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Single particle sum rules for nuclei
Volume 26B, volume 12
PHYSICS
SINGLE
PARTICLE
Theoretical
Physics
LETTERS
SUM
13 Mav 1968
RULES
FOR
C. F. CLEMENT Division, A.E.R.E., Harwell,
NUCLEI
Didcot,
Berks,
UK
Received 24 March 1968
The exchange contribution to the exact model-independent sum rules is examined. It is shown to be related to the spectroscopic factors for single particle pickup reactions. Consequences to Butler’s new stripping theory are noted.
Model independent single particle sum rules have recently been derived by Clement [l] and similar, but not exact, sum rules have been given by Butler et al. [2,3]. In the former work the physical importance of antisymmetrization was pointed out. We now examine its consequences in detail. For simplicity we restrict the discussion to the case of a neutron, quantum numbers jl, added to a spin zero nucleus whose wave function is x0(1. ..A).Antisymmetric bound states of the (A+l)-particle system are denoted by qB(l. . .A+l). Their projections onto x0 define the spectroscopic factors, S,, and the normalized form factors, $,, by: (A+&,,(l.
. . A)l$Jl..
. A+l))
= 6’n$n ,
(1)
where S, = / 8, / 2 and Gn is a function of r, the relative distance of particle A+1 from the centre of mass of the first A particles. Strictly speaking, since all particles are being treated as identical, we should include isospin indices and variables. The right hand side of eq. (1) should contain a vector coupling coefficient for isospin, but this is unity in the case being considered and initially reference to isospin will be suppressed. Taking a general normalized single particle wave function $, we form the antisymmetric combinations: @J(l.. . A+l) = (A+l+ where PiA+I is the permutation operator We consider the overlap function
(1 - &
interchanging (+I4
where it can easily be verified
PiA+l)x,(l..
. A) @ (A+l) ,
all coordinates
of the two particles
i, A+l.
= l-A,
(2)
that the exchange term A may be written as: A =A((x,(l.
- .A+#dA))
bNA)Ix,U..
.A)))
(3)
.
We shall consider this term later. The sum rules are obtained by inserting the unit operator for the (A+l)-particle Hamiltonian into the matrix element (2). The terms are identified as in ref. 1 except that the continuum wave functions are now also antisymmetrized. For example, for the neutron elastic scattering channel with incident relative moment k, the wave function is: *(o+)(k) = (2n)-f(A+i)-f[l
-Z$I
Pti+l]
[[l
+E_A+it
$ viA+I]xo(l...A) 2=1
exp(ik’r)X(‘JA+19
*A+1
where H is the (A+l)-particle Hamiltonian, and the incident plane wave and spin wave function of the incoming neutron are given explicitly. The single-particle elastic scattering wave function with the correct asymptotic behaviour is:
710
)I ’
Volume
26B.
number
12
PHYSICS
LETTERS
13 Mav 1968
&+)(k,r; uA+1 rA+l) = (A+l)fb&,+)(k)). In general these single-particle ment in ref. 2. The generalised
wave functions are not continuum normalized, sum rules are then:
contrary
to the state-
where f(E), which is expected to lie between 1 and 2 numerically, takes account of the other continuum channels. On choosing Cpas each $n in turn we obtain the previous sum rules [l]. The presence of the exchange terms A, and the other continuum channels, represented by the functions f(E), invalidate the theorem stated in ref. 3. In fact the situation is considerably more complex in that one needs to know the ground state wave function as well as the contributions of the other channels to determine the spectroscopic factors in principle. A treatment of the latter contributions will be given elsewhere, but the exchange terms have a simple physical interpretation which we now examine. The quantity A can be regarded as the probability that the single particle state 6 is occupied in xo. This can easily be seen if x0 in eq. (3) takes the form of a sum of Slater determinants. To relate A to experiment we need the spectroscopic factors defined for pickup reactions. If the bound states of the (A-1)-particle system are denoted by Go (1.. .A-1), the usual definition is: A&(1..
