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Deviations from the Rutherford scattering due to the ground state quadrupole moment
ANNALS
OF PHYSICS
78, 518-534 (1973)
Deviations From the Rutherford Scattering Due To the Ground State Quadrupole Moment K. ALDER AND F. ROESEL Institute of Theoretical Physics, University of Basel, Switzerland
AND
U. SMILANSKY Department
of Nuclear Physics, Weizman Institute of Sciences, Rehovot, Israel
ReceivedOctober 29, 1971 The deviations from the pure Rutherford scattering due to the finite quadrupole moment in the ground state has been recalculated to second order in the quadrupole moment. It is shown that for unpolarized targets the deviations are small once the excitations into higher states are taken into account. The smallness of the effect is due to a strong cancellation between the two second order terms in the perturbation expansion. This fact was not realized by the authors who dealt previously with this problem. For polarized or aligned targets the hrst order amplitude contributes most of the effect and deviations of the order of 10 % are expected. Finally, a convenient parameterization of the deviations is presented.
I. INTRODUCTION It was recognized as early as 1952 by Bohr and Mottelson [l, 21 that large deformations can occur in nuclei. One of the earliest suggestions for detecting these large quadrupole moments was to look for deviations [3] from the Rutherford scattering cross section. However, for unpolarized targets the average first order term vanishes and the deviation from the pure Coulomb scattering is due to a second order term and thus proportional to the square of the quadrupole moment. One expects that these quadratic terms are small so that they can usually be neglected [4]. If the target nuclei are polarized the deviation will be proportional to the quadrupole moment in the ground state. It may be noted that for such an experiment a vector polarization is not necessary. It is sufficient that the target nuclei are aligned and have a nonvanishing tensor polarization so that deviations from the pure Rutherford scattering are obtained. 518 Copyright All rights
Q 1973 by Academic Press, Inc. of reproduction in any form reserved.
RUTHERFORD
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519
In recent papers by C. F. Clement [5] and Y. N. Kim [6] it was pointed out that the magnitude of the quadratic scattering effect is of the same order as the first order effect. Their results show that particularly if heavy ions are used as projectiles the second order term is dominating. This would suggest a simple experimental determination of the quadrupole moments in the ground state by measuring the elastic cross section of an unpolarized target. We should like to point out that the above mentioned result is misleading and is due to the fact that the scattering amplitude was computed only up to the first order in the quadrupole moment. However, to obtain correct results for the cross section up to second order, the scattering amplitude has to be expanded to the same order. Then it turns out that the square of the first order terms nearly cancels the interference term between the Rutherford amplitude and the second order amplitude. (In the semiclassical limit this cancellation is even completel). This strong cancellation will therefore diminish strongly the importance of the second order deviations compared to the leading first order term. These results make experiments with unpolarized targets unfeasible and one must use polarized or aligned targets for a measurement of sign and magnitude of the quadrupole moment in the ground state. One should, however, keep in mind that there are other effects which give deviations from the pure Rutherford scattering. First, all the higher order contributions which we shall neglect in the following may become important, especially if heavy ions are used as projectiles. Second, the virtual excitation of higher lying nuclear states will also give rise to deviations of the scattering cross section. This virtual intermediate excitation can be considered as being due to the polarization of the nucleus. An estimate of this effect has recently been carried out [7] with the result that it does not give rise to an observable contribution. Third, real excitation will, of course, also alter the elastic scattering cross section. However, this contribution can be computed exactly and one may correct for it. Finally, such effects as the influence of the main nuclear potential or of the atomic electrons-which may also give small contributions -must be estimated. In the following we should like to present the results of our calculations. In Section II the theoretical considerations of the deviation from the Rutherford scattering are collected and a convenient parameterization is introduced. The next section presents the numerical results which are illustrated in figures and tables. These are applied to some important examples. The last section gives a summary of the conclusions which can be drawn from these new results, 1 C. F. Clement shows by dimensional analysis that the two quadratic terms have similar behavior as functions of the physical parameters which define the problem. But, since an exact evaluation was not carried out, he missed the fact that the two contributions actually cancel each other.
520
ALDER, ROESEL, SMILANSKY
II. THEORY The scattering process of a charged projectile in a pure Coulomb field can be solved exactly in a quanta1 treatment. The Coulomb scattering amplitude fc is given by [8] fc
=
e/2) ~-in10~sin*e/2+2io0,
-(a/2)(l/sin2
(1)
where a is half the distance of closest approach in a head-on collision, 8 the scattering angle in the center of mass system, 77the Coulomb strength parameter and u0 the Coulomb phase shift. These quantities are defined by the following relations: Z,Z2e21Av
(2)
a = ZlZ2e2/mv2
(3)
7j
=
and u. = arg r(l
+ i$,
(4)
where Z, and Z, are the charge numbers of target and projectile, respectively. The relative velocity is denoted by v and the reduced mass between target and projectile by m. We consider now a scattering process where the pure Rutherford scattering is disturbed by additional nuclear multipole fields. Usually, we can restrict ourselves to the influence of quadrupole fields only. The total scattering amplitude from the ground state j r&Vi) to a final state 1IfMf) is given by
where the second term in Eq. (5) is due to the influence of the electric multipole interaction. Spin and magnetic quantum number in the initial and final state are denoted by JiMi and I,Mf . To get the deviation from the pure Coulomb scattering we must, of course, compute the scattering amplitude for li = If = I. The additional scattering amplitude f can be computed in various ways. If the multipole interaction is sufficiently small, a perturbation expansion can be carried out [9]. For stronger interactions,fmay also be computed by means of the coupled channel method [lo] where care is taken automatically of the effects of virtual and real excitations. Finally, it is also possible to compute f in the semiclassical limit. In the following, we will mainly discuss the results of the perturbation expansion which we carried out up to the second order. However, let us first make some
RUTHERFORD
521
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qualitative investigations with the semiclassical limit. in this limit the amplitude is given by &+‘,M,
= fC(~IFMFKm)
It can be shown [lo] that
- ~,JFhfiM,)
where a ,,M,(ZJ4i) is the classical excitation amplitude computed in a system where the direction of the incoming projectile coincides with The x-axis is then conveniently chosen in the plane of the orbit with component of the projectile velocity. The second term in Eq. (6) cancels first term in Eq. (5) leading to
(6)
coordinate the z-axis. a positive exactly the
= f,a,,,u,UMih f 'iMi-+'FMF The elastic cross section from an initial number Mi is thus
(7)
nuclear state with magnetic quantum
(g),, = ($)RUthg Ia~,~,(ZiMd12 F
where the pure Rutherford
cross section is given by the well-known formula du/dQRUth = (l/4) a2 sin4( e/2).
