- Email: [email protected]
Optical model analysis of neutron elastic scattering from nitrogen
I
2.E
[
Nuclear Physics 67 (1965) 289--295; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or m i c r o f i l m without written permission from the publisher
O P T I C A L M O D E L ANALYSIS OF N E U T R O N E L A S T I C S C A T T E R I N G F R O M N I T R O G E N BORJE JOHANSSON t Research Institute of National Defence, Stockholm 80, Sweden Received 15 October 1964 Abstract: Optical model calculations of neutron scattering for the nucleus N 1~have been made. The energy region examined was 4.5-11.6 MeV. At the lower energies use was made of the HauserFeshbach formalism in order to estimate the compound elastic scattering. Reasonable fits were obtained, especially at higher energies.
1. Introduction In an earlier paper by L. Wallin and myself 1) a preliminary report on the application of the optical model to the light nuclei N 14 and 016 was given. In the present paper a more thorough study o f the optical model predictions of neutron scattering data for the element N 14 is carried out. The results obtained are good in the energy region 7-11.6 MeV. N o try was made to fit experimental data at energies higher than 14 MeV. The fact that the parameter set obtained at 4.5 MeV is rather unexpected m a y indicate that this is a lower energy limit for the application of the optical model. This is further supported by the fact that efforts to find parameter sets which fit data at energies < 4.50 MeV were not successful. The good agreement in the interval 7-14 MeV can be utilized to fill the gap of existing data in this energy region, especially as the parameters obtained show a smooth variation with energy. It should be stated that such predictions will be approximate only.
2. The Optical Potential The employed potential was of the following form: V(r) = -- VcRVcr(r ) - i V I M v i m ( r ) - Vsovso(r)(a . l),
(1)
where the function V~r(r) is given by Vow(r) =
1 1 +exp[r--R1/a]
t Present address: NORDITA, Copenhagen. 289
(Woods-Saxon form),
(2)
290 and
e. JOHANSSON
Vim(r)
is the surface absorption form factor (Ganssian form).
(3)
The spin-orbit term is given by v,o(r)
=
--
-r
vr°(r)
(Thomas form).
(4)
The quantities R 1 and R 2 are given by R 1 = ror At,
(5)
R 2 ---- roi h t,
(6)
where A is the mass number of the target nucleus. In the calculations reported here the parameters rot and roi were given the same value r 0. The quantity R = ro A t is often identified with the nuclear radius. Some remarks on this interpretation will be given later. With the potential (1), the form factors (2)-(4) and the restriction Rx = R2, we are left with the following set o f parameters: Vat, VIM, Vso, a, b and r o.
3. Fitting Procedure Single parameter variations were first performed in order to find how different cross sections were influenced and especially how the differential cross section ael(0 ) was affected. It was found very difficult to draw definite conclusions about this influence as the parameters are dependent on each other. Moreover the parameters have not always the same effect at different energies. The latter difficulty is avoided by performing the same parameter variations at every energy, where a fitting procedure is intended. For the calculations the programme ABACUS 2, coded for an IBM 7090 computer was used 2). In this programme is incorporated art automatic parameter search routine which tries to minimize the quantity X2 given by /,.rgale ~exp\ 2 1 N
where Aa is the experimental error of the measured quantity a. The search routine is used to furnish the best parameter set at each energy where a fit is sought. Here the information obtained from the single parameter variations cart be useful when determining the parameters which ought to be varied in the search procedure, thereby reducing the total computing time. ABACUS 2 also contains the Hauser-Feshbach formalism. In these calculations no use was made o f this formalism for energies > 7.0 MeV as the compound elastic scattering is believed to be unimportant at such energies 3).
OPTICAL MODEL ANALYSIS OF NEUTRON ELASTIC SCATTERING
291
The fitting procedure as described above was fully carried out at the energies 4.50, 6.53 and 11.6 MeV. Then a linear interpolation was made between the parameter sets obtained in order to get starting values of the parameters for X2 runs at intermediate energies.
4. Experimental D a t a
At the energies 6.02, 6.53, 8.0 and 11.6 MeV data f r o m Chase et al. 4) were used for comparison. At 7.0 MeV the differential cross section was taken f r o m Phillips 5) and combined with the value of trr = 1.35 f r o m B N L 325. At 4.50 MeV the differential cross section measured by BostrSm et aL 6) was taken as the comparison values for the optical model predictions. The total cross sections vary heavily in certain energyregions [4.8, 5.3 and 8.0 MeV (BNL 325)] indicating that the optical model predictions should not be expected to be valid in these regions. For energies lower than 7 MeV the Hauser-Feshbach formalism was applied and the following energy-levels were used 7): E(MeV)
J
rr
0 2.312 3.945 4.91 5.10 5.69 5.83 6.05 6.23 6.44
1 0 1 0 2 1 3 1 1 3
+ + + ----
where the values J = 1 o f the level 6.05 and 7r = - at the levels 6.05 and 6.23 were ascribed without any theoretical or experimental justification.
