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The effects of quadrupole deformations on hexadecapole moments
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
THE EFFECTS OF QUADRUPOLE DEFORMATIONS ON HEXADECAPOLE MOMENTS L. ZAMICK 1 Serin Physics Laboratory, Piscataway, NJ 08854, USA Received 23 October 1979
To explain large E4 effective charges needed in the l s - 0 d shell we consider two effects. The first is that a closed shell which experiences a quadrupole deformation has a hexadecapole moment. Second we consider valence polarization which is extremely important in 2°Ne.
It has been noted by Brown and Wildenthal [ I - 4 ] that the isoscalar hexadecapole polarization charge in the s - d shell is about one (thus making the total effective charge about two). First-order core polarization calculations yield much smaller values [5]. To explain the large E4 effective charge we consider a second-order effect arising from the fact that when the core experiences a quadrupole deformation, it will also acquire a hexadecapole moment. We also consider the change in the E4 m o m e n t o f a valence particle due to anisotropy of the one-body potential. For convenience we write the E4 operator as r4 Y4,0 = 1 ~
Q4,0,
where Q4,0 = 35z4 - 30z2r2 + 3r4 = 8z 4 _ 24z2(x2 + y 2 ) + 3(x 2 + y 2 ) 2 = r 4 [8 cos40 - 24(cos20 - cos40) + 3(1 - 2 cos20 + cos40)]. Using an anisotropic harmonic oscillator with cox = COy ~ coz the single-particle matrix element o f this operator can be expressed in terms o f the Nilsson quantum numbers
1 Supported by the National Science Foundation.
(NM3AJQ4,0[NM3A) ___8b 4 [(M3 + ~)2
+ ¼(M3 + a)(M3 + 2) + ¼M3(M 3 - a)l - 2 4 b 2 b 2 ( M 3 + ½ ) ( N - M 3 + 1) + 364[(N-M3 + ¼(N-M 3-A
+ I) 2 + 2)(N-M 3+A+2)
+ ¼(N - i 3 - A ) ( N - i 3 + A ) ] . In the above b x = by and b z are the oscillator length parameters. We also define b via b 3 = bxbyb z. If we neglect the anisotropy, that is if we make b x = by = b z = b then this expression reduces to one given b y Bohr and Mottelson [6, p. 140] :
(NM3A[Q4,o [ARM3A) 3 23 - 2 2 M 3 N + 3 N 2 = ~-(27M
_ A 2 - 6 M 3 - 2 N ) b 4.
We now consider a closed shell, say 160, plus several valence nucleons. The valence nucleons polarize the core causing it to deform. We assume that we can use a Slater determinant with deformed harmonic oscillator wavefunctions to describe the situation. A given singleparticle wavefunction will have the structure tk(X/bx, y / b y , z/bz). We make the transformation o f variables .it = x b/bx, fJ = y b/by, ~ = z bib z . In terms of these variables the wavefunction o f the 160 core will look "spher ical" but the operator gets changed to 23
Volume 92B, number 1,2
PHYSICS LETTERS
Q4,0 = r4 [8(b4z/b4) c°s40
5 May 1980
Table 1
_ 24(b2z/bZ)(b2x/b2)(cos20 _ cos40) 170 2°Ne 2aMg
+ 3(b4/b4)(1 - 2c°s20 + c°s40)] " In these coordinates the operator has a scalar, quadrupole and hexadecapole part. For the closed core only the scalar part will contribute, and we can project this out by integrating with (I/47r) f dr2:
~-~fQ4,0dg2= bs 2 / b4 2}(bz/b ) 22. - g2( r
-
The value for the closed shell is then (b2z/b 2 - b /b2) 2 X Zi(i r 4 i}, where the sum is over occupied states. For the harmonic oscillator we have
bz
bx
~e2,c
6e2,v
1.0337b 1.0951b 1.0933b
0.98356b 0.95559b 0.95638b
0.607 0.429 0.337
0.093 0.271 0.363
there is some evidence that in the sd shell a value e ° = 1.7 is better so we shall adjust b z (keeping b 2 = b3/ bz) to this lower value. We consider three cases: (1) 170 with a particle in the (220) orbit, (2) 20Ne with four particles in (220), (3) 24Mg with four particles in (220) and four particles in (211). The equations we obtain are:
(nllr21nl) = (2n +l + -~)b 2, 170:
(nllr41nl) = [(2n +l +_~)2
(29b 2 - 25b2)/4b 2 = e 0 --- 1.7,
20Ne:
(44b 2 - 28b2)/16b 2 = e 0 = 1.7,
+(n + 1)(n + l + ~3 ) + n ( n +l+ 1)] b 4, where n is the number of nodes in the radial wavefunction and l the orbital angular momentum (n starts from zero). Thus we obtain for the 160 core: (Q4,0) = 192 b4(b2/b2 - b2x/b2) 2. We estimate (b 2 - b 2) by fitting the quadrupole properties of nuclei. The expectation value of the mass quadrupole operator Q2,0
= (2z 2 _ x 2 _ y 2 ) i s (Q2,o)(bx , b z) = ~
(2M 3 + 1)b 2 - ( N - M
3 + 1)b 2.