(9)
The result of Eq. (8) can also be expressed in terms of the total excitation probability Pex by (dW2)el
= (d~/dQ),,t,(1
One can now express the total excitation probability semiclassical expression [ 131.
- Pex).
(IO)
P,, using the “symmetrized”
(11) where du,,/dQ is the differential Coulomb excitation cross section to the state Zf . v, is the asymptotic velocity of the projectile after leaving the target excited in state I. (da/d@&, is properly symmetrized Rutherford cross section corresponding to the excitation from the ground state Zi to the excited state It. Inserting Eq. (11) into Eq. (10) one obtains the effect of both real and virtual excitation of higher states on the elastic scattering cross section. In most cases of practical interest the inaccuracy introduced by using Eq. (11) is negligible even when compared to the small effect discussed here. Pex is typically of the order of 10 % and the semiclassical approximation deviates at most by 1 % from the exact result. Thus an upper limit for the error introduced by applying Eq. (11) would be 0.1 % as compared to a total effect of a few percent. 595/7W-15
522
ALDER,
ROESEL,
SMILANSKY
The result of Eqs. (10) and (11) is, however, changed for finite values of the Coulomb parameter 7. A finite, quanta1 interference between fc and j which is of the order I/T will become important. The classical amplitude 3 in Eq. (6) leading from the state 1ZM) to the same state, i.e.,
3,bf44 =
.Lh&w - 1) (12) will in a correct quanta1 treatment be changed in its absolute value and phase. We may therefore write fiM-IM
=~,(QGW
- 1) Cd+hd
(13) The factor C,, is of the order unity. This value is approached asymptotically as l/q2 for large values of T.The nonvanishing phase #IM which behaves as l/q for large values of v, gives rise to a strong interference and thus to a deviation from the pure Coulomb cross section. We thus get for the cross section from Eq. (7) x (1 + 2C,,Re((a,,(IM)
-1) ei”ZM) + 1C,, I21a,,(ZM)
--I I”). (14)
In first order perturbation theory, this result can be further simplified because we get for the excitation amplitude a,,(ZM) assuming only quadrupole interaction a,,(ZM)
- 1 = -i&1(21
+ l)lj2 (-lYM
(_‘,
0’ ;,
R,,(&
0)
(15)
where x,+{ is the quadrupole strength parameter which is connected with the reduced matrix element of the quadrupole operator &‘(E2) or with the quadrupole moment in the ground state by Xr I = (167W2
+
&WI
I-4’(E2)1 I 0
15fizXz2(21+ 1)1/Z
1 - __ 36
2,~~ Q ((Z + 1)(2Z + 3))‘/” - #iv -a2 (Z(2Z - 1))1/2 _
The quantity Rzo(f9, 0) is the classical orbital integral in the incident coordinate system discussed above. It can be expressed explicitly by elementary functions as follows R,,(B, 0) = 3(d5/4){Pg(O/2)(1
- tg(e/2) (77 - 8)/2) - sin2 (e/2)
From Eqs. (14) and (15) we obtain for the deviation scattering keeping terms only linear in xr+, (dddQ),l
=
cos
ej.
(17)
from the pure Coulomb
(d+%ltll
x 11 + 2xd2z
+ 01’2 (-Yf
( -zM 0” h) R,,(e, 0) sin 3bpo/.
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523
SCATTERING
Here we have chosen for the correction factor C,, = 1 which is actually very well fulfilled. It is also noted that the phase t,&,(e) is nearly independent of the scattering angle 8 and behaves as l/q as a function of 7. Thus, the phase $&B) can approximately be expressed by [ 121 (19)
sLzo(Q ‘v --0.6/71. Let us now express the semiclassical expansion. We get in second order qM(ZM)
= 1 - i~~-,~(--)‘-~ -
12 ZXId
Z
(
-M
;
amplitude
a,,(ZM)
(-zM
;j
;j’
;
in the perturbation
(2Z + l)1!a R,,
R;,(2Z $ 1).
(20)
I is then easily seen that in the main part of the semiclassical elastic cross section, i.e., in the expression Ifc 121~,.dz~)12,
(21)
the square of the second term cancels exactly the interference term between the first and the third term. This is the very point which Clement and Kim missed. It is, therefore, not permissible to leave out the second order term in Eq. (20). One expects that in a quanta1 treatment the third term in Eq. (20) must also be corrected in the absolute value and by a phase. This leads again to quantum mechanical correction terms which are proportional to x2 and behave as l/q2 for large values of 7). The above discussed strong cancellation is, however, also correct for finite values of 7). The scattering amplitude fliMi+,fA,, for a transition from the nuclear state I Z,Mi) to 1ZfMj) can be decomposed in contributions from different partial waves waves Ii , If and from different total angular momenta J. One gets [9, lo] hiMi
IRMA
=
(i/K)
. c (21, + l)lj2 izi-‘f(ZiOZ&Zi j JMi?
+I2
* (Zfm,ZfM,
where the Coulomb
1JMi){61i,,6,,L,
- ~i~uz~~n~~-u~~~~f~~~j:li’li2i) Y,,,,(o)
(22)
phase shift of the partial wave I is given by (~~(7) = arg Z?Z + 1 + iq)
(23)
The rapid oscillatory part of the s-matrix due to the Coulomb phases uzl + aEf has been removed leading to the rJ matrix in Eq. (22). This r-matrix can be computed
524
ALDER,
ROESEL,
SMILANSKY
by solving a set of coupled differential equations [lo] or by a perturbation expansion [9]. Keeping only quadratic terms in x, we get in the latter case
(24)
where the geometrical factor W is defined by
W:l.Il= 1 f
(3 G/2)(21
+ l)l’2 (24 + 1)“2 (21, + 1)1’2 (-I)J+’
($ ‘ot $1;
; ‘;1, (25)
while the dimensionless Coulomb function F,(h)
radial integral Z is expressed through the regular radial as follows
The symbol a, is the proper symmetrized expression for half the distance of closest approach in a head on collision, i.e., a, = Z,Z2f?2/FTlViVf.