5. Results
In table I the optical model predictions o f a T and O'el are given together with the corresponding experimental values. The agreement is seen to be good, especially for energies _~ 7 MeV. Figs. l a - f show the theoretical differential elastic cross section together with the experimental data points. The fits are fairly good at all energies, but also here with some preference for the higher energies.
292
B. JOHANSSON
As remarked in sect. 4 the rapid variation of a r around 8 MeV indicates that the optical model predictions could not be expected to agree with the experimental data at this energy. However, as seen in table 1 and fig. le the fit obtained is very good. For the same reason a good fit at 4.99 MeV could not be expected and here it was impossible to get an agreement in spite o f great efforts. The conclusion is that in regions where a r exhibits rapid variation the possibility of a good local fit cannot be excluded. TAnLE I The experimental and calculated values of a T and ¢rel
En
crT(©xp)
O'T(Calc)
~el(exp)
4.50 6.02 6.53 7.0 8.0 11.6
1.424-0.06 1.484-0.05 1.35 1.354-0.05 1.444-0.07
1.42 1.47 1.35 1.37 1.44
1.51 1.054-O.16 1.04=I=0.16 0.85 0.874-0.10 0.994-0.10
o'ex(calc) 1.39 1.20 1.16 0.85 0.87 0.94
The curious value o f VCR obtained at 4.50 MeV has already been discussed in ref. 1). The investigations made since then give no indication that a more reliable value of Vex can provide a good fit. At 11.6 MeV the parameter set obtained at 14.0 MeV by Lutz et aL s) was used as the initial values in a Xz search. It was found that their values o f a, r o and b preferentially could be carried over to 11.6 MeV. It is remarkable how well aT was predicted for the higher energies by the previously mentioned linear interpolation. However, parameter variations were necessary to fit also the differential cross sections. At 6.02 and 6.53 MeV the experimental values of a,o,-el ~ 0.37 and 0.44 b, respectively, seem a little too high to be entirely reliable. The experimental values o f anen-,1 have been obtained by the expression
O'non.el = O"T- f O'*l(0)d~' and this favours the calculated values of acl given in table 1 because of the lower accuracy of cre~compared with a r. In fig. 2 the obtained parameter values and those found by Lutz et al. s) at 14 MeV are given in diagrams. We see that the parameter to, which is usually considered to be intimately associated with the nuclear radius, has been allowed to vary with energy. At the beginning of these calculations much work was devoted to keep ro at
293
OPTICAL MODEL ANALYSIS OF N E U T R O N ELASTIC SCATTERING
a fixed value, but without success. All parameters seem to have a rather smooth variation at energies higher than 7 MeV, but to have a more irregular behaviour at lower energies.
En=4.SD HeY
En = 6.02HEY Vcr .34.o
/-00
(o)
Vcr = 37.7
s°°fa
V i m = 2.3
Vim - 10.0
/-00
(b)
Vso = g,0 r 0 = 1.25 O b
300
= 032 = O,gO
300
200
20
100
101
1,0
I
I
I
0.8
0,5
0.4
I
02
I o
I I 1 I - 0 2 -o.c - o s -ae
I -1Jo
1.0
I
I
I
I
I
I
O.l
0.6
0./-
n2
0
-0.2
Vso =
?.s
ro
•
1,/-0
O b
= 0.50 : 0.9#
I
I
-0./. -0.6 -0.0
I
-1.0
E n = 7.0 H e Y
E n = 6.53 H e Y 300
300
Vet =/-2.4
Ycr : 3 7 . 0 VV i m ffi 2.1 so = 7.7
(C) I,,.
I
200
re
100
V i m = 2.3
(d)
200
7.3
vs o = r e • 1.30
• 1./-0
a
= 0.50
b
= 0.50
~k
o
= 0.52 = o.g5
b
100
'0
1.0
I
I
I
I
I
I
0.41
0.6
0./-
0.2
0
-0.2
I
I
I
I
-0./- -0.6 -0.0
-1.0
1.0
I 0.II
I
I
I
I
I
0.6
O./.
