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Effects of imaginary potentials on spurious resonances in resonating-group calculations
V01ume468, num6er 3
PHY51C5 LE77ER5
1 0ct06er 1973
EFFEC750F 1MA61NARY P07EN71AL50N 5PUR10U5 RE50NANCE51N RE50NA71N6-6R0UP CALCULA710N5* F.5. CHW1ER07H, Y.C. 7AN6 and D.R. 7H0MP50N
5ch0010f Phy51c5,Un1ver51ty0f M1nne50ta,M1nneap0115,M1nne50ta55455, U5A Rece1ved 12 Au9u5t 1973 8y 5tudy1n9the c0up1ed-channe1d + 3He and p + a re50nat1n9-9r0upca1cu1at10n,1t 15f0und that the 1ntr0duct10n 0f phen0men01091ca11ma91naryp0tent1a1515a 51mp1eway t0 e11m1nate5pur10u5re50nance5fr0m re50nat1n9-9r0upand perhap5 0ther m1cr05c0p1creact10n ca1cu1at10n5. 1n nuc1ear react10n the0r1e5 wh1ch 501ve the pr0jec. t10n e4uat10n (54¢1H-E14•) = 0
(1)
1n a truncated funct10n 5pace, 5pur10u5 m0de1 re50nance5 w111appear 1n ener9y re910n5 where 0pen react10n channe15 are n0t c0n51dered [1]. 5uch 5pur10u5 re50nance5 have 6een f0und, f0r examp1e, 1n 0ne- and tw0-channe1 re50nat1n9-9r0up ca1cu1at10n5 0f many 119ht nuc1ear 5y5tem5 [2]. 1n th15 c0mmun1cat10n, we 5ha115h0w 6y a 5pec1f1cexamp1e that even w1th a crude c0n51derat10n 0f the5e 0pen react10n channe15 6y the 1ntr0duct10n 0f phen0men01091ca1 1ma91nary p0tent1a15, the5e 5pur10u5 re50nance5 w111 d15appear fr0m the ca1cu1at10n. 0 n the 0ther hand, there are true re50nance5 1n the ca1cu1at10n wh1ch are 11tt1e affected 6y the pre5ence 0f the 1ma91nary p0tent1a15. 7he5e 1atter re50nance5 are th05e wh1ch are we11 de5cr16ed 6y the c0n51dered truncated funct10n 5pace and have 6een ca11ed p0tent1a1 re50nance5 6y 8en0hr and W11dermuth [3]. 7he 5pec1f1cexamp1e t0 6e c0n51dered here 15the tw0-channe1 (d + 3He and p + a channe15) 5tudy 0f the f1ve-nuc1e0n 5y5tem 1n the 5 = • 5tatet ~w1th the tr1a1 funct10n = A {~6h~0dF(r f) ~ + ¢a 6(r9)~),
(2)
where A 15 an ant15ymmetf12at10n 0perat0r and ~ 15 an appr0pr1ate 5p1n-1505p1nfunct10n f0r 5 = • and 7 = •. 7he funct10n5 ~ de5cr16e the 5pat1a1 6ehav10ur * W0rk 5upp0rted 1npart 6y the U.5. At0m1cEner9y C0mm15510n. j.1 51neea pure1ycentra1 nuc1e0n-nuc1e0np0tent1a115u5ed 1n the ca1cu1at10n,the t0ta15p1nan9u1arm0mentum 15e0n5erved.
