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Strutinsky smoothing and the partition function approach
l
I.E.2
I
Nuclear Physics A207 (1973) 538--544; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
STRUTINSKY S M O O T H I N G AND T H E P A R T I T I O N F U N C T I O N A P P R O A C H B. K. J E N N I N G S Physics Department, McMaster University, Hamilton, Ontario, Canada t Received 7 March 1973
Abstract: The Strutinsky shell correction is analytically shown to be equivalent to the shell correction obtained from the partition function approach, provided certain restrictions on the Strutinsky smoothing parameter are met.
I. Introduction
Recently considerable interest has been shown in calculating shell correction energies 1) of nuclei. This is usually done by using the Strutinsky prescription 2, 3). In this method one first considers the single-particle quantum mechanical density of states. One then finds a smoothed single-particle density of states by replacing each delta function in the single-particle quantum mechanical density of states by a Gaussian multiplied by a curvature function. The shell correction is found by taking the difference between the energy calculated from the single-particle quantum mechanical density of states and the energy calculated from the smoothed single-particle density of states. Other methods have been proposed for finding the shell corrections. They include the entropy method 4), the Green function method 5, 6), the large-volume method v) and an extended Thomas-Fermi m e t h o d 8,16,17). Another method, the partition function method 9), is somewhat similar to the Strutinsky method in that they both deal with the single-particle density of states. In the partition function method one finds the smoothed density of states from the corresponding semi-classical singleparticle partition function, which we define in a precise way later in this paper. In this paper it will be shown analytically that the shell correction obtained from the partition function approach is the same as the shell correction obtained by the Strutinsky method. This was demonstrated numerically 9) for a few model cases where the semi-classical partition function is analytically known; our work is a general proof of this equivalence, and lends some insight into the Strutinsky smoothing procedure. The proof will be given in two steps. The first step consists of showing that the same density of states is obtained from applying the Strutinsky smoothing to the quantum mechanical single-particle density of states and to the semi-classical single-particle density of states if the smoothing parameter is large enough. Work supported by the National Research Council of Canada.
538
STRUTINSKY SMOOTHING
539
The second step consists of showing that the same energy is obtained from the semiclassical single-particle density of states as from the Strutinsky smoothed semi-classical density, if the smoothing parameter is small enough. Consequently, within certain bounds of the smoothing parameter, it follows that the same shell correction to energy is obtained fi;om the Strutinsky procedure as from the semi-classical partition function approach. 2. Smoothing and the partition function The Strutinsky smoothed density of states, denoted ~(e), is obtained from a density of states, 9(~), by a Gaussian smoothing modulated with a curvature function: =
L
l _
7\,/~
In this expression g(e) can be either the single-particle quantum mechanical density' of states, denoted by gq.m.(g), o r the single-particle semi-classical density of states, denoted gs.c.(e). The L,~ is an associated Laguerre polynomial 13); 7 is the smoothing parameter. The Strutinsky smoothed quantum mechanical density of states is denoted by 9q.m.(e), and the Strutinsky smoothed semi-classical density of states is denoted by We now take the two-sided Laplace transform of 0(~) (see appendix) and obtain:
-oc = Z(/~
e_~.,, e
L1[y2]dy '
(2)
where 13 is the transform variable and Z(13) is the Laplace transform of 9(e), thus it is just the partition function corresponding to 9(e). The partition function corresponding to 9~.~.(~) will be denoted Z~.~.(13), and the partition function obtained from ffq.m.(£:) will be denoted Zq.m,(13). The integral in (2) is done in the appendix and the following expression for 5¢[~(e)] obtained: S[0(e)] Z(/?)e+''2~P(137). =
Here P(flT) is a polynomial of order 2n in the variable/~7. We now consider the identity: - l[~_(~[Oq.m.(/;)]]
:
gq.m.(B),
where ~ ' - ~ denotes the inverse Laplace transform. The inverse Laplace transform can be written 10) as:
1 fc+i~
540
B.K. JENNINGS
where c is a constant chosen such that the line o f integration lies in the strip o f convergence o f the Laplace transform. In this case c can be chosen to be any small positive number. Writing out 5('- 1 [~('[~q.m.(e)]] explicitly we have: 1
~c+ioo Zqm(fl)e~2"~2P(flY)e$~dfl"
9q'm'(8) = ~ - l[~O[Oq'm'(8)]] = ~Je-loo
(3)
In (3) we make the substitution ]3 = c+ix and obtain:
0q.m.(qg) --
d_Zq.m.(C+ix)e-:ix~'2e~ixC'2e'X~P[(c+ix)/;~]dx.(4)
e~.C~+ct f ~
D u e to the presence of the factor e - ~ the only significant contribution to the integral comes from x near zero s). Also since the positive p a r a m e t e r e can be chosen as small as we like as long as it is not zero, we see that the integral depends only on Zq.m.(]~ ) near/3 = O. We n o w need an expansion of the q u a n t u m mechanical partition function valid for small ft. One such expansion that has been used 9) is the semi-classical expansion given by: Zs.c.(fl ) --= Zc(fl)(1 + h)~l(fl) +
h2z2(fl) +
...).
