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Effect of nuclear surface diffuseness on diffraction scattering
Nuclear Physics 63 (1965) 6 8 9 - - 6 9 4 ; (~) North-Holland Publishin# Co., Amsterdam Not to be reproduced by photoprlnt or microfilm without written permission from the publisher
E F F E C T OF NUCLEAR SURFACE DIFFUSENESS ON DIFFRACTION SCATTERING E. V. INOPIN and Yu. A. BEREZHNOY Physical-Technical Institute of the Academy of Sciences of the Ukrainian Republic, Kharkov, USSR
Received 3 February 1964 Abstract: A simple method is proposed for taking into account nuclear surface diffuseness in diffractional scattering. Theoretical predictions are compared with experimental data on the scattering of ~-particles by Mgs4 nuclei. The range of ~-particles in nuclear matter is estimated. 1. Introduction The scattering o f particles on the black nucleus with excitation of rotational and vibrational levels was considered by Drozdov 1) and Inopin 2). Comparison o f the results o f such theoretical treatment with experimental data yields information about nuclei (deformation, nuclear size) as was shown by Blair 3) (see also refs. 4, 5)). However, despite a qualitative agreement of the theoretical predictions and experimental data there are considerable discrepancies in the cross sections for large angles. It is assumed in the above papers that the nuclear surface is sharp. It is natural to suppose that consideration o f nuclear surfaces diffuseness will introduce essential changes in the region o f large angles and will perhaps provide a better fit with experiment. Besides, the investigation o f nuclear surface diffuseness may furnish interesting data on the free path o f particles in the nucleus. The diffraction scattering o f particles by spherical nuclei with diffuse surface was considered in ref. 7). Another method o f taking into account nuclear surface diffuseness is proposed in this paper- it hardly makes the calculations more complicated and yields simple results. It is shown that the free path of scattered particles in nuclear matter can also be estimated from a comparison of the results of the calculation and experimental data. 2. Scattering Amplitude The amplitude of the diffraction scattering of a point particle by a non-spherical nucleus is given by the experssion
f(~, O) = i~nf og(p)e-"'° dp, 689
(1)
690
E.V.
INOPIN AND Yu. A. BEREZHNOY
where 0cis the deformation parameter, 0 the scattering angle, K the wave vector o f the incident particle and x the momentum transfer; the integration is performed in a plane normal to the direction of the wave vector o f the incident particle. The function co(p) describes the properties of the nucleus as an absorbing medium. In a sharp surface nucleus we have co(p) = coo(P), where the function coo(P) equals unity if the end of the radius vector p lies inside the projection o f the nucleus on the plane normal to the direction of the wave vector o f the incident particle and is zero if the end o f p lies outside this projection. Nuclear surface diffuseness can be described by the function co(P) = f coo(U)~(lu - pl)du, d
(2)
where the function ~(z) is a positive quantity and must decrease rapidly at large values o f the argument and pass into 6(z) if the diffuseness width tends to zero. Substituting eq. (2) into eq. (1) we obtain
f(~, 0) = fo(~, 0)F(0),
(3)
wherefo(~, 0) is the amplitude of scattering on a sharp surface nucleus (i.e. co(p) = COo(p)) . The expression
F(O) = t ~ ( z ) e - ~'' "dz d
(4)
describes the nuclear surface diffuseness. Since the nucleus is regarded as a strong absorbent the condition
co(O) = ~ coo(U)~(u)du = 1 d
(5)
must be imposed on co(p). This condition means that for a particle passing near the centre of the nucleus the probability for capture is unity. I f the diffuseness is characterized by the width A, i.e. it is assumed that ~(z) is an appreciable quantity only when z < A and that A <
f ,I)(u)du
= 1.
(6)
From eqs. (4) and (6) it follows that F(0) = 1, i.e. diffuseness has no effect, as can well be expected, on scattering through small angles. Hence it follows in particular that the total cross section for all the processes a t remains constant. The function F(O) describing the diffuseness is obviously a real quantity decreasing as 0 increases. Since the function F(O) does not depend on the collective variables ~ , it alters in the same way both elastic and inelastic scattering. Thus the diffuseness has no effect on the ratio o f these cross sections.