0f the 3He, deuter0n and a c1u5ter5; tch1r 5pec1f1c f0rm5 are 91ven 1n ref5. [2] and [4]. 7he funct10n5 F(rf) and 6(r9) de5cr16e the re1at1ve m0t10n5 0f the d + 3He and p + t~ c1u5ter5, re5pect1ve1y, and are determ1ned 6y 501v1n9e4.(1). F1na11y. 1t 5h0u1d 6e ment10ned that, 1n e4.(1), E 15 the t0ta1 ener9y 0f the 5y5tem and H 15 the Ham11t0n1an, 91ven 6y 712 5
5
1=1
(3)
1>j=1
w1th V116e1n9 the nuc1e0n-nuc1e0n p0tent1a1 0f ref. [4] w1th0ut the 5p1n-0r61t c0mp0nent. 5u65t1tut10n 0f e4. (2) 1nt0 e4. (1) 1ead5 t0 the f0110w1n9 c0up1ed 1nte9r0d1fferent1a1 e4uat10n5 f0r the re1at1ve-m0t10n funct10n5:
[~fV2+Ef-VDf(rf)-VCf(rf)]F(rf) f
=
•
,
Kff(rf, rf) F~rf) dtf +
(4a)
,
(rf, r9) 6(r9
[ ~9 V29+E9-- VD9(r9)-- VC9(•9)] 6(r9)
(46)
= fK9f(r9, r•f)F( • ~f)dr~f+fK99r9, )6(r•9) dr•9 , where the 5u65cr1pt5 f and 9 refer t0 the d + 3He and p + ~, channe15, re5pect1ve1y. 7he expre5510n5 f0r the d1rect nuc1ear p0tent1a15 1•D, the d1rect C0u10m6 p0tent1a15 VC, and the kerne1 funct10n5 are very 1en9thy and are 91ven 1n ref. [5]. 7he 4uant1t1e5 Ef and E9 are the re1at1ve k1net1c ener91e5 0f the c1u5ter5 at 1ar9e 5eparat10n 1n the c.m. 5y5tem; they are re1ated 301
v01ume 468, num6er 3
PHY51c5 LE77ER5
6y the e4uat10n
E9 = Ef+ Eth .
(5)
W1th the ch05en f0rm5 f0r ~6, Eth 15 ca1cu1ated t0 6e 20.45 MeV, wh1ch 15 1n fa1r a9reement w1th the va1ue 0f 18.35 MeV determ1ned exper1menta11y. Fr0m the 501ut10n5 0f e45. (4a) and (46), we determ1ne the 5catter1n9 matr1x wh05e d1a90na1 e1ement5 are wr1tten a5 5~f = r/f exp(216/f) , (6)
5199 = r19 exp(21619) w1th 7"/f= 719.7he re5u1t5 f0r 6/f w1th 1 = 0 t0 3 are 5h0wn a5 a funct10n 0 f E f 6y the 5011d curve5 1n f19. 1. Here 0ne 5ee5 that rather 5harp re50nance 5tructure5 are pre5ent at ar0und 25 MeV 1n the 1 = 0 and 1 part1a1 wave5, wh11e 50mewhat 6r0ader re50nance 5tructure appear f0r 1 = 2 and 3.51nce the ener91e5 at wh1ch the5e re50nance5 appear are a60ve the three-60dy 6reakup thre5h01d5, a 4ue5t10n ar15e5 a5 t0 whether the5e ca1cu1ated re50nance5 are 5pur10u5 re50nance5 200
1
r
r
,,°1 100h
100
/
50
1
)
~
)
1
1
1
1
1>
50
1
•
0
83f
32f
50
0 0
0 Ef (MeV)
1
1
113 20
F
30
1
40
1
50
Ef (MeV)
F19. 1. Pha5e 5h1ft561f ca1cu1ated w1th (5011dd0t5) and w1th0ut (5011d 11ne5)1ma91naryp0tent1a15. 302
+ 4 exp(r~)
[1 +ex
,
,7,
R9 = 2.25 f m ,
a f = a 9 = 0.5 fm .(8)
81~ 150
0~L
W1(r1)=--W01{[1+exp(r1-R1~] -1
Rf = 3.5 f m ,
80f
Lk
