- Email: [email protected]
A new impulse approximation in the α(d, np)α reaction
Nuclear Physics A137 (1969) 51-64;
@
North-Holland Publishing Co., Amsterdam
Not to be reprodueed by photoprint or mierotibn without written permission from the publisher
A NEW IMPULSE
APPROXIMATION
IN THE a(d, np)& REACTION
HIROSHI NAKAMURA College of Science and Engineering, Aoyama Gakuin University, 576 Megurisawa, Setagaya-ku, Tokyo Received 22 May 1969 Abstract: An improved impulse approximation in the a(d, np)a reaction is introduced. The effects of the final state interaction of the spectator nucleon and the initial state interaction of the incoming deuteron are partly taken into account in the calculation of differential cross sections. The results give a rather good fit to the experimental analysis.
1. Introduction
The problem of the cr(d, np)cr reaction has recently been analysed by several authors using the impulse approximation ’ -“) and R-matrix theory “). The prominent peaks and bumps in the energy spectra which were observed in the experimental analyses [refs. 3- “)I are well understood as the final state interaction peaks due to the formation of ‘Li or 5He, or the pole peaks due to the deuteron poles in the matrix element, which were pointed out by Chew and Low “). However, these theoretical analyses may not be satisfactory. In fact, the differential cross section near the deuteron pole, which was calculated using the impulse approximation, was 4 x 6 times larger than the experimental value ‘). As discussed in the previous papers iv ‘), this discrepancy may be due to the effects of the final state interaction between the spectator nucleon and alpha particle as well as the initial state interaction of the incoming deuteron. In the present article, we shall try a crude estimation of such effects in order to obtain the outlook on our theory, which was introduced in ref. 7), before going to the more exact numerical analysis. 2. The improved impulse approximation In the present analysis, the same notations are used as in ref. ‘). The matrix element which corresponds to a pole term To” can be written as &I U’“‘lk) ,
(2.1)
where + (2.2)++ t When the spectator nucleon is the proton (neutron), we attach a superscript n (p) to the concerned quantities. tt See formula (7.2) of ref. ‘). 51
52
H.
NAKAMURA
Notations Ih,) and Ik) represent the plane waves for the outgoing proton and incoming deuteron, respectively. As in the DWBA method, the final state interaction of the outgoing proton and the initial state interaction of the incoming deuteron change the matrix element (2.1) to (hy’l UC”)1 kin) , where Ihy’) given by
P-3)
and Ik’“) represent the distorted waves. The whole matrix element is
(2.4)
We take a partial wave expansion
where Then, the matrix element (2.3) can be written as (/$“I U’“‘lk’“) = @“‘I U(“C)lki”)+ (h~“lW’“‘lk’“>,
(2.6)
where
UC"') =(a,+a,x,)(a
- p),
W’“’ ={~(ql)-(ao+alx,)(a.p)}.
(2.7) (2-g)
Since the strong final and initial state interactions occur mainly in the first term of (2.6) as will be seen in appendix 2, we obtain an approximate formula (h’$“‘lW’“‘(k’“) = (h,lW’“‘jk).
(2.9)
Here, we neglect the first term in formula (2.6), i.e. (h”,“‘lU’“‘~k’“) =
(2.10)
The accuracy of this approximation will be discussed in sect. 3. The matrix element (h,(W’“‘lk) is obtained from (h,(U’“‘lk) simply by a substitution 9%)
+ 4(‘I1)-(%+%
X1)(@* PI.
(2.11)
The same approximation is adopted for another matrix element (ho;ltlU(P)lk’“). Using formulas (2.4), (2.10) and (2.1 l), the differential cross section is calculated by almost the same procedure as in the impulse approximation. The results are illustrated in figs. l-7, together with the experimental data obtained by Warner and Bercaw ‘) and also by Tanabe “). It should be mentioned that there is no parameter and the differential cross section was evaluated without normalization. The matrix elements
&=42t.w
S_=S.S’ Bp=40’
Q,=9.8BP’ 30
6,.
9 8’
$= 24
16
I
I
24
32
E,,W’V)
Fig. 1. Absolute differential cross section, in mb/sr z * MeV, for a+d -+ a+p+n, which were calculated by the use of approximation (2.10). Notations E,, and Ed represent the energies of the incoming and outgoing alpha particles in laboratory system, respectively, &(t?,) the angle formed by the momenta of the outgoing alpha particle (proton) and the incoming alpha particle. The alpha particle, neutron and proton are emitted coplanar. The arrows represent the points where the differential cross section is kept unchanged by the correction S (e). The experimental points were given by Warner and Bercaw 4).