0.2
0
-02
400
En = 6 . 0 H e Y
Vim =
20(
o b
•
= :
I
I -1.0
Vcr • 40.1
3,3
Vim •
)00
vso = s.0 r° = 1.35
(e)
I
En = 11.6HeV
Vcr = /,2.0 300
I
-O.t, -0.6 - 0 9
0.55 0.90
(f)
300
. 100
.__e
5,/,
Vso: /-.1 ro
.
1.20
o
=
0.6/-
b
:
0.79
100
f 1.0
0.0
0.6
0./-
0.2
0
-0.2 -0.4
-0.5 -0.8 cos
-1,0
10
OR
0.5
0.~
11.2
0
-0.2 -0./- -0.5 -0.0 -1.0
eC.M.
Figs. la-f. The optical model differential elastic cross section compared with the experimental points at energies given in the figures. The found values o f the parameter sets are also included.
294
B.
JOHANSSON
6. Discussion As we have seen, the agreement between the predictions and the experimental data is better at the higher than at the lower energies. This is in conformity with the expected applicability of the optical model. At lower energies the compound elastic cross section has to be accounted for. This has been done by aid of the Hauser-Feshbach
Vcr
50 f
•
30
|
•
•
,
.
,
,
t
,
S.O
10.1
Vim
e••
0.or
• *
10`0[
•
•
,
|
!
16.0
,
,
,
i
,
10.0
n
a
• • •
•
En
o
1S,0 r n
l
l
a
• l
l
,
i,
,
•
,
15.0 En
,.. ,.. : , , , ' ,
1-00t 1
,
10.0
: i
5.0
10.0
•
15.0 Eit
•
0.901
•
o.oo~
.
n ro
.
•
7.0[
0.70I
t
•
5.0
o
t
•
5.0
/..0
..
10,0
•
1.C
Vso
•
&o
lJ.O I 1,]0
•
.
.
n
51.0 t •
•
• ,
.
,
•
• I
5`0
i
, 1S.0 E n
1.20 u
?
10.0
i
I
l
I
10.0
I
• i
I
I
15.0 E n
Fig. 2. The found parameter values plotted vs. energy. The parameter set at 14 M e V is that found by Lutz et al. a).
formalism, which however introduces an additional difficulty, if a bad fit should be related to discrepancies either of the optical model or to the Hauser-Feshbacb formalism. The interpretation of R = ro,4 ~ as the nuclear radius does not seem to be entirely valid, as ro was permitted to vary with energy. As mentioned before, no good agreement could be obtained with a fixed value of ro. Some trials with roi # ro, have also been performed, but this did not improve the results.
OPTICAL MODEL ANALYSIS OF NEUTRON ELASTIC SCATTERING
295
The finite size of the neutron can be neglected when compared to a heavy nucleus but may have some importance for a light nucleus such as N 14. This circumstance can perhaps explain the energy variations of Vca and r o (see fig. 2). However, when the neutron energy increases, the wavelength associated with the neutron becomes shorter and, in this way, the importance of the finite size of the neutron diminishes. This may be compared with the similar effect found by Lutz et aL s), when they compared the optical model predictions with neutron scattering by different light nuclei. They found that r o becomes higher and VCR lower the larger the size of the neutron was in comparison with the target nucleus. The increment of VXMwith energy is in accordance with the theoretical predictions, based on the effect of the exclusion principle. 7. Conclusions
The application of the optical model to the light nucleus N 14 has been much more successful than could be expected 1). However, the calculations seem to indicate an energy limit of about 5 MeV, under which an application of the optical model cannot be successful. Moreover, one cannot always exclude the existence of more than one solution because of the high order of the parameter space. But even so it might be valuable if a vast body of experimental data can be summarized in a few parameters. I wish to thank Dr. M. Leimddrfer for proposing this work and for his continued interest. I am also indebted to L. Wallin for many interesting and stimulating discussions and for his careful reading of the manuscript. References 1) 2) 3) 4) 5) 6) 7) 8)
B. Johansson and L. Wallin, F O A 4 rapport A4375-411 Auerbach et aL, KAPL-M-NCF-4 (1963) M. H. McGregor, UCRL-5230 Chase et aL, AFSWC-TR-61-15 D. D. Phillips, BNL-400, 7-0-6 N. A. Bostrdm, BNL-400, 7-0-4 Landolt-B6rnstein, New Series, group 1, vol. 1, p. 1-45 Lutz et aL, Nuclear Physics 47 (1963) 521
2.E
[
Nuclear Physics 67 (1965) 289--295; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or m i c r o f i l m without written permission from the publisher
O P T I C A L M O D E L ANALYSIS OF N E U T R O N E L A S T I C S C A T T E R I N G F R O M N I T R O G E N BORJE JOHANSSON t Research Institute of National Defence, Stockholm 80, Sweden Received 15 October 1964 Abstract: Optical model calculations of neutron scattering for the nucleus N 1~have been made. The energy region examined was 4.5-11.6 MeV. At the lower energies use was made of the HauserFeshbach formalism in order to estimate the compound elastic scattering. Reasonable fits were obtained, especially at higher energies.