.A-1;
JaMa;
T,T30)/~0(1...A;
OO,T,T,)) =
where we have included explicitly the spin and isospin indices, and the spectroscopic factors are defined by So = / Qcr12. The form factors $o are normalised to unity. To obtain an expansion for A we insert into matrix element (3) the unit operator for the (A-l)-partitle Hamiltonian. The wave function @(A) selects the given quantum numbers, jZm, and we obtain: A =
c /(4 /4,) / ‘C2Sa /(2i+l)
+A, ,
(Y
where C2 = (T~ T 3a$giTo To)'and A, is the contribtuion from continuum states of the (A -l)-particle system. The spin and isospin factors appear because C2S, is to be interpreted as the total number of neutrons in the single particle state, whereas S, is the probability that the state is occupied. The definitions used merely differ in the factor (2j+l). We are now, of course, perfectly at liberty to choose 4 = $o instead of @,, thus obtaining another set of formal sum rules. The entire set of sum rules are evidently the exact analogues of the MacFarlane and French sum rules [4] which state that the sum of spectroscopic factors for stripping and pickup reactions is unity. Physically this is generally a valid point of view because (1) single particle states with different principal quantum numbers are well separated in energy and (2) the widths of single particle states are small enough so that continuum contributions (apart from isolated resonances) can be neglected. The first statement implies that the single particle overlaps ($14,) a;ld (4 / 6,) are both near unity when $ is chosen to be one of either set, and the second implies that the continuum contribution in eq. (4) and A, are both negligible. These conditions will not always be satisfied so that further examination of the exact sum rules will be fruitful. Returning to the exchange term we observe that it almost never vanishes, even at supposedly closed shells. In the case of 40Ca, considered in ref. 2, experimental evidence from pickup reactions has been summarised by Hubbard and Jolly [5] (their S is our C2S). For f; neutron pickup the value of C2S is not yet uniquely determined experimentally but may be as large as 0.53 giving a maximum value of A of about 0.06. The value of S for the f; state of 41Ca determined from the new Butler stripping theory [6], is therfore likely to be affected. The basic relation of this theory (eq. (6) of ref. 1 and eq. (7) of ref. 2) needs an exchange term. This can be determined starting from eq. (7) of ref. 1 by exactly the same antisymmetrization process as that used above for the sum rule. In the notation of ref. 1 the re711
Volume 26B, number 12
PHYSICS
LETTERS
13 May 1968
sult is:
F(l-S,)= c n
q*n
Mel sqcr#Ibq)g+Mc+Mc,+M*, 4
where II~~is the direct stripping matrix element to final state n, M, is the elastic continuum term evaluated’in the new theory (MS of ref. 2) and M, t is the contribution from the residual continuum states. The new exchange term, MA, is:
MA = A((x,(l..
.A)(~dn(A))(~n(A)j~o(l...A))) ,
03)
where $& is the neutron stripping wave function defined in ref. 1. By again inserting the unit operator for the (A-1)-particle Hamiltonian we can relate MA to effective matrix elements for pickup reactions. Neglecting continuum terms and using definition (5) we obtain: n n/m=c--- ‘“‘cY /y 2j+l
($n /$‘a, )b,
1@dn) .
To apply this to 40Ca we take @o - r&, when the final matrix element becomes (@, / Qh) Thus, if there are no f+ states in 4lCa other than the ground state, eq. (‘7) becomes: M, = e,M,(l
-Sn -
A)-1 .
= Mn/Bn. (10)
The value of S for the ft level found from the same formula without A is 0.6 [6]. With a value of 0.06 for A the above formula would predict S to be 0.57. This is not a significant change and the ommission of exchange in the original theory can be ruled out as an explanation for the small value of S. The inclusion of another fl level in the 41Ca spectrum is unlikely to increase S for the ground state except if the state is in the continuum [2]. The separation of the ff strength into two major parts 8 MeV or more apart would be very hard to understand. The most likely explanation for this small value of S is that the approximations made in the evaluation of M, [6] are inadequate. In conclusion we have seen that it is necessary to include the exchange terms both in the sum rules and in the new stripping theory. The former provide powerful constraints on the values of spectroscopic factors found in stripping and pickup reactions. The author would like to thank Dr. J. K. Perring
for critically
reading the manuscript.
References 1. C.F.Clement. Phps. Rev. Letters 20 (1968) 22. 2. S.T.Butler. R.G.L.Hewitt and J.S.Truelove, Phys. Letters 26B (1968) 264. 3. S.T.Butler. R.G.L.Hewitt and J.S.Truelove. Phys. Letters 26B (1968) 267. 4. M.H.MacFarlane and J.B.French. Rev. Mod. Phys. 32 (1960) 567. 5. L.B.Hubbard and H.P.Jolly. Phys. Rev. 164 (1967) 1434. 6. S. T.Butler, R.G. L.Hewitt and J.S. Truelove. Phys. Rev. 162 (1967) 1061. *****
712
PHYSICS
SINGLE
PARTICLE
Theoretical
Physics
LETTERS
SUM
13 Mav 1968
RULES
FOR
C. F. CLEMENT Division, A.E.R.E., Harwell,
NUCLEI
Didcot,
Berks,
UK
Received 24 March 1968
The exchange contribution to the exact model-independent sum rules is examined. It is shown to be related to the spectroscopic factors for single particle pickup reactions. Consequences to Butler’s new stripping theory are noted.
