Page 1

PHYSICAL REVIE%' C

VOLUME 33, NUMBER 5

MAY 1986

Elastic scattering of protons from

Ca from 20 to 50 MeV and nuclear matter radii

R. H. McCamis,'T. N. Nasr, J.Birchall, N. E.Davison,

%.T. H. van Oers, and P.J.T. Verheijen~

Department ofPhysics,

Winnipeg,

University ofManitoba, Manitoba,

Canada R3T2N2

R.F. Carlson and A. J.Cox

University ofRed!ands, Red!ands, California 92374

Department ofPhysics,

B.C. Clark

Department ofPhysics, The Ohio State Uniuersity,

Columbus,

Ohio 43210

E.D. Cooper

Nuclear Theory Group, TRIUMI', Vancouver, British Columbia,

Canada V6T233

S.Hama

Department ofPhysics, The Florida State Uniuersity,

Tallahassee, Florida 32306

R. L. Mercer

(Received 15 July 1985)

Differential cross sections for proton elastic scattering from

energies in the laboratory

25.0 and 45.0 MeV. Measurements

and at 5.0'intervals

4%; the scale errors are also 3%. The data have been analyzed

tic optical models. The optical potentia1 parameters

formation for the calcium isotopic series, which is compared to previous results.

~'Ca have been measured at seven

energy range from 21.0 to 48.4 MeV, and for~Ca at six energies between

were made at 2.5'intervals

between 90'and 170'. Relative errors in the data are typically between 3% and

from 10'to90'(laboratory

angles),

by both nonrelativistic

and relativis-

were used to calculate nuclear matter radius in-

I. INTRODUCTION

The determination

problem

in nuclear physics.

able to give precise and meaningful

clear matter and charge distributions

the state of our understanding

issues, such as the nature of the interactions

ous types of particles and the role of these interactions

scattering and reaction phenomena.

The calcium

isotopic series

excellent set of nuclei for testing sundry

The large number of stable isotopes which are available,

the large

variation

in N/Z,

"double-magic"

nuclei are all important

popularity.

In particular,

in recent years, the calcium iso-

topes

have

been

widely

utilized

methods of extracting

nuclear size information

perimental

data. To give only some examples,

vestigations

have used pion elastic and inelastic

ing,"

intermediate

energy

scattering,

elastic and inelastic scattering of alpha par-

ticles,'

pionic atom information,

tions,'

and low energy

to gain information

on the sizes of calcium nuclei.

Of course, nuclear charge radii for the calcium isotopes

have been obtained to high accuracy from electron scatter-

ing

measurements,

and

from

of nuclear

sizes is an important

The extent to which we are

statements

within nuclei reveals

of much more fundamental

about nu-

between vari-

in

(

"Ca) provides

nuclear theories.

an

and

the existence of two

exponents of this

in

exploring

various

from ex-

recent in-

scatter-

inelastic

proton

elastic

and

'

pion total cross sec-

( ~ 100 MeV) proton scattering"'

'

studying

transitions

of

muonic calcium atoms,'

ic probes are less certain, since the strong interaction

not as well understood

as is the electromagnetic

tion.

nucleon

interaction

at the energies

by the well-known

hs3 resonance.

Bethe'~ point out that the pion interacts at resonance en-

ergies primarily

in the low-density

density

distribution,

so that nuclear matter radii may be

difficult

to study

in this way.

Boyer et al.'shows a dependence of the radii on the pion

energy, which the authors attribute to shortcomings

model which they used to analyze their data.

analysis of intermediate

ton elastic scattering from nuclei appears to give the most

reliable

values for neutron

density

elements of the analysis still need to be tested more fully.

The recently

developed

relativistic

tion'

's(RIA) inay well provide

the

nonrelativistic

treatment

Unfortunately,

the nucleon-nucleon

input to some analyses are still not well determined

appropriate

Alpha particles have the advan-

tage that the analysis is simplified

nature of the projectile,

but complications

structure of the alpha particle, such as particle exchange

between the projectile and the target, may be significant.'

