Coulomb excitation parameters

NUCLEAR

INSTRUMENTS

AND METHODS

146 ( 1 9 7 7 )

293-299

; O

NORTH-HOLLAND

PUBLISHING

CO.

COULOMB EXCITATION PARAMETERS* JORRIT DE BOER

Sektion Physik, Universitiit Miinchen, Miinchen, W. Germany

The parameters entering the computation of the Coulomb excitation process are explained starting from simplified descriptions.

1. Introduction Heavy ions, as they are available from modern tandems, supply projectiles suited to study nuclear properties by Coulomb excitation, thereby supplementing information obtained by other means. In contrast to heavy-ion induced compound (HI, xn) reactions, Coulomb excitation populates the nucleus starting from a well-defined state, the ground state. Although the excitation process with heavy projectiles is a rather complicated one, the level of complexity can be" gradually controlled by choosing appropriate bombarding conditions. The tandem accelerator with its specific quality of easily variable projectile species and bombarding energy is especially suited for this type of experiment. This paper aims at giving a feeling for the Coulomb excitation process. It turns out that most of the parameters entering a detailed computation appear already in a classical treatment and have here easily understandable meanings. The classical picture also gains in accuracy for heavier projectiles. 2. Simplifying assumptions In order to simplify the discussion it is assumed that a structureless projectile excites the rotational motion of a deformed nucleus. The interaction beSupported in part by the Bundesministerium for Forschung und Technologie, W. Germany. b I

tween projectile and target is of electric monopole (E0) and quadrupole (E2) type only (no nuclear forces). The relevant quantities are given in table 1.

3. Coulomb excitation theory Table 2 gives descriptions of the excitation whereby the ease of acquiring a feeling for the process decreases as the level of sophistication and accuracy increases (fig. 1). For full details the listed references (1-5) have to be studied. The excitation process is characterized by a set of parameters, collected in table 3. The result of the excitation process, i.e. the final state, can again be expressed in terms of parameters listed in table 4. It TABLE 1 Projectile and target parameters•

Projectile properties mass charge energy

mp = hp amu Zp e Ep = ½mpV~

e = electron charge

Target properties mass charge shape charge distribution mass distribution

t = A t amu Z, e axially symmetric ellipsoid intrinsic quadrupole moment Qo moment of inertia J m

Quantum-properties of target beam direction

t'v2

rotational spin sequence of even-even nucleus olion

//

1~ = 0 +,2 +,4 + , . . . rotational energy spectrum

symme ry

ax~s

E, = (h2/2J) 1(1+ 1) rotational E2 matrix elements

Fig. 1. Pictorial representation of the Coulomb excitation process.

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TOPICS

J. DE B O E R

294 TABLE

2

Descriptions o f excitation process.

1) Primitive

Projectile comes close to target; d e f o r m e d target is at r a n d o m initial orientation flo-

While in vicinity o f deformed target (fig. I).

It is closer to one tip o f target t h a n t, other. Target feels torque.

2) Qualitative

Projectile, accelerated to velocity v~, is in vicinity o f target when distance d(t) <~ 4a.

T i m e scale with d(t = 0) = dm~; Vicinity o f target specified by d(h < t < t2)~ 4a:

T o r q u e = derivative o f orientatioJ energy EQ(,8) with respect to fl il field gradient p r o d u c e d by Zt at Eo.M~v(d, fl) = 72 x x Zp Qo, b,,, cos 2 fl/dtam

ZJ/vtcinity = t2 -- tl •

E o . . . . = E o ( d = 2a, fl = 0 °) rma, = IOEo/Ofl (drain, '8 = 45°)1 EQ, max

Projecule orbit r(t) is calculated numerically for a large n u m b e r o f initial conditions (,81, b . . . . ). Orbit is approximately hyperbolic if E2 interaction is small compared to E0 interaction (q,~r/). (B) Semi-classical Projectile m o v e s on hyperbolic orbit specified by v~, a, and e or oq or I.

3) Quantitative (A) Classical

The distance is a well-defined function o f time d(t).

T o r q u e changes with time. For head. on collision a n d small interaction: T(t) oc d - a (t) (fig. 2).