(27)
A polarized or aligned ensemble of target nuclei can be characterized either by the density matrix = (ZM 1p [ ZM’) PMM’ (28) or by the corresponding
statistical tensor p&Z), defined by
pdl) = M;, (;
-‘Mt --kKjPWh44-)‘-“’ w + lY2.
(29)
The polarized target will usually be produced by cryogenic methods or by optical pumping. Therefore, in some definite direction the density matrix p will be diagonal and the diagonal elements denote just the occupation probability of the state 1ZM). However, this special direction will usually not coincide with the direction of the incoming projectile beam. The density matrix has therefore to be transformed from the original system to the incident beam system used earlier in this paper. By this transformation the density matrix will have the most general form of Eq. (28) and contain also nondiagonal matrix elements.
RUTHERFORD
525
SCATTERING
The elastic cross section can now be written in terms of the density matrix and scattering amplitude by
from the definition of the statistical tensor Eq. (29) and the expression of the excitation amplitude Eqs. (22) and (24) we obtain for the elastic scattering cross section up to the second order in the quadrupole moment
x (-)“’
+ (xd
pZKfy*fy(2k
+ l)l’2
(21 + 1) ; C-1”’
1;
‘I
k’\ (31)
The first term is due to the interference between the Coulomb scattering and the first order amplitude, the second term is the square of the first order correction, while the third term is due to the interference of the Coulomb amplitude with the second order excitation amplitude. For the first order amplitudef,“’ one obtains from Eqs. (22) and (24)
(32)
The second order amplitudef,‘,2’
is similarly given by
526
ALDER, ROESEL, SMILANSKY
Expressions (29), (30), and (31) reduce the problem to the computation of first order Coulomb excitation matrix elements. These matrix elements can, however, be computed by the well-known series expansion of hypergeometric function [I l] and by recursion relations. In Eq. (31), the sum over k includes the even value k = 0,2,4 as well as the odd values k = 1, 3. However, the contribution of the odd values is about a factor of 100 smaller than the corresponding contribution of even k. In the semiclassical limit, the contributions of the odd values of k vanish even exactly. We can, therefore, safely leave out these contributions and restrict ourselves to even k. It is then convenient to write the final expression in the following form
where the universal functions fza and F,, are defined by
fzK = (4 - 26,,) -& Re(f,!)f,*)
(35)
and FkK = (4 - 26,,) 1; Re (1
K’K”
(-)“’
(_“,,
z,, tj
f$*j”‘(2k
+ 1)1’2)
They depend only on the scattering angle 0 and the Coulomb parameter r). It is noted that the deviation from the pure Coulomb scattering depends only on the statistical tensors with even k, that is on the tensor polarization. It is, therefore, not necessary to use polarized targets for such experiments. An aligned target can be used and will yield the same information.
III.
NUMERICAL
RESULTS
The deviations of the elastic cross section due to a finite quadrupole moment are given by Eq. (34). Here, all quantities are expressed with respect to the incident coordinate system, the z-axis being in the direction of the incoming projectile and the x-axis in the plane of the orbit. Let us consider the case that by cryogenic methods in a certain z’-direction we
RUTHERFORD
527
SCATTERING
get different population probabilities PM for the states with magnetic quantum number M. The statistical tensor with respect to the (x’y’z’) system is then given by
PkK=
+ lY ; (-I’-”
h3w
(;
-‘zM “,) PM = b17, .
(37)
The statistical tensor transforms under a rotation R in the same way as the spherical harmonics, and it is therefore easy to compute the statistical tensor in any new system. We get P Lx ~- D:,(R) 17, .
(38)
The D-function in Eq. (38) can be expressed by a spherical harmonic depending on the Eulerian angles 01,/3, y of the rotation. The statistical tensor can then be written
Two special experimental situations are apparent. The incoming beam might be in direction of the original direction of polarization or it might be perpendicular to it. For these two special cases, we obtain for the statistical tensor pk,c
=
PkK
=
(40)
6,,17k
or 2k+1 4ir
-lj2 1
Y&I/2,
0) PY7~
(41)
where 01is the azimuthal angle of the initial polarization direction with respect to the x-axis. In order to compute the deviation from the Rutherford cross section, it is necessary to know the functions f2K and F,, as a function of the scattering angle 8 and of 7. It is expected that f2K asymptotically behave as 1/q and 1/T” as a function of 7. These functions have been computed numerically and are illustrated in different figures. In Fig. 1, the function f2Jrj, 0) is illustrated as a function of the scattering angle for ~7= 4 and r) = 8. The Figs. 2, 3, and 4 demonstrate the strick l/q dependence of the function f2K(q, 6) for K = 0, 1, and 2. The quadratic term proportional to x2 is in our considerations of minor importance. In practice the contribution of this term is at most of the order of 10 % of the linear terms. In order to compute the necessary corrections, the functions F2K and FdKare given in Figs. 5 and 6 as a function of the scattering angle for 7 = 4 and 7 = 8. These functions can be obtained for other 17 values by interpolating with the help of the I/r12-rule. The applicability of this rule is illustrated in Fig. 7
7-4
/ -
-.25
f*o
-.20
-.I5
-.I0
-.05
120
140
160
8
FIG. 1. The function f&v, 0) is shown as a function of the scattering angle 8 and for 7 = 4 and 7 = 8. The different values of K are indicated.
-. 4
-.3
-. 2
-.I
l/l2
l/6
l/4
l/3
+i
The functionf,,(q, 0) is shown as a function of 1/T for different values of the scattering angle 0. The strict linear dependence with l/7 should be noted. FIG.
2.
529
RUTHERFORD SCATTERING
e
f 21 .3
-
.2
-
IZOO loo0 140°
160°
t l/12
FIG. 3. The functionf,,(T, angle 8.
l/a
l/4
113
i
YT
0) is shown as a function of 1/T for different values of the scattering
160'
FIG. 4. The functionf,,(v, angle e.
