AEc. This technique is used in the present experiment. The measured shifts in (3He, t) Q-values with respect to the precisely known ground state (p, n) Q-values for targets of 42Ca, “Cr, 54Fe and “Ni are used to determine AEc for all the other targets studied. The region 40 < A < 70 was chosen for study since it includes nuclei in a relatively pure shell (If+) as well as a mixed shell (lf-2p). The 1& shell is of particular interest because effects such as the Coulomb spin-orbit interaction are enhanced due to the large orbital angular momentum involved. Most of the ground-state isobaric analogue states are bound and have been studied using the (p, n) and (3He, p) reactions for several isotopic sequences are known [refs. s-11)]. Also the charge distributions 14*’ ‘). from electron scattering 12, r3) and muonic X-ray measurements

COULOMB

DISPLACEMENT

2. Experimental 2.1.

ENERGIES

153

procedure

TARGETS

Targets for all of the stable isotopes between 42Ca and 68Zn inclusive were prepared from natural, mixed and enriched isotopic material using a variety of techniques. The majority of targets were prepared by evaporation onto 25 pg/cm* carbon backing. Targets likely to deteriorate by oxidation were sealed with a 3 pg/cm* coating of Formvar. The remaining targets were prepared by sputtering or electroplating. Approximate target thicknesses were calculated from the energy loss of 241Am u-particles passing through the foils. The thicknesses of the different components in of yields most of the targets (‘*C, ’ 6O isotope) were determined from a comparison of elastically scattered 3He a; 18” and 24.6 MeV incident energy, with optical model predictions. The optical model potentials were extrapolated from potentials which fit 3He elastic scattering data in the same energy region 16-18). We estimate the error in the target thickness obtained by such a procedure as f 15 %. This was checked by measuring carbon and nickel foils of known thicknesses. The relative isotopic abundances in the enriched targets used in the present measurements are given in table 1. TABLE

Relative

Target

A

isotopic

abundances

Relative abundance

A

Target

(%) 42Ca 43Ca 44Ca 46Ca

48Ca 46Ti 47Ti

42 40 43 44

94.4 12.0 81.1 98.6

40 44 46 40 48 46 48 47 48

61.9 5.8 24.7 15.7 81.9 86.1 10.6 80.3 15.7

EXPERIMENTAL

enriched

Relative abundance

targets

“)

Target

Relative abundance

A

(%) 48Ti

48 48 49 48 50

49Ti 50Ti

53. 54cr

54Fe 56Fe s’Fe 58Fe

SET-UP

61Ni 62Ni 64Ni 64Zn 66Zn 67Zn

61 62 64 64 66 67

92.1 98.7 97.9 99.9 99.4 89.6

see fig. 4 54 97.1 56 99.7 56 11.7 57 87.6 56 15.8 58 82.2

68Zn

68

98.5

were supplied

AND

( %)

99.4 18.5 76.1 12.4 83.2

50.52

“) The isotopic enrichments listed abundances > 10 % are listed. 2.2.

1

of isotopically

ENERGY

by the manufacturer.

Only

those

isotopes

with

CALIBRATION

The beam of 24.6 MeV 3He++ ions was produced by the J. H. Williams Laboratory MP Tandem Van de Graaff accelerator. The absolute beam energy was determined I’) to an accuracy of k 5 keV with a spread of 5 2 keV FWHM. The scattering geometry is shown in fig. 1.

F. D. BECCHETTI

154

et al.

The outgoing reaction products were detected in a 500 pm thick, 30 mm long by 10 mm high position-sensitive detector 20)t mounted in the focal plane of an Enge split-pole spectrometer ‘l). The position times energy loss and the energy loss signals, XE and E, respectively, were digitalized and routed through a direct memory inter-

Fig. 1. The scattering

geometry.

The quantity

X is the distance

along

the focal plane.

face ““) to an on-line CDC 3100 computer. The position spectrum was obtained by dividing XE by E digitally in the computer 23). The position resolution width obtained using standard electronics was about 0.3 mm FWHM. A triton energy resolution width of 8 keV FWHM was obtained for the thinnest targets used. Different types of particles having the same Bp values were identified by their energy loss, E, in the detector. The detector length of 30 mm included a triton energy range of about 400 keV. Photographic plates could not be used effectively due to the strong deuteron groups usually present at the Bp values of the tritons from the (3He, t) reaction. To obtain accurate particle energies the detectors were calibrated using triton energies which we deduced from the precisely measured (p, n) threshold energies of Freeman et al. 24, 2 ‘). The positions of the triton groups in the spectra were determined both by finding the median and by using a Gaussian peak fitting routine 26). Measurements using different calibration energies were overlapped whenever possible and checked for self-consistency. Corrections for energy losses in the targets were included using the measured target composition. Since the (3He, t) Q-values to analogue states within an isotopic sequence are constant to within a few hundred keV, most measurements could be made with little or no change in the magnetic field. The errors in the (3He, t) Q-values determined relative to a calibration energy are estimated to be +3 keV for targets less than 100 ng/cm2 thick and slightly more than this for thicker targets. t The detectors used in the present experiment were supplied by Nuclear Diodes, Inc.

COULOMB 2.3. TRITON

SPECTRA

DISPLACEMENT

ENERGIES

155

AND CROSS SECTIONS

Typical triton spectra from the (3He, t) reaction are shown in figs. 2-8. The 18”(lab) angle at which ail observations were made corresponds to the second maximum in

5’ ‘. ,.;

..

,+. . . .

20

40

. . . . ..

60

. :.

80

CHANNEL NUMBER

,’

. ..%+a+

100

* *\.

.z 120 (40 XE_ E .&I..

_

160

;8;

Fig. 2. Triton position spectra for the Ca(3He, t) reaction. Excitation energies in keV for states in Q 43*44*46*48Sc near the energy expected for the IAS are indicated by arrows and are accurate to +13 keV or better (see table 2). The relative triton energies between different isotopes are accurate to &3 keV. The calibration of XE/E is about 1.65 keV per channel. All of the spectra were obtained using the same detector position, magnetic field, and incident beam energy except that for 46Sc which was obtained using a slightly lower beam energy.

the AL = 0 angular distribution for analogues of J” = 0’ states 27r2a*33). The absolute (3He, t) cross sections at 18”(lab) were determined to an accuracy of + 12 % relative to each other and to an absolute accuracy of 5 20%. The results are presented in table 2.