We define the isoscalar quadrupole effective charge as the ratio
e 0 = (Q2,0)(b x, bz)/(Q2,0)(b, b). With the popular prescription of a neutron E2 effective charge of 1/2 and a proton effective charge of 1.5, e 0 would be equal to two (e 0 = e n + ep). However,
(56b 2 - 36b2x)/20b 2= e 0 = 1.7. 24Mg: The values obtained for b z and b x are listed in columns 1 and 2 of table 1. The polarization charge can be divided into a core part and a valence part. We call these 6e2, c and 8e2,v, which are listed in columns 3 and 4 of table 1. We now proceed to the E4 polarization. The values of (Q4,0)(b, b) are listed in the first column of table 2. Since the 24Mg value is zero we cannot define a polarization charge, we must simply list the contribution of our mechanism. For the other two nuclei we can define the e 4 effective charge in a fashion analogous to the e 2 case:
e° = ~. (iQ4,oi}(bx'bz)/~i
(iQ4,0 i)(b'b)"
Table 2 gives the results. Using the values of b z and
Table 2
170 2°Ne 24Mg
24
(Q4,o)(b, b)
(Q4,o)c/b 4
(Q4,o)v/b 4
8e4,c
8e4,v
e]
24b 4 96b 4 0
1.964 15.715 15.122
32.651 205.904 30.0689
0.0818 0.1637 -
0.3605 1.14483 -
1.442 2.309 -
Volume 92B, number 1,2
PHYSICS LETTERS
b x which were fitted to the E2 properties, we define 6e4, c and 6e4, v as the core and valence polarization charges. The total effective charge e 0 = 6e4, c + 6e4, v
+1. The very strong polarization charge in 20Ne comes mainly from the valence part. The expression for Q4,0 for the (220) orbit is 78 b 4 - 60 b2b 2 + 6 b 4. The value for b z = b x = b is 24 b2. It is remarkable that this value is more than doubled b y changing b z / b from 1 to 1.095. One usually argues that a small deformation can have large consequences because there are many nucleons involved. Here we have an example where even one nucleon suffers a large change. If one neglects anisotropy the 24Mg moment is zero. Although we cannot assign an effective charge here we can note that the calculated ratio o f (Q4,0) in 24Mg relative to 20Ne is 0.21. A nice intuitive reason for why E4 matrix elements go through zero, changing from positive to negative is given by Bertsch [7].
5 May 1980
The effects we have calculated here would be very difficult to calculate in a spherical basis using perturba. tion theory. One would clearly have to go to very high orders. I would like to thank C. Glashausser and B.A. Brown for comments. References
[1 ] [2] [3] [4]
B.A. Brown, private communication. R. de Swiniarski et ah, Phys. Rev. Lett. 23 (1969) 317. H. Rebel et al., Nueh Phys. A182 (1972) 145. A.L. Goodman, G.L. Struble, J. Bar-Touv and A. Goswami, Phys. Rev. C2 (1970) 380. [5] H. Sagawa, Phys. Rev. C19 (1979) 506. [6] A. Bohr and B. Mottelson, Nuclear structure, Vol. 2 (New York, 1975). [7] G.F. Bertsch, Phys. Lett. 26B (1968) 130.