0) is shown as a function of l/v for different values of the scattering
530
ALDER,
ROESEL,
SMILANSKY
where Fdo is shown for different scattering angles as a function of l/+. It may be noted that Foe and Fk, for odd k are very small and can be safely neglected in our considerations. In Table I, the order of magnitude of the effect is demonstrated for some typical examples. The first and second order deviations in percent are listed together with the total effect. It may be noted that for unpolarized targets the total effect
120
140
160
8
FIG. 5. The function FsK(q, 0) is shown as a function of the scattering angle 0 for 7 = 4 and 7 = 8. The different values of K and the sign, with which it should be multiplied are indicated.
practically vanishes. Only the quadratic terms with k = 0 will contribute and give deviations of the order of 1O-4 to 10-5. If the incoming projectile is perpendicular to the initial direction of polarization, we expect, according to Eq. (41), an azimuthal dependence of the scattering cross section of the form da do [I + A@) + W) dsz = t-1dL? Ruth
cos 201+ c(e) cos 4a],
(42)
where the last term with the cos ~CXdependence originates from the second order effect and is expected to be small. For a possible experiment 175L~(~, a) 175Lu, the deviations in percent are shown in Fig. 8. Both possibilities for the incoming
RUTHERFORD
531
SCATTERING
0.8
0.6
0.4
0. 2
0.
J 6. The function F&T, 6) is shown as a function of the scattering angle 0 for 7 =m-4 and 7 = 8. The different values of K and the sign, with which it should be multiplied are indicated. FIG.
TABLE
Target and projectile
Energy (MeV)
Q barns
13’Gd p
11.4
2.0
6.4
I
oW
“I”,
-7.31 -2.34
0.004
0.0001
o’Gl
12.8 7.2
4.0
l=Eu
11.0
2.5
-2.21
0.05
-2.16
“SLU
14.0
5.68
-5.70
0.49
-5.21
-___-
2.80 0.32
,;: 10-S
-7.31 -2.34
16’Er p
-15.53 -4.97
$JNPOL
-12.73 -4.65
<10-s -
10-4 -.: 1 o-5
The table describes for some proposed reactions the deviation from the pure Rutherford scattering cross section in percent. The columns denote the type of reaction, the projectile energy. Also the first and second order deviations as well as the total deviation are given in percent for a complete polarization and for a scattering angle of 180 degrees. It has been assumed that the particle beam is in the same direction as the direction of polarization. The last column gives the small deviations for an unpolarized target.
1.6
r
,
FIG. 7. The function F&v, 0) is plotted as a function of l/v” for different scattering angles 0. The strict linear dependence with l/v2 should be noted.
% -5.C ,-
-4.c
,-
-30
-2.C
I-
b
-1.c
,/
c 120
140
100
8
FIG. 8. The deviations from the pure Coulomb scattering for the 176Lu(~, ~r)“~Lu reaction with an a-energy of 14 MeV are plotted as a function of the scattering angle 8. The polarization of the target nuclei has been assumed to be complete. One curve shows the deviation in percent if the incoming particle beam is in direction of the polarization. The functions A(B), B(B), and C(B) describe the deviation in percent, according to Eq. (42), if the projectile beam is perpendicular to the direction of polarization.
RUTHERFORD SCATTERING
533
projectile, parallel or perpendicular to the direction of polarization have been considered. It has been assumed that the polarization is complete, i.e., that only the state with A4 = Z populated.
IV.
CONCLUSIONS
The results of this work show that for an unpolarized target the deviations from the pure Rutherford scattering are negligible provided that possible excitations into higher states are taken into account. If polarized or aligned targets are available, the quadrupole moments of the ground state can be measured due to the interference between the Coulomb amplitude and the first order amplitude. For the large quadrupole moments in the rare earth region, deviations of the order of 5 to 10 % are expected. The second order corrections of the first order term are, in contradiction to Clement and Kim, small and of the order of 10 ‘A of the first order term in some typical examples. It is noted that it is sufficient that the target is aligned. The largest effect is obtained if the incoming projectile moves in the direction of polarization. If the projectile moves perpendicularly to the polarization, the effect is roughly cut by a factor of two. On the other hand, an azimuthal dependence of cos 2or is expected for scattering angles different from 180 degrees. With such polarized targets it should therefore be possible to measure the quadrupole moments including the sign. The quadrupole moments in the ground state are usually known from spectroscopic data to about 20 % accuracy. The main source of uncertainty in extracting quadrupole moments from these experiments is the calculation of the quadrupole field induced by the atomic electrons. Reorientation experiments which are aimed at measuring the ground state quadrupole moments are also very difficult to analyze. Therefore, it seems that the method proposed in this work can supply independent information on ground state quadrupole moments. In spite of the experimental difficulties involved (smallness of the effect and the necessity to work with polarized targets or beams) such experiments can yield results of the same order of accuracy as the atomic spectroscopy methods and serve as a cross checking tooi. REFERENCES 1. A. BOHR, Mat. Fys. Me&. Danske Vid. Selsk. 26 (1952), p. 1. 2. A. BOHR AND B. R. MO~EL.WN, Mat. Fys. Medd. Danske Vid. Se/k. 27 (1953), p. 1. 3. B. R. MOTELSON, International Physics Conference, Copenhagen, June 1952, in “Coulomb Excitation”
(K. Alder and A. Winther, Eds.) Academic Press, New York, 1966, p. 11.
534
ALDER,
ROESEL,
SMILANSKY
4. U. SMILANSKY, Nucl. Phys. A 118 (1968), 529. 5. C. F. CLEMENT, Ann. Physics 56 (1970), 198. 6. Y. N. Kul, Phys. Rev. C 4 (1971), 650. 7. R. BECKER AND M. KLEBER, private communication. 8. N. F. Morr AND H. S. W. MOSSEY, “The Theory of Atomic Collisions,” Oxford University Press, London, 1949. 9. K. ALDER, F. R~SEL, AND R. MORF, Nucl. Phys. A 106 (1972), 449. 10. K. ALDER AND H. K. PAULI, Nucl. Phys. A 128 (1969), 193. 11. K. ALDER, A. BOHR, T. Huus, B. MOTTEL.Y.ON, AND A. WINTHER, Rev. Mod. Phys. 28 (1956), 432. 12. K. ALDER AND F. R&EL, private communication to be published. 13. J. DE BOER AND A. WINTHER, in “Coulomb Excitation” (K. Alder and A. Winther, Eds.) Academic Press, New York, 1966, p. 305.
