Doppler-broadened γ-ray lineshape analysis in multiple Coulomb excitation

NUCLEAR INSTRUMENTS AND METHODS IZ 3 0 9 7 5 ) 529-539;

DOPPLER-BROADENED

© NORTH-HOLLAND PUBLISHING CO.

~,-RAY L I N E S H A P E A N A L Y S I S I N M U L T I P L E

COULOMB EXCITATION T. 1NAMURA*, F. KEARNS and J. C. LISLE Department of Physics, Schuster Laboratory, The University, Manchester M13 9PL, England

Received 14 October 1974 A procedure is described for extracting nuclear lifetimes from the Doppler-broadened v-ray lineshapes in multiple Coulomb excitation. It is shown that in the region of deformed nuclei the statistical tensors calculated using rotational matrix elements according to the Winther-de Boer multiple Coulomb excitation computer programme give a lineshape practically independent of the values of the matrix elements used because the inelastic particle angular distribution remains virtually unchanged for variations of about 20% in the values of the matrix elements concerned. It is found that for high recoil velocities (up to 6.5%

of the velocity of light) better fits to the experimental lineshapes are obtained using Northcliffe-Schilling electronic stopping power data coupled with Lindhart, Scharff and Schiott (LSS) theoretical nuclear stopping power, compared to those produced using LSS theoretical stopping powers only. The procedure we have used also takes into account the effects of multiple nuclear scattering, photon aberration, and the change in detection efficiency of Ge(Li) with the Doppler shift. Extracted lifetimes of several Coulomb excited states are compared with those independently reported and are found to be in good agreement with them.

1. Introduction

reduced E2 matrix elements between all pairs of nuclear states involved are available. It may be noted that the Winther-de Boer semi-classical theory with rotational model matrix elements should give g o o d relative values for the physical quantities associated with HI multiple C o u l o m b excitation of deformed nuclei. This makes it possible to apply the D S A M lineshape analysis to singles 7-ray spectra observed in H I multiple C o u l o m b excitation because this requires only a knowledge of the relative populations of excited states and the angular distributions of the inelastic scattering particles and of the de-excitation 7-rays. The experimental simplicity and high counting rates make singles y-ray measurements in multiple C o u l o m b excitation very attractive as a method of extracting nuclear lifetimes of high spin rotational states.

The Doppler-shift attenuation method ( D S A M ) has been used by m a n y investigators 1-6) for determining nuclear lifetimes in the region 1 0 - 1 4 - i 0 -xl s. This method was initially used in studies of light nuclei, primarily by measuring centroid shifts of y-ray lines. With the advent o f heavy mass projectiles it is now possible to employ D S A M to determine the lifetimes of excited states in heavy mass nuclei by the analysis of Doppler-broadened y-ray lineshapes. The usefulness of this technique in the study of C o u l o m b excited states in heavy mass nuclei was first demonstrated by the Yale groupS). In C o u l o m b excitation, unlike the (HI, xn) reaction, there is no reaction feeding time to mask nuclear lifetimes [the feeding time inherent in the (HI,xn) reaction can be as long as 10ps7)], and therefore C o u l o m b excitation is particularly well adapted to the measurement of short lifetimes. Using this technique we have determined lifetimes (1-10 ps) of high spin rotational states in a large n u m b e r of even deformed nuclei, and we reported those results in our previous publicationS). We now present a study of the method of ),-ray lineshape analysis in multiple C o u l o m b excitation. Using the Winther-de Boer computer p r o g r a m m e for multiple C o u l o m b excitation9), one can calculate excitation cross-sections, inelastic scattering-particle and de-excitation y-ray angular distributions provided * Present address: Cyclotron Laboratory, The Institute of Physical and Chemical Research, Saitama, Japan. 529

Very recently a similar approach to the Dopplerbroadened y-ray lineshape in C o u l o m b excitation has been reported~°), but use was made of the tabulation, based on first order theory, given by Alder et al. ~ ) to evaluate differential excitation cross-sections and y-ray angular distributions. This limits its application to single (and possibly double) C o u l o m b excitation. We describe a computer p r o g r a m m e for the analysis of Doppler-broadened y-ray lineshapes observed in singles measurements in multiple C o u l o m b excitation, which is used in conjunction with a modified Winther-de Boer C o u l o m b excitation programme, and we examine in detail its application. We also discuss the slowingdown characteristics o f recoil-ions in target materials, the differential detection efficiency of Ge(Li) and photon aberration.

