Isomeric cross-section ratio for (n, 2n) reactions induced by 14.7 MeV neutrons

I

2.F

[ NuclearPhysics A l l 8 t

(1968) 449--~60; (~)

North-HollandPublishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

I S O M E R I C CROSS-SECTION RATIO FOR (n, 2n) REACTIONS INDUCED BY 14.7 MeV NEUTRONS B. MINETTI and A. PASQUARELLI

Istituto dt Fisica Sperimentale, Istituto di Fisica Tecnica e lmpianti Nucleari, laboratorio reattore, Politecnicodi Torino Received 2 May 1968 Abstract: The activation cross sections for (n, 2n) reactions induced by 14.7 MeV neutrons have been

measured in SSRb, S?Rb, sssr, ssSr, 9°Zr, S~Mo, l°Spd, 1°?At and ]°SAg. For all the reactions, it is possible to measure the isomeric cross-section ratio. This ratio has been analysed using the Huizenga-Vandenbosch method. Estimates have been obtained for the parameter a which characterizes the spin dependence of the nuclear level density.

INUCLEARREACTIONSS~,S?Rb,Se, SSSr,S°Zr,92Mo,lOsPd,lO?,1°~Ag(n,2n),E=14.7MeV,[ E

measured a, isomeric yield ratios. Deduced spin distribution parameter. Natural targets.

1. Introduction The isomeric cross-section ratio was related by Huizenga and Vandenbosch 1,2) to the spin cut-off parameter a, which characterizes the spin dependence of the level density. To obtain information about the a-values, the isomeric cross-section ratios were measured for reactions induced by 14.7 MeV neutrons. Only (n, 2n) reactions were investigated, because these reactions take place primarily via the compound nucleus mechanism.

2. Experimental procedure The cross-section measurements were carried out by the activation method as described in refs. 3-5). In our measurements of fl--particles, the conversion electrons from the isomeric transitions were not observed because their energy is so small that they were completely absorbed in the target and in the window of the Geiger-Mtiller counter.

3. Measurements Natural mixtures of isotopes of Rb, Sr, Zr, Mo, Pd 14.7 MeV neutrons. In table 1, the reactions are listed life, the values used for the fl- and y-energies, and the radiation. These values were taken from Nuclear Data 449

and Ag were irradiated with with the corresponding halfprobabilities of the observed Sheets.

B. MINETrIAND A. PASQUARELLI

450

TABLE 1 Half-lives and fl and ?-energies Reaction

Halfqife

/~- max energy (MeV)

7-energy (MeV)

Internal conversion coefficient

86Rb(n, 2n)"=Rb 85Rb(n, 2n)84gRb

20 mill 33 d

0.465 (35 %) if+ (20.6%)

0.1

8~Rb(n, 2n)SemRb STRb(n, 2n)86BRb

1.04 min 19 d

0.56 (100 %) 1.084 (9 %)

0.38

s6Sr(n, 2n)s6mSr 86Sr(n, 2n)s6gSr

70 rain 64 d

0.225 (86 %) 0.514(100 %)

0.024

ssSr(n, 2n)87mSr

2.8 h

0.388(100 %)

0.33

°°Zr(n, 2n)s°mZr DoZr(n, 2n)sggZr

4.3 rain 79 h

0.588 (93 %) 0.915(100 %)

0.09 0.011

°2Mo(n, 2n)gXmMo °lMo(n, 2n)°ZgMo

66 sec 16 rain

0.658 (57 %) fl+ (94 %)

0.055

z°sPd(n, 2n)l°~mPd

22 see

0.22 (100 %)

0.4

X°VAg(n,2n)l°*mAg aS~Ag(n, 2n)t°egAg

8.3 d 24 min

Z°lAg(n, 2n)Z°BgAg

2.4 min

0.513 (86 %) /~- 1.45 ( 7 %) fl- 1.96 (54 %) fl+ 0.69 (0.2 %) p- 1.65 (94 %) /~- 1.02 (1.9 ~o)

