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Angular correlations for E2 and E3 transitions in Coulomb excitation
[
2~.B
Nuclear Physics 85 (1966) 537--544; ( ~ North-Holland Publishin# Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
A N G U L A R C O R R E L A T I O N S F O R E2 AND E3 T R A N S I T I O N S IN COULOMB EXCITATION R. J. KEDDY Nuclear Physics Research Unit, University of the Witwatersrand, Johannesbury, South Africa and Y. YOSHIZAWA Faculty of Science, University of Osaka, Toyonaka, Japan Received 17 March 1966
Abstract: A first-order perturbation theory calculation is made of the particle-gamma angular correlation coefficients used in Coulomb excitation. Both the E2 and E3 cases are treated and are separately compared to the results obtained from the sudden approximation theory for tA2/tAO = O. 1. Introduction Coulomb excitation, whereby a target nucleus is excited due to the interaction of its own electromagnetic field with the electromagnetic field of an accelerated charged particle, can be conveniently investigated by detecting the g a m m a photon emitted due to the decay of the excited nucleus in coincidence with the inelastically scattered particle which produced the excitation. I f absolute Coulomb excitation yields are to be determined from such experiments it is necessary to establish the angular correlation between the inelastically scattered particle and the g a m m a ray. General discussions of the Coulomb excitation process have been presented in a number of review articles 1, z) but it is ref. 1) to which we shall refer as A B H M W , which contains most of the pertinent theory, particularly with regard to first- and second-order perturbation arguments. Angular correlations have been treated by first-order methods only. 2. Theory I f we consider the target nucleus prior to excitation to be in state Ii and that it decays to state Iff after being Coulomb excited to state If (fig. 1), then the angular distribution of the emitted g a m m a radiation can be described by W0p,~Op(QT) =
~
1~-~,bif(lff~/lffln~(Qr, tr)]IfMf)l 2,
(1)
f f M i M f f Mr
where bif are the excitation amplitudes from state I i to If and the other various parameters are defined in A B H M W . I f in eq. (1) one expresses the bif in the matrix 537
538
R.J.
KEDDY AND Y. YOSHIZAWA
form and considers the excitation to take place by a transition of pure multipole order 2, then we may write for the angular distribution of the gamma rays
Wop, ~,p(~2~) = Z a~(Ov, q~p, ¢)A~k~)rk~(O~),
(2)
kr
where A~ka) are the usual gamma-gamma angular distribution coefficients which would describe the cascade Ii --+ If ~ Iff and ak~ which are independent of the nuclear states If
E>,
EL
Iff
Fig. 1. E x c i t a t i o n a n d d e c a y m o d e s .
involved, are the particle - gamma correlation coefficients. These correlation coefficients may be further defined in the form
a (op,
4) =
bk~/boo,
(3)
where, if we express the angular distribution of the gamma quanta with reference to a Z-axis in the direction of the incident beam of particles
x Yz,,(½7r,0)Iz,(0p, ~)Iz~,(0p, ¢)Dk,,(½7~+ ½0p, ½7z,tpp).
(4)
In eq. (4) the Yz~(½7r,0) are as defined in ABHMW; the Iz,(O, ~) which we shall abbreviate to Iz~ are the orbital integrals tabulated by Alder and Winther 3) and the k are the rotational matrices. After calculating the spherical harmonics and D~,~ rotational matrices and substituting the results into eq. (4) we are able to write the gamma-ray angular distribution expression in terms of the orbital integrals. 3. Results
The particle and gamma-ray angles are indicated pictorially in fig. 2.
539
COULOMB EXCITATION
3.1. THE E2 CALCULATIONS We expand eq. (2) to
W(Ov, q)r) = l + A(22) E aZ2,,Y2,,+ A~2, ~ aa,, 2K
(5)
4x
by assuming that the level Iff is also the level Ii. The expressions from eqs. (3) and (4) are substituted into (5) where the form of the matrix d2,m(½rc)required for the second term of eq. (5) has been published 4) and the matrix d4,m(½7r) is presented as an appendix to this paper. We are able then to express the gamma-ray angular distribution in terms of the orbital integrals, Legendre polynomials and associated Legendre polynomials.
Fig. 2. Detector geometry.
a = 120"
/"
Fig. 3. Particle detector arrangement. It can be shown that if we have for the detection of the particles an annular counter o r a system of three or five or more evenly spaced counters around the path of the
540
R, J. KEDDY
AND
Y. Y O S H I Z A W A
incoming beam of particles as in fig. 3 that if we integrate over the particle counting system around the beam direction then all the terms containing the associated Legendre polynomials reduce to zero and we eliminate the cpv and the ~0pdependence.