156

F. D. BECCHETTI

al.

et

Experimental and theoretical studies have shown that the AL = 0 (3He, t) cross sections to isobaric analogue states should be a simple function of the neutron excess [refs. ‘**30-3”)]. If the neutron excess is within the same shell for a series of isotopes,

F

4 .

0” u IO t

.-a 0

.

**

_-&.

20

40

CHANNEL

60 NUMBER

80

100

. .

120 .I _ 140 s, E

I

f.

160

*

180

Fig. 3. Triton spectra for the Ti(3He, t) reaction, (Calibration: ;=: X.6.5 keV per channel.) AI1 of the spectra were obtained for the same detector position and magnetic field except that for 49V which was obtained during a different series of runs using a slightly different detector position.

then an (N-Z)fA’ cross-section dependence is expected for 0* to 0’ IAS transitions “* 33). The absolute cross sections at 18” for targets of even mass have been and the results are plotted in fig. 9. With the normalized by the quantity (N-Z)/A*, exception of 56Co, 58Co and 64Ni, there is good agreement with the (N--2)/A*

COULOMB

DISPLACEMENT

0

ENERGIES

a

0

::

l:NNWGod SINIl03

$

8

0

$

13NNWi3

CJ

&I% 51Nfl03

0

$

2

0

18”

“_

.

C"61

MHz

40

. - ....

60

00

NUMBER

100

-.*. . t *.

.

1

1 .5: .

CHANNEL

20

120 XE_ E

.. . __‘.,.:,‘_‘:-.

‘.T

C~~~(25.17)

MHz

: .. .

‘: .

1

1643

cPc202)

F-46.161

.. . . ..: :. I. . . . _a.. I . ,~-.-_-:.“,‘:.

F=46.160

140

160

I

cu5Gtg s)

180

w 1.50

Cli6’

: .. ...-._*., * . ._.__.

CP

6610 .. 1 . ..'.. * . . .: '.: :"_ :*. :; .: **s. .,'.'.... _.t:a.e*_. _*.._-_.:.. .* ...._..: . ... .:-" ..s.. . ..__.. * _a... L . . w.. .. ... ... . '....A

eL=

.I ...

6826

Ni(He?t)Cu

Fig. 6. Triton spectra for the Ni(3He, t) reaction. (Calibration: keV per channel.)

0

20

0

0

IO

MHz

EHe3 = 24.6 MeV

F-46.161 MeV

-

.

6130

*-*-"-.-_ e.. - . -',

.

I1

6170

6007

.. . . . e-.'_' .I:.'

. "_‘..

GO67

.-. ... *. -* *..* .:.

1

6031

Ga6*

Zn(He?t!Ga

.”

.

*. .

. ..t

40

-

#..

.

.

60

:

. ‘. . .

NUMBER

. .

'. .

I

2025

“_.-;.A..-,

*.

MHz

..L.

..:-..

CHANNEL

20

I

. . I..._

F=45.500

_.I_..

.*

.

*.

.

.

.

:

:.

00

100

.* .-_. .

. . . ..-*

.

.

.

120 XE -F--

.

-. .

‘:,-. *

. :

140

.

. *

.

-

G&

.

180

.

. ._.. _._

Go66

160

“.w.mLL_.

.

1

I 1

.: . **.-... .

3600

3655

3960

MHz

Fig. 7. Triton spectra for the Zn(3He, t) reaction. (Calibration: w 1.45 keV per channel.)

.

.

F=45.500

I .. :. . . *. * 2.. ..a . . m... ..‘_._ _. . .. :.. _ ..:*.-... . ..' *. * _ .~__....--,._C._._._._-_....~._. ;_...*L....L..L

MHz

. *:*_.*. -'-a..: ..,.,,,....,'

F=45.500

.. . _..

.9,=18”

MHz

F-45498 EHe3=24.6

COULOMB

DISPLACEMENT

ENERGIES

159

dependence for the Of to Of transitions. Cross sections for odd-mass targets have not been included in this comparison because transitions with AL =f=0 are allowed for these nuclei 28*3‘*3“). The triton spectra obtained from the targets 56Fe, 58Fe, 61Ni and 64Ni corresponding to states in the nuclei 56Co, 58Co, (jlCu and 64Cu contain several peaks in the F=45.808 MHz EH$=24.6 MeV IO

!

e,=I8" . .

0 i

. .

-...

‘A..

zn= (7&323

.-0

t

.*

Cu(H&)Zn

zn6%96)

. .*

.

‘a

.11.*.

I

.* .

I.

.*.’ ,,.L.

.

.

.

.

“* .**

*-

. *_.* t*.C&

.“_._y

*

e

.

.

*.

*.*.*w _ _. .*‘

F=46.487 MHz 7319

. . .

“:.*“_._-_.*

. .

. . . ,-w’.*~

CofHe?t)Ni Ni-

i 10

* . _.‘._*... 0

::. ..e’%‘..t . . . . ..

. .*_.;*... *

.

. .

‘A.-.

o..

F=47.i30

‘.,A . .

.

. ..e.* . ..‘w’ ..

*

‘a.,*

. .

.

: - ..,. l““-.* . . ..“L”o......... . . . d&.

MHz

M~(H~~t) Fe XL0

Fe55

.I

e. * - a.

II... ---

F=42922

. *. t. .. : ” .,:w* *:*** * . . ..-..,.-*w** _ .m**-a.. *_ a“-.* -*.a’e..*... w..._----; 1 5611 MHz V(He? t)Cr * .

l

*

Cl-51

.

l** l

-

l*. ,

.*

:

*- 5

0

* .s’.-*&,.*...****,** ‘&p&*.** F =48.680

; *~-,_*-*~&~~~~~*“*~-*-~‘i’-~~* l

l

6730

MHz

Sc(He?t )Ti

I . 20 -

*

l

l l

Or

Tib5

l

*-

20

l

-“c *z* a l. 40

CHANNEL

60

80

NUMBER

100

120

140

_ ,m * ‘-.160

l

180

XE -E

Fig. 8. Triton position spectra for the (“He, t) reaction on several natural targets. Each channel in XE/E corresponds to about 1.45, 1.52, 1.56, 1.62 and 1.68 keV for the Zn, Ni, Fe, Cr and Ti spectra, respectively.

