- Email: [email protected]
Do the medium effects exist for the (3He, t) reaction at intermediate energies?
V01ume 226, num6er 3,4
PHY51C5 LE77ER5 8
10 Au9u5t 1989
D 0 7 H E M E D 1 U M E F F E C 7 5 EX157 F 0 R 7 H E (3He, t) R E A C 7 1 0 N A7 1 N 7 E R M E D 1 A 7 E ENER61E5• v.F. DM17R1EV 1n5t1tute 4f Nuc1ear Phy51c5, 630 090 N0v05161r5k-90, u55R
Rece1ved 16 March 1989; rev15edmanu5cr1pt rece1ved 30 May 1989
D15t0rt10n effect5 f0r 1ne1a5t1c5catter1n9 0f heavy 10n5fr0m nuc1e1at 1ntermed1ate ener91e5 are d15cu55ed1n term5 0f 61au6er mu1t1p1e5catter1n9 the0ry. Effect1ve1mpact parameter5 are 1ntr0duced f0r 0ne-5tep 1ne1a5t1cc0111510n5.7he penetrat10n 0fa pr0ject11e 1n51dethe tar9et 159reater than f0r e1a5t1c5catter1n9 creat1n9 med1um effect5 5tr0n9 en0u9h t0 exp1a1nthe p051t10n0f the A-peak 1n the (3He, t) react10n.
1n recent year5 much 1ntere5t ha5 6een 6r0u9ht t0 5119ht1y 1ne1a5t1c nuc1eu5-nuc1eu5 c0111510n5. A c1ear examp1e 0f th15 15 the (3He, t) react10n where var10u5 react10n channe15 have 6een 065erved. Name1y, the exc1tat10n 0f 6am0w-7e11er tran51t10n5, nuc1e0n kn0ck0ut and A-pr0duct10n have c0mpara61e cr055 5ect10n5 [ 1-4 ]. 7he parameter5 0 f t h e A-peak 065erved 1n the react10n 0n nuc1e1 d1ffer fr0m th05e 065erved 1n the react10n 0n a pr0t0n. 7 h e peak 15 5h1fted d0wn 6y a60ut 40 MeV and 15 6r0adened up t0 150 MeV. 7 h e p055161e exp1anat10n 0f th15 5h1ft 15 u5ua11y 6a5ed 0n med1um effect5 [5,6 ]. 7he5e effect5 6ec0me 519n1f1cant at den51t1e5 0f a60ut 0.5n0, where n0 15 the avera9e nuc1ear den51ty [ 7 ]. 0 n the 0ther hand, ca1cu1at10n5 0 f t h e cr055 5ect10n 6a5ed 0n the nuc1ear re5p0n5e funct10n and the c0nv01ut10n m0de1 f0r the d15t0rt10n fact0r 91ve extreme per1phera1 re5u1t5 0f the pr0ce55 and can exp1a1n ne1ther the 065erved 5h1ft n0r the ma9n1tude 0f the cr055 5ect10n [ 8 ]. 7he 0r191n 0fth15 fa11ure 11e5 1n u51n9 an 1nappr0pr1ate d15t0rt10n fact0r. 1ndeed, the d15t0rt10n fact0r exp[-•2(r)1=exp
-~aNN f d2r ~p A ( r - r
)pm,(
)
(1)
appear5 1n the de5cr1pt10n 0f e1a5t1c nuc1eu5-nuc1eu5 c0111510n5 [9 ]. 1t depend5 0n the 1mpact parameter wh1ch f0r e1a5t1c 5catter1n915 the d15tance 6etween the center5 0fma55 0 f t h e c0111d1n9 nuc1e1.7heref0re, 1t te115 n0th1n9 a60ut the p051t10n 0f the nuc1e0n 1n a pr0ject11e under901n9 char9e exchan9e. 7he ma9n1tude 0f the 1ne1a5t1c cr055 5ect10n depend5, h0wever, 0n the tar9et den51ty at the p051t10n 0fth15 very nuc1e0n and n0t 0n the p051t10n 0f the center 0f ma55 0f a pr0ject11e. 51nce the tar9et den51ty var1e5 rap1d1y at the 5urface the d1fference 0f the p051t10n5 mu5t 6e 1mp0rtant f0r the cr055 5ect10n. 1n th15 1etter the d15t0rt10n fact0r f0r the ( 3He, t ) react10n w1th exc1tat10n 0f 1p - 1h 0r A-h 5tate5 1n the tar9et nuc1eu5 15 06ta1ned. 70 de5cr16e the d15t0rt10n, the 61au6er appr0ach f0r acc0unt1n9 re5catter1n9 w1116e u5ed. Let 7 , . ~ ( 5 - r ) 6e the 7-matr1x amp11tude f0r the char9e exchan9e react10n 1n nuc1e0n-nuc1e0n c0111510n5. 1t can 6e e1ther e1a5t1c char9e exchan9e 0r 1ne1a5t1c w1th A-pr0duct10n. 70 06ta1n the nuc1eu5-nuc1eu5 amp11tude we 5ha11 f0110w5 the 11ne5 0fref. [9 ] f0r e1a5t1c nuc1eu5-nuc1eu5 c0111510n5. F1r5t, we 1ntr0duce the pr0f11e funct10n y~/(5~-rj) f0r e1a5t1c nuc1e0n-nuc1e0n 5catter1n9 6etween nuc1e0n5 0~ andj: y ~ j ( 5 ~ - r j ) = ~ 1 f d2c1exp[14(5~-r J)1f~J(4) ~ .
