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Shell theory and Pb208 (II)
I 1.D.I : [ 1.E.4:
I
Nuclear Physics 37 (1962) 312--318; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
S H E L L T H E O R Y A N D P b z°s (II) W. T. PINKSTON t Vanderbilt University, Nashville, Tennessee Received 21 May 1962 Abstract: The 3- and 4 + energy levels predicted for Pb 2°8 by the shell model in the usual Tamm-Dan-
coff approximation are studied. The energy of the 3- level was reported in a previous calculation, and is not in agreement with experiment. The predicted wave function is used to calculate the E3 transition rate to the ground state. The rate is found to be too small by a factor of about 20. The energies and wave functions for the 4+ levels are calculated. A level of the correct energy is predicted by the E4 transition rate to the ground state is 30 to 100 times too small.
1. I n t r o d u c t i o n The systematic a p p e a r a n c e o f 2 + first excited states in n o n - d e f o r m e d nuclei is consistent w i t h the d e s c r i p t i o n o f such states as arising f r o m q u a d r u p o l e surface v i b r a t i o n s a b o u t a spherical e q u i l i b r i u m shape 1) C o n s i d e r a b l e evidence z,3) has recently accrued for the existence o f octupole a n d L = 4 v i b r a t i o n s in a d d i t i o n to L = 2 oscillations. I n the experiments o f C r a n n e l l et aL 3) strong E2, E3 a n d E4 t r a n s i t i o n s were observed resulting f r o m the inelastic scattering o f high energy electrons f r o m m e d i u m weight a n d h e a v y nuclei. A c o m p a r i s o n o f the e x p e r i m e n t a l t r a n s i t i o n rates to single-particle estimates i n d i c a t e d enhancements as large as 200, verifying the collective n a t u r e o f the excited states. A l t h o u g h these excited states are often described in terms o f surface oscillations, a m o r e interesting a n d f u n d a m e n t a l p r o s p e c t is that o f giving an e x p l a n a t i o n o f collective p h e n o m e n a in terms o f the m o t i o n s o f i n d i v i d u a l nucleons. The usual starting p o i n t o f such calculations is the nuclear shell model. The collective states are t h e n t h o u g h t o f as consisting o f r a t h e r c o m p l i c a t e d s u p e r p o s i t i o n s o f excited configurations. A d o u b l y m a g i c nucleus such as Pb 2°8 is o f p a r t i c u l a r interest for such considerations since the large energy g a p between m a j o r shells will restrict these configurations to a relatively small n u m b e r o f 1-hole 1-particle (1H-1P) configurations. T w o m e t h o d s exist for calculating the energies a n d excitation p r o b a b i l i t i e s ofcoUect i r e states. T h e usual p r o c e d u r e o f shell theory, sometimes referred to as the T a m m D a n c o f f a p p r o x i m a t i o n or the m e t h o d o f configuration interaction, consists o f d i a g o n a l i z i n g that p a r t o f the H a m i l t o n i a n which connects 1H-1P states to one another. This results in a n u n m o d i f i e d g r o u n d state wave function for a d o u b l e m a g i c nucleus a n d excited state wave functions which are linear s u p e r p o s i t i o n s o f t Supported by U. S. National Science Foundation. 312
SHELL-THEORY
AND
313
Pb~°8(lI)
1H-1P states. This procedure has been carried out for the odd parity states of Pb 2°s by Carter, Pinkston and True,4)t, but the energy of the lowest 3- level was predicted about one MeV higher than observed. Recently much work has been done 5, 6) on a more accurate method known as the r a n d o m phase approximation, a method first applied to nuclear shell theory calculations by Fallieros and Ferrell 7). ]For a nucleus such as Pb 2°s this procedure essentially admixes 2H-2P states into the ground state wave function and 3H-3P states into the excited state functions in addition to the 1H-1P states. In the present work the E3 transition rate from the first excited state of Pb 2°s to the ground state is calculated with the wave functions resulting from the calculations of (I) and is compared to experiment 3). Little if any enhancement is obtained by using a superposition of wave functions rather than a single 1H-1P state. The d = 4 + states of Pb 2°s are predicted on the T a m m - D a n c o f f approximation. The predicted wave functions fail to predict a sufficiently large E4 transition probability, but the theory does predict an energy level at about 4 MeV, consistent with experiment. 2. The
J = 3-
Level
The hole-particle configurations contributing to the J = 3 - states of Pb 2°8 were assumed in (I) to be f~ t gb f~-i i~, d~ 1 h~, s~-1 f~ and d~ I f~. The single particle energy levels involved are shown in fig. 1 and were taken from experimental data
¢
0.71 ¸
¢ 0.90
0.79 gg/~
i
1
f 3.44 t
i t 0.57 0.33 .f
4.26
Pl/2 f~/~
S
d3/2
0.35 1
1.49
0.73 h,,/~ NEUTRONS
....
