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Single-particle behaviour in fast neutron (n, 2n) reactions
2.D
[
Nuclear Physics A125 (1969) 593--612; (~) North-Holland Publishiny Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
SINGLE-PARTICLE BEHAVIOUR
IN FAST N E U T R O N (n, 2n) REACTIONS Miss SWAPNA CHATTERJEE and APARESH CHATTERJEE
Calcutta University and Saha Institute of Nuclear Physics, Calcutta Received 17 April 1968 Abstract: A recent survey of the fast neutron (n, 2n) reaction cross sections at excitation energies of 3, 6 and 7 MeV reveal a few interesting overlapping trends. Plotted against neutron excess, a structureless gross trend is seen throughout the mass region, while the second trend corroborates with the limited observations of Csikai and Peto; the third trend is one showing features of single-particle behaviour (shell effects). The gross trend has been analysed in terms of a Levkovskii type equation. Further, an attempt has been made to understand all these three trends semi-quantitatively as a compound nuclear decay process where the available residual excitation of a suitably shifted Fermi gas is properly treated; the observed single-particle behaviour (shell closure minima and the two Csikai-Peto trends) has been compared in details with the description in terms of the modified Rosenzweig shift functions. It is found that the predicted behaviour agrees reasonably well with all the observations reported here in the medium weight and heavy nuclei.
1. Introduction
There have been recent attempts 1-4) at the systematic understanding of the (n, 2n) cross-section behaviour at 14 MeV. Cindro et al. 1) experimentally demonstrated the existence of the neutron shell effects as minima around mass 90 region (target neutron number 50). Bormann 2) and Manero 3) have qualitatively discussed the neutron shell effects in these cross sections. Little analysis of the pairing effects in these reactions has been done as yet, although pairing corrections in the excitation energy depending on the nature of the target nucleus is now customary. Recently Csikai and Peto 4) have reported a neutron-excess dependence of the (n, 2n) reaction cross sections. They have used their own experimental data at about 3.0 MeV of residual excitation for the light and medium weight nuclei and have also used the reduced data at 3.0 MeV excitation from other excitation energies for the heavy nuclei using the Blatt and Weisskopf formula 5); it was found that a linear plot of these experimental and inferred cross sections against the neutron excess 2( = N - Z of the target nucleus (for constant even values of the neutron number N ranging from 24 to 82) gives a family of straight lines with approximately parallel slopes (except tbr N = 28) irrespective of Z being odd or even 4). In the present work, we re-examine the neutron excess trend on the basis of more recent cross section and excitation function data and notice that the real behaviour is more complicated. It shows three separable distinct trends. A gross trend may be 593
594
S. C H A T T E R J E E A N D
A.
CHATTERJEE
expressed in terms of the two-neutron decay probability (Levkovskii equation). Further understanding of the three trends comes from the detailed analysis of the compound nuclear decay process; we reformulate the single-particle excitation characteristics in the framework of the modified Rosenzweig model (combinatorial occupational degeneracy model in subshells shifting the Fermi surface with respect to suitable doubly magic nuclei). An expression for the relative level density as a function of the excitation variables is compared with the three observed neutron excess trends. 2. Observations The available data 6) on the fast (n, 2n) reaction cross sections have been used here at the three selected excitation energies U of 3.0, 6.0 and 7.0 MeV in two different ways. In cases where the excitation functions of the (n, 2n) reactions have been accurately measured, it has been possible to use the interpolated cross sections at these excitations within an error A U = _0.1 MeV; in the other cases where only the "14 MeV" data exist, an attempt to use the same excitations resulted in the excitation energy uncertainties A U = +0.8 MeV. The reported cross-section error Aan,2n in most cases is within ___15700. These combined data are summarized in table 1. In the analysis to be presented, no attempt has been made to include the 14 MeV data exclusively. In a few cases, e.g., l°6Ag, 114In, 12°Sb, 133Ba, a3SBa, 137Ba, ~SSTb and 164Ho residual nuclei, formation cross sections are known only to their metastable states. In these cases, an estimate of their total cross sections have been made by using the following procedure: The spin cut-off parameters were first estimated from the known isomeric and ground state cross sections a m and a g in the nearest neighbour nuclei (e.g., 115mCd, 115gCd, 129mTe, 129gTe, etc.). Suitable average values of these cut-off parameters (consistent with the fact that the (n, 2n) cross sections should never exceed the geometrical cross sections) were chosen, and the isomer ratio equation was used to estimate the ground-state cross sections a g. In practically all cases, the total was less than twice the isomer cross section a m. For 92Nb, a g was assumed to be approximately equal to the total cross section because the isomeric states decay by 7-emission to the longer lived ground state. 2.1. THE GROSS BEHAVIOUR A plot of the cross sections listed in table 1 is shown in the semilog scale in fig. 1 as a function of the residual neutron excess 2( at two values of approximately constant excitation U of 3 and 6 MeV. 3-he points for the residual isotopes and isotones have been joined by solid and dotted straight lines respectively in most cases for comparison with the reported Csikai-Peto trends 4). Confining our attention to the excitation pattern at 6.0 MeV for the most abundant target elements only, we notice that a gross smoothed behaviour is suggested for all such nuclei by the trend of the curve marked A. This trend is one of (i) a very rapid
(n, 2n) REACTIONS
595
rise of the reaction cross section fin, 2n from 2( = 0 to 2( = 4, (ii) a tapering off of °'n, 2n to a value ~ 1.5 b at A R ~ 90 (2( = 12) and then (iii) a very slowly rising continuation till the highest available 2( ,~ 150 at the end of the nuclear table where cr,,2n reaches ~ 2.2 b. The fewer data on the 3 MeV excitation in fig. 1(c) shows a similar gross trend (also marked as A); the maximum in Crn~2nhere reaches a lower value ~ 1.0 b. A comparison with the limited data at 7.0+0.8 MeV excitation (not presented here) suggests a similar gross trend. 2.2. T H E I S O T O N I C A N D I S O T O P I C B E H A V I O U R : C S I K A I - P E T O T R E N D S
Associated with the gross trend A shown in fig. 1, there are relative isotonic and isotopic trends. To be able to see these interesting features, we now consider all nuclei listed in table 1 irrespective of their abundancy. In our presentation (where the residual nuclei are being considered), all isotonic lines (solid straight lines) belong to the odd isotones; these correspond to the even isotones of Csikai and Peto. The isotopic lines (dotted straight lines), of course, occur with both odd and even atomic numbers. The lines of constant N (residual isotones) seem to align themselves approximately parallel to the gross trend A in figs. 1(a) and l(b) in the rapidly rising part near the lowest values of 2(. Thus the general features for 44Sc and 45Ti (N = 23), 57Ni and 54Mn (N = 29), 63Zn and 62Cu (N = 33), 69Ge and 68Ga (N = 37), 83Sr and 8°Br (N = 45), seem to be approximately parallel. These features up to AR ~ 100 have already been noted by Csikai and Peto 4) from their own experimental data. We call this trend the "Csikai-Peto trend" for isotones. The trend shows that generally an isotone of lower residual Z-value has a higher cross section than its higher isotope member. We may similarly define a Csikai-Peto trend for isotopes (not observed or reported in ref. 4)); a heavier isotope usually has a higher cross section than the lighter one. This is clearly seen for 62Cu and 64Cu ( Z = 29), 69Ge and VSGe (Z = 32), 12°Sb and 122Sb (Z = 51) residual nuclei. This is an "inverse Gardner trend"; Gardner 35) has shown that in the case of 14 MeV (n, p) reactions, the cross section of the liohter target isotope is higher (roughly by a factor of 1.5 to 2.0) than that of the heavier target. Thus, in regions where the gross trend has positive slopes, these isotopic and isotonic lines seem to obey the two Csikai-Peto trends reasonably satisfactorily. The trends are observed in all cases of the normalized excitation energies considered here, viz., at 3.0, 6.0 and 7.0 MeV. 2.3. D E P A R T U R E