(2)
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. ( N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n )
219
V 0 1 u m e 2 2 6 , n u m 6 e r 3,4
PHY51C5 LE77ER5
8
10 A u 9 u 5 t 1989
1n the f0110w1n9 the 51mp1e5t ca5e w1116e a55umed f0r the pr0f11e funct10n 7~j(5~-rj), name1y, 1t w1116e taken 5p1n and 1505p1n 1ndependent. M0re0ver, the 5ame y(5-r) w1116e u5ed f0r A-nuc1e0n e1a5t1c 5catter1n9 a5 we11. W1th the5e appr0x1mat10n5 the amp11tude 0f the char9e-exchan9e react10n 6etween nuc1e1 8 and A can 6e wr1tten a5 •1
8
8
A
7.,= y d3Rexp(14R) ~ ~. 7c.~.(R+5~-r,) 131-1 [1-y(R+5e-r )1. 1=1 a•=1
(3)
/3=1 /,•=1
(f1¢-0~)A(k¢-/)
7he 5tructure 0f the amp11tude (3) 15 06v10u5. Every re5catter1n9 91ve5 a fact0r 1 - 7(R + 5 - r ) and 0ne mu5t exc1ude re5catter1n9 6etween 1ne1a5t1ca11y 5cattered nuc1e0n5 t0 av01d d0u61e c0unt1n9. 7he next 5tep 15 the ca1cu1at10n 0f the matr1x e1ement5 0f the amp11tude (3) 6etween nuc1ear 5tate5. [t w1116e d0ne f0r (3He, t) react10n5 0n car60n. At th15 5tep 1t 151mp0rtant what k1nd 0ff1na15tate515 exc1ted 1n the tar9et nuc1eu5. We 5ha11 re5tr1ct 0ur5e1ve5 t0 1p-1h 0r A-h 5tate5 that can 6e exc1ted 6y 0ne 1ne1a5t1c 5tep. 1n th15 ca5e the matr1x e1ement 6etween tar9et 5tate5 can 6e fact0r12ed: f
,-1
3
d3Rexp(14R)(f] ~ 7~.~.(R+~-r,)[0)(0[
7,,,•f=
/=1
,1
]-11-[ [1-7(R+5/~-r~)]]0>
(4)
f1=1 k = 1 (f1# 1 ) A ( k ~ D
7h15 appr0x1mat10n 15 expected t0 6e 900d 1n the re910n5 • t h e nuc1e0n 0r the A 4ua51free peak5.1n 0ther re910n5 0fthe tr1t0n 5pectrum mu1t15tep exc1tat10n5 5h0u1d 6e taken 1nt0 acc0unt. 70 ca1cu1ate the d1a90na1 matr1x e1ement 0f re5catter1n95 1n (4) the u5ua1 appr0x1mat10n f0r the tar9et wave funct10n a5 a pr0duct 0f 51n91e-part1c1e den51t1e5 w1116e u5ed. F0r the d1a90na1 matr1x e1ement 0ne 06ta1n5 3
<01
1
H1-1 [1-~(R+5j,-r~)110>
f1=~ k = 1
(f1# ~ ) A ( k ~ ] )
(•f
= 1-~
d2rf1A(r).[y(R+52--r)+7(R+53--r)-~(R+52-r)7(R+53-r)]
×
d2r/~4(r)
1-~
)
~ ~(R+5~-r)-- 2 )~(R+5~--r)7(R+5/~-r) ~=1
0e >13
.4 - 1
+,(8+81--r),(R+52--r)y(R+53--r))]
,
(5)
where/~A (r) = f °2~ d2 PA( r, 2) 15 the th1Ckne55 funct10n. 7he pr0f11e fUnCt10n 7(r) 15 narr0W-peaked C0mpared t0 the 512e5 0f60th the tar9et nuC1eu5 A and 3He 0r t. 1t Can 6e U5ed t0 06ta1n the 0pt1Ca1 11m1t that 91ve5 a C1ear p1CtUre 0f the pr0Ce55. F1r5t, 1nte9rat10n 0ver r 1n ( 5 ) Can 6e perf0rmed exp11C1t1y:
f 7(R+5--r)f1A(r) d2r~7f1.a(R+5) ,
(6)