_
_
i_ _ _
PROTONS
Fig. I. Single particle energy levels in Pb region (not to scale). T h e h~k level is d a s h e d indicating that its position is experimentally u n k n o w n . T h e value given is t a k e n f r o m a theoretical calculation 18). * Referred to as (I).
314
W. T. PINKSTON
on the levels of nuclei resulting from one proton or neutron added to or subtracted from Pb 2°8. The energy levels and wave functions were calculated for the following choices of the interparticle potential: a) a Gauss shape with Serber space exchange and a triplet strength 1.5 times the singlet strength, b) a Gauss shape with Rosenfeld exchange, c ) a 6-function interaction with triplet strength 1.5 times the singlet strength. Harmonic oscillator radial wave functions were used in calculating the radial integrals of the potential. The results are listed in table 1. The lowest energy level is predomiT~L~ 1 Energy level and eigenfunction predictions for the lowest 3- level in Pb 2°8 for 3 choices of potential Eigenfunction
Potential
Energy (MeV)
f~- 1 g~
f~ 1 i~
d~- 1 h}
s~ t f~
d~ 1 f}
Gauss-Serber Gauss-Rosenfeld Zero Range
3.66 3.38 3.13
0.287 0.124 0.239
--0.576 --0.616 --0.617
--0.711 --0.737 --0.687
--0.261 --0.237 --0.274
+0.106 +0.082 +0.122
nantly d~" ~ ~ due to the attractive Coulomb force between proton holes and particles. This brings it below the f~- 1 N neutron level which is lowest in zero order. Its energy is predicted to be 3.13 MeV, 0.5 MeV higher than the experimental value of 3.6 MeV. For the more realistic potentials the discrepancyis closer to 1.0 MeV. It was, therefore, concluded in (I) that the superposition of the 5 configurations above is not equivalent to an L = 3 surface oscillation. N o calculation of the E3 transition probability was made, however, and it is reported here for completeness. Expressions for the transition rates for electromagnetic multipole transitions are given by Blatt and WeisskopfS). For E3 transitions the rate in sec -1 can be expressed as F(E3) = 318E 7 KJfllrafa(O)llJi>12
[J,]
(1)
Here energy is expressed in MeV and length in units of 1 fm = 10 - l a cm. The notation [n] -= 2n + 1 is used. The quantity C3M is an unnormalized spherical harmonic tensor as defined by Racah 9) and Racah's definition of a reduced matrix element is used. The matrix element for the de-excitation o f a particle-hole state in a closed shell is given by Brink and Satchler lo). With the Racah convention for reduced matrix elements, we have
= ( - )t'-t2Ol l~ (lt IrL112).
(2)
A choice must be made for the radial functions. The calculations here were made with the radial functions for nucleons moving in a Saxon potential well as tabulated by Blomquist and Wahlborn 11).
SHELL-THEORY AND Pb~°8(II)
315
If the excited state wave function is written
0 = ~, B(jlJz)O(J; ~JzJ),
(3)
then eq. (1) reduces to r = 3.65 x 10s I 2.07B(d~h0+2.17B(s~f~) - 1.18B(d~f~)] 2.