0,2)

(l,l2)

(oJ)*l+) d+

%-

59

58 60 __

Co-Ni

Ni-Cu

1 2

8

0+ 0+ ^

H-

0’

54 56

Fe-Co

t

$

8 3

(2.3)

5s

Mn-Fe

0’ 0+

1 2

(2,4;:

50 52 53 54

Cr-Mn

%0’

% 3

-zr-

;;

2

t

0+

:

58

51

V-Cr

%-

a

I?Q- “1 o+

46 41 48 49 50

Of 0+

3 4

4

45

Sc-Ti

Ti-V

pi

ii2

T”)

57

42 43 44 46 48

A

Ca-Sc

Analogue pair

(9) (9) (7) (7) (8) (8)

0 3522 3592 7274 7288 5739 5759 (8)

(5)

7640

(3) (9) II”,

(7) (6)

6969 6141

202 2547 /-‘.*

(7)

292:

7349

(5)

6611

(9) (5) (5) (6)

(7)

4730 0 4168 3022 6432 4805

(8) (6) (11) (13)

0 4234 2785 so12 6691

rABLE

64 110 r_

180

139 14 90 98

75 55 115

137

190 197

160 12s

338

113 172 246 415 313

310

200 352 368 520 700

2

A&

9454 ,.*n_

9205

8880 8950 8927 8893 8893 8830 8850 8841

8654

8414 8350 8302

8145

7836 7867 7818 7822 7802

7571

7238 7214 7177 7184

(6) l,\ .,,

(8)

(6) (6) (7) ‘) (7)z) (8) =) (7) (7) (7)‘)

(6)

(5) (4) (4)

(5)

(4) (4) (4) (4) (4)

(4)

(4) (4) (4) (6)

9552 9451

9151

8770

8939

9033.4

8641

8406

8413.6

‘1

J)

(S) .a.“1 (6) ‘1

(20) ‘)

(SO) “)

(19) J)

(4.8) =*=‘)

GO)

(20) ‘1

(2.6) =*‘)

(40)

(30) *) (30) ‘)

7730 7740 8184

(40) “) (21) ‘)

(2.2)

(21) ‘)

(2.3) “*‘)

7810 7840

7836.7

7584

7213.6

from cp, n)

9481

8818

8308

8396

7846 7798 7807 7778

‘) ‘) ‘) ‘)

(25) m*“*r)

(20) rn.0)

(40) c*m)

(40) f. “)

(18) (18) (18) (18)

d) p) p) p)

(P, n)

(P, d)

(3He, (3He, (“He, (3He,

(P, n)

(P, n)

(P, n>

p) p) p) d)

(3He, t)

(P, n) (3He, t) (3He, t)

(p, n)

(3He, (“He, (3Ha, (“He,

geh) b, b, b,

(30) “)

(30) k,

(30) k,

(40) ‘*“‘)

(40) “) (40) rsrn) (40) f*m)

(40) ‘)

(15) (15) (15) (15)

9543 9462 nlrlt

9140

(7) ‘1; (15) “) IIC\ “\

(SO) ‘)

8868 (18) P) 8934 (18) ‘) 8912 lpp) 8900 (30) q,

8530

8290

8040

7807

7800 7787 7787

7580

7246 7228 7209 7175

Previous measurements of dEc (keV) ~_~ from from additional experiments (‘He, t)

displacement energies, 40 < A < 70

rr(l So) “) present &b/sr) exp. d, (keV)

A listing of Coulomb

3 f?.

8 5:

8

?’ p

64 66 67 68

Zn-Ga

0”

s-

o*

0’

!:

2

(7) (4)

2025 (30) 3855 (7) 8007 (7) 603X (7)

5496 1432

6398 (7) 4643 (11) 6810 (6) 61326 (6)

72 81 120 X28

110 183

35 172 37 100

9879 9813 9787 9733

9644 9564 (6) (6) (6) (6)

f6b (6)

9350 (6) “) 9360 (6) 9271 (6) 9287 (6) 9282 (7) “)

9890 9800 9190 9760

9550 9400

9292

9361

(3 ‘1 (25) m*orF)

(25) m,I)*‘)

9356

9273

(15) “)

9292

(p, n)

(P. n)

9740

9810

(8) ‘1

(8) ‘1

(JHe, co 9610 (50) “1 9570 (50) “)