Pionic x-ray experiments

yield only two pieces of data,

but investigations

using hadron-

is

interac-

Pion studies''ohave the advantage

that the pion-

is dominated

studied

However, Johnson and

region of the nuclear-

Further,

the analysis of

in the

While the

energy (=500—1000 MeV) pro-

distributions,

several

impulse

an improvement

at intermediate

amplitudes

approxima-

over

energies.

needed as

at the

energies.'

by the spin-0, isospin-0

due to the

1624

1986 The American

Physical Society

Page 2

33

ELASTIC SCATTERING OF PROTONS FROM

'

'

'Ca. . .

1625

strong interaction

the matter radii very well.

ied the elastic scattering of low energy polarized

from a wide range of nuclei

including

the calcium isotopes.

limited angular

range at only one energy, 65 MeV. Lom-

bardi

studied

with energies between

10.8 and 16.3 MeV, which Lerner

et al.

argue is too low an energy for the analysis which

was used.

The motivation

for this work is to study the reliability

of the extraction of nuclear

dard optical model analyses of a very comprehensive

of low energy proton elastic scattering data.

it is of interest to compare the results of this nonrelativis-

tic treatment

with an analysis

gy

II. EXPERIMENT

shifts and widths, and do not constrain

A previous experiment"

stud-

protons

(=25 targets were examined),

However,

they covered a

et al.'

'

'

Ca, using

polarized

protons

size information

from stan-

set

In addition,

using Dirac phenomenolo-

21

The proton beam facility of the University of Manitoba

sector focussed cyclotron was used to measure

scattering

differential

cross sections for protons interact-

ing with the calcium isotopes.

ed for the isotopes

45.0, and 48.4 MeV.

Due to there being previous

surements

on ~Ca at 21.0, 23.5, 26.3, and 48.0 MeV,2

measurements

were taken

only at 25.0, 27.5, 30.0, 35.0,

40.0, and 45.0 MeV for that isotope.

energies were known with an uncertainty

48 MeV) from a calibration of the momentum-analyzing

magnetic

field using cross-over measurements;

gy spread of the proton

beam

(FWHM) at 45.0 MeV. The incident beam was collected

in a well-shielded

Faraday

current integrator.

Enriched targets of

iments;

their

thicknesses

and

shown

in Table I. Target thicknesses

combining

accurate weighing

mined target areas.

All handling

formed

in an inert atmosphere

tion.

Scattered protons were simultaneously

angles to the right of the beam and at nine angles to the

left of the beam using arrays of NaI(T1) detector assem-

blies. The two arrays were separated

30.00 deg, while the detectors in each array were spaced at

intervals of 10.00+0.02 deg. By an appropriate

turntable

positions

(which could be set to an accuracy of

+0.01 deg), data were accumulated

the elastic

Experiments

were conduct-

''Ca at 21.0, 25.0, 30.0, 35.0, 40.0,

mea-

The incident proton

of +200 keV (at

the ener-

200 keV

was less than

cup and integrated

using

a

'''

Ca were used for the exper-

enrichment

were measured

and photographically

of the targets was per-

to minimize

factors

are

by

deter-

contamina-

detected at eight

by a fixed angle of

choice of

at laboratory

angles

III. DATA REDUCTION

The energy

computer program to integrate the elastic scattering peaks

and to subtract

any background

under the peak of interest.

The typical energy resolution

of the NaI(Tl) detectors

used was 300 keV at 30 MeV.

Occasionally,

however,

when

spectra

collected

were

analyzed

using

a

which might

have been

the peak

due to inelastic

IO4

]

25,0 MeV

IO3

IO2

IO

lO

10

—2

IO

from 10 to 90 deg in steps of 2.5 deg, and from 90 to 170

deg in 5.0 deg steps.

The angular

detector was approximately

2.2 deg vertically

horizontally.

Data were collected using standard electron-

ic systems,

and spectra were stored on magnetic

subsequent

analysis.

Two

fixed

monitor

detectors

geometries

were accurately

aligned to be at equal angles

(15.0 deg) left and right of the incident beam.

justments

to the beam transport

data taking, were made to bring the ratio of the number of

elastic scattering

counts in the two monitor

1.00, while at the same time keeping

through

the center of the scattering

sured that the incident beam passed along the zero degree

axis of the scattering

chamber.

counts was checked at frixluent

lection at angles larger than 90 deg to ensure beam stabili-

ty.

acceptance of each

by 0.8 deg

tape for

with

nearly

identical

Slight ad-

system, just before final

detectors to

the beam passing

chamber.

