Two-photon decay and the lifetime of giant resonance states

,I2.I : 3 . A ]

Nuclear Physics 28 (1961) 5 2 4 - - 5 2 8 ; (~) North-Holland Publishing Co., Amsterdam

I

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T W O - P H O T O N D E C A Y A N D T H E L I F E T I M E OF G I A N T R E S O N ANCE STATES t3. M A R G O L I S t

Physics Department, The Ohio State University, Columbus, Ohio t¢ Received 6 J u n e 1961 A b s t r a c t : T h e relative p r o b a b i l i t y for t w o - p h o t o n d e c a y in a low e n e r g y 0+-0 + t r a n s i t i o n is s h o w n to be related to t h e lifetime of t h e l - s t a t e s of t h e s e nuclei in t h e g i a n t r e s o n a n c e region. E s t i m a t e s of t h i s lifetime are m a d e u s i n g n u c l e a r r e a c t i o n t h e o r y . C o m p a r i s o n of t h e r e s u l t i n g t w o - p h o t o n d e c a y p r o b a b i l i t y is m a d e w i t h e x p e r i m e n t .

Recently 1, 2) Ericson has pointed out the connection between the lifetime of excited states of a nucleus and fluctuations in reaction cross sections in the continuum region of the compound nucleus. If one knows the lifetimes, one can infer the fluctuations and conversely. The object of this note is to point out an independent method of getting information about the lifetimes of states in the continuum region. Moreover, the method to be described relates to the lifetimes of states of a particular spin and parity, namely, the i - states in the region of the giant resonance observed in photonuclear processes. We consider a low energy 0+-0 + transition to the ground state of a nucleus (e.g. Zrg°). A small fraction of these transitions will proceed by two-photon decay. This is a second order process and will proceed predominantly b y electric dipole transitions through the 1- intermediate nuclear states. Since electric dipole matrix elements are very weak to low lying levels and very strong to states in the giant dipole region ( ~ 20 MeV excitation), we find that the process proceeds predominantly through the giant dipole states despite the larger energy denominators involved. The process of two-photon decay has been considered previously by several authors 3-G). We can write this probability as rlrd~o df2d.Q'

-

ded3?'dJ2

( e2~ 2

c

× p,xp' xn L

M2c 4

+

h (~Ol--e)n-- o~') J

* N o w a t McGiI1 U n i v e r s i t y , ~IontreaI, C a n a d a ** \Vork s u p p o r t e d in p a r t b y t h e U.S. A t o m i c E n e r g y C o m m i s s i o n . 524

TXVO-PHOTON

DECAY

525

I n the above the p h o t o n s h a v e energies, m o m e n t a and polarizations /ho, k, %(k) and ?io/, k', %,(k'). The m a t r i x elements are e(r)t~ ~ ( f [ ~ e~r~[n), 8

where e is the p r o t o n charge and e, and r~ are the effective charge and position v e c t o r of the stn nucleon. In eq. (1), e = tio~/Mc 2, where M is the nucleon mass. The electric dipole m a t r i x elements are v e r y w e a k for low e n e r g y transitions a n d v e r y strong for transitions between the g r o u n d state and states in the giant resonance region. One also expects large m a t r i x elements for transitions f r o m the giant resonance region to low lying excited states when the excitation of these states is considerably less t h a n the width of the giant resonance. This is c e r t a i n l y the case for the first excited 0 + states of Zr 9° a n d Ge 72. F o r the o t h e r two cases where the first excited state has spin and p a r i t y 0% Ca 4° a n d O ~6, the excitations are of the order of the giant resonance width or larger. Since we get the strong c o n t r i b u t i o n s to the sum in (1) from the terms corresponding to co~--coI >> a~ we drop o) a n d o)' in these sums. S u m m i n g over polarizations of the p h o t o n and integrating over p h o t o n energies and angles, we get the transition p r o b a b i l i t y for t w o - p h o t o n d e c a y ~

105x

(~c) a \?ic] (Ei--E')'

~(o),,--coi)

W e are t h e n led to consider sums of the form

W e now c o m p a r e this sum with the ( - - 2 ) - m o m e n t of the p h o t o n u c l e a r absorpt i o n cross section 7). This q u a n t i t y is given b y a_ 2 ~

~

d W ~ 4 ~ 2 hc

~(wn--o~,) '

(4)

w h e r e a is t h e t o t a l p h o t o n absorption cross section for p h o t o n e n e r g y W. W e now write

(5)

Isl = 122 We have then

w.

=

( E , - E , 1 7 u c4(

c

I t is to be n o t e d t h a t a n y c o n v e n i e n t mass can be used in formula (6).