0
1.5--
1.0
0,5
0,C
/
-0.5,
/
I
-,ct 20 40 60 80 100 120 140 160 180 ep Fig. 4. Particle-gamma angular correlation coefficients a~(0p, ~). Note that a system of four counters around the beam direction is excluded. Without the associated Legendre polynomials the angular distribution reduce~ to the simple form
Vvo~,(O~) = 1 + A(22)a~ P2(cos Or) + A~2)a~ P4(cos Or),
(6)
where a~ and a~ are the particle-gamma angular correlation coefficients and are given
541
COULOMB EXCITATION
by 2
2
a 2 = 3122 -- 2/20 + 312- 2 -- 6(122 I20 + I2o I2- 2) cos 0p 2
2
2
3•22 + 2•20 + 3 1 2 - 2
a2=_
9(I22 q-4120+12_2)--60(I2212o + I2oI2_2)COS 0p+21012212_ 2 COS 20p 32(3122 + 21220+ 312- 2)
(7)
These coefficients computed by the IBM 1620 computer of the University of the Witwatersrand are plotted in figs. 4 and 5. All the orbital integral values as
K
I'I'I'I I l'l'l'll
o.5
:"
I!
",%
o.c
4.0 /
/
/
-05_ O.
;.0
°\\2,o
- 1.C
- 1 . =_
20
Fig. 5.
40
Particle-gamma
60
80 angular
100
120
correlation
140
160
coefficients
180
Op
a42(0p, ~).
published by Alder and Winther a) have been used so that both coefficients vary with 0 v and 4. 3.2. THE E3 CALCULATIONS
The assumption made here is that we have pure octupole excitation and pure
R. J. KEDDY AND Y.
542
YOSHIZAWA
dipole de-excitation and the expansion of eq. (2) leads to
W(O~,q~,) = l+A(3) E a3~y2~.
(8)
2•
With similar substitutions and a particle detecting system as in the E2 case the distribution simplifies to
Wo,(O,) = 1 + A~z3)a]P2(cos 0,), where the correlation coefficient is a 3
25(123 +I32_3)- 9(I32t+I~_1)-(3613113_1+30(133131 +13-113-3)) i 8 ( I ] 1 + 132 - 1) + 30(133 2 + I32 3)
cos 0p.
(9)
These coefficients again calculated on the 1620 computer are shown in fig, 6. All Alder and Winther's Is~(0, ¢) values have been used as input data.
' ['
II
I'1'
I'111'l
'
o~ k©
m O.~_
OC
/
/
/
/
J•/
/
40
60
-05
G¢
20 F i g . 6.
80
10o
120
140
360
Particle-gamma angular correlation coefficients
180
Op
a2a(0p, ~).
4. The Sudden Approximation Theory Alder and Winther s) have shown, that if the Coulomb reaction between the initial and final states characterized by spin and magnetic quantum number Ii, Mi
COULOMB EXCITATION
543
and If, Mf has the character of a sudden impact and that the impinging particle has zero spin, then simple trigonometrical expressions depending upon the matrix element (IfMfkflTII~Miki) give the particle-gamma angular correlation coefficients. Here T is the reaction matrix and ki, kf the wave numbers of the projectile in the initial and final states. Further, the matrix elements of T can be expanded in multipole components
(If Mr kf] TII i M~ k~) = ~ (lfMf AI~]I~Mi)tA~, Aft
where taa = (2A + 1)(2Ii + 1) -1 ~ (If Mr A.ulIi Mi)(IfMf kr[ TII~ M i kl), MiMe
and (lfMfA#[Ii Mi) is a Clebsch-Gordan coefficient. I f it can be shown that tau = 0 e.g. 0p = 0 ° or 180 ° and ~t # 0 we arrive at particularly simple expressions for the correlation coefficients defined as Gk/Go E2 Gz/G o = ½(1 - 3 cos 0p),
G,/Go = - 1 3 8 ( 9 - 2 0 cos 0p+35 cos 20p), E3 G2/Go = ½ ( 1 - 3 cos 0p). For 0p -- 180 ° it is noted that all the calculated results agree with the sudden approximation values i.e. for E2 excitation a 2 = 2.0 and a 2 = 1.5 and for E3 excitation aza = 1.33. In figs. 4-6 the sudden approximation results for ta~ = 0 are also included.