OF PHYSICS
78, 518-534 (1973)
Deviations From the Rutherford Scattering Due To the Ground State Quadrupole Moment K. ALDER AND F. ROESEL Institute of Theoretical Physics, University of Basel, Switzerland
AND
U. SMILANSKY Department
of Nuclear Physics, Weizman Institute of Sciences, Rehovot, Israel
ReceivedOctober 29, 1971 The deviations from the pure Rutherford scattering due to the finite quadrupole moment in the ground state has been recalculated to second order in the quadrupole moment. It is shown that for unpolarized targets the deviations are small once the excitations into higher states are taken into account. The smallness of the effect is due to a strong cancellation between the two second order terms in the perturbation expansion. This fact was not realized by the authors who dealt previously with this problem. For polarized or aligned targets the hrst order amplitude contributes most of the effect and deviations of the order of 10 % are expected. Finally, a convenient parameterization of the deviations is presented.
I. INTRODUCTION It was recognized as early as 1952 by Bohr and Mottelson [l, 21 that large deformations can occur in nuclei. One of the earliest suggestions for detecting these large quadrupole moments was to look for deviations [3] from the Rutherford scattering cross section. However, for unpolarized targets the average first order term vanishes and the deviation from the pure Coulomb scattering is due to a second order term and thus proportional to the square of the quadrupole moment. One expects that these quadratic terms are small so that they can usually be neglected [4]. If the target nuclei are polarized the deviation will be proportional to the quadrupole moment in the ground state. It may be noted that for such an experiment a vector polarization is not necessary. It is sufficient that the target nuclei are aligned and have a nonvanishing tensor polarization so that deviations from the pure Rutherford scattering are obtained. 518 Copyright All rights
Q 1973 by Academic Press, Inc. of reproduction in any form reserved.
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519
In recent papers by C. F. Clement [5] and Y. N. Kim [6] it was pointed out that the magnitude of the quadratic scattering effect is of the same order as the first order effect. Their results show that particularly if heavy ions are used as projectiles the second order term is dominating. This would suggest a simple experimental determination of the quadrupole moments in the ground state by measuring the elastic cross section of an unpolarized target. We should like to point out that the above mentioned result is misleading and is due to the fact that the scattering amplitude was computed only up to the first order in the quadrupole moment. However, to obtain correct results for the cross section up to second order, the scattering amplitude has to be expanded to the same order. Then it turns out that the square of the first order terms nearly cancels the interference term between the Rutherford amplitude and the second order amplitude. (In the semiclassical limit this cancellation is even completel). This strong cancellation will therefore diminish strongly the importance of the second order deviations compared to the leading first order term. These results make experiments with unpolarized targets unfeasible and one must use polarized or aligned targets for a measurement of sign and magnitude of the quadrupole moment in the ground state. One should, however, keep in mind that there are other effects which give deviations from the pure Rutherford scattering. First, all the higher order contributions which we shall neglect in the following may become important, especially if heavy ions are used as projectiles. Second, the virtual excitation of higher lying nuclear states will also give rise to deviations of the scattering cross section. This virtual intermediate excitation can be considered as being due to the polarization of the nucleus. An estimate of this effect has recently been carried out [7] with the result that it does not give rise to an observable contribution. Third, real excitation will, of course, also alter the elastic scattering cross section. However, this contribution can be computed exactly and one may correct for it. Finally, such effects as the influence of the main nuclear potential or of the atomic electrons-which may also give small contributions -must be estimated. In the following we should like to present the results of our calculations. In Section II the theoretical considerations of the deviation from the Rutherford scattering are collected and a convenient parameterization is introduced. The next section presents the numerical results which are illustrated in figures and tables. These are applied to some important examples. The last section gives a summary of the conclusions which can be drawn from these new results, 1 C. F. Clement shows by dimensional analysis that the two quadratic terms have similar behavior as functions of the physical parameters which define the problem. But, since an exact evaluation was not carried out, he missed the fact that the two contributions actually cancel each other.
520
ALDER, ROESEL, SMILANSKY
II. THEORY The scattering process of a charged projectile in a pure Coulomb field can be solved exactly in a quanta1 treatment. The Coulomb scattering amplitude fc is given by [8] fc
=
e/2) ~-in10~sin*e/2+2io0,
-(a/2)(l/sin2
(1)
where a is half the distance of closest approach in a head-on collision, 8 the scattering angle in the center of mass system, 77the Coulomb strength parameter and u0 the Coulomb phase shift. These quantities are defined by the following relations: Z,Z2e21Av
(2)
a = ZlZ2e2/mv2
(3)
7j
=
and u. = arg r(l
+ i$,
(4)
where Z, and Z, are the charge numbers of target and projectile, respectively. The relative velocity is denoted by v and the reduced mass between target and projectile by m. We consider now a scattering process where the pure Rutherford scattering is disturbed by additional nuclear multipole fields. Usually, we can restrict ourselves to the influence of quadrupole fields only. The total scattering amplitude from the ground state j r&Vi) to a final state 1IfMf) is given by
where the second term in Eq. (5) is due to the influence of the electric multipole interaction. Spin and magnetic quantum number in the initial and final state are denoted by JiMi and I,Mf . To get the deviation from the pure Coulomb scattering we must, of course, compute the scattering amplitude for li = If = I. The additional scattering amplitude f can be computed in various ways. If the multipole interaction is sufficiently small, a perturbation expansion can be carried out [9]. For stronger interactions,fmay also be computed by means of the coupled channel method [lo] where care is taken automatically of the effects of virtual and real excitations. Finally, it is also possible to compute f in the semiclassical limit. In the following, we will mainly discuss the results of the perturbation expansion which we carried out up to the second order. However, let us first make some
RUTHERFORD
521
SCATTERING
qualitative investigations with the semiclassical limit. in this limit the amplitude is given by &+‘,M,
= fC(~IFMFKm)
It can be shown [lo] that
- ~,JFhfiM,)
where a ,,M,(ZJ4i) is the classical excitation amplitude computed in a system where the direction of the incoming projectile coincides with The x-axis is then conveniently chosen in the plane of the orbit with component of the projectile velocity. The second term in Eq. (6) cancels first term in Eq. (5) leading to
(6)
coordinate the z-axis. a positive exactly the
= f,a,,,u,UMih f 'iMi-+'FMF The elastic cross section from an initial number Mi is thus
(7)
nuclear state with magnetic quantum
(g),, = ($)RUthg Ia~,~,(ZiMd12 F
where the pure Rutherford
cross section is given by the well-known formula du/dQRUth = (l/4) a2 sin4( e/2).