where 7= YY(r) d2r. 5eC0nd1y, the term5 1n (5) c0nta1n1n9 the pr0duct 0f y•5 w1th d1fferent 5~ can 6e 0m1tted 6ecau5e 1n the 1nte9rat10n 0ver a115 w1th wave funct10n5 0f 3He and t the5e term5 w111c0nta1n the 5ma11 fact0r (r0/R H0)2, where r015 the w1dth 0f 7 (r). 0m1tt1n9 the5e term5 and tak1n9 the 11m1tA >> 1 0ne 06ta1n5
(0]
H []
[1-7(R+5/~-r~)]
]0> ~ e x p
-~2--~.~(R+~)-7f1~(R+52)-~7~.,(R+53)
f1=1 /,=1 (f1# 1 )A ( k # j )
N0w the matr1x e1ement 0fthe react10n amp11tude 15 220
•
(7)
V01ume226, num6er 3,4
7~=
f d3Rd35,
PHY51C5 LE77ER5 8
10 Au9u5t 1989
d35aexp(14R)(f1 ~ 7~.~(R+5t-r1)10) j=1
×exp -j-~-6.~(R+5,
) -~.,(R+52)
- f~..~(R+53)
)
~,~(5~, 52, 5 3 ) ~ ( 5 , , 52, 53),
(8)
where ~(51, 52, 53) 15 the 1nterna1 wave funct10n 0f3He 0r t depend1n9 0n 1nterna1 c00rd1nate5 5~ (53 = - 5 j -52 ). 7he amp11tude (8) ha5 a c1ear 1nterpretat10n. 7he part1c1e5 emer9e fr0m 60und 3He w1th pr06a6111t1e5 91ven 6y the wave funct10n ~H~(5~). Ne91ect1n9 the exc1tat10n ener91e5 0f 3He and t 1n 1ntermed1ate 5tate5, every part1c1e re5catter5 1ndependent1y w1th 1t5 0wn fact0r exp [ - 7f1~4(R + 5,) ]. 7he part1c1e under901n9 char9e exchan9e ha5 0ne re5catter1n9 1e55 than the 0ther5. 7he pr06a6111ty 0f c011ect1n9 them 1nt0 60und tr1t1um 15 de5cr16ed 6y the wave funct10n ~,~(5,). 7he amp11tude 0f char9e exchan9e depend5 0n the c00rd1nate 5~, thu5 c0up11n9 the 1ne1a5t1c 5tep w1th e1a5t1c re5catter1n9. 1n 0rder t0 5eparate them, 1et u5 1ntr0duce the effect1ve 1mpact parameter R0ff=R+5~. 7h15 15, 1n fact, the d15tance 6etween the pr0t0n 1n 3He under901n9 char9e exchan9e and the tar9et. 7he amp11tude 0f the react10n can 6e repre5ented a5
7He•.,=
d3Refrexp(14R~f~.)(f[ ~ 7~.~.(R~--rj)10)exp[-~2m(R~fr, 4)],
(9)
1~1
where the 1ne1a5t1c e1k0na1 fact0r 15 1ntr0duced:
exP [ - ~21n( R~-, 4 ) ] =exP( - ~ ~ ×exp[
p.,( R~,.,-)) ~ d 35, d 352 exP ( -145, ) V/*~( 51, 52, ,3 )
-~p.~(R~f~.+5~ -5t ) -~.1(R~f~.+53
- 5 t ) ] ~u~ (5t,
52, 53)
•
(10)
F0r 9au551an wave funct10n5 0f 3He and t the 1nte9rat10n 0ver 10n91tud1na1 c00rd1nate5 can 6e fact0r12ed and perf0rmed exp11c1t1y 91v1n9 the 10n91tud1na1 f0rm fact0r 0f the (3He, t) vertex. 7he rema1n1n9 tran5ver5a1 e1k0na1 fact0r f0r 0 ° react10n5 0n car60n 15 5h0wn 1n f19. 1. 1t 15 06v10u51y 1e55 5teep at the nuc1ear 5urface a5 c0mpared t0 the e1a5t1c e1k0na1 fact0r 1n the c0nv01ut10n m0de1 and ha5 a 9reater pr06a6111ty 0f penetrat10n 1n51de a nuc1eu5 t0 h19her den51ty. 7he 5m00th1n9 ar05e fr0m add1t10na1 1nte9rat10n 0ver 5j 1n (10). 