(15) “1

9375

“) The quantities T and J” are the isospin and the total spin and parity of the parent analogue states in the Z,: nucleus, When there is a possibility for isospin mixing in the Z:, nucieus between states of isospin T and T-f , both values of isospin are given in parenthesis. b, The quantity I& is the excitation energy in the final nucleus of the state listed and is ded.uced from the t3He, t) Q-values measured in the present experiment and the (p, n) Q-value to the ground state, which are taken either from the tables of Mattauch et al. 2Q) or from the values of Freeman (It ai. 2J* 2s1. “j The quantity ~(18’) is the c.m. (jHe, t) cross section at 18” (lab) for 3He particles with 24.6 MeV incident energy. The relative standard deviation errors are estimated to be & 12 % and absolute errors &20 %, except for the chromium isotopes which have errors 120 % relative and &:JO “/:,absolute. df The quantities listed are Coulomb displacement energies deduced using the measured c3He, t) Q-values and eq. (2.2). These values have been obtained using the calibration energies fisted in footnote *). The standard deviation error in the Coulomb energy shift between isotopes is estimated to be &3 keV. Absolute errors are given in parentheses. “) The 42Ca(3He, t)42S~ reaction was used as a cahbration reaction for ah targets between “%a and 50Ti. The reactions ‘eCr(“He, t)“‘Mn and “‘FX3He, t)44Co were used as calibrations for the chromium and iron targets, respectively. The Coulomb displacement energies for 5s* Go,a1L42,64Ni, b3*6BC~, and 6**68*G7+6aZnhave been calculated assuming a Q-value of -_9349.6&5.0 keV for the S8Ni(p n)58Cu(g.s.) reaction 2s) and an excitation energy of 202&2 keV for the 0’ IAS in ssCu. The energy splitting between the 58Cu g.s. and 0’ IAS is an average value obtained from three different (jHe, t) spectra. 1) Ref. 45). k) Ref. 46). r) The value listed is a cross+section weighted average* ‘) Ref. 24), hf Ref. 4J). @) Ref. 43)s “f Ref. 49). “) Ref. 48). a) Ref. 47). ‘) Ref. 48). Of Ref I 21) . “) ltefs, 8*-=). “) Ref. 7). ‘) Ref. s0). The results are measured relative to a S”Ni(p, r~)~~Cu(g.s.) Q-value of -9349.6 and does not include the experime~~~al error of the latter (i5 keV), ref. *$). ‘) Ref. $4)* “) Ref. 52). “) Ref. f3). “1 Ref. jl). ‘) Ref. 6). “) The cross sections and Coulomb displacement energies quoted have been obtained assuming that the strongest transition seen at 18‘ (lab) is the analogue of the ground state of the parent nucleus, Other levels which could correspond to anaiogues of low-lying excited states of the target nucleus arc aiso listed together with the spins and parities of those levels and the Coulomb energies deduced from the measured Q-values and the excitation energies in the target nucleus. The other Levels seen, hawever, may also be TX non-analogue states or T c states mixed with the analogue state (set subsect. 2.4).

63 65

Cu-Zn

62 64

F. D. BECCHETTI

162

et al,

region of excitation energy expected for the ground-state analogues. These peaks are not due to isotopic impurities. Moreover, fig. 9 indicates that the 18” cross section to either member of the doublets in 56Co, “*Co or 64Cu is iess than expected from the systematic (N-Z)/A’ dependence found for O* to Ot TAS transitions,

Zn-Ga

a

i6

$

T

bl: l_J___L I 1 I i L-I-_I--i1 22 26 30 36 38 NEWRON NUMBER Fig. 9. 2-b i8”,,& f3He, t) cr5ss section to states near the excitation energy expected for the fAS of several doubly even target nuciei (see table 2). The cross sections have been ~orrnalj~ed by the quantity I@(N-2)/A’. An (N-Z)/A’ dependence is expected for ilL = 0, (%e, t) O* -* Gt IAS transitions between isotopes having a neutron excess within the same shell (see subsect. 2,3). For nuclei in which two states were strongly excited, the individual cross sections (L, C’) and the summed cross sections (0) are shown. The straight lines are least-squares fits to the summed cross sections.

2.4. FRAGMENTING

OF BOUND

ANALOGUE

STATES

A plausibie explanation of these results is that a charge-dependent part of the nuclear Hamiltonian such as the Coulomb interaction causes the analogue state to mix with a state of Iower isospin which has the same spin and parity. Such phenomena have frequently been observed for unbound analogue states in resonance experiments Thedoub~ets observed in 56Co and 58Co have been studied previously [refs. “-“)]. and the results reported elsewhere 40). It was determined that at least in 5sCo, and probably in 56Co the 7” = O”, T = T,, analogue state is mixed with one or more nearby J” = O’, .i = To - 1 states resulting in a fragmentation of the f3He, t) 0’ IAS strength. It was shown that the cross section to the analogue state is the sum of cross sections to the states involved, provided the transition strength to T, states of the

COULOMB

same spin and parity is negligible

DISPLACEMENT

‘O). The summed

ENERGIES

cross sections to the doublets

163

ob-

served in 56Co, 58Co and 64Cu are shown in fi g. 9. It can be seen that these values are in much better agreement with the (N--2)/A’ systematics than are the cross sections to individual states. No conclusion can be made about isobaric spin-mixing in 6’Cu. In order for isospin mixing to occur it is necessary that the T, state mixing with the analogue state have the same spin and parity which for 56Co, 58Co and 64Cu is J” = O+. In the work reported earlier, it was concluded that the members of the doublets in 56Co and 58Co are J” = O+, although a l+ assignment for one of the states seen in 56Co could not be ruled out 40). In an experiment in which the 16 keV doublet in 64Cu was not resolved 35), the (3He, t) differential cross section to these states was found to have a shape typical of AL = 0. This would indicate J” = 0’ or possibly 1’ for these states. The work of Bruge et al., however, indicates that the (3He, t) angular distribution for a O+ to l+ transition would differ significantly from a AL = 0 shape 34), so that the states observed in 64Cu are most likely J” = O+. From the known systematics of level densities one can estimate that the average energy spacing of J” = O+ levels near the analogue states in 56Co, 58Co and 64Cu are 900 keV, 120 keV and 25 keV, respectively, so that there is a reasonable probability of having a J” = O+, T, state close in energy to the analogue state 41). If the members of the doublets observed in ’ 6Co, 58Co and 64Cu are taken to be J” = O+ states, the magnitude of the matrix elements of the charge-dependent parts of the Hamiltonian coupling the analogue and T, states must have values 33, 10 and 7 keV, respectively, in order to explain the energy splittings and (3He, t) cross sections observed in the present experiment 40). Th e sp ins of some of these states have not been definitely established, and therefore the hypothesis of isospin mixing for these states is still a tentative one. The present results indicate, however, that in medium weight nuclei, bound isobaric analogue states can be fragmented among several states. 2.5.