Target has q u a n t u m states

Solve t i m e - d e p e n d e n t SchrOdinger equation for initial conditions (t = - o e ) corresponding to target in g r o u n d state 10). C o m p u t e r calculates a m p l i t u d e s aim (t).

ILM/5

=

[sS.

Electric" m o n o p o l e part o f interaction is accounted for by hyperbolic orbit. Electric q u a d r u p o l e matrix element is time-dependent:

ri VEz(t)ls'). (C) Q u a n t u m mechanical

T h e process is described by a stationary wave ~ff.

Vt2 couples incoming to outgoing waves, labelled by J, l, 1.

is the aim of this section to emphasize the relation between the parameters entering the different computational methods. 3.1. PRIMITIVE DESCRIPTION This description uses no formulae, just words. One can learn from it that the target picks up angular m o m e n t u m only when it is deformed. 3.2. QUALITATIVE VERSION This version should elucidate why the parameters introduced play a role in the quantitative calculations of the excitation process. It assumes a head-on collision between projectile and target. It further assumes that the projectile motion is not

C o m p u t e r solves time-independent SchrOdinger equation fulfilling b o u n d a r y conditions at origin and infinity.

affected by the transfer of energy and angular momentum during the excitation (E0) interaction much larger than E2 interaction). This is equivalent to requiring that q <~ n or Eq (dmin,/30 = 45 ) ~ Ep and Ez=q ~ Ep., A comparison of the qualitative value of the angular m o m e n t u m Tm,~ Atv~cini~,,with the exact integral ~Tdt shows that t~ and )2 have to be chosen approximately as ++_4(a/v=)= +_4 tco~ for an E2 excitation (fig. 2). The adiabaticity parameter ~ compares the time behaviour of the projectile with that of the target. = 0 means: target nucleus does not move while projectile flies by. ~ >~ 1 means: target nucleus orientation is always adjusted to have the smallest orientation energy for a given projectile position.

COULOMB

EXCITATION

295

PARAMETERS

After collision, target has picked up a n g u lar m o m e n t u m hi,

and rotates with a n g u l a r velocity 09 a n d rotational energy Erot.

W h a t a b o u t q u a n t u m - m e c h a n i c a l effects ?

At t = + oo: ang. morn. ~ torque x X z J t v l e l n i t y ; m a x i m u m value h/max m TmaxAtvicinity •

hI=~oJ; E, o t = h 2 1 2 / 2 J ; angle of rotation during collision Aft = rco. o = r c o , x

T h e y will be small when: q u a n t u m n u m b e r s are large; - q u a n t u m - m e c h a n i c a l averages are meaningful.

MAt =

t~;

~

dflmax~q~; r~uc] - (o -~ ( I = 2); = r¢o,/zn,¢l, adiabaticity. Small x+__+large J: target does not m o v e during collision f l ( t ) = flo, sudden collision. Large ~+--+ small J: target orientation ,8(t) follows projectile m o t i o n adiabatically ,~ (t = + oo) = co ~ 0, no excitation.

Integrate torque over time hltBo)=S+_~ T(t) dt (see fig. 3). Obtain P ( I ) by appropriate averaging over ,8o (see fig. 4).

In example o f fig. 3 finite adiabaticity ( ~ = 0 . 0 2 ) a n d finite interaction (q/rI = 0.045) cause shift o f curves m a x i m u m f r c m ,8o = 45 ° and /max from 2q.

Effect o f excitation on orbit exactly accountable, a n d Q M interference effects connected with double-vatuedness o f l(flo) accountable (fig. 4), target H a m i l t o n i a n Ho is explicitly required.

State o f target is specified by set o f amplitudes a1M(t = + oo), e.g.:

(E, ot> = E Erot(I) P~

Ho not explicitly needed, o n l y eigenvalues E~ a n d matrix elements M ( E 2 ) , ' , . Q M interference exactly accountable. Effect o f excitation on orbit a p p r o x i m a t e l y accountable by using symmetrized orbits. Accurate results for I , ~ r/.

l

(see fig. 5)

e,,~ = la,,..,l~; P, = z la..,l~; ( M ) = Z MPIM. M IM

(see fig. 5)

C o m p u t e r calculates scattering matrix rl{ specifying scattering process

Excitation probabilities m a y be defined as PI(,9) = da1(~l)/Z, do-1(6t)

C o m p u t e r code expensive to run, limited to 1_< 10, q_<20.