530

T. I N A M U R A

in the form

2. Formalism for the D S A M in multiple Coulomb excitation

There have been several reports on the computation of Doppler-broadened ?-ray lineshapes 3-6'12,~a). The present programme is based on a formalism similar to that of Sieet al. 6) and Hoffman et a1.13), except for a part which is associated with multiple Coulomb excitation. We shall not repeat the details of the standard kinematics involved which can be found, for example in refs. 12 and 13. In Coulomb excitation the observed number of deexcitation 7-rays, with a Doppler shift A E.,, from the state N to the state M between time t and t+At for projectiles with an energy between Ep and E~+AEp can be written (in the coordinate system fixed on the moving nuclei) as

AN(AEy) oc e(O~, ~p,;) x

d 2

a(N ~ M) dQp dQ:.

AEp dEp/dx

A.QpA(2,~At,

x (1)

AE, = e , ( a t ) . , , , ) / ~ ,

v(t)

(dE/dx)-l dr,

dQ~ (3)

d~iv~M(Eo, Op, 0~, (p~., t) dO,. = (4n)-"

Ak~(E o, Op, t) x

~ k=0,2,4

k~t<
× Fk(IM, Is) Yk~(O~,, ~Pr),

(4)

in the coordinate system with the z-axis in the beam direction. The coefficients Ak~(Ep, Op, t) are given by

+~k~(N+l) Gk(Is,ls+l)[ ~ 1

exp( -- t/zn) +

[-TN - - TN + 1

,

] +cq~(N+2)

+--exp(-t/zs+i) TN+I--T

×

N

x Gk(IN,IN+2)

--1

exp(--t/zs)+--x

N - - "L'N+ 2

~ N + 2 --2"N

exp ( - t/rN) +

x rN--~N+ ,) ( ~ N - rN+~)

+

ZN+ I

"on+2

In semiclassical Coulomb excitation theory the particle-7 double differential cross-section can be written

e x p ( - t / r u + 2 ) ] / + .... (S)

(r,, + 2 - zN)(rN + 2 - r N + ,)

_]/

where the quantities ek~(N) are the statistical tensors of the state N with spin IN for a given E v and 0p, and rN is the lifetime of the state N. The coefficients Fk(IM,IN) and Gk(IM,I~) are as defined by Winther-de Boer9). From a practical viewpoint we take only dipole and quadrupole ),-transitions into consideration, i.e. ]I M I~1 <2. Hence we have -

(dE/dx) -I dr.

exp( -- I/ZN+ 1) +

( r s + ~ - rN)(T~+ 1 -- ~N+ ~)

+

where v(0) is the initial recoil velocity induced by the projectile with the energy Ep, M~ is the atomic mass of the recoil ions, and dE/dx is the stopping power for them in the target material. The fraction of unshifted (AE~ = 0) ),-rays is given by the integration of eq. (l) over the time t from t~ to infinity; t~ is the time at which the recoil ions with the initial velocity v(0) come to rest,

i v(O) °

dQp

where according to Winther-de Boer 9) the angular distribution WN~ M for the de-excitation ),-rays is given by

(2)

Jr(O)

tr = M x

_ dO'R,the~fo~ddWN~M(Ep, Op, O,l, (p~., t)

dQp dQ?.

× exp(--t/zN+e)] + Gk(IN, IN+l) Gk(IN+~, IN+2) ×

where c is the velocity of light and E;, is the unshifted ),-ray energy. If the target is thick enough to stop the recoil ions completely, the time t is related to the velocity of the recoils by

i

d2 a(N~M)

Ak~(E p, 0p, t) = ~k~(N) zN 1 exp(--t/rN) +

where d2e(N--*M)/d.Qpd[2~ is the "time-dependent" particle-7 double differential cross-section given below, dEv/dx is the stopping power of the target material for the projectiles, and e(0~.,~pT) is the differential detection efficiency of the y-ray detector. The energy shift AE~ of ),-rays emitted in the direction n 7 from recoiling nuclei with velocity v(t) is given, to first order in v/c, by

t = M1

e t al.