EFFECT OF NUCLEAR SURFACE DIFFUSENESS
691
If the function
1 e_:/a~
(7)
~A 2
is taken as ~(z), we have e(0)
= e
.
(8)
The function og(p) corresponding to the function ~(z) chosen according to eq. (7) is plotted in fig. 4 (solid line) in the spherical case for particular values of the parameters R o and A.
3. Comparison with Experiment The scattering of ~-particles on Mg 24 nuclei is considered below as an example for the application o f the above theory. The experimental data on the scattering of ~particles in the energy range of 28 to 41 MeV by Mg 24 nuclei are taken from ref. 5). The elastic scattering of ~-particles and the scattering with the excitation of the first vibrational level of the Mg 24 nucleus are considered (Q = - 1 . 3 7 MeV, I = 2+). The differential cross sections are given by the expression dcr
where the respective cross sections for elastic and inelastic scattering by a sharp surface nucleus (da/dfa)o are
=
VJI(2KRosin ½0)]2 L 2- o
J
oo)
The values o f the deformation parameter fl = 0.24 and the radius R o = 5.97 fm found in ref. 5) from the scattering o f ~-particles by a sharp surface nucleus are used without modification. The cross sections given by eq. (9) reproduce well the experimental angular distributions but prove smaller than these approximately by a factor o f 1.6. Fig. 1 gives the differential cross section of elastic scattering. The continuous solid curve corresponds to the experimental cross section, the solid curve with breaks is calculated by eq. (9) with subsequent multiplication by 2 = 1.6 and the dashed curve corresponds to the scattering cross section for a sharp surface nucleus. The diffuseness width is found to be A = 0.79 fm. Fig. 2 gives the differential cross section for inelastic scattering (notations are the same as in fig. 1).
692
E.V.
INOPIN A N D Yu. A. B E R E Z H N O Y
From figs. 1 and 2 it is clear that the assumption about the nuclear surface diffuseness changing the elastic and inelatic cross section by the same factor yields a good fit in the region of large values.
fO =
•(frn") fO
fO
1
/-
//-'\
fO-I
,10"~
f0"3
6
to
f4
tO"
2Kl~tn-~O
Fig. 1. Differential cross section for elastic scattering (see text).
,
*
*
,
eKR,an~O
Fig. 2. Differential cross section for inelastic scattering (see text).
F~ to
0
2
6
f0
f4
2KRjLn~ 0
Fig. 3. Comparison o f the ratio (dtr/dO)exp/Z(da/d£2)o with F2(O).
Fig. 3 compares the ratio (da/dl2)exp/2(dtr/dI2)o at the points corresponding to the maxima in figs. 1 and 2 (circles denote elastic and triangles inelastic scattering) with the quantity F2(0) giveing the nuclear surface diffuseness. The choice of different functions (besides the form (7), ¢(z) was taken as step function and as an exponential) has a very small effect on the function F(O). 4. Free Path of a Particle in the Nucleus
A point particle moving along the axis x is described by the wave function ~k0 = e x p ( i K x ) . According to the optical model the wave factor after the scattering is K ' = K 1 + i K 2. The quantity l = 1/K 2 describes the damping of the plane wave ~k0
after the scattering and can be called the free path of a particle in the nucleus. According to the diffraction theory the wave function of a particle after scattering is
EFFECT OF NUCLEAR SURFACE DIFFUSENESS
693
~k = ~ko[1-co(p)]. Assuming that Kt ~ K (the scattering inside the nucleus is neglected) we obtain ~k = ~boexp(-x//). In the case of a finite size nucleus we have l = l(x) and the ratio x/l must be replaced by the integral S+ ~ dx/l(x). Comparing both expressions for co(p) we obtain the following formula for o~(p): co(p) = 1 - e x p
-
~
(12)
.
Bearing in mind that x(r) --- ( r 2 - p 2 ) ½, co(p) can conveniently be represented as
co(p) = 1--exp [
(13)
.]p x/r2~p-pZ/(r)J
To estimate the free path of ~-particles in nuclear matter, l(r) was chosen as O,
r < R 1,
r-R1 ----, R2--R 1 1
1
l(r)
lo
R1
1,
< r <
RE, (14)
R 2 < r < R 3,
R 4 - - 1"
-
-
R 4 - R3 0,
,
R 3 <
r <
R4,
R4 < r,
where lo describes the free path of ~-particles in the nucleus. ~0171 tO
0.5
I
i
I~
f
i
Fig. 4. F u n c t i o n co(p) i n t h e s p h e r i c a l case for t h e M g ~4 n u c l e u s .
