REACTION CROSS SECTIONS FROM EIASTIC SCATIERING DATA
by
Felix Sannes
A thesis submitted to the Faculty of Graduate Studies
and Research in partial fulfilment of the requirements
for the degree of Master of Science.
Foster Radiation Laboratory
McGill University
June
Montreal
@
Felix Sann
1967
1966
Au thor:
Ti tle:
SANNES
REACTION CROSS SECTIONS FROM
l~LASnC
SCAT1ERING DATA
- i -
ABSTHACT
A method for obtaining proton-nucleus reaction cross sections
from the elastic scattering data is described.
The procedure involves
the determination of the forward nuclear scattering amplitude from
the elastic data and the calculable Coulomb seattering amplitude.
The interference of the nuclear amplitude with the known Coulomb
amplitude at small scattering angles makes it possible to determine
both the magnitude and phase of the nuclear amplitude.
Application
of the optical theorem then gives the total nuclear cross section
from which the reaction cross section can be deduced.
Existing elastie seattering data is analysed and severa!
reaction cross sections are computed.
Good agreement between the
computed and existing experimentally measured reaction cross sections
i s obtained.
- ii -
ACKNOWI.EDGEMENTS
I wish to thank
nr.
D. G. Stairs, my research director,
who offered many helpful suggestions during the course of this work.
His guidance and assistance were greatly appreciated.
I am also
grateful for the encouragements I received from my director and
the staff and students of the Foster Radiation Laboratory.
F'inally, I acknowledge the financial assistance in the
form of a scholarsbip from the Canadian Kodak Company tbat was held
during part of this work.
TABLE OF CON1EN'fS
Page
Abstract
i
Acknowledgements
ii
I.
Introduction
1
II.
Theoretical Background
5
III.
The Reaction Cross Section from the
Elastic Scattering Data
lC
IV.
Calculations and Results
18
v.
Conclusion
30
Appendix A.
Measured Form Factors
33
Appendix B.
Detai led Computations, Carbon 143 Mev
36
References
37
- 1 -
I
INTRODUCTION
Cbarged particles incident on atomic nuclei are scattered
Q, two forces, the short range nuclear force and the weaker, long
range Coulomb force.
The scattering amplitude for such a process,
say proton scattering from light nuclei, can be written to a good
approximation (Goldberger and Watson, 1964) as the sum of the Coulomb
amplitude and the nuclear amplitude:
In scattering experiments which attempt to stuqy the nature of the
nuclear force, the presence of Coulomb scattering is often unwelcome
since it complicates the analysis.
presence of the
r~ulomb
In sorne instances however the
force can be used to advantage.
In partieular,
the observed interference of the Coulomb and nuelear amplitudes at
small angles gives information about the phase of the nuelear amplitude
since the Coulomb foree is known and the
caleulated.
r~ulomb
amplitude can be
Tbus if the charged particle differentiai cross section
is known, both the magnitude and phase of the nuclear amplitude ean
be
determined.
Neutra! partiele differentiai cross sections on the
other band would give no information about the phase of the nuclear
amplitude.
- 2 -
The formula for determining the nuclear amplitude
its interference with the
Bethe (1958).
r~ulomb
~
observing
amplitude was first obtained by
He calculated the real and imaginary parts of the
forward nuclear amplitude for the scattering of 313 Mev protons off
carbon using the elastic scattering data of Chamberlain (1956).
The
formula was derived using nonrelativistic quantum mechanics of an
extended particle.
The corresponding formula starting from relativistic
quantum field theory was later derived by Solov'ev (1966).
This method for obtaining the nuclear scattering amplitude
was also used
~
HOldeman and Thaler (1965) and extended to yield
an expression for the total reaction cross section.
The procedure
is based on the optical theorem and applies only to charged particle
scattering.
Once the nuclear amplitude is known, the optical theorem
can be applied to obtain the total cross section
where fnuc(O) is the forward nuclear amplitude.
Tb obtain the reaction
cross section, it is necessary only to subtract the total nuclear
elastic cross section which is computed from the measured elastic
differentiai cross section and the calculated
C~ulomb
differentiai
cross section.
Thus a detailed examination of the elastic scattering data
and the charge distribution of the target nucleus yields the reaction
cross section (the charge distribution or form factor of the scattering
nucleus must be known to enable the accurate calculation of the Coulomb
scattering amplitude).
- 3 -
The following sections of this paper will describe the method
in detail and severa! proton reaction cross sections for light elements
will be calculated.
One pleasing feature of this method for obtaining the reaction
cross section is that it is model independent.
~at
is, its accuracy
does not depend on uncertainties in the nuclear potential.
The form
of the nuclear interaction does not enter the calculations at ali.
Only the form of the Coulomb interaction is needed and this is well
known.
The method however bas sorne limitations.
The elastic
differentiai cross section must be known to very small angles,
at !east as small as the Coulomb interference minimum (i.e. that
region in which the nuclear and Coulomb scattering amplitudes interfere
destructively causing a dip in the differentiai cross section).
This
minimum occurs at about 5° for medium energy (say 100 Mev) protons
scattered from light nuclei.
It is difficult to obtain accurate elastic
data for small angles and bence the accuracy of the computed reaction
cross section is limited
by
the uncertainty in the elastic data.
Furthermore, the method eannot be applied to scattering
from heavy nuclei.
In this instance, the form factor derived
by
first
Born approximation is not aceurate since only one-photon exchange
processes are included in this approximation.
In the strongelectro-
magnetic field of heavy nuclei two-photon and higher orocesses contribute
appreciably and higher order Born terms must be included.
No simple
- 4 -
expression for the Coulomb scattering amplitude can be obtained and
bence it is not possible to calculate the nuclear scattering amplitude
nor the reaction cross section.
In the region wbere the method is applicable however, the
predicted reaction cross sections lie well within the experimental
uncertainties of the measured reactions cross sections.
- 5-
II
TfEORETICAL BACKGROUND
The standard technique for treating scattering problems is
that of partial phase shifts.
