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On the form factor in inelastic deuteron scattering
Volume 28B. number 2
ON
THE
PHYSICS
FORM
FACTOR
LETTERS
IN INELASTIC
11 November
DEUTERON
1968
SCATTERING
J. M. BANG and 0. SAETHRE The Niels
Bohr Institute,
University
Received
of Copenhagen.
15 September
Effects of the finite size of the deuteron and the diffuseness potentials in inelastic deuteron scattering are calculated.
The excitation of collective nuclear states in inelastic scattering finds a simple description in DWBA or coupled channel calculations. However, in some cases of scattering of charged particles, discrepancies are found. Particularly, it has been difficult to reproduce the deuteron scattering data with one set of parameters for elastic and inelastic channels and for different energies. It is argued here that the inclusion of two finite size effects, those connected with the finite size of the deuteron and with the extension of the nuclear charge, will change the form factors of the DWBA calculations so as to remove these discrepancies. Several attempts have been made to deduce the deuteron nucleus potential from the proton- and neutron- nucleus potentials [l-3]. In a first approximation, the deuteron potential is calculated as V(R)
= ~dr]$d(‘)~2[V,(]R-
frl)
+ Vp(jR+
+r])]
(1) where @d is the intrinsic deuteron wave function, usually taken as the Hulhen wave function. Yn and VP are the neutron and proton potentials. We now generalize the monopole expression to arbitrary multipoles
Denmark
1968
of the target charge density on the coupling
l@+w12 = (3)
where Q and p are the parameters in the Hulthe’n wave function, usual1 taken as (Y = 0.23 fm-l, p = 1.58 fm-l, and N 3 = ap(o+P)/(o -p)2 277. The angular integration of eq. (2) is simple to perform by using the identity
(4) x2
Y,m*(i,)ynm(i2)
To calculate iVin)) and (Vi’)>> we change integration variables from the intrinsic deuteron coordinate to the coordinate of the nucleon considered. This gives VA(R) = - 8n N2 ,[R1
2dR1 (Vp)(Rl)
+ Vi”
(RI))
x
2(ff+P) V,(I?)
X
= VA(R) Yx(CZ) = (2)
=./dr
I~d(Y)/2[v~)(l~-~r/)+v~)(I~
1
_/
I-lW
jh(WR<)hA
+$rl)] +
where Vi”’ and Vi” are multiple components of order X of the neutron and proton interactions, respectively. Using the Hulthen wave function, the deuteron density function can be written as
(l)(i@
,) +
(5)
4P 2(o+P) I*dp j~(i/.&)h(l)(ipR-.) s h 4cY
I
By performing the multiplication jx (inReZ)h’,l) (iyR>) this can be written in a form more convenient for numerical integration VA(R) =J dR#+‘(R1)
+V~P)(R#~(R,R1)
(6)
99
Volume 28B. number 2
PHYSICS
11 November
LETTERS
1968
Table 1 The functions fg, 2
An1 0
4
5
rl
1
1 il
4
where
1
-a
7
8 and 9
-a
*3a
3(a2 * a)
-6a
+3(5a2 +- 2~)
rlOa
5(9a2 + 2U)
9a2
9a2
- 15(a3 + 3~2)
- 45(2a3 + a2)
- 225a3
-225a3
+ 15(7U3 t 9u2)
l5(7u4*42u3+9u2)
525(2u4 + 3~3)
1575(%4
dA(R, RI) is a function peaked around R
dA(RlR1)
= 4rN2z
{PlYP2!P3)fi
Rl
2x+1 21
[&In{Pi,
p2, ~.j}~ -t
= P2p-“e~-pdp+f2p-nexp-pdp p3 Pl
pi
= 4a(R> f R<) ,
P; = 2(a + p)(R, * R,) , _I
P; = 4P(R, * R<) and f iln are functions of (R \ - R<) and (R> + R,:), respectively. These functions are given in table 1, for multipoles up to X = 4. The functions dA( R,Rl) can be calculated once for different R and RI) and the integration (6) can be performed in an interval around R. The effect of the integration is shown in table 2. In these calculations (VA@)( RI) + VA@)( RI)) is re placed by 1. The results are given for X = 2 and 4. We see a reduction of the multipole field at small R, with greater effects for higher multi-
fi,,
poles. The
multipole moments given by eq. (6) can be used in inelastic deuteron scattering, with e.g. the nucleon multipoles given by Tamura [4] in the excitation of vibrational or rotational states. There is, however, an ambiquity in choosing the neutron and proton optical potentials which generate the multipoles. We could use neutron and proton optical potentials which will reproduce the deuteron optical potential (the X = 0 component of eq. (6)) describing the elastic scattering. However. we know that the optical potential to a
100
6
1
2 3
3
n (a is given by (R, *R.)2/R,R<).