5. Conclusion
Figs. 4-6 indicate that for ~ < 1 the error introduced in the correlation coefficient values for particle angles between about 0p = 120 ° a n d 180 ° will not be greater than 10 70 if the simple expression of Alder and Winther for t t ~ = 0 is used. For other values of ~ and 0p however, the error can be much greater and the sudden approximation results should be used with discretion.
544
R. J. KEDDY AND Y. YOSHIZAWA
Appendix The matrix
m'/m 4
d4m,m(½~) 4
3
2
~ :--:
,/2 T
3 2 1 0 -- I
-2
,/7
,/14
T
,/14
8
,/7___00 16 ,/14
-8-
4
-1
-2
,/7
,/14
,/70
,/14
,/7
,/2
T
-T-
16
8
T
T
,/7
0
,/7
,/14
1 4#2
- - -4,/2
,/14
¼
_,/7 -
8
-
0 ,/7
~`/t4
--~-
_,/2 8
_~
t
_d2 T
1--~
0
_~
T
`/7 T
-3
-
,/2
1
1
442 -
-
,/5 4,/2 1
4,/~ ¼ ,/14 8 ,/7 8-
45
1 4,/2
-3
-4
,/2 ,/14
¼ 1
- -8
,/7 8-
,/7 8
414 T
0
,/70 16
~
0
-~
442
0
~
0
,/5 442
-~
0
~
4,/2
_ 45 4`/---i
1 4`/2
¼
_4__7 8
`/14 8
_~
,/14 8
47 "8-
,/2 ~-
1 44--i 4_._77 8
0
V14 -8-
d70 16
1
1 z-:
-
-
47
8
,/14 8
,/14 8
,/7 T `/2 8 1 1-~
References 1) K. Alder et a/., Revs. Mod. Phys. 28 (1956) 43i 2) P. H. Stelson and F. K. McGowan, Ann. Rev. Nucl. Sci. 13 (1963) 163 3) K. Alder and A. Winther, Mat. Fys. Medd.. Dan. Vid. Selsk. 31, No. 1 (1956) 4) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, N.J., 1957) p. 129 5) K. Alder and A. Winther, Nuclear Physics 37 (1962) 194
2~.B
Nuclear Physics 85 (1966) 537--544; ( ~ North-Holland Publishin# Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
A N G U L A R C O R R E L A T I O N S F O R E2 AND E3 T R A N S I T I O N S IN COULOMB EXCITATION R. J. KEDDY Nuclear Physics Research Unit, University of the Witwatersrand, Johannesbury, South Africa and Y. YOSHIZAWA Faculty of Science, University of Osaka, Toyonaka, Japan Received 17 March 1966
Abstract: A first-order perturbation theory calculation is made of the particle-gamma angular correlation coefficients used in Coulomb excitation. Both the E2 and E3 cases are treated and are separately compared to the results obtained from the sudden approximation theory for tA2/tAO = O. 1. Introduction Coulomb excitation, whereby a target nucleus is excited due to the interaction of its own electromagnetic field with the electromagnetic field of an accelerated charged particle, can be conveniently investigated by detecting the g a m m a photon emitted due to the decay of the excited nucleus in coincidence with the inelastically scattered particle which produced the excitation. I f absolute Coulomb excitation yields are to be determined from such experiments it is necessary to establish the angular correlation between the inelastically scattered particle and the g a m m a ray. General discussions of the Coulomb excitation process have been presented in a number of review articles 1, z) but it is ref. 1) to which we shall refer as A B H M W , which contains most of the pertinent theory, particularly with regard to first- and second-order perturbation arguments. Angular correlations have been treated by first-order methods only. 2. Theory I f we consider the target nucleus prior to excitation to be in state Ii and that it decays to state Iff after being Coulomb excited to state If (fig. 1), then the angular distribution of the emitted g a m m a radiation can be described by W0p,~Op(QT) =
~
1~-~,bif(lff~/lffln~(Qr, tr)]IfMf)l 2,
(1)
f f M i M f f Mr
where bif are the excitation amplitudes from state I i to If and the other various parameters are defined in A B H M W . I f in eq. (1) one expresses the bif in the matrix 537
538
R.J.
