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Gamma-ray angular distribution emitted after excitation of nuclei
3.A
I
Nuclear Physics 81 (1966) 215--219; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
GAMMA-RAY A N G U L A R DISTRIBUTION E M I T T E D AFTER EXCITATION OF NUCLEI V. A. D J R B A S H I A N
Institute of Physics, Yerevan, USSR Received 13 September 1965
Abstract: Formulae are derived for the angular distribution of ~-rays following the electromagnetic
excitation of nuclei by particles with arbitrary spin.
1. Introduction
The quantal calculation of orbital electromagnetic excitation is presented by Biedenharn, McHale and Thaler 1) and by Alder et al. z), where together with the total cross sections, the angular distributions of the emitted y-rays for pure multipole excitations are given. However, in the angular distribution of y-rays, unlike in the total cross sections, the possibility of mixed electromagnetic excitation arises. For the particular case of M1-E2 this is considered by Biedenharn and Thaler 3). They have also shown that in case of an M1 transition and particle spin ½, the contribution of the spin magnetic moment of the particle together with the orbital magnetic moment is significant. Naturally, this contribution will have more significance for particles with spin i > ½. Previously 4), we have obtained cross sections of electromagnetic excitation taking into account the contribution of the nuclear forces as well as the spin magnetic moment of the particle with an arbitrary spin. It has been established that the cross section of the 2 ~ pole magnetic spin excitation is proportional to i(i + 1)#~, where 2Z 1 #p i is the spin magnetic moment of the particle (in nuclear magnetons) and is expressed in terms of the cross section of the 2 ~+ 1 pole electric excitation. In the present paper it is shown that an analogous situation arises when considering the angular distribution of y-rays. Expressions are given which take into account the multipole excitation due to the spin magnetic moment of the particle with an arbirary spin together with the mixed multipole orbital excitation. Various particular cases are considered. For the mixture M1-E2 an expression is obtained, which differs by a coefficient ½ from the corresponding expression of Biedenharn and Thaler 3). For the case of a pure M1 transition and for particle spin ½, the particle parameters b~ Spin~ coincide 4) with the ones obtained in ref. 3) at v = 0 and differ from them by a coefficient ¼ at v = 2. 215
216
V. A. DJRBASHIAN 2. General Formulae
The angular distribution of the y-quanta emitted at the de-excitation of nuclei may be written in the form
W(8v) = ~
(I'M~IE(2)II'M'~XI'MiIV(*)II'MI),
(2.1)
M'IM' 2
where 8~ is the angle between the directions of the incident particle and the emitted y-quantum. The spin of the nucleus and its projection are denoted by I and M, respectively. Here and hereafter the unprimed, single primed and double primed magnitudes denote the initial, the intermediate (when the nucleus is excited) and the final states of the system, respectively. The matrix elements (I'M~IV(1)II'M~) and (I'M.~[E(2)II'M;) describe the electromagnetic excitation of the nucleus of the particle with an arbitrary spin and the emission of a y-quantum, respectively: , , I, V (IM
cl)
~ v~.i,M l i ,M a, ) =JM~W<,
., .#.~. . iVlllM#k)(I'M'2#'k']VlilM#k)*df2,k ,,
(I'M'2IE(Z)II'MI) = J ~, (I'M'2IH2II"M")(I'MI]H2II"M")*.
(2.2) (2.3)
M
In eq. (2.2), # and k are the projection of the spin i and the wave vector of the particle. The matrix elements under the integral over the solid angle f2k,,, produced by the rotation of the final direction of the particle around the initial one, are given in ref. 4). Substituting the expressions (2.2)-(2.4) and (2.10) of ref. 4) into eq. (2.2), we obtain
(I'UilVmlZ'Mi) _-
~ ( - 1 ) ~-z+M''+*2 16n 2 ~.,~z,&k,~m,',,**~ (2)., + 1)(23~z + 1)
x ( - 1) m -a.,)- *(I'll~, 2, IlI)( - 1) m -*~=)- *(I'll ~2 22 i ll)(2k + 1) ×
-
M',
M 2' lClka 1
x (#'k'li - x ~ + m - ~ ) -
--0"2 /~/t~ 2 21
113
1~(~(82,~2,cr2)l/zk>*dY2,e,.