This en-

This ratio of monitor

intervals

during data col-

Isotope

Ca

Ca

"Ca

"Ca

TABLE I. Target details.

Thickness

(mg/cm

}

6.57+0.13

5.87+0.11

6.71+0.10

7.12+0.12

Enrichment

factor

{Fo)

99.97

93.70

98.55

97.69

0

30

60

Bc ~

90

120

l50

l80

(deg)

FIG. 1. Present elastic scattering differential

25.0 MeV, for the four calcium isotopes.

best fit differential

cross sections using the nonrelativistic

cal model.

cross sections at

The sohd lines are the

opti-

Page 3

1626

R. H. McCAMIS et aL

33

TABLE II. Contributions

to the relative uncertainties

in the differential

cross sections.

Type of uncertainty

Counting

Dead time correction

Correction

Finite geometry

Detector angle

Incident

proton

Impurity

subtraction

statistics

for reaction

losses in NaI(Tl)

correction

energy

(~orst case)

Value

(1. 0%%uo

g0.1%%uo

(0. 3%%uo

&1. 0%%uo

g 1.5%

2.0%

~ 1. 0%%uo

scattering to the first excited state and the elastic scatter-

ing peak partially

overlapped

targets, because of the lower excitation

first excited states2 ), the analysis

peaks, using the elastic scattering peak (obtained at anoth-

er angle)

as the characteristic

were made for any possible oxygen contamination

targets

as previously

described,

ferential

cross

sections

ing

Corrections

to the data were also made for computer

dead time, the contributions

in the target, finite geometry,

in the NaI(Tl) detector material.i'

magnitudes

for the above corrections

(except

at

very

forward

angles,

correction

was

several

percent),

(usually

for ~2Caor ~Ca

energies of their

program

unfolded

the

peak shape.

Corrections

of the

dif-

scatter-

utilizing

available

for proton-' 0 elastic

26

29

from other calcium isotopes

and nuclear reaction losses

It may be noted that

were less than 1%

where

the

approximately

dead

time

5—10%

(worst case value), less than 2%, and less than 3%, respec-

tively.

In most cases, multiple

at a given beam

energy

and scattering

both left and right detector arrays.

tions were then taken as a weighted

dual measurements.

The differential

cross sections

errors between 3% and 4%. The various contributions

the relative

errors in the cross sections are indicated

Table II. The absolute

uncertainty

distribution

is approximately

uncertainty

are given in Table III.

Representative

angular

distributions

differential

cross sections for all four isotopes

35.0, and 45.0 MeV are shown in Figs. 1—3, respectively.

Differential

cross sections

in tabular

upon requests

or from the authors (W.T.H.v.O.).

measurements

were made

and with

angle,

The final cross sec-

average of the indivi-

generally

have relative

to

in

for a given angular

3%,the contributions

to this

of the measured

at 25.0,

form are available

IO

lQ

/

I

I

I

I

I

I

I

l

[

l

IO

35.0 MeV

IQ3

IO2

2

(n

IO

IO

1

IO

I

L

CA

IO0

E

a

Oo

IO'—

IO

-2

IO

-3

IO

IO

IO

I

I

30

60

90

I20

I50

I80

Hc~(deg)

IO&

(

l

(

I

(

I

~

I

i

I

0

50

60

8c ~

90

I20

l50

l80

(deg)

FIG. 2. As in Fig. 1, for 35.0 MeV.

FIG. 3. As in Fig. 1, for 45.0 MeV.

Page 4

33

ELASTIC SCATTERING OF PROTONS FROM

'

'

'Ca. . .

TABLE III. Contributions

scale of the differential cross sections.

to the uncertainty

in the absolute

Type of uncertainty

Detector solid angle

Target rotation

Target thickness

Beam current

angle

integration

Value

1.0%

(1. 0%

(see Table I)

1.0%

IV. NONRELATIVISTIC

OPTICAL MODEL ANALYSIS

U(r)=V,(r)

Vf(xo— )+i4ar IVY„f(xl)

iWf(xr— )+(V +iW

)

—f(x

dr

) S L

4

r

d

The data were analyzed

version of the program sEEK.

was of the form:

using a modified

The optical potential used

and extended

other weighting

data points.

of analyzing

tions was introduced.