FROM THE CSIKAI-PETO TRENDS
A detailed look at a few selected mass regions, however, reveals a serious departure from the isotopic and isotonic trends discussed so far. For AR between 50 and 70 (values of 2( in fig. 1 between 4 and 8), the positive-going isotopic and isotonic slopes
TABLE 1 Total (n, 2n) reaction cross sections at excitation energies of 3.0 and 6.0 MeV Target nucleus
Residual nucleus
Neutron excess residual
A
AR
(N--Z)R
Q-value (MeV)
1
2
3
4
18F
0
19F
crn,2n in mb at an excitation of
--10.44
5
6
7
8
49.5±7
7)
79
4-7
7)
29 ±2.2
8)
72
±7
8)
37 4-24
7)
87.34-6.1
lo)
112.54-4.5
9)
58.6 4-14
7)
23 11Na
22 11Na
0
36.45_23.5
11)
69
31 15 P
30 15 P 38 19 K
0
--12.32
29 4-3
12)
55.254-6
0
--13.08
11.34-1.2
7
9.93
1070
4-360
13)
--1t.32
4-18
14)
~K 48 20 Ca ~Sc
47 20 Ca 44 218c
2
46 22 TI•
45 22 TI-
1
~Cr
~Cr
1
--12.41
U = 3.0:~0.8 reference U = 6.04-0.8 reference (MeV) (MeV)
--13.2
8) 14) 14)
224
4-12
14)
144 :~12
17)
214
4-25
17)
--12.93
48 ± 4
17) 13)
,CrS'
3
-12.05
4
--10.22
793 4-48
15)
~Fe
26Fe53
1
--13.62
50.45:5
16)
26Fe
3
--11.21
440 4-88
13)
~Co
27Co58
4
--10.46
232 ±30
19)
~Cu
62 29Cu
4
--11.92
--10.84
8)
120 4-7
54 25Mn
1
12)
296.34-7
~Mn
57 • 28N1
4-2
11)
490
280 -L50
58 • 28N1
24
4-13
586 4-100
18)
563 i 3 4
15)
39.3±2
14)
84.5 4-6.5
17)
1030
4-62
15)
640
±68
19)
795
±100
18)
778
4-47
15)
67.6 ±3.4
14)
39 4-4
19)
24
±3
19)
38 ±2.7
20)
55
4-3.9
20)
420 4-90
21)
820
±90
21)
438 4-28
22)
530
4-50
22)
378 4-34
23)
758
+68
23)
430 ±55
18)
823
±70
18)
424 ::[:22
24)
,m387
25)
512 4-38
20)
~635 842
25) 4- 57
20)
597
(n, 2n) REACTIONS TABLE 1 (continued) Target nucleus A 1
695Cu
~Zn
Residual nucleus
Neutron excess residual
AR
(N--Z) R
Q-value (MeV)
2
3
4
5
64 29Cu
6
--9.91
650 ~31
14)
990
±50
14)
739 4-25 747 4-52
26) 27)
1003 975
4-60 ±85
26) 12)
1076
4-75
27)
377
4-30
28)
63 3oZn
3
an, 2n in mb at an excitation of
11.86
U = 3.04-0.8 reference U = 6.0-4-0.8 reference (MeV) (MeV)
293 ~25 ~199
69 31Ga ~]Ga 70~e 32~ ]~Ge
68 31Ga 7O 31Ga
6
10.23
8
--9.2
69 32 Ge
5
--11.62
11
]~As 74 34 se
32Ge75 74 33As 73Ne 34-
8 5
--10.24 --12.04
~Se
81 3,Se
13
--9.19
22) 32)
216 ± 16
20)
174 4-17.4
29)
688 ~66
8)
664 dz20
14)
666 ~233
31)
858 4-36
14)
383 4-29
38)
358 ±29
33)
--9.45
7
8
~358 404
32) 4-28
20)
1171
:]:_115
8)
2180
4-218
30)
883
4-45
14)
1200
4-240
3O)
1195
4-61
14)
1500
4-345
31)
1119
4-89
33)
788 ±61
34)
793 4-48
35)
615 ~-75
18)
810
4-85
18)
--10.53
1240 ~75
14)
1660
± 166
14)
379Br
35Br78
8
--10.60
81 35 Br
8o 35 Br 84 37Rb
l0
-- 10.12
10
875Rb 87 37 Rb
6
86 Rb 37 83 Sr 38
12
--9.92
890 ~53
14)
1300
! 60
14)
84 38 Sr
7
11.91
180.64-9
14)
255
~35
14)
386Sr
85 38Sr
9
-- 11.46
592 ~_51
1)
388Sr
87 38Sr
11
--11.12
215 ~24
X)
89y
88 39 Y
10
--11.69
1173
±59
14)
685 4-68.5 542 ~_58
90 4oZr
89 4oZr
9
-- 11.95
36) 1)
~400
37)
677 ± 5
38)
856 4-26
14)
TABLE 1 (continued) Neutron excess
crn, 2n in mb at an excitation of
Target
Residual
nucleus
nucleus
residual
Q-value
A
AR
(N--Z) R
(MeV)
3
4
10
--8.82
1
~Nb
2
92 41Nb
U = 3.04-0.8 reference U = I 6 . 0 4 - 0 . 8 reference (MeV) (MeV) 5
270
6
4-27
36)
~Mo
43Mo91
7
--13.14
632
4-130
2a)
~Ru
44Ru95
7
--10.20
616
4-50
34)
1031/h 45 ~ ll0pd 46 ~
102Rh 45--~ 109pd 46~--
12
--9.33
580 4-58
36)
17
--9.37
109 A_ 47.~g
108 _ 47Ag
14
--9.18
106Cd
108Cd
9
--8.50
48~-l t49 5 .m 112 50Sn-
llScd 48~ 1~4I49 -n 111 5oSa-
19
--8.64
1442 4-102
16
--9.02
1132 4-57
11
--11.09
1400 4-110
34)
725 4-73
36)
1508 4-117
38)
48~116Cd
121gh 51~ 123gh 51~ 128~ 521e
45~-
120gh 51-122gh 51-127 52Te
18
--9.29
~725
7
8
420
4-42
36)
560
4-62
37)
499
4-91
39)
464
4-23
4o)
939
4-138
21)
790
4-80
36)
2570
4-160
41)
710
4-110
34)
710
4-106
30)
883
4-88
42)
827
4-63
a8)
14)
1642
4-117
14)
14)
1503
4-76
14)
25)
~1053
25)
1180
4-180
3o) 31)
20
--8.98
1245
4-300
23
--8.43
779
4-230
31)
640
4-23
41)
599
4-120
at)
130~ 52 l e
129 52Te
25
--7.96
127I 53" 133~ 55 ~s
126I 53132~ 55 ~s
20
--9.15
22
--6.74
134~ 56 ~a l~Ba
133 56 B~ l~Ba
21
--9.28
23
138n 56 Da 140~ 58 ~e
137 ~ 56 Ba 139 58 C~_
141~ 59rr
140 59Pr_
580
4-133
41)
1290
4-140
18)
1200
4-110
18)
940
+80
44)
--9.20
700
±80
44)
25
8.58
1250
± 100
44)
23
--9.06
3000
4-400
44)
22
--9.37
1640
i150
23)
1378
4-207
3o)
~.900
1231
4-111
43)
23)
TABLE 1 (continued) Neutron Target nucleus
Residual nucleus
A 1
O'n,2nin mb at an excitation of
excess
AR
residual (N--Z) R
Q-value (MeV)
2
3
4
144~ 62am
143 __ 62Slu
19
--10.78
1 62 54
_ SIll
1 56 23 ~~ m
29
--7.99
151Eu 63
150~ 63 ~u
24
~8.05
U = 3.0/:0.8 reference U = 6.0-4-0.8 reference (MeV) (MeV) 5
6
1484 ±120
38)
1670 ±400
45)
7
8
1801
4-135
38)
2100
±300
44)
2250
-4-900 34)
1500
4-30
44)
500
1200
44)
4-64
30)
480
±62.4
46)
750
-4-200 44)
640 153~63~U
152Eu63
26
--8.65
159Th 65~ 165~_
28
--8.15
>1250
67 ~
158 65 Tb 164ml_l_ 6 7 ~.u
30
--8.12
166 68 E".
165 68 Et_
29
--8.43
181T~
73--a 182W
180
73T~ 181W
34 33
--7.65 --8.00
2300
~200
49)
186W
185W
37
--7.28
2290
±230
49)
185-75~e 187R75 ~c 198 78 Pt
184-75Ke 186R75 ~ 197Or 79-*
36 36
--7.73
1910
±600
5o)
--7.24
1440
±410
51)
41
--7.9l
2770
±1500
31)
197--
196A.
79~u
38
--8.07
1900
::k190
36)
1722
4-460
31)
203T! 81--"
202TI 81--"
40
--8.80
74"'
74'"
79~U
209~.
8351
232Th 90~--
238H 92 v
74
73""
208R:
83-1
231Th 90---
237H 92~
42
5l
53
1620 ! 1 6 2
36)
--6.10
47)
2100
±210
3o)
2760
4-55
4t)
1000
±400
44)
1800
~300
48)
1210 ~121
36)
1596
-4-165 36)
867 ±43
14)
1300
±70
14)
2420
4-200
28)
2300
-4-300 48)
1820
-}-182 52)
1646
4-175
36)
1540
4-80
5a)
--7.39
--6.43
4-300
1840 ±40
1800 4-100
53)
54)
1740
-4-176 a4)
1860
±100
54)
/ O
[
I
38
?
I
4s/
....
I
I
/ /,/~" "
I S
GE
/-l.-;~ X-"
ISOTOPICLINE
ISOTONIC LINF
I
Cu
47AS
I
IM0
I
t~'v/
~5
Y N ~" t1~j -
68
U = 6 Mev [~= MOST ABUNDANT TARGET DATA (~= LESS )~ ~) I)
1 I0
"~.
I
wH
I
k'B--~,-
(N -Z) R
1
- ~'~57
~N6_
~ ~-
89
_64
k
~, ~
~1) ~
I
l
I 15
,GO
%7
!/S I
.-:-,
------
I
I
I
I 20
@p~o~ ,,,, t SM I ,,~A N=Sl
I
o
I
I
- . ,.
I
.I-ISO
Y-. ',,-.~ Z12T~.."
..
PilL/#'
//~
~4e/t~
139 6CE
I 25
_
I
- ~
I
is
S~3
I
1
30
IE'I~-
IHo[
T ,64.
]
Fig. 1 (a). Experimentally measured (n, 2n) reaction cross sections plotted against n e u t r o n excess o f the residual nuclei. (a) T h e neutron excess range up to 2~ = 30 (up to about mass 160) a n d (b) the s a m e f r o m 2~ = 25 to 55 (till the end o f the nuclear t a b l e ) . T h e residual excitation is 6 MeV in b o t h the graphs. Notice the three distinct trends: (i) a gross t r e n d A d r a w n t h r o u g h the m o s t a b u n d a n t target elements, (ii) the isotopic lines a n d the isotonic lines drawn as solid straight lines dotted straight lines respectively when all isotopes a n d isotones irrespective o f their a b u n d a n c y are considered a n d (iii) a n d a few distinct m i n i m a s h o w n as a dotted curve s t e m m i n g out o f the trend A whose positions are roughly labelled as A1, A2, etc. See text.
I0
P
CK
I
04
Z
I-,1
P q-v
$
%
E
E
137
,d
26
IO 3 -
i BA
I
T~P
1
U=6t~ev
i
1
3o
I
I
I
IA
I
35
I
' PE-I--~
I
¢..¢~'s
_~W T
18.5
I
,~6
I
I
t
I
I
OBP
p~7
Fig. l ( b ) . See c a p t i o n to fig. l ( a ) .
40
__0
( N - Z) R ---,.-
W z. .:.7. .4 r . . . . . A:-
181
I
I
4,5
I
I
1
I
I
,50
I
I
I
T~ ~'-0U2~7-
I
55
z
.,..,
?
O
4'7 11
K
38
I
I
r#o%,-
IT
I/.//~ s,
I
I
mlkU/.LR
r~B4 ~111
")')
74 . , . o ~ B~ N
33
i 5
i
i
~/
,,
^;,~.,
i
I
i IO
\. 92~ NB ~
i
I
I
I is
/~J
~
(N - Z )~---,--
i
"~s#'
.'~*L o',,\~ q
86"
I
I
I
I
i2o nSB
I 20
I
ii26
u l N _ _ _ ~
i
I
R
i
i 25
I
i
i
i 30
Fig. 1 (c). A plot o f the experimental trn, 2n against 2C at 3 MeV excitation. The same three trends as discussed in figs. 1 (a) a n d 1 (b) are also observed here. See text.