(9)
The result of Eq. (8) can also be expressed in terms of the total excitation probability Pex by (dW2)el
= (d~/dQ),,t,(1
One can now express the total excitation probability semiclassical expression [ 131.
- Pex).
(IO)
P,, using the “symmetrized”
(11) where du,,/dQ is the differential Coulomb excitation cross section to the state Zf . v, is the asymptotic velocity of the projectile after leaving the target excited in state I. (da/d@&, is properly symmetrized Rutherford cross section corresponding to the excitation from the ground state Zi to the excited state It. Inserting Eq. (11) into Eq. (10) one obtains the effect of both real and virtual excitation of higher states on the elastic scattering cross section. In most cases of practical interest the inaccuracy introduced by using Eq. (11) is negligible even when compared to the small effect discussed here. Pex is typically of the order of 10 % and the semiclassical approximation deviates at most by 1 % from the exact result. Thus an upper limit for the error introduced by applying Eq. (11) would be 0.1 % as compared to a total effect of a few percent. 595/7W-15
522
ALDER,
ROESEL,
SMILANSKY
The result of Eqs. (10) and (11) is, however, changed for finite values of the Coulomb parameter 7. A finite, quanta1 interference between fc and j which is of the order I/T will become important. The classical amplitude 3 in Eq. (6) leading from the state 1ZM) to the same state, i.e.,
3,bf44 =
.Lh&w - 1) (12) will in a correct quanta1 treatment be changed in its absolute value and phase. We may therefore write fiM-IM
=~,(QGW
- 1) Cd+hd
(13) The factor C,, is of the order unity. This value is approached asymptotically as l/q2 for large values of T.The nonvanishing phase #IM which behaves as l/q for large values of v, gives rise to a strong interference and thus to a deviation from the pure Coulomb cross section. We thus get for the cross section from Eq. (7) x (1 + 2C,,Re((a,,(IM)
-1) ei”ZM) + 1C,, I21a,,(ZM)
--I I”). (14)
In first order perturbation theory, this result can be further simplified because we get for the excitation amplitude a,,(ZM) assuming only quadrupole interaction a,,(ZM)
- 1 = -i&1(21
+ l)lj2 (-lYM
(_‘,
0’ ;,
R,,(&
0)
(15)
where x,+{ is the quadrupole strength parameter which is connected with the reduced matrix element of the quadrupole operator &‘(E2) or with the quadrupole moment in the ground state by Xr I = (167W2
+
&WI
I-4’(E2)1 I 0
15fizXz2(21+ 1)1/Z
1 - __ 36
2,~~ Q ((Z + 1)(2Z + 3))‘/” - #iv -a2 (Z(2Z - 1))1/2 _
The quantity Rzo(f9, 0) is the classical orbital integral in the incident coordinate system discussed above. It can be expressed explicitly by elementary functions as follows R,,(B, 0) = 3(d5/4){Pg(O/2)(1
- tg(e/2) (77 - 8)/2) - sin2 (e/2)
From Eqs. (14) and (15) we obtain for the deviation scattering keeping terms only linear in xr+, (dddQ),l
=
cos
ej.
(17)
from the pure Coulomb
(d+%ltll
x 11 + 2xd2z
+ 01’2 (-Yf
( -zM 0” h) R,,(e, 0) sin 3bpo/.
RUTHERFORD
523
SCATTERING
Here we have chosen for the correction factor C,, = 1 which is actually very well fulfilled. It is also noted that the phase t,&,(e) is nearly independent of the scattering angle 8 and behaves as l/q as a function of 7. Thus, the phase $&B) can approximately be expressed by [ 121 (19)
sLzo(Q ‘v --0.6/71. Let us now express the semiclassical expansion. We get in second order qM(ZM)
= 1 - i~~-,~(--)‘-~ -
12 ZXId
Z
(
-M
;
amplitude
a,,(ZM)
(-zM
;j
;j’
;
in the perturbation
(2Z + l)1!a R,,
R;,(2Z $ 1).
(20)
I is then easily seen that in the main part of the semiclassical elastic cross section, i.e., in the expression Ifc 121~,.dz~)12,
(21)
the square of the second term cancels exactly the interference term between the first and the third term. This is the very point which Clement and Kim missed. It is, therefore, not permissible to leave out the second order term in Eq. (20). One expects that in a quanta1 treatment the third term in Eq. (20) must also be corrected in the absolute value and by a phase. This leads again to quantum mechanical correction terms which are proportional to x2 and behave as l/q2 for large values of 7). The above discussed strong cancellation is, however, also correct for finite values of 7). The scattering amplitude fliMi+,fA,, for a transition from the nuclear state I Z,Mi) to 1ZfMj) can be decomposed in contributions from different partial waves waves Ii , If and from different total angular momenta J. One gets [9, lo] hiMi
IRMA
=
(i/K)
. c (21, + l)lj2 izi-‘f(ZiOZ&Zi j JMi?
+I2
* (Zfm,ZfM,
where the Coulomb
1JMi){61i,,6,,L,
- ~i~uz~~n~~-u~~~~f~~~j:li’li2i) Y,,,,(o)
(22)
phase shift of the partial wave I is given by (~~(7) = arg Z?Z + 1 + iq)
(23)
The rapid oscillatory part of the s-matrix due to the Coulomb phases uzl + aEf has been removed leading to the rJ matrix in Eq. (22). This r-matrix can be computed
524
ALDER,
ROESEL,
SMILANSKY
by solving a set of coupled differential equations [lo] or by a perturbation expansion [9]. Keeping only quadratic terms in x, we get in the latter case
(24)
where the geometrical factor W is defined by
W:l.Il= 1 f
(3 G/2)(21
+ l)l’2 (24 + 1)“2 (21, + 1)1’2 (-I)J+’
($ ‘ot $1;
; ‘;1, (25)
while the dimensionless Coulomb function F,(h)
radial integral Z is expressed through the regular radial as follows
The symbol a, is the proper symmetrized expression for half the distance of closest approach in a head on collision, i.e., a, = Z,Z2f?2/FTlViVf.