When the scattering potential is of
finite range, as is the case far the short range nuelear potential,
the solution of the
Sehroding~r
equation has the asymptotic form
(Messiah, 1961)
(2.1)
<t'
~k\f
= et..\t'i +
fte} ~
where k is the wave number of the incident particle and f(e) is the
seattering amplitude which may be expressed in terms of the partial
phase shifts ôt as
~
calculating the incident and seattered quantum mechanical
partiele current, it may be shown that the elastic differentiai cross
section is
(2.3)
<r le) ~
l f'(e) l ~
The total cross section is
(2.4)
- 6-
where f(O) is the forward scattering amplitude and otot includes both
the total number of scattered particles and the number of absorbed
particles.
F.q. (2.4) is known as the optical theorem and represents
no more than the requirement of particle conservation.
understood as follows:
only in a
decreas~
It can be
The scattered oarticles can have their orLgin
of strength of
tb~
incident plane wave.
This decrease
is accomplished by means of a destructive interference between the
plane wave and the scattered wave which takes place in the forward
direction.
The optical theorem expresses the balance between this
destructive interference and the scattered and absorbed particles.
rbe above analysis can be applied only to scattering from
a potential of finite range and not to the long range Coulomb potential.
In the case of Coulomb scattering the incident and scattered waves
cannot be considered as plane.
long range of the force.
They are distorted because of the
Also the phase sbifts do not approach a
constant but are a function of "r".
The asymptotic form of the scattered
part of the wave in the case of coulomb scattering is (Messiah, 1961)
fc(6) is the
(2.6)
r.~ulomb
fe.(<il) ~
scattering amplitude and is given by
l: . .
'S l.IA.
e e.xf>
?:
li. (.2. <>. +ïT - là' .Q.... si."' ~ )11j
or in terms of partial phase shifts,
cQ
<2. 7)
..ç~(.&)
=-
\C'
L l 'lll+l) exi> t \. <ï.st.) s-~"'" ~ f>.R..<...
JL
œt
e)
- 7-
The parameter y is given
(2.8)
'(
~
-
and the COulomb phase shifts ot
~
(2.9)
When charged particles such as protons are scattered from
nuclei, the effects of both the Coulomb and the nuclear forces must
be considered.
Goldberger and Watson (1964) have shown that for nearly
ali nuclear scattering experiments the total scattering amplitude
in the case of nuelear plus Coulomb scattering is given to a very
good approximation
~
(2.10)
~
The term fc(e) is the Coulomb amplitude given
F.q. (2,6) and
fn(~)
is the nuclear amplitude
oO
<2.11>
('
.,"'(e)
_,)
=(.2 ~ '..) L
)
L2R..+ \ e
2\..(S"...
~;. ,,.
( e
_ \)
~
,-JL ( ~ e)
.(...
The amplitude fn(e) describes the scattering
as modified
~
the Coulomb force.
~
the nuclear potential
This modification however is only
very slight due to the relatively greater strength of the nuclear
forces.
1he preceding analysis treats the source of the Ooulomb
field as a point charge.
Since the nuclear charge is in faet
- 8-
distributed over a finite region in space, a correction must be made
to the COulomb differentiai cross section derived from Eq. (2.6).
This correction becomes more important as the energy of the incident
particle increases.
At high energies the incident oarticle appreciably
penetrates the nucleus and no longer sees a Llr2 force field.
The
effect of the finite charge distribution is best illustrated if the
problem is treated in the first Born approximation.
Hofstadter (1957)
bas shown that the Coulomb differentiai cross section in tllis approximation
is
(2.12)
where
(2.13)
fc(Q) is calcu1ated from Eq. (2.6).
The quantity F(q) is called the
charge form factor and is given by
(2.14)
Ft'\-\
~ ~
ç~ c. ...hi."'<."\-~)
v- ..tv
p(r) is the charge distribution (spherieal1y symmetric in this caser
and q is the momentum transfer given
by
(2.15)
where k is the wave number of the incident partic1e in the centre
of mass coordinates.
Thus in the first Born approximation the
- 9-
actual COulomb differentiai cross section for a finite nucleus is
obtained from the point charge cross section by multiplying by the
square of a form factor appropriate to the particular nucleus under
consideration.
'T'his approximation works weil for ligbt nuelei but not for
medium and heavy nuelei.
'T'bis is beeause in the first Born approximation
the incident and scattered waves are considered plane waves.
For
beavy nuelei the waves are actually distorted by the strong
~lectromagnetic
field and the plane wave approximation is no longer valid.
Another
way of saying this is that the first Born approximation considers
only one-photon exchange processes while in the exact scattering
(especially from the heavier elements) two-photon processes contribute
appreciably and higher order Born terms must be included.
In view
of these difficulties, only seattering from ligbt nuclei will be
considered in this paper.
The nuclear structure (i.e. charge distribution and form
factor) is determined
expe~imentally
by electron scattering.
For
electron energies of lOO Mev and higher, the first Born approximation
gives accurate results.
The procedure involves fitting the exnerimental
scattering data with a mode! for the charge distribution.
Severa!
form factors relevant to the calculations in this paper are given
in Appendix A.
- 10-
III
Tffi REACTION GROSS SEGTTON FROM Tffi El.ASTIG DATA
The results of the preceding section enable the calculation
of the reaction cross section from the elastic scattering data.
Proton
scattering data in the energy region from 90 Mev to 300 Mev will be
analysed.
Only scattering from the light elements (up to aluminum)
will be considered.
The procedure to be outlined follows the work
of Holdeman and Thaler (1965).
The elastic scattering amplitude for proton-nucleus scattering
can be written according to Eq. (2.10) as the sum of the Ooulomb and
nuclear scattering amplitudes:
where the Coulomb amplitude is (Eq. (2.6))
ol='
(3.2)
.çt. (e) ~ ~-l
I
C.lSl.+ 1) e)c'f ( \..
~JL) S'~IA a-.._ t>..\. (c..os e)
.S2.