The internal conversion coefficients were obtained f r o m Nuclear D a t a Sheets and f r o m the table of Sliv and Band 6). In table 2 are listed the experimental cross section for each reaction and the literature values. F o r the SlBr(n, 2n) s °Br and a 1opd(n ' 2n)1 ogpd reactions, experimental cross sections were obtained f r o m refs. 7,a). 4. Discussion The theoretical cross sections for (n, 2n) reactions were calculated by means o f the Blatt-Weisskopf formula 3 o). The cross-section for c o m p o u n d nucleus formation was taken f r o m ref. 3~). F o r the zero-spin level-density parameter a for nuclear temperature calculation, the following expression 32) was used: a = 0.095(Jn + Jp + 1)A ~, where Jn and Jp are the effective single-particle angular m o m e n t u m q u a n t u m numbers and A the nucleus mass number. The calculated cross sections are listed in table 2.

(n, 2n) REACTIONS

451

TABLE 2 E x p e r i m e n t a l a n d theoretical cross sections Reaction

(am)ex p (mb)

SXBr(n, 2n)*°mBr 81Br(n, 2n)8°SBr

7404- 40

aSRb(n, 2 n ) ~ m R b 8BRb(n, 2 n ) ~ S R b

9264- 61

aVRb(n, 2n)SemRb 87Rb(n, 2n)86gRb

9324-150

86Sr(n, 2n)a6mSr a6Sr(n, 2n)s*gSr

2224- 25'

8sSr(n, 2n)87mSr

3654- 30

' ° Z r ( n , 2n)a'mZr

1914- 15

T o t a l experimental cross section (mb)

T o t a l calculated cross section (rob)

2904- 25

10304- 65

1493

4)

7564-161

16824-222

1495

a T = 15204- 70 b) a T = 6874- 74 e)

16194-250

25514-350

1540

a T = 11944- 60 b) a T = 8354- 36 e)

9104- 80

11324-105

1256 1394

9°Zr(n, 2n)8'gZr

74-

163-1- 12

5174- 80

11°Pd(n, 2 n ) l ° ' m p d n ° P d ( n , 2n)l°gsPd

5104- 30

19TAg(n, 2n)l°emAg

653 4- 30

615

am = as = aT =

3124- 50 e) 2804- 10 e) 7014-1100 )

am = am =

2154- 24 e) 2134- 12 e)

am am as aT aT aT

= = = = = =

168474 470 768450246774-

12 t) g) s) 23 b) 36 o) 51 u)

am as as aT aT aT aT aT aT

= = = = = = = = =

5142054158421141904132431043154188

10 t) 251 ) 50 16 h) 29 k) 211) 87 m) 35 e) ~)

15904-140

1770

o)

1523-t- 70

1763

am as ~rg ag as as as as ag as

1771

crg = 3 1 1 ± 1 5 0 k) crs = 1000-4-100 r) ag = 710q-105 t)

1080 4-110 a m = 6500 8704- 40

1°gAg(n, 2n)l°SgAg

797-t- 50

o) ~) g) h)

1704- 14

L i t e r a t u r e values (mb)

1731

X°TAg(n, 2n)l°6gAg

¢). 9). xo). x0 .

1038

2

x°sPd(n, 2n)l°Tmpd

Ref. Ref. Ref. Ref.

5934- 97 762 4- 82

' S M o ( n , 2n)'XmMo ' l M o ( n , 2n)'XSMo

a) b) e) a)

(as)ex p (mb)

Ref. Ref. Ref. Ref.

x2). 19). 10 . 15).

l) J) k) 1)

Ref. Ref. Ref. Ref.

16). 3¢). 1¢). 18).

m) n) o) P)

Ref. Ref. Ref. Ref.

xs). 80). 8). 20.

q) r) B) t)

Ref. Ref. Ref. Ref.

~). 2a). 14). ~s).

= = = = = = = = = =

v)

600 q) 662 P) 740 q) 519-4-260 k) 5604- 5 6 0 8 8 9 ± 65 h) 5374- 1 5 0 6574-100 t) 7344- 44 u) 360 v)

u) Ref. 26). v) Ref. 27).