7h15 1nte9rat10n c0rre5p0nd5 t0 avera91n9 0ver d1fferent p051t10n5 0f the 3He center 0f ma55 w1th the effect1ve 1mpact parameter 6e1n9 f1xed. 7he e1k0na1 fact0r (10) ha5 6een u5ed t0 ca1cu1ate the a6501ute cr055 5ect10n 1n the A-re910n f0r a 0 ° ~2C(3He, t) react10n at 2 6 e V k1net1c ener9y 0f3He. At th15 ener9y the m0mentum tran5fer 1n the A-re910n 15 rather 1ar9e c0mpared t0 the 512e 0f t~-C. 7hu5, a 10ca1 den51ty appr0x1mat10n f0r the nuc1ear re5p0n5e ha5 6een u5ed [ 7 ]. F0r the nuc1ear re5p0n5e funct10n at a 91ven den51ty the m0de1 deve10ped 1n ref. [ 6 ] ha5 6een u5ed. 7he re5u1t5 0f the ca1cu1at10n are 5h0wn 1n f19. 2 6y the 5011d 11ne. F0r c0mpar150n, tw0 0ther re5u1t5 are pre5ented 1n the 5ame f19ure. 7he da5hed 11ne repre5ent5 the tr1t1um 5pectrum ca1cu1ated u51n9 the e1a5t1c e1k0na1 fact0r 1n the c0nv01ut10n m0de1 (c0rre5p0nd1n9 t0 the da5hed 11ne 1n f19. 1 ). 7he d1fference 15 c1ear, the cr055 5ect10n 15 pr0p0rt10na1 t0 p.~(r) e x p [ - 2 , , ( r , 4)1 .
(11)
7he e1a5t1c e1k0na1 fact0r 91ve5 the extreme per1phera11ty 0f the react10n. 7he tar9et den51ty ••5een•• 6y a pr0ject11e 15 5ma11 re5u1t1n9 1n a 5ma11er cr055 5ect10n and 5ma11er 5h1ft 0fthe A-peak. 7he d0tted 11ne 15 the 4ua51free ca1cu1at10n w1th0ut the effect5 0f p10n pr0Pa9at10n 1n51de the tar9et. 1n th15 ca5e the peak p051t10n rema1n5 unchan9ed and c01nc1de5 w1th the 0ne 1n the react10n p(3He, t)A ++ 1t 15 1n5truct1ve t0 n0te that the peak p051t10n determ1ne5 t0 a 1ar9e extent the ma9n1tude 0fthe A-peak. 7he 10w-ener9y w1n9 0fthe tr1t1um 5pectrum 1n f19.2 15 c0mm0n f0r a11 three curve5. 1t 15 determ1ned 6y the 510pe 0f 221
V01ume 226, num6er 3,4
PHY51C5 LE77ER5 8
10 Au9u5t 1989
1.0 /
1
f 1
//
0.8
0.6
/ / / /~
0.4
1111 1600
f~00
1800
1900
[M0v]
0,2 1
r/Rc
F19. 1. E1k0na1 fact0r5 f0r ~2C. Fu1111ne: 1ne1a5t1c e1k0na1 fact0r f0r the (3He, t) react10n. Da5hed 11ne: the e1k0na1 fact0r f0r e1a5t1c 5catter1n9 1n the c0nv01ut10n m0de1.
F19. 2. 7he cr055 5ect10n d20/dE, d~2 1n m 6 / 5 r / M e V f0r the L~C(3He, t) react10n. 7he 11ne5 are ca1cu1at10n5 6a5ed 0n ref5. [6,7] f0r the nuc1ear re5p0n5e funct10n. 5011d 11ne: the ca1cu1at10n w1th the 1ne1a5t1c e1k0na1 fact0r. Da5hed 11ne: the ca1cu1at10n w1th the e1a5t1c e1k0na1 fact0r. D0tted 11ne: 4ua51free pr0duct10n w1th0ut med1um effect5. 7he 5h0rt ran9e 1nteract10n c0n5tant5 are 92 =0.5, 9 ~ = -0.2.