* a3)
11025a4
certain degree includes the effects of strong couplings. It would therefore be reasonable to start from neutron and proton potentials, reproducing the deuteron optical potential for a nearby spherical nucleus, where the couplings are much weaker [5]. When the deuteron- or proton energy is not very high compared with the Coulomb barrier, the usual expression for the Coulomb multipole fields [4] may be too inaccurate [6]. Instead of a sharp charge surface we could use a continuous charge distribution and introduce deformed equi-charge density-surfaces p(r)
=p(r[l
+6@,(p)]-l
=p(r[l
-6
+62...])
(7)
Assuming, for simplicity, a static quadrupole deformation, this will, to first order in p2 - give V2(R) = 21 Z2e2b R3 ~:(r)~4dr/lmp(~)rd~
(8)
0
Using a Fermi charge distribution, and comparing the quadrupole field well outside the nucleus with the quadrupole field from a sharp surface q(O), we have v2
F
N $0)
(1 + 23 (a/Rc)2)
(9)
a is the diffuseness and R, the charge radius. With a,‘R = 0.1, we see that the quadrupole field is enhanced some 25% with a Fermi charge
where
Table 2 The reduction of the multipole fields at small R, R is given in fm. See the text for further explanation.
0.087
0.290
0.599
0.761
0.846
0.894
0.96
Volume 28B. number 2
PHYSICS
LETTERS
distribution. Similar expressions can be given for the octupole and “hexadecapole” fields. The corresponding enhancements are some 40% and ‘70%, respectively. This may be an explanation of the discrepancy between deformation parameters found from Coulomb excitation and more recent experiments with higher energies. Calculations with form factors based on eq. (6) have been performed by J. Bang et al. [7], applying a second-order Born approximation in lnelastic scattering. We show in fig. 1, the result
of DWBA for excitation of a 2+ and 3- level in the reaction 1 50Sm (d,d’) [8]. The optical potential used reproduces the elastic deuteron scattering [9]. The coupling potentials are based on (6) and [8]. We have chosen neutron and proton parameters which will reproduce the deuteron optical potential for the spherical 148Srn nucleus [lo]. The main difference in the two optical potentials is a smaller radius parameter for the imaginary part in the latter. The improvements in the angular distributions are due partly to this fact, and partly to the consideration of the finite size of the deuteron. The authors wish to express their gratitude to the experimental group of N.B.I., and particularly to Dr. B. Elbek, for many helpful discussions. One of us (0.S.) thanks the Niels Bohr Institute and especially Professor A. Bohr for his hospitality and for providing excellent working conditons.
2 2 E
1
.c
11 November 1968
~0s 0
0.2 1. S. Watanabe, Nucl. Phys.O (1958) 484. 2. J. R.Rook, Nucl. Phys. 61 (1965) 219. 3. J.Testoni and L.C.Gomes, Nucl.Phys.89 (1966) 288. 4. T. Tamura, Rev. Mod. Phys. 37 (1965) 679. 5. N. K.Glendennina, D. L. Hendrie and 0. N. Jarvis. Phys. Letters 26% (1968) 131. 6. L. W.Owen and G. R. Satchler, Nucl. Phys. 51 (1964) 155. 7. J.Bang et al., submitted to Nucl. Phys. 8. B.Zeidman et al., Nucl.Phys.86 (1966) 471. 9. B. Elbek, private communication. 10. G. Ldvhdiden, private communication.
: I
300
I 60°
I SO’
I 120°
I 150°
8
Fig. 1. Inelastic deuteron scattering to the 2+ and 3levels of 150Sm. The full lines are results of the new form factors; the dashed lines are results of ordinary DWBA calculations using the optical deuteron potential which will reproduce the elastic scattering. The deformation parameters used to fit the data are/32 = 0.15 and /33 = 0.085.