COULOMB

Coulomb

DISPLACEMENT

displacement

energies

ENERGIES

have been deduced

AE, = 764 keV-Q-E’

using the relation (2.2)

and the results are listed in table 2. The quantity Q is the measured (3He, t) Q-value to the analogue of a state in the target nucleus which has an excitation energy E’. The quantity 764 keV is the sum of the 3He-triton and neutron-proton mass differences. For the doublets in 56Co, ‘*Co and 64Cu, the Coulomb displacement energies are listed in table 2 for both members as if each were the analogue of the target ground state. For these nuclei, the cross section weighted average energies are also listed. The latter quantity would be the correct Coulomb displacement energy if the T, component of each doublet had zero strength in the (3He, t) reaction. The errors listed for the weighted averages do not include the possibility that the T, component may have some spectroscopic strength. In the (3He, t) spectra for the nuclei 57Co and 6’Cu, other peaks were seen (see

164

F. D. BECCHETTI

et al.

figs. 5 and 6) which could correspond to transitions to analogues of low-lying firstexcited states in 57Fe (P = 3-, E’ = I4 keV) and 61Ni (f” = s-, E’ = 67 keV), respectively. The cross sections and Coulomb displacement energies deduced from the Q-values to these levels are listed. It is also possible that the states seen are non-

0

-80 z 1 -120 F7 R I 120 N 80 2 + LO s: 0 G. I Q -LO &J a 80 10

-80 22

2L NEUTRON

26 NUMBER

28

30

Fig. 10. Experimental Coulomb displacement energies as a function of 2 and N of the parent nucleus. The solid lines connect exper~ment~ points. The solid points and data taken from table 2. The open circles represent data taken from the following references: Q1Ca-41Sc, ref. 2p); 4*Ca-45Sc, ref. 7’); 47ca-47Sc, ref. 44); 49Ca-4qSc, ref. 42)* 43Sc-“3Ti, ref. 72); 46Sc-46Ti, ref. 73); 47Sc-47Ti, ref. 74); 4sSc-QqTi ref. ‘4): J1Ti-51V, ref. 75). Values of A& calculated using the model of Hecht and &nedke for seniority zero and one nuclei (see text) are indicated by crosses.

analogue T, states or T, states of the proper spin and parity, which are mixed with the ground-state analogues and share the (3He, t) strength. No peak which would correspond to the analogue of the first excited state in 4 ‘SC (J” = 8+, E’ = 13 keV) was seen (see fig. 8). Several checks on the present data can be made which help establish their validity.

COULOMB

DISPLACEMENT

ENERGIES

165

For example, the present value for 46Ti-46V of 7826 i_ 4 keV agrees very well with the precisely measured 7826.7 + 2.2 keV of Freeman et al. ‘“). In three separate runs, the spacing between the 1’ ground state and 0” analogue state in 58Cu was within 1 keV of 202 keV. This agrees with 202+ 2 keV determined by Harchol el al. in a (p, ny)

26

28 NEUTRON

30

32

3.4

NUMBER

Fig. 11. Experimental Coulomb displacement energies as a function of 2 and N of the parent nucleus. The solid lines connect experimental points. The solid points are data, taken from table 2. The open circles represent data taken from the following references: 49V-4gCr, ref. ‘e); s2V-a*Cr > ref * 76); 53V-53Cr, ref. 76); 53Mn-53Fe, ref. 53); 55Fe-SJCo, ref. 77). Values of -lEc calculated using the model of Hecht and Jlnecke for seniority zero and one nuclei (see text) are indicated by crosses.

experiment using a Ge(Li) y-ray detector 4g), After a series of runs, the ‘sNi calibration point was repeatable within 2 keV, and the 42Ca point was repeatable within 2.7 keV. Several months before the titanium measurements quoted in table 2 were made, a pair of targets of mixed isotopic composition were tested. One contained equal amounts of 46Ti, 48Ti, “Ti and the other contained equal proportions of 46Ti, 47Ti

F. D. BECCHETTI

166

et al.

and 4gTi. That set of results differs from those of table 2 by no more than 2 keV. Hence, one has some confidence in the 4 to 7 keV accuracy quoted for the present results. Coulomb displacement energies calculated using eq. (2.2) are listed in table 2 to-

-80

-160 Co-Ni

-240 F 0 > P =: z

- 160

&

-240

3 c s:

-320

2=2-l

-80

Ni -Cu a?=28

-400

, -280

Cu-Zn 2 = 29

g

-360

s

-140 rF

\,

I1

II

30

I!

32

I

34 NEUTRON

I

36

I

I

38

I

I

I

40

NUMBER

Fig. 12. Experimental Coulomb displacement energies as a function of 2 and N of the parent nucleus. The solid lines connect experimental points. The solid points are data taken from table 2. The open circles represent data taken from the following references: 57Co-57Ni, ref. 48); 61Co-6’Ni, ref. ‘a); 63Co-63Ni, ref. 78); 59Ni-59Cu, ref. 5’); 63Ni-63Cu, ref. 79); 65Ni-65Cu, ref. ae); 66Ni-66Cu, ref. 5’); 66Cu-66Zn, ref. 81); 67Cu-67Zn, ref. 53); 6qZn-69Ga, ref. j4); 70Zn-7*Ga, ref. 54). The scale is different from that used in fig. 1I.

gether with excitation energies deduced from the measured Q-values and the 1964 mass table ““). The present measurements of AE, are shown in figs. lo-12 as a function of neutron number, N. Values of AEc obtained from other measurements are shown for isotopes not included in the present experiment, The quantity 7200 + 300 (Z-20) keV has been subtracted from the experimental values of AE, in order to show the details in the data.

COULOMB

DISPLACEMENT

3. Theoretical 3.1. BASIC

EXPRESSION

FOR

ENERGIES

167

interpretation

AE,

Several review articles on Coulomb displacement energies can be found in recent literature 51*56. 57). Most of the existing analyses use a phenomenological model to predict a functional form of the Coulomb displacement energies. They require several free parameters to fit the experimental data. Nolen and Schiffer ‘92*51, “) have used a simple expression relating Coulomb displacement energies, AE,, to the overlap of the neutron excess with the proton distribution “). In their formulation AE,

= AE;+ AEFch+ AE;“‘+ AE;” = direct + exchange + spin-orbit

(3.1) + correction

terms,

where AE; = ;T

(3.2)