Here Z~(fl) is the classical partition function and the X are defined in terms of the oneb o d y potential. Other expansions valid for small fl, such as a Laurent expansion, :3 . 0 4 ! - ¢:
I
"
.o3!
.02=
20
I
/
t
/ 4// ;' , ]
i i
.}0~
-.2:2i
i
-.-~3i
I
~Oa L -
L
3
Fig. 1. A plot of AE, representing/~--/~s.¢. (solid line), and /~s.c.--Es.¢. (dotted line), as a function of 7 for 4, 20 and 100 particles in an axially symmetric harmonic oscillator well with a deformation parameter of 0.4. Both and ~, are expressed in units of ht~. The range of y for which /~ = Es.c. can be seen by comparing the solid and dotted line.
AE
STRUTINSKY
SMOOTHING
541
could also be used, but we will use the semi-classical expansion in this paper for the sake of definiteness. Eq. (4) now becomes: e-,[-c2~,2 + c~:~ oG
gq.m.(~)
-
-
~f
.:-- dxZs.~.(c+ix)e-+":~'2e~i~':~e'X~P[(c+ix)/7].
The right-hand side of this equation is the Strutinsky smoothed semi-classical density of states. Thus we have:
and hence the energy, E, calculated from gq.m.(e)is the same as the energy, E ..... calculated from j~.c.(e). F r o m the form of Zs.~.(/~)we see that ~.¢.(e) is expressed as a series which has an infinite number of terms. Only terms that do not make a significant contribution to the energy calculated from gs.~.(e) can be neglected. For example, the first three terms must be kept for a harmonic oscillator although the third term is a derivative of a delta function. It should be noted that we have not implied that the two partition functions are approximately equal for all/~. In fact they will differ considerably for large ~. The equivalence of E and E .... was checked numerically for the case of an axially symmetric deformed harmonic oscillator well. The solid line in fig. 1 shows E - E .... as a function of ~, for three different particle numbers. The lower bound on ~, in this case is seen to be about 1 hco.
3. Strutinsky smoothed semi-classical density of states In this section it will be shown that the same energy is obtained from ~.c.(~) and gs.c.(e). This is done in two steps. First it is shown that the chemical potential obtained in both cases is the same and then it is shown that the energies are the same. The chemical potential is obtained from the equation N = ~ y(e)de, where N i s the number of particles and/x the chemical potential. In the case of a Strutinsky smoothed density of states this becomes N = J'~_~ ~(~)d~. To show that the same chemical potential is obtained from both ~s.~.(~) and g .... (~) it must be shown that: o gs.¢.(,~)de = 0. The difference of these two integrals will be denoted by IN. Substituting for .~.¢. in IN from (1) we obtain:
IN
=f oO~.~.(e)de-f"- odef f gs.o.(e')e-(~-~')2/'2L~[(,-e')2/~,2]dd. yx/Tr
In the second term the order of the integration is changed and the two dummy vari-
542
B. K. JENNINGS
ables interchanged. The two terms can be rearranged to give :
1
iN =j'fgs.o.(e)deIl_ f"-~ Tx/TC -
f u,3gs.~.(e)deft*
e- (~:-~:')2/~2 , L~E(e - e')2/y2]de '.