(2.4)
The values 8 = 0, 1, 2 designate the electric, the magnetic-orbital and the magneticspin excitation, respectively; EC0) is the integer part of the number p. The operators of the transition connected with the particle are given in ref. *). In eq. (2.3), J is a summation over all the unmeasurable characteristics of the radiation. When one observes only the direction of the y-quantum, the matrix dement (2.3) may be written in the form s)
(I'MIIE(2)II'MI) =
~
(--1)I'+~+M'2aL, C~L~(2I' +1) -=*
LtL2vp
x ( 2 v + l ) ~ Mi
--M2,
p
L2I I)Dov(R
),
(2.5)
V-RAY A N G U L A R
DISTRIBUTION
2]7
where the real magnitude t 6L is 2,5) the reduced matrix element (I'[[L[[I'). The argument R-i of the rotation matrix D is a rotation, bringing the radiation coordinate system to a coordinate system with a quantization axis along the direction of the incident beam. The coefficients F v are known in the theory of angular correlations and are tabulated 5)
F,,(L 1 L f l " I ' ) = (-- 1) I''+1'- i[(2~-1-1)(2/t-l- 1)(2L1 + I)(2Lz + 1)] +
-1
0/iv
z' r ' "
(2.6)
Substituting the expressions (2.4) and (2.5) into the eq. (2.1), summing over M~, M~ and integrating over f~k,', using the obvious form of the matrix elements (#'k'li-~+E(*-io-i~(s2, o-)]#k) (the formulae (2.11)-(2.13) of ref. 4)) for the angular distribution of 7-quanta, we finally obtain
W(O,) =
Z
(°',,~Ji(°~:2) ~ Z < : l ~:~F~(,~l,~2U')
x E 6L,6L~F~(L1L2I"I')P*(c°s 0~),
L1LZ
(2.7)
where o-~ is the cross section of the excitation 4) of the type ~ and multipolarity 2; the sign of the square root coincides with the sign of the reduced matrix element
(l'll:cllO. The parameters of the particle in eq. (2.7) are given by the following expressions: _,,a"'~* ==~2 --
/~gl~lB2~2
~
(2.8)
b; ~ = b~~"z,
(2.9)
b~i~*~2 = h~v~'~*~** ° v
(2.10)
For the orbital excitation corresponding to the values Q, e2 = 0, 1 the parameters equal to
bel~.lt:2~,2 are
-i
Z(-
lll2l't
x(Zl~+l)(Zlz+l)(Zl~+l)expi[~h,-qt~]
(21 1~2 v t ( l , + e Iz It 1'~1\ 0
l[ 0
201)
X (/2: £ '~0 "~2~ (llO]\O '20 ;)C@lIllrl~l)C(82121'l~2) -*~-l-~*~x-x2-*-~2 X Mhl, , "~'*tat', ,
(2.11)
t The ratio of the total intensity of the 2Lz pole radiation to the intensity of the 2L~pole radiation is equal to (dLz/6L~)~.
218
V. A. DJRBASHIAN
where c(0tl'~)
(2.12)
= 1,
C(lll'it) = [-it-1(2 + 1)(22 + 1)(/+ 1)(21+ 1)(21 + 3)1 ~ {2/
2
1+1
~1
"
(2.13)
~Ihe spin orbital interferences give no contribution: b2~,~;.~
=
0
h~t~.i222 =
for
81, ~2 -# 2.