Fig. 4 gives the function co(p) in the spherical case for the Mg 24 nucleus (solid curve). The dashed curve corresponds to the choice of parameters R 1 = Rz = O, Ra = 4.81 fm and R4 --- 6.81 fm. This case corresponds to the trapezoidal nuclear density p(r) with surface layer thickness 1.6 fm if it is assumed that 1/l(r) oc p(r).
694
E. V. INOPIN AND Yu. A. BEREZHNOY
The dot-and-dash line passing into the dashed curve when p = 4.81 fm corresponds to the choice of parameters R 1 = 2.81 fro, R 2 = R 3 = 4.81 fm and R 4 = 6.81 fm and belongs to the surface absorption case. In both cases the free path o f 0t-particles in the Mg 24 nucleus was found to be equal to l o = 2.0 fm which is in good agreement with other estimates of this quantity 6). The idea of taking into account nuclear surface diffuseness in the diffraction scattering of particles by a non-spherical nuclei has been expressed earlier by Professor A. G. Sitenko, which the authors acknowledge with gratitude. References 1) 2) 3) 4) 5) 6) 7)
S. I. Drozdov, JETP 28 (1955) 734, 736 (translation: JETP (Soviet Physics) 1 (1955) 588) Ye. V. Inopin, JETP 31 (1956) 901 (translation: JETP (Soviet Physics) 4 (1957) 764) J. S. Blair, Phys. Rev. 115 (1959) 928 D. K. McDaniels, J. S. Blair, S. W. Chert and G. W. Farwell, Nuclear Physics 17 (1960) 614 J. S. Blair, G. W. Farwell and D. K. McDaniels, Nuclear Physics 17 (1960) 641 C. E. Porter, Phys. Rev. 99 (1955) 1400 A. G. Sitenko and V. K. Tartakovsky, Ukrain. Fiz. Zhur. 6 (1961) 12
E F F E C T OF NUCLEAR SURFACE DIFFUSENESS ON DIFFRACTION SCATTERING E. V. INOPIN and Yu. A. BEREZHNOY Physical-Technical Institute of the Academy of Sciences of the Ukrainian Republic, Kharkov, USSR
Received 3 February 1964 Abstract: A simple method is proposed for taking into account nuclear surface diffuseness in diffractional scattering. Theoretical predictions are compared with experimental data on the scattering of ~-particles by Mgs4 nuclei. The range of ~-particles in nuclear matter is estimated. 1. Introduction The scattering o f particles on the black nucleus with excitation of rotational and vibrational levels was considered by Drozdov 1) and Inopin 2). Comparison o f the results o f such theoretical treatment with experimental data yields information about nuclei (deformation, nuclear size) as was shown by Blair 3) (see also refs. 4, 5)). However, despite a qualitative agreement of the theoretical predictions and experimental data there are considerable discrepancies in the cross sections for large angles. It is assumed in the above papers that the nuclear surface is sharp. It is natural to suppose that consideration o f nuclear surfaces diffuseness will introduce essential changes in the region o f large angles and will perhaps provide a better fit with experiment. Besides, the investigation o f nuclear surface diffuseness may furnish interesting data on the free path o f particles in the nucleus. The diffraction scattering o f particles by spherical nuclei with diffuse surface was considered in ref. 7). Another method o f taking into account nuclear surface diffuseness is proposed in this paper- it hardly makes the calculations more complicated and yields simple results. It is shown that the free path of scattered particles in nuclear matter can also be estimated from a comparison of the results of the calculation and experimental data. 2. Scattering Amplitude The amplitude of the diffraction scattering of a point particle by a non-spherical nucleus is given by the experssion
f(~, O) = i~nf og(p)e-"'° dp, 689
(1)
690
E.V.