-

Fk(l,w, IN)

=

62~M(L) Fk(L,L, [M, IN)+ -2 + ON~M(L+ I) Fk(L+ 1, L + I, I M, IN) +

+ 6N~M(L ) (SN~M(L+ 1) x

531

])-RAY L I N E S H A P E ANALYSIS

× [Fk(L, L + I , IM, IN) + + F k ( L + I , L, IM, IN)I,

(6)

fk (L, L', IM, IN) being the usual y-7 correlation function

I SHAPE~

(see, for example, ref. 14), and

Gk(IM, IN) = (--1) I M + I u + L + k

DISC

X

DISC MT

x [(2IM+ 1) ( 2 I u + 1)] ~ x

× {a~M(L) (l + ~ ( L ) ) x W(IMIMINI N ; kL) -- (~2~M(L+

~TATISTI!AL TENSORS

x

--

1) [1 + ~ N + M ( L +

x W(IMIMININ ; k L + 1)},

~- RAY SPECTRUM

Fig. 1. Organization of the present computer programme for the analysis of the Doppler-broadened 7-ray lineshapes in multiple Coulomb excitation.

1)] X (7)

where L = [IN--&I for lug=IN, L = 1 for IM=IN, and ,SN._,~(L ) = 0 if L > 2 . The quantity c52~M is the L-pole (N-+M) transition fraction of the total y-decay from the state N, and ~N-,M(L) is the total internal conversion coefficient for the L-pole (N--+M) transition. We have chosen the phase definition of the transition matrix elements given by Winther-de Boer 9) in such a way that the sign of the cross term a(L) a(L+ 1) for (M1 + E 2 ) transitions is positive. Reduced E2 matrix elements between all pairs of nuclear states are needed as input parameters for the Winther-de Boer multiple Coulomb excitation computer programme9). Since we are concerned with Coulomb excited states in deformed nuclei, we may make use of rotational model E2 matrix elements. It will be shown in section 5 that the calculated lineshape is only weakly dependent on the matrix elements used.

3. Computation We have adopted the following procedure for computation. Firstly, the statistical tensors of Coulomb excited states are calculated for a sufficient number of projectile energies Ep and scattering angles 0p, using Winther-de Boer multiple Coulomb excitation programme (COULEX), and a permanent file of the statistical tensors is created. Secondly, using this file, the time-dependent angular distributions given by eq. (4) are evaluated, and shifted and unshifted components are obtained by integrating eq. (1) with respect l:o Ep, 0p, q~p and y-ray trajectories (SHAPES); the integrations over the y-ray trajectories includes the geometry, construction and relative differential detection efficiency of the Ge(Li) detector. In this procedure the time t is calculated for a given velocity v [0 < t,_< v(0)] by using eq. (2), and At in eq. (i) is replaced by At, for convenience. The transformation from the coordinate

system fixed on the recoil ions to the lab system is carried out as described in subsection 5.3. A schematic diagram of the organization of our computer programme is given in fig. 1. The programme can accommodate ten states, including the ground state, so that in the case of even deformed nuclei excited states up to J~ - 18 + can be taken into account in the analysis. The least-squares method is employed to fit the experimental lineshapes. The basic fitting parameters are five: the lifetime of the state in question, the peak position of the unshifted y-rays, the fwhm o f a Gaussian fit to the unshifted photopeak, the skewness of its lower tail, and the energy per channel in the fitting region. However, it should be noted that all the parameters except the lifetime can be accurately estimated from calibration data. The fit is first carried out for the highest excited state observed to obtain the correct feeding time. Other possible higher excited states which decay into this state should be taken into account and the lifetimes of these unknown states which may be estimated approximately according to the rotational model become additional fitting parameters. The number of degrees of freedom in this fitting is therefore changed according to the number of the unknown higher excited states taken into consideration. In practice the number of the effective higher excited states can be limited to two or three because the population of the higher excited states is in general negligibly small compared with that of the state of interest. In this analysis the y-ray detector can be at any angle with respect to the beam direction.

4. Experimental procedures A large number of rare earth isotopes have been bombarded with the Kr beam of 4.14 MeV per nucleon from the Manchester HILAC. The metallic targets

532

T. I N A M U R A

used were isotopically enriched and thick enough to stop the recoil ions completely (~>50mg/cm2). A natural Ta foil (85 mg/cm 2) was also bombarded with the Kr beam. Singles de-excitation 7-rays from Coulomb excited states were detected simultaneously at 0 ° and 90 ° with respect to the beam direction using Ge(Li) detectors. The detector at 0 ° was an ORTEC 70 cm 3 true coaxial Ge(Li) and its energy resolution was 3.0keV at 1274.5 keV photopeak (2ZNa y-rays). The 0 ° spectra were used to extract lifetimes from the Doppler-shifted 7-ray lineshapes, while the energies of 7-rays were deduced from the 90 ° spectra. 5. Results and discussion 5.1. INELASTICPARTICLEANGULARDISTRIBUTION Inelastic scattering-particle (projectile) angular distributions can be written in the form of the product of the Rutherford scattering cross-section and the 0-th rank statistical tensor ~oo in eq. (5). In singles measurements the inelastic particle angular distribution

1.0 / a

,

\\

/

x

05

},,' . . . .