E,,=42NeV $=14
8”
ep= 60’
E,(MeV)
Fig. 2. Same as fig. 1.
Let us compare the present results with that of the off-energy-shell and on-energy shell impulse approximations j, One typical example (E,, = 42 MeV, 0, = 9.8”, t The impulse approximation with formula (3.3) (formula (3.6)) in ref. ‘) is called the off-energyshell (on-energy-shell) impulse approximation.
ax[d, np)a
REACTION
55
L,=LZMeV e&=19.84 80
ep=60*
30-
G-19
0
8’
e*=30*
&=19.8’ e,= 20’
8
12
16
20
24
28
32
L, WeV)
Fig. 3. Same as fig. I.
8, = 20”) is illustrated in fig. 8. The predicted differential cross section in the present
analysis (solid curve) is almost three times smaller than that for the off-energy-shell impulse appro~mation (broken curve), giving a better fit to the data. A considerable change of shape is observed between the curves for the off-energy-shell impulse approximation and the on-energy-shell one (dashed and dotted curve).
56
H. NAKAMURA
E,,=29.2h!ev &=lO. 9,;
60
45’
&.=ll' 9,-W
40
20
9,=16* &=45
80 e,:2v 8.145’
60
Fig. 4. Same as fig. 1, but the experimental points were given by Tanabe s).
a(d,
np)a
57
REACTION
800600 -
Fig. 5. Same as fig. 4.
3. The estimation of the residual matrix element As discussed in ref. ‘), the general form of the matrix element M is given by M = S(b’p)+i(V.p)+A.(aXp)+
C 7~jaipj. ij
(3.1)
58
l-i. NAKAMURA
E,i= 29.2t.w &=lcr 0,=45' Yp'O.
Yp=32'
E*,(MeV)
15
20
25
Fig. 6. Same as fig. 4, but the outgoing particles are not emitted coplanar in general. Notation qP represents the azimuthal angle of the outgoing proton (see ref. ‘)) .
The main contribution to the differential cross section comes from the first term S(@ * p) in general, in which the scalar S is called the “scalar amplitude”. We shall show in this section that the differential cross section obtained in sect. 2 does not change much by the correction due to the scalar amplitude S(n*pc)of the matrix element
cc(d, np)a
59
REACTION
E,, =29 2MeV 8*=1129 f3,=30 ‘9,-o
Fig. 7. Same as fig. 6.
The following approximations are adopted to calculate ,SCnc). (i) The terms which are proportional to (h, x A;) - t7 are neglected in calculating the matrix element (ii)
60
H. NAKAMURA
Edi = 42MeV Q,=9.8 e,=zo’
100 -
50
24
16
E,,(MeV)
Fig. 8. Absolute differential cross sections at Oa = 92, Or, = 20”, E,, = 42 MeV which were calculated by the present new impulse approximation (solid curve), the off-energy-shell impulse approximation (broken curve) and the on-energy-shell impulse approximation (dashed and dotted curve). They have been reduced by the indicated factors.
where
A(“) = B’“’
=
&;
$“‘(O,f: h, h,),
(.f’“‘(l, 4: h, h,)+W”‘(L 3: hl Ml.
The first and second terms of (3.2) represent
the
(3.3)
S- and P-wave amplitudes,respectively.
(iii) h’ w k-h,.
(3.4)
Then, we get a partial wave expansion U’“” x {A’“’E T(L,) + B’“)L;dT(L,, P
where
the definitions
of the amplitudes
T(L,)
and T(L,,
&)}(a 15,)
* P),
(3.5)
are given in appendix 1.
a(d,
61
nptff REACTiON
fn the process which corresponds to the first (second) term T(I;,)(T(L,, &)), the outgoing neutron, proton and the incoming deuteron have the orbital angular momenta 0 (I), Ld(LP) and Ld, respectively. Let us take the approximation +
@p)(LpJ,: S,(L”J:
r,) % eUiapsin (h, rP+6&, for t = L”-J, etaasin (kr, + a,), rd) m o for t # L”-J, ,
(3.6)
where 6, and 6, represent the phase shifts of rx-pand or-dscatterings, respectively. Then, near the deuteron pole (h z $I), we can adopt the following approximate formula. (iv)
cfnf& J,, , L, J, , Ld J) w D’“‘(L, Jp , Ld J)C’“‘(L,, J,, , L, Jp , I&J),
(3-7)
@(L,
(3.8)
where J,, LdJ)
= ei@pfad) (COS(Sp-6d)-R(h-3k)sin(6,-S,)).