(2.14)
The parameters for magnetic spin excitation are expressed by the parameters of the electric excitation through: b2Z, 222 = _.}i(i + 1)/, 2 [(2, + 1)(22, + 1.)(221 + 3)(22 + 1)(222 + 1)(222 + 3)3~* (it; x
22
;)-*(2,?1
22+1
-- 1
-- 1
v~/,~,+l O] [ 22
22+1 2,
;}hOa,+,0a=+, vv
(2.15)
Let us note that the total cross sections of the electromagnetic excitation 4) may be presented in the form " h \2[1-E(1-½e)]
a~
64= z
= (met)
b~Z'
(2.16)
where boEa is given by eqs. (2.9), (2.11) and (2.15). 3. D i s c u s s i o n
The particular cases b °~°~ = bye*,~v/~12~l~"= vv/~M°rb'~ of the particle parameters (2.11) corresponding to pure 2 ~ pole transitions, coincide with the ones given in the literature *' 2). The particular case ;L1 = 1, 22 = 2 of the interference orbital parameter b~ **°x2 is considered by Biedenharn and Thaler 3). From eqs. (2.7) and (2.11) it is not difficult to obtain the angular distribution for the mixed E2-M1 excitation, which will differ from the expression (7) given by Biedenharn and Thaler 3) by the fact that the second term, corresponding to the contribution 2Re(b 1102), will be a factor of two smaller. It is known that in the electromagnetic excitation, the E2 transitions are most important. It is however not excluded that in some cases (for example due to the selection rule) other types of transitions, in particular M1, may be seen. It is shown by Biedenharn and Thaler that in such cases, the contribution of the spin magnetic moment is important for proton bombardments. But since according to (2.15) it is proportional to i(i+ 1)/*2, it may for the particles with spin i > ½ (e.g. for deuterons) become the main one. For the 24 pole magnetic-spin transition, denoting the parameter of the particle b~~2. = b~ spinx one may from eq. (2.15) get the following expression for the ratio
0-RAY ANGULAR DISTRIBUTION
219
bM spinZ/hE2 + 1 y /~y b~ ~"~ bfa +1
_ ,~(2~ + .~ + 2)(2,~ - ~, + 1)l-v(v + 1) - 2(.~ + 1)(,~ + 2)1 iO + 1",/..,~
(3.1)
6(2 + 2)[v(v + 1 ) - 22(2 + 1)3
I n the case o f 2 = 1, i = ½, o n e finds f r o m (3.1) bMspin i 0
~
_3, 2hE2 2/~p t"O
bMspin 1 _ 3.2qE2 2 -- -- 4/*p '-'2 •
(32) (3.3)
T h e result (3.2) c o i n c i d e s 4) w i t h eq. (4) g i v e n i n ref. 3). H o w e v e r , o u r e x p r e s s i o n (3.3) differs f r o m eq. (5) o f ref. 3) b y a coefficient ¼.
References 1) 2) 3) 4) 5)
L. C. Biedenharn, J. L. McHale and R. M. Thaler, Phys. Rev. 100 (1955) 376 K. Alder et aL, Revs. Mod. Phys. 28 (1956) 432 L. C. Biedenharn and R. M. Thaler, Phys. Rev. 104 (1956) 1643 V. A. Djrbashian, Nuclear Physics 65 (1965) 167 L. C. Biedenharn and M. E. Rose, Revs. Mod. Phys. 25 (1953) 729
I
Nuclear Physics 81 (1966) 215--219; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
GAMMA-RAY A N G U L A R DISTRIBUTION E M I T T E D AFTER EXCITATION OF NUCLEI V. A. D J R B A S H I A N
Institute of Physics, Yerevan, USSR Received 13 September 1965
Abstract: Formulae are derived for the angular distribution of ~-rays following the electromagnetic
excitation of nuclei by particles with arbitrary spin.
1. Introduction
The quantal calculation of orbital electromagnetic excitation is presented by Biedenharn, McHale and Thaler 1) and by Alder et al. z), where together with the total cross sections, the angular distributions of the emitted y-rays for pure multipole excitations are given. However, in the angular distribution of y-rays, unlike in the total cross sections, the possibility of mixed electromagnetic excitation arises. For the particular case of M1-E2 this is considered by Biedenharn and Thaler 3). They have also shown that in case of an M1 transition and particle spin ½, the contribution of the spin magnetic moment of the particle together with the orbital magnetic moment is significant. Naturally, this contribution will have more significance for particles with spin i > ½. Previously 4), we have obtained cross sections of electromagnetic excitation taking into account the contribution of the nuclear forces as well as the spin magnetic moment of the particle with an arbitrary spin. It has been established that the cross section of the 2 ~ pole magnetic spin excitation is proportional to i(i + 1)#~, where 2Z 1 #p i is the spin magnetic moment of the particle (in nuclear magnetons) and is expressed in terms of the cross section of the 2 ~+ 1 pole electric excitation. In the present paper it is shown that an analogous situation arises when considering the angular distribution of y-rays. Expressions are given which take into account the multipole excitation due to the spin magnetic moment of the particle with an arbirary spin together with the mixed multipole orbital excitation. Various particular cases are considered. For the mixture M1-E2 an expression is obtained, which differs by a coefficient ½ from the corresponding expression of Biedenharn and Thaler 3). For the case of a pure M1 transition and for particle spin ½, the particle parameters b~ Spin~ coincide 4) with the ones obtained in ref. 3) at v = 0 and differ from them by a coefficient ¼ at v = 2. 215
216
V. A. DJRBASHIAN 2. General Formulae
The angular distribution of the y-quanta emitted at the de-excitation of nuclei may be written in the form
W(8v) = ~
(I'M~IE(2)II'M'~XI'MiIV(*)II'MI),
(2.1)
M'IM' 2
where 8~ is the angle between the directions of the incident particle and the emitted y-quantum. The spin of the nucleus and its projection are denoted by I and M, respectively. Here and hereafter the unprimed, single primed and double primed magnitudes denote the initial, the intermediate (when the nucleus is excited) and the final states of the system, respectively. The matrix elements (I'M~IV(1)II'M~) and (I'M.~[E(2)II'M;) describe the electromagnetic excitation of the nucleus of the particle with an arbitrary spin and the emission of a y-quantum, respectively: , , I, V (IM
cl)
~ v~.i,M l i ,M a, ) =JM~W<,
., .#.~. . iVlllM#k)(I'M'2#'k']VlilM#k)*df2,k ,,
(I'M'2IE(Z)II'MI) = J ~, (I'M'2IH2II"M")(I'MI]H2II"M")*.