T

/ /

O.OI

I

2

~6°Dy* 320 MeV 4Kr

I

I iOO °

J

I

I

/ ~,

I

I

I

150 °

SCATTERING ANGLES OF Kr-IONS 0(C.M.)

Fig. 2. I n e l a s t i c p a r t i c l e a n g u l a r d i s t r i b u t i o n s for the 12 + s t a t e in the a 6 ° D y + 3 2 8 M e V S4Kr r e a c t i o n c a l c u l a t e d w i t h (a) r o t a t i o n a l m a t r i x e l e m e n t s u p to 18 + s t a t e a n d (b) r o t a t i o n a l v a l u e s u p to 18 + s t a t e e x c e p t for t h o s e o f 8 +, 10 + a n d 12 + s t a t e s w h i c h are r e d u c e d by 1 0 % . E a c h c u r v e w a s o b t a i n e d b y a n o r m a l i z a t i o n to the o p t i m u m differential c r o s s - s e c t i o n . W h e n the m a t r i x e l e m e n t s a s s o c i a t e d w i t h these t h r e e s t a t e s are inc r e a s e d by 1 0 % , the r e s u l t is m u c h the s a m e b u t in the o p p o s i t e direction.

et al.

becomes important because it determines the probability of Doppler shift A E 7 at t = 0 as a function of Ep, 0p and q~v. We shall show that for deformed nuclei the inelastic particle angular distribution is not sensitive to the E2 matrix elements used in the multiple Coulomb excitation computation. In the case of Kr ions with an initial energy 348 MeV, we have made calculations for 9 energies in steps of 20 MeV (for low-lying states in 18~Ta, in steps of 35 MeV) and for 18 scattering angles from 10° to 180° (c.m.) in steps of 10° for each energy. Fig. 2 shows the difference in the inelastic Kr angular distributions (c.m.) for the 12 + state in the 16°Dy+328 MeV84Kr reaction calculated with two different sets of input matrix elements: (a) rotational matrix elements up to 18 ÷ state and (b) rotational values up to 18 + state except for those of 8 +, 10 + and 12 + states which are changed by 10%, i.e. their lifetimes differ from the rotational model prediction by about 20%. The choice of the incident energy parameter of 328 MeV was made because it was the most effective energy for the excitation of these three states. It should also be mentioned that the difference shown in fig. 2 is larger than that found for a 20% change in the matrix elements associated with the 12 + state alone. Thus the comparison shown in fig. 2 is an extreme example to demonstrate the effect of a probable deviation in the rotational model matrix elements from the actual nuclear matrix elements. In terms of absolute values, the differential crosssections for the 12 + state are different by about 30% between (a) and (b). Nevertheless, the two angular distributions are quite similar and the difference between the average scattering angles is only 2.7 ° in c.m. Taking into account the finite solid angle of a Ge(gi) detector and its inert core, it can be seen that even in this extreme case the average initial Doppler shift obtained using the modified matrix elements differs by less than 2.2% from that obtained using the rotational matrix elements. In practice this small difference can hardly be noticed in the fitting procedure. Therefore, it is concluded that in the deformed nuclear region the rotational model E2 matrix elements are good enough to provide the inelastic Kr angular distribution which is a principal factor in the determination of the lineshape of Doppler-shifted 7-rays in our singles measurements. It should be borne in mind that it is the relative excitation cross-sections for the states concerned that affect the feeding from higher states. This fact together with the above conclusion makes the calculated lineshape insensitive to the absolute values of the matrix elements used.

7-RAY LINESHAPE ANALYSIS 5.2. STOPPING POWER The stopping power dE/dx is composed of an electronic part (dE/dx)~ and a nuclear part (dE/dx) n, that is, =

+

.