ar151n9 a5 a c0n5e4uence 0f 0ur m0de1 a55umpt10n 0f 0m1tt1n9 three- and m0re-60dy channe15 1n the ca1cu1at10n, 0r they d0 1n fact 1nd1cate the pre5ence 0f re1at1ve1y 10n9-11ved re50nance 5tructure5 1n the c0mp0und nuc1eu5 5L1. 1n 0rder t0 an5wer th15 4ue5t10n, a ca1cu1at10n 1n wh1ch a11 0pen channe15 are 1nc1uded 5h0u1d 6e perf0rmed. 5uch a ca1cu1at10n 15, 0f c0ur5e, n0t pract1ca1 at the pre5ent m0ment. 7heref0re, a5 a f1r5t 5tep, we 5ha11 51mp1y exam1ne the 6ehav10ur 0f the5e re50nance5 6y a ca1cu1at10n 1n wh1ch the 0m1tted 0pen channe15 are appr0x1mate1y taken 1nt0 acc0unt 6y the 1ntr0duct10n 0f 1ma91nary p0tent1a15. 1ma91nary p0tent1a15 are 1ntr0duced 1nt0 the f0rma115m 6y 51mp1y rep1ac1n9 VDf6y VDf+ 1Wf and VD9 6y VD9+1W9 1n e45. (4a) and (46). F0r the f0rm5 0f Wf and W9, We U5e
w1th 1 den0t1n9 e1ther f 0r 9, and
200
r
1 0ct06er 1973
At ener91e5 where exper1menta1 data [6, 7] ex15t, we have adju5ted W0f and W09 50 that 0ur ca1cu1at10n y1e1d5 the 6e5t a9reement w1th mea5ured d1fferent1a1 e1a5t1c and part1a1 react10n cr055 5ect10n5, a5 we11 a5 t0ta1 react10n cr055-5ect10n va1ue5t 2. 7he a9reement 50 06ta1ned 15, 1n 9enera1, 4u1te 900d [5, 8], wh1ch 15 an 1nd1cat10n that 0ur pr0cedure 0f tak1n9 1nt0 acc0unt the 0pen channe15 15 a rea50na61e 0ne. Further, 6y a55um1n9 that W0f and W09, and c0n5e4uent1y the t0ta1 react10n cr055 5ect10n, vary 5m00th1y w1th ener9y, we can a150 06ta1n 1nterp01ated va1ue5 f0r W0f and W09 at ener91e5 where n0 exper1menta1 data are ava11a61et3. 7he pha5e 5h1ft5 6/1-f0r 1 = 0 t0 3 ca1cu1ated w1th 1ma91nary p0tent1a15t 4 are 5h0wn 6y 5011d d0t5 1n f19. 1. Fr0m th15 f19ure, we n0te the f0110w1n9 1mp0rtant feature5: t 2 F0r th15 c0mpar150n we have further c0n51dered the 5 = 3/2, d+3He channe1. t 3 F0r Ff ar0und 25 MeV, W0f and W09 are a60ut e4ua1 t0 1.4 and 0.8 MeV, re5pect1ve1y. W1th 1ma91naryp0tent1a15 1n the f0rmu1at10n, the 5 matr1x 15n0t un1tary and r1f 15n0 10n9er e4ua1 t0 ~-/9.
V01ume 468, num6er 3 100
80 E~ v
9 60 L
PHY51C5 LE77ER5
.. 1
1
1
804
40 1
20
1
25 Ef (MeV)
1
30
35
F19. 2. Pha5e 5h1ft5609 ca1cu1ated w1th (5011dd0t5) and w1th0ut (5011d 11ne5)1ma91naryp0tent1a15.
(1) 1n c0ntra5t w1th the ca5e 0f n0 1ma91nary p0tent1a15, the pha5e5 ~0f and 8 1f n0 10n9er 5h0w any re50nant 6ehav10ur. (11) At ener91e5 far fr0m the re50nance5, the 1ntr0duct10n 0f 1ma91nary p0tent1a15 ha5 0n1y m1n0r effect5 0n the pha5e 5h1ft5 80f and 8 1f. (111) F0r 1 = 2 and 3, the va1ue5 0f the pha5e 5h1ft5 ca1cu1ated w1th and w1th0ut 1ma91nary p0tent1a15 are very 51m11ar0ver the ent1re ener9y ran9e. Fr0m the5e re5u1t5, we c0ne1ude that the 1 = 0 and 1 re50nance5 at ar0und 25 MeV are 1n fact 5pur10u5, wh11e the 1 = 2 and 3 re50nance5 are n0t. 7he pha5e 5h1ft5 509 ca1cu1ated w1th (5011d d0t5) and w1th0ut (5011d 11ne5) 1ma91nary p0tent1a15 are 5h0wn a5 a funct10n 0 f E f 1n f19. 2. Here a150 0ne f1nd5 that the rap1d-chan91n9 6ehav10ur 1n the ca5e w1th W1 = 0 15 5m00thed 0ut 6y the 1nc1u510n 0f 1ma91nary p0tent1a15.