(6)
We now m a k e the substitution y = ( e ' - e ) / 7 , dy = de'/y and do the g-integration by parts. After rearranging terms we obtain: IN=
fo gs.c.(e)de- fode(fo g~.¢.(e')de')
7x/rc C½,[(It-e)2/72].
e-('-~)~/'~
The Gaussian in the second integral d a m p s out the integrand for e far from p; consequently it follows that if ~ can be a p p r o x i m a t e d by a polynomial near IL it can be replaced by a polynomial everywhere without appreciably affecting the value of the integral. Similarly the lower limit can be replaced by - o0. I f the polynomial used to approximate ~) is of order less than or equal to 2 N + l, it follows immediately that I N is zero by the self-consistency condition 2, a, 11) for the Strutinsky smoothing. Thus the same p is obtained from both gs.¢.(e) and ~ .... (e) if ~ is small enough. N o w consider the energies Es.~. = S~ and E .... = S~ e//s.~.(e)de for equal Fermi energies. The difference E~.~.-E .... will be denoted by I E. We n o w repeat the procedure done for I N on I E to obtain an expression analogous to (6) which is integrated by parts to give:
gs.¢.(e')de'
gs.¢.(e')de'
egs.~.(e)de
!E = //
fi'g .... ()e dE --,u fo (fo g~.¢.(e')de' ) 7~/x1 e-("~-"?/';2L~[(Ia-e)2/~2]de + +
-¢
g~.¢.(e')dd g .... (e')de'
-I+J_~
./- ~'_
\/rC
v/re
e
y2)dy de.
g~.¢.(e)
The first two terms are just /~ times I N while the last two terms are I N with replaced by .[~)gs.c.(e')de'. It thus follows from the result obtained for I N that I E is zero if j'~ can be a p p r o x i m a t e d by a polynomial of order less than 2 N + 1 for e near/~ and ~ is small enough. We now have the result E .... - E~.,. = 0. This was checked numerically for the case o f an axially symmetric deformed harmonic oscillator, and a plot of E .... - Es.~. against y for three different particle numbers is shown by the dashed line in fig. 1. The upper limit is seen to be less than 3hco and to increase with particle number.
g~.¢.(e')de'
STRUTINSKY SMOOTHING
543
4. C o n c l u s i o n
It has n o w been shown that if certain restrictions on ), and g~.~.(e) are met, E E~.~ and E .... ~ E~.¢.. It follows immediately that E ~ Es.~. and hence the Strutinsky procedure yields the same shell corrections as the partition function approach. It should be noted that two restrictions are placed on ),. The first is a lower b o u n d associated with part of one the proof. The second is an upper bound associated with the second part of the proof. I f these two restrictions are met, the Strutinsky shell correction will be independent of 7 and equal to the shell corrections obtained f r o m the partition function approach. The author would like to thank Professor R. K. Bhaduri and Mr. C. K. Ross for useful discussions. Appendix
We here consider the Laplace transform of: =
- ~-~
~/Tr
Taking the Laplace transform of both sides we have: c~[j(e)] =
-
-(e)da =
e-P~de de'g(e') e - ~o ~0 ~/~
Lj[(e_e,)2/),2l,
where the two-sided Laplace transform is used because ~(e) is not zero for e < 0. Interchanging the orders o f integration and making the change of variable e = z' + 7Y in the e integration, we obtain ~5~'[~#(e)] --
=
g ( d ) d d ~-oo e -p(~'' +~'' e-'~ x/zr
fo °
L~"(y2)dy
f ~o ~ oo _ y2 o e_~.,g( ,)d e, J - ~ e_p~y ex/~ L~(y2)dy"
The second integration is independent of d. The first integral is just the one-sided Laplace transform of g(e'); that is the partition function Z(fl). Thus d [ O ( e ) ] = Z(fl)
- ~oe -~'" e-'~ ~/~
L~(y2)dY"
N o w consider the y-integration:
f
o9
--y2
Iy = •~-~ e _ ~ y ex/~ L~(y2)dy. Using identities 12) involving the associated Laguerre polynomial we obtain:
L-2(x 2) = L~(:,=). m=O
(A.I)
544
B . K . JENNINGS
Also we have 12):
L: (x
_ ( - 1)"H2.(x). n!2 2"
Combining these two equations we see that:
L (x = [ ( - 0 2 H:.(x). .,:o m!2 2m
Substituting this into (A.1) we have:
Iy = ~ (__l)m f~:
-- ~,2
e -t~''e
m=o r n ! 2 2 m , j - ~
H2m(y)dy . \,/n