1

15

i

i

]

i

I

'

I

4

Ap~q '

I 0

O

'

I

(dram

i

i

l

10

'

'

i

i

I

i

,

i

i

13

,

,

i

i

I

'

i

,

,

I

'

i

[

'

I

/

I

i----r"'T--,,

-5

i tl

i

I

L

- Toot 0 ÷Too,,

&t VICINIty

t2

5

10

a )

t {~

Fig. 2. The change of the E2 interaction with time for the case of a head-on collision and q,~ ~/.

0

I

0o

_

L

L

]

30 °

~o

60°

\1 90°

Fig. 3. Classical calculation of the angular m o m e n t u m ,~/, transferred in a head-on collision of 170 MeV 4°At with 238U (Q0 = 11 b) taken from ref. 4. T h e parameters are q = 5.7, r / = 127, ~ 2 0 = 0 . 0 2 , and a = 8 . 2 fro. IV.

SELECTED

TOPICS

296

J. DE BOER

TAB3LE Parameters characterizing excitation process.

Initial conditions time projectile velocity impact parameter target orientation

target rotaUon quantum state

t =

--oo

v(t=-m) = vo~ b /3(t = - o o ) = / 3 o , /3 is the angle between the target symmetry axis and the incident beam direction. (For finite impact parameters orientation is given by two angles.)

/~(t = -oo) = ¢Oo= 0 I/o=0, Mo=0)

= [0)

Collision parameters generai orbit hyperbolic orbit

eccentricity deflection angle collision time typical value (Ep close to Coulomb barrier) distance typical value (Ep close to Coulomb barrier) impact parameter head-on collision minimum distance Sommerfeld parameter orbital angular momentum angular momentum of 90 ° orbit relative strength of E2 and E0 interaction

r(t) x = a (coshw+e) y = a .¢(e 2 - 1 ) s i n h w z=0 t = (alva) (~ sinh w + w ) e = I/sin ½0 0 Teoll =

a/v~c

Z¢oU = 2 x 10 22 s a = (I +Ap/A,) ZpZ, e~-/(Ep 2) a = 0.72 (i +Ap/At) Zp/fl/'~>MeV(fm ) 2a = 15 fm b = a ctg ½0 d(t) = tr(t)l d,,,i. = d O = O ) = 2a ~l = ZpZte2/hv., q = 0.1575ZpZ, ./(Ap/EpM~V)

3,3.1. Classical cak'ulation Classical calculations have been performed for realistic bombarding conditions and idealized, but realistic rotational nuclei (ref. 4) with the restriction to zero impact parameter b (/= 0). The angular momentum hi which the target picks up in a head-on collision has been calculated (ref. 4) for the case of 170 MeV 4°Ar on 238U solving the classical equations of motion by a computer. Fig. 3 shows the way in which hi depends on the target's initial orientation /30. For infinitely large 1/and moment of inertia J the maximum value would be reached at /30=45 ° and would be equal to 2q. The small but finite values of the adiabaticity parameter C02--0.02 and of the ratio q/q = 0.04 account for the small shift in the maximum of the curve. Giving each initial orientation a weight appropriate to the three-dimensional process (ref. 4) the classical probability distribution P(1) has been calculated and is shown in fig. 4 for the same example. Quantum-mechanical effects like the excitation of classically forbidden spin-states ( l > 2 q ) and interference phenomena can be approximately accounted for by an appropriate extension of the classical treatment (see ref. 1 and papers quoted in this reference). 3,3.2. Semiclassical calculation In the semiclassical treatment it is assumed that I

i

i

i

I

[

I

I

hi = hq ctg ½,9 hq q/q =

QoE2p 2

0t

3 5

Z p Z t e (I+A~/A,) z AEQ . . . . /E . . . . = ½(q/~l)

3.3. QUANTITATIVE COMPUTER CODES Quantitative treatments generally require extensive calculations by large computers. In s o m e limits (e.g. perturbation limit or sudden approximation limit) tables can be used conveniently. C o m puter programs have been published for the subgroups B and C o f this section (refs. 2 and 3). The d i m e n s i o n l e s s parameters used in the computer codes have their counterparts in the qualitative description.