1. Introduction In an earlier paper by L. Wallin and myself 1) a preliminary report on the application of the optical model to the light nuclei N 14 and 016 was given. In the present paper a more thorough study o f the optical model predictions of neutron scattering data for the element N 14 is carried out. The results obtained are good in the energy region 7-11.6 MeV. N o try was made to fit experimental data at energies higher than 14 MeV. The fact that the parameter set obtained at 4.5 MeV is rather unexpected m a y indicate that this is a lower energy limit for the application of the optical model. This is further supported by the fact that efforts to find parameter sets which fit data at energies < 4.50 MeV were not successful. The good agreement in the interval 7-14 MeV can be utilized to fill the gap of existing data in this energy region, especially as the parameters obtained show a smooth variation with energy. It should be stated that such predictions will be approximate only.
2. The Optical Potential The employed potential was of the following form: V(r) = -- VcRVcr(r ) - i V I M v i m ( r ) - Vsovso(r)(a . l),
(1)
where the function V~r(r) is given by Vow(r) =
1 1 +exp[r--R1/a]
t Present address: NORDITA, Copenhagen. 289
(Woods-Saxon form),
(2)
290 and
e. JOHANSSON
Vim(r)
is the surface absorption form factor (Ganssian form).
(3)
The spin-orbit term is given by v,o(r)
=
--
-r
vr°(r)
(Thomas form).
(4)
The quantities R 1 and R 2 are given by R 1 = ror At,
(5)
R 2 ---- roi h t,
(6)
where A is the mass number of the target nucleus. In the calculations reported here the parameters rot and roi were given the same value r 0. The quantity R = ro A t is often identified with the nuclear radius. Some remarks on this interpretation will be given later. With the potential (1), the form factors (2)-(4) and the restriction Rx = R2, we are left with the following set o f parameters: Vat, VIM, Vso, a, b and r o.
3. Fitting Procedure Single parameter variations were first performed in order to find how different cross sections were influenced and especially how the differential cross section ael(0 ) was affected. It was found very difficult to draw definite conclusions about this influence as the parameters are dependent on each other. Moreover the parameters have not always the same effect at different energies. The latter difficulty is avoided by performing the same parameter variations at every energy, where a fitting procedure is intended. For the calculations the programme ABACUS 2, coded for an IBM 7090 computer was used 2). In this programme is incorporated art automatic parameter search routine which tries to minimize the quantity X2 given by /,.rgale ~exp\ 2 1 N
where Aa is the experimental error of the measured quantity a. The search routine is used to furnish the best parameter set at each energy where a fit is sought. Here the information obtained from the single parameter variations cart be useful when determining the parameters which ought to be varied in the search procedure, thereby reducing the total computing time. ABACUS 2 also contains the Hauser-Feshbach formalism. In these calculations no use was made o f this formalism for energies > 7.0 MeV as the compound elastic scattering is believed to be unimportant at such energies 3).
OPTICAL MODEL ANALYSIS OF NEUTRON ELASTIC SCATTERING
291
The fitting procedure as described above was fully carried out at the energies 4.50, 6.53 and 11.6 MeV. Then a linear interpolation was made between the parameter sets obtained in order to get starting values of the parameters for X2 runs at intermediate energies.
4. Experimental D a t a
At the energies 6.02, 6.53, 8.0 and 11.6 MeV data f r o m Chase et al. 4) were used for comparison. At 7.0 MeV the differential cross section was taken f r o m Phillips 5) and combined with the value of trr = 1.35 f r o m B N L 325. At 4.50 MeV the differential cross section measured by BostrSm et aL 6) was taken as the comparison values for the optical model predictions. The total cross sections vary heavily in certain energyregions [4.8, 5.3 and 8.0 MeV (BNL 325)] indicating that the optical model predictions should not be expected to be valid in these regions. For energies lower than 7 MeV the Hauser-Feshbach formalism was applied and the following energy-levels were used 7): E(MeV)
J
rr
0 2.312 3.945 4.91 5.10 5.69 5.83 6.05 6.23 6.44
1 0 1 0 2 1 3 1 1 3
+ + + ----
where the values J = 1 o f the level 6.05 and 7r = - at the levels 6.05 and 6.23 were ascribed without any theoretical or experimental justification.