IO
i
d,7#,/J-¢.,,
So []
FE ,~Cu~. 1 ,,
u . s4 ..... .~ ,,
3)
64 AS ~ / <>',""': Cu ;,B~ j . ~ ..%s 6 8 ~ n E " 7.7~-8s _ _ . : ~ o ~ ~ r , ~ ' t C ~ - 9 5 u u r ~ O - S R ,a-49,_ #/
ISOTONE LINE ISOTOPE LINE
....
-~,E~C~~
22
|T,O
iO 2'-.
I0
5
_
,/) LESS
~=
~ = DATA FOR MOST ABUNDANT TARGET ELEMENT
U = 3 MeV
C) ;= ,..] ,.-]
>
Z
C3 = > ,..]
t-J
(n, 2n) REACTIONS
603
appear to align with the gross curve A. This can be seen by comparing the an, 2n of 89Zr and 86Rb (N = 49), and 95Ru and 92Nb (N = 51), where the gross trend A tapers off considerably. Slightly beyond 2~ ~ 8, (in the region of AR between 80 to 100) there is even a region showing negative slopes, e.g., in the S3Sr and 87Sr (Z = 38) and 8aRb and 86Rb (Z = 37) isotopes, and in the aSy and STSr isotones (N = 49); in the particular case of the 95Ru and 92Nb isotones (N = 51), there is indeed an extremely large drop in the cross section at 2( = 10. This dip is labelled as curve A1 in figs. l(a) and l(c). The nuclei where this large abrupt dip exist 1) are near Z = 40 and N = 50. Cross sections just beyond this dip (shown as the dotted curve A 2 for two different excitations in fig. 1) again show positive isotopic and isotonic slopes at the beginning but with much larger than the average Csikai-Peto slope trends; e.g., S7Sr and 86Rb (N = 49) define a much larger slope than that for N = 23, 29 or 33; they then merge gradually into the gross trend A. There appears to be a slight deviation from the gross trend near AR ~ 110 (2ff ~ 14) and near A R ~ 135 (2~ ~ 24). This is shown as curves Aa, A4 and As, A6 respectively in figs. 1(a) and l(c). The isotopic and isotonic lines could not be drawn accurately in the region of A 3 A4 due to lack of data, but the small hump near A 5 A 6 seems to be real from the rapidly varying behaviour and the adjoining negative isotonic slopes e.g., in 14°pr and 13VBa (N = 81), 129Te, ~33Ba and 132Cs (N = 77) and the negative isotopic slopes e.g., at 127Te and 129Te (Z = 52) and 133Ba and 35Ba (Z = 56). We note that in the Ba isotopes, the isomeric cross section shows a dip in 13SBa relative to its lighter and heavier isotope; it is expected therefore that the same dip could be observed for the total cross section in these isotopes. The data are few and far between beyond 164Ho. A detailed study of the local behaviour is therefore not possible but there seems to be some evidence of one more dip near the 2~ ~ 36 final nucleus (shown as curve A7 in fig. 1(b). In the original Csikai-Peto graph, the region of the heavy nuclei (AR ~ 150, 2~ ~ 25) have been covered only from the reduced data estimated from the Blatt-Weisskopf estimates 5) and not from direct experiments. The existing data for the nuclei in this mass region seem to suggest a different trend than that conjectured in ref. 4). In regions where these dips are approached, the usually inverted Gardner trend is changed into a " n o r m a l " Gardner trend, i.e., the (n, 2n) cross sections for the heavier isotopes or isotones are lower 55) than those of the lighter ones. It is thus clear that although the two (isotopic and isotonic) Csikai-Peto trends of approximately parallel slopes seem to exist in a few selected regions, the real crosssection behaviour is more complicated. There is a gross trend A resembling a "growth curve"; superimposed on this are the relative Csikai-Peto trends in a few selected regions; other regions of this gross trend A exist where the slopes are zero or negative; the regions of negative slope seem to approach positions of distinct minima, beyond which the rising Csikai-Peto type trends are again clearly seen on the heavier mass side. The minima observed in the present work correspond t o the following values of
604
s. CHATTERJEE AND A. CHATTERJEE
2( for the residual nucleus: 2( = 10, 14, 24 and 36. l h e r e also appears to be weak evidence of a minimum at 2( ~ 4. As the volume and precision of the experimental information increases, it is possible that more detailed character of the crudely drawn gross trend A may reveal itself and a more detailed study of the three crudely observed effects may then be possible.
3. Analysis of the gross trend The gross trend of the cross section as a function of the neutron excess 2( or of the mass number AR seems to have a similarity with the behaviour of the neutron absorption cross section ac(n ) and is reminescent of a giant excitation function covering the whole mass region (as shown previously s6) in the case of 14 MeV (n, p) reactions. Levkovskii s7) has expressed the 14 MeV (n, p) reaction cross section an, p in the form O'n, p = O ' c ( n ) ~ p )
(1)
where ~p is the probability of proton decay of the compound nucleus. This may be rewritten in terms of the nucleon radius and the de Broglie wavelength 2 as
a.. p = nr2o(A* + >~/ro)Z~p.
(2)
The quantity % was chosen by Levkoskii in terms of N, Z and A of the target nucleus to be ~p : exp [fl(N-Z)/A], (3) where fl is a constant (fl = - 0 . 3 3 for 14 MeV (n, p) reactions). We analogously try to express the (n, 2n) reaction cross section in the form a , , 2 . = a~(n)~2, = ~zrg(A ~ + ~/ro) z exp [?(U- Z)/A ],
(4)
in terms of the relative (n, 2n) decay probability c%, and a constant T. Now, ~/ro = 1.01 for 14 MeV neutrons with a value of r 0 = 1.2 fro. Normalizing the observed cross sections at the 141Pr and 19apt target nuclei (2( = 24 and 42) at 6 MeV excitation, we obtain a good fit with 7 - - 0 . 5 0 . Eq. (4) may now be written as
a , . 2 , ~ 45.2 (A*+ 1) z exp [ - 0 . 5 0 ( U - Z ) / A ] m b .
(5)
Clearly, T represents the excitation energy dependent term in c~2n; at 3 MeV excitation, y = - 2 . 6 0 fits the experimental data. Total cross section calculations using (5) at 3.0 and 6.0 MeV excitations are shown graphically in fig. 2; they may be compared with the gross trends A of fig. 1 (reproduced as dotted lines). It is noticed that a real difference is indicated in the lightest nuclei, persisting up to A ~ 60, 2( ~ 4. Beyond this, there is little difference between eq. (5) and the observed gross trend A.
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NEUTRON EXCESS (N-Z)RESiDUA L
Fig. 2. A comparison of the semi-empirically calculated values of the (n, 2n) reaction cross sections (after Levkovskii) with the experimentally observed gross trends of fig. 1; the fit is reasonably good for N - - Z > 4 (A R > 60) for both 3 and 6 MeV excitations.
606
S. C H A T T E R J E E A N D A. C H A T T E R J E E
The semi-empirical eq. (5) thus predicts a structureless behaviour. The observed Csikai-Peto type of isotcpic and isotonic trends are averaged out and the clips reported in subsect. 2.3 are also smoothed out; these cannot be reproduced from eq. (5). We need a more detailed cross section equation to analyse the trends reported in subsect. 2.2 and subsect. 2.3. This is attempted in sect. 4 where a more detailed character of ~2. is studied.
4. Detailed analysis The (n, 2n) reaction cross section in terms of the compound nuclear formation probability a~(n), is 5a, 59)
a., 2. = a¢ganP2.Pzn(U2.,J)/~ gil~,p,(V,,j),
(6)
i
where g, # and p(U,j) are the spin weight factors, reduced mass and level densities at excitation U respectively. We are not concerned, with the barrier effects in this case. The sum i is over all possible decay channels including the (n, 2n) channel occurring in the numerator in eq. (6). In the two Fermion gas model, the level density in the usual notation is 59)
p(U,j) ~ A-2k(2j+ 1)ep(U, A),
(7)
where
p(U, A) ~ U -2 exp rr(AU/Q~:, and k is a constant. We assume here that (i) the isotopic and isotonic behaviour and (ii) the low (n, 2n) cross sections in the selected regions are both due to an effect causing an excitation energy shift from U to U' through a suitable shift f u n c t i o n f
U= U'+f
( f < U).
(8)
The shifted level density then is 59)
p(U',j) ~ kA-Z(2j+ 1),. p(U, A)[exp ½7~(A/~U)~f}]/(1 + 2f/U) P o [exp½rr(A/e V )~f} 1/(1 + 2f/U),
(9)
where Po may be normalized at the midshell if necessary, remains constant for a particular shell, but varies from shell to shell. Rosenzweig shift functions were used to account for the shell effects in 14 MeV (n, ~) and (n, p) reactions 56, s9). It is of interest to see if the same shift function 6~)
f = ~1. ,/1. ,M ~ 2 - t_- y ~1 p,4 ~7
'2
- ~ld , ( n - ~ 1N t ) 2 -½dp(P-½Z') 2,
(10)
is able to explain and reproduce other features of the single-particle behaviour in the present case also. In eq. (10), dn and dp are the neutron and proton level spacings, n and p are the extracore neutrons and protons and N ' and Z ' are the maximum occupation numbers of the neutrons and protons of the unfilled sub-shells in question.
(n, 2n) REACTIONS
607
We adopt the notation of fig. 3. Here Ao = No+Zo represents the magic core nucleus, AR, N, Z are the mass, neutron and atomic numbers of a particular residual nucleus in question and ~ o = No - Zo is a measure of the shell filling orders of neutrons relative to protons. In terms of these variables, eq. (10) can be easily modified after some algebra to
F = f / d = --4-~[(N'+Z')2+(N'-Z')Z]+¼[(N'+Z')(AR--Ao)-(AR--Ao) z] + ¼[ ( N ' - Z')(2~ - ~ o ) - ( 2 ~ - ~o)2 ] •
(11)
We notice that eq. (11 ) is essentially of the form
F = -a+F(A)+F(~),
(12)
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I
l
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where a = ,-~[(N'+Z')Z+(N'-Z') 2] is a constant and depends only on the degeneracies N', Z ' of the shell model states, F(A) contains the mass dependent terms alone and F(~) depends only on the neutron excess N - Z ; both F(A) and F(~) contain linear and quadratic terms in contrast with the simple Levkovskii equation (5). Combining eqs. (9), (10) and (11), we get
p(U',j) ~, Po exp [ - zrF~+/36(UA)+]/(1+eF/9UA),
(13)
which is essentially of the form
p(U',j) ,,~ Po{ exp(-CIF/A)+}/( 1 +C2F/A).