25
PHYSICS LETTERS
5 May 1980
THE EFFECTS OF QUADRUPOLE DEFORMATIONS ON HEXADECAPOLE MOMENTS L. ZAMICK 1 Serin Physics Laboratory, Piscataway, NJ 08854, USA Received 23 October 1979
To explain large E4 effective charges needed in the l s - 0 d shell we consider two effects. The first is that a closed shell which experiences a quadrupole deformation has a hexadecapole moment. Second we consider valence polarization which is extremely important in 2°Ne.
It has been noted by Brown and Wildenthal [ I - 4 ] that the isoscalar hexadecapole polarization charge in the s - d shell is about one (thus making the total effective charge about two). First-order core polarization calculations yield much smaller values [5]. To explain the large E4 effective charge we consider a second-order effect arising from the fact that when the core experiences a quadrupole deformation, it will also acquire a hexadecapole moment. We also consider the change in the E4 m o m e n t o f a valence particle due to anisotropy of the one-body potential. For convenience we write the E4 operator as r4 Y4,0 = 1 ~
Q4,0,
where Q4,0 = 35z4 - 30z2r2 + 3r4 = 8z 4 _ 24z2(x2 + y 2 ) + 3(x 2 + y 2 ) 2 = r 4 [8 cos40 - 24(cos20 - cos40) + 3(1 - 2 cos20 + cos40)]. Using an anisotropic harmonic oscillator with cox = COy ~ coz the single-particle matrix element o f this operator can be expressed in terms o f the Nilsson quantum numbers
1 Supported by the National Science Foundation.
(NM3AJQ4,0[NM3A) ___8b 4 [(M3 + ~)2
+ ¼(M3 + a)(M3 + 2) + ¼M3(M 3 - a)l - 2 4 b 2 b 2 ( M 3 + ½ ) ( N - M 3 + 1) + 364[(N-M3 + ¼(N-M 3-A
+ I) 2 + 2)(N-M 3+A+2)
+ ¼(N - i 3 - A ) ( N - i 3 + A ) ] . In the above b x = by and b z are the oscillator length parameters. We also define b via b 3 = bxbyb z. If we neglect the anisotropy, that is if we make b x = by = b z = b then this expression reduces to one given b y Bohr and Mottelson [6, p. 140] :
(NM3A[Q4,o [ARM3A) 3 23 - 2 2 M 3 N + 3 N 2 = ~-(27M
_ A 2 - 6 M 3 - 2 N ) b 4.
We now consider a closed shell, say 160, plus several valence nucleons. The valence nucleons polarize the core causing it to deform. We assume that we can use a Slater determinant with deformed harmonic oscillator wavefunctions to describe the situation. A given singleparticle wavefunction will have the structure tk(X/bx, y / b y , z/bz). We make the transformation o f variables .it = x b/bx, fJ = y b/by, ~ = z bib z . In terms of these variables the wavefunction o f the 160 core will look "spher ical" but the operator gets changed to 23
Volume 92B, number 1,2
PHYSICS LETTERS
Q4,0 = r4 [8(b4z/b4) c°s40
5 May 1980
Table 1
_ 24(b2z/bZ)(b2x/b2)(cos20 _ cos40) 170 2°Ne 2aMg
+ 3(b4/b4)(1 - 2c°s20 + c°s40)] " In these coordinates the operator has a scalar, quadrupole and hexadecapole part. For the closed core only the scalar part will contribute, and we can project this out by integrating with (I/47r) f dr2:
~-~fQ4,0dg2= bs 2 / b4 2}(bz/b ) 22. - g2( r
-
The value for the closed shell is then (b2z/b 2 - b /b2) 2 X Zi(i r 4 i}, where the sum is over occupied states. For the harmonic oscillator we have
bz
bx
~e2,c
6e2,v
1.0337b 1.0951b 1.0933b
0.98356b 0.95559b 0.95638b
0.607 0.429 0.337
0.093 0.271 0.363
there is some evidence that in the sd shell a value e ° = 1.7 is better so we shall adjust b z (keeping b 2 = b3/ bz) to this lower value. We consider three cases: (1) 170 with a particle in the (220) orbit, (2) 20Ne with four particles in (220), (3) 24Mg with four particles in (220) and four particles in (211). The equations we obtain are:
(nllr21nl) = (2n +l + -~)b 2, 170:
(nllr41nl) = [(2n +l +_~)2
(29b 2 - 25b2)/4b 2 = e 0 --- 1.7,
20Ne:
(44b 2 - 28b2)/16b 2 = e 0 = 1.7,
+(n + 1)(n + l + ~3 ) + n ( n +l+ 1)] b 4, where n is the number of nodes in the radial wavefunction and l the orbital angular momentum (n starts from zero). Thus we obtain for the 160 core: (Q4,0) = 192 b4(b2/b2 - b2x/b2) 2. We estimate (b 2 - b 2) by fitting the quadrupole properties of nuclei. The expectation value of the mass quadrupole operator Q2,0
= (2z 2 _ x 2 _ y 2 ) i s (Q2,o)(bx , b z) = ~
(2M 3 + 1)b 2 - ( N - M
3 + 1)b 2.