(4)
All three of the wave functions of table 1 yield about the same result, F ~ 2 x 109 see - j , which is too small compared to the experimental value a) of (3.80__ 1.4)x 10 ~° sec -~ by a factor of about 20. The calculated value is almost exactly the value which one would obtain for a pure h~ particle to d~-hole transition, indicating that the superposition over several configurations does not enhance the transition here. One can in fact very easily calculate the maximum value of F which can arise from an expression such as eq. (4). An expression of the form ( ~ , a(i) x(i)) 2, in which the a(i) are known constants and the x(i) are variables subject to the constraint ~ x ( i ) 2 = 1, has as its maximum ~ a ( i ) 2. Applied to eq. (4) this results in a maximum of 3.8 x 1 0 9 sec- 1, 10 times smaller than F~p. Thus it is impossible to reproduce the experimental value with any linear combination o f the proton I H - I P configurations d~- ~ h~, s~ ~ f~ and d~-~ f~. 3. The J - - 4
+ Level
The inelastic scattering results of ref. 3)imply strongly excited 4 + states at 4.3 MeV in both Pb 2°s and Bi 2°9. The E4 tritnsition rate for the de-excitation of this level in Pb 2°s was found to be F = (2.23___0.7)x 10 s sec -~. In this section we extend the methods of (I) to even parity 1H-1P states and compare the results to this data. In the case of even parity 1H-IP states there is some uncertainty as to the zero order energies of the proton states. A J = 4 + proton level can arise either from the excitation of an h~ proton hole state and an odd parity proton particle state such as or f~ or to the excitation o f a d~ proton hole and an i~ particle. Due to the sparcity of data on the levels of T12°7 no even parity hole state is known. The i~ particle state has not been identified in Bi 2°9 but probably corresponds to the 1,61-MeV level o f that nucleus. We must be guided here by theoretical calculations. Blomquist and Wahlborn ~~) have calculated the single particle energy levels and wave functions for nucleons moving in a Saxon well with parameters appropriate to Pb region. Their energy calculations agree well with the known level spectra of Pb 2°7, T12°7, Pb 2°9 and Bi ~°9. We therefore assume that they will be fairly reliable for the as yet unobserved levels. These calculations predict the h~ hole level 1.84 MeV below the s½ ground state ofT1 z°7 and the d I hole level 0.65 MeV deeper. The h÷ level is listed in fig. 1 but is dashed to indicate that its position is uncertain. We shall ignore the dl level since the d21 i~ will be 1.36 MeV higher than h;~ 1 f,,_and 2.26 MeV higher than h;~ 1 hl. We therefore take as the dominant proton configurations, hT~1 f~ and h;~ ~ f,_. F r o m fig. 1 it is clear that the important neutron configurations are i2~~ gl andi;~ 1 i÷, The only other possibility is f~- 1 j .~ which is probably several MeV higher.
316
W. T. PINKSTON
The energy levels, assuming these four configurations, have been calculated using a zero range internucleon potential without exchange. Previous investigations '~, 22) in this region indicate that the main features of the energy spectrum are not sensitive either to the shape or to the exchange character of the forces. The matrix elements for such a potential can be evaluated using the methods of Brink and Satchler lo) and the two body results o f Newby and Konopinski 22). For identical particles the hole particle interaction is given by a sum of direct and exchange terms:
(j ? lje JI VIj3 tj, d) = [Jl]~[J2-l~[J3]~[J*]¢ (j~j2½-½}dO)(j3j,½-½[dO}F°(d+x).
2Es]
(5)
The quantity F ° is the Slater integral of a delta function 12), and d and x are given by d ~--- ( _ _ ) / 2 + / 4 + 1
I(__)jl+j3+l,+13+1 + ([Jl ~ + ( -- .)Y'+j~ + J[J2])([Ja] 4 J ( J + 1) + ( - )j~ +J" +J[J*])l-,_~ (6) x = 2 ( - ) "h+i~+l.
For unlike particles, as was noted in (I), the matrix element is the same as the like particle exchange term (d = 0). (An interesting consequence of these formulas is that the diagonal matrix elements are attractive for i7~~ ~ and h~ ~ f~. The usual statement that hole-particle forces have the opposite sign of particle-particle forces is of course rigorously true only in the long range limit in which exchange integrals are negligible compared to direct integrals.) For the proton configurations there is also a Coulomb potential. The Coulomb matrix elements can be estimated very easily. The Coulomb force is a long range force; therefore the off-diagonal matrix elements are expected to be negligible. The diagonal elements can be approximated by the Coulomb energy of 2 protons uniformly distributed over a sphere of radius R = ro A ~ with ro = 1.2 fm. Then we have
(Vc> ~ 6eZ/R = 0.24 MeV. This is in good agreement with more accurate calculations 4, 22). The results of the diagonalization of the energy matrix are plotted in fig. 2. The dependence of the results on the location of the hq hole state was checked by varying the h~-s~ separation from 1.84 MeV to 0.64 MeV in units o f 0.3 MeV. The lowest level has an energy close to the observed value of 4.3 MeV, but it is predominantly i~ ~ g~, a neutron state. The wave functions of this level is listed in table 2 for several values of the h~-s½ separation. The transition rate in sec- ~ for an E4 transition is given by F(E4) = 1.22 x 10-*E 9 I(Jfllr*C*llJi)lz
[Y,]
SHELL-THEORY AND Pb~°8(II)
31'~
A g a i n the units o f energy used are M e V a n d the u n i t o f length is I fm. The radial wave f u n c t i o n s o f ref. l l ) were used, as before, for the calculation o f (12fr*fll). The result is F = 6.79 x 10610.783 B ( h ~ ~ ) + 1.502 B ( h ~ f~)]2.