KEDDY AND Y. YOSHIZAWA
form and considers the excitation to take place by a transition of pure multipole order 2, then we may write for the angular distribution of the gamma rays
Wop, ~,p(~2~) = Z a~(Ov, q~p, ¢)A~k~)rk~(O~),
(2)
kr
where A~ka) are the usual gamma-gamma angular distribution coefficients which would describe the cascade Ii --+ If ~ Iff and ak~ which are independent of the nuclear states If
E>,
EL
Iff
Fig. 1. E x c i t a t i o n a n d d e c a y m o d e s .
involved, are the particle - gamma correlation coefficients. These correlation coefficients may be further defined in the form
a (op,
4) =
bk~/boo,
(3)
where, if we express the angular distribution of the gamma quanta with reference to a Z-axis in the direction of the incident beam of particles
x Yz,,(½7r,0)Iz,(0p, ~)Iz~,(0p, ¢)Dk,,(½7~+ ½0p, ½7z,tpp).
(4)
In eq. (4) the Yz~(½7r,0) are as defined in ABHMW; the Iz,(O, ~) which we shall abbreviate to Iz~ are the orbital integrals tabulated by Alder and Winther 3) and the k are the rotational matrices. After calculating the spherical harmonics and D~,~ rotational matrices and substituting the results into eq. (4) we are able to write the gamma-ray angular distribution expression in terms of the orbital integrals. 3. Results
The particle and gamma-ray angles are indicated pictorially in fig. 2.
539
COULOMB EXCITATION
3.1. THE E2 CALCULATIONS We expand eq. (2) to
W(Ov, q)r) = l + A(22) E aZ2,,Y2,,+ A~2, ~ aa,, 2K
(5)
4x
by assuming that the level Iff is also the level Ii. The expressions from eqs. (3) and (4) are substituted into (5) where the form of the matrix d2,m(½rc)required for the second term of eq. (5) has been published 4) and the matrix d4,m(½7r) is presented as an appendix to this paper. We are able then to express the gamma-ray angular distribution in terms of the orbital integrals, Legendre polynomials and associated Legendre polynomials.
Fig. 2. Detector geometry.
a = 120"
/"
Fig. 3. Particle detector arrangement. It can be shown that if we have for the detection of the particles an annular counter o r a system of three or five or more evenly spaced counters around the path of the
540
R, J. KEDDY
AND
Y. Y O S H I Z A W A
incoming beam of particles as in fig. 3 that if we integrate over the particle counting system around the beam direction then all the terms containing the associated Legendre polynomials reduce to zero and we eliminate the cpv and the ~0pdependence.
0
1.5--
1.0
0,5
0,C
/
-0.5,
/
I
-,ct 20 40 60 80 100 120 140 160 180 ep Fig. 4. Particle-gamma angular correlation coefficients a~(0p, ~). Note that a system of four counters around the beam direction is excluded. Without the associated Legendre polynomials the angular distribution reduce~ to the simple form
Vvo~,(O~) = 1 + A(22)a~ P2(cos Or) + A~2)a~ P4(cos Or),
(6)
where a~ and a~ are the particle-gamma angular correlation coefficients and are given
541
COULOMB EXCITATION
by 2
2
a 2 = 3122 -- 2/20 + 312- 2 -- 6(122 I20 + I2o I2- 2) cos 0p 2
2
2
3•22 + 2•20 + 3 1 2 - 2
a2=_
9(I22 q-4120+12_2)--60(I2212o + I2oI2_2)COS 0p+21012212_ 2 COS 20p 32(3122 + 21220+ 312- 2)
(7)
These coefficients computed by the IBM 1620 computer of the University of the Witwatersrand are plotted in figs. 4 and 5. All the orbital integral values as
K
I'I'I'I I l'l'l'll
o.5
:"
I!
",%
o.c
4.0 /
/
/
-05_ O.
;.0
°\\2,o
- 1.C
- 1 . =_
20
Fig. 5.
40
Particle-gamma
60
80 angular
100
120
correlation
140
160
coefficients
180
Op
a42(0p, ~).
published by Alder and Winther a) have been used so that both coefficients vary with 0 v and 4. 3.2. THE E3 CALCULATIONS
The assumption made here is that we have pure octupole excitation and pure
R. J. KEDDY AND Y.
542
YOSHIZAWA
dipole de-excitation and the expansion of eq. (2) leads to
W(O~,q~,) = l+A(3) E a3~y2~.