(14)
The constants C~ and Cz depend only on the Fermi level and excitation energies of the system. With e ~ 24 MeV and U between 3 and 6 MeV, the value of C 2 lies
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N E U T P O N EXCESS ( N - Z ) RESIDUAL Fig. 4. Plots o f the modified Rosenzweig shift function F a n d its modified c o m p o n e n t functions F(A) and F ( O against the residual neutron excess N - - Z . T h e main ordinate scales (from --10 to + 1 0 0 ) belong to F a n d F(A). The F(~) scale (from --20 to + 2 0 ) is shifted to coincide with the value + 6 0 o f F or F(A). Note the striking similarity in the shapes o f the total function F a n d the m a s s dependent part o f the shift function F(A). Note also that F ( 0 averages out to zero, These two features can be seen by plotting also against the residual m a s s AR.
(n, 2n) REACTIONS
609
between 1.0 and 0.5. The m a x i m u m value of F then reaches ~ 40 for A ~ 200. These substitutions show that
C2F/A ,~ 0.1, and hence is negligible compared to unity in the denominator of eq. (14). Again, C1/A"~ is constant for the "fixed-A assumption" within a shell but varies from shell to shell. Disregarding the detailed forms of the normalizing factors Po and C 1, the variation of the shift function F itself is expected to account well for the relative variation of the cross sections with the neutron excess N - Z or with the mass number AR. For a semiquantitative comparison of the observed variation of the cross sections, a plot of F against N - Z is considered together with its component functions F(A) and F(() in fig. 4. We notice that F(() averages out to zero for most nuclei with only a few exceptions; a gross average of F(~) may be taken as F(~) ~ 0. As is expected from the form of F ( ( ) in eq. (11), nuclei with one fixed value of (o lie on one parabola but those with different values of (o lie on different parabolas. The variation of F(A) with ( N - Z ) is much more prominent than that of F((). The total function F has thus a variation which is closely matched with that of the parabolic function F(A). Both these curves show dips at 2( = 4, 10, 16, 23, 28 and 43, only a few of which (discussed before) are observed at the present state of experimental precision. Different nuclei lie on the distorted parabolas of F with corresponding values of A0. The isotopic and isotonic lines are essentially guided by the envelope of these parabolas and their slopes are determined from the relative positions on these parabolas.
5. Discussion and conclusions
In this work we have used a crude comparative procedure throughout to understand a limited set of observations. The gross behaviour of the observed (n, 2n) reaction cross section shows a structureless form, similar to that of the neutron absorption cross-section behaviour. A semi-empirical Levkovskii-type analysis agrees fairly well with the observed trend for mass number A > 60. The discrepancy in the region A R < 60 (2~ < 8) is believed to be not due to single-particle effects and a detailed optical model analysis has not been attempted here to study the neutron absorption behaviour. One of the most important structure effects - the nuclear shell effect - has been described 56. s9) in terms of a suitably shifted Fermi gas model in the fast neutron (n, ~) and (n, p) channels. We essentially see the same effect in the (n, 2n) channel here against isospin or neutron excess as a variable parameter. The description of the singleparticle behaviour in terms of the shifted gas seems to be a fair representation in the (n, 2n) channel as well. In the 14 MeV (n, ~) and (n, p) channels, the average residual excitation energies U are high (approximately g 17 MeV and g 14 MeV corresponding to a compound
610
S. CHATTERJEE AND A. CHATTERJEE
nuclear excitation ~ 20 MeV) and the average small relative Q-corrections, to the first order, have negligible effects for comparison purposes and for studies in structure effects. In the present case of (n, 2n) reactions, the highly variable mass effect Q(n, 2n) reduces the residual excitation U within the range of 3 to 8 MeV approximately; it is therefore extremely important to normalize the excitation energies for similar comparison purposes and for detailed studies in the nuclear structure effects in this channel. One should thus attempt to select the data from available excitation function studies, and should not use the 14 MeV (n, 2n) data alone as has been done by Hille 60). The single-particle effects, viz., the shell effects, are apt to get lost in a cross-section comparison at unDormalized low residual excitation energies in accordance with Hille's observations; we have here a variable f-correction (0 to ~ 3 MeV) superimposed on a fluctuating residual excitation ( ~ 2 to ~ 8 MeV). It is interesting that the Rosenzweig model formolation used here seems to work within the domain of low lying structure (up to about half the binding energy of the last nucleon, i.e., > 3 MeV). The parabolic form of the Rosenzweig shift function within the respective particle subshells orients the different residual isotopes and isotones on the envelope of the (-parabolas fixed by (o. It is by chance that some of the isotopic and isotonic slopes 4) are aligned parallel after subshell closures (the Csikai-Peto effects). The shift function within a shell is strongly mass dependent. It also contains a relative mass variation across the shells. It is interesting that the average (-variation in the shift function is negligible. Thus, the residual excitation energies are strongly correlated with the ground state energies of different nuclei as a first approximation. Historically, the ( N - Z ) / A dependence in the 14 MeV (n, 2n) cross sections have been previously studied by Barr et al. 6~) and Perlstein 62) and was briefly reported by Breunlich et al. at the Antwerp conference 64). The subshell effects in the cross sections are rather indistinctly observed here by using 2( = N - Z as a variable parameter. For light nuclei 2( -- 0, and all cross section values lie on the ordinate scale in fig. l; for the heavier (medium weight and heavy) nuclei, there is the eflect of non-identical neutron and proton subshell interferences. These reduce the total observable magnitudes of the single-particle effects. Use of ( N - Z)/A as a parameter has an additional effect of the overlapping individual particle bebaviour introduced through the mass number AR -- N+Z. The net result of these two "averaging" processes is an almost structureless variation of the cross section with ( N - Z ) / A in accordance with the previous observations 6 0 - 6 2 , 6 4 ) . We are thankful to Professor J. Csikai and Dr. I. Angeli of Debrecen, Hungary, to Dr. M. Bormann of Hamburg, Germany and to Dr. P. Hille of Vienna, Austria, for their kind interest, comments and criticisms about this work. We are particularly indebted to Csikai and Hille for pointing out a few mistakes in the data used in the original manuscript.
(n, 2n) REACTIONS
611
References 1) P. Strohal, N. Cindro and B. Eman, Nucl. Phys. 30 (1962) 49 2) M. Bormann, Nucl. Phys. 65 (1965) 257 3) F. Manero, JEN Internal Report (unpublished) and Intern. Conf. on the study of Nuclear Structure with Neutrons (Antwerp, 1965) Vol. II, paper No. 120; private communication 4) J. Csikai and G. Peto, Phys. Lett. 20 (1966) 52 5) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (John Wiley, New York, 1952) p. 484, eq. 2.21 6) S. Chatterjee and A. Chatterjee, A table of fast neutron (n, 2n) excitation functions (unpublished) 7) J. Picard and C. Willamson, Nucl. Phys. 63 (1965) 673 8) M. Bormann, E. Fretwtirst, P. Schehka, G. Wrege, H. Buttner, A. Lindner and H: Meldner, Nucl. Phys. 63 (1965) 438 9) O. D. Brill, N. A. Vlasov, S. P. Kalinir and L. S. Sokolov, Doklady (Soy. Phys.) 6 (1961) 24 10) H. Liskien and A. Paulsen, Nucl. Phys. 63 (1965) 393 11) J. Picard and C. Willamson, J. Phys. Radium 24 (1963) 813 12) M. Bormann, S. Cierjacks, E. Fretwiirst, K. J. Giesecke, H. Neuert and H. Pollehn, Z. Phys. 174 (1963) 1 13) P. Hille, H. Vonach and R. Wenusch, ref. H 62 I of Tables of cross-sections of nuclear reactions with neutrons in the 14-15 MeV energy range by H. Neuert and H. Pollehn, EUR 122 e 14) R. J. Prestwood and B. P. Bayhurst, Phys. Rev. 121 (1961) 1438 15) A. Paulsen and H. Lieskien, J. Nucl. Energy, part A/B 19 (1965) 907 16) S. R. Salisbury and R. A. Chalmers, Phys. Rev. 140 (1965) B305 17) P. Jessen, M. Bormann, F. Dreyer and H. Neuert, private communication (1965); Nuclear Data 1 (1966) 103 18) M. Bormann, S. Cierjacks, R. Langkau and H. Neuert, Z. Phys. 166 (1962) 477 19) J. M. E. Jeronymo, G. S. Mani, J. Olkowsky, A. Sadeghi and C. F. Williamson, Nucl. Phys. 47 (1963) 157 20) A. Paulsen and H. Liskien, Nukleonik, part 3, 7 (1965) 117 21) J. E. Brolley, J. L. Fowler and L. R. Schlacks, Phys. Rev. 88 (1962) 618 22) A . V . Cohen and P. H. White, Nucl. Phys. 1 (1956) 73 23) J. M. Ferguson and W. E. Thompson, Phys. Rev. 118 (1960) 228 24) R. M. Glover and E. Weigold, Nucl. Phys. 29 (1961) 309 25) L. A. Rayburn, Phys. Rev. 130 (1963) 731 26) J. P. Butler and D. C. Santry, Priv. communication to "Compilation of experimental excitation functions of some fast neutron reactions up to 20 MeW' by P. Jessen, M. Bormann, F. Dreyer, H. Neuert, Hamburg, Germany, Jan. 1965 (unpublished) 27) H. Liskien and A. Paulsen, J. Nucl. Energy, part A/B 19 (1965) 73 28) D. R. Kochler and W. L. Alford, Phys. Rev. 119 (1960) 311 29) A. M. Ghosh and B. Mitra, Proc. Symp. Nuclear Physics, Kanpur (1967); private communication (1966). 