the (3He, t) vertex f0rm fact0r. 7heref0re, 1fthe p051t10n 0fthe A-peak 1n the nuc1ear re5p0n5e funct10n 15 h19her than - 3 0 0 MeV 0f the exc1tat10n ener9y, a c0n51dera61e part 0f 1t w111 6e cut 1n the tr1t1um 5pectrum 6y the (3He, t) f0rm fact0r at 2 6 e V k1net1c ener9y 0f a 3He. 7he 5011d 11ne 1n f19. 2 5t111 m155e5 50me part 0f the 5h1ft 0f the A-peak. Neverthe1e55, th15 51tuat10n 15 5at15fact0ry due t0 the appr0x1mat10n5 u5ed 1n the ca1cu1at10n5 pre5ented. F1r5t 0f a11, 0n1y 1A-1 h and 1p-1 h f1na1 5tate5 were taken 1nt0 acc0unt - th1515 n0t true f0r the ta11 0fthe 4ua51e1a5t1c peak wh1ch 151mp0rtant t0 repr0duce the r19ht w1n9 0fthe A-peak; 5ec0nd1y, the react10n take5 p1ace at the nuc1ear 5urface where the tar9et den51ty var1e5 rap1d1y. 1n the 10ca1 den51ty appr0x1mat10n u5ed 1n the ca1cu1at10n pre5ented the den51ty 9rad1ent5 were n0t acc0unted f0r. 1n 5ummary, 1t ha5 6een 5h0wn that the e1k0na1 fact0r f0r 1ne1a5t1c 0ne-5tep react10n5 1n nuc1eu5-nuc1eu5 c0111510n5 d1ffer5 fr0m the 0ne f0r e1a5t1c nuc1eu5-nuc1eu5 5catter1n9. 1t 91ve5 6etter penetrat10n 0f the pr0ject11e 1n51de the tar9et thu5 revea11n9 the med1um effect5 re5p0n5161e f0r the A-peak p051t10n 1n the (3He, t) react10n. 7he auth0r ackn0w1ed9e5 the d15cu5510n5 0fthe 5u6ject w1th V.L. K0r0tk1ch and L.C. Max1m0n
Reference5 [ 1 ] 1.8er94u15t et a1., Nuc1. Phy5. A 469 ( 1987 ) 648. [2] C. 6aarde, Nuc1. Phy5. A 478 (1988) 475c. [3] V.6. A61eev et a1., 50v. Phy5. JE7P Lett. 40 (1984) 763. [4] P. C0ntard0 et a1., Phy5. Lett. 8 198 (1986) 331. [5] 6. Chanfray and M. Er1c50n, Phy5. Len. 8 141 (1984) 163. [6 ] V.F. Dm1tr1ev and 7. 5u2uk1, Nuc1. Phy5. A 348 ( 1985 ) 697. [7] V.F. Dm1tr1ev, Yad. F12.46 (1987) 770; N0v05161r5k prepr1nt 1NP 118-86 (1986). [ 8 ] H. E56en5en and 7.H.5. Lee, Phy5. Rev. C 32 ( 1985 ) 1966. [9] W. C2y2 and L.C. Max1m0n, Ann. Phy5.52 (1969) 59.
222
PHY51C5 LE77ER5 8
10 Au9u5t 1989
D 0 7 H E M E D 1 U M E F F E C 7 5 EX157 F 0 R 7 H E (3He, t) R E A C 7 1 0 N A7 1 N 7 E R M E D 1 A 7 E ENER61E5• v.F. DM17R1EV 1n5t1tute 4f Nuc1ear Phy51c5, 630 090 N0v05161r5k-90, u55R
Rece1ved 16 March 1989; rev15edmanu5cr1pt rece1ved 30 May 1989
D15t0rt10n effect5 f0r 1ne1a5t1c5catter1n9 0f heavy 10n5fr0m nuc1e1at 1ntermed1ate ener91e5 are d15cu55ed1n term5 0f 61au6er mu1t1p1e5catter1n9 the0ry. Effect1ve1mpact parameter5 are 1ntr0duced f0r 0ne-5tep 1ne1a5t1cc0111510n5.7he penetrat10n 0fa pr0ject11e 1n51dethe tar9et 159reater than f0r e1a5t1c5catter1n9 creat1n9 med1um effect5 5tr0n9 en0u9h t0 exp1a1nthe p051t10n0f the A-peak 1n the (3He, t) react10n.