The integral is the two-sided Laplace transform of (e (yfl)2me+r2a2. Hence we obtain:
_y2
/~/n)Hzm(y ), which is t4)
m=0 n!22
where P(yfl) is a polynomial of order 2n in yfl. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11 ) 12) 13) 14) 15) 16) 17)
J. R. Nix, Ann. Rev. Nucl. Sci. 22 (1972), and references therein V. M. Strutinsky, Nucl. Phys. A95 (1967) 420 V. M. Strutinsky, Nuel. Phys. A122 (1968) 1 V. S. Ramamurthy and S. S. Kapoor, preprint R. Balian and C. Bloch, Ann. of Phys. 63 (1971) 592 R. Balian and C. Bloch, Ann. of Phys. 60 (1970) 401 Ph. J. Siemens and A. Sobiczewski, Phys. Lett. 41B (1972) 16 D. H. E. Gross, Phys. Lett. 42B (1972) 41 R. K. Bhaduri and C. K. Ross, Phys. Rev. Lett. 27 (1971) 606 Van der Pol and H. Bremmer, Operational calculus based on the two-sided Laplace integral (Cambridge University Press, London, 1955) p. 16 C. F. Tsang, Ph.D. thesis, University of California; Lawrence Radiation Lab. report UCRL18899 (1969) M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, New York, 1965) p. 785 M. Abramowitz and l. A. Stegun, op. cit., p. 775 Van der Pol and H. Bremmer, op. cit., p. 83 M. Abramowitz and I. A. Stegun, op. cit., p. 786 A. S. Tyapin, Yad. Fiz. 11 (1970) 98; Soy. J. Nucl. Phys. 11 (1970) 53 A. S. Tyapin, Yad. Fiz. 14 (1971) 88; Soy. J. Nucl. Phys. 14 (1972) 50
I.E.2
I
Nuclear Physics A207 (1973) 538--544; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
STRUTINSKY S M O O T H I N G AND T H E P A R T I T I O N F U N C T I O N A P P R O A C H B. K. J E N N I N G S Physics Department, McMaster University, Hamilton, Ontario, Canada t Received 7 March 1973
Abstract: The Strutinsky shell correction is analytically shown to be equivalent to the shell correction obtained from the partition function approach, provided certain restrictions on the Strutinsky smoothing parameter are met.
I. Introduction
Recently considerable interest has been shown in calculating shell correction energies 1) of nuclei. This is usually done by using the Strutinsky prescription 2, 3). In this method one first considers the single-particle quantum mechanical density of states. One then finds a smoothed single-particle density of states by replacing each delta function in the single-particle quantum mechanical density of states by a Gaussian multiplied by a curvature function. The shell correction is found by taking the difference between the energy calculated from the single-particle quantum mechanical density of states and the energy calculated from the smoothed single-particle density of states. Other methods have been proposed for finding the shell corrections. They include the entropy method 4), the Green function method 5, 6), the large-volume method v) and an extended Thomas-Fermi m e t h o d 8,16,17). Another method, the partition function method 9), is somewhat similar to the Strutinsky method in that they both deal with the single-particle density of states. In the partition function method one finds the smoothed density of states from the corresponding semi-classical singleparticle partition function, which we define in a precise way later in this paper. In this paper it will be shown analytically that the shell correction obtained from the partition function approach is the same as the shell correction obtained by the Strutinsky method. This was demonstrated numerically 9) for a few model cases where the semi-classical partition function is analytically known; our work is a general proof of this equivalence, and lends some insight into the Strutinsky smoothing procedure. The proof will be given in two steps. The first step consists of showing that the same density of states is obtained from applying the Strutinsky smoothing to the quantum mechanical single-particle density of states and to the semi-classical single-particle density of states if the smoothing parameter is large enough. Work supported by the National Research Council of Canada.