30) C. S. Khurana and H. S. Hans, Nucl. Phys. 28 (1961) 560 31) E. B. Paul and R. L. Clarke, Can. J. Phys. 31 (1953) 267 32) L. A. Rayburn, Bull. Am. Phys. Soc. 6 (1961) 462; see also ref. 25) 33) P. V. Rao and R. A. Fink, Phys. Rev. 154 (1967) 1023 34) L. A. Rayburn, Bull. Am. Phys. Soc. 4 (1959) 288 35) M. Cevolani and S. Petralia, Nuovo Cim. 26 (1962) 1328 36) Tewes, Caretto, Miller and Meltraway, WASH 1028 (1960), p. 66 37) V. L. Glagolev and P. A. Yampolsku, JETP (Soy. Phys.) 13 (1961) 520 38) L. A. Rayburn, Phys. Rev. 122 (1961) 168 39) E. T. Bramlitt and R. W. Fink, Phys. Rev. 131 (1963) 2649 40) R. Reider and H. Munzer, E A N D C 505 Vol. II, p. 123 41) G. C. Bonazzola, P. Brovetto, E. Chiavassa and R. Spinoglio, Nucl. Phys. 51 (1964) 337 42) S. K. Mukherjee, A. K. Ganguly and N. K. Majumdar, Proc. Phys. Soc. A77 (1961) 508 43) H. C. Martin and R. F. Taschek, Phys. Rev. 89 (1953) 1302 44) R. G. Wille and R. W. Fink, Phys. Rev. 118 (1960) 242
612 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63)
S. CHATTERJEE AND A. CHATTERJEE
W. L. Alford and D. R. Koehler, Phys. Rev. 129 (1963) 703 H. Spenke, Nucl. Phys. 51 (1964) 329 R. Prasad, D. C. Sarkar and C. S. Khurana, Nucl. Phys. 88 (1966) 349 L. Rosen and L. Stewart, Phys. Rev. 107 (1957) 824 A. A. Druzhinin, A. A. Lbov and L. P. Bilibin, Sov. J. Nucl. Phys. 4 (1967) 366 R. A. Karam, T. F. Parkinson and W. E. Ellis, Nucl. Sc. Abstr. 17, paper No. 24380 R. A. Karam, T. F. Parkinson and F. J. Munuo, Nucl. Sc. Abstr. 16, paper No. 13960 R. J. Howerton, UCRL 5226 (1956), WASH 193 (1957) J. P. Butler and D. C. Santry, Can. J. Chem. 39 (1961) 689 J. P. Knight, R. K. Smith and B. Warren, Phys. Rev. 112 (1958) 259 D. G. Gardner, Nucl. Phys. 29 (1962) 373 A. Chatterjee, Nucl. Phys. 60 (1964) 273 V. N. Levkovskii, JETP (Sov. Phys.) 18 (1964) 213 N. B. Gove and R. Nakasima, Nuovo Cim. 22 (1961) 158 A. Chatterjee, Phys. Rev. 134 (1964) B374; Nucl. Phys. 41 (1963) 192; 47 (1963) 511 P. Hille, Nucl. Phys. A107 (1968) 49 D. W. Barr, C. I. Browne and J. S. Gilmore, Phys. Rev. 123 (1961) 859 S. Pearlstein, Nucl. Sc. Eng. 23 (1965) 238 T. Ericson, Adv. in Phys. 9 (1960) 425; N. Rosenzweig, Phys. Rev. 108 (1957) 817 64) W. Breunlich, P. Hille, B. Karlik and S. Tegesen, Intern. Conf. on the Study of Nuclear Structure with Neutrons, (Antwerp, 1965) paper no. 117
[
Nuclear Physics A125 (1969) 593--612; (~) North-Holland Publishiny Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
SINGLE-PARTICLE BEHAVIOUR
IN FAST N E U T R O N (n, 2n) REACTIONS Miss SWAPNA CHATTERJEE and APARESH CHATTERJEE
Calcutta University and Saha Institute of Nuclear Physics, Calcutta Received 17 April 1968 Abstract: A recent survey of the fast neutron (n, 2n) reaction cross sections at excitation energies of 3, 6 and 7 MeV reveal a few interesting overlapping trends. Plotted against neutron excess, a structureless gross trend is seen throughout the mass region, while the second trend corroborates with the limited observations of Csikai and Peto; the third trend is one showing features of single-particle behaviour (shell effects). The gross trend has been analysed in terms of a Levkovskii type equation. Further, an attempt has been made to understand all these three trends semi-quantitatively as a compound nuclear decay process where the available residual excitation of a suitably shifted Fermi gas is properly treated; the observed single-particle behaviour (shell closure minima and the two Csikai-Peto trends) has been compared in details with the description in terms of the modified Rosenzweig shift functions. It is found that the predicted behaviour agrees reasonably well with all the observations reported here in the medium weight and heavy nuclei.
1. Introduction
There have been recent attempts 1-4) at the systematic understanding of the (n, 2n) cross-section behaviour at 14 MeV. Cindro et al. 1) experimentally demonstrated the existence of the neutron shell effects as minima around mass 90 region (target neutron number 50). Bormann 2) and Manero 3) have qualitatively discussed the neutron shell effects in these cross sections. Little analysis of the pairing effects in these reactions has been done as yet, although pairing corrections in the excitation energy depending on the nature of the target nucleus is now customary. Recently Csikai and Peto 4) have reported a neutron-excess dependence of the (n, 2n) reaction cross sections. They have used their own experimental data at about 3.0 MeV of residual excitation for the light and medium weight nuclei and have also used the reduced data at 3.0 MeV excitation from other excitation energies for the heavy nuclei using the Blatt and Weisskopf formula 5); it was found that a linear plot of these experimental and inferred cross sections against the neutron excess 2( = N - Z of the target nucleus (for constant even values of the neutron number N ranging from 24 to 82) gives a family of straight lines with approximately parallel slopes (except tbr N = 28) irrespective of Z being odd or even 4). In the present work, we re-examine the neutron excess trend on the basis of more recent cross section and excitation function data and notice that the real behaviour is more complicated. It shows three separable distinct trends. A gross trend may be 593
594
S. C H A T T E R J E E A N D
A.
CHATTERJEE
expressed in terms of the two-neutron decay probability (Levkovskii equation). Further understanding of the three trends comes from the detailed analysis of the compound nuclear decay process; we reformulate the single-particle excitation characteristics in the framework of the modified Rosenzweig model (combinatorial occupational degeneracy model in subshells shifting the Fermi surface with respect to suitable doubly magic nuclei). An expression for the relative level density as a function of the excitation variables is compared with the three observed neutron excess trends. 2. Observations The available data 6) on the fast (n, 2n) reaction cross sections have been used here at the three selected excitation energies U of 3.0, 6.0 and 7.0 MeV in two different ways. In cases where the excitation functions of the (n, 2n) reactions have been accurately measured, it has been possible to use the interpolated cross sections at these excitations within an error A U = _0.1 MeV; in the other cases where only the "14 MeV" data exist, an attempt to use the same excitations resulted in the excitation energy uncertainties A U = +0.8 MeV. The reported cross-section error Aan,2n in most cases is within ___15700. These combined data are summarized in table 1. In the analysis to be presented, no attempt has been made to include the 14 MeV data exclusively. In a few cases, e.g., l°6Ag, 114In, 12°Sb, 133Ba, a3SBa, 137Ba, ~SSTb and 164Ho residual nuclei, formation cross sections are known only to their metastable states. In these cases, an estimate of their total cross sections have been made by using the following procedure: The spin cut-off parameters were first estimated from the known isomeric and ground state cross sections a m and a g in the nearest neighbour nuclei (e.g., 115mCd, 115gCd, 129mTe, 129gTe, etc.). Suitable average values of these cut-off parameters (consistent with the fact that the (n, 2n) cross sections should never exceed the geometrical cross sections) were chosen, and the isomer ratio equation was used to estimate the ground-state cross sections a g. In practically all cases, the total was less than twice the isomer cross section a m. For 92Nb, a g was assumed to be approximately equal to the total cross section because the isomeric states decay by 7-emission to the longer lived ground state. 2.1. THE GROSS BEHAVIOUR A plot of the cross sections listed in table 1 is shown in the semilog scale in fig. 1 as a function of the residual neutron excess 2( at two values of approximately constant excitation U of 3 and 6 MeV. 3-he points for the residual isotopes and isotones have been joined by solid and dotted straight lines respectively in most cases for comparison with the reported Csikai-Peto trends 4). Confining our attention to the excitation pattern at 6.0 MeV for the most abundant target elements only, we notice that a gross smoothed behaviour is suggested for all such nuclei by the trend of the curve marked A. This trend is one of (i) a very rapid