The data used in the analysis

ferential cross sections, the previous data of Bray et al.i

on p- Ca elastic scattering

26.3, and 48.0 MeV, the available experimental

tal reaction cross sections

power

data

available

for protons

21.0i26.3,730.0,s35.8,s940.0,~45.5,sand 49.0 (Ref.

41) MeV. No analyzing

power data are available for the

other calcium isotopes.

The initial optical model parameters

data

were

taken

from

a previous

scattering

by van Oers.

For the analysis of the p- Ca

data, the initial

values for the optical model parameters

were the results of the present analysis for

were then made, in turn, over the potential

geometrical

parameters,

both

geometry

(except for the exchange term), and finally the

exchange potential

parameters.

at zero for all of the work, although

than the relative error of the individual

Consequently

no arbitrary

powers with respect to differential

relative weighting

cross sec-

included

the present dif-

cross sections at 21.0, 23.5,

proton to-

for

Ca, and the analyzing

scattered

by

Ca at

for the p-~

analysis

~Ca

of p-~Ca

Ca. Searches

strengths,

strengths

the

and

potential

Note that W„was fixed

exploratory

searches

—V,„(—1)'f(x,„).

V, is the usual Coulomb

by the potential

ly:

potential,

which is represented

charged sphere, name-

due to a uniformly

3——,

R

I

1

0.8—Zl. Q MeV

0.4—

-0.4—

t

I

I

t

l

fag

~

'a

~g

0.8

V,(r)=

r &R,

Z8

2R

P

Ze

I')R

T

Here R is the Coulomb

radius parameter

experiments,

and 1.240 fm for~' 2'~'~sCa,respectively.'i

form factorsf(x, ) are Woods-Saxon (Fin'ini) functions:

f(x, )=[1+exp(x, )]

where

radius R =r,A'~, and r, is the

as determined

from electron scattering

which

was taken to be 1.316, 1.306, 1.285,

The radial

(3b)

300 MeY

—gV

0.4 —35.8 MeV

I

.,

v

-0.4—

40.0 MeV

0 8

45.5 MeV

0,4

T

T

T

'T

L~

-Q 4

0.8

0.4

-Q 4

The exchange, or l-dependent

order to improve

particular,

as

guarantee

that the optical model parameters

unduly

influenced

by problems in Atting the backward an-

gle data. This potential contained eight adjustable geome-

trical parameters,

namely

a,„and six dynamical

parameters:

and

V,„. These parameters,

them, are adjusted to

ive the best fit to the experimental

data, using standard X minimization

is the sum of the X values for the individ~!al data points

(differential

cross section and analyzing

term, has been included

in

the fits to the backward

suggested

by

angle data in

studies,

would not be

several

and

to

ro, ao, rI, al, r, a, r,„,and

V, 8, 8'I, V,8'

or any

desired

subset of

procedures.

Here X

power) with no

49.0 Mev

,:,TV

0.4

0

}

l

I

I

I

l

i

I

0

20 40 60

80 l00

l20

F40 l60

)80

Gc ~

(deg)

FIG. 4. The best fit analyzing

nonrelativistic

optical madel, compared to previous

tal measurements

at 21.0 MeV {Ref. 36), 30.0 MeV (Ref. 38),

35.8 MeV (Ref. 39), 40.0 MeV (Ref. 40), 45.5 MeV (Ref. 39),

and 49.0 MeV (Ref. 41).

powers for p- Ca from the

experimen-

Page 5

1628

R. H. McCAMIS et al.

33

over W„did not result in either large values of W„or

improved

fits to the data. It is also worthwhile

that, although

the strength of the exchange potential

showed

limited

systematic

energy

porary deletion of that term from the potential

crease the total X for the differential

points for a given case by a factor of 2 or more.