Notation R represents a constant which is related to the radius of deuteron. Two choices of R were tried in the present analysis, namely R = 0 and 3 fm. From formulas (3.5), (3.7) and (3.8), we get PC) = A’“’ c, I?“)&,
Ld)T(Ld) -I-i?q_
B(“)(Lp, Ld)T(Lp , Id&,
(3.9)
where
F”‘(L* IL,) =
1 z: (2J,+ l)(W+ l)D’“‘&, Jp, r, J). 6(2Lp + 1)(2&j + 1) J,f
(3.10)
The scalar SCpc)is calculated by the same procedure and the total scalar amplitude St’) is given by SW = SW + c$PC) (3.11) As we have discussed in sect. 2, the matrix element in the present new impulse approximation is given by (h,[W’“‘lk>I-(h,IW’P’lk>. (3.12) Let St’) be the scalar amplitude of the matrix element (3.12). The absolute value of SC’) is comparable to that of SCT’in general. However, the inequality I, < I < 12,
or
I, >I>],,
(3.13)
holds in many regions indicating that the differential cross section does not change much by the correction S(O), where J I- IS(T)12 I, = IS’T’;s~~#, I2 = (S(T)+Sg!.3fm12 _ .
(3.14)
t The d&nitions of .F(pj, SC, c@) and tYn) are given in ref. ‘). Refer to the formulas (3.4). (6.1) and (7.4).
H.
62
NAKAMURA
The arrows with circles in figs. 1-7 represent the positions where the inequality (3.13) holds satisfying another inequality (3.15)
0.8 < IZ/li < 1.2.
At the positions indicated by the arrows with crosses, the inequalities (3.13) and 1,/I, < 0.5,
or
(3.16)
Iz/I, > 2,
are satisfied. For the purpose of reference, some of the actual values of SCT’and S@) are given in table 1. TABLE 1 Some of the typical examples of the scalar arnplitu~~ ST) and S@) 4% = 42 MeV
4&Q.)
Wkw)
&(MeV)
R(fm)
Sn (fm*>
S(@(fm*) 33.0-152.3i 69.8-126.63.
9.8
20
28
0 3
46.4+127.4i
14.8
50
22
0 3
41.1+
3O.li
14.8
10
24
0 3
-
3.2-
15.3i
19.8
60
20
0 3
-24.5-
19.8
40
14
0 3
-
1.2+
68.81 0.2i
-lS.O-10.2-k
16.Oi 29.73.
27.2-122.25 73.093.6i -
4.9f 27.6i lOO.S+ 7LOi 69.8+152.Oi 146.8-k157.4i
In the present analysis, the phase shift Bdwas obtained by the simple extrapolation of the experimental value from the region E_, 6 7 MeV [ref. ‘)I. 4. Discussion The strong final state interaction of the spectator nucleon and the initial state interaction of the incoming deuteron occur mainly in the residual matrices Ufn*pc),while the matrices WC”* p) include only the effect of the subtraction of such partial waves due to the absorption and the phase change. Then, if we take an example in the theory of eiastic scattering, the present new impulse approximation introduced in sect. 2 may correspond to the analysis in which only the effect of the diffraction scattering is taken into account, The calculated differential cross section has given a rather good agreement with the experimental data especially in the region close to the deuteron poIe. It has been shown that the correction for the present theory due to the residual matrices UCnSpc) is not small but it changes only the phase of the scalar amplitude SCT)in many regions and, therefore, the differential cross section may not be changed much. In fact, we have
a(d, np)a REACTION
63
obtained fairly good results in such regions. As expected, we have got very poor results in the region IS( >> jS’V]. ’ The simple impulse approximation has been improved considerably in the present analysis as we have seen in fig. 8. We expect that more excellent agreement will be obtained in the coming more exact analysis which is now in progress. Appendix 1 The definitions of T(&) and T(L,, L,). T(0) = a,, T(l) = a, Xl,
T(0, 1) =
(a0 k-34
h,)hz, ,
T&O) = -(aokh
khy,
T&2)
= a, kh(x, ~1 -+Y,),
T(2,l)
= -a,
Yl -+I).
bh(x1
The other T(Ld), T(L,, L,) = 0, where Yl =A*&,
3
z,=A4.