1n recent year5 much 1ntere5t ha5 6een 6r0u9ht t0 5119ht1y 1ne1a5t1c nuc1eu5-nuc1eu5 c0111510n5. A c1ear examp1e 0f th15 15 the (3He, t) react10n where var10u5 react10n channe15 have 6een 065erved. Name1y, the exc1tat10n 0f 6am0w-7e11er tran51t10n5, nuc1e0n kn0ck0ut and A-pr0duct10n have c0mpara61e cr055 5ect10n5 [ 1-4 ]. 7he parameter5 0 f t h e A-peak 065erved 1n the react10n 0n nuc1e1 d1ffer fr0m th05e 065erved 1n the react10n 0n a pr0t0n. 7 h e peak 15 5h1fted d0wn 6y a60ut 40 MeV and 15 6r0adened up t0 150 MeV. 7 h e p055161e exp1anat10n 0f th15 5h1ft 15 u5ua11y 6a5ed 0n med1um effect5 [5,6 ]. 7he5e effect5 6ec0me 519n1f1cant at den51t1e5 0f a60ut 0.5n0, where n0 15 the avera9e nuc1ear den51ty [ 7 ]. 0 n the 0ther hand, ca1cu1at10n5 0 f t h e cr055 5ect10n 6a5ed 0n the nuc1ear re5p0n5e funct10n and the c0nv01ut10n m0de1 f0r the d15t0rt10n fact0r 91ve extreme per1phera1 re5u1t5 0f the pr0ce55 and can exp1a1n ne1ther the 065erved 5h1ft n0r the ma9n1tude 0f the cr055 5ect10n [ 8 ]. 7he 0r191n 0fth15 fa11ure 11e5 1n u51n9 an 1nappr0pr1ate d15t0rt10n fact0r. 1ndeed, the d15t0rt10n fact0r exp[-•2(r)1=exp
-~aNN f d2r ~p A ( r - r
)pm,(
)
(1)
appear5 1n the de5cr1pt10n 0f e1a5t1c nuc1eu5-nuc1eu5 c0111510n5 [9 ]. 1t depend5 0n the 1mpact parameter wh1ch f0r e1a5t1c 5catter1n915 the d15tance 6etween the center5 0fma55 0 f t h e c0111d1n9 nuc1e1.7heref0re, 1t te115 n0th1n9 a60ut the p051t10n 0f the nuc1e0n 1n a pr0ject11e under901n9 char9e exchan9e. 7he ma9n1tude 0f the 1ne1a5t1c cr055 5ect10n depend5, h0wever, 0n the tar9et den51ty at the p051t10n 0fth15 very nuc1e0n and n0t 0n the p051t10n 0f the center 0f ma55 0f a pr0ject11e. 51nce the tar9et den51ty var1e5 rap1d1y at the 5urface the d1fference 0f the p051t10n5 mu5t 6e 1mp0rtant f0r the cr055 5ect10n. 1n th15 1etter the d15t0rt10n fact0r f0r the ( 3He, t ) react10n w1th exc1tat10n 0f 1p - 1h 0r A-h 5tate5 1n the tar9et nuc1eu5 15 06ta1ned. 70 de5cr16e the d15t0rt10n, the 61au6er appr0ach f0r acc0unt1n9 re5catter1n9 w1116e u5ed. Let 7 , . ~ ( 5 - r ) 6e the 7-matr1x amp11tude f0r the char9e exchan9e react10n 1n nuc1e0n-nuc1e0n c0111510n5. 1t can 6e e1ther e1a5t1c char9e exchan9e 0r 1ne1a5t1c w1th A-pr0duct10n. 70 06ta1n the nuc1eu5-nuc1eu5 amp11tude we 5ha11 f0110w5 the 11ne5 0fref. [9 ] f0r e1a5t1c nuc1eu5-nuc1eu5 c0111510n5. F1r5t, we 1ntr0duce the pr0f11e funct10n y~/(5~-rj) f0r e1a5t1c nuc1e0n-nuc1e0n 5catter1n9 6etween nuc1e0n5 0~ andj: y ~ j ( 5 ~ - r j ) = ~ 1 f d2c1exp[14(5~-r J)1f~J(4) ~ .
(2)
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. ( N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n )
219
V 0 1 u m e 2 2 6 , n u m 6 e r 3,4
PHY51C5 LE77ER5
8
10 A u 9 u 5 t 1989
1n the f0110w1n9 the 51mp1e5t ca5e w1116e a55umed f0r the pr0f11e funct10n 7~j(5~-rj), name1y, 1t w1116e taken 5p1n and 1505p1n 1ndependent. M0re0ver, the 5ame y(5-r) w1116e u5ed f0r A-nuc1e0n e1a5t1c 5catter1n9 a5 we11. W1th the5e appr0x1mat10n5 the amp11tude 0f the char9e-exchan9e react10n 6etween nuc1e1 8 and A can 6e wr1tten a5 •1