The nuclear amplitude (as modified qy the Coulomb force) is given by
(3.3)
~\1\.'e) ~
fc...e)
~ (_'). i. \.Y'
l-
Z
-f~t.a)
(.2 .~a+-t) '"l< il
(_ '2.;. o-... )(e~ f' (2 i. ~... ) - ~ ~ (<.o,eJ
}l.
where bt is the phase shift due to the nuclear force only and at is
- 11 -
the Coulomb phase shift defined
~
Eq. (2.9).
The sum in Eq. (3.3)
needs to be taken only up to L where L is large enough so that ail
higher order partial waves are outside the finite range of the nuclear
For t > L, the phase shifts ôt are zero and ali the t
force.
~
L
terms of Eq. (2.11) are also zero.
The total cross section is infinite because the Coulomb
cross section is infinite but it is assumed that their difference,
the cross section due to the nuclear force alone, is finite.
Taking
the imaginary part of the forward amplitudes and ignoring infinities,
Eq. (3.1) becomes
which
~
the optical theorem is just
(3. 5)
The total nuclear cross section is thus
(3.6)
The total cross section atot ean bo written as the sum of the reaction
cross section and the total elastic cross section so that Eq. (3.6)
bee ornes
Cï l'"
(3. 7)
-
q-..
o~-)J..n.l{<-C.e} +t,..C.e) \ ... - f..tJl.lfc.<.e)\"'
+
r
t
~JI, 1f ..<.e) \"' + 2 ~ -t;<.e) .ç',..le~
- 12 -
r~mbining
Eqs. (3.6) and (3.7), the reaction cross section is then
given by
(3.1:3)
where from Eq. (3.7) ael is
(3.9)
<r....t. ~ ~ ot V\.
( \
_ ~ ol~ [
"',.Le.) \-.. t-'l '4
f.,."' la) .f~Le )l
\r"._ t.~)}
(l"";te) -
a~Ît(e) is the experimentally observed differentia! elastie
cross section which includes both
r~ulomb
and nuclear scattering.
The Coulomb amplitude fc(e) and cross section a(e)c can be calculated
from Eq. (2.6) and the measured form factor.
Thus to obtain the reaction
cross section ar from Sq. (3.8) it remains only to determine the forward
nuclear amplitude fn(O).
Because the sum in Rq. (3.3) is finite, fn(e) may be expressed
for small angles as
(3.10)
For angles very much smaller than e
and fn(e) - fn(O).
The
= 1/L,
fn H:H is nearly constant
weak dependence of f 0 (6) one at these small
angles may be expressed with the aid of a form factor F(q).
Bethe (1958)
bas shown that fn(e) is given approximately for small scattering angles
by
- 13 -
(3.11)
This result follows from the first Born approximation for fn(e) if
the nuclear potential is assumed proportional to the nuclear density
p(r) ( F(q) is calculated from p(r) as in Eq. (2.14)).
Bethe further
assumed that the proton density distribution is the same as the nucleon
distribution.
excess.
This is certainly true for light nuclei with no neutron
Thus the Coulomb and nuclear amplitudes are both proportional
to the same form factor F(q).
f 0 (0) cao now be determined from the measured elastic scattering
cross section with the aid of Eq. (3.1).
section is given
The elastic scattering cross
~
fc(e) is the COulomb amplitude for a point charge and can be calculated
from the equations of Section II.
Tbus if the elastic scattering cross
section and charge form factor are known, Eq. (3.12) can be solved
to give fn(O).
The cross section at two angles must be known since
f 0 (0) is complex.
When calculating the COulomb amplitude for a distributed
charge., the "point" ('.oulomb amplitude of Eq. (2.6) is multiplied by
the charge form factor F(q).
The form factor obtained from first
Born approximation is real and bence it has no effect on the phase
of
fe(~).
It changes only the magnitude of fc(e). It is not
- 14 -
immediately obvious that only the magnitude and not the phase of fc(6)
should be changed
~
the extended charge distribution of the nucleus.
In actuality, the phase probably is affected but to a lesser extent
than is detectable in the first Born approximation.
This is one
limitation of this method and the reason why accurate results are
obtained only for scattering from light nuclei.
The best region in which to solve Eq. (3.12) for fn(O) is
in the very small angle region because here the approximation for
fn(6), Eq. (3.11), is very good.
But it is difficult to measure
accurately the elastic cross section at small angles beeause of
multiple scattering.
Also in this region both a~ît<e> and fe(9) are
mueh larger than fn(e) and the determination of fn{O) from Eq. (3.12)
depends on the small difference of two large numbers one of which,
a~Ît(e),
bas a large uncertainty associated with it.
rhis last
consideration suggests the best region in whieh to evaluate fn(O)
is where \fn(6)
minimum.
l-
lfe(9)
l•
This occursc. near the interference
If fn(O) is evaluated in this region, the uncertainty
in fn(O) will not be much larger than that of a~~t(6).
Hbwever,
if the interference minimum occurs at too large an angle, and if
fn(O) is evaluated at this large angle, then the approximation for
fn(e) Bq. (3.11) no longer holds.
Thus there is a limited region
in whieh fn (0) may be calculated wi thout too great an error.
It should be noted that there are two solutions of Eq. (3.12)
since it is a second order equation.
The fact that the COulomb and
- 15 -
nuclear amplitudes interfere destruetively indicates that the two
amplitudes have opposite phases.
Thus the correct physical solution
of Eq. (3.12) is that numerieal solution fn(O) which differs in phase
~
about w from the known phase of the Coulomb amplitude fc(e).
It remains now only to ca1culate the quantity ael given by
To evaluate Eq. (3.13) it will prove helpful to write it as
3 14
<• >
\r.a~.