e.c
i;'/~i,,/~ 0
~
0
7 ° ~ E(4+)
.....
O.
,
o
o
o
~ ,-~ , ~ , - t h,~/~hg/a
~
00 ~ ~
~
h,,/z%/z
0 0
EXP 4.0
0 . . . .
0
0
~
0
~
0
l,~/zgg/z --~
I
1.5
&E
,I
1.0
0.5
Fig. 2. Theoretical predictions for the or = 4+ levels of Pb2°L The abscissa, AE, is the h~-s~ separation in MeV and is varied from 1.84 MeV down to 0.64 MeV. The m a x i m u m o f this expression, subject to the n o r m a l i z a t i o n constraint, is 1.95 x 107 sec -1, too small by a factor o f a b o u t 10. T h e wave f u n c t i o n o f table 2 give values r a n g i n g from 2 x 106 sec -~ to 6 x 10a sec -1, 30 to 100 times too small. TABLE 2 Eigenfunctions for the lowest 4+ level in Pb ~°8 for different values of the s½-h~ separation. s~-h~k separation in MeV 1.84 1.54 1.24 0.94 0.64
. 1. g~. 1¥ 0.916 0.895 0.868 0.833 0.790
.
Eigenfunction .be 1 1¥ h~ 1 h~ 0.043 0.048 0.053 0.059 0.065
0.036 0.039 0.043 0.047 0.050
h~- i f~ 0.398 0.442 0.492 0.548 0.608
4. Conclusions It was s h o w n i n (I) that the T a m m - D a n c o f f shell theory failed to predict a 3 level anywhere near the k n o w n level at 2.6 MeV. The present calculation shows that this theory c a n n o t a c c o u n t for the large E3 t r a n s i t i o n rate of this level to the g r o u n d state. I n the case of the 4 + level the T a m m - D a n c o f f theory predicts a level at the cor-
318
W. T. PINKSTON
rect energy, but its E4 transition rate is far too small. Tamura and Udagawa 13) have treated the 3- and 4 + levels in Pb 2°8 using the random phase approximation. With the interparticle potential parameters adjusted to give the correct energy values, they find the transition rates to be in good agreement with experiment. The superiority of the random phase approximation over the Tamm-Dancoff approximation is thus clearly established, but a calculation of both the energy levels and the transition rates, making use of two-body forces takeff from analyses of scattering data, is needed in order to determine the quantitative success of the theory. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) A. M. Lane and E. D. Pendlebury, Nuclear Physics 15 (1960) 39 Crannell, Helm, Kendall, Oeser and Yearian, Phys. Rev. 123 (1961) 923 Carter, Pinkston and True, Phys. Rev. 120 (1960) 504 M. Baranger,, Phys. Rev. 120 (1960) 957 and references contained therein T. Marumori, Progr. Theoret. Phys. (Kyoto) 24 (1960) 331; D. J. Thouless, Nuclear Physics 22 (1961) 78 S. Fallieros, Ph. D. Thesis, University of Maryland, 1959; S. Fallieros and R. A. Ferrell, Phys. Rev. 116 (1960) 660 J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, New York, 1952) pp. 595--599 G. Racah, Phys. Rev. 62 (1942) 438 D. M. Brink and G. R. Satchler, Nuovo Cim. 4 (1956) 549 J. Blomquist and S. Wahlborn, Arkiv fOr Fysik 16 (1960) 545 N. Newby and E. J. Konopinski, Phys. Rev. 115 (1959) 434 T. Tamura and T. Udagawa, unpublished.