8
8
A
7.,= y d3Rexp(14R) ~ ~. 7c.~.(R+5~-r,) 131-1 [1-y(R+5e-r )1. 1=1 a•=1
(3)
/3=1 /,•=1
(f1¢-0~)A(k¢-/)
7he 5tructure 0f the amp11tude (3) 15 06v10u5. Every re5catter1n9 91ve5 a fact0r 1 - 7(R + 5 - r ) and 0ne mu5t exc1ude re5catter1n9 6etween 1ne1a5t1ca11y 5cattered nuc1e0n5 t0 av01d d0u61e c0unt1n9. 7he next 5tep 15 the ca1cu1at10n 0f the matr1x e1ement5 0f the amp11tude (3) 6etween nuc1ear 5tate5. [t w1116e d0ne f0r (3He, t) react10n5 0n car60n. At th15 5tep 1t 151mp0rtant what k1nd 0ff1na15tate515 exc1ted 1n the tar9et nuc1eu5. We 5ha11 re5tr1ct 0ur5e1ve5 t0 1p-1h 0r A-h 5tate5 that can 6e exc1ted 6y 0ne 1ne1a5t1c 5tep. 1n th15 ca5e the matr1x e1ement 6etween tar9et 5tate5 can 6e fact0r12ed: f
,-1
3
d3Rexp(14R)(f] ~ 7~.~.(R+~-r,)[0)(0[
7,,,•f=
/=1
,1
]-11-[ [1-7(R+5/~-r~)]]0>
(4)
f1=1 k = 1 (f1# 1 ) A ( k ~ D
7h15 appr0x1mat10n 15 expected t0 6e 900d 1n the re910n5 • t h e nuc1e0n 0r the A 4ua51free peak5.1n 0ther re910n5 0fthe tr1t0n 5pectrum mu1t15tep exc1tat10n5 5h0u1d 6e taken 1nt0 acc0unt. 70 ca1cu1ate the d1a90na1 matr1x e1ement 0f re5catter1n95 1n (4) the u5ua1 appr0x1mat10n f0r the tar9et wave funct10n a5 a pr0duct 0f 51n91e-part1c1e den51t1e5 w1116e u5ed. F0r the d1a90na1 matr1x e1ement 0ne 06ta1n5 3
<01
1
H1-1 [1-~(R+5j,-r~)110>
f1=~ k = 1
(f1# ~ ) A ( k ~ ] )
(•f
= 1-~
d2rf1A(r).[y(R+52--r)+7(R+53--r)-~(R+52-r)7(R+53-r)]
×
d2r/~4(r)
1-~
)
~ ~(R+5~-r)-- 2 )~(R+5~--r)7(R+5/~-r) ~=1
0e >13
.4 - 1
+,(8+81--r),(R+52--r)y(R+53--r))]
,
(5)
where/~A (r) = f °2~ d2 PA( r, 2) 15 the th1Ckne55 funct10n. 7he pr0f11e fUnCt10n 7(r) 15 narr0W-peaked C0mpared t0 the 512e5 0f60th the tar9et nuC1eu5 A and 3He 0r t. 1t Can 6e U5ed t0 06ta1n the 0pt1Ca1 11m1t that 91ve5 a C1ear p1CtUre 0f the pr0Ce55. F1r5t, 1nte9rat10n 0ver r 1n ( 5 ) Can 6e perf0rmed exp11C1t1y:
f 7(R+5--r)f1A(r) d2r~7f1.a(R+5) ,
(6)
where 7= YY(r) d2r. 5eC0nd1y, the term5 1n (5) c0nta1n1n9 the pr0duct 0f y•5 w1th d1fferent 5~ can 6e 0m1tted 6ecau5e 1n the 1nte9rat10n 0ver a115 w1th wave funct10n5 0f 3He and t the5e term5 w111c0nta1n the 5ma11 fact0r (r0/R H0)2, where r015 the w1dth 0f 7 (r). 0m1tt1n9 the5e term5 and tak1n9 the 11m1tA >> 1 0ne 06ta1n5
(0]
H []
[1-7(R+5/~-r~)]
]0> ~ e x p
-~2--~.~(R+~)-7f1~(R+52)-~7~.,(R+53)
f1=1 /,=1 (f1# 1 )A ( k # j )
N0w the matr1x e1ement 0fthe react10n amp11tude 15 220
•
(7)
V01ume226, num6er 3,4
7~=
f d3Rd35,
PHY51C5 LE77ER5 8
10 Au9u5t 1989
d35aexp(14R)(f1 ~ 7~.~(R+5t-r1)10) j=1
×exp -j-~-6.~(R+5,
) -~.,(R+52)
- f~..~(R+53)
)
~,~(5~, 52, 5 3 ) ~ ( 5 , , 52, 53),
(8)