452

B. M I N E T T I A N D A . P A S Q U A R E L L I

When it is possible to measure only a m (or ag), crg (or am) was evaluated by subtracting the experimental a m (or ;¢rg)from the theoretical total cross section because of the fairly good agreement between theoretical and experimental total cross sections. This occurs for the reactions 10Spd(n ' 2n): °78Pd and 109Ag(n ' 2n)l 0gmAg where the residual nucleus has such a long half-life that the residual activity cannot be measured and for the reaction SSSr(n, 2n)S78Sr where the residual nucleus is stable. Isomeric cross-section ratios were calculated with the Huizenga- and Vandenbosch method 1,2). For the spin-dependent level density, the following expression was used: p(]) oc p(0)(2J + 1) exp [-- ( ] + ½)/2a2], where p (0) is the level density of zero-spin levels and l the nuclear spin. The spin cut-off parameter a was assumed in our calculations to be energy independent in neutron evaporation and during the y-ray de-excitation. The transmission coefficients were taken from ref. 33); for the nuclear radius, the value r o --- 1.5 fm was used. In the y-ray de-excitation, only dipole emission was considered. The average number of y-rays emitted can be evaluated 38)

where E is the excitation energy. The values obtained for P are listed in table 3. It was assumed that average energy of the neutrons is given by E , = 2T, where the nuclear temperature T is related to the excitation energy U by aT2-4T

= U,

where a is the zero-spin level density parameter given in ref. 32). The excitation energy'U is corrected for the pairing energy. This method gives too large a value for the energy of the neutrons when the threshold of the reaction is very high as in the case of 9°Zr and 92Mo. For these reactions, the average energy of the two emitted neutrons was calculated using the statistical model. Calculations were performed for many values of the spin cut-off parameter and of the number ~ of the emitted y-rays. The results are given in fig. 1. In order to test the experimental results in a convenient way, it is helpful to use the isomeric cross-section ratio, i.e. the ratio of the reaction cross-section a~ leaving the residual nucleus in the lower spin state and the reaction cross-section o-h leaving the residual nucleus in the higher-spin state. Experimental results and spin cut-off parameter values obtained by comparison of experimental values and calculations are listed in table 3. It can be seen that no value of the cr-parameter can reproduce the isomeric crosssection ratio for the reaction 92Mo(n, 2n)glMo. For the reaction 9°Zr(n, 2n)Sgzr, a high value of the spin cut-off parameter is required to reproduce the isomeric ratio.

453

(n, 2n) REACTIONS TABLE 3

Experimental and theoretical cross section Reaction

Jm

SlBr(n, 2n)S°mBr SlBr(n, 2n)8°~Br

5

SSRb(n, 2n)S4mRb 85Rb(n, 2n)S4gRb

6

STRb(n, 2n)8OmRb STRb(n, 2n)S6gRb

6

8°Sr(n, 2n)sSmSr s6Sr(n, 2n)s~gSr

½

88Sr(n, 2nWmSr ssSr(n, 2n)8~Sr

½

9°Zr(n, 2n)s'mZr ~°Zr(n, 2n)s°sZr

½

Jg

a~ all

Literature values

F

a

2

6 +1

756 4-161 0.8 4-0.2

1

4.94-0.5

16194-250 1.704-0.5

1

2.94-0.5

1

3.0±0.5

2

3.44-0.5

2904- 25 0.39±0.05 9 2 6 t 61

2

0.724-0.26 ~) 0.564-0.09 b)

932 4-150 2 ~

2224- 25

9104- 80 0.244-0.05

3654- 30 ~

1029

xosPd(n, 2n)l°Tmpd 4/l°sPd(n, 2n)l°TgPd

~

11°Pd(n, 2n)l°~mpd ~ n°Pd(n, 2n)X°°gPd

~

1°TAg(n, 2n)X°emAg 6 1°TAg(n, 2n)l°~Ag

1

1°gAg(n, 2n)l°SmAg 6 1°gAg(n, 2n)l°SgAg

1

74- 2

762~ 82 0.254-0.05

0.39±0.11 s) 0.164-0.01 a)

0

3.54-0.5

1634- 12 0.044-0.01

0.254-0.08 e)

0

6 4-1

2

2.84-0.5

5174-80

2.34

1214 5104-30

10804-110

2.14-0.3

2.024-0.45 ~)

2

34-0.5

8704- 40 1.304-0.1

9.8 f) 0.814-0.19 g)

2

2.84-0.5

3

44-0.5

6534-30 974 797 4- 50 e) Ref. lo).