Appenilix 2 The same analysis as in formulas (3.5)~(3.10) can be carried out for W(O).One can write W’“’ z {A(“)Z T’“‘(L3+8’n)~~~T’“‘(LP, LJ}(b * p), P
where T”(L,) and T4(L,, L,) represent the contributions partial waves. For Ld 5 2 they are given by T“” (2) =
T’“‘(2,l)
T’“‘(3,2)
T’“‘(3) = u,P,(x,),
Mzh),
= +Q 4(3x,
T’“‘(1,2) = -4% =
h,{+z,
from the corresponding
Y, -xl),
vh(3x1
k-&zz
Zl -Y1)9
h,){5x;y,
-2x,
z1 -yl}.
Since the phase shifts 6, and 6, are large in S-, P- and S-, D-waves ‘), respectively, significant corrections occur only in terms T(‘) (1,2), T(“) (3,2) and T’“’ (2). They can be calculated by the similar formulas to (3.9) and (3.10). The results show that the amplitude of WC”’ increases only by 20 % at most by the correction for T’“’ (1,2) and T(‘) (3,2) in the neighbourhood of the deuteron pole. The correction for T(“) (2) gives smaller change. Then, the approximation (2.9) must be reasonable. t See table 1.
64
H. NAKAMURA
References 1) 2) 3) 4) 5)
6) 7) 8) 9)
H. Nakamura, Phys. Lett. 24B (1967) 509 D. A. Burress, Oberlin College (1966) unpublished K. Nagatani, T. A. Tombrello and D. A. Bromley, Phys. Rev. 140 (1965) B824 R. E. Warner and R. W. Bercaw, Phys. Lett. 24B (1967) 517; Nucl. Phys. A109 (1968) 205 T. Tanabe, J. Phys. Sot. Japan 25 (1968) 21 G. F. Chew and F. E. Low, Phys. Rev. 113 (1959) 1640 H. Nakamura and K. Nagatani, Nucl. Phys. Al01 (1967) 557 J. L. Gammell and R. M. Thaler, Phys. Rev. 109 (1958) 2041 L. C. McIntyre and W. Haeberli, Nucl. Phys. A91 (1967) 382
@
North-Holland Publishing Co., Amsterdam
Not to be reprodueed by photoprint or mierotibn without written permission from the publisher
A NEW IMPULSE
APPROXIMATION
IN THE a(d, np)& REACTION
HIROSHI NAKAMURA College of Science and Engineering, Aoyama Gakuin University, 576 Megurisawa, Setagaya-ku, Tokyo Received 22 May 1969 Abstract: An improved impulse approximation in the a(d, np)a reaction is introduced. The effects of the final state interaction of the spectator nucleon and the initial state interaction of the incoming deuteron are partly taken into account in the calculation of differential cross sections. The results give a rather good fit to the experimental analysis.
1. Introduction
The problem of the cr(d, np)cr reaction has recently been analysed by several authors using the impulse approximation ’ -“) and R-matrix theory “). The prominent peaks and bumps in the energy spectra which were observed in the experimental analyses [refs. 3- “)I are well understood as the final state interaction peaks due to the formation of ‘Li or 5He, or the pole peaks due to the deuteron poles in the matrix element, which were pointed out by Chew and Low “). However, these theoretical analyses may not be satisfactory. In fact, the differential cross section near the deuteron pole, which was calculated using the impulse approximation, was 4 x 6 times larger than the experimental value ‘). As discussed in the previous papers iv ‘), this discrepancy may be due to the effects of the final state interaction between the spectator nucleon and alpha particle as well as the initial state interaction of the incoming deuteron. In the present article, we shall try a crude estimation of such effects in order to obtain the outlook on our theory, which was introduced in ref. 7), before going to the more exact numerical analysis. 2. The improved impulse approximation In the present analysis, the same notations are used as in ref. ‘). The matrix element which corresponds to a pole term To” can be written as &I U’“‘lk) ,
(2.1)
where + (2.2)++ t When the spectator nucleon is the proton (neutron), we attach a superscript n (p) to the concerned quantities. tt See formula (7.2) of ref. ‘). 51
52
H.