533

and

(dE/dx)~ (NS) + (dE/dx). (LSS). In each case the multiple nuclear scattering has been la)'

(8) iO3

e

~ 0 , 9 2 6 ps

According to the theory of Lindhart, Scharff and Schiott (LSS)~5), the two terms can be written in the form 0.965 p

d(~x) = 4 n a N Z I Z 2 e 2 M1 (dpp) , e,MI+M 2 e,n

(9)

zto 2

where N is the number of stopping atoms per cm 3, Z~ and Z2, M~ and M 2 refer to the moving ions and stopping material atomic numbers and the atomic mass, respectively, and a is the screening parameter given by a = 0.8853 ao (Z~ + Z2~)- ½.

e = aE

M~ M 2 (M I + M2) 2'

M2 Zl Z2 e2 (M1 + M2)"

'& ::.;,":L2,v'::;7

0

Io

The parameter a o is the first Bohr radius. The parameters p and e are dimensionless measures of distance x and energy E, defined by

p = 4na2xN

~

I

o

I

IO

= n

~ 0.67+2.07e+0.03e 2"

d(~_~ = ke~, \op/ ¢

8*-"-~6" from

[

4O

50

,b, I

3 4 2 3 KeV ~-rays

tC'=Dy* 348 MeV e4Kr

ps

tO''

(11)

(12)

,__

Fig. 3. (a) Examples of the best least-squares fits to the line-shapes of Doppler-shifted 7-rays in Coulomb excitation 104Dy+348 MeV 84Kr: (a) 544.7 KeV (14+--+12+) ),-rays.

(10)

This expression has been used throughout. For recoil velocities v<(cZ~)/137 the LSS theory leads to the electronic stopping power

1

2O 30 CHANNEL NUMBER

It is known 5) that the LSS nuclear stopping power is given approximately by (dp)

,

1

ol tZ O L)

ps

iO 3

where k = 0.0793 Z~ (Z, Z2) ½ (A, + A2) ~

(z +zb•

AIA2

One can also make use of the extensive tables of electronic stopping power data presented by Northcliffe and Schilling (NS)16). Consequently computations have been made with two different sets of the stopping powers,

(dE/dx)e (LSS) + (dE/dx). (LSS),

O

IO

20 30 CHANNEL NUMBER

40

Fig. 3. (b) 342.5 keV (8+--+6+) 7-rays. Solid curves were obtained by using (dE/dx)e(NS), and dotted curves by using (dE]dx)e(LSS) ; and for the nuclear part the same LSS-Blaugrund theoretical presentation was used.

534

T. INAMURA et al. t a k e n into a c c o u n t a c c o r d i n g to B l a u g r u n d ' s formalisml7). As s h o w n in figs. 3 a n d 4, it is f o u n d that (dE/dx)e (NS) p r o d u c e s better fits to the lineshapes o b s e r v e d with the 348 MeV K r b e a m t h a n (dE/dx)e (LSS); with the former, in terms o f Z 2, the g o o d n e s s o f fit is generally i m p r o v e d by a factor o f a b o u t 2 at the m i n i m u m . T h e r e a s o n why the latter does n o t give such a satisfactory fit is p r o b a b l y because eq. (12) is likely to u n d e r e s t i m a t e the electronic s t o p p i n g p o w e r for larger recoil velocities, while o v e r e s t i m a t i n g it for small recoil velocities (l~/c < 3%). E m p i r i c a l l y it is f o u n d that better lineshape fits m a y be o b t a i n e d by i n t r o d u c i n g into eq. (12) a linear term

3.0

u

~2.O

.= .c

u

\

~o

//

l.c

(al-ith( )NS 0:7

0.8

O:O~O--

Nuclear Lifetime

Iii

1.2

'c ps

Values of e = 0 . 7 a n d / ? = 0 . 0 5 have been f o u n d satisfactory in the p r e s e n t case where v/c reaches 6.5%. However, the electronic s t o p p i n g (dE/dx)e(NS) still seems better t h a n the corrected LSS electronic s t o p p i n g

Fig. 4. Reduced Z 2 curves as a function of lifetime parameter T in the fit to the 14+----~12+ de-excitation 7-ray lineshape. The curve (a) corresponds to the fit with (dE/dx)e(NS) and the curve (b) to the fit with (dE/dx)e(LSS). ; Slowing-down time

ps ,

4 . 0 3.0 I

2.s

2.0

,.s

,.o

l

~

I

I

l

'

'

'

'

i

I

'

lla if

i0?~ IC~20 J lib

o.5 ' b~.c

a~c

L

I0

b-c

0

a+c

1

1.0

2.0

3.0 4.0 vlc in PER CENT

5.0

6.0

Fig. 5. Slowing-down characteristics for Dy ions in Dy which were obtained for the case 164Dy+ 348 MeV S4Kr. The time at which the initial recoil velocity of v(t = O)/c = 6.5 % becomes a given value of v(t)/c (bottom scale) is indicated at the top as the slowingdown time in ps. Multiple nuclear scattering effects were taken into account to estimate the slowing-down time according to Blaugrund's formalism17) with a correction factor 0.85 (see subsection 5.2). The time scale (a) is for the use of (dE/dx)e(LSS) and the time scale (b) for (dE/dx)e(NS).