I’,-(r)P,,..(r)d3r. I

The quantity Vc(r) is the Coulomb potential due to the protons in the parent nucleus and pexc(r) is the charge distribution of the protons in the analogue state obtained by the T- operation on the neutron excess distribution of the parent nucleus. The exchange term, AEeXCh, arises from the antisymmetrization of the extra proton with the core proton distribution and is always negative. The term AEZ”. is the Coulomb spin-orbit displacement energy. Typically, the ratio AE~ch/AE~ is -4 %, while AE&“./AEg is +2 %, depending on the configuration of the neutron excess. Exact expressions and numerical approximations for AEGxch and AEC”’ can be found in refs. 51P“). The quantity AEF” includes several terms. The most important terms are energy shifts due to core polarization, isospin impurities, the Thomas-Ehrman effect for unbound analogues ‘, ‘, 51*5 7- 60). Most of these corrections are estimated to be less than 2 ‘A of AE: with some having positive and some having negative signs. The direct term thus accounts for more than 90 % of the total Coulomb displacement energy. 3.2. CHARGE

RADII

A functional form for the direct and exchange contributions to AE, for mirror nuclei can be obtained by assuming that the charge distribution due to all protons is uniform within a radius R, (see refs. G1,‘j*) and references therein). The statistical model was used to estimate the exchange term; the resulting expression is of the form [ref. ““)I

AE; = [a(2Z + 1) - bZ” - cf(Z,

T)] ;.

,

(3.3)

” where a = 0.60, b = 0.613, c = 0.30 and f(2, T = 4) = (- 1)“. It was suggested [refs. “, ““)I that for odd Z and T > 3 the proton pairing effect should decrease with

F. D. BECCHETTI

168

et al.

increasing T. This was confirmed by experiment ‘j3). This T-dependence of the pairing term may be combined with the previous term by using for f(2, T) the following form **): f(Z,

T)

=

1-e

1-(-1)z_ 2T

It is customary

to assume that the radius R, varies with A as R, = rJ+,

(3.4)

although this expression is known 61) to represent only the gross variation of R, as a function of A. The coefficients a, b, c,f(Z, T) and eqs. (3.3) and (3.4) define the parameters R, and r, in terms of Coulomb displacement energies. A least-squares fit to the experimental values of AEc listed in table 2 with the coefficients a = 0.60, b = 0.613, c = 0.30 yields r, = 1.264 fm. This is in very good agreement with the value r, = 1.270 fm as determined by fitting the charge radii for the same nuclei which were measured by electron scattering and mu-mesic X-rays ’ 5S“* 64). Although substantial differences between experimental AE, values and the AE,U from eq. (3.3) are observed for individual nuclei, particularly those within the If% shell, we have deduced approximate values for charge radii of individual nuclei, from the inverse of eq. (3.3) by setting AE: = AEc. The resulting R, values compare favorably with the charge radii determined by other means 6’s 64) (rms deviation approximately +0.06 fm). The isotopic shifts of R, tend to be less than those observed in electron scattering or mu-mesic X-ray measurements; however, the qualitative features are similar. The relation between the isotopic shifts in the measured Coulomb displacement energies and the shifts in the radii of j&(r) is discussed in detail in subsect. 3.4. The value of the coefficients a, b and c depend on the assumptions made about the radial form of the charge distribution and p_(r), and the overlap between them (eq. (3.2)). Furthermore, it is known 51) that the exchange term as given by eq. (3.3) with a = 0.60, b = 0.613 and c = 0.30, overestimates the true exchange term by about 35 % for A = 50. Thus the agreement with the known charge radii may be fortuitous. However, in the absence of direct measurements, eq. (3.3) can be used to estimate values for R, in the mass region 40 < A < 70 to an accuracy of better than +O.l fm. It is also useful to hnd a simple functional form AE,(A, Z) to parameterize the AE, values; e.g., Long et al. 65) have used AE, = aZA-+--P

(3.5)

in the region 34 < A < 165, and found CI = 1429 keV and fi = 849 keV. We obtain from an analysis of the data between A = 40 and A = 70 CI = 1464 keV and /I = 1049 keV.

COULOMB 3.3. MODEL

OF HECHT

DISPLACEMENT

169

ENERGIES

AND JANECKE

If the neutron excess can be considered as occupying a single shell the general expression for d& reduces to a simple form. The lf; shell is a particularly good example. Hecht has derived 66S67) expressions for AE, in terms of T, the isospin of the anaIogue pair, and v, the seniority of the neutron excess of the parent state. If one assumes the lowest possible seniority (v = 0 or l), then the ordinary Coulomb displacement energies for the neutron excess confined to the lfi shell reduce to the form “) v = 0:

I+ =

1:

AE,

= cr+pZ’-yz

121-(n-8)2

2T-I$

2T+3

[ AE,

-

1 ’

n-8

= a+jKZ’+y,

2T(T+l)

where it = A-40, 2’ = &z-T = Z-20. The quantities CI,p, y1 and yz are related to Coulomb matrix elements “). In the absence of a Coulomb spin-orbit interaction when y1 = y2, Janecke has shown “) that, for a uniformly charged sphere with a radius R, = r,A*, the expression (3.6) reduces to a form similar to that of eq. (3.3), provided the quantity Z”A-* is approximately constant 5s). The introduction of the Coulomb spin-orbit interaction

Parameterizations Parameterization of d& (keV>

TABLE3 of Coulomb displacement

energies rms xa/Pc) deviationdf WV)

Source

Range of A& data fitted

A 1429.Z4-1’3-849

ref. 65)

34 < A < 165

B 1464ZA-“3-1049

this exp. 40 < A <

70

59

2

160

86

C eq. 3.3 with R, = 1.265A-*‘3 fm

thisexp.40

< A <

70

59

1

355

114

20
28 28

22

5

“’

10

E Same as D but using present data this exp. 20 < N < (42Ca-42Sc, 47Ti-47V excluded) 2otz< %O = 7344 key, &, 3: 329 keV, a = 0.239 ~01 = 2.94 key, yoz = 3.58 keV

28 28 ?I

5

5

16

D Hecht-Janecke (eq. (3.6)), lf$ shell, I = 0 or 1 only (42Ca-42Sc exciuded) ref. 4, a, = 7353 keV, & = 326keV, ,I = 0.236 701 = 2.89 keV, yol = 3.70 keV

“) b, ‘) ‘? A%.

n *) W)