NAKAMURA
Notations Ih,) and Ik) represent the plane waves for the outgoing proton and incoming deuteron, respectively. As in the DWBA method, the final state interaction of the outgoing proton and the initial state interaction of the incoming deuteron change the matrix element (2.1) to (hy’l UC”)1 kin) , where Ihy’) given by
P-3)
and Ik’“) represent the distorted waves. The whole matrix element is
(2.4)
We take a partial wave expansion
where Then, the matrix element (2.3) can be written as (/$“I U’“‘lk’“) = @“‘I U(“C)lki”)+ (h~“lW’“‘lk’“>,
(2.6)
where
UC"') =
- p),
W’“’ =
(2.7) (2-g)
Since the strong final and initial state interactions occur mainly in the first term of (2.6) as will be seen in appendix 2, we obtain an approximate formula (h’$“‘lW’“‘(k’“) = (h,lW’“‘jk).
(2.9)
Here, we neglect the first term in formula (2.6), i.e. (h”,“‘lU’“‘~k’“) =
(2.10)
The accuracy of this approximation will be discussed in sect. 3. The matrix element (h,(W’“‘lk) is obtained from (h,(U’“‘lk) simply by a substitution 9%)
+ 4(‘I1)-(%+%
X1)(@* PI.
(2.11)
The same approximation is adopted for another matrix element (ho;ltlU(P)lk’“). Using formulas (2.4), (2.10) and (2.1 l), the differential cross section is calculated by almost the same procedure as in the impulse approximation. The results are illustrated in figs. l-7, together with the experimental data obtained by Warner and Bercaw ‘) and also by Tanabe “). It should be mentioned that there is no parameter and the differential cross section was evaluated without normalization. The matrix elements
&=42t.w
S_=S.S’ Bp=40’
Q,=9.8BP’ 30
6,.
9 8’
$= 24
16
I
I
24
32
E,,W’V)
Fig. 1. Absolute differential cross section, in mb/sr z * MeV, for a+d -+ a+p+n, which were calculated by the use of approximation (2.10). Notations E,, and Ed represent the energies of the incoming and outgoing alpha particles in laboratory system, respectively, &(t?,) the angle formed by the momenta of the outgoing alpha particle (proton) and the incoming alpha particle. The alpha particle, neutron and proton are emitted coplanar. The arrows represent the points where the differential cross section is kept unchanged by the correction S (e). The experimental points were given by Warner and Bercaw 4).
E,,=42NeV $=14
8”
ep= 60’
E,(MeV)
Fig. 2. Same as fig. 1.
Let us compare the present results with that of the off-energy-shell and on-energy shell impulse approximations j, One typical example (E,, = 42 MeV, 0, = 9.8”, t The impulse approximation with formula (3.3) (formula (3.6)) in ref. ‘) is called the off-energyshell (on-energy-shell) impulse approximation.
ax[d, np)a
REACTION
55
L,=LZMeV e&=19.84 80
ep=60*
30-
G-19
0
8’
e*=30*
&=19.8’ e,= 20’
8
12
16
20
24
28
32
L, WeV)
Fig. 3. Same as fig. I.
8, = 20”) is illustrated in fig. 8. The predicted differential cross section in the present
analysis (solid curve) is almost three times smaller than that for the off-energy-shell impulse appro~mation (broken curve), giving a better fit to the data. A considerable change of shape is observed between the curves for the off-energy-shell impulse approximation and the on-energy-shell one (dashed and dotted curve).
56
H. NAKAMURA
E,,=29.2h!ev &=lO. 9,;
60
45’
&.=ll' 9,-W
40
20
9,=16* &=45
80 e,:2v 8.145’
60
Fig. 4. Same as fig. 1, but the experimental points were given by Tanabe s).
a(d,
np)a
57
REACTION
800600 -
Fig. 5. Same as fig. 4.
3. The estimation of the residual matrix element As discussed in ref. ‘), the general form of the matrix element M is given by M = S(b’p)+i(V.p)+A.(aXp)+
C 7~jaipj. ij
(3.1)
58
l-i. NAKAMURA
E,i= 29.2t.w &=lcr 0,=45' Yp'O.
Yp=32'
E*,(MeV)
15
20
25
Fig. 6. Same as fig. 4, but the outgoing particles are not emitted coplanar in general. Notation qP represents the azimuthal angle of the outgoing proton (see ref. ‘)) .