1.124-0.18 e)

0.35

1914- 15 ] ]

b) Ref. 29).

ag

7404- 40 1

~ZMo(n, 2n)91mMo ½ S2Mofn, 2n)91gMo

s) Ref. ~s).

am

d) Ref. a0.

0.82 e) Ref. 16).

f) Ref~0.

g) Ref. 22).

T h e low e x p e r i m e n t a l isomeric cross-section ratio for these two reactions is p r o b a b l y due to the finite energy difference between the two isomers. A s a consequence o f the high t h r e s h o l d for these reactions, after n e u t r o n e v a p o r a t i o n , the residual nucleus has a high p r o b a b i l i t y to have a smaller excitation energy t h a n the energy difference between the isomers; in such a case only the g r o u n d state can be p o p u l a t e d i n d e p e n d ently o f the spin values. The result is an increase o f the g r o u n d state cross section. This effect can be e v a l u a t e d when the p r o b a b i l i t y to o b t a i n a residual nucleus with given excitation energy is known. This p r o b a b i l i t y was c a l c u l a t e d with the statistical model. The p r o b a b i l i t y t h a t a n e u t r o n with energy E is e m i t t e d f r o m the c o m p o u n d nucleus o f excitation energy U a n d spin Jc so t h a t the residual nucleus is left with excitation energy Ef a n d spin Jf

B. MINETTI AND A. PASQUARELLI

454 7.5

Oi/O'h

1511

II

5.

V=5 V=4

V,4

V=3

V,3

V~

81 Br (n.2n) 80 Br

V=2

85 Rb(n.2n)84R b

---

i~/~h(~Xp)

J

O~

1.5

~

4

1.5 V=5

T Gj/0"h

~.+

V,4 V.

'~.4 V=3

I

~J,2

~=2 ~I =I

8Sr (n.2n)87Sr

90Zr (n.2n)89Zr

V, )5

Q5

0

3

0

4

15

1

~/0"h

3

4

5

0..6

Z5

'9=5 V,4

10-

_V=2

2

110Pd (n.2n)lO9pd

V,I

......

o~

3

0-[/0~h ( ~ p )

4

5

0"~

o~

107Ag(n.2n)106Ag

455

(n, 2n) REACTIONS

0.75 t

i/~h

V___~,_4

0.5

87Rb In, 2 n)86Rb

86Sr (n,2n)85Sr

V-1

~),1 g.25

5~ _ _

_C~l/O.,h(exp) 3

1.5 ll

~.4 %1,3

0"4

d~/~h '9,3

1 ~:

5

,.5/tl

'9"5

Ci/O-h

4

92Mo(n.2n)91Mo

05

'9,2 '9 =I

108p d (n.2n)107

0.5

~--~/O~h (~p) 0

~/O'h(ex p) . . . . . . . . . . . . . . . . . . . 3

01

~l

0-~

3

~,

5

0"4

15

~/O-h I0

V=5

~0=4 x),3

~,2

109Ag (n,2n)108Ag

0"~

Fig. 1. C o m p a r i s o n o f the m e a s u r e d isomeric crosssection ratios with the theoretical curves plotted in t e r m s o f t h e spin cut-off p a r a m e t e r ¢~ a n d the multiplicity ~.

456

B. MINETTI AND A. PASQUARELLI

is proportional to a4) (2Je+ 1)f2(Ef, Je) (2s + l)l~Etri.,(Jf , Jc, E) - - - --,

(2So + 1)a(ts, so)

where # is the reduced mass of the neutron, s the neutron spin, ainv the cross section for the inverse reaction and fI(U, Jc) and fI(Ef, Jr) the level densities of the compound and residual nucleus, respectively. If P (Jo) is the spin distribution of the compound nucleus, the probability of emission of a neutron having energy E is proportional to ~, P(S¢)Eal,v(J ~ , Je, E) (2St + 1)fI(Ef, Se).