538
STRUTINSKY SMOOTHING
539
The second step consists of showing that the same energy is obtained from the semiclassical single-particle density of states as from the Strutinsky smoothed semi-classical density, if the smoothing parameter is small enough. Consequently, within certain bounds of the smoothing parameter, it follows that the same shell correction to energy is obtained fi;om the Strutinsky procedure as from the semi-classical partition function approach. 2. Smoothing and the partition function The Strutinsky smoothed density of states, denoted ~(e), is obtained from a density of states, 9(~), by a Gaussian smoothing modulated with a curvature function: =
L
l _
7\,/~
In this expression g(e) can be either the single-particle quantum mechanical density' of states, denoted by gq.m.(g), o r the single-particle semi-classical density of states, denoted gs.c.(e). The L,~ is an associated Laguerre polynomial 13); 7 is the smoothing parameter. The Strutinsky smoothed quantum mechanical density of states is denoted by 9q.m.(e), and the Strutinsky smoothed semi-classical density of states is denoted by We now take the two-sided Laplace transform of 0(~) (see appendix) and obtain:
-oc = Z(/~
e_~.,, e
L1[y2]dy '
(2)
where 13 is the transform variable and Z(13) is the Laplace transform of 9(e), thus it is just the partition function corresponding to 9(e). The partition function corresponding to 9~.~.(~) will be denoted Z~.~.(13), and the partition function obtained from ffq.m.(£:) will be denoted Zq.m,(13). The integral in (2) is done in the appendix and the following expression for 5¢[~(e)] obtained: S[0(e)] Z(/?)e+''2~P(137). =
Here P(flT) is a polynomial of order 2n in the variable/~7. We now consider the identity: - l[~_(~[Oq.m.(/;)]]
:
gq.m.(B),
where ~ ' - ~ denotes the inverse Laplace transform. The inverse Laplace transform can be written 10) as:
1 fc+i~
540
B.K. JENNINGS
where c is a constant chosen such that the line o f integration lies in the strip o f convergence o f the Laplace transform. In this case c can be chosen to be any small positive number. Writing out 5('- 1 [~('[~q.m.(e)]] explicitly we have: 1
~c+ioo Zqm(fl)e~2"~2P(flY)e$~dfl"
9q'm'(8) = ~ - l[~O[Oq'm'(8)]] = ~Je-loo
(3)
In (3) we make the substitution ]3 = c+ix and obtain:
0q.m.(qg) --
d_Zq.m.(C+ix)e-:ix~'2e~ixC'2e'X~P[(c+ix)/;~]dx.(4)
e~.C~+ct f ~
D u e to the presence of the factor e - ~ the only significant contribution to the integral comes from x near zero s). Also since the positive p a r a m e t e r e can be chosen as small as we like as long as it is not zero, we see that the integral depends only on Zq.m.(]~ ) near/3 = O. We n o w need an expansion of the q u a n t u m mechanical partition function valid for small ft. One such expansion that has been used 9) is the semi-classical expansion given by: Zs.c.(fl ) --= Zc(fl)(1 + h)~l(fl) +
h2z2(fl) +
...).
Here Z~(fl) is the classical partition function and the X are defined in terms of the oneb o d y potential. Other expansions valid for small fl, such as a Laurent expansion, :3 . 0 4 ! - ¢:
I
"
.o3!
.02=
20
I
/
t
/ 4// ;' , ]
i i
.}0~
-.2:2i
i
-.-~3i
I
~Oa L -
L
3
Fig. 1. A plot of AE, representing/~--/~s.¢. (solid line), and /~s.c.--Es.¢. (dotted line), as a function of 7 for 4, 20 and 100 particles in an axially symmetric harmonic oscillator well with a deformation parameter of 0.4. Both and ~, are expressed in units of ht~. The range of y for which /~ = Es.c. can be seen by comparing the solid and dotted line.
AE
STRUTINSKY
SMOOTHING
541
could also be used, but we will use the semi-classical expansion in this paper for the sake of definiteness. Eq. (4) now becomes: e-,[-c2~,2 + c~:~ oG
gq.m.(~)
-
-
~f
.:-- dxZs.~.(c+ix)e-+":~'2e~i~':~e'X~P[(c+ix)/7].