*****
101
ON
THE
PHYSICS
FORM
FACTOR
LETTERS
IN INELASTIC
11 November
DEUTERON
1968
SCATTERING
J. M. BANG and 0. SAETHRE The Niels
Bohr Institute,
University
Received
of Copenhagen.
15 September
Effects of the finite size of the deuteron and the diffuseness potentials in inelastic deuteron scattering are calculated.
The excitation of collective nuclear states in inelastic scattering finds a simple description in DWBA or coupled channel calculations. However, in some cases of scattering of charged particles, discrepancies are found. Particularly, it has been difficult to reproduce the deuteron scattering data with one set of parameters for elastic and inelastic channels and for different energies. It is argued here that the inclusion of two finite size effects, those connected with the finite size of the deuteron and with the extension of the nuclear charge, will change the form factors of the DWBA calculations so as to remove these discrepancies. Several attempts have been made to deduce the deuteron nucleus potential from the proton- and neutron- nucleus potentials [l-3]. In a first approximation, the deuteron potential is calculated as V(R)
= ~dr]$d(‘)~2[V,(]R-
frl)
+ Vp(jR+
+r])]
(1) where @d is the intrinsic deuteron wave function, usually taken as the Hulhen wave function. Yn and VP are the neutron and proton potentials. We now generalize the monopole expression to arbitrary multipoles
Denmark
1968
of the target charge density on the coupling
l@+w12 = (3)
where Q and p are the parameters in the Hulthe’n wave function, usual1 taken as (Y = 0.23 fm-l, p = 1.58 fm-l, and N 3 = ap(o+P)/(o -p)2 277. The angular integration of eq. (2) is simple to perform by using the identity
(4) x2
Y,m*(i,)ynm(i2)
To calculate iVin)) and (Vi’)>> we change integration variables from the intrinsic deuteron coordinate to the coordinate of the nucleon considered. This gives VA(R) = - 8n N2 ,[R1
2dR1 (Vp)(Rl)
+ Vi”
(RI))
x
2(ff+P) V,(I?)
X
= VA(R) Yx(CZ) = (2)
=./dr
I~d(Y)/2[v~)(l~-~r/)+v~)(I~
1
_/
I-lW
jh(WR<)hA
+$rl)] +
where Vi”’ and Vi” are multiple components of order X of the neutron and proton interactions, respectively. Using the Hulthen wave function, the deuteron density function can be written as
(l)(i@
,) +
(5)
4P 2(o+P) I*dp j~(i/.&)h(l)(ipR-.) s h 4cY
I
By performing the multiplication jx (inReZ)h’,l) (iyR>) this can be written in a form more convenient for numerical integration VA(R) =J dR#+‘(R1)
+V~P)(R#~(R,R1)
(6)
99
Volume 28B. number 2
PHYSICS
11 November
LETTERS
1968
Table 1 The functions fg, 2
An1 0
4
5
rl
1
1 il
4
where
1
-a
7
8 and 9
-a
*3a
3(a2 * a)
-6a
+3(5a2 +- 2~)
rlOa
5(9a2 + 2U)
9a2
9a2
- 15(a3 + 3~2)
- 45(2a3 + a2)
- 225a3
-225a3
+ 15(7U3 t 9u2)
l5(7u4*42u3+9u2)
525(2u4 + 3~3)
1575(%4
dA(R, RI) is a function peaked around R
dA(RlR1)
= 4rN2z
{PlYP2!P3)fi
Rl
2x+1 21
[&In{Pi,
p2, ~.j}~ -t
= P2p-“e~-pdp+f2p-nexp-pdp p3 Pl
pi
= 4a(R> f R<) ,
P; = 2(a + p)(R, * R,) , _I
P; = 4P(R, * R<) and f iln are functions of (R \ - R<) and (R> + R,:), respectively. These functions are given in table 1, for multipoles up to X = 4. The functions dA( R,Rl) can be calculated once for different R and RI) and the integration (6) can be performed in an interval around R. The effect of the integration is shown in table 2. In these calculations (VA@)( RI) + VA@)( RI)) is re placed by 1. The results are given for X = 2 and 4. We see a reduction of the multipole field at small R, with greater effects for higher multi-
fi,,
poles. The
multipole moments given by eq. (6) can be used in inelastic deuteron scattering, with e.g. the nucleon multipoles given by Tamura [4] in the excitation of vibrational or rotational states. There is, however, an ambiquity in choosing the neutron and proton optical potentials which generate the multipoles. We could use neutron and proton optical potentials which will reproduce the deuteron optical potential (the X = 0 component of eq. (6)) describing the elastic scattering. However. we know that the optical potential to a
100
6
1
2 3
3
n (a is given by (R, *R.)2/R,R<).