7-RAY LINESHAPE ANALYSIS

power. It should be mentioned that this correction is in the opposite direction to Northcliffe's c o m m e n t 18) that for relatively light moving ions (up to Ne) experimental wdues of the constant k are approximately 20% larger than the predicted one. Fig. 5 shows the slowing-down characteristics for D y ions in D y which were obtained for the case, 164Dy + 348 MeV 84Kr, using the two sets of the stopping powers. It is worth pointing out that in D S A M experiments reported so far in this mass region the m a x i m u m value of v/c was just above 3% where the curve o f (dE/dx)~(NS) crosses that of (dE/dx)e(LSS),

535

536

T. I N A M U R A et al.

predicts somewhat higher probability for small shifts than the data show. It is difficult to probe uncertainties inherent in the stopping power data at present. However, it is worth testing how much effect a change of the stopping power by a constant factor has on the nuclear lifetimes extracted from the lineshape analysis. Although such a correction to the stopping power may not be realistic for high recoil velocities, this test should give an idea about the size of uncertainties in the stopping power used when the lineshape lifetimes are compared with the lifetimes obtained independently of the stopping power. To carry out this test we chose (dE/dx)¢(NS) coupled with (dE/dx)n(LSS) and Blaugrund's scattering term because this combination (referred to as " s t a n d a r d " hereafter) has proved to be the most successful in the present analysis. The electronic and nuclear parts were changed by 20% separately and the lineshape fits were made for the two experimental examples shown in fig. 3. The goodness of fit was the same as that of the standard fit in each case. The lifetimes extracted with different stopping powers are presented in table 1. It seems that increasing the nuclear stopping power has little effect in extracting short lifetimes. This, however, may be due to the experimental data used; in other words, account should be taken of a statistical error (see subsection 5.5). Comparing these results with the recoil distance lifetimes reported by Sayer et al. 2°) (see table 2), we feel the uncertainty in (dE/dx)e(NS) to be less than 15%. 5.3. PHOTON ABERRATION When the velocity of the recoils reaches a few percent of the velocity of light, it becomes necessary to correct for photon aberration, i.e. the difference between the direction of propagation of a photon in the laboratory and its direction of propagation in a coordinate system moving with the recoils. This correction can be made as follows. When a Lorentz transformation is carried out between a moving system fixed on the recoils and a lab system (note that for the present purpose one can rotate each system so that the z-axis is in the recoil direction), the following relation can easily be derived,

Comparison between the present data and others. Nuclear lifetimes (ps) Present a Other

Excited states

164Dy

8+ 10 + 12 +

9.804-0.10 3.36-/-0.04 1.62±0.02

10.0 4-0.8 e 3.4 +0.7 e 1.9 -I- 1.0 e

16°Dy

2+

1.95 ± 0.04

aStTa

l 1/2 +

22.9 4- 3.9 b

2.03 4- 0.009 d 21.4 e

a Errors quoted are statistical only (see text). Small differences between the present 164Dy data and those published previously s) are mainly due to the exact calculation of inelastic scattering particle and de-excitation y-rays angular distributions, and multiple cascade feedings in the present procedure; and the differences are likely to become larger with decreasing the spin values of excited states. b This value is tentative because of background which remained unsubtracted (for details see text, subsection 5.5). e Ref. 20, recoil distance. d Ref. 21, lineshape in coincidence with backscattered particles; the value quoted is an average. e Ref. 22, B(E2)~'; the value cited here was obtained from tl/,z which was given in this reference, but there was no error quoted.

is the angle between the directions of the recoils and y-rays in the lab system. For a given inelastic particle the actual number of y-rays into A-Q.; whose energy is shifted by AE~. can be written in the form

Since

A N (AE~) ~

df2.~,

as defined by eq. (l), the registered number of y-rays is given by

N'(AE~) ~ [1 + 2(AE.~/E.~)] N(AE~).