1 0ct06er 1973
We have a150 065erved 5pur10u5 re50nance5 1n a 51n91e-channe1 ca1cu1at10n 0f 3He + t~ 5catter1n9 [9] when n0 a650rpt10n 15 1nc1uded. 1n th15 ca5e, c0n51dera61e exper1menta1 data ex15t 1n the ener9y re910n ar0und the5e re50nance5 50 that the 5tren9th 0f the 1ma91nary p0tent1a1 can 6e read11y determ1ned. W1th the 1nc1u510n 0f the 1ma91nary p0tent1a1, we f1nd a150 that the5e 5pur10u5 re50nance5 are 5m00thed 1n a way wh1ch 15 51m11art0 that d15cu55ed a60ve f0r the c0up1ed-channe1 ca1cu1at10n 1n the ma55-5 5y5tem. 1n c0nc1u510n, 1t 15 0ur 0p1n10n that the 1ntr0duct10n 0f phen0men01091ca1 1ma91nary p0tent1a15 pr0v1de5 a c0nven1ent way n0t 0n1y t0 acc0unt appr0x1mate1y f0r react10n effect5, 6ut a150 t0 e11m1nate 5pur10u5 re50nance5 fr0m re50nat1n9-9r0up and perhap5 0ther m1cr05c0p1c react10n ca1cu1at10n5.
Reference5 [1] E.W. 5chm1d, 1n Pr0c. 11th 1nt. Un1ver51t~t5w0chenfur Kernphy51k, 5ch1adm1n9,Au5tr1a(5pr1n9er, W1en, 197 2). [2] F.5. Chw1er0th, Y.C. 7an9 and D.R. 7h0mp50n, Nuc1. Phy5. A189 (1972) 1; F.5. Chw1er0th, R.E. 8r0wn, Y.C. 7an9 and D.R. 7h0mp50n, Phy5. Rev., t0 6e pu6115hed. [3] H.C. 8en6hr and K. W11dermuth,Nuc1. Phy5. A128 (1969) 1. [4] 1. Re1ch5te1nand Y.C. 7an9, Nuc1. Phy5. A158 (1970) 529. [5] F.5. Chw1er0th, Ph.D. the515,Un1ver51ty0f M1nne50ta(1973). [6] 7.R. K1n9and R. 5mythe, Nuc1. Phy5. A183 (1972) 657; N.5. Chant, pr1vate c0mmun1cat10n. [7] 6.E. 7h0mp50n, M.8. Ep5te1nand 7. 5awada, Nuc1. Phy5. A142 (1970) 571. [8] F.5. Chw1er0th, Y.C. 7an9 and D.R. 7h0mp50n, t0 6e pu6115hed. [9] LA. K0epke, R.E. 8r0wn, Y.C. 7an9 and D.R. 7h0mp50n, t0 6e pu6115hed.
303
PHY51C5 LE77ER5
1 0ct06er 1973
EFFEC750F 1MA61NARY P07EN71AL50N 5PUR10U5 RE50NANCE51N RE50NA71N6-6R0UP CALCULA710N5* F.5. CHW1ER07H, Y.C. 7AN6 and D.R. 7H0MP50N
5ch0010f Phy51c5,Un1ver51ty0f M1nne50ta,M1nneap0115,M1nne50ta55455, U5A Rece1ved 12 Au9u5t 1973 8y 5tudy1n9the c0up1ed-channe1d + 3He and p + a re50nat1n9-9r0upca1cu1at10n,1t 15f0und that the 1ntr0duct10n 0f phen0men01091ca11ma91naryp0tent1a1515a 51mp1eway t0 e11m1nate5pur10u5re50nance5fr0m re50nat1n9-9r0upand perhap5 0ther m1cr05c0p1creact10n ca1cu1at10n5. 1n nuc1ear react10n the0r1e5 wh1ch 501ve the pr0jec. t10n e4uat10n (54¢1H-E14•) = 0
(1)
1n a truncated funct10n 5pace, 5pur10u5 m0de1 re50nance5 w111appear 1n ener9y re910n5 where 0pen react10n channe15 are n0t c0n51dered [1]. 5uch 5pur10u5 re50nance5 have 6een f0und, f0r examp1e, 1n 0ne- and tw0-channe1 re50nat1n9-9r0up ca1cu1at10n5 0f many 119ht nuc1ear 5y5tem5 [2]. 1n th15 c0mmun1cat10n, we 5ha115h0w 6y a 5pec1f1cexamp1e that even w1th a crude c0n51derat10n 0f the5e 0pen react10n channe15 6y the 1ntr0duct10n 0f phen0men01091ca1 1ma91nary p0tent1a15, the5e 5pur10u5 re50nance5 w111 d15appear fr0m the ca1cu1at10n. 0 n the 0ther hand, there are true re50nance5 1n the ca1cu1at10n wh1ch are 11tt1e affected 6y the pre5ence 0f the 1ma91nary p0tent1a15. 7he5e 1atter re50nance5 are th05e wh1ch are we11 de5cr16ed 6y the c0n51dered truncated funct10n 5pace and have 6een ca11ed p0tent1a1 re50nance5 6y 8en0hr and W11dermuth [3]. 7he 5pec1f1cexamp1e t0 6e c0n51dered here 15the tw0-channe1 (d + 3He and p + a channe15) 5tudy 0f the f1ve-nuc1e0n 5y5tem 1n the 5 = • 5tatet ~w1th the tr1a1 funct10n = A {~6h~0dF(r f) ~ + ¢a 6(r9)~),
(2)
where A 15 an ant15ymmetf12at10n 0perat0r and ~ 15 an appr0pr1ate 5p1n-1505p1nfunct10n f0r 5 = • and 7 = •. 7he funct10n5 ~ de5cr16e the 5pat1a1 6ehav10ur * W0rk 5upp0rted 1npart 6y the U.5. At0m1cEner9y C0mm15510n. j.1 51neea pure1ycentra1 nuc1e0n-nuc1e0np0tent1a115u5ed 1n the ca1cu1at10n,the t0ta15p1nan9u1arm0mentum 15e0n5erved.