(n, 2n) REACTIONS
595
rise of the reaction cross section fin, 2n from 2( = 0 to 2( = 4, (ii) a tapering off of °'n, 2n to a value ~ 1.5 b at A R ~ 90 (2( = 12) and then (iii) a very slowly rising continuation till the highest available 2( ,~ 150 at the end of the nuclear table where cr,,2n reaches ~ 2.2 b. The fewer data on the 3 MeV excitation in fig. 1(c) shows a similar gross trend (also marked as A); the maximum in Crn~2nhere reaches a lower value ~ 1.0 b. A comparison with the limited data at 7.0+0.8 MeV excitation (not presented here) suggests a similar gross trend. 2.2. T H E I S O T O N I C A N D I S O T O P I C B E H A V I O U R : C S I K A I - P E T O T R E N D S
Associated with the gross trend A shown in fig. 1, there are relative isotonic and isotopic trends. To be able to see these interesting features, we now consider all nuclei listed in table 1 irrespective of their abundancy. In our presentation (where the residual nuclei are being considered), all isotonic lines (solid straight lines) belong to the odd isotones; these correspond to the even isotones of Csikai and Peto. The isotopic lines (dotted straight lines), of course, occur with both odd and even atomic numbers. The lines of constant N (residual isotones) seem to align themselves approximately parallel to the gross trend A in figs. 1(a) and l(b) in the rapidly rising part near the lowest values of 2(. Thus the general features for 44Sc and 45Ti (N = 23), 57Ni and 54Mn (N = 29), 63Zn and 62Cu (N = 33), 69Ge and 68Ga (N = 37), 83Sr and 8°Br (N = 45), seem to be approximately parallel. These features up to AR ~ 100 have already been noted by Csikai and Peto 4) from their own experimental data. We call this trend the "Csikai-Peto trend" for isotones. The trend shows that generally an isotone of lower residual Z-value has a higher cross section than its higher isotope member. We may similarly define a Csikai-Peto trend for isotopes (not observed or reported in ref. 4)); a heavier isotope usually has a higher cross section than the lighter one. This is clearly seen for 62Cu and 64Cu ( Z = 29), 69Ge and VSGe (Z = 32), 12°Sb and 122Sb (Z = 51) residual nuclei. This is an "inverse Gardner trend"; Gardner 35) has shown that in the case of 14 MeV (n, p) reactions, the cross section of the liohter target isotope is higher (roughly by a factor of 1.5 to 2.0) than that of the heavier target. Thus, in regions where the gross trend has positive slopes, these isotopic and isotonic lines seem to obey the two Csikai-Peto trends reasonably satisfactorily. The trends are observed in all cases of the normalized excitation energies considered here, viz., at 3.0, 6.0 and 7.0 MeV. 2.3. D E P A R T U R E
FROM THE CSIKAI-PETO TRENDS
A detailed look at a few selected mass regions, however, reveals a serious departure from the isotopic and isotonic trends discussed so far. For AR between 50 and 70 (values of 2( in fig. 1 between 4 and 8), the positive-going isotopic and isotonic slopes
TABLE 1 Total (n, 2n) reaction cross sections at excitation energies of 3.0 and 6.0 MeV Target nucleus
Residual nucleus
Neutron excess residual
A
AR
(N--Z)R
Q-value (MeV)
1
2
3
4
18F
0
19F
crn,2n in mb at an excitation of
--10.44
5
6
7
8
49.5±7
7)
79
4-7
7)
29 ±2.2
8)
72
±7
8)
37 4-24
7)
87.34-6.1
lo)
112.54-4.5
9)
58.6 4-14
7)
23 11Na
22 11Na
0
36.45_23.5
11)
69
31 15 P
30 15 P 38 19 K
0
--12.32
29 4-3
12)
55.254-6
0
--13.08
11.34-1.2
7
9.93
1070
4-360
13)
--1t.32
4-18
14)
~K 48 20 Ca ~Sc
47 20 Ca 44 218c
2
46 22 TI•
45 22 TI-
1
~Cr
~Cr
1
--12.41
U = 3.0:~0.8 reference U = 6.04-0.8 reference (MeV) (MeV)
--13.2
8) 14) 14)
224
4-12
14)
144 :~12
17)
214
4-25
17)
--12.93
48 ± 4
17) 13)
,CrS'
3
-12.05
4
--10.22
793 4-48
15)
~Fe
26Fe53
1
--13.62
50.45:5
16)
26Fe
3
--11.21
440 4-88
13)
~Co
27Co58
4
--10.46
232 ±30
19)
~Cu
62 29Cu
4
--11.92
--10.84
8)
120 4-7
54 25Mn
1
12)
296.34-7
~Mn
57 • 28N1
4-2
11)
490
280 -L50
58 • 28N1
24
4-13
586 4-100
18)
563 i 3 4
15)
39.3±2
14)
84.5 4-6.5
17)
1030
4-62
15)
640
±68
19)
795
±100
18)
778
4-47
15)
67.6 ±3.4
14)
39 4-4
19)
24
±3
19)
38 ±2.7
20)
55
4-3.9
20)
420 4-90
21)
820
±90
21)
438 4-28
22)
530
4-50
22)
378 4-34
23)
758
+68
23)
430 ±55
18)
823
±70
18)
424 ::[:22
24)
,m387
25)
512 4-38
20)
~635 842
25) 4- 57
20)
597
(n, 2n) REACTIONS TABLE 1 (continued) Target nucleus A 1
695Cu
~Zn
Residual nucleus
Neutron excess residual
AR
(N--Z) R
Q-value (MeV)
2
3
4
5
64 29Cu
6
--9.91
650 ~31
14)
990
±50
14)
739 4-25 747 4-52
26) 27)
1003 975
4-60 ±85
26) 12)
1076
4-75
27)
377
4-30
28)
63 3oZn
3
an, 2n in mb at an excitation of
11.86
U = 3.04-0.8 reference U = 6.0-4-0.8 reference (MeV) (MeV)
293 ~25 ~199
69 31Ga ~]Ga 70~e 32~ ]~Ge
68 31Ga 7O 31Ga
6
10.23
8
--9.2
69 32 Ge
5
--11.62
11
]~As 74 34 se
32Ge75 74 33As 73Ne 34-
8 5
--10.24 --12.04
~Se
81 3,Se
13
--9.19
22) 32)
216 ± 16
20)
174 4-17.4
29)
688 ~66
8)
664 dz20
14)
666 ~233
31)
858 4-36
14)
383 4-29
38)
358 ±29
33)
--9.45
7
8
~358 404
32) 4-28
20)
1171
:]:_115
8)
2180
4-218
30)
883
4-45
14)
1200
4-240
3O)
1195
4-61
14)
1500
4-345
31)
1119
4-89
33)
788 ±61
34)
793 4-48
35)
615 ~-75
18)
810
4-85
18)
--10.53
1240 ~75
14)
1660
± 166
14)
379Br
35Br78
8
--10.60
81 35 Br
8o 35 Br 84 37Rb
l0
-- 10.12
10
875Rb 87 37 Rb
6
86 Rb 37 83 Sr 38
12
--9.92
890 ~53
14)
1300
! 60
14)
84 38 Sr
7
11.91
180.64-9
14)
255
~35
14)
386Sr
85 38Sr
9
-- 11.46
592 ~_51
1)
388Sr
87 38Sr
11
--11.12
215 ~24
X)
89y
88 39 Y
10
--11.69
1173
±59
14)
685 4-68.5 542 ~_58
90 4oZr
89 4oZr
9
-- 11.95
36) 1)
~400
37)
677 ± 5
38)
856 4-26
14)
TABLE 1 (continued) Neutron excess
crn, 2n in mb at an excitation of
Target
Residual
nucleus
nucleus
residual
Q-value
A
AR
(N--Z) R
(MeV)
3
4
10
--8.82
1
~Nb
2
92 41Nb
U = 3.04-0.8 reference U = I 6 . 0 4 - 0 . 8 reference (MeV) (MeV) 5
270
6
4-27
36)
~Mo
43Mo91
7
--13.14
632
4-130
2a)
~Ru
44Ru95
7
--10.20
616
4-50
34)
1031/h 45 ~ ll0pd 46 ~
102Rh 45--~ 109pd 46~--
12
--9.33
580 4-58
36)
17
--9.37
109 A_ 47.~g
108 _ 47Ag
14
--9.18
106Cd
108Cd
9
--8.50
48~-l t49 5 .m 112 50Sn-
llScd 48~ 1~4I49 -n 111 5oSa-
19
--8.64
1442 4-102
16
--9.02
1132 4-57
11
--11.09
1400 4-110
34)
725 4-73
36)
1508 4-117
38)
48~116Cd
121gh 51~ 123gh 51~ 128~ 521e
45~-
120gh 51-122gh 51-127 52Te
18
--9.29
~725
7
8
420
4-42
36)
560
4-62
37)
499
4-91
39)
464
4-23
4o)
939
4-138
21)
790
4-80
36)
2570
4-160
41)
710
4-110
34)
710
4-106
30)
883
4-88
42)
827
4-63
a8)
14)
1642
4-117
14)
14)
1503
4-76
14)
25)
~1053
25)
1180
4-180
3o) 31)
20
--8.98
1245
4-300
23
--8.43
779
4-230
31)
640
4-23
41)
599
4-120
at)
130~ 52 l e
129 52Te
25
--7.96
127I 53" 133~ 55 ~s
126I 53132~ 55 ~s
20
--9.15
22
--6.74
134~ 56 ~a l~Ba
133 56 B~ l~Ba
21
--9.28
23
138n 56 Da 140~ 58 ~e
137 ~ 56 Ba 139 58 C~_
141~ 59rr
140 59Pr_
580
4-133
41)
1290
4-140
18)
1200
4-110
18)
940
+80
44)
--9.20
700
±80
44)
25
8.58
1250
± 100
44)
23
--9.06
3000
4-400
44)
22
--9.37
1640
i150
23)
1378
4-207
3o)
~.900
1231
4-111
43)
23)
TABLE 1 (continued) Neutron Target nucleus
Residual nucleus
A 1
O'n,2nin mb at an excitation of
excess
AR
residual (N--Z) R
Q-value (MeV)
2
3
4
144~ 62am
143 __ 62Slu
19
--10.78
1 62 54
_ SIll
1 56 23 ~~ m
29
--7.99
151Eu 63
150~ 63 ~u
24
~8.05
U = 3.0/:0.8 reference U = 6.0-4-0.8 reference (MeV) (MeV) 5
6
1484 ±120
38)
1670 ±400
45)
7
8
1801
4-135
38)
2100
±300
44)
2250
-4-900 34)
1500
4-30
44)
500
1200
44)
4-64
30)
480
±62.4
46)
750
-4-200 44)
640 153~63~U
152Eu63
26
--8.65
159Th 65~ 165~_
28
--8.15
>1250
67 ~
158 65 Tb 164ml_l_ 6 7 ~.u
30
--8.12
166 68 E".
165 68 Et_
29
--8.43
181T~
73--a 182W
180
73T~ 181W
34 33
--7.65 --8.00
2300
~200
49)
186W
185W
37
--7.28
2290
±230
49)
185-75~e 187R75 ~c 198 78 Pt
184-75Ke 186R75 ~ 197Or 79-*
36 36
--7.73
1910
±600
5o)
--7.24
1440
±410
51)
41
--7.9l
2770
±1500
31)
197--
196A.
79~u
38
--8.07
1900
::k190
36)
1722
4-460
31)
203T! 81--"
202TI 81--"
40
--8.80
74"'
74'"
79~U
209~.
8351
232Th 90~--
238H 92 v
74
73""
208R:
83-1
231Th 90---
237H 92~
42
5l
53
1620 ! 1 6 2
36)
--6.10
47)
2100
±210
3o)
2760
4-55
4t)
1000
±400
44)
1800
~300
48)
1210 ~121
36)
1596
-4-165 36)
867 ±43
14)
1300
±70
14)
2420
4-200
28)
2300
-4-300 48)
1820
-}-182 52)
1646
4-175
36)
1540
4-80
5a)
--7.39
--6.43
4-300
1840 ±40
1800 4-100
53)
54)
1740
-4-176 a4)
1860
±100
54)
/ O
[
I
38
?