(15)

This correction has been incorporated in the programme. Ge(Li) In the present calculation the differential detection efficiency of a true coaxial Ge(Li) is given in the form 5.4. DIFFERENTIAL DETECTION EFFICIENCY OF

dl2 v = 1 -- (v/c) 2 df~ [1 - (v/c) cos ~k']2 1 + 2(v/e) cosff' = 1 + 2(AEJEy)

TABLE 2

(14)

for v/c < 0.1, where .c2~,and -Q'vare the solid angles in the moving system and the lab system, respectively, and ~p'

e(E~., 07) ~ ~(E~, 0r) expE-P(E~) Z(0~)] x

× {I - e x p [ - / ~ ( E , )

X(0y)]},

(16)

537

y-RAY LINESHAPE ANALYSIS

where # is the total linear absorption coefficient, X is the path length of photons in the active region of the detector, Z is the path length in the inert core, and is the correction factor which should be determined by experiments. By using our experimentally observed differential efficiency, a simplified form with ~ = 1 was found to be good enough to reproduce the Dopplerbroadened lineshapes in the present measurements where the solid angle subtended by the 70 cm a Ge(Li) was nearly 0.15 n. Care should be exercised in handling ~-factor in eq. (16) for a detector subtending a large solid angle. When the recoil velocity is large, it is necessary to take into account the variation of detection efficiency with the Doppler shift. In the present model the correction for this can easily be made as follows. The absorption coefficient /~ can be written in the form p( e'7)/p(ET) = (E'~/E~) k

(17)

in a small range of energy such that E ~ - AE~_ E~_
~(E~+AE~) =

/~(Er) (1 +AE/ET)k.

(18)

It is interesting to note that this correction is comparable but in the opposite direction to the photon aberration correction. I

2~--~ 0;.. 964.5 I(~V B-rays from 16°Dy*348MeV ~Kr • - 1.95ps

5.5. COMPARISON WITH OTHER MEASUREMENTS AND ERROR ESTIMATION

Firstly, to demonstrate the application of the present method to 7-vibrational 2 + states and odd deformed nuclei, we show in figs. 7 and 8 the fits to the Dopplerbroadened lineshape of 2 + ~ 0+,d de-excitation 7-rays observed in the 16°Dy+348 MeV 84Kr reaction and to the lineshape of 19/2 + ~ 15/2 + de-excitation 7-rays in the l alTa + 348 MeV S4Kr reaction. I n table 2 the present lineshape lifetimes are compared with the lifetimes obtained previously using different techniques. It can be seen that the present results are in good agreement with these independent results: lifetimes of the 8 ÷, 10 + and 12 ÷ states in 164Dy which were reported by Sayer et al. 2°) were obtained by means of the recoil distance method; the lifetime of the 7-vibrational 2 ÷ state in 16°Dy was derived from the lineshape of de-excitation 7-rays (2+~0~+,a) observed in coincidence with backscattered 4°Ar ions2~); the lifetime of the 11/2 + state in ~atTa was derived from the B(E2)I" value in Coulomb excitation with an ~-beam22). It should be stated that the lifetime of the 11/2 + state in ~SlTa obtained in the present analysis is tentative because it was obtained in a different manner rather than the normal procedure in which the full experimental 7-ray lineshapes were included in the analysis. In the shifted part of the 11/2+~7/2 + deexcitation 7-ray lineshape there was background which

J

19/2*'~eTa --~lS+/2348 ÷from 522. MeV7KcVa4Kr N-rays T = 1.62ps

I0~

~ 103

102l

1 1020

10

20CHANNtegOuMBE R 40

SO

Fig. 7. Doppler-broadened lineshape of the 2~ --~0+na deexcitation y-rays observed in the 160Dy + 348 MeV S4Kr reaction. Apparent background 7,-rays on the lower tail of unshifted photopeak were neglected in the fitting procedure.

10 ~q-.I 0

, IO

I 20 ao CHANNEL NUMBER

I 40

s0

Fig. 8. Doppler-broadened lineshape of the 19/2+-->15/2 + deexcitation 7'-rays observed in the 181Ta + 348 MeV 84Kr reaction.

538

T. I N A M U R A et al.

remained unsubtracted since a Compton edge and a Doppler-broadened cascade transition ?,-ray line occurred in the middle of the lineshape, so that the fit to the lineshape was made neglecting the considerable part affected by these intruders. Because of this unfortunate situation the lineshape is more dependent on uncertainties in background subtraction than is normally the case. The lifetime quoted is the weighted average of the lifetimes extracted using different background subtraction criteria. The errors quoted on our results are statistical only (standard deviations). The statistical errors were deduced from the curvature of a parabolic expansion of the Zz hypersurface z3) with respect to nuclear lifetime parameter z since the variation of Z2 with respect to z is almost independent of the values of the other parameters near the minimum. Of course there are other errors due to uncertainties inherent in the present analysis: errors due to uncertainties in the stopping power data and background subtraction, an error due to the assumed unshifted (intrinsic) photopeak shape, i.e. the modified Gaussian function, and an error due to the theoretical relative populations (feeding correction). For the 8 +, 10 + and 12 + states in 164Dy, the lifetimes obtained in the present study are very similar to those measured using the recoil distance method2°). This fact suggests that our standard stopping power combination gives a good approximation to the actual slowing-down process for high recoil velocities. However, we cannot exclude uncertainties in the standard stopping power combination because of the size of errors quoted on the recoil distance results. At present we should take a probable error due to the uncertainties in the stopping power used to be as large as the error quoted on the recoil distance results. Averaging the latter errors, we have a probable error of 10.5%, which should be taken into account in the present analysis. Information on the actual probable error is still to be obtained by accumulating experimental data. On the basis of a series of trials, we estimated the other errors as follows:

of singles 7-rays from Coulomb excitation with heavy ions is a useful method for the determination of nuclear lifetimes of high spin rotational states. In order to fit the singles 7-ray lineshape, one has to know the relative inelastic particle differential cross-sections and the de-excitation ),-ray angular distributions for the excited states as a function of impinging particle energies and scattering angles. The statistical tensors of Coulomb excited states which provide these quantities can be calculated according to Winther-de Boer's multiple Coulomb excitation computer programme. We have shown that the rotational model E2 matrix elements are good enough for the present analysis, and that the calculated lineshapes are insensitive to the values of the nuclear matrix elements. When the model E2 transition matrix element used and the one calculated from the extracted lifetime are different by > 20%, iteration should be made using the latter value. For high recoil velocities, the Northcliffe-Schilling electronic stopping power coupled with the Lindhart, Scharff and Schiott nuclear stopping power is likely to give much more realistic slowing-down characteristics than the LSS theory only. The NS and the LSS electronic stopping power are very different as a function of v/c. This difference becomes crucial to the fits to the Doppler-broadened ?,-ray lineshapes when the recoil velocity becomes larger than 3% of the velocity of light. The photon aberration correction and detection efficiency correction due to the Doppler shift both were taken into account in the present analysis. It has been found that these two corrections tend to cancel each other.

background subtraction < 3%, unshifted photopeak shape < 5%, feeding < 2%. Details of these estimations will be discussed in ref. 24.

The authors wish to thank Prof. J. C. Willmott for his valuable discussion and encouragement during the course of this study. They gratefully thank Prof. W. R. Phillips for providing them with the Coulomb excitation computer programme based on the Winther-de Boer multiple Coulomb excitation code. It is a pleasure to express their thanks to Mr G. Varley for his constant cooperation during this work, the staff of the Manchester H I L A C for the excellent operation of the machine, and Mr T. Morgan who prepared the isotopically enriched metallic foil targets. Thanks are also due to Dr J. Mo for his discussion about differential detection efficiency and cooperation in measuring it. One of the authors (T.1.) would like to acknowledge award of a fellowship by the Science Research Council.

6. Summary The DSAM Doppler-broadened lineshape analysis

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Error source

Error

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539

11) K. Alder, A. Bohr, T. Huns, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432. 12) D. Schwalm, A. Bamberger, P. G. Bizzeti, B. Pouh, G. A. P. Engelbertine, J. W. Olness and E. K. Warburton, Nucl. Phys. 192 (1972) 449. 13) E. J. Hoffman, D. M. Van Patter, D. G. Sarantites and J. H. Barker, Nucl. Instr. and Meth. 109 (1973) 3. 14) A. J. Ferguson, Angular correlation methods in gamma-ray spectroscopy (North-Holland Publ. Co., Amsterdam, 1965). 15) j. Lindhart, M. Scharff and H. E. Schiott, Mat. Fys. Medd. Vid. Selsk. 33, No. 14 (1963). 16) L. C. Northcliffe and R. F. Schilling, Nucl. Data Tables 7, nos. 3-4 (1970). 17) A. E. Blaugrund, Nucl. Phys. 88 (1966) 501. ~8) L. C. Northcliffe, Ann. Rev. Nucl. Sci. 13 (1963) 67. 19) W. M. Currie, Nucl. Instr. and Meth. 73 (1969) 173. 20) R. O. Sayer et al., Bul. Am. Phys. Soc. Ser. 2 19 (1974) 524. 21) G. B. Hagemann, private communication (1973). 22) F . K . McGowan and P. H. Stelson, Phys. Rev. 109 (1958) 901. 23) p. R. Bevington, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969) p. 245. 24) F. Kearns, Thesis (University of Manchester, 1974).