2

< 150

n is the number of A& values which were used to determine the parameters. M is the number of free parameters which were varied to fit the experimental AE, data. x2/-P is the total chi-square value divided by the number of degrees of freedom, P (= n-Mf. The numbers listed are the rms deviations between the calculated and experimental values of

170

F. D. BECCHETTI et al.

requires that y1 # yz and gives rise to an odd-A pairing term in addition to the odd-Z term included “) in eq. (3.3). Janecke has used Hecht’s model to analyse Coulomb displacement energies in severai different mass regions 4, 55r’ “). Within the If; shell it was assumed that the quantities c(, fi, y1 and y2 all had an A-dependence of the form a = c&&4)““, etc. Jgnecke performed a least-squares analysis of twenty-three experimental Coulomb displacement energies (excluding 42Ca-42Sc) using five free parameters: go, PO, yol, yo2 and A and obtained an rms deviation of about +8.5 keV. Lowest seniority was assumed, i.e., v = 1 for odd A, J” = se; v = 0 for even A, $A - T = even, and J” = O+. These calculations have been repeated using the results of present measurements (table 2) together with other data in this region (see figs. 10 and 11). The accuracy of the data used in the present analysis is typically a factor of four greater than the data used in the original analysis of Janecke. The results are listed in table 3 together with Janecke’s results. The rms deviation using the best-fit parameter set obtained in the present analysis of 21 AE, values is 16 keV. The calculated values of dEc using these parameters: cl0 = 7344 keV, PO = 329 keV, yol = 2.94 keV, yoz = 3.58 keV and I = 0.239 are shown in figs. 10 and 11. The values of y o1 and yoz obtained in the present analysis are in good agreement with Hecht’s values if the pair 42Ca-42Sc is excluded from the analysis. The origin of the anomaly for A = 42 has been discussed by Janecke and others 4, 56), b u t no satisfactory explanation has appeared. The calculated value of AE, for 47Ti-47V agrees quite well with the experimental value if we assume that the state at 4300 keV in 47V (see fig. 3) is the analogue to the J” = $- (v = 1) first excited state of “Ti. The 47Ti ground-stats analogue (J’ = s-) must have v > 1 SO that eqs. (3.6) are no longer valid. Hecht’s model reproduces much of the fine structure that is observed in the COUlomb displacement energies for If; shell nuclei. Hecht’s model and also the Carlson-Talmi model 6g) have been extended to the region Z > 28, N > 28 but the large amount of con~guration mixing increases the number of parameters, and the resulting expressions for AEc become quite complicated 70). In this case, it may be meaningful to apply eq. (3.1) directly and fit the A& data in terms of parameters of p,(r) and p@(r). 3.4. ISOTOPIC

SHWTS

OF THE NEUTRON

EXCESS RADII

Several authors have employed eq. (3.2) to calculate Coulomb displacement energies using wave functions obtained from various model calculations 51, 57- 60). The results can be expressed in terms of the difference in rms radii of the neutrons and protons: (ri)” - {r,2>” or the rms radius of the neutron excess ((r&>*). Although the approximations used in these calculations differ, the results indicate (r~)h-+ E 0.0i0.2 fm in the region 40 .c A < 70. It is possible, however, to circumvent the numerical evaluation of eq. (3.2) and still obtain information about the neutron excess. Eq. (3.2) implies 51) the following approximate relations between the isotopic

COULOMB

shifts of the quantity

AE:

DISPLACEMENT

and

+W?) M

the rms radii

_

AE; 6(AE:) AE;

1 2

z

of the neutron

W~A

constant

excess distributions:

p,(r),

” ’

_ 1 d(’

constant

P_(Y).

(3.7)

z 8((r&)*). Th ese relations can be used to obif the isotopic shifts in charge radii are known.

We have assumed here that S((r,‘)*) tain an estimate of ~((re~,)3)/(re~,)~ Using eqs. (3.2) and (3.7) one finds

~(+> x

171

ENERGIES

S(

_

+2

[

<&,,>’ where AE,d = AE,-AE~h-AE~o.-AE~;

(t-,2)+

1’

64~34) x

AE, is the experimental

(3.8) Coulomb

dis-

placement energy. The usefulness of eq. (3.8) is limited by uncertainties in the isotopic shifts in AEph, AEF”. and the other correction terms, AEFrr. However, these shifts can be expected to be small between nuclei with protons and neutrons in the same shells. Therefore numerical approximations 51) for AEFh and AEC” are probably sufficiently accurate for use in eq. (3.8). We have used AEFch = [0.6-0.6132”-0.3f(Z,

r)] $

,

(3.9)

K”

with f(Z, T) as defined previously, e2 = 1440 keV * fm and R,, the charge radius. For the Coulomb spin-orbit term we have used the approximate relationship

A-%“~= -

‘g Rzj C (nj-pj)(l u

where the nj and pj are the neutron angular momentum j; (I * CT)= I ifj

* c) keV * fm3,

(3.10)

J

and proton occupation numbers in an orbit of = I+* and (I . CJ) = -I-- 1 if j = 1-J.

Isotopic shifts of (ret,)* between pairs of nuclei (Z, N) and (Z, N+2) have then been calculated using eqs. (3.8)-(3.10) and the experimental AE, data listed in table 2. Charge radii and isotopic shifts in charge radii were taken from electron scattering or mu-mesic X-ray work. The results are shown in fig. 13 as a function of ~+2, the neutron number of the heavier isotope. Also shown are the isotopic shifts in the radii of the proton distributions. Curve a shows the isotopic shifts in the neutron excess radii assuming LIE: = AE,, i.e., neglecting all correction terms. Curve b shows the shifts when eqs. (3.9) and (3.10) are used to estimate the exchange and spin-orbit corrections. In calculating AE:“. mixed lf-2p proton and neutron configurations were used. The error bars include only the experimental errors in the shifts of (rl)+ and in AE,. The calculated values at N+ 2 = 28, 30 and 36 for different pairs of isotopes are very similar

172

F. D. BECCUETTI

ei al.

except for the two points at N+2 = 30 which include the Coulomb spin-orbit corrections. The discrepancy between the two values reflects the uncertainties in the calculation of the term AE$“’ which are mainly due to uncertainties in the occupation numbers for the If% and lf+ shelis.