I
4s/
....
I
I
/ /,/~" "
I S
GE
/-l.-;~ X-"
ISOTOPICLINE
ISOTONIC LINF
I
Cu
47AS
I
IM0
I
t~'v/
~5
Y N ~" t1~j -
68
U = 6 Mev [~= MOST ABUNDANT TARGET DATA (~= LESS )~ ~) I)
1 I0
"~.
I
wH
I
k'B--~,-
(N -Z) R
1
- ~'~57
~N6_
~ ~-
89
_64
k
~, ~
~1) ~
I
l
I 15
,GO
%7
!/S I
.-:-,
------
I
I
I
I 20
@p~o~ ,,,, t SM I ,,~A N=Sl
I
o
I
I
- . ,.
I
.I-ISO
Y-. ',,-.~ Z12T~.."
..
PilL/#'
//~
~4e/t~
139 6CE
I 25
_
I
- ~
I
is
S~3
I
1
30
IE'I~-
IHo[
T ,64.
]
Fig. 1 (a). Experimentally measured (n, 2n) reaction cross sections plotted against n e u t r o n excess o f the residual nuclei. (a) T h e neutron excess range up to 2~ = 30 (up to about mass 160) a n d (b) the s a m e f r o m 2~ = 25 to 55 (till the end o f the nuclear t a b l e ) . T h e residual excitation is 6 MeV in b o t h the graphs. Notice the three distinct trends: (i) a gross t r e n d A d r a w n t h r o u g h the m o s t a b u n d a n t target elements, (ii) the isotopic lines a n d the isotonic lines drawn as solid straight lines dotted straight lines respectively when all isotopes a n d isotones irrespective o f their a b u n d a n c y are considered a n d (iii) a n d a few distinct m i n i m a s h o w n as a dotted curve s t e m m i n g out o f the trend A whose positions are roughly labelled as A1, A2, etc. See text.
I0
P
CK
I
04
Z
I-,1
P q-v
$
%
E
E
137
,d
26
IO 3 -
i BA
I
T~P
1
U=6t~ev
i
1
3o
I
I
I
IA
I
35
I
' PE-I--~
I
¢..¢~'s
_~W T
18.5
I
,~6
I
I
t
I
I
OBP
p~7
Fig. l ( b ) . See c a p t i o n to fig. l ( a ) .
40
__0
( N - Z) R ---,.-
W z. .:.7. .4 r . . . . . A:-
181
I
I
4,5
I
I
1
I
I
,50
I
I
I
T~ ~'-0U2~7-
I
55
z
.,..,
?
O
4'7 11
K
38
I
I
r#o%,-
IT
I/.//~ s,
I
I
mlkU/.LR
r~B4 ~111
")')
74 . , . o ~ B~ N
33
i 5
i
i
~/
,,
^;,~.,
i
I
i IO
\. 92~ NB ~
i
I
I
I is
/~J
~
(N - Z )~---,--
i
"~s#'
.'~*L o',,\~ q
86"
I
I
I
I
i2o nSB
I 20
I
ii26
u l N _ _ _ ~
i
I
R
i
i 25
I
i
i
i 30
Fig. 1 (c). A plot o f the experimental trn, 2n against 2C at 3 MeV excitation. The same three trends as discussed in figs. 1 (a) a n d 1 (b) are also observed here. See text.
IO
i
d,7#,/J-¢.,,
So []
FE ,~Cu~. 1 ,,
u . s4 ..... .~ ,,
3)
64 AS ~ / <>',""': Cu ;,B~ j . ~ ..%s 6 8 ~ n E " 7.7~-8s _ _ . : ~ o ~ ~ r , ~ ' t C ~ - 9 5 u u r ~ O - S R ,a-49,_ #/
ISOTONE LINE ISOTOPE LINE
....
-~,E~C~~
22
|T,O
iO 2'-.
I0
5
_
,/) LESS
~=
~ = DATA FOR MOST ABUNDANT TARGET ELEMENT
U = 3 MeV
C) ;= ,..] ,.-]
>
Z
C3 = > ,..]
t-J
(n, 2n) REACTIONS
603
appear to align with the gross curve A. This can be seen by comparing the an, 2n of 89Zr and 86Rb (N = 49), and 95Ru and 92Nb (N = 51), where the gross trend A tapers off considerably. Slightly beyond 2~ ~ 8, (in the region of AR between 80 to 100) there is even a region showing negative slopes, e.g., in the S3Sr and 87Sr (Z = 38) and 8aRb and 86Rb (Z = 37) isotopes, and in the aSy and STSr isotones (N = 49); in the particular case of the 95Ru and 92Nb isotones (N = 51), there is indeed an extremely large drop in the cross section at 2( = 10. This dip is labelled as curve A1 in figs. l(a) and l(c). The nuclei where this large abrupt dip exist 1) are near Z = 40 and N = 50. Cross sections just beyond this dip (shown as the dotted curve A 2 for two different excitations in fig. 1) again show positive isotopic and isotonic slopes at the beginning but with much larger than the average Csikai-Peto slope trends; e.g., S7Sr and 86Rb (N = 49) define a much larger slope than that for N = 23, 29 or 33; they then merge gradually into the gross trend A. There appears to be a slight deviation from the gross trend near AR ~ 110 (2ff ~ 14) and near A R ~ 135 (2~ ~ 24). This is shown as curves Aa, A4 and As, A6 respectively in figs. 1(a) and l(c). The isotopic and isotonic lines could not be drawn accurately in the region of A 3 A4 due to lack of data, but the small hump near A 5 A 6 seems to be real from the rapidly varying behaviour and the adjoining negative isotonic slopes e.g., in 14°pr and 13VBa (N = 81), 129Te, ~33Ba and 132Cs (N = 77) and the negative isotopic slopes e.g., at 127Te and 129Te (Z = 52) and 133Ba and 35Ba (Z = 56). We note that in the Ba isotopes, the isomeric cross section shows a dip in 13SBa relative to its lighter and heavier isotope; it is expected therefore that the same dip could be observed for the total cross section in these isotopes. The data are few and far between beyond 164Ho. A detailed study of the local behaviour is therefore not possible but there seems to be some evidence of one more dip near the 2~ ~ 36 final nucleus (shown as curve A7 in fig. 1(b). In the original Csikai-Peto graph, the region of the heavy nuclei (AR ~ 150, 2~ ~ 25) have been covered only from the reduced data estimated from the Blatt-Weisskopf estimates 5) and not from direct experiments. The existing data for the nuclei in this mass region seem to suggest a different trend than that conjectured in ref. 4). In regions where these dips are approached, the usually inverted Gardner trend is changed into a " n o r m a l " Gardner trend, i.e., the (n, 2n) cross sections for the heavier isotopes or isotones are lower 55) than those of the lighter ones. It is thus clear that although the two (isotopic and isotonic) Csikai-Peto trends of approximately parallel slopes seem to exist in a few selected regions, the real crosssection behaviour is more complicated. There is a gross trend A resembling a "growth curve"; superimposed on this are the relative Csikai-Peto trends in a few selected regions; other regions of this gross trend A exist where the slopes are zero or negative; the regions of negative slope seem to approach positions of distinct minima, beyond which the rising Csikai-Peto type trends are again clearly seen on the heavier mass side. The minima observed in the present work correspond t o the following values of
604
s. CHATTERJEE AND A. CHATTERJEE
2( for the residual nucleus: 2( = 10, 14, 24 and 36. l h e r e also appears to be weak evidence of a minimum at 2( ~ 4. As the volume and precision of the experimental information increases, it is possible that more detailed character of the crudely drawn gross trend A may reveal itself and a more detailed study of the three crudely observed effects may then be possible.
3. Analysis of the gross trend The gross trend of the cross section as a function of the neutron excess 2( or of the mass number AR seems to have a similarity with the behaviour of the neutron absorption cross section ac(n ) and is reminescent of a giant excitation function covering the whole mass region (as shown previously s6) in the case of 14 MeV (n, p) reactions. Levkovskii s7) has expressed the 14 MeV (n, p) reaction cross section an, p in the form O'n, p = O ' c ( n ) ~ p )
(1)
where ~p is the probability of proton decay of the compound nucleus. This may be rewritten in terms of the nucleon radius and the de Broglie wavelength 2 as
a.. p = nr2o(A* + >~/ro)Z~p.
(2)
The quantity % was chosen by Levkoskii in terms of N, Z and A of the target nucleus to be ~p : exp [fl(N-Z)/A], (3) where fl is a constant (fl = - 0 . 3 3 for 14 MeV (n, p) reactions). We analogously try to express the (n, 2n) reaction cross section in the form a , , 2 . = a~(n)~2, = ~zrg(A ~ + ~/ro) z exp [?(U- Z)/A ],
(4)
in terms of the relative (n, 2n) decay probability c%, and a constant T. Now, ~/ro = 1.01 for 14 MeV neutrons with a value of r 0 = 1.2 fro. Normalizing the observed cross sections at the 141Pr and 19apt target nuclei (2( = 24 and 42) at 6 MeV excitation, we obtain a good fit with 7 - - 0 . 5 0 . Eq. (4) may now be written as
a , . 2 , ~ 45.2 (A*+ 1) z exp [ - 0 . 5 0 ( U - Z ) / A ] m b .
(5)
Clearly, T represents the excitation energy dependent term in c~2n; at 3 MeV excitation, y = - 2 . 6 0 fits the experimental data. Total cross section calculations using (5) at 3.0 and 6.0 MeV excitations are shown graphically in fig. 2; they may be compared with the gross trends A of fig. 1 (reproduced as dotted lines). It is noticed that a real difference is indicated in the lightest nuclei, persisting up to A ~ 60, 2( ~ 4. Beyond this, there is little difference between eq. (5) and the observed gross trend A.
10 4
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CALCULA3-ED POINTS FOR
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NEUTRON EXCESS (N-Z)RESiDUA L
Fig. 2. A comparison of the semi-empirically calculated values of the (n, 2n) reaction cross sections (after Levkovskii) with the experimentally observed gross trends of fig. 1; the fit is reasonably good for N - - Z > 4 (A R > 60) for both 3 and 6 MeV excitations.