0f the 3He, deuter0n and a c1u5ter5; tch1r 5pec1f1c f0rm5 are 91ven 1n ref5. [2] and [4]. 7he funct10n5 F(rf) and 6(r9) de5cr16e the re1at1ve m0t10n5 0f the d + 3He and p + t~ c1u5ter5, re5pect1ve1y, and are determ1ned 6y 501v1n9e4.(1). F1na11y. 1t 5h0u1d 6e ment10ned that, 1n e4.(1), E 15 the t0ta1 ener9y 0f the 5y5tem and H 15 the Ham11t0n1an, 91ven 6y 712 5
5
1=1
(3)
1>j=1
w1th V116e1n9 the nuc1e0n-nuc1e0n p0tent1a1 0f ref. [4] w1th0ut the 5p1n-0r61t c0mp0nent. 5u65t1tut10n 0f e4. (2) 1nt0 e4. (1) 1ead5 t0 the f0110w1n9 c0up1ed 1nte9r0d1fferent1a1 e4uat10n5 f0r the re1at1ve-m0t10n funct10n5:
[~fV2+Ef-VDf(rf)-VCf(rf)]F(rf) f
=
•
,
Kff(rf, rf) F~rf) dtf +
(4a)
,
(rf, r9) 6(r9
[ ~9 V29+E9-- VD9(r9)-- VC9(•9)] 6(r9)
(46)
= fK9f(r9, r•f)F( • ~f)dr~f+fK99r9, )6(r•9) dr•9 , where the 5u65cr1pt5 f and 9 refer t0 the d + 3He and p + ~, channe15, re5pect1ve1y. 7he expre5510n5 f0r the d1rect nuc1ear p0tent1a15 1•D, the d1rect C0u10m6 p0tent1a15 VC, and the kerne1 funct10n5 are very 1en9thy and are 91ven 1n ref. [5]. 7he 4uant1t1e5 Ef and E9 are the re1at1ve k1net1c ener91e5 0f the c1u5ter5 at 1ar9e 5eparat10n 1n the c.m. 5y5tem; they are re1ated 301
v01ume 468, num6er 3
PHY51c5 LE77ER5
6y the e4uat10n
E9 = Ef+ Eth .