PROTONS

NEUTRON EXCESS

22

24

26

28

30

NEUTRON

32 34 NUMBER

36

:

Fig. 13. The percent shift in + (top) and 3 (bottom) between the isotope pairs (Z, N) and (2, N+2) as a function of the neutron number of the heavier isotope (= N-I-2). The values shown for the proton radii have been deduced from the charge radii and isotopic shifts given in the following references: 40-42Ca, refs. 12*r3); 42--44Ca, refs. 12*r3); 46--48Ti, refs. rz*az); 48-_5Ti, rcfs. 12. 82). SO-SCr, ref 83). 52-S44~r, ref. 83). S4-Sfif?e, r&q 14); SS-LONj, refs. 13.14)* 7 9 63-6S(& refs* 1:. 83 6O--6ZNi, refs 13.14 ). In some cases average values obtained from electron )‘; scattering and mu-mesic X-ray measurements have been used. The values shown for the neutron excess have been obtained from the proton shifts and the measured shifts in the Coulomb displacement energies (table 2) using eq. (3.8). The curve labelea %” connects points which have been calculated without including corrections for isotopic shifts in the exchange and Coulomb spin-orbit terms in the expression for aEc. The curve labeled “b” connects points which include corrections using the numerical approximations given in ref. sl). Error bars estimated from experimental uncertainties are shown for the points associated with curve a only. The broken lines have been calculated assuming that the radii are proportional to A+ and that the diffuseness is zero, SO that

There are several interesting features in the isotopic shifts of the radii of the proton and of the neutron excess distributions as shown in fig. 13. The rms radius of the proton distribution, (ri)“, as determined by mu-mesic X-ray and electron scattering work, shows essentially no net increase between N-t-2 = 22 and N+2 = 28. In contrast, the rms radius of the neutron excess distribution, (r&>*, increases, but at

COULOMB

DISPLACEMENT

ENERGIES

173

less than the rate expected from an A* dependence of (r&J*. The addition of a pair of If+ -2p shell neutrons to the closed lf: neutron shell results in a substantial increase in (ri)% and (r_&)*. This increase is equal to or, for the neutron excess, probably exceeds the A* increase. With the further addition of neutron pairs to N-l- 2 = 30, ~ associated with curve a of fig. 13 have been performed and yielded isotopic shifts in the rms radii of the proton distributions which are very similar to the other known shifts in (t-i)* for the same neutron number. A term in AEprr which is linear in 2T = N-Z and thus may contribute significantly to the isotopic dependence of the displacement energy is due to the charge dependence of nuclear forces ‘*). Estimates of the size of this effect, however, vary by a factor of 10. Therefore this term was not included in the present analysis. It is possible to use S((r&,J3) and 6((rt)*) to obtain information about d(
However, this requires assumptions about the shift in the radii of the core neutrons: ~(?r) z ~({r~}~) we can evaluate eq. (3.11) using the results on ~(“)/” as plotted in fig. 13. The term

was calculated by using the approximate value of 0.05 for 62Ni as given in ref. 57). Values of this quantity at lower N were then estimated from the shifts 6((r&>*> and d((ri)*). The results using d((r,“,,>t) of fig. 13, bare shown in fig. 14 as a function of N4 2. (The rms radius of the nuclear matter distribution may be obtained as a weighted average of (ri)* and (r:)*, but again the results depend on the assumptions made for the radius of the core neutron distribution.) It is not possible to draw definite conclusions about the isotopic shifts in the rms

174

F. D. BECCHETTI

et al.

radii of the neutron distribution from accurate Coulomb energy differences unless the rms radii of the core neutron distributions are known. The Coulomb energy differences provide a measure only of the neutron excess distribution if the charge distributions

are known

from mu-mesic

24

26

X-ray and electron

28

30

32

34

36

38

scattering

experiments.

How-

40

NEUTRON NUMBER

Fig. 14. The percent shift in the rms radius of the neutron distribution between pairs of isotopes (Z, N) and (Z, N+Z) assuming S() = &t)/
ever, optical model studies 84** 5), rc-mesic scattering and absorption and K- mesic X-ray studies * ‘) may yield neutron radii.

experiments

86)

The targets used in this experiment were prepared by Mrs. C. Harrison at Princeton University and by Mr. G. Ott at the University of Minnesota. Dr. H. Ohnuma, Dr. W. Makofske, Mr. W. Dykoski, and Mr. R. Wallen are to be thanked for their assistance in taking data. The authors are grateful to Dr. R. Sherr for useful suggestions in planning the experiment and to Drs. G. W. Greenlees and J. P. Schiffer for valuable discussions. The authors are also indebted to Drs. E. Romberg and H. J. Kim for making experimental data available prior to their publication.

References 1) J. A. Nolen, Jr., J. P. Schiffer and N. Williams, Phys. Lett. 27B (1968) 1 2) J. P. Schiffer et al., Lett. 29B (1969) 396, 399 3) H. A. Bethe, Phys. Rev. 54 (1938) 436; H. A. Bethe and R. F. Bather, Rev. Mod. Phys. 8 (1936) 82 4) J. Janecke, Nucl. Phys. All4 (1968) 433 5) J. D. Anderson and C. Wong, Phys. Rev. Lett. 8 (1962) 442 6) J. D. Anderson, C. Wong and J. W. McClure, Phys. Rev. 138 (1965) B615 7) R. Sherr, A. G. Blair and D. D. Armstrong, Phys. Lett. 20 (1966) 392 8) T. A. Belote, W. E. Dorenbusch and J. Rapaport, Nucl. Phys. A109 (1968) 666 9) J. M. Loget and J. Gastebois, Nucl. Phys. Al22 (1968) 431 10) J. A. Cookson, Phys. Lett. 24B (1967) 570 11) C. J. Batty, B. E. Bonner, E. Friedman, C. Tschalar, L. E. Williams, A. S. Clough and J. B. Hunt, Nucl. Phys. All6 (1968) 643

COULOME 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60)

DISPLACEMENT

ENERGIES

175

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