The main contribution to the differential cross section comes from the first term S(@ * p) in general, in which the scalar S is called the “scalar amplitude”. We shall show in this section that the differential cross section obtained in sect. 2 does not change much by the correction due to the scalar amplitude S(n*pc)of the matrix element
cc(d, np)a
59
REACTION
E,, =29 2MeV 8*=1129 f3,=30 ‘9,-o
Fig. 7. Same as fig. 6.
The following approximations are adopted to calculate ,SCnc). (i) The terms which are proportional to (h, x A;) - t7 are neglected in calculating the matrix element
60
H. NAKAMURA
Edi = 42MeV Q,=9.8 e,=zo’
100 -
50
24
16
E,,(MeV)
Fig. 8. Absolute differential cross sections at Oa = 92, Or, = 20”, E,, = 42 MeV which were calculated by the present new impulse approximation (solid curve), the off-energy-shell impulse approximation (broken curve) and the on-energy-shell impulse approximation (dashed and dotted curve). They have been reduced by the indicated factors.
where
A(“) = B’“’
=
&;
$“‘(O,f: h, h,),
(.f’“‘(l, 4: h, h,)+W”‘(L 3: hl Ml.
The first and second terms of (3.2) represent
the
(3.3)
S- and P-wave amplitudes,respectively.
(iii) h’ w k-h,.
(3.4)
Then, we get a partial wave expansion U’“” x {A’“’E T(L,) + B’“)L;dT(L,, P
where
the definitions
of the amplitudes
T(L,)
and T(L,,
&)}(a 15,)
* P),
(3.5)
are given in appendix 1.
a(d,
61
nptff REACTiON
fn the process which corresponds to the first (second) term T(I;,)(T(L,, &)), the outgoing neutron, proton and the incoming deuteron have the orbital angular momenta 0 (I), Ld(LP) and Ld, respectively. Let us take the approximation +
@p)(LpJ,: S,(L”J:
r,) % eUiapsin (h, rP+6&, for t = L”-J, etaasin (kr, + a,), rd) m o for t # L”-J, ,
(3.6)
where 6, and 6, represent the phase shifts of rx-pand or-dscatterings, respectively. Then, near the deuteron pole (h z $I), we can adopt the following approximate formula. (iv)
cfnf& J,, , L, J, , Ld J) w D’“‘(L, Jp , Ld J)C’“‘(L,, J,, , L, Jp , I&J),
(3-7)
@(L,
(3.8)
where J,, LdJ)
= ei@pfad) (COS(Sp-6d)-R(h-3k)sin(6,-S,)).
Notation R represents a constant which is related to the radius of deuteron. Two choices of R were tried in the present analysis, namely R = 0 and 3 fm. From formulas (3.5), (3.7) and (3.8), we get PC) = A’“’ c, I?“)&,
Ld)T(Ld) -I-i?q_
B(“)(Lp, Ld)T(Lp , Id&,
(3.9)
where
F”‘(L* IL,) =
1 z: (2J,+ l)(W+ l)D’“‘&, Jp, r, J). 6(2Lp + 1)(2&j + 1) J,f
(3.10)
The scalar SCpc)is calculated by the same procedure and the total scalar amplitude St’) is given by SW = SW + c$PC) (3.11) As we have discussed in sect. 2, the matrix element in the present new impulse approximation is given by (h,[W’“‘lk>I-(h,IW’P’lk>. (3.12) Let St’) be the scalar amplitude of the matrix element (3.12). The absolute value of SC’) is comparable to that of SCT’in general. However, the inequality I, < I < 12,
or
I, >I>],,
(3.13)
holds in many regions indicating that the differential cross section does not change much by the correction S(O), where J I- IS(T)12 I, = IS’T’;s~~#, I2 = (S(T)+Sg!.3fm12 _ .
(3.14)
t The d&nitions of .F(pj, SC, c@) and tYn) are given in ref. ‘). Refer to the formulas (3.4). (6.1) and (7.4).
H.
62
NAKAMURA
The arrows with circles in figs. 1-7 represent the positions where the inequality (3.13) holds satisfying another inequality (3.15)
0.8 < IZ/li < 1.2.
At the positions indicated by the arrows with crosses, the inequalities (3.13) and 1,/I, < 0.5,
or
(3.16)
Iz/I, > 2,
are satisfied. For the purpose of reference, some of the actual values of SCT’and S@) are given in table 1. TABLE 1 Some of the typical examples of the scalar arnplitu~~ ST) and S@) 4% = 42 MeV
4&Q.)