606
S. C H A T T E R J E E A N D A. C H A T T E R J E E
The semi-empirical eq. (5) thus predicts a structureless behaviour. The observed Csikai-Peto type of isotcpic and isotonic trends are averaged out and the clips reported in subsect. 2.3 are also smoothed out; these cannot be reproduced from eq. (5). We need a more detailed cross section equation to analyse the trends reported in subsect. 2.2 and subsect. 2.3. This is attempted in sect. 4 where a more detailed character of ~2. is studied.
4. Detailed analysis The (n, 2n) reaction cross section in terms of the compound nuclear formation probability a~(n), is 5a, 59)
a., 2. = a¢ganP2.Pzn(U2.,J)/~ gil~,p,(V,,j),
(6)
i
where g, # and p(U,j) are the spin weight factors, reduced mass and level densities at excitation U respectively. We are not concerned, with the barrier effects in this case. The sum i is over all possible decay channels including the (n, 2n) channel occurring in the numerator in eq. (6). In the two Fermion gas model, the level density in the usual notation is 59)
p(U,j) ~ A-2k(2j+ 1)ep(U, A),
(7)
where
p(U, A) ~ U -2 exp rr(AU/Q~:, and k is a constant. We assume here that (i) the isotopic and isotonic behaviour and (ii) the low (n, 2n) cross sections in the selected regions are both due to an effect causing an excitation energy shift from U to U' through a suitable shift f u n c t i o n f
U= U'+f
( f < U).
(8)
The shifted level density then is 59)
p(U',j) ~ kA-Z(2j+ 1),. p(U, A)[exp ½7~(A/~U)~f}]/(1 + 2f/U) P o [exp½rr(A/e V )~f} 1/(1 + 2f/U),
(9)
where Po may be normalized at the midshell if necessary, remains constant for a particular shell, but varies from shell to shell. Rosenzweig shift functions were used to account for the shell effects in 14 MeV (n, ~) and (n, p) reactions 56, s9). It is of interest to see if the same shift function 6~)
f = ~1. ,/1. ,M ~ 2 - t_- y ~1 p,4 ~7
'2
- ~ld , ( n - ~ 1N t ) 2 -½dp(P-½Z') 2,
(10)
is able to explain and reproduce other features of the single-particle behaviour in the present case also. In eq. (10), dn and dp are the neutron and proton level spacings, n and p are the extracore neutrons and protons and N ' and Z ' are the maximum occupation numbers of the neutrons and protons of the unfilled sub-shells in question.
(n, 2n) REACTIONS
607
We adopt the notation of fig. 3. Here Ao = No+Zo represents the magic core nucleus, AR, N, Z are the mass, neutron and atomic numbers of a particular residual nucleus in question and ~ o = No - Zo is a measure of the shell filling orders of neutrons relative to protons. In terms of these variables, eq. (10) can be easily modified after some algebra to
F = f / d = --4-~[(N'+Z')2+(N'-Z')Z]+¼[(N'+Z')(AR--Ao)-(AR--Ao) z] + ¼[ ( N ' - Z')(2~ - ~ o ) - ( 2 ~ - ~o)2 ] •
(11)
We notice that eq. (11 ) is essentially of the form
F = -a+F(A)+F(~),
(12)
No+N' Zo+Z' Z'
Z
+
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~
++_o__J ', Zo--
I
l
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Fig. 3. A sketch o f the notation used in sect. 4. Starting from the bottom of the nuclear well, a given nucleus of mass A contains Z protons and N neutrons. The numbers Zo, No refer to the last filled shells and Z o + Z ' and No+N" to the magic shells when p protons and n neutrons equal the total shell model degeneracies Z ' and N ' of the partially occupation shell. By definition 2~ = N - - Z and No--Zo = ¢o"
where a = ,-~[(N'+Z')Z+(N'-Z') 2] is a constant and depends only on the degeneracies N', Z ' of the shell model states, F(A) contains the mass dependent terms alone and F(~) depends only on the neutron excess N - Z ; both F(A) and F(~) contain linear and quadratic terms in contrast with the simple Levkovskii equation (5). Combining eqs. (9), (10) and (11), we get
p(U',j) ~, Po exp [ - zrF~+/36(UA)+]/(1+eF/9UA),
(13)
which is essentially of the form
p(U',j) ,,~ Po{ exp(-CIF/A)+}/( 1 +C2F/A).
(14)
The constants C~ and Cz depend only on the Fermi level and excitation energies of the system. With e ~ 24 MeV and U between 3 and 6 MeV, the value of C 2 lies
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N E U T P O N EXCESS ( N - Z ) RESIDUAL Fig. 4. Plots o f the modified Rosenzweig shift function F a n d its modified c o m p o n e n t functions F(A) and F ( O against the residual neutron excess N - - Z . T h e main ordinate scales (from --10 to + 1 0 0 ) belong to F a n d F(A). The F(~) scale (from --20 to + 2 0 ) is shifted to coincide with the value + 6 0 o f F or F(A). Note the striking similarity in the shapes o f the total function F a n d the m a s s dependent part o f the shift function F(A). Note also that F ( 0 averages out to zero, These two features can be seen by plotting also against the residual m a s s AR.
(n, 2n) REACTIONS
609
between 1.0 and 0.5. The m a x i m u m value of F then reaches ~ 40 for A ~ 200. These substitutions show that
C2F/A ,~ 0.1, and hence is negligible compared to unity in the denominator of eq. (14). Again, C1/A"~ is constant for the "fixed-A assumption" within a shell but varies from shell to shell. Disregarding the detailed forms of the normalizing factors Po and C 1, the variation of the shift function F itself is expected to account well for the relative variation of the cross sections with the neutron excess N - Z or with the mass number AR. For a semiquantitative comparison of the observed variation of the cross sections, a plot of F against N - Z is considered together with its component functions F(A) and F(() in fig. 4. We notice that F(() averages out to zero for most nuclei with only a few exceptions; a gross average of F(~) may be taken as F(~) ~ 0. As is expected from the form of F ( ( ) in eq. (11), nuclei with one fixed value of (o lie on one parabola but those with different values of (o lie on different parabolas. The variation of F(A) with ( N - Z ) is much more prominent than that of F((). The total function F has thus a variation which is closely matched with that of the parabolic function F(A). Both these curves show dips at 2( = 4, 10, 16, 23, 28 and 43, only a few of which (discussed before) are observed at the present state of experimental precision. Different nuclei lie on the distorted parabolas of F with corresponding values of A0. The isotopic and isotonic lines are essentially guided by the envelope of these parabolas and their slopes are determined from the relative positions on these parabolas.
5. Discussion and conclusions
In this work we have used a crude comparative procedure throughout to understand a limited set of observations. The gross behaviour of the observed (n, 2n) reaction cross section shows a structureless form, similar to that of the neutron absorption cross-section behaviour. A semi-empirical Levkovskii-type analysis agrees fairly well with the observed trend for mass number A > 60. The discrepancy in the region A R < 60 (2~ < 8) is believed to be not due to single-particle effects and a detailed optical model analysis has not been attempted here to study the neutron absorption behaviour. One of the most important structure effects - the nuclear shell effect - has been described 56. s9) in terms of a suitably shifted Fermi gas model in the fast neutron (n, ~) and (n, p) channels. We essentially see the same effect in the (n, 2n) channel here against isospin or neutron excess as a variable parameter. The description of the singleparticle behaviour in terms of the shifted gas seems to be a fair representation in the (n, 2n) channel as well. In the 14 MeV (n, ~) and (n, p) channels, the average residual excitation energies U are high (approximately g 17 MeV and g 14 MeV corresponding to a compound
610
S. CHATTERJEE AND A. CHATTERJEE
nuclear excitation ~ 20 MeV) and the average small relative Q-corrections, to the first order, have negligible effects for comparison purposes and for studies in structure effects. In the present case of (n, 2n) reactions, the highly variable mass effect Q(n, 2n) reduces the residual excitation U within the range of 3 to 8 MeV approximately; it is therefore extremely important to normalize the excitation energies for similar comparison purposes and for detailed studies in the nuclear structure effects in this channel. One should thus attempt to select the data from available excitation function studies, and should not use the 14 MeV (n, 2n) data alone as has been done by Hille 60). The single-particle effects, viz., the shell effects, are apt to get lost in a cross-section comparison at unDormalized low residual excitation energies in accordance with Hille's observations; we have here a variable f-correction (0 to ~ 3 MeV) superimposed on a fluctuating residual excitation ( ~ 2 to ~ 8 MeV). It is interesting that the Rosenzweig model formolation used here seems to work within the domain of low lying structure (up to about half the binding energy of the last nucleon, i.e., > 3 MeV). The parabolic form of the Rosenzweig shift function within the respective particle subshells orients the different residual isotopes and isotones on the envelope of the (-parabolas fixed by (o. It is by chance that some of the isotopic and isotonic slopes 4) are aligned parallel after subshell closures (the Csikai-Peto effects). The shift function within a shell is strongly mass dependent. It also contains a relative mass variation across the shells. It is interesting that the average (-variation in the shift function is negligible. Thus, the residual excitation energies are strongly correlated with the ground state energies of different nuclei as a first approximation. Historically, the ( N - Z ) / A dependence in the 14 MeV (n, 2n) cross sections have been previously studied by Barr et al. 6~) and Perlstein 62) and was briefly reported by Breunlich et al. at the Antwerp conference 64). The subshell effects in the cross sections are rather indistinctly observed here by using 2( = N - Z as a variable parameter. For light nuclei 2( -- 0, and all cross section values lie on the ordinate scale in fig. l; for the heavier (medium weight and heavy) nuclei, there is the eflect of non-identical neutron and proton subshell interferences. These reduce the total observable magnitudes of the single-particle effects. Use of ( N - Z)/A as a parameter has an additional effect of the overlapping individual particle bebaviour introduced through the mass number AR -- N+Z. The net result of these two "averaging" processes is an almost structureless variation of the cross section with ( N - Z ) / A in accordance with the previous observations 6 0 - 6 2 , 6 4 ) . We are thankful to Professor J. Csikai and Dr. I. Angeli of Debrecen, Hungary, to Dr. M. Bormann of Hamburg, Germany and to Dr. P. Hille of Vienna, Austria, for their kind interest, comments and criticisms about this work. We are particularly indebted to Csikai and Hille for pointing out a few mistakes in the data used in the original manuscript.
(n, 2n) REACTIONS
611
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S. CHATTERJEE AND A. CHATTERJEE
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