5. Results
In table I the optical model predictions o f a T and O'el are given together with the corresponding experimental values. The agreement is seen to be good, especially for energies _~ 7 MeV. Figs. l a - f show the theoretical differential elastic cross section together with the experimental data points. The fits are fairly good at all energies, but also here with some preference for the higher energies.
292
B. JOHANSSON
As remarked in sect. 4 the rapid variation of a r around 8 MeV indicates that the optical model predictions could not be expected to agree with the experimental data at this energy. However, as seen in table 1 and fig. le the fit obtained is very good. For the same reason a good fit at 4.99 MeV could not be expected and here it was impossible to get an agreement in spite o f great efforts. The conclusion is that in regions where a r exhibits rapid variation the possibility of a good local fit cannot be excluded. TAnLE I The experimental and calculated values of a T and ¢rel
En
crT(©xp)
O'T(Calc)
~el(exp)
4.50 6.02 6.53 7.0 8.0 11.6
1.424-0.06 1.484-0.05 1.35 1.354-0.05 1.444-0.07
1.42 1.47 1.35 1.37 1.44
1.51 1.054-O.16 1.04=I=0.16 0.85 0.874-0.10 0.994-0.10
o'ex(calc) 1.39 1.20 1.16 0.85 0.87 0.94
The curious value o f VCR obtained at 4.50 MeV has already been discussed in ref. 1). The investigations made since then give no indication that a more reliable value of Vex can provide a good fit. At 11.6 MeV the parameter set obtained at 14.0 MeV by Lutz et aL s) was used as the initial values in a Xz search. It was found that their values o f a, r o and b preferentially could be carried over to 11.6 MeV. It is remarkable how well aT was predicted for the higher energies by the previously mentioned linear interpolation. However, parameter variations were necessary to fit also the differential cross sections. At 6.02 and 6.53 MeV the experimental values of a,o,-el ~ 0.37 and 0.44 b, respectively, seem a little too high to be entirely reliable. The experimental values o f anen-,1 have been obtained by the expression
O'non.el = O"T- f O'*l(0)d~' and this favours the calculated values of acl given in table 1 because of the lower accuracy of cre~compared with a r. In fig. 2 the obtained parameter values and those found by Lutz et al. s) at 14 MeV are given in diagrams. We see that the parameter to, which is usually considered to be intimately associated with the nuclear radius, has been allowed to vary with energy. At the beginning of these calculations much work was devoted to keep ro at
293
OPTICAL MODEL ANALYSIS OF N E U T R O N ELASTIC SCATTERING
a fixed value, but without success. All parameters seem to have a rather smooth variation at energies higher than 7 MeV, but to have a more irregular behaviour at lower energies.
En=4.SD HeY
En = 6.02HEY Vcr .34.o
/-00
(o)
Vcr = 37.7
s°°fa
V i m = 2.3
Vim - 10.0
/-00
(b)
Vso = g,0 r 0 = 1.25 O b
300
= 032 = O,gO
300
200
20
100
101
1,0
I
I
I
0.8
0,5
0.4
I
02
I o
I I 1 I - 0 2 -o.c - o s -ae
I -1Jo
1.0
I
I
I
I
I
I
O.l
0.6
0./-
n2
0
-0.2
Vso =
?.s
ro
•
1,/-0
O b
= 0.50 : 0.9#
I
I
-0./. -0.6 -0.0
I
-1.0
E n = 7.0 H e Y
E n = 6.53 H e Y 300
300
Vet =/-2.4
Ycr : 3 7 . 0 VV i m ffi 2.1 so = 7.7
(C) I,,.
I
200
re
100
V i m = 2.3
(d)
200
7.3
vs o = r e • 1.30
• 1./-0
a
= 0.50
b
= 0.50
~k
o
= 0.52 = o.g5
b
100
'0
1.0
I
I
I
I
I
I
0.41
0.6
0./-
0.2
0
-0.2
I
I
I
I
-0./- -0.6 -0.0
-1.0
1.0
I 0.II
I
I
I
I
I
0.6
O./.