Representative

fits to the cross sections are shown

Figs.

powers are shown in Fig. 4. The optical model parame-

ters which resulted

in the optimal fits are available

bular form (Ref. 32 or from the authors).

to note

V,„

dependence,

the tem-

could in-

cross section data

in

1—3, and the fits to the available

p- Ca analyzing

in ta-

V. RELATIVISTIC OPTICAL MODEL ANALYSIS

Modern

within a relativistic

relativistic

approaches

et al.

and Serot and Walecka.

at intermediate

Nuclear

optical model

treatments

of the nuclear many-body

framework

indicate the importance of

Relativistic

quantum

been

reviewed

recently

The success of the RIA

energies

is now

well

studies

using the Dirac equation

problem

effects.

have

field theoretical

by Anastasio

documented.'

I

containing

four-vector

standard

In this paper we extend these studies and consider calcium

isotopes at lower energies than previously

In most work employing

Dirac phenomenology,

local, spherically

symmetric

case, the Dirac equation

contains potentials

scalar, S, Lorentz

four-vector,

only), and tensor, T, components,

[a.p+P(m +S)—(E—V)

large cancelling

potentials

Schrodinger

Lorentz

scalar and Lorentz

its superiority

based

phenomenology.

have shown

equation

to the

considered.

static,

In this

potentials

are used.

with Lorentz

component

V (timelike

and may be written

iPa—

rT)/=0 .

(4)

The vector contains the static Coulomb potential for pro-

tons obtained

from the empirical

the tensor contains a small contribution

tion of the anomalous

moment of the proton or neutron

with the charge distribution

large spin-orbit

term required

include at least two of the three potentials

becomes

obvious

when

one considers

Dirac equation

for the upper two component

tion given by,

charge distribution

from the interac-

and

of the target.

by experiment

To obtain the

we need to

in Eq. (4). This

the second-order

wave func-

V +(E—V)—(m +S)

T— T— +— —

r

T— —(r p)T—

r

2

2

2

1

BA

Br

i

A

BA

Br

1

aA

Br

ro +

rArA

2T

(o"L) $„=0,

where

A =(m +S+E—V)l(E+m) .

The standard

(several

tentials (the SV model) to produce the spin-orbit

ment.

However,

one could consider other combinations

which

rely on the tensor to obtain

strength.

Scalar plus tensor (ST),vector plus tensor

or all three (SVT) potential

data at intermediate

ploy the standard (SV) model.

potentials

given by

model of Dirac phenomenology

MeV), cancelling

uses large

hundred

scalar and vector po-

enhance-

the large spin-orbit

( VT)

models have been used to fit

energies.~In this work we will em-

In this case we use optical

Fig. 5. In this figure the large real vector and scalar po-

tentials for the best fit to the p-~Ca data at 35 MeV (see

Fig. 7) are shown

along

with

imaginary

scalar

potential

and the negative

imaginary

vector potential.

the positive,

short-range

long-range

produce the

These potentials

200

S=Vf,(r)+i Wg,(r),

(7)

i~/&PYÃPi PZ/iP~

V=V„f„(r)+iW„g„(r),

where the form factors are two parameter

The model

contains

standard

nonrelativistic

model to analyze

the calcium

cross section data and available

at those energies where analyzing

able.

The imaginary

central

narily has a surface peaked geometry

this while using a volume form for the imaginary

and vector potentials

through

Fermi shapes.

the same as in the

%'e used

isotope elastic differential

Ca analyzing po~er data

power data were avail-

12 parameters,

phenomenology.

this

potential

at low energies

We accom. plished

ordi-

scalar

the cancellation

sho~n in

I Scalar

-400

-600

t

I

4

R (fm)

FIG. 5. Real and

determined

tentials have been multiplied

imaginary

scalar

and vector potentials

from p- Ca at 35 MeV. The plotted imaginary

by a factor of 10.

po-

Page 6

33

M

ELASTIC SCATTERING OF PROTONS FRO

IO

-20

Ca

-40

IO"

y)lo'

JD

E

!0'

40C

55 MeV

I

R (urn)

FIG. 6. Effective central spin orbit andp

from the potentials of Fig. 5.

and

otentials calculated

b,o,

IO—

effective central

and spin orbit terms

ed character of the central absorption

The imaginary

spin-orbit

to the real part, just as at higher

opposite in sign to

e r

ies~~It is worth emphasizing

'

1

sequence of the relativistic

orbit potential

is a consequence

shown

in Fig.

is

evident.

term is very

ener-

p'-

treat-

that the imaginary

'

- Ca data

thi m

ep

resent

Figure 7 shows a typical fit to p-

ion. The analyzing

as the cross section bu

tivistic

analysis

at every

quality

spin observables

f the lack of systematics

someo

t e ac

ciency.

e

Th

agreement

persists throug

out

h ut this energy region, cannot be obtai

hod'

r

henomenology

in the Schrodinger

pe

dependence

and

additional

model.