'tr
-: 2rr
5[ ~~·"<.e)
- G"'ç_L.e)1 s\."'fa ote
G«>
e,[
+'ln [
1-Ç,.t.e)
\"'o~-'l'il...-Ç~"'c_..,Jt.Je)~ (,,.acl.ld
0
Now if B0 is so sma11 that fn(6) is nearly constant fore 490 then
Eq. (3.14) becomes
(3.15)
ll'...t
~
1r
':1. tr ~ (
(J"~.. )
- I:S""t.
ce)i ~~....e .I.e
s.
+ L-hr l ,.{') l c \
VI.
\"1. ,
1.
So
s "'"" 2:"
+ 4'lT (4. [.ç>,..lo) ~'~o \.
eo
.çt!'u,) s.:.... ~c~.al
E:.
The first integral can be eva1uated from the measured elastic data
.--
and the computed Coulomb cross section (correeted for the finite charge
- 16 -
distribution of the nucleus
factor).
~
multiplication
~
the appropriate form
Since fn(O) is already known, it remains only to determine
the limit of the last integral.
To accomplish this, Eq. (2.7) for
the Coulomb amplitude is integrated from 0 to rr and it is seen that
only the t
=0
term contributes:
(3.16)
Also
ti
(3.17)
Th us
) .ç:(.e.) ».:... ~.le ~ ~
u..,o (-li.'<.)[ 1 -
"-K~ C.:l i. ~ .~~,.. \'"' 9
-i)l
eo
e~~
(3.!6l
~:o )~!la) ~;,.oa ..l<a
Eq. (3.15) may now be written as
(3.19)
\r...... •
\1\c\) t~
~:tr ).
.a.1. (_")
e-.
+ ~ I .... ~ f,. to) \_1 - Q.~f C.- 'l i.<r; -t-'2.; ~JI-~.:.... ~~1
where the last integral of Eq. (3.15) has been multiplied by i and
the imaginary part taken instead of the real.
- 17 -
Combining Eqs. (3.8) and (3.19), the reaction cross section
finally becomes
(3. 20)
<:J.. •
~ !-( f,,to)
e.<f> ( -~ .:.". +'li. t )...
rr
Ç( \ï:1b)
:l..,.
- 11., L~)\ ".:... e
'iÏ."'-
~~
.tQ
e ..
Eq. (3.20) may be used to compute the reaction cross section
for charged particles incident on atomic nuclei.
This cannot be done
for neutral incident particles because in this case fn(O) cannat be
determined.
If the charge is zero, Eq. (3.20) reduces to the optical
theorem since if Z
= 0,
then y and a0 go to zero and taking
zero, Eq. (3.20) becomes
(3.21)
\..oltr"
~
r
\Mo
f-"" lo)
which is identical to Eq. (3.8).
e0 to
- 18 -
IV
CALCUlA TI ONS AND RESULTS
The caleulations of the reaction cross sections from the
elastic scattering data were performed on the McGill
I~d
7044 computer.
The results for proton scattering from carbon at 143 Mev are reproduced
in Appendix
B.
The Coulomb differentiai cross section is calculated from
Eq. (2.6) and the appropriate form factor (see Appendix A).
The total
Coulomb scattering into angles greater than e0 is then obtained by
numerical integration using Simpson's rule.
scattering into angles greater than
10 obtain the total elastic
e0 from the differentiai elastic
data, the trapezoïdal approximation for integration is used.
Next, Eq. (3.12) is solved for fn(O), the forward nuclear
amplitude.
The computations are simplified by writing Eq. (3.12)
in the form
-(;qi'\:
~AA.. C.e)
(4.1)
F~C.c:v)
The observed differentiai cross section a!~t(9) is divided by the
square of the form factor F2(q) to give the cross section for a
"point nucleus" <Bethe, 1958).
Eq. (4.1) is solved at successive
pairs of points in the smail angle region (two points are necessary
since fn(O) is complex).
At each pair of points the two equations
- 19-
are reduced to one equation (with one unknown) which is solved by
Newton's method of successive approximations.
Sq. (3.20) then gives
the total reaction cross section.
As was mentioned earlier, the most accurate reaction cross
section should be obtained if Eq. (4.1) is solved for fn (0) near the
interference minimum.
The r"'sults of the calculat:ions bear this out.
'T'he computed reaction cross sections for 143 Mev protons incident
on carbon toqether wi th the pairs of angles at which fn (0) was
evaluated are gi ven in 'T'able I.
The same resul ts are plotted as a
function of angle in figure 1 together with the experimentally measured
differentiai cross sections (Steinberg et al., 1964).
'!'able I.
Computed proton-carbon reaction cross
sections err at 143 r.iev and angles at which fn (0)
was evaluated.
el
e2
err (mb)
4.34
4.80
227.20
4.55
5.42
225.71
5.42
5.97
229.56
5.97
6.51
229.ô9
6.51
7.59
231.73
7.59
U.68
240.55
8.63
9.76
252.52
- 20 -
Figure 1.
The proton-carbon reaction cross section
at 143 Mev as a function of angle together with the
elastic differentiai cross section from which the
reaction cross section was eomputed.
The elastic
data is that of Steinberg et al. (1964).
FIGURE
700
CARBON
1.
143 MEV
\
0
600
MEASURED
der
-dn
CROSS
ELASTIC
SECTION
500
0
\
400
0
"-o---o------o------o______ o-
300~----~------~----~~----~------~------
5
6
e (c.m .)
7
8
9
250
240
{mb}
-o- o_..,.,
230
/
0
0
COMPUTED
.,/'0
CROSS
REACTION
SECTION
220
4
5
6
7
8 (c.m.)
8
9
- 21 -
The reaction cross section shou1d of course be independant
of where f 0 {0) is eva1uated and this is the case near the interference
minimum where the curve of Fig. 1 is flat.
At small angles the computed
cross section is low, due probably to uncertainties in the measured
elastic data caused by multiple scattering.
COrmack {1964) bas made
caleulations based on the Molière theory of multiple scattering Qy
a screened Coulomb field to determine the contribution of multiple
scattering to the counting rate.