0.2
0
-02
400
En = 6 . 0 H e Y
Vim =
20(
o b
•
= :
I
I -1.0
Vcr • 40.1
3,3
Vim •
)00
vso = s.0 r° = 1.35
(e)
I
En = 11.6HeV
Vcr = /,2.0 300
I
-O.t, -0.6 - 0 9
0.55 0.90
(f)
300
. 100
.__e
5,/,
Vso: /-.1 ro
.
1.20
o
=
0.6/-
b
:
0.79
100
f 1.0
0.0
0.6
0./-
0.2
0
-0.2 -0.4
-0.5 -0.8 cos
-1,0
10
OR
0.5
0.~
11.2
0
-0.2 -0./- -0.5 -0.0 -1.0
eC.M.
Figs. la-f. The optical model differential elastic cross section compared with the experimental points at energies given in the figures. The found values o f the parameter sets are also included.
294
B.
JOHANSSON
6. Discussion As we have seen, the agreement between the predictions and the experimental data is better at the higher than at the lower energies. This is in conformity with the expected applicability of the optical model. At lower energies the compound elastic cross section has to be accounted for. This has been done by aid of the Hauser-Feshbach
Vcr
50 f
•
30
|
•
•
,
.
,
,
t
,
S.O
10.1
Vim
e••
0.or
• *
10`0[
•
•
,
|
!
16.0
,
,
,
i
,
10.0
n
a
• • •
•
En
o
1S,0 r n
l
l
a
• l
l
,
i,
,
•
,
15.0 En
,.. ,.. : , , , ' ,
1-00t 1
,
10.0
: i
5.0
10.0
•
15.0 Eit
•
0.901
•
o.oo~
.
n ro
.
•
7.0[
0.70I
t
•
5.0
o
t
•
5.0
/..0
..
10,0
•
1.C
Vso
•
&o
lJ.O I 1,]0
•
.
.
n
51.0 t •
•
• ,
.
,
•
• I
5`0
i
, 1S.0 E n
1.20 u
?
10.0
i
I
l
I
10.0
I
• i
I
I
15.0 E n
Fig. 2. The found parameter values plotted vs. energy. The parameter set at 14 M e V is that found by Lutz et al. a).
formalism, which however introduces an additional difficulty, if a bad fit should be related to discrepancies either of the optical model or to the Hauser-Feshbacb formalism. The interpretation of R = ro,4 ~ as the nuclear radius does not seem to be entirely valid, as ro was permitted to vary with energy. As mentioned before, no good agreement could be obtained with a fixed value of ro. Some trials with roi # ro, have also been performed, but this did not improve the results.
OPTICAL MODEL ANALYSIS OF NEUTRON ELASTIC SCATTERING
295
The finite size of the neutron can be neglected when compared to a heavy nucleus but may have some importance for a light nucleus such as N 14. This circumstance can perhaps explain the energy variations of Vca and r o (see fig. 2). However, when the neutron energy increases, the wavelength associated with the neutron becomes shorter and, in this way, the importance of the finite size of the neutron diminishes. This may be compared with the similar effect found by Lutz et aL s), when they compared the optical model predictions with neutron scattering by different light nuclei. They found that r o becomes higher and VCR lower the larger the size of the neutron was in comparison with the target nucleus. The increment of VXMwith energy is in accordance with the theoretical predictions, based on the effect of the exclusion principle. 7. Conclusions
The application of the optical model to the light nucleus N 14 has been much more successful than could be expected 1). However, the calculations seem to indicate an energy limit of about 5 MeV, under which an application of the optical model cannot be successful. Moreover, one cannot always exclude the existence of more than one solution because of the high order of the parameter space. But even so it might be valuable if a vast body of experimental data can be summarized in a few parameters. I wish to thank Dr. M. Leimddrfer for proposing this work and for his continued interest. I am also indebted to L. Wallin for many interesting and stimulating discussions and for his careful reading of the manuscript. References 1) 2) 3) 4) 5) 6) 7) 8)
B. Johansson and L. Wallin, F O A 4 rapport A4375-411 Auerbach et aL, KAPL-M-NCF-4 (1963) M. H. McGregor, UCRL-5230 Chase et aL, AFSWC-TR-61-15 D. D. Phillips, BNL-400, 7-0-6 N. A. Bostrdm, BNL-400, 7-0-4 Landolt-B6rnstein, New Series, group 1, vol. 1, p. 1-45 Lutz et aL, Nuclear Physics 47 (1963) 521