I
Nuclear Physics 37 (1962) 312--318; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
S H E L L T H E O R Y A N D P b z°s (II) W. T. PINKSTON t Vanderbilt University, Nashville, Tennessee Received 21 May 1962 Abstract: The 3- and 4 + energy levels predicted for Pb 2°8 by the shell model in the usual Tamm-Dan-
coff approximation are studied. The energy of the 3- level was reported in a previous calculation, and is not in agreement with experiment. The predicted wave function is used to calculate the E3 transition rate to the ground state. The rate is found to be too small by a factor of about 20. The energies and wave functions for the 4+ levels are calculated. A level of the correct energy is predicted by the E4 transition rate to the ground state is 30 to 100 times too small.
1. I n t r o d u c t i o n The systematic a p p e a r a n c e o f 2 + first excited states in n o n - d e f o r m e d nuclei is consistent w i t h the d e s c r i p t i o n o f such states as arising f r o m q u a d r u p o l e surface v i b r a t i o n s a b o u t a spherical e q u i l i b r i u m shape 1) C o n s i d e r a b l e evidence z,3) has recently accrued for the existence o f octupole a n d L = 4 v i b r a t i o n s in a d d i t i o n to L = 2 oscillations. I n the experiments o f C r a n n e l l et aL 3) strong E2, E3 a n d E4 t r a n s i t i o n s were observed resulting f r o m the inelastic scattering o f high energy electrons f r o m m e d i u m weight a n d h e a v y nuclei. A c o m p a r i s o n o f the e x p e r i m e n t a l t r a n s i t i o n rates to single-particle estimates i n d i c a t e d enhancements as large as 200, verifying the collective n a t u r e o f the excited states. A l t h o u g h these excited states are often described in terms o f surface oscillations, a m o r e interesting a n d f u n d a m e n t a l p r o s p e c t is that o f giving an e x p l a n a t i o n o f collective p h e n o m e n a in terms o f the m o t i o n s o f i n d i v i d u a l nucleons. The usual starting p o i n t o f such calculations is the nuclear shell model. The collective states are t h e n t h o u g h t o f as consisting o f r a t h e r c o m p l i c a t e d s u p e r p o s i t i o n s o f excited configurations. A d o u b l y m a g i c nucleus such as Pb 2°8 is o f p a r t i c u l a r interest for such considerations since the large energy g a p between m a j o r shells will restrict these configurations to a relatively small n u m b e r o f 1-hole 1-particle (1H-1P) configurations. T w o m e t h o d s exist for calculating the energies a n d excitation p r o b a b i l i t i e s ofcoUect i r e states. T h e usual p r o c e d u r e o f shell theory, sometimes referred to as the T a m m D a n c o f f a p p r o x i m a t i o n or the m e t h o d o f configuration interaction, consists o f d i a g o n a l i z i n g that p a r t o f the H a m i l t o n i a n which connects 1H-1P states to one another. This results in a n u n m o d i f i e d g r o u n d state wave function for a d o u b l e m a g i c nucleus a n d excited state wave functions which are linear s u p e r p o s i t i o n s o f t Supported by U. S. National Science Foundation. 312
SHELL-THEORY
AND
313
Pb~°8(lI)
1H-1P states. This procedure has been carried out for the odd parity states of Pb 2°s by Carter, Pinkston and True,4)t, but the energy of the lowest 3- level was predicted about one MeV higher than observed. Recently much work has been done 5, 6) on a more accurate method known as the r a n d o m phase approximation, a method first applied to nuclear shell theory calculations by Fallieros and Ferrell 7). ]For a nucleus such as Pb 2°s this procedure essentially admixes 2H-2P states into the ground state wave function and 3H-3P states into the excited state functions in addition to the 1H-1P states. In the present work the E3 transition rate from the first excited state of Pb 2°s to the ground state is calculated with the wave functions resulting from the calculations of (I) and is compared to experiment 3). Little if any enhancement is obtained by using a superposition of wave functions rather than a single 1H-1P state. The d = 4 + states of Pb 2°s are predicted on the T a m m - D a n c o f f approximation. The predicted wave functions fail to predict a sufficiently large E4 transition probability, but the theory does predict an energy level at about 4 MeV, consistent with experiment. 2. The
J = 3-
Level
The hole-particle configurations contributing to the J = 3 - states of Pb 2°8 were assumed in (I) to be f~ t gb f~-i i~, d~ 1 h~, s~-1 f~ and d~ I f~. The single particle energy levels involved are shown in fig. 1 and were taken from experimental data
¢
0.71 ¸
¢ 0.90
0.79 gg/~
i
1
f 3.44 t
i t 0.57 0.33 .f
4.26
Pl/2 f~/~
S
d3/2
0.35 1
1.49
0.73 h,,/~ NEUTRONS
....