The right-hand side of this equation is the Strutinsky smoothed semi-classical density of states. Thus we have:
and hence the energy, E, calculated from gq.m.(e)is the same as the energy, E ..... calculated from j~.c.(e). F r o m the form of Zs.~.(/~)we see that ~.¢.(e) is expressed as a series which has an infinite number of terms. Only terms that do not make a significant contribution to the energy calculated from gs.~.(e) can be neglected. For example, the first three terms must be kept for a harmonic oscillator although the third term is a derivative of a delta function. It should be noted that we have not implied that the two partition functions are approximately equal for all/~. In fact they will differ considerably for large ~. The equivalence of E and E .... was checked numerically for the case of an axially symmetric deformed harmonic oscillator well. The solid line in fig. 1 shows E - E .... as a function of ~, for three different particle numbers. The lower bound on ~, in this case is seen to be about 1 hco.
3. Strutinsky smoothed semi-classical density of states In this section it will be shown that the same energy is obtained from ~.c.(~) and gs.c.(e). This is done in two steps. First it is shown that the chemical potential obtained in both cases is the same and then it is shown that the energies are the same. The chemical potential is obtained from the equation N = ~ y(e)de, where N i s the number of particles and/x the chemical potential. In the case of a Strutinsky smoothed density of states this becomes N = J'~_~ ~(~)d~. To show that the same chemical potential is obtained from both ~s.~.(~) and g .... (~) it must be shown that: o gs.¢.(,~)de = 0. The difference of these two integrals will be denoted by IN. Substituting for .~.¢. in IN from (1) we obtain:
IN
=f oO~.~.(e)de-f"- odef f gs.o.(e')e-(~-~')2/'2L~[(,-e')2/~,2]dd. yx/Tr
In the second term the order of the integration is changed and the two dummy vari-
542
B. K. JENNINGS
ables interchanged. The two terms can be rearranged to give :
1
iN =j'fgs.o.(e)deIl_ f"-~ Tx/TC -
f u,3gs.~.(e)deft*
e- (~:-~:')2/~2 , L~E(e - e')2/y2]de '.
(6)
We now m a k e the substitution y = ( e ' - e ) / 7 , dy = de'/y and do the g-integration by parts. After rearranging terms we obtain: IN=
fo gs.c.(e)de- fode(fo g~.¢.(e')de')
7x/rc C½,[(It-e)2/72].
e-('-~)~/'~
The Gaussian in the second integral d a m p s out the integrand for e far from p; consequently it follows that if ~ can be a p p r o x i m a t e d by a polynomial near IL it can be replaced by a polynomial everywhere without appreciably affecting the value of the integral. Similarly the lower limit can be replaced by - o0. I f the polynomial used to approximate ~) is of order less than or equal to 2 N + l, it follows immediately that I N is zero by the self-consistency condition 2, a, 11) for the Strutinsky smoothing. Thus the same p is obtained from both gs.¢.(e) and ~ .... (e) if ~ is small enough. N o w consider the energies Es.~. = S~ and E .... = S~ e//s.~.(e)de for equal Fermi energies. The difference E~.~.-E .... will be denoted by I E. We n o w repeat the procedure done for I N on I E to obtain an expression analogous to (6) which is integrated by parts to give:
gs.¢.(e')de'
gs.¢.(e')de'
egs.~.(e)de
!E = //
fi'g .... ()e dE --,u fo (fo g~.¢.(e')de' ) 7~/x1 e-("~-"?/';2L~[(Ia-e)2/~2]de + +
-¢
g~.¢.(e')dd g .... (e')de'
-I+J_~
./- ~'_
\/rC
v/re
e
y2)dy de.
g~.¢.(e)
The first two terms are just /~ times I N while the last two terms are I N with replaced by .[~)gs.c.(e')de'. It thus follows from the result obtained for I N that I E is zero if j'~ can be a p p r o x i m a t e d by a polynomial of order less than 2 N + 1 for e near/~ and ~ is small enough. We now have the result E .... - E~.,. = 0. This was checked numerically for the case o f an axially symmetric deformed harmonic oscillator, and a plot of E .... - Es.~. against y for three different particle numbers is shown by the dashed line in fig. 1. The upper limit is seen to be less than 3hco and to increase with particle number.