Model independent single particle sum rules have recently been derived by Clement [l] and similar, but not exact, sum rules have been given by Butler et al. [2,3]. In the former work the physical importance of antisymmetrization was pointed out. We now examine its consequences in detail. For simplicity we restrict the discussion to the case of a neutron, quantum numbers jl, added to a spin zero nucleus whose wave function is x0(1. ..A).Antisymmetric bound states of the (A+l)-particle system are denoted by qB(l. . .A+l). Their projections onto x0 define the spectroscopic factors, S,, and the normalized form factors, $,, by: (A+&,,(l.
. . A)l$Jl..
. A+l))
= 6’n$n ,
(1)
where S, = / 8, / 2 and Gn is a function of r, the relative distance of particle A+1 from the centre of mass of the first A particles. Strictly speaking, since all particles are being treated as identical, we should include isospin indices and variables. The right hand side of eq. (1) should contain a vector coupling coefficient for isospin, but this is unity in the case being considered and initially reference to isospin will be suppressed. Taking a general normalized single particle wave function $, we form the antisymmetric combinations: @J(l.. . A+l) = (A+l+ where PiA+I is the permutation operator We consider the overlap function
(1 - &
interchanging (+I4
where it can easily be verified
PiA+l)x,(l..
. A) @ (A+l) ,
all coordinates
of the two particles
i, A+l.
= l-A,
(2)
that the exchange term A may be written as: A =A((x,(l.
- .A+#dA))
bNA)Ix,U..
.A)))
(3)
.
We shall consider this term later. The sum rules are obtained by inserting the unit operator for the (A+l)-particle Hamiltonian into the matrix element (2). The terms are identified as in ref. 1 except that the continuum wave functions are now also antisymmetrized. For example, for the neutron elastic scattering channel with incident relative moment k, the wave function is: *(o+)(k) = (2n)-f(A+i)-f[l
-Z$I
Pti+l]
[[l
+E_A+it
$ viA+I]xo(l...A) 2=1
exp(ik’r)X(‘JA+19
*A+1
where H is the (A+l)-particle Hamiltonian, and the incident plane wave and spin wave function of the incoming neutron are given explicitly. The single-particle elastic scattering wave function with the correct asymptotic behaviour is:
710
)I ’
Volume
26B.
number
12
PHYSICS
LETTERS
13 Mav 1968
&+)(k,r; uA+1 rA+l) = (A+l)fb&,+)(k)). In general these single-particle ment in ref. 2. The generalised
wave functions are not continuum normalized, sum rules are then:
contrary
to the state-
where f(E), which is expected to lie between 1 and 2 numerically, takes account of the other continuum channels. On choosing Cpas each $n in turn we obtain the previous sum rules [l]. The presence of the exchange terms A, and the other continuum channels, represented by the functions f(E), invalidate the theorem stated in ref. 3. In fact the situation is considerably more complex in that one needs to know the ground state wave function as well as the contributions of the other channels to determine the spectroscopic factors in principle. A treatment of the latter contributions will be given elsewhere, but the exchange terms have a simple physical interpretation which we now examine. The quantity A can be regarded as the probability that the single particle state 6 is occupied in xo. This can easily be seen if x0 in eq. (3) takes the form of a sum of Slater determinants. To relate A to experiment we need the spectroscopic factors defined for pickup reactions. If the bound states of the (A-1)-particle system are denoted by Go (1.. .A-1), the usual definition is: A&(1..