Figure 8 shows

our results

4sCa at 35 MeV; again

represented.

In order to compare

the resu ts o

anal sis with

those of Dirac phenomenology

tral and spin orbit terms from ~.

gives

in

power is not as w

t is much better than the nonrela-

i

gy.

ener

. The need

cannot be overemp asiz, an

can

is traceable to t is

at large ang es, a

s e

without

parameters

ults for

the

large

angle

well r

or

'g e

e i-

o

a feature which

obtained

introduc'

th

e

Ca,

data

data

g

optical

a, an

is well

is we

in

its of the nonrelativistic

e

it is cus-

l0'

I.O

50

100

!50

l

05—

50

I00

8c ~ (deg)

l

l50

FIG. 7.

powers for p- Ca at

est

8

fit differential

35 M V using the relativistic optical mode .

cross sections

and

analyzing

model.

e

u

'

Z aA

ar

3

aA

U =

cen

2EV+2mS— V2+$2+ Ti+2

1

8

dr

BA

Br

2riA

and

—1 aA

rA

r

r

+2-

(&0)

2E

k h

8r

red

oes no ap-

Furthermore,

as the

In the wor

contribution

reciably

the corresponding

tic analysis.

The ac o

1

considered

ere

f th

ect. t e ex

the tensor potential

e

anomalous

affect the extracted

a,

effect is omitted

is ignored,nota-

term

o

moment

paraineters.

1

rom

at the energies

k of

s stematic

lack

o

ig

k f hi h quality spin observables at the

here

produces

g'

some

y

behavior of the scalar and vecto

ch energy.

Integrated

eac

en

of these potentials

p

r

otentials determined

such as the rms ra ii

in

g

e inte rais and rms radii

at

~ ~

quantities,

or the volume

r

- Ca for the relativistic

or p-

a or

d the nonrelativistic

real J/A from Dirac phenomenology

varies

linearly

with

— —447+2.2T

while the nonrelativistic

pear somewhat

less systematic.

volume integrds per nucleon

analysis

(open circ es

(closed circles).

(dashed

line)

an

1

)

ana ysis

e r

Th

ener

energy

with

wi

values ap-

'

in Fi . 9

The sohd line

'g.

Page 7

1630

R. H. McCAMIS et al.

33

IO5

I00—

Irnag in(3ry

G

I

l0

20

30

40

50

60

70

80

90

100

IO'

IO

FIG. 9. Volume integral per nucleon for the central potential

from nonrelativistic

{closed circles) and relativistic

analyses of p-~Ca data.

{open circles)

IO-

i

tials determined

given in Table IV. These average values have been used in

calculating

b,r~ values from the various isotopic differ-

ences as described

beloved.

from

the analysis

of each isotope are

VI. EXTRACTION

OF THE NUCLEAR MAT;j.ER RADII

50

ec.~.(deg)

Ioo

I50

The mean square radius of a potential,

by:

V(r), is given

FIG. 8. Best fit differential

and p- Ca at 35 MeV using the relativistic optical model.

cross sections for p-42Ca, p-44Ca,

shows J/A obtained

tion.

The J/A values from the relativistic

the other isotopes are in good agreement

by open circles in Fig. 9. The imaginary

of the effective central potential for p-~Ca are essentially

constant.

In Fig. 9 the 80 MeV point is from the relativis-

tic analysis given in Ref. 45.