He found that the ratio of the observed
counting rate (including COulomb and multiple scattering) to the counting
rate due to COulomb scattering alone was about 2 for proton-carbon
scattering at the smallest angles considered qy Steinberg.
Hence
the reaction cross section calculated from the small angle data cannot
be expected to be accurate.
At large angles the calculated reaction cross section is
high.
At these angles the approximation for the nuclear ampli tude
Eq. (3.12) is no longer sufficiently accurate to be applicable.
Thus when determining the average reaction cross section, only those
values on the plateau of Fig. 1 {i.e. the interference region) are
included.
The computed reaction cross sections were also found to be
independant of the angle Q0 of Eq. (3.20).
This angle was varied
throughout the interference region with no appreciable effeet on the
cross section (as indeed there should not be).
The method of successive approximations used to solve Eq. {4.1)
- 22-
introduces almost no error into the final results since two successive
results were required to differ by no more than .01% before the
calculations were terminated.
The main source of error in the final
computed reaction cross sections is the error in the measured elastic
scattering data.
by
An estimate of the size of this error
m~
be obtained
writing the differentiai elastic cross section as
where the interference term has been neglected.
(4.3)
J.~
"='
Then
l f"'" at~~
and
(4.4)
J.~
q-
--
2.. -fYI. ~f~
i c.oz. + s; "'. . . .
Since Eq. (4.1) is solved in the region where f .... f 1 Eq. (4.4) becomes
n
c
(4.5)
J.r;-a"'
::
2 .ç..... ol.P"'"
~ f ""-""
-
otf
-T"'"
Thus the percentage error in the nuclear ampli tude is of
the same order as the percentage error in the elastie differentiai
cross section.
This error appears also in the total nuclear cross
section since it is directly proportional to the imaginary part of
the nuclear amplitude.
No further error is introduced when the total
elastie cross section is subtracted from the total nuclear cross section
to obtain the reaction cross section.
This is because both the nuclear
- 23-
ampli tude and the total elastic cross section are calculated from
the same elastic data.
~he
errors in both have the same sign thus
partially cancelling, leaving the percentage error in the reaction
cross section unchanged.
Hence if the expression for the Coulomb
amplitude Eq. (2.6) is assumed correct and to introduce no error,
t'ben the percent error in the computed reaction cross section should
not be much larger than the percent error in the elastic scattering
data.
HELIUM;:, LI'l'HIIJM AND BElNLLHJM
The helium scattering data of Cormack and collaborators (1959)
at 147 Mev was analysed and a reaction cross section of 86 mb was
obtained.
~he
elastic data was first increased·:by
ll~·~
to correct for
an error in the original measurements (see Palmieri et al., 1963).
The reaction cross section has not been measured for helium at this
energy however the total cross section for incident protons not to
leave the target within a cone (half-angle 6°) in the forward direction
was measured by Palmieri and Goloskie (1964).
of ll2.4 mb at 149 Mev.
~bey
found a value
When the elastic scatt.ering into angles greater
than 6° is subtracted, a value of 83 mb remains for the reaction cross
section which is 3.5% smaller than the computed value of 86mb.
~he
lithium elastic scattering data of Johansson and collaborators
(1960) at 160 Mev gave a reaction cross section of 170mb.
Johansson et al.
(1961) have also measured the reaction
- 24 -
Table II. Experimenta11y measured reaction cross sections
.
Energy
Mev
Nucleus
:t
Pa1mieri, Goloskie (1964)
149
He
83 :!-5
Johansson et al. (1961)
IBO
Li
149 ±3
Kirkby, Link (1966)
99.3
Be
231 ±7
Johansson et al. (1961)
180
Be
185.5 ±4.5
Kirschbaum (1954)
185
Be
172 ±17
77
...,
219 ±a
tt
95
232.5 ±7
tt
113
lt
133
Au thor
Goloskie, Strauch (1962)
("
Kirkby, Unk (1966)
99.1
Johansson et al. (1961)
180
c
c
c
c
c
Kirschbaum (1954)
185
c
204 ±20
Millburn et al. (1954)
290
c
199 ±20
Goloskie, Strauch (1962)
77
Al
444 ±'14
"
95
Al
415 ±13
lt
113
Al
408 ±i3
u
133
Al
424 ±13
Kirkby, Link (1966)
99.7
Al
430 ±13
Jobansson et al. (1961)
180
Al
390 ±ro
Kirschbaum (1954)
185
Al
408 ±40
219 ±7
223 ±6
245 ±7
212.3 ±5
- 25-
cross section of lithium but at 180 Mev (see '!"able
of 149 mb was extrapolated to 160 Mev using an
rn.
~mergy
Their
value
dependence simi lar
to that for beryllium and a value of 162 mb was obtained which differs
from the comput.ed value by
5~6.
The small angle e1astie seatt.ering data of Steinberg, Pa1mieri
and Cormack (1965) for beryllium at 141.5 Mev (together with the 1arger
angle data of Taylor and
t~od
(1961) was analysed and a reaction cross
section of 218 mb was obtained.
The proton reaction cross section
for beryllium has been measured at severa! energies (see "'able II).
Interpolation between these values results in a reaction cross
section of 205mb at 141.5 i\Iev which differs by 6% from the comput.ed
value of 218 mb.
CAROON
'l'he reaction cross section for carbon was computed for severa!
energies.
<at al.
A value of 241 mb was obtained from the data of Gerstein
0957) at 91.8 ilfev.
'!"his is 3. 5% higher than the measured value
of 233 mb obtained by Goloskie and Strauch (1962) at 95 Mev.
Steinberg 's elastic data together with 'l"aylor 's data at
larger angles was used to obtain a reaction cross section of 229 mb
at 143 ;,1ev (see Appendix 8).
Unear interpolation between the measured
reaction cross sections of Goloski e and Strauch (223 mb at 133 r.1ev),
Jo hans son (212 mb at 180 Mev) and Kirschbaum (204 mb at 185 Mev) gi ves
- 26-
a value of 222 mb at 143 Mev.
""he computed value is thus about 3::6
higher than the experimental value.