(5)
W1th the ch05en f0rm5 f0r ~6, Eth 15 ca1cu1ated t0 6e 20.45 MeV, wh1ch 15 1n fa1r a9reement w1th the va1ue 0f 18.35 MeV determ1ned exper1menta11y. Fr0m the 501ut10n5 0f e45. (4a) and (46), we determ1ne the 5catter1n9 matr1x wh05e d1a90na1 e1ement5 are wr1tten a5 5~f = r/f exp(216/f) , (6)
5199 = r19 exp(21619) w1th 7"/f= 719.7he re5u1t5 f0r 6/f w1th 1 = 0 t0 3 are 5h0wn a5 a funct10n 0 f E f 6y the 5011d curve5 1n f19. 1. Here 0ne 5ee5 that rather 5harp re50nance 5tructure5 are pre5ent at ar0und 25 MeV 1n the 1 = 0 and 1 part1a1 wave5, wh11e 50mewhat 6r0ader re50nance 5tructure appear f0r 1 = 2 and 3.51nce the ener91e5 at wh1ch the5e re50nance5 appear are a60ve the three-60dy 6reakup thre5h01d5, a 4ue5t10n ar15e5 a5 t0 whether the5e ca1cu1ated re50nance5 are 5pur10u5 re50nance5 200
1
r
r
,,°1 100h
100
/
50
1
)
~
)
1
1
1
1
1>
50
1
•
0
83f
32f
50
0 0
0 Ef (MeV)
1
1
113 20
F
30
1
40
1
50
Ef (MeV)
F19. 1. Pha5e 5h1ft561f ca1cu1ated w1th (5011dd0t5) and w1th0ut (5011d 11ne5)1ma91naryp0tent1a15. 302
+ 4 exp(r~)
[1 +ex
,
,7,
R9 = 2.25 f m ,
a f = a 9 = 0.5 fm .(8)
81~ 150
0~L
W1(r1)=--W01{[1+exp(r1-R1~] -1
Rf = 3.5 f m ,
80f
Lk
ar151n9 a5 a c0n5e4uence 0f 0ur m0de1 a55umpt10n 0f 0m1tt1n9 three- and m0re-60dy channe15 1n the ca1cu1at10n, 0r they d0 1n fact 1nd1cate the pre5ence 0f re1at1ve1y 10n9-11ved re50nance 5tructure5 1n the c0mp0und nuc1eu5 5L1. 1n 0rder t0 an5wer th15 4ue5t10n, a ca1cu1at10n 1n wh1ch a11 0pen channe15 are 1nc1uded 5h0u1d 6e perf0rmed. 5uch a ca1cu1at10n 15, 0f c0ur5e, n0t pract1ca1 at the pre5ent m0ment. 7heref0re, a5 a f1r5t 5tep, we 5ha11 51mp1y exam1ne the 6ehav10ur 0f the5e re50nance5 6y a ca1cu1at10n 1n wh1ch the 0m1tted 0pen channe15 are appr0x1mate1y taken 1nt0 acc0unt 6y the 1ntr0duct10n 0f 1ma91nary p0tent1a15. 1ma91nary p0tent1a15 are 1ntr0duced 1nt0 the f0rma115m 6y 51mp1y rep1ac1n9 VDf6y VDf+ 1Wf and VD9 6y VD9+1W9 1n e45. (4a) and (46). F0r the f0rm5 0f Wf and W9, We U5e
w1th 1 den0t1n9 e1ther f 0r 9, and
200
r
1 0ct06er 1973
At ener91e5 where exper1menta1 data [6, 7] ex15t, we have adju5ted W0f and W09 50 that 0ur ca1cu1at10n y1e1d5 the 6e5t a9reement w1th mea5ured d1fferent1a1 e1a5t1c and part1a1 react10n cr055 5ect10n5, a5 we11 a5 t0ta1 react10n cr055-5ect10n va1ue5t 2. 7he a9reement 50 06ta1ned 15, 1n 9enera1, 4u1te 900d [5, 8], wh1ch 15 an 1nd1cat10n that 0ur pr0cedure 0f tak1n9 1nt0 acc0unt the 0pen channe15 15 a rea50na61e 0ne. Further, 6y a55um1n9 that W0f and W09, and c0n5e4uent1y the t0ta1 react10n cr055 5ect10n, vary 5m00th1y w1th ener9y, we can a150 06ta1n 1nterp01ated va1ue5 f0r W0f and W09 at ener91e5 where n0 exper1menta1 data are ava11a61et3. 7he pha5e 5h1ft5 6/1-f0r 1 = 0 t0 3 ca1cu1ated w1th 1ma91nary p0tent1a15t 4 are 5h0wn 6y 5011d d0t5 1n f19. 1. Fr0m th15 f19ure, we n0te the f0110w1n9 1mp0rtant feature5: t 2 F0r th15 c0mpar150n we have further c0n51dered the 5 = 3/2, d+3He channe1. t 3 F0r Ff ar0und 25 MeV, W0f and W09 are a60ut e4ua1 t0 1.4 and 0.8 MeV, re5pect1ve1y. W1th 1ma91naryp0tent1a15 1n the f0rmu1at10n, the 5 matr1x 15n0t un1tary and r1f 15n0 10n9er e4ua1 t0 ~-/9.