~

z

0 o~

00o,

J

~

Fig. 4. The P ( I ) for a (Q0 11 b). and a = 8.2

L

L

I [

I

I

I0

I~

classical and semiclassical excitation probabilities head-on collision of 170MeV 4°Ar with 238U The parameters are q = 5 . 7 , r/ 127, ~2o 0.02, fro.

COULOMB

EXCITATION

PARAMETERS

297

TAm.E 4 Parameters characterizing result of excitation. Angular momenta hi q = ZpeQo/4 hvooa 2

transferred angular momentum typical value of transferred angular momentum hq

.l

q = 7.6 Qob.... ~/AI EI~M,v (1 + Ao/At) 2 ZpZ 2 1/tl or q/q I/q ~ 10%

relative angular momemum loss typical value for highest experimentally observable I Energies

transferred energy typical value of transferred energy relative energy loss typical value for highest experimentally observable 1

El

E(I = q) = (hq)2/2j EI/Ep EI/E p ~ 1%

Adiabaticity

collision time target time adiabaticity parameter

l'coll

~ a/Uoo

c,~d = h / ( E r - E~) ~l' I

= l'¢oll/'Cnu¢l

~rl = ZpZt x/Ap (1 + Ap/At) ( E r - E t ) M , v 12.70 Ep~ M¢V

typical value (Ep close to Coulomb barrier)

~rl "~" 0.5 ( E r - E i ) u , v

Perturbation theory (PI , o ~ 1)

excitation probability

e~ = (Z~o22)2 RJ~0, O

relative excitation probability

R2(,9, ~) (fig. 7)

strength parameter

~o-2 = 14.36

relation to q for rotational target

Xo~2 = x/(16/45) q

a quantum-mechanical system, the target nucleus with Hamiltonian H0 satisfying

H o JIM5 = E, IIM), is exposed to the time-dependent quadrupole interaction VE2(t) engendered by the projectile moving on a classical orbit r(t). The orbit may be symmetrized ~n) to approximately account for the transfer of energy and angular momentum from projectile to target. The target wavefunction ~(t) is expanded in terms of IIM) by time-dependent amplitudes am (t): lip(t)) =

~

am(t)[IM).

I,M

A computer (ref. 2) is used to solve the time-dependent Schr~dinger equation ih~ = [ H o + VE2(t)] ~P.

x/AP E~M'v M(E2)2o ZpZZt (1 + Ap/A,) 2

In terms of am(t) it takes the form ihdrM,(t ) =

~

(I'M']VE2(t)]IM)

x

1, M

x e x p [ i ( E r - E t ) t/h] am(t), and has to be solved for the initial conditions a 0 0 ( t = - o~)= 1.0 and all other a ~ u ( t = - oo)=0. The matrix element can be written as

(I' M']VE2(t)IIM 5 = q f (t) n(IM, I'M'), w h e r e f ( t ) has the geneial shape of [dmJd(t)] 3 (see fig. 2) and the numerical factor n is of order unity. The computer solves this problem starting at an appropriately large negative time and proceeding in small steps to a large positive time. The development of the nuclear wavefunction with time can be observed. As an example fig. 5 shows the time dependence of the average rotational energy (E) and the average angular momentum in the z-diIV. S E L E C T E D

TOPICS

298

J. DE B O E R

rection (NO when 2 3 8 U is excited by a 605 MeV '32Xe projectile moving on a 90 ° orbit. Fig. 6 shows P,=~la/M(+Oo)]2 for the same case and

where R~ (u, 4) is normalized to unity for t9 = 180° and ~ = 0 (see fig. 7), so Z 2, which is proportional to q2, measures the excitation probability for back-

M

I = 0, 6, 12, 18. The z-direction is perpendicular to the reaction plane. Figs. 4 and 8 show Pz calculated for typical bombarding conditions. The relevance of the value q is seen as a steep drop in P / f o r l>2q (classically unreachable angular momenta). Table 5 lists the magnitude of some of the parameters for these cases.