Theelastic data of Johansson et al. (1961) at 132.7 Mev
gave a reaction cross section of 20U mb.
This value may not be too
reliable since the elastic data did not extend into the small angle
region (the sma1lest angle was about ~ which is just barely in the
interference region) •
.,he hi gh energy (313 Mev) el asti c scattering data of
Chamberlain et al. 0956) was used to obtain a reaction cross section
of 207 mb.
This value is 4% higher than Mi llburn 's measured value
of 199 mb at 290 Mev.
The computed cross sections for carbon as well as the
experimentally measured values are plotted as a function .of energy
in Fig. 2.
AWMINUM
Gerstein's elastic data for aluminum at 92 Mev gives a
reaction cross section of 395 mb whi ch is 6. 5/~ lower than the measured
value of 422 mb obtained by interpolation of the measured values of
Table II.
A value of 408 mb was obtained from Steinberg's data at
143 Mev and 414 mb from Johansson 's data at 160 Mev.
'T'hese are 2% lower
and 2% higher respectively than the values obtained by interpolation
-·
-~-
Figure 2.
Measured and computed reaction cross
sections for protons incident on carbon.
e
e
e
FIGURE
2.
MEASURED
FOR
AND
PROTONS
COMPUTED
INCIDENT
ON
REACTION
CROSS
SECTIONS
CARBON
260
rr
1
lT
240
0
O"r
220
(mb)
T
1
1
T
0
0
l
T
0
1
T
A
1
o
measured
!::,.
computed
bT
16
10
200
180
100
120
PROTON ENERGY
140
(Mev)
160
180
- 2û -
between the measured cross sections.
A summary of the computed reaction cross sections is given
in Table III together with the measured cross sections (obtained from
table II
~
linear interpolation).
The errors given with the computed
cross sections are equivalent to the errors in the elastic data used
in the computations.
- 29-
~able
III.
Computed and measured reaction cross sections ar.
'T'he errors in 'ar computed' are those of the elastic data. ""'he
values of 'ar measured' were obtained from Table II Qv linear
interpolation.
Snergy
Mev
Nucleus
O"r (mb)
computed
ar (mb)
measured
Cormack et al.
0959)
147
He
36 +5
U3 +5
Johansson et al.
(1960)
!6ü
u
170 +51
162 +9
Steinberg et al.
(1964)
141.5
Be
218 +7
-
205 +10
Gerstein et al.
0957)
91.8
c
241 +8
-
233 +8
Au thor
c data)
(~!asti
-
-
-
-
-
-
Steinberg et al.
(1964)
143
c
229 +8
222 +8
Johansson et al.
(1961)
182.7
c
205 +8
210 +6
Chamberlain et al.
(1956)
313
c
207 +40
199 +20
Gerstein et al.
(1957)
92
Al
395 +12
422 +18
Steinberg et al.
(1964)
143
Al
408 +13
-
415 +20
Johansson et al.
(1960)
160
Al
414 +124
407 +2(!
-
-
-
-
-
-
-
-
-
- 30-
v
CONCWSION
The computed reaction cross sections in most cases agree
quite well with the measured values.
uncertainties.
AU are within the experimental
'T'he good agreement in the case of carbon at 182.7 Mev
is partly fortui tous in view of the fact that the elastic scattering
data did not extend far into th<a small angle region.
'l'he application of this method to heavy nuclei has not.
been attempted because of the unavai labi li ty of a form factor as
defined in first Born approximation.
In principle the reaction cross section for the heavy
elements should of course be calculable from the elastic data since
the charge distribution is known and a knowledge of the charge
distribution determines the Coulomb ampli tude.
.1-bwever the form
factor multiplying the Coulomb ampli tude of Eq. (2. 6) would no
longer be the simple one of Eq. (2.14) since higher Born terms
would have t.o be included.
Indeed, "''he form factor has no
meaning outside of the first Born
- 31 -
approximation.
Thus in order to compute the reaction cross section
for the heavy nuclei, a method must first be developed for calculating
the Coulomb amplitude for these nuelei.
The method used
~
Bethe {1958) and later
~
Jarvis (1966)
for computing the reaction cross section from the elastic data depends
strongly on the radius of the nucleus for which the cross section
is sought.
In Bethe's formula, the reaction cross section is
proportional to the square of the radius and bence this radius
must be very weil known in
to give accurate results.
ord~r
The
method of Holdeman and Thaler which is used in this paper to calculate
the reaction cross section, on the other band, does not depend
nearly so strongly on the radius.
The radius enters only into the
form factor which at the small angles considered in this paper bas
the form
F(ae)
= 1 + O(a2e2).
TO first order the calculations do
not depend on the nuclear radius.
Therefore this method bas a definite
advantage over Bethe's method since only the elastic scattering needs
to be known accurately.
Although only proton-nucleus seattering is eonsidered in
this paper, there is no reason why the method should not apply to
other charged particle scattering, such as
for example.
w t - proton scattering
The method eould also perhaps be extended to include
proton scattering from the heavier nuclei and so predict reaction
cross sections when only the elastic scattering data is available.
When both the differentiai and reaction cross sections are available,
- 32 -
the method serves as a consistency check on these two very different
types of measurements.
- 33 -
Appendix A
MEASURED FORM FACTûHS
Electron scattering experiments indicate (Hofstadter, 1956)
that the helium data is best fitted with a Gaussian charge distribution
of the form
where "a" is the root-rnean-square radius of the charge distribution
defined by
This charge distribution gives rise to a Gaussian form factor
The best value of the rms charge radiuswas found to be 1.61 fermi.
Lithium and beryllium are fitted with a modified exponential
charge distribution (Hofstadter, 1956)
and form factor
- 34 -
Th~
rms charge radii of lithium 7 and beryllium 9 are 2.71 and
3.04 fermi respectively.
The effects of the first p-shell nucleons on the charge
distribution {in sucb nuclei as carbon) may be described
a parabolic weil for the shell mode!.