(27)
A polarized or aligned ensemble of target nuclei can be characterized either by the density matrix = (ZM 1p [ ZM’) PMM’ (28) or by the corresponding
statistical tensor p&Z), defined by
pdl) = M;, (;
-‘Mt --kKjPWh44-)‘-“’ w + lY2.
(29)
The polarized target will usually be produced by cryogenic methods or by optical pumping. Therefore, in some definite direction the density matrix p will be diagonal and the diagonal elements denote just the occupation probability of the state 1ZM). However, this special direction will usually not coincide with the direction of the incoming projectile beam. The density matrix has therefore to be transformed from the original system to the incident beam system used earlier in this paper. By this transformation the density matrix will have the most general form of Eq. (28) and contain also nondiagonal matrix elements.
RUTHERFORD
525
SCATTERING
The elastic cross section can now be written in terms of the density matrix and scattering amplitude by
from the definition of the statistical tensor Eq. (29) and the expression of the excitation amplitude Eqs. (22) and (24) we obtain for the elastic scattering cross section up to the second order in the quadrupole moment
x (-)“’
+ (xd
pZKfy*fy(2k
+ l)l’2
(21 + 1) ; C-1”’
1;
‘I
k’\ (31)
The first term is due to the interference between the Coulomb scattering and the first order amplitude, the second term is the square of the first order correction, while the third term is due to the interference of the Coulomb amplitude with the second order excitation amplitude. For the first order amplitudef,“’ one obtains from Eqs. (22) and (24)
(32)
The second order amplitudef,‘,2’
is similarly given by
526
ALDER, ROESEL, SMILANSKY
Expressions (29), (30), and (31) reduce the problem to the computation of first order Coulomb excitation matrix elements. These matrix elements can, however, be computed by the well-known series expansion of hypergeometric function [I l] and by recursion relations. In Eq. (31), the sum over k includes the even value k = 0,2,4 as well as the odd values k = 1, 3. However, the contribution of the odd values is about a factor of 100 smaller than the corresponding contribution of even k. In the semiclassical limit, the contributions of the odd values of k vanish even exactly. We can, therefore, safely leave out these contributions and restrict ourselves to even k. It is then convenient to write the final expression in the following form
where the universal functions fza and F,, are defined by
fzK = (4 - 26,,) -& Re(f,!)f,*)
(35)
and FkK = (4 - 26,,) 1; Re (1
K’K”
(-)“’
(_“,,
z,, tj
f$*j”‘(2k
+ 1)1’2)
They depend only on the scattering angle 0 and the Coulomb parameter r). It is noted that the deviation from the pure Coulomb scattering depends only on the statistical tensors with even k, that is on the tensor polarization. It is, therefore, not necessary to use polarized targets for such experiments. An aligned target can be used and will yield the same information.
III.
NUMERICAL
RESULTS
The deviations of the elastic cross section due to a finite quadrupole moment are given by Eq. (34). Here, all quantities are expressed with respect to the incident coordinate system, the z-axis being in the direction of the incoming projectile and the x-axis in the plane of the orbit. Let us consider the case that by cryogenic methods in a certain z’-direction we
RUTHERFORD
527
SCATTERING
get different population probabilities PM for the states with magnetic quantum number M. The statistical tensor with respect to the (x’y’z’) system is then given by
PkK=
+ lY ; (-I’-”
h3w
(;
-‘zM “,) PM = b17, .
(37)
The statistical tensor transforms under a rotation R in the same way as the spherical harmonics, and it is therefore easy to compute the statistical tensor in any new system. We get P Lx ~- D:,(R) 17, .
(38)
The D-function in Eq. (38) can be expressed by a spherical harmonic depending on the Eulerian angles 01,/3, y of the rotation. The statistical tensor can then be written
Two special experimental situations are apparent. The incoming beam might be in direction of the original direction of polarization or it might be perpendicular to it. For these two special cases, we obtain for the statistical tensor pk,c
=
PkK
=
(40)
6,,17k
or 2k+1 4ir
-lj2 1
Y&I/2,
0) PY7~
(41)
where 01is the azimuthal angle of the initial polarization direction with respect to the x-axis. In order to compute the deviation from the Rutherford cross section, it is necessary to know the functions f2K and F,, as a function of the scattering angle 8 and of 7. It is expected that f2K asymptotically behave as 1/q and 1/T” as a function of 7. These functions have been computed numerically and are illustrated in different figures. In Fig. 1, the function f2Jrj, 0) is illustrated as a function of the scattering angle for ~7= 4 and r) = 8. The Figs. 2, 3, and 4 demonstrate the strick l/q dependence of the function f2K(q, 6) for K = 0, 1, and 2. The quadratic term proportional to x2 is in our considerations of minor importance. In practice the contribution of this term is at most of the order of 10 % of the linear terms. In order to compute the necessary corrections, the functions F2K and FdKare given in Figs. 5 and 6 as a function of the scattering angle for 7 = 4 and 7 = 8. These functions can be obtained for other 17 values by interpolating with the help of the I/r12-rule. The applicability of this rule is illustrated in Fig. 7
7-4
/ -
-.25
f*o
-.20
-.I5
-.I0
-.05
120
140
160
8
FIG. 1. The function f&v, 0) is shown as a function of the scattering angle 8 and for 7 = 4 and 7 = 8. The different values of K are indicated.
-. 4
-.3
-. 2
-.I
l/l2
l/6
l/4
l/3
+i
The functionf,,(q, 0) is shown as a function of 1/T for different values of the scattering angle 0. The strict linear dependence with l/7 should be noted. FIG.
2.
529
RUTHERFORD SCATTERING
e
f 21 .3
-
.2
-
IZOO loo0 140°
160°
t l/12
FIG. 3. The functionf,,(T, angle 8.
l/a
l/4
113
i
YT
0) is shown as a function of 1/T for different values of the scattering
160'
FIG. 4. The functionf,,(v, angle e.