_
_
i_ _ _
PROTONS
Fig. I. Single particle energy levels in Pb region (not to scale). T h e h~k level is d a s h e d indicating that its position is experimentally u n k n o w n . T h e value given is t a k e n f r o m a theoretical calculation 18). * Referred to as (I).
314
W. T. PINKSTON
on the levels of nuclei resulting from one proton or neutron added to or subtracted from Pb 2°8. The energy levels and wave functions were calculated for the following choices of the interparticle potential: a) a Gauss shape with Serber space exchange and a triplet strength 1.5 times the singlet strength, b) a Gauss shape with Rosenfeld exchange, c ) a 6-function interaction with triplet strength 1.5 times the singlet strength. Harmonic oscillator radial wave functions were used in calculating the radial integrals of the potential. The results are listed in table 1. The lowest energy level is predomiT~L~ 1 Energy level and eigenfunction predictions for the lowest 3- level in Pb 2°8 for 3 choices of potential Eigenfunction
Potential
Energy (MeV)
f~- 1 g~
f~ 1 i~
d~- 1 h}
s~ t f~
d~ 1 f}
Gauss-Serber Gauss-Rosenfeld Zero Range
3.66 3.38 3.13
0.287 0.124 0.239
--0.576 --0.616 --0.617
--0.711 --0.737 --0.687
--0.261 --0.237 --0.274
+0.106 +0.082 +0.122
nantly d~" ~ ~ due to the attractive Coulomb force between proton holes and particles. This brings it below the f~- 1 N neutron level which is lowest in zero order. Its energy is predicted to be 3.13 MeV, 0.5 MeV higher than the experimental value of 3.6 MeV. For the more realistic potentials the discrepancyis closer to 1.0 MeV. It was, therefore, concluded in (I) that the superposition of the 5 configurations above is not equivalent to an L = 3 surface oscillation. N o calculation of the E3 transition probability was made, however, and it is reported here for completeness. Expressions for the transition rates for electromagnetic multipole transitions are given by Blatt and WeisskopfS). For E3 transitions the rate in sec -1 can be expressed as F(E3) = 318E 7 KJfllrafa(O)llJi>12
[J,]
(1)
Here energy is expressed in MeV and length in units of 1 fm = 10 - l a cm. The notation [n] -= 2n + 1 is used. The quantity C3M is an unnormalized spherical harmonic tensor as defined by Racah 9) and Racah's definition of a reduced matrix element is used. The matrix element for the de-excitation o f a particle-hole state in a closed shell is given by Brink and Satchler lo). With the Racah convention for reduced matrix elements, we have
(2)
A choice must be made for the radial functions. The calculations here were made with the radial functions for nucleons moving in a Saxon potential well as tabulated by Blomquist and Wahlborn 11).
SHELL-THEORY AND Pb~°8(II)
315
If the excited state wave function is written
0 = ~, B(jlJz)O(J; ~JzJ),
(3)
then eq. (1) reduces to r = 3.65 x 10s I 2.07B(d~h0+2.17B(s~f~) - 1.18B(d~f~)] 2.