We define the isoscalar quadrupole effective charge as the ratio
e 0 = (Q2,0)(b x, bz)/(Q2,0)(b, b). With the popular prescription of a neutron E2 effective charge of 1/2 and a proton effective charge of 1.5, e 0 would be equal to two (e 0 = e n + ep). However,
(56b 2 - 36b2x)/20b 2= e 0 = 1.7. 24Mg: The values obtained for b z and b x are listed in columns 1 and 2 of table 1. The polarization charge can be divided into a core part and a valence part. We call these 6e2, c and 8e2,v, which are listed in columns 3 and 4 of table 1. We now proceed to the E4 polarization. The values of (Q4,0)(b, b) are listed in the first column of table 2. Since the 24Mg value is zero we cannot define a polarization charge, we must simply list the contribution of our mechanism. For the other two nuclei we can define the e 4 effective charge in a fashion analogous to the e 2 case:
e° = ~. (iQ4,oi}(bx'bz)/~i
(iQ4,0 i)(b'b)"
Table 2 gives the results. Using the values of b z and
Table 2
170 2°Ne 24Mg
24
(Q4,o)(b, b)
(Q4,o)c/b 4
(Q4,o)v/b 4
8e4,c
8e4,v
e]
24b 4 96b 4 0
1.964 15.715 15.122
32.651 205.904 30.0689
0.0818 0.1637 -
0.3605 1.14483 -
1.442 2.309 -
Volume 92B, number 1,2
PHYSICS LETTERS
b x which were fitted to the E2 properties, we define 6e4, c and 6e4, v as the core and valence polarization charges. The total effective charge e 0 = 6e4, c + 6e4, v
+1. The very strong polarization charge in 20Ne comes mainly from the valence part. The expression for Q4,0 for the (220) orbit is 78 b 4 - 60 b2b 2 + 6 b 4. The value for b z = b x = b is 24 b2. It is remarkable that this value is more than doubled b y changing b z / b from 1 to 1.095. One usually argues that a small deformation can have large consequences because there are many nucleons involved. Here we have an example where even one nucleon suffers a large change. If one neglects anisotropy the 24Mg moment is zero. Although we cannot assign an effective charge here we can note that the calculated ratio o f (Q4,0) in 24Mg relative to 20Ne is 0.21. A nice intuitive reason for why E4 matrix elements go through zero, changing from positive to negative is given by Bertsch [7].
5 May 1980
The effects we have calculated here would be very difficult to calculate in a spherical basis using perturba. tion theory. One would clearly have to go to very high orders. I would like to thank C. Glashausser and B.A. Brown for comments. References
[1 ] [2] [3] [4]
B.A. Brown, private communication. R. de Swiniarski et ah, Phys. Rev. Lett. 23 (1969) 317. H. Rebel et al., Nueh Phys. A182 (1972) 145. A.L. Goodman, G.L. Struble, J. Bar-Touv and A. Goswami, Phys. Rev. C2 (1970) 380. [5] H. Sagawa, Phys. Rev. C19 (1979) 506. [6] A. Bohr and B. Mottelson, Nuclear structure, Vol. 2 (New York, 1975). [7] G.F. Bertsch, Phys. Lett. 26B (1968) 130.
25