(2.2) (2.3)
M
In eq. (2.2), # and k are the projection of the spin i and the wave vector of the particle. The matrix elements under the integral over the solid angle f2k,,, produced by the rotation of the final direction of the particle around the initial one, are given in ref. 4). Substituting the expressions (2.2)-(2.4) and (2.10) of ref. 4) into eq. (2.2), we obtain
(I'UilVmlZ'Mi) _-
~ ( - 1 ) ~-z+M''+*2 16n 2 ~.,~z,&k,~m,',,**~ (2)., + 1)(23~z + 1)
x ( - 1) m -a.,)- *(I'll~, 2, IlI)( - 1) m -*~=)- *(I'll ~2 22 i ll)(2k + 1) ×
-
M',
M 2' lClka 1
x (#'k'li - x ~ + m - ~ ) -
--0"2 /~/t~ 2 21
113
1~(~(82,~2,cr2)l/zk>*dY2,e,.
(2.4)
The values 8 = 0, 1, 2 designate the electric, the magnetic-orbital and the magneticspin excitation, respectively; EC0) is the integer part of the number p. The operators of the transition connected with the particle are given in ref. *). In eq. (2.3), J is a summation over all the unmeasurable characteristics of the radiation. When one observes only the direction of the y-quantum, the matrix dement (2.3) may be written in the form s)
(I'MIIE(2)II'MI) =
~
(--1)I'+~+M'2aL, C~L~(2I' +1) -=*
LtL2vp
x ( 2 v + l ) ~ Mi
--M2,
p
L2I I)Dov(R
),
(2.5)
V-RAY A N G U L A R
DISTRIBUTION
2]7
where the real magnitude t 6L is 2,5) the reduced matrix element (I'[[L[[I'). The argument R-i of the rotation matrix D is a rotation, bringing the radiation coordinate system to a coordinate system with a quantization axis along the direction of the incident beam. The coefficients F v are known in the theory of angular correlations and are tabulated 5)
F,,(L 1 L f l " I ' ) = (-- 1) I''+1'- i[(2~-1-1)(2/t-l- 1)(2L1 + I)(2Lz + 1)] +
-1
0/iv
z' r ' "
(2.6)
Substituting the expressions (2.4) and (2.5) into the eq. (2.1), summing over M~, M~ and integrating over f~k,', using the obvious form of the matrix elements (#'k'li-~+E(*-io-i~(s2, o-)]#k) (the formulae (2.11)-(2.13) of ref. 4)) for the angular distribution of 7-quanta, we finally obtain
W(O,) =
Z
(°',,~Ji(°~:2) ~ Z < : l ~:~F~(,~l,~2U')
x E 6L,6L~F~(L1L2I"I')P*(c°s 0~),
L1LZ
(2.7)
where o-~ is the cross section of the excitation 4) of the type ~ and multipolarity 2; the sign of the square root coincides with the sign of the reduced matrix element
(l'll:cllO. The parameters of the particle in eq. (2.7) are given by the following expressions: _,,a"'~* ==~2 --
/~gl~lB2~2
~
(2.8)
b; ~ = b~~"z,
(2.9)
b~i~*~2 = h~v~'~*~** ° v
(2.10)
For the orbital excitation corresponding to the values Q, e2 = 0, 1 the parameters equal to
bel~.lt:2~,2 are
-i
Z(-
lll2l't
x(Zl~+l)(Zlz+l)(Zl~+l)expi[~h,-qt~]
(21 1~2 v t ( l , + e Iz It 1'~1\ 0
l[ 0
201)
X (/2: £ '~0 "~2~ (llO]\O '20 ;)C@lIllrl~l)C(82121'l~2) -*~-l-~*~x-x2-*-~2 X Mhl, , "~'*tat', ,
(2.11)
t The ratio of the total intensity of the 2Lz pole radiation to the intensity of the 2L~pole radiation is equal to (dLz/6L~)~.