Jo,~

(2so + 1)u(v, Sc)

If it is assumed that the inverse cross section is spin independent and that the level density may be expressed as the product of functions of spin and energy, respectively, a ( e , J) = pj(J)p(E), the emission probability of a neutron with energy E is proportional to eainv(e)P(e,), where p(E~) is the energy-dependent part of level density of the residual nucleus. In the (n, 2n) reaction induced by neutrons of energy Eo, the probability of emission of the first neutron with energy E' is /(Co, e') ~ e'a,°v(E')p(eo-E'). The probability of emission of the second neutron having energy E " is f ( E o + Q - E', E") oc E"ai.v(E")p(Eo + Q - e ' - E' 3,

where (E o + Q - E') is the maximum kinetic energy carried out by the second neutron. The total energy carried by the two neutrons is e = E ' + E " , and the excitation energy of the residual nucleus is E¢ = E o + Q - ( E ' + E "

) = Eo+Q-e.

The probability of having a residual nucleus of excitation energy Ec equals the probability of having two neutrons of total energy e and is given by P(E¢) = • f ( e o , E')f(Eo + Q - E', E"). E',F,"

The sum is extended over all values of E ' and E " such that E ' + E " = e. Assuming that the sum may be replaced by an integral, one obtains

(n, 2n) REACTIONS

457

P(Ec) = f~f(Eo, E')f(E o +Q-E', e-E')OE'

~

oc E' ( ~-E' ) tri,v( E' ) trinv(e-E' )(Eo-E')( p p Eo+Q-e ) dE' O0

= p(Eo +Q-~)~(~),

In order to evaluate q~(e), the following expressions 35) were used: O-inv(E) oc (1 q_ f l ) ,

f l = 2"12A-*-0"05 0.76 + 2.2A- ~ '

and for the level density of the first residual nucleus

where T is the nuclear temperature. This approximation yields for the integral

)lE x-ex (If E m is the energy of the isomeric state, the probability to find the residual nucleus with energy less than E m is

f 2mP(E¢)dE~ P = ~Eo+Q

L

P(Ec)dE~

Taking into account the effect of the finiteenergy interval between the isomeric and ground state, the probability that the residual nucleus is left in the isomeric state is

(1 --P)Pm, while the probability that the residual nucleus is left in the ground state is

p+(1-p)Pg, where Pm and Pg are the probabilities given by the Huizenga-Vandenbosch method. The isomeric cross-section ratio is given by (7m trs

(1 --P)Pm (1 --p)Ps-l-p

(1--p)Pm 1--(1 --P)Pra

If E~ << E o + Q, the probability p goes to zero, and trm/trg is given correctly by the Huizenga-Vandenbosch method.

458

B. MINETTI AND A. PASQUARELLI

i =

61/ O~h V=5

M 4\ 0.4

Vi3 I I

]

Oo

90Zr (n,2n) 8 9 Z r

! 1 I

iiil 3

4

5

ff_~=.

6

Fig. 2. Comparison of the measured isomeric cross-section ratios with the theoretical curves for the special casesof ~=Mo and g°Zr (soc text).

(n, 2n) REACTIONS

459

These corrections have been applied to all the reactions for which E 0 + Q is less t h a n 3 MeV; these reactions are 9°Zr(n, 2n)SgZr,

92Mo(n, 2n)91Mo.

The Q-values were t a k e n from ref. 36). The probability depends strongly o n the expression used for the level density of the residual nucleus at low excitation energy; we used for this level density

p(E) = exp (2x/a--E),

E = at2-t.

(E+t) n Calculations were performed for n = 2 a n d n = ¼; the results are given in fig. 2. It can be seen that a better fit is o b t a i n e d with n = 2. The a-values listed in table 3 are derived after corrections for the finite energy interval between the isomeric a n d g r o u n d states. The authors wish to t h a n k Professor G. Lovera for helpful discussion d u r i n g this work, Professors C. Codegone a n d C. A r n e o d o for their k i n d hospitality a n d interest in this work a n d Ing. G. Gaggero for p r o g r a m m i n g a n d calculating the isomeric crosssection ratios. References

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)

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AND

A. PASQUARELLI

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