INOPIN AND Yu. A. BEREZHNOY
where 0cis the deformation parameter, 0 the scattering angle, K the wave vector o f the incident particle and x the momentum transfer; the integration is performed in a plane normal to the direction of the wave vector o f the incident particle. The function co(p) describes the properties of the nucleus as an absorbing medium. In a sharp surface nucleus we have co(p) = coo(P), where the function coo(P) equals unity if the end of the radius vector p lies inside the projection o f the nucleus on the plane normal to the direction of the wave vector o f the incident particle and is zero if the end o f p lies outside this projection. Nuclear surface diffuseness can be described by the function co(P) = f coo(U)~(lu - pl)du, d
(2)
where the function ~(z) is a positive quantity and must decrease rapidly at large values o f the argument and pass into 6(z) if the diffuseness width tends to zero. Substituting eq. (2) into eq. (1) we obtain
f(~, 0) = fo(~, 0)F(0),
(3)
wherefo(~, 0) is the amplitude of scattering on a sharp surface nucleus (i.e. co(p) = COo(p)) . The expression
F(O) = t ~ ( z ) e - ~'' "dz d
(4)
describes the nuclear surface diffuseness. Since the nucleus is regarded as a strong absorbent the condition
co(O) = ~ coo(U)~(u)du = 1 d
(5)
must be imposed on co(p). This condition means that for a particle passing near the centre of the nucleus the probability for capture is unity. I f the diffuseness is characterized by the width A, i.e. it is assumed that ~(z) is an appreciable quantity only when z < A and that A <
f ,I)(u)du
= 1.
(6)
From eqs. (4) and (6) it follows that F(0) = 1, i.e. diffuseness has no effect, as can well be expected, on scattering through small angles. Hence it follows in particular that the total cross section for all the processes a t remains constant. The function F(O) describing the diffuseness is obviously a real quantity decreasing as 0 increases. Since the function F(O) does not depend on the collective variables ~ , it alters in the same way both elastic and inelastic scattering. Thus the diffuseness has no effect on the ratio o f these cross sections.
EFFECT OF NUCLEAR SURFACE DIFFUSENESS
691
If the function
1 e_:/a~
(7)
~A 2
is taken as ~(z), we have e(0)
= e
.
(8)
The function og(p) corresponding to the function ~(z) chosen according to eq. (7) is plotted in fig. 4 (solid line) in the spherical case for particular values of the parameters R o and A.
3. Comparison with Experiment The scattering of ~-particles on Mg 24 nuclei is considered below as an example for the application o f the above theory. The experimental data on the scattering of ~particles in the energy range of 28 to 41 MeV by Mg 24 nuclei are taken from ref. 5). The elastic scattering of ~-particles and the scattering with the excitation of the first vibrational level of the Mg 24 nucleus are considered (Q = - 1 . 3 7 MeV, I = 2+). The differential cross sections are given by the expression dcr
where the respective cross sections for elastic and inelastic scattering by a sharp surface nucleus (da/dfa)o are
=
VJI(2KRosin ½0)]2 L 2- o
J
oo)
The values o f the deformation parameter fl = 0.24 and the radius R o = 5.97 fm found in ref. 5) from the scattering o f ~-particles by a sharp surface nucleus are used without modification. The cross sections given by eq. (9) reproduce well the experimental angular distributions but prove smaller than these approximately by a factor o f 1.6. Fig. 1 gives the differential cross section of elastic scattering. The continuous solid curve corresponds to the experimental cross section, the solid curve with breaks is calculated by eq. (9) with subsequent multiplication by 2 = 1.6 and the dashed curve corresponds to the scattering cross section for a sharp surface nucleus. The diffuseness width is found to be A = 0.79 fm. Fig. 2 gives the differential cross section for inelastic scattering (notations are the same as in fig. 1).
692
E.V.
INOPIN A N D Yu. A. B E R E Z H N O Y
From figs. 1 and 2 it is clear that the assumption about the nuclear surface diffuseness changing the elastic and inelatic cross section by the same factor yields a good fit in the region of large values.
fO =
•(frn") fO
fO
1
/-
//-'\
fO-I
,10"~
f0"3
6
to
f4
tO"
2Kl~tn-~O
Fig. 1. Differential cross section for elastic scattering (see text).