218
V. A. DJRBASHIAN
where c(0tl'~)
(2.12)
= 1,
C(lll'it) = [-it-1(2 + 1)(22 + 1)(/+ 1)(21+ 1)(21 + 3)1 ~ {2/
2
1+1
~1
"
(2.13)
~Ihe spin orbital interferences give no contribution: b2~,~;.~
=
0
h~t~.i222 =
for
81, ~2 -# 2.
(2.14)
The parameters for magnetic spin excitation are expressed by the parameters of the electric excitation through: b2Z, 222 = _.}i(i + 1)/, 2 [(2, + 1)(22, + 1.)(221 + 3)(22 + 1)(222 + 1)(222 + 3)3~* (it; x
22
;)-*(2,?1
22+1
-- 1
-- 1
v~/,~,+l O] [ 22
22+1 2,
;}hOa,+,0a=+, vv
(2.15)
Let us note that the total cross sections of the electromagnetic excitation 4) may be presented in the form " h \2[1-E(1-½e)]
a~
64= z
= (met)
b~Z'
(2.16)
where boEa is given by eqs. (2.9), (2.11) and (2.15). 3. D i s c u s s i o n
The particular cases b °~°~ = bye*,~v/~12~l~"= vv/~M°rb'~ of the particle parameters (2.11) corresponding to pure 2 ~ pole transitions, coincide with the ones given in the literature *' 2). The particular case ;L1 = 1, 22 = 2 of the interference orbital parameter b~ **°x2 is considered by Biedenharn and Thaler 3). From eqs. (2.7) and (2.11) it is not difficult to obtain the angular distribution for the mixed E2-M1 excitation, which will differ from the expression (7) given by Biedenharn and Thaler 3) by the fact that the second term, corresponding to the contribution 2Re(b 1102), will be a factor of two smaller. It is known that in the electromagnetic excitation, the E2 transitions are most important. It is however not excluded that in some cases (for example due to the selection rule) other types of transitions, in particular M1, may be seen. It is shown by Biedenharn and Thaler that in such cases, the contribution of the spin magnetic moment is important for proton bombardments. But since according to (2.15) it is proportional to i(i+ 1)/*2, it may for the particles with spin i > ½ (e.g. for deuterons) become the main one. For the 24 pole magnetic-spin transition, denoting the parameter of the particle b~~2. = b~ spinx one may from eq. (2.15) get the following expression for the ratio
0-RAY ANGULAR DISTRIBUTION
219
bM spinZ/hE2 + 1 y /~y b~ ~"~ bfa +1
_ ,~(2~ + .~ + 2)(2,~ - ~, + 1)l-v(v + 1) - 2(.~ + 1)(,~ + 2)1 iO + 1",/..,~
(3.1)
6(2 + 2)[v(v + 1 ) - 22(2 + 1)3
I n the case o f 2 = 1, i = ½, o n e finds f r o m (3.1) bMspin i 0
~
_3, 2hE2 2/~p t"O
bMspin 1 _ 3.2qE2 2 -- -- 4/*p '-'2 •
(32) (3.3)
T h e result (3.2) c o i n c i d e s 4) w i t h eq. (4) g i v e n i n ref. 3). H o w e v e r , o u r e x p r e s s i o n (3.3) differs f r o m eq. (5) o f ref. 3) b y a coefficient ¼.
References 1) 2) 3) 4) 5)
L. C. Biedenharn, J. L. McHale and R. M. Thaler, Phys. Rev. 100 (1955) 376 K. Alder et aL, Revs. Mod. Phys. 28 (1956) 432 L. C. Biedenharn and R. M. Thaler, Phys. Rev. 104 (1956) 1643 V. A. Djrbashian, Nuclear Physics 65 (1965) 167 L. C. Biedenharn and M. E. Rose, Revs. Mod. Phys. 25 (1953) 729