* a3)
11025a4
certain degree includes the effects of strong couplings. It would therefore be reasonable to start from neutron and proton potentials, reproducing the deuteron optical potential for a nearby spherical nucleus, where the couplings are much weaker [5]. When the deuteron- or proton energy is not very high compared with the Coulomb barrier, the usual expression for the Coulomb multipole fields [4] may be too inaccurate [6]. Instead of a sharp charge surface we could use a continuous charge distribution and introduce deformed equi-charge density-surfaces p(r)
=p(r[l
+6@,(p)]-l
=p(r[l
-6
+62...])
(7)
Assuming, for simplicity, a static quadrupole deformation, this will, to first order in p2 - give V2(R) = 21 Z2e2b R3 ~:(r)~4dr/lmp(~)rd~
(8)
0
Using a Fermi charge distribution, and comparing the quadrupole field well outside the nucleus with the quadrupole field from a sharp surface q(O), we have v2
F
N $0)
(1 + 23 (a/Rc)2)
(9)
a is the diffuseness and R, the charge radius. With a,‘R = 0.1, we see that the quadrupole field is enhanced some 25% with a Fermi charge
where
Table 2 The reduction of the multipole fields at small R, R is given in fm. See the text for further explanation.
0.087
0.290
0.599
0.761
0.846
0.894
0.96
Volume 28B. number 2
PHYSICS
LETTERS
distribution. Similar expressions can be given for the octupole and “hexadecapole” fields. The corresponding enhancements are some 40% and ‘70%, respectively. This may be an explanation of the discrepancy between deformation parameters found from Coulomb excitation and more recent experiments with higher energies. Calculations with form factors based on eq. (6) have been performed by J. Bang et al. [7], applying a second-order Born approximation in lnelastic scattering. We show in fig. 1, the result
of DWBA for excitation of a 2+ and 3- level in the reaction 1 50Sm (d,d’) [8]. The optical potential used reproduces the elastic deuteron scattering [9]. The coupling potentials are based on (6) and [8]. We have chosen neutron and proton parameters which will reproduce the deuteron optical potential for the spherical 148Srn nucleus [lo]. The main difference in the two optical potentials is a smaller radius parameter for the imaginary part in the latter. The improvements in the angular distributions are due partly to this fact, and partly to the consideration of the finite size of the deuteron. The authors wish to express their gratitude to the experimental group of N.B.I., and particularly to Dr. B. Elbek, for many helpful discussions. One of us (0.S.) thanks the Niels Bohr Institute and especially Professor A. Bohr for his hospitality and for providing excellent working conditons.
2 2 E
1
.c
11 November 1968
~0s 0
0.2 1. S. Watanabe, Nucl. Phys.O (1958) 484. 2. J. R.Rook, Nucl. Phys. 61 (1965) 219. 3. J.Testoni and L.C.Gomes, Nucl.Phys.89 (1966) 288. 4. T. Tamura, Rev. Mod. Phys. 37 (1965) 679. 5. N. K.Glendennina, D. L. Hendrie and 0. N. Jarvis. Phys. Letters 26% (1968) 131. 6. L. W.Owen and G. R. Satchler, Nucl. Phys. 51 (1964) 155. 7. J.Bang et al., submitted to Nucl. Phys. 8. B.Zeidman et al., Nucl.Phys.86 (1966) 471. 9. B. Elbek, private communication. 10. G. Ldvhdiden, private communication.
: I
300
I 60°
I SO’
I 120°
I 150°
8
Fig. 1. Inelastic deuteron scattering to the 2+ and 3levels of 150Sm. The full lines are results of the new form factors; the dashed lines are results of ordinary DWBA calculations using the optical deuteron potential which will reproduce the elastic scattering. The deformation parameters used to fit the data are/32 = 0.15 and /33 = 0.085.
*****
101