,
*
*
,
eKR,an~O
Fig. 2. Differential cross section for inelastic scattering (see text).
F~ to
0
2
6
f0
f4
2KRjLn~ 0
Fig. 3. Comparison o f the ratio (dtr/dO)exp/Z(da/d£2)o with F2(O).
Fig. 3 compares the ratio (da/dl2)exp/2(dtr/dI2)o at the points corresponding to the maxima in figs. 1 and 2 (circles denote elastic and triangles inelastic scattering) with the quantity F2(0) giveing the nuclear surface diffuseness. The choice of different functions (besides the form (7), ¢(z) was taken as step function and as an exponential) has a very small effect on the function F(O). 4. Free Path of a Particle in the Nucleus
A point particle moving along the axis x is described by the wave function ~k0 = e x p ( i K x ) . According to the optical model the wave factor after the scattering is K ' = K 1 + i K 2. The quantity l = 1/K 2 describes the damping of the plane wave ~k0
after the scattering and can be called the free path of a particle in the nucleus. According to the diffraction theory the wave function of a particle after scattering is
EFFECT OF NUCLEAR SURFACE DIFFUSENESS
693
~k = ~ko[1-co(p)]. Assuming that Kt ~ K (the scattering inside the nucleus is neglected) we obtain ~k = ~boexp(-x//). In the case of a finite size nucleus we have l = l(x) and the ratio x/l must be replaced by the integral S+ ~ dx/l(x). Comparing both expressions for co(p) we obtain the following formula for o~(p): co(p) = 1 - e x p
-
~
(12)
.
Bearing in mind that x(r) --- ( r 2 - p 2 ) ½, co(p) can conveniently be represented as
co(p) = 1--exp [
(13)
.]p x/r2~p-pZ/(r)J
To estimate the free path of ~-particles in nuclear matter, l(r) was chosen as O,
r < R 1,
r-R1 ----, R2--R 1 1
1
l(r)
lo
R1
1,
< r <
RE, (14)
R 2 < r < R 3,
R 4 - - 1"
-
-
R 4 - R3 0,
,
R 3 <
r <
R4,
R4 < r,
where lo describes the free path of ~-particles in the nucleus. ~0171 tO
0.5
I
i
I~
f
i
Fig. 4. F u n c t i o n co(p) i n t h e s p h e r i c a l case for t h e M g ~4 n u c l e u s .
Fig. 4 gives the function co(p) in the spherical case for the Mg 24 nucleus (solid curve). The dashed curve corresponds to the choice of parameters R 1 = Rz = O, Ra = 4.81 fm and R4 --- 6.81 fm. This case corresponds to the trapezoidal nuclear density p(r) with surface layer thickness 1.6 fm if it is assumed that 1/l(r) oc p(r).
694
E. V. INOPIN AND Yu. A. BEREZHNOY
The dot-and-dash line passing into the dashed curve when p = 4.81 fm corresponds to the choice of parameters R 1 = 2.81 fro, R 2 = R 3 = 4.81 fm and R 4 = 6.81 fm and belongs to the surface absorption case. In both cases the free path o f 0t-particles in the Mg 24 nucleus was found to be equal to l o = 2.0 fm which is in good agreement with other estimates of this quantity 6). The idea of taking into account nuclear surface diffuseness in the diffraction scattering of particles by a non-spherical nuclei has been expressed earlier by Professor A. G. Sitenko, which the authors acknowledge with gratitude. References 1) 2) 3) 4) 5) 6) 7)
S. I. Drozdov, JETP 28 (1955) 734, 736 (translation: JETP (Soviet Physics) 1 (1955) 588) Ye. V. Inopin, JETP 31 (1956) 901 (translation: JETP (Soviet Physics) 4 (1957) 764) J. S. Blair, Phys. Rev. 115 (1959) 928 D. K. McDaniels, J. S. Blair, S. W. Chert and G. W. Farwell, Nuclear Physics 17 (1960) 614 J. S. Blair, G. W. Farwell and D. K. McDaniels, Nuclear Physics 17 (1960) 641 C. E. Porter, Phys. Rev. 99 (1955) 1400 A. G. Sitenko and V. K. Tartakovsky, Ukrain. Fiz. Zhur. 6 (1961) 12