.A-1;
JaMa;
T,T30)/~0(1...A;
OO,T,T,)) =
where we have included explicitly the spin and isospin indices, and the spectroscopic factors are defined by So = / Qcr12. The form factors $o are normalised to unity. To obtain an expansion for A we insert into matrix element (3) the unit operator for the (A-l)-partitle Hamiltonian. The wave function @(A) selects the given quantum numbers, jZm, and we obtain: A =
c /(4 /4,) / ‘C2Sa /(2i+l)
+A, ,
(Y
where C2 = (T~ T 3a$giTo To)'and A, is the contribtuion from continuum states of the (A -l)-particle system. The spin and isospin factors appear because C2S, is to be interpreted as the total number of neutrons in the single particle state, whereas S, is the probability that the state is occupied. The definitions used merely differ in the factor (2j+l). We are now, of course, perfectly at liberty to choose 4 = $o instead of @,, thus obtaining another set of formal sum rules. The entire set of sum rules are evidently the exact analogues of the MacFarlane and French sum rules [4] which state that the sum of spectroscopic factors for stripping and pickup reactions is unity. Physically this is generally a valid point of view because (1) single particle states with different principal quantum numbers are well separated in energy and (2) the widths of single particle states are small enough so that continuum contributions (apart from isolated resonances) can be neglected. The first statement implies that the single particle overlaps ($14,) a;ld (4 / 6,) are both near unity when $ is chosen to be one of either set, and the second implies that the continuum contribution in eq. (4) and A, are both negligible. These conditions will not always be satisfied so that further examination of the exact sum rules will be fruitful. Returning to the exchange term we observe that it almost never vanishes, even at supposedly closed shells. In the case of 40Ca, considered in ref. 2, experimental evidence from pickup reactions has been summarised by Hubbard and Jolly [5] (their S is our C2S). For f; neutron pickup the value of C2S is not yet uniquely determined experimentally but may be as large as 0.53 giving a maximum value of A of about 0.06. The value of S for the f; state of 41Ca determined from the new Butler stripping theory [6], is therfore likely to be affected. The basic relation of this theory (eq. (6) of ref. 1 and eq. (7) of ref. 2) needs an exchange term. This can be determined starting from eq. (7) of ref. 1 by exactly the same antisymmetrization process as that used above for the sum rule. In the notation of ref. 1 the re711
Volume 26B, number 12
PHYSICS
LETTERS
13 May 1968
sult is:
F(l-S,)= c n
q*n
Mel sqcr#Ibq)g+Mc+Mc,+M*, 4
where II~~is the direct stripping matrix element to final state n, M, is the elastic continuum term evaluated’in the new theory (MS of ref. 2) and M, t is the contribution from the residual continuum states. The new exchange term, MA, is:
MA = A((x,(l..
.A)(~dn(A))(~n(A)j~o(l...A))) ,
03)
where $& is the neutron stripping wave function defined in ref. 1. By again inserting the unit operator for the (A-1)-particle Hamiltonian we can relate MA to effective matrix elements for pickup reactions. Neglecting continuum terms and using definition (5) we obtain: n n/m=c--- ‘“‘cY /y 2j+l
($n /$‘a, )b,
1@dn) .
To apply this to 40Ca we take @o - r&, when the final matrix element becomes (@, / Qh) Thus, if there are no f+ states in 4lCa other than the ground state, eq. (‘7) becomes: M, = e,M,(l
-Sn -
A)-1 .
= Mn/Bn. (10)
The value of S for the ft level found from the same formula without A is 0.6 [6]. With a value of 0.06 for A the above formula would predict S to be 0.57. This is not a significant change and the ommission of exchange in the original theory can be ruled out as an explanation for the small value of S. The inclusion of another fl level in the 41Ca spectrum is unlikely to increase S for the ground state except if the state is in the continuum [2]. The separation of the ff strength into two major parts 8 MeV or more apart would be very hard to understand. The most likely explanation for this small value of S is that the approximations made in the evaluation of M, [6] are inadequate. In conclusion we have seen that it is necessary to include the exchange terms both in the sum rules and in the new stripping theory. The former provide powerful constraints on the values of spectroscopic factors found in stripping and pickup reactions. The author would like to thank Dr. J. K. Perring
for critically
reading the manuscript.
References 1. C.F.Clement. Phps. Rev. Letters 20 (1968) 22. 2. S.T.Butler. R.G.L.Hewitt and J.S.Truelove, Phys. Letters 26B (1968) 264. 3. S.T.Butler. R.G.L.Hewitt and J.S.Truelove. Phys. Letters 26B (1968) 267. 4. M.H.MacFarlane and J.B.French. Rev. Mod. Phys. 32 (1960) 567. 5. L.B.Hubbard and H.P.Jolly. Phys. Rev. 164 (1967) 1434. 6. S. T.Butler, R.G. L.Hewitt and J.S. Truelove. Phys. Rev. 162 (1967) 1061. *****
712