V01ume 468, num6er 3 100
80 E~ v
9 60 L
PHY51C5 LE77ER5
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804
40 1
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25 Ef (MeV)
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F19. 2. Pha5e 5h1ft5609 ca1cu1ated w1th (5011dd0t5) and w1th0ut (5011d 11ne5)1ma91naryp0tent1a15.
(1) 1n c0ntra5t w1th the ca5e 0f n0 1ma91nary p0tent1a15, the pha5e5 ~0f and 8 1f n0 10n9er 5h0w any re50nant 6ehav10ur. (11) At ener91e5 far fr0m the re50nance5, the 1ntr0duct10n 0f 1ma91nary p0tent1a15 ha5 0n1y m1n0r effect5 0n the pha5e 5h1ft5 80f and 8 1f. (111) F0r 1 = 2 and 3, the va1ue5 0f the pha5e 5h1ft5 ca1cu1ated w1th and w1th0ut 1ma91nary p0tent1a15 are very 51m11ar0ver the ent1re ener9y ran9e. Fr0m the5e re5u1t5, we c0ne1ude that the 1 = 0 and 1 re50nance5 at ar0und 25 MeV are 1n fact 5pur10u5, wh11e the 1 = 2 and 3 re50nance5 are n0t. 7he pha5e 5h1ft5 509 ca1cu1ated w1th (5011d d0t5) and w1th0ut (5011d 11ne5) 1ma91nary p0tent1a15 are 5h0wn a5 a funct10n 0 f E f 1n f19. 2. Here a150 0ne f1nd5 that the rap1d-chan91n9 6ehav10ur 1n the ca5e w1th W1 = 0 15 5m00thed 0ut 6y the 1nc1u510n 0f 1ma91nary p0tent1a15.
1 0ct06er 1973
We have a150 065erved 5pur10u5 re50nance5 1n a 51n91e-channe1 ca1cu1at10n 0f 3He + t~ 5catter1n9 [9] when n0 a650rpt10n 15 1nc1uded. 1n th15 ca5e, c0n51dera61e exper1menta1 data ex15t 1n the ener9y re910n ar0und the5e re50nance5 50 that the 5tren9th 0f the 1ma91nary p0tent1a1 can 6e read11y determ1ned. W1th the 1nc1u510n 0f the 1ma91nary p0tent1a1, we f1nd a150 that the5e 5pur10u5 re50nance5 are 5m00thed 1n a way wh1ch 15 51m11art0 that d15cu55ed a60ve f0r the c0up1ed-channe1 ca1cu1at10n 1n the ma55-5 5y5tem. 1n c0nc1u510n, 1t 15 0ur 0p1n10n that the 1ntr0duct10n 0f phen0men01091ca1 1ma91nary p0tent1a15 pr0v1de5 a c0nven1ent way n0t 0n1y t0 acc0unt appr0x1mate1y f0r react10n effect5, 6ut a150 t0 e11m1nate 5pur10u5 re50nance5 fr0m re50nat1n9-9r0up and perhap5 0ther m1cr05c0p1c react10n ca1cu1at10n5.
Reference5 [1] E.W. 5chm1d, 1n Pr0c. 11th 1nt. Un1ver51t~t5w0chenfur Kernphy51k, 5ch1adm1n9,Au5tr1a(5pr1n9er, W1en, 197 2). [2] F.5. Chw1er0th, Y.C. 7an9 and D.R. 7h0mp50n, Nuc1. Phy5. A189 (1972) 1; F.5. Chw1er0th, R.E. 8r0wn, Y.C. 7an9 and D.R. 7h0mp50n, Phy5. Rev., t0 6e pu6115hed. [3] H.C. 8en6hr and K. W11dermuth,Nuc1. Phy5. A128 (1969) 1. [4] 1. Re1ch5te1nand Y.C. 7an9, Nuc1. Phy5. A158 (1970) 529. [5] F.5. Chw1er0th, Ph.D. the515,Un1ver51ty0f M1nne50ta(1973). [6] 7.R. K1n9and R. 5mythe, Nuc1. Phy5. A183 (1972) 657; N.5. Chant, pr1vate c0mmun1cat10n. [7] 6.E. 7h0mp50n, M.8. Ep5te1nand 7. 5awada, Nuc1. Phy5. A142 (1970) 571. [8] F.5. Chw1er0th, Y.C. 7an9 and D.R. 7h0mp50n, t0 6e pu6115hed. [9] LA. K0epke, R.E. 8r0wn, Y.C. 7an9 and D.R. 7h0mp50n, t0 6e pu6115hed.
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