In first-order perturbation theory P2 is given b / )

10 V

'

"

'

"0" 180 °

_

'

I

'

'

'

'

,

i

2

R2(~',~

%05 ~

P2 = I z o ~ l ~ R~(,9, O,

i

i flnal value- ~

MeV

¢:6oo o

A

,

h

of final a

i

f

\\

as ~

o

Fig. 7. Relative excitation probabilities R~(,9, ~j), taken from ref. 1. In first-order perturbation theory, the excitation probability is given by P 2 - (Zoo2) 2 R~(,9, ~20).

haft of fllhQI

~°AI -2

-I

0 W

I

86K I arrows st

2

Fig. 5. Average values of the excitation energy (E) and of the angular momentum projection
605 MeV

i0 -I

~n

\

<,

l0- 2 -

\

\

\

\ ~-

i

1

i

F

t

+

Z

~P~'l

:,-..

\

,o-~

\ 3LOMeV

\ "~K~

\

o,

185MeV\

~OA \ I0 ~ -

00% -

-,

,I0

/

w

*I

*2

,

*3

i

Fig. 6. Excitation probabilities for the example of fig. 5 as a

function of the time parameter w, related to t by t=(a/v~)x(e sinh w+w). The parameters are q = 13, r/= 365, ~2o=0.02, a = 9 . 2 f m , e 1/¢2.

0

10

\

73.~v

\

\

,00

\

\

]

20

Fig. 8. Semiclassically calculated excitation probabilities for various ions exciting 238U in a ,9= 165° orbit. The relevant parameters are given in table 5. The values of 2q are indicated by arrows starting at the top of the figure. The sharp drop in P(I) for classically unreachable l>2q is evident.

COULOMB EXCITATION TABLE 5 Excitation parameters for z38U. Q0 = 11.12 eb, E2+ = 0.045 MeV. Projectile

1608 40 18Ar22 ~6Kr50 132v~ 54Aw78 2381 r 92w146

Ep (MeV)

q

,7

~20

a (fro)

73

2.7

54

0.018

7.7

185

6.5

121

0.017

7.5

340

8.7

262

0.024

9.5

605

13.0

365

0.021

9.2

1300

19.6

570

0.020

9.4

scattering in the perturbation limit. The set of strength parameters Xz-r takes the role of q for a general, i.e. non-rotational, nucleus. 3.3.3. Quantum-mechanical calculation A computer program for this case is described in ref. 3. The coupling potential responsible for the excitation can again be expressed as V~r,n(r) = q (2a/r) 3 n(J,11, I'l'), where J is the channel spin, and II and l'l' are the

PARAMETERS

299

target and orbital angular momentum quantum numbers in the entrance and exit channels, respectively. The numerical factor n is of order unity. The number of channel spins Jm,x, for which the time-independent Schr~Sdinger equation has to be solved, is governed by r/ with Jm~x~ 4r/. The reaction matrix should show a regular behaviour for J>~q. The relative deviations between semiclassically and quantum-mechanically calculated quantities are expected to be proportional to I/~l. Some of these features are borne out by recent calculationsS). References 1) K. Alder and A. Winther, Electromagnetic' excitation (NorthHolland PuN. Co., Amsterdam, 1975). 2) A. Winther and J. de Boer, in Coulomb excitation, Perspectives in physics (eds. K. Alder and A. Winther; Academic Press, New York, 1966) p. 303. 3) F. R~Ssel, J. X. Saladin and K. Alder, Comp. Phys. Comm. 8 (1974) 35. 4) H. Massmann and J. O. Rasmussen, Nucl. Phys. A 243 (1975) 155. 5) j. de Boer, H. Massmann and A. Winther, Proc. Int. Workshop on Gross properties o[ nuclei and nuclear excitations HI, Hirschegg, Austria (1975) p. 89.

IV. S E L E C T E D TOPICS