(4)
All three of the wave functions of table 1 yield about the same result, F ~ 2 x 109 see - j , which is too small compared to the experimental value a) of (3.80__ 1.4)x 10 ~° sec -~ by a factor of about 20. The calculated value is almost exactly the value which one would obtain for a pure h~ particle to d~-hole transition, indicating that the superposition over several configurations does not enhance the transition here. One can in fact very easily calculate the maximum value of F which can arise from an expression such as eq. (4). An expression of the form ( ~ , a(i) x(i)) 2, in which the a(i) are known constants and the x(i) are variables subject to the constraint ~ x ( i ) 2 = 1, has as its maximum ~ a ( i ) 2. Applied to eq. (4) this results in a maximum of 3.8 x 1 0 9 sec- 1, 10 times smaller than F~p. Thus it is impossible to reproduce the experimental value with any linear combination o f the proton I H - I P configurations d~- ~ h~, s~ ~ f~ and d~-~ f~. 3. The J - - 4
+ Level
The inelastic scattering results of ref. 3)imply strongly excited 4 + states at 4.3 MeV in both Pb 2°s and Bi 2°9. The E4 tritnsition rate for the de-excitation of this level in Pb 2°s was found to be F = (2.23___0.7)x 10 s sec -~. In this section we extend the methods of (I) to even parity 1H-1P states and compare the results to this data. In the case of even parity 1H-IP states there is some uncertainty as to the zero order energies of the proton states. A J = 4 + proton level can arise either from the excitation of an h~ proton hole state and an odd parity proton particle state such as or f~ or to the excitation o f a d~ proton hole and an i~ particle. Due to the sparcity of data on the levels of T12°7 no even parity hole state is known. The i~ particle state has not been identified in Bi 2°9 but probably corresponds to the 1,61-MeV level o f that nucleus. We must be guided here by theoretical calculations. Blomquist and Wahlborn ~~) have calculated the single particle energy levels and wave functions for nucleons moving in a Saxon well with parameters appropriate to Pb region. Their energy calculations agree well with the known level spectra of Pb 2°7, T12°7, Pb 2°9 and Bi ~°9. We therefore assume that they will be fairly reliable for the as yet unobserved levels. These calculations predict the h~ hole level 1.84 MeV below the s½ ground state ofT1 z°7 and the d I hole level 0.65 MeV deeper. The h÷ level is listed in fig. 1 but is dashed to indicate that its position is uncertain. We shall ignore the dl level since the d21 i~ will be 1.36 MeV higher than h;~ 1 f,,_and 2.26 MeV higher than h;~ 1 hl. We therefore take as the dominant proton configurations, hT~1 f~ and h;~ ~ f,_. F r o m fig. 1 it is clear that the important neutron configurations are i2~~ gl andi;~ 1 i÷, The only other possibility is f~- 1 j .~ which is probably several MeV higher.
316
W. T. PINKSTON
The energy levels, assuming these four configurations, have been calculated using a zero range internucleon potential without exchange. Previous investigations '~, 22) in this region indicate that the main features of the energy spectrum are not sensitive either to the shape or to the exchange character of the forces. The matrix elements for such a potential can be evaluated using the methods of Brink and Satchler lo) and the two body results o f Newby and Konopinski 22). For identical particles the hole particle interaction is given by a sum of direct and exchange terms:
(j ? lje JI VIj3 tj, d) = [Jl]~[J2-l~[J3]~[J*]¢ (j~j2½-½}dO)(j3j,½-½[dO}F°(d+x).
2Es]
(5)
The quantity F ° is the Slater integral of a delta function 12), and d and x are given by d ~--- ( _ _ ) / 2 + / 4 + 1
I(__)jl+j3+l,+13+1 + ([Jl ~ + ( -- .)Y'+j~ + J[J2])([Ja] 4 J ( J + 1) + ( - )j~ +J" +J[J*])l-,_~ (6) x = 2 ( - ) "h+i~+l.
For unlike particles, as was noted in (I), the matrix element is the same as the like particle exchange term (d = 0). (An interesting consequence of these formulas is that the diagonal matrix elements are attractive for i7~~ ~ and h~ ~ f~. The usual statement that hole-particle forces have the opposite sign of particle-particle forces is of course rigorously true only in the long range limit in which exchange integrals are negligible compared to direct integrals.) For the proton configurations there is also a Coulomb potential. The Coulomb matrix elements can be estimated very easily. The Coulomb force is a long range force; therefore the off-diagonal matrix elements are expected to be negligible. The diagonal elements can be approximated by the Coulomb energy of 2 protons uniformly distributed over a sphere of radius R = ro A ~ with ro = 1.2 fm. Then we have
(Vc> ~ 6eZ/R = 0.24 MeV. This is in good agreement with more accurate calculations 4, 22). The results of the diagonalization of the energy matrix are plotted in fig. 2. The dependence of the results on the location of the hq hole state was checked by varying the h~-s~ separation from 1.84 MeV to 0.64 MeV in units o f 0.3 MeV. The lowest level has an energy close to the observed value of 4.3 MeV, but it is predominantly i~ ~ g~, a neutron state. The wave functions of this level is listed in table 2 for several values of the h~-s½ separation. The transition rate in sec- ~ for an E4 transition is given by F(E4) = 1.22 x 10-*E 9 I(Jfllr*C*llJi)lz
[Y,]
SHELL-THEORY AND Pb~°8(II)
31'~
A g a i n the units o f energy used are M e V a n d the u n i t o f length is I fm. The radial wave f u n c t i o n s o f ref. l l ) were used, as before, for the calculation o f (12fr*fll). The result is F = 6.79 x 10610.783 B ( h ~ ~ ) + 1.502 B ( h ~ f~)]2.