g~.¢.(e')de'
STRUTINSKY SMOOTHING
543
4. C o n c l u s i o n
It has n o w been shown that if certain restrictions on ), and g~.~.(e) are met, E E~.~ and E .... ~ E~.¢.. It follows immediately that E ~ Es.~. and hence the Strutinsky procedure yields the same shell corrections as the partition function approach. It should be noted that two restrictions are placed on ),. The first is a lower b o u n d associated with part of one the proof. The second is an upper bound associated with the second part of the proof. I f these two restrictions are met, the Strutinsky shell correction will be independent of 7 and equal to the shell corrections obtained f r o m the partition function approach. The author would like to thank Professor R. K. Bhaduri and Mr. C. K. Ross for useful discussions. Appendix
We here consider the Laplace transform of: =
- ~-~
~/Tr
Taking the Laplace transform of both sides we have: c~[j(e)] =
-
-(e)da =
e-P~de de'g(e') e - ~o ~0 ~/~
Lj[(e_e,)2/),2l,
where the two-sided Laplace transform is used because ~(e) is not zero for e < 0. Interchanging the orders o f integration and making the change of variable e = z' + 7Y in the e integration, we obtain ~5~'[~#(e)] --
=
g ( d ) d d ~-oo e -p(~'' +~'' e-'~ x/zr
fo °
L~"(y2)dy
f ~o ~ oo _ y2 o e_~.,g( ,)d e, J - ~ e_p~y ex/~ L~(y2)dy"
The second integration is independent of d. The first integral is just the one-sided Laplace transform of g(e'); that is the partition function Z(fl). Thus d [ O ( e ) ] = Z(fl)
- ~oe -~'" e-'~ ~/~
L~(y2)dY"
N o w consider the y-integration:
f
o9
--y2
Iy = •~-~ e _ ~ y ex/~ L~(y2)dy. Using identities 12) involving the associated Laguerre polynomial we obtain:
L-2(x 2) = L~(:,=). m=O
(A.I)
544
B . K . JENNINGS
Also we have 12):
L: (x
_ ( - 1)"H2.(x). n!2 2"
Combining these two equations we see that:
L (x = [ ( - 0 2 H:.(x). .,:o m!2 2m
Substituting this into (A.1) we have:
Iy = ~ (__l)m f~:
-- ~,2
e -t~''e
m=o r n ! 2 2 m , j - ~
H2m(y)dy . \,/n
The integral is the two-sided Laplace transform of (e (yfl)2me+r2a2. Hence we obtain:
_y2
/~/n)Hzm(y ), which is t4)
m=0 n!22
where P(yfl) is a polynomial of order 2n in yfl. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11 ) 12) 13) 14) 15) 16) 17)
J. R. Nix, Ann. Rev. Nucl. Sci. 22 (1972), and references therein V. M. Strutinsky, Nucl. Phys. A95 (1967) 420 V. M. Strutinsky, Nuel. Phys. A122 (1968) 1 V. S. Ramamurthy and S. S. Kapoor, preprint R. Balian and C. Bloch, Ann. of Phys. 63 (1971) 592 R. Balian and C. Bloch, Ann. of Phys. 60 (1970) 401 Ph. J. Siemens and A. Sobiczewski, Phys. Lett. 41B (1972) 16 D. H. E. Gross, Phys. Lett. 42B (1972) 41 R. K. Bhaduri and C. K. Ross, Phys. Rev. Lett. 27 (1971) 606 Van der Pol and H. Bremmer, Operational calculus based on the two-sided Laplace integral (Cambridge University Press, London, 1955) p. 16 C. F. Tsang, Ph.D. thesis, University of California; Lawrence Radiation Lab. report UCRL18899 (1969) M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, New York, 1965) p. 785 M. Abramowitz and l. A. Stegun, op. cit., p. 775 Van der Pol and H. Bremmer, op. cit., p. 83 M. Abramowitz and I. A. Stegun, op. cit., p. 786 A. S. Tyapin, Yad. Fiz. 11 (1970) 98; Soy. J. Nucl. Phys. 11 (1970) 53 A. S. Tyapin, Yad. Fiz. 14 (1971) 88; Soy. J. Nucl. Phys. 14 (1972) 50