The average mean square radii of the real central poten-

from a relativistic

Hartree calcula-

analysis of

with those shown

volume integrals

r2V r

IV(r)dr

r

For the relativistic

tential is used.

optical

potential

(coro'~ '"),&r'„& ts

(rot1 )—

analysis

the real effective central po-

In the nonrelativistic

given

by Eq. (3) is used.

case, a real central

To order

gro+

g2r 0o

(12)

where

model parameters

calculated for each isotope at each proton energy studied,

ro and ao are obtained

for that potential.

from the best fit optical

This parameter

was

TABLE IV. Numerical

the text.

results of the present analyses; for various nuclear mean-square,

or root-mean-square

radii, as discussed in

Nucleus

{fm )

(r')

{fm }

(r2)1/2a

P

{fm)

(r2)1/2

{fm)

hr„P

{fm)

Nonrelativistic

analysis

Ca

Ca

44Ca

Ca

17.01+0.33

17.31%0.24

17.54+0.51

17.73+0.19

11.60+0.23

11.90+0.47

12.13+0.65

12.32+0.44

3.39

3.42

3.42

3.38

3.42+0.06

3.48+0.13

3.53+0.17

3.60+0.11

0.03+0.05'

0.06+0.13

0.11+0.17

0.22+0.11

Relativistic

analysis

40Ca

42Ca

44Ca

4'Ca

15.58+0.48

15.67+0.56

16.11+0.39

16.60+0.47

11.60+0.23

11.69+0.77

12.13+0.66

12.62+0.71

3.39

3.42

3.42

3.38

3.42+0.06

3.42+0.22

3.53%0.17

3.67+0.17

0.03+0.05b

0.00+0.22

0.11+0.18

0.29+0.17

From Ref. 13, with a correction for the finite size of the proton. The uncertainties

This value of hr,~for

Ca was used as input to our analysis (see the text for discussion).

in these values a&ere taken to be +1%.

Page 8

33

ELASTIC SCATTERING- OF PROTONS FROM """ea.. .

1631

and then the average values of (r,~& were calculated for

each isotope (see Table IV). It should be noted that the

value of (r,~&for

Ca at 21.0 MeV was omitted

the averaging

process, since that value was greater than

seven standard

deviations

froin the average of the remain-

ing results for that isotope.

Greenlees

et aI.

have shown that, to a first approxi-

mation, the mean square radius of the matter distribution

and the mean square radius of the optical potential are re-

lated by:

&»' &=( . ',&—( '&pi, ,

where (r

nucleon

interaction.

The nuclear

can be calculated from the formula:

p~(r)=—

from

(13)

&zb is the mean square radius of the nucleon-

matter

density, p~(r),

p&(r)+—p„(r),

Z

where p~(r) is the proton density distribution

the neutron density distribution,

ty distributions

are normalized

la, it can be seen that:

&=—

and p„(r) is

and ~here all three densi-

to one. From this formu-

„&r~p&+—

„&r.& .

(15)

(

2&1/2

(

2&1/2

(16)

From Eq. (13), one obtains for the radius difference

tween

Ca and

Ca, for example,

be-

&r. 'p&42 (r. 'p&4c=&r'&4/

Thus, knowing (r,~& and (r~ & for each isotope, one

(17)

The values of (r~~&,the mean square radius for point pro-

ton distributions,

are taken from electron scattering exper-

iments.'

The errors on (rz & are 2.0%.

Because of the severity of the approximations

been used in deriving Eq. (13), and because of the result-

ing uncertainty

concerned only with the relative quantity br„~,the differ-

ence between the rms radii for the neutron and proton dis-

tributions,

given by:

that have

in the value for (r &2s,here we will be

would like to be able to solve Eqs. (15)—(17) for (r„&and

then for hr„~. However,

this is not possible without

further

piece of information,

the analysis of high energy proton-nucleus

the following:

b,r„z( Ca)=0.03+0.05 fm .

Then, Eqs. (15)—(17) were solved for the values of b,r„

fori'

Ca, which are shown in Table IV.

The uncertainties

in (r» &,„,given

obtained by combining,

in quadrature,

tion from the average

value, and two smaller

tions:

an uncertainty

the absolute

normalization

cross sections (taken to be +0.09 fm ),and an uncertainty

in (r,~&,due to possible ambiguities of the optical model

in fitting

the experimental

taken to be

0.08 fm, since such a variation in (r,

sulted in an increase in X by approximately

made a fit noticeably

worse than the optimal fit. Uncer-

tainties in the absolute

energy calibration

of the angular

scale were studied and were found to can-

cel, to first order, due to the experimental

methods

used. The other uncertainties

follow from the uncertainties

~.„,1~Ca).

one

which

we have taken from

scattering to be

in Table IV, were

the standard

devia-

contribu-

in (r,~&,due to the uncertainty

of the ineasured

in

differential

data.