~
assuming
With this potential (oscillator)
the charge distribution becomes {Ehrenberg et al., 1959)
cJ...
"1..
~ (. \)) ( \ + o.~ )
"&..
2-)t
f ( -
C:..t\
~
0.. c.\\e~." ~ e.
)
where
The form factor for this type of charge distribution is
For carbon, the best fit to the experimental data is obtained for
a= 4/3, ac.m.
= 1.66
fermi and acbarge = 1.71 fermi.
For the slightly beavier elements, Helm {1956) bas introduced
a "folded" charge distribution given
~
This formulation bas the advantage that the resultant form factor
is the product of two individual form factors
- 35-
Helm calls this model the "Gaussian uniform" distribution since it
consists of a uniform distribution
IS·C.~) • ~ 4~ -R~
and a Gaussian distribution
~
1 (...-)
"'
(:<_~ ';)1. ) ~
'1.
Ui>
~ - ;~.. l
These distributions give rise to form factors
and
For aluminum the values R = 3.36 and g
= 1.12
fermi are used.
- 36 -
Appendix
tl
HESULTS FOR CAUBON AT 143 MEV.
The fol1owing pages contain the output data of the program
used with the McGill IBv'l 7044 computer.
On1y the results of proton-
carbon scattering at 143 Mev are given.
The other elements were
treated in a similar manner.
CARRON 12, 143.0 MEV
C.AlCULATICN OF NUU.E:J\C". <iCATTfPING AMPLITUDE AND
REACTION CRQSS StCTIUN
DATA.
All
FRO~
~LAt;T[C
DATA, CAlCULAfiONS M'ID
SCATTERINr,
~E:SULTS
4RE IN
CENTER OF HASS. ANGLES ARE IN DEGREES, LENGT~S IN
CENTIME:TERS, CROSS SECTlOI\IS lf\J BA;H.jS (UNLESS
OTHERWISE: SPECiflfD).
TARGfT
ATO~!C
NO l
ATUMIC
~T
INCIDENT PROTON
~
=
F~E~GY
PROTON VELOCITY V=
A.O
= 12.0
(M~V)
O.l~90E
SIGMA C (COUL PHASF SHiff) =
THETA 0
=
l
143.0
11
-0.505~E-Ol
5.97
lNTEGRATED ELA<;TIC =
ie
=
0.76RlE-Ol
•~
AR tLJ 1-J
l2,
l4
~
• ,<
!.4 f V
MEA5UREO CROSS SECTIJN 1, 2
FOR~
FACTOR 1 1 2 =
0.9638f 00
POINT CROSS SECTION 1, ?
A8C:
VALUE,
ARS VAtUF,
=
0.9545E 00
Cnul
CALCULATEn CROSS
=
A!1PilTU'1F ? =
S~CTtnN
0.5085
0.7145
AKG COUl A,_\PLIP.I'J[ l
~R.G
0.4632
0.6~23
z
=
l, 7
0.1236[-ll
0.3617F 01
G.9766E-l?
0.3596E 01
ABS vALUE, ARG r-:UCL AMPLITUDE (F{Q)) =
RFAL,
lM f(O) =
o. 5085
0. 734 5
0.8643E-l2
O.l241E 01
0.8l78f-12
0.21<J5f-12
=
TJTAL REACTION CROSS SECTION (Mlltl8ARNS)
227.20
CARBON 12. 143.0 MfV
r4FASIJRFIJ CROSS
FOQ~
SH.TlON
FACTOR 1, ?
PCINl CRûS$
2
0.4632
=
O.Q545F 00
StCTlUN l, 2 =
0.3fll0
0.944lf 00
o.5or.~
0.427'5
1\BS VALUE,
1\RG r:fliiL
M•PlJHifif 1
~
o.q766F-l?.
0.35Q6f 01
Af'.S VALUF,
ARC. Cf111L
f\MPLilUIH
=
0.7912E-12
0.3578f 01
CALCULATEO CROSS
ABS 'VALUE,
e
r
1,
REAL,
SFCTIU~
1, 2
AftG NUCL AMPliTUOf:
lh\ f(O}
:::
C.?l3341:-l2
2
=
0.4275
0.5085
(ffO))
=
0.87llE-12
0.82J7f-12
i
TOTAL REACTlGN CROSS SE(TIGN fMJLllBARNS} :
228.71
O.l239E 01
e
(.AR f\ 0\
i
~
NG lE
1,
2
l?
=
~
4 '• "-';. 0
5 • 4 'l
'> • '.J 1
MEA:ïURFD CRClSS
Sc(T ItlN
FORM FACTOR 1,
4.::
=
ArH~
=
l
r.93.?7E 00
1
=
G.74l?f-ll
0.3578E 01
(.OUt J'.f'IPU f'lqf_
2
--
0.6540E-l?
ù.35blf Ol
1\RG NliCl
RfAL, lM FtOJ
0.4124
0 .. 4275
A~Pt
=
AMPliTUDE
0.29?5F-12
o. 412't
0.4275
CALCULATFO CROSS SfCTtUN l, ? =
ABS V.l.lUt:,
~.3587
0.'3RlC
l TU,)t
ABS VAUif:, ARu COUt.
t
2 =
1,
fl.q44lf:: 00
POINT CROSS SfCTiON lt
ABS Vbi.Uf
Mf V
(f(Cll
=
0.~616E-12
n.fl3!f,f-12
=
TOT.\l !(FACliUN CROSS SfCl ION OHLllBARNS}
MFASUREn CROSS
S~CflON
FORM FACTOR lt 2
=
POINT CPUSS SECllllN
1\BS VAl
ABS
ur:,
VALUt:,
le 2 :
J.9~?TF
l,
? "'
A~Pl!
0.3587
or
A'\G (1lUL MWl ITUOf
1\RG CIHJL
Tll'îf
AjjS VALUE, ARG NliCL AMPlJTUOF:
=
0.2ij73f-l?