e.c
i;'/~i,,/~ 0
~
0
7 ° ~ E(4+)
.....
O.
,
o
o
o
~ ,-~ , ~ , - t h,~/~hg/a
~
00 ~ ~
~
h,,/z%/z
0 0
EXP 4.0
0 . . . .
0
0
~
0
~
0
l,~/zgg/z --~
I
1.5
&E
,I
1.0
0.5
Fig. 2. Theoretical predictions for the or = 4+ levels of Pb2°L The abscissa, AE, is the h~-s~ separation in MeV and is varied from 1.84 MeV down to 0.64 MeV. The m a x i m u m o f this expression, subject to the n o r m a l i z a t i o n constraint, is 1.95 x 107 sec -1, too small by a factor o f a b o u t 10. T h e wave f u n c t i o n o f table 2 give values r a n g i n g from 2 x 106 sec -~ to 6 x 10a sec -1, 30 to 100 times too small. TABLE 2 Eigenfunctions for the lowest 4+ level in Pb ~°8 for different values of the s½-h~ separation. s~-h~k separation in MeV 1.84 1.54 1.24 0.94 0.64
. 1. g~. 1¥ 0.916 0.895 0.868 0.833 0.790
.
Eigenfunction .be 1 1¥ h~ 1 h~ 0.043 0.048 0.053 0.059 0.065
0.036 0.039 0.043 0.047 0.050
h~- i f~ 0.398 0.442 0.492 0.548 0.608
4. Conclusions It was s h o w n i n (I) that the T a m m - D a n c o f f shell theory failed to predict a 3 level anywhere near the k n o w n level at 2.6 MeV. The present calculation shows that this theory c a n n o t a c c o u n t for the large E3 t r a n s i t i o n rate of this level to the g r o u n d state. I n the case of the 4 + level the T a m m - D a n c o f f theory predicts a level at the cor-
318
W. T. PINKSTON
rect energy, but its E4 transition rate is far too small. Tamura and Udagawa 13) have treated the 3- and 4 + levels in Pb 2°8 using the random phase approximation. With the interparticle potential parameters adjusted to give the correct energy values, they find the transition rates to be in good agreement with experiment. The superiority of the random phase approximation over the Tamm-Dancoff approximation is thus clearly established, but a calculation of both the energy levels and the transition rates, making use of two-body forces takeff from analyses of scattering data, is needed in order to determine the quantitative success of the theory. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) A. M. Lane and E. D. Pendlebury, Nuclear Physics 15 (1960) 39 Crannell, Helm, Kendall, Oeser and Yearian, Phys. Rev. 123 (1961) 923 Carter, Pinkston and True, Phys. Rev. 120 (1960) 504 M. Baranger,, Phys. Rev. 120 (1960) 957 and references contained therein T. Marumori, Progr. Theoret. Phys. (Kyoto) 24 (1960) 331; D. J. Thouless, Nuclear Physics 22 (1961) 78 S. Fallieros, Ph. D. Thesis, University of Maryland, 1959; S. Fallieros and R. A. Ferrell, Phys. Rev. 116 (1960) 660 J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, New York, 1952) pp. 595--599 G. Racah, Phys. Rev. 62 (1942) 438 D. M. Brink and G. R. Satchler, Nuovo Cim. 4 (1956) 549 J. Blomquist and S. Wahlborn, Arkiv fOr Fysik 16 (1960) 545 N. Newby and E. J. Konopinski, Phys. Rev. 115 (1959) 434 T. Tamura and T. Udagawa, unpublished.