Wkw)
&(MeV)
R(fm)
Sn (fm*>
S(@(fm*) 33.0-152.3i 69.8-126.63.
9.8
20
28
0 3
46.4+127.4i
14.8
50
22
0 3
41.1+
3O.li
14.8
10
24
0 3
-
3.2-
15.3i
19.8
60
20
0 3
-24.5-
19.8
40
14
0 3
-
1.2+
68.81 0.2i
-lS.O-10.2-k
16.Oi 29.73.
27.2-122.25 73.093.6i -
4.9f 27.6i lOO.S+ 7LOi 69.8+152.Oi 146.8-k157.4i
In the present analysis, the phase shift Bdwas obtained by the simple extrapolation of the experimental value from the region E_, 6 7 MeV [ref. ‘)I. 4. Discussion The strong final state interaction of the spectator nucleon and the initial state interaction of the incoming deuteron occur mainly in the residual matrices Ufn*pc),while the matrices WC”* p) include only the effect of the subtraction of such partial waves due to the absorption and the phase change. Then, if we take an example in the theory of eiastic scattering, the present new impulse approximation introduced in sect. 2 may correspond to the analysis in which only the effect of the diffraction scattering is taken into account, The calculated differential cross section has given a rather good agreement with the experimental data especially in the region close to the deuteron poIe. It has been shown that the correction for the present theory due to the residual matrices UCnSpc) is not small but it changes only the phase of the scalar amplitude SCT)in many regions and, therefore, the differential cross section may not be changed much. In fact, we have
a(d, np)a REACTION
63
obtained fairly good results in such regions. As expected, we have got very poor results in the region IS( >> jS’V]. ’ The simple impulse approximation has been improved considerably in the present analysis as we have seen in fig. 8. We expect that more excellent agreement will be obtained in the coming more exact analysis which is now in progress. Appendix 1 The definitions of T(&) and T(L,, L,). T(0) = a,, T(l) = a, Xl,
T(0, 1) =
(a0 k-34
h,)hz, ,
T&O) = -(aokh
khy,
T&2)
= a, kh(x, ~1 -+Y,),
T(2,l)
= -a,
Yl -+I).
bh(x1
The other T(Ld), T(L,, L,) = 0, where Yl =A*&,
3
z,=A4.
Appenilix 2 The same analysis as in formulas (3.5)~(3.10) can be carried out for W(O).One can write W’“’ z {A(“)Z T’“‘(L3+8’n)~~~T’“‘(LP, LJ}(b * p), P
where T”(L,) and T4(L,, L,) represent the contributions partial waves. For Ld 5 2 they are given by T“” (2) =
T’“‘(2,l)
T’“‘(3,2)
T’“‘(3) = u,P,(x,),
Mzh),
= +Q 4(3x,
T’“‘(1,2) = -4% =
h,{+z,
from the corresponding
Y, -xl),
vh(3x1
k-&zz
Zl -Y1)9
h,){5x;y,
-2x,
z1 -yl}.
Since the phase shifts 6, and 6, are large in S-, P- and S-, D-waves ‘), respectively, significant corrections occur only in terms T(‘) (1,2), T(“) (3,2) and T’“’ (2). They can be calculated by the similar formulas to (3.9) and (3.10). The results show that the amplitude of WC”’ increases only by 20 % at most by the correction for T’“’ (1,2) and T(‘) (3,2) in the neighbourhood of the deuteron pole. The correction for T(“) (2) gives smaller change. Then, the approximation (2.9) must be reasonable. t See table 1.
64
H. NAKAMURA
References 1) 2) 3) 4) 5)
6) 7) 8) 9)
H. Nakamura, Phys. Lett. 24B (1967) 509 D. A. Burress, Oberlin College (1966) unpublished K. Nagatani, T. A. Tombrello and D. A. Bromley, Phys. Rev. 140 (1965) B824 R. E. Warner and R. W. Bercaw, Phys. Lett. 24B (1967) 517; Nucl. Phys. A109 (1968) 205 T. Tanabe, J. Phys. Sot. Japan 25 (1968) 21 G. F. Chew and F. E. Low, Phys. Rev. 113 (1959) 1640 H. Nakamura and K. Nagatani, Nucl. Phys. Al01 (1967) 557 J. L. Gammell and R. M. Thaler, Phys. Rev. 109 (1958) 2041 L. C. McIntyre and W. Haeberli, Nucl. Phys. A91 (1967) 382