(8)
2•
With similar substitutions and a particle detecting system as in the E2 case the distribution simplifies to
Wo,(O,) = 1 + A~z3)a]P2(cos 0,), where the correlation coefficient is a 3
25(123 +I32_3)- 9(I32t+I~_1)-(3613113_1+30(133131 +13-113-3)) i 8 ( I ] 1 + 132 - 1) + 30(133 2 + I32 3)
cos 0p.
(9)
These coefficients again calculated on the 1620 computer are shown in fig, 6. All Alder and Winther's Is~(0, ¢) values have been used as input data.
' ['
II
I'1'
I'111'l
'
o~ k©
m O.~_
OC
/
/
/
/
J•/
/
40
60
-05
G¢
20 F i g . 6.
80
10o
120
140
360
Particle-gamma angular correlation coefficients
180
Op
a2a(0p, ~).
4. The Sudden Approximation Theory Alder and Winther s) have shown, that if the Coulomb reaction between the initial and final states characterized by spin and magnetic quantum number Ii, Mi
COULOMB EXCITATION
543
and If, Mf has the character of a sudden impact and that the impinging particle has zero spin, then simple trigonometrical expressions depending upon the matrix element (IfMfkflTII~Miki) give the particle-gamma angular correlation coefficients. Here T is the reaction matrix and ki, kf the wave numbers of the projectile in the initial and final states. Further, the matrix elements of T can be expanded in multipole components
(If Mr kf] TII i M~ k~) = ~ (lfMf AI~]I~Mi)tA~, Aft
where taa = (2A + 1)(2Ii + 1) -1 ~ (If Mr A.ulIi Mi)(IfMf kr[ TII~ M i kl), MiMe
and (lfMfA#[Ii Mi) is a Clebsch-Gordan coefficient. I f it can be shown that tau = 0 e.g. 0p = 0 ° or 180 ° and ~t # 0 we arrive at particularly simple expressions for the correlation coefficients defined as Gk/Go E2 Gz/G o = ½(1 - 3 cos 0p),
G,/Go = - 1 3 8 ( 9 - 2 0 cos 0p+35 cos 20p), E3 G2/Go = ½ ( 1 - 3 cos 0p). For 0p -- 180 ° it is noted that all the calculated results agree with the sudden approximation values i.e. for E2 excitation a 2 = 2.0 and a 2 = 1.5 and for E3 excitation aza = 1.33. In figs. 4-6 the sudden approximation results for ta~ = 0 are also included.
5. Conclusion
Figs. 4-6 indicate that for ~ < 1 the error introduced in the correlation coefficient values for particle angles between about 0p = 120 ° a n d 180 ° will not be greater than 10 70 if the simple expression of Alder and Winther for t t ~ = 0 is used. For other values of ~ and 0p however, the error can be much greater and the sudden approximation results should be used with discretion.
544
R. J. KEDDY AND Y. YOSHIZAWA
Appendix The matrix
m'/m 4
d4m,m(½~) 4
3
2
~ :--:
,/2 T
3 2 1 0 -- I
-2
,/7
,/14
T
,/14
8
,/7___00 16 ,/14
-8-
4
-1
-2
,/7
,/14
,/70
,/14
,/7
,/2
T
-T-
16
8
T
T
,/7
0
,/7
,/14
1 4#2
- - -4,/2
,/14
¼
_,/7 -
8
-
0 ,/7
~`/t4
--~-
_,/2 8
_~
t
_d2 T
1--~
0
_~
T
`/7 T
-3
-
,/2
1
1
442 -
-
,/5 4,/2 1
4,/~ ¼ ,/14 8 ,/7 8-
45
1 4,/2
-3
-4
,/2 ,/14
¼ 1
- -8
,/7 8-
,/7 8
414 T
0
,/70 16
~
0
-~
442
0
~
0
,/5 442
-~
0
~
4,/2
_ 45 4`/---i
1 4`/2
¼
_4__7 8
`/14 8
_~
,/14 8
47 "8-
,/2 ~-
1 44--i 4_._77 8
0
V14 -8-
d70 16
1
1 z-:
-
-
47
8
,/14 8
,/14 8
,/7 T `/2 8 1 1-~
References 1) K. Alder et a/., Revs. Mod. Phys. 28 (1956) 43i 2) P. H. Stelson and F. K. McGowan, Ann. Rev. Nucl. Sci. 13 (1963) 163 3) K. Alder and A. Winther, Mat. Fys. Medd.. Dan. Vid. Selsk. 31, No. 1 (1956) 4) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, N.J., 1957) p. 129 5) K. Alder and A. Winther, Nuclear Physics 37 (1962) 194