This uncertainty

was

&re-

and

50%,

and in the zero

and analytical

given in Table IV

in (r,~&,„,(r~&'/i,

and

VII. DISCUSSION

A summary

along

ments, and the results of two theoretical

presented

in Table V. It should Q noted that a similar,

but more extensive

summarized

the situation to that time.

present

work, recent

values of b,r„~, given

have been obtained from an analysis of 800 MeV proton

elastic scattering,

a study of pion elastic scattering

three energies

between

116 and 293 MeV analyzed

Kisslinger-type

optical-model

of 104 MeV alpha particle elastic scattering,

of the b,r„~results of the present analyses,

the results of the analyses of other experi-

with

calculations,

are

table,"published

in 1979, has ably

In addition to the

in Table V,

at

by a

potential,

and an analysis

using a fold-

TABLE V. Values of br„~, in fm, for the isotopes of calcium.

table, published

in 1979,is given in Ref. 4.

The methods

are discussed further

in the text. A more extensive

Method

(p,p)20—50 MeV

(p,p)20— 50 MeV

(m,m.) 116 MeV

(m, n) 180 MeV

(n,m') 292.5 MeV

(p,p) 800 MeV

(a,a) 104 MeV

HFB

DME

0.03+0.05'

0.03+0.05'

0.15+0.07

— 0.02+0.04

—0.09+0.02

0.01%0.08

—0.016+004b

—0.04

—0.05

Ar„p (fm)

"'

-Ca

0.06+0.13

0.00+0.22

— 0.04+0.03

—0.11+0.05

0.08+0.08

0.03+0.04

0.01

0.03

"Ca

0.11+0.17

0.11~0.18

0.21+0.10

0.11+0.04

0.05+0.10

0.10+0.08

0.02+0.04

0.06

0.09

"Ca

0.22+0. 11

0.29+0.17

0.33+0.09

0.18+0.04

0.16+0.04

0.18+0.08

0.17+0.05

0.14

0.19

Ref.

This work (nonrelativistic)

This work

(relativistic)

1

1

1

4

49

50

51

'Input to analysis.

Calibration

nucleus (see original reference for details).

Page 9

R. H. McCAMIS et al.

33

ing model approach.

representative

Hartree-Fock-Bogoliubov

other using the density matrix expansion (DME) approach

to Hartree-Fock theory.

While the relativistic optical model is better able to fol-

low the p- Ca analyzing

powers (see Figs. 4 and 7), the

values of b,r„~deduced from the nonrelativistic

tivistic

analyses

are seen to be in reasonable

with each other, as each follows the general trends from

nucleus to nucleus as observed

in Table V. Although

the errors in the present

larger than most other quoted errors, they are comparable

to the scatter in the hr,~values from the various sources.

It should

be noted

that the results of the pion elastic

scattering

experiment'

show a definite

for all isotopes as the pion energy rises, which is attribut-

ed by the authors

to insufficiencies

model used to analyze their data.

Also shown

calculations,

(HFB) formalism,

in Table V are two

one

theoretical

using

and

the

the

'

and rela-

agreement

in the other studies given

work are

decrease

in hr,~

in the theoretical

%e conclude

equation

phenomenology.

represent

is the agreement

rms radii, and volume integrals, are at least as systematic

as in the nonrelativistic

case. The relativistic

provides

a testing ground for various

structure

calculations;

however,

tests of Schrodinger

versus Dirac phenomenologies

energies are hampered

at present by the lack of high qual-

ity spin data.

that

viable

Fewer

the approach

adjunct

parameters

based on the Dirac

Schrodinger

are

needed

is

a

to

based

to

the data in the Dirac case; especially noticeable

at larger angles.

Integrated

quantities,

approach

nuclear

relativistic

we stress that definitive

at low

This work was supported

ences and Engineering

Research Corporation,

ence Foundation

U.S.Department of Energy.

in part by the Natural

Research

Council of Canada,

U.S.A., by the U.S. National

under Grant Phy. -8306268, and by the

Sci-

by

Sci-

'Present address:

Manitoba, Canada ROE ILO.

Present address:

ta, Canada T6G 2C2.

~Present address:

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