TOTAL NfACTION CROSS SECTION
22Q.f\b
0-.3582
0.9204f 00
{1.4124
0.4?2P
l
=
O.é540f-12
0.35&1E OL
2
-=-
0. '>497f-12
O. 3546E
U\lCULATt::O CHOSS SECTlflN 1, 2 -
RFAlt tM Fl01
O.l.?33E 01
0.4124
(f(O)):
01
0.422R
0.8772E-l2
O.RlH~F-12
(~llLJ8A~NSt
:
229.R9
O.l237E 01
CARBON 12. 143.0
ANGLE
lt 2 :..:
MEASU~fü
o.')l
~[V
7.r)q
CR.USS c::.c:CTllHJ 1,
fORM FACTOR
l,
7
=
O.'l2C4f 00
POINT CROSS SECTION 1,
2
=
0.8931F 00
0.422P.
1 =
ABS VALUt:,
ARG COUl
A~PlliUOt
ABS VAlUE,
AR(. f!JIJt
A11PI. 1 TU'Jf 2
LALCUL.ATED CkG)') "'=Cfl J"l
AtlS \11\I.U::,
RtAL,
lM
A:(,G
f(l),
'IIJCL
=
:.otF AS UR F D
C~
G ') <;
fORM fACTrJK lt
7.51
PUINT CRUSS SFCTION
ABS
VALUF, ARG CJUL
IU3S VALUE,
VAL\JL,
REAL,
TOT~L
0.3546E 01
o.4o~of-l2
o.3519E nt
0.4621
O.R595E-l2
O.l262f 01
n.~l.Ri\f-12
Sf(TIO~i
=
(MILLIBARNSt
231.73
R.bq
l ,
2 -
0.3675
O.R626f IJO
'J.I:i-:rHE fJU
1,
2
=
0.4621
0.4939
~~PllTU)f
1 =
0.4040F-12
0.3519F 01
ARG (>.j'Il AMPLITUDE
'l =
0.3095f-12
0.3495E 01
CALCULATEO CROSS
AdS
0.54q7E-12
{FIOl)""
c,•Wt!Titllt
S t= C T 1 UN
2 -=
0.4-621
0.4?2B
2 --
0.?611[-ll
TOTAL REACTtfJN CiW"iS
ANGLF lt 2 =
i,
0.3686
0.1c:;82
2"'"
SfCTI t: 1,
ARr; NU(.L
(M F(O)
=
l'
\MPL l f!I!)F
O.t?05E-l?
··
0.4621
( F( !Jt)
=
0.493q
O.B2"i2E-l2
O.RC29f-l?
REACTION CROSS SECTION (MJLLISARNSJ
c
240.85
O.l338E 01
C:AK.RO'J
A ~J GL F
1 .,
2
12,
=
145.0 ._,[V
'3 • 6 ri
ML\SUlH:O CPO.;;S
..J •
Sff.T IùN
POINT CkOSS SFCTION L,
r6
l,
2
2:::::
ABS VALIJf,
ARG COUt. AMPLITlJOF 1 =
ABS VALUE,
ARG CDUL <\MPLJTUDF 2
CALCULATtf1 CRfJSS <;FCTION l,
ABS VALUE,
~EAL,
td•,G
=
lM FCO)
TOTAl. RE.\CTION
CAQRQ~
e
!
NUfL
12,
2
1\MPI_ITIJDF
O.!~O~F-12
c;;,nss
Q.3525
0. 36 75
S~CTiùN
141.0 MEV
=
=
C.309'5E-12
0.3495E 01
0.2447E-12
0.347'iE 01
0.5125
(,.4939
fFtO))
=
0.7903E"-12
0.783~~-12
OHLl JRARNS)
=
?52.52
O.l443E 01
- 37-
REFERENŒS
Bethe, H.. A., Ann. of Phys., ,21 190 (1958).
o.,
Chamberlain,
et al., Phys. Rev., 102, 1659 (1956).
Cormack, A. M., et al., Phys. Re v., 115, 599 (1959).
r.I.,
Cormack, A.
Nu cl. Phys.,
g,
236 (1964).
Ebrenberg, H. F., et al., Phys. Rev.,
ill' 666 (1959).
Gerstein, G., et al., Phys. Rev., lOB, 427 (1957).
Goldberger, M. L., Watson, K• .M., 'Collision Theory ', John Wiley and
Sons Inc., New York, 1964.
Goloskie,
R.,
He lm, R. IL,
Strauch,
K.,
Nue!. Phys., 29, 474 (1962).
Phys. Rev., 104, 1466 (1956).
lbfstadter, R. H., Rev. Mod. Pbys.,
~~
Ann. Rev. Nue!. Sei.,
214 0956),
1, 231 (1957).
Holdeman, J. T., Thaler, R. M., Pbys. Rev. Let., 14, 81 (1965).
Jarvis, O. N., Nucl. Phys., ]2, 305 {1966).
Johansson, A., et al., Nucl. Phys., 21, 383 (1960),
Ark. Fys., 19, 527 (1961),
Ark. Fys.,
Kirkby, P., Link,
w.,
12
1
541 (1961).
To be publisbed, Can. J. of Phys. (1966).
Kirschbaum, A. J., tbesis, Univ. of Cal. Report UCRL-1967 (unpub1isbed),
(The results of Kirscbbaum are quoted by Millburn et al., 1954).
!'riessiab, A., 'Quantum Mechanics, Vol. I ', Nortb-Iblland Publisbing Co.,
Amsterdam, 1961.
- 38-
Millburn, G.
P.,
et.
al., Phys. Hev., 95, 1268 (1954).
Palmieri, J. N., et al., Phys. Le tt. , !!.__, 289 (1963).
Nucl. Phys., 59, 253 (1964).
Solov'ev, L.
D.,
Sov. Phys. JE'f'P, 22, 205 (1966}.
Steinberg, D. J., et al., Nucl. Phys., 56, 46 (1964).
Taylor, A. E., et al., Nuclear Phys., 25, 642 (1961).