0) is shown as a function of l/v for different values of the scattering
530
ALDER,
ROESEL,
SMILANSKY
where Fdo is shown for different scattering angles as a function of l/+. It may be noted that Foe and Fk, for odd k are very small and can be safely neglected in our considerations. In Table I, the order of magnitude of the effect is demonstrated for some typical examples. The first and second order deviations in percent are listed together with the total effect. It may be noted that for unpolarized targets the total effect
120
140
160
8
FIG. 5. The function FsK(q, 0) is shown as a function of the scattering angle 0 for 7 = 4 and 7 = 8. The different values of K and the sign, with which it should be multiplied are indicated.
practically vanishes. Only the quadratic terms with k = 0 will contribute and give deviations of the order of 1O-4 to 10-5. If the incoming projectile is perpendicular to the initial direction of polarization, we expect, according to Eq. (41), an azimuthal dependence of the scattering cross section of the form da do [I + A@) + W) dsz = t-1dL? Ruth
cos 201+ c(e) cos 4a],
(42)
where the last term with the cos ~CXdependence originates from the second order effect and is expected to be small. For a possible experiment 175L~(~, a) 175Lu, the deviations in percent are shown in Fig. 8. Both possibilities for the incoming
RUTHERFORD
531
SCATTERING
0.8
0.6
0.4
0. 2
0.
J 6. The function F&T, 6) is shown as a function of the scattering angle 0 for 7 =m-4 and 7 = 8. The different values of K and the sign, with which it should be multiplied are indicated. FIG.
TABLE
Target and projectile
Energy (MeV)
Q barns
13’Gd p
11.4
2.0
6.4
I
oW
“I”,
-7.31 -2.34
0.004
0.0001
o’Gl
12.8 7.2
4.0
l=Eu
11.0
2.5
-2.21
0.05
-2.16
“SLU
14.0
5.68
-5.70
0.49
-5.21
-___-
2.80 0.32
,;: 10-S
-7.31 -2.34
16’Er p
-15.53 -4.97
$JNPOL
-12.73 -4.65
<10-s -
10-4 -.: 1 o-5
The table describes for some proposed reactions the deviation from the pure Rutherford scattering cross section in percent. The columns denote the type of reaction, the projectile energy. Also the first and second order deviations as well as the total deviation are given in percent for a complete polarization and for a scattering angle of 180 degrees. It has been assumed that the particle beam is in the same direction as the direction of polarization. The last column gives the small deviations for an unpolarized target.
1.6
r
,
FIG. 7. The function F&v, 0) is plotted as a function of l/v” for different scattering angles 0. The strict linear dependence with l/v2 should be noted.
% -5.C ,-
-4.c
,-
-30
-2.C
I-
b
-1.c
,/
c 120
140
100
8
FIG. 8. The deviations from the pure Coulomb scattering for the 176Lu(~, ~r)“~Lu reaction with an a-energy of 14 MeV are plotted as a function of the scattering angle 8. The polarization of the target nuclei has been assumed to be complete. One curve shows the deviation in percent if the incoming particle beam is in direction of the polarization. The functions A(B), B(B), and C(B) describe the deviation in percent, according to Eq. (42), if the projectile beam is perpendicular to the direction of polarization.
RUTHERFORD SCATTERING
533
projectile, parallel or perpendicular to the direction of polarization have been considered. It has been assumed that the polarization is complete, i.e., that only the state with A4 = Z populated.
IV.
CONCLUSIONS
The results of this work show that for an unpolarized target the deviations from the pure Rutherford scattering are negligible provided that possible excitations into higher states are taken into account. If polarized or aligned targets are available, the quadrupole moments of the ground state can be measured due to the interference between the Coulomb amplitude and the first order amplitude. For the large quadrupole moments in the rare earth region, deviations of the order of 5 to 10 % are expected. The second order corrections of the first order term are, in contradiction to Clement and Kim, small and of the order of 10 ‘A of the first order term in some typical examples. It is noted that it is sufficient that the target is aligned. The largest effect is obtained if the incoming projectile moves in the direction of polarization. If the projectile moves perpendicularly to the polarization, the effect is roughly cut by a factor of two. On the other hand, an azimuthal dependence of cos 2or is expected for scattering angles different from 180 degrees. With such polarized targets it should therefore be possible to measure the quadrupole moments including the sign. The quadrupole moments in the ground state are usually known from spectroscopic data to about 20 % accuracy. The main source of uncertainty in extracting quadrupole moments from these experiments is the calculation of the quadrupole field induced by the atomic electrons. Reorientation experiments which are aimed at measuring the ground state quadrupole moments are also very difficult to analyze. Therefore, it seems that the method proposed in this work can supply independent information on ground state quadrupole moments. In spite of the experimental difficulties involved (smallness of the effect and the necessity to work with polarized targets or beams) such experiments can yield results of the same order of accuracy as the atomic spectroscopy methods and serve as a cross checking tooi. REFERENCES 1. A. BOHR, Mat. Fys. Me&. Danske Vid. Selsk. 26 (1952), p. 1. 2. A. BOHR AND B. R. MO~EL.WN, Mat. Fys. Medd. Danske Vid. Se/k. 27 (1953), p. 1. 3. B. R. MOTELSON, International Physics Conference, Copenhagen, June 1952, in “Coulomb Excitation”
(K. Alder and A. Winther, Eds.) Academic Press, New York, 1966, p. 11.
534
ALDER,
ROESEL,
SMILANSKY
4. U. SMILANSKY, Nucl. Phys. A 118 (1968), 529. 5. C. F. CLEMENT, Ann. Physics 56 (1970), 198. 6. Y. N. Kul, Phys. Rev. C 4 (1971), 650. 7. R. BECKER AND M. KLEBER, private communication. 8. N. F. Morr AND H. S. W. MOSSEY, “The Theory of Atomic Collisions,” Oxford University Press, London, 1949. 9. K. ALDER, F. R~SEL, AND R. MORF, Nucl. Phys. A 106 (1972), 449. 10. K. ALDER AND H. K. PAULI, Nucl. Phys. A 128 (1969), 193. 11. K. ALDER, A. BOHR, T. Huus, B. MOTTEL.Y.ON, AND A. WINTHER, Rev. Mod. Phys. 28 (1956), 432. 12. K. ALDER AND F. R&EL, private communication to be published. 13. J. DE BOER AND A. WINTHER, in “Coulomb Excitation” (K. Alder and A. Winther, Eds.) Academic Press, New York, 1966, p. 305.