(6)

EI--Et (MeV)

6.07

3.35

0.69

1.73

Nucleus

016

Ca4O

GeV2

mr 9o

B

12.0

~10

15.8

15.6

(l~eV)

15.8

~20

19.3

21.7

~7 4.3

4.2

3.4

(MeV)

and experimental

(~eV)

U

Theoretical

7.3 × 10 ~

5.7 × 10*

5.1 × lO s

6.7 × 109

(sec-1)

W~,/l ~

two photon

(exp)

1.1 × 10 7

3.4 × 10 ~

2.9 × 10 8

1.4 × 101°

(sec -1)

W

1.5 × 10 19

4 × 1 0 -~°

1 × 10 -*°

2 × 10 -21

rc (sec)

10 8

7 × 10 ~

1.3 × 10 2

< 7 . 9 × 10 a

<6.4×

(sec-1)

Wry (theor)

decay probabilities and related quantities

TABLE 1

.2

6 × 10 -8

4 × 10 .5

< 3 × 10 . 2

<5×10

W(exp)

W~ (theor)

-a

1.1 × 10 -3

<6×10

W(exp)

Wvr (exp)

T W O - P H O T O N D E CA Y

5~7

Table 1 gives two-photon decay probabilities divided by/s, calculated byeq. (6). The values for ~-2 have been taken from the paper of Levinger 7); ~-s ~ 7A# #b/MeV for O 16 and ~-2 ~ 3.5 A# /~b/MeV for the other nuclei. The total measured decay probabilities W(exp) are also tabulated s). The predominant mode of decay is by monopole transitions. In order to have a theoretical number for W~7 we must know [. For cases when Ei--E ~is considerably less than the giant resonance width, we expect that for the states E n within the giant resonance I(z),~l~l(z)~ll. These matrix elements are expected to change sign randomly at energy intervals ~/zc, where ~ is the lifetime of a state in this region. We have then I s ~ ~I (re F), where re is the lifetime of a state in the giant resonance region and 7~ is the width of the giant resonance. We see then that for a nucleus like Zr 90 a measurement of W ~ will yield an estimate of re. Conversely, we m a y use the estimate of Ericson 1) for zc based on neutron reaction theory, to estimate the probability of two-photon decay:

pc(U) r° ~ p n ( U - - B )

2~s 2M T" -- T~ c

(7)

In the above ~o is the average cross section for formation of a compound nucleus with neutrons of energy a few MeV; B is the binding energy of the last neutron; U is the giant resonance excitation energy; pc(E) and pn(E) are the level densities for the compound and residual nuclei at excitation E. We take pn(E)----pe(E+b) to compensate for the odd-even effect in nuclear densities, with ~ ---- 34/A~ MeV (the nuclear pairing energy). We use the form pc(E) ~ Cexp(~/aE) with a = 7, 10, 4, 1 MeV-1 for Zr 9°, Ge 7s, Ga 4° and 016, respectively 9). The nuclear temperature of the residual nucleus is given by T -~ [(U--B--~)

/a]~.

The values of ro obtained from (7) are given in table 1 along with the value of W ~ calculated using these values of ze- Values of WT~/W(exp) are also tabulated. For Ca 4° and especially for 016 it is expected t h a t the calculations of W ~ based on the above considerations will be overestimates since the spacing between the two 0+ states is not considerably less than the giant resonance width. We indicate this in table 1 by affecting our estimates with < signs. The upper limit or value of W~,~(exp)/W(exp) is also listed in table 1 in the cases in which experiments have been performed s, 10). Reide, Thieberger and Alvager 10) have recently reported detecting two-photon decay in Zr 9°, giving a value of W,~/W ~ 1.1 × 10-~. This leads to zc ~ 9 × 10 -19 sec, in good agreement with our theoretical value which is expected to be of the right order of magnitude.

528

B. MARGOLIS

The author wishes to thank Dr. A. Schwarzschild for discussions concerning the experimental data. He also wishes to thank Dr. M. K. Ramaswamy for drawing his attention to this problem. References 1) 2) 3) 4) 5) 6) 7) 8)

T. Ericson, Advances in Physics 9 (1960) 425 T. Ericson, Phys. Rev. Left. 5 (1960) 439 M. Goeppert-Mayer, Ann. Physik 9 (1931) 273 R. G. Sachs, Phys. l~ev. 57 (1940) 194 J. Eichler and G. Jacob, Zeits. f. Phys. 157 (1959) 286 D. P. Grechukhin, J E T P 5 (1957) 846 J. S. Levinger, Phys. Rev. 107 (1957) 554 M. Nessin, Report CU(PNPL)--201 (available from the Office of Technical Services, Department of Commerce, W*ashington 25, D.C.) 9) K. L. Le Couteur and D. W. Lang, I'~'uclear Physics 13 (1959) 32 10) Reide, Thieberger and Alvager, Phys. Rev. Lett. 6 (1961) 475