where ~(51, 52, 53) 15 the 1nterna1 wave funct10n 0f3He 0r t depend1n9 0n 1nterna1 c00rd1nate5 5~ (53 = - 5 j -52 ). 7he amp11tude (8) ha5 a c1ear 1nterpretat10n. 7he part1c1e5 emer9e fr0m 60und 3He w1th pr06a6111t1e5 91ven 6y the wave funct10n ~H~(5~). Ne91ect1n9 the exc1tat10n ener91e5 0f 3He and t 1n 1ntermed1ate 5tate5, every part1c1e re5catter5 1ndependent1y w1th 1t5 0wn fact0r exp [ - 7f1~4(R + 5,) ]. 7he part1c1e under901n9 char9e exchan9e ha5 0ne re5catter1n9 1e55 than the 0ther5. 7he pr06a6111ty 0f c011ect1n9 them 1nt0 60und tr1t1um 15 de5cr16ed 6y the wave funct10n ~,~(5,). 7he amp11tude 0f char9e exchan9e depend5 0n the c00rd1nate 5~, thu5 c0up11n9 the 1ne1a5t1c 5tep w1th e1a5t1c re5catter1n9. 1n 0rder t0 5eparate them, 1et u5 1ntr0duce the effect1ve 1mpact parameter R0ff=R+5~. 7h15 15, 1n fact, the d15tance 6etween the pr0t0n 1n 3He under901n9 char9e exchan9e and the tar9et. 7he amp11tude 0f the react10n can 6e repre5ented a5
7He•.,=
d3Refrexp(14R~f~.)(f[ ~ 7~.~.(R~--rj)10)exp[-~2m(R~fr, 4)],
(9)
1~1
where the 1ne1a5t1c e1k0na1 fact0r 15 1ntr0duced:
exP [ - ~21n( R~-, 4 ) ] =exP( - ~ ~ ×exp[
p.,( R~,.,-)) ~ d 35, d 352 exP ( -145, ) V/*~( 51, 52, ,3 )
-~p.~(R~f~.+5~ -5t ) -~.1(R~f~.+53
- 5 t ) ] ~u~ (5t,
52, 53)
•
(10)
F0r 9au551an wave funct10n5 0f 3He and t the 1nte9rat10n 0ver 10n91tud1na1 c00rd1nate5 can 6e fact0r12ed and perf0rmed exp11c1t1y 91v1n9 the 10n91tud1na1 f0rm fact0r 0f the (3He, t) vertex. 7he rema1n1n9 tran5ver5a1 e1k0na1 fact0r f0r 0 ° react10n5 0n car60n 15 5h0wn 1n f19. 1. 1t 15 06v10u51y 1e55 5teep at the nuc1ear 5urface a5 c0mpared t0 the e1a5t1c e1k0na1 fact0r 1n the c0nv01ut10n m0de1 and ha5 a 9reater pr06a6111ty 0f penetrat10n 1n51de a nuc1eu5 t0 h19her den51ty. 7he 5m00th1n9 ar05e fr0m add1t10na1 1nte9rat10n 0ver 5j 1n (10). 7h15 1nte9rat10n c0rre5p0nd5 t0 avera91n9 0ver d1fferent p051t10n5 0f the 3He center 0f ma55 w1th the effect1ve 1mpact parameter 6e1n9 f1xed. 7he e1k0na1 fact0r (10) ha5 6een u5ed t0 ca1cu1ate the a6501ute cr055 5ect10n 1n the A-re910n f0r a 0 ° ~2C(3He, t) react10n at 2 6 e V k1net1c ener9y 0f3He. At th15 ener9y the m0mentum tran5fer 1n the A-re910n 15 rather 1ar9e c0mpared t0 the 512e 0f t~-C. 7hu5, a 10ca1 den51ty appr0x1mat10n f0r the nuc1ear re5p0n5e ha5 6een u5ed [ 7 ]. F0r the nuc1ear re5p0n5e funct10n at a 91ven den51ty the m0de1 deve10ped 1n ref. [ 6 ] ha5 6een u5ed. 7he re5u1t5 0f the ca1cu1at10n are 5h0wn 1n f19. 2 6y the 5011d 11ne. F0r c0mpar150n, tw0 0ther re5u1t5 are pre5ented 1n the 5ame f19ure. 7he da5hed 11ne repre5ent5 the tr1t1um 5pectrum ca1cu1ated u51n9 the e1a5t1c e1k0na1 fact0r 1n the c0nv01ut10n m0de1 (c0rre5p0nd1n9 t0 the da5hed 11ne 1n f19. 1 ). 7he d1fference 15 c1ear, the cr055 5ect10n 15 pr0p0rt10na1 t0 p.~(r) e x p [ - 2 , , ( r , 4)1 .
(11)
7he e1a5t1c e1k0na1 fact0r 91ve5 the extreme per1phera11ty 0f the react10n. 7he tar9et den51ty ••5een•• 6y a pr0ject11e 15 5ma11 re5u1t1n9 1n a 5ma11er cr055 5ect10n and 5ma11er 5h1ft 0fthe A-peak. 7he d0tted 11ne 15 the 4ua51free ca1cu1at10n w1th0ut the effect5 0f p10n pr0Pa9at10n 1n51de the tar9et. 1n th15 ca5e the peak p051t10n rema1n5 unchan9ed and c01nc1de5 w1th the 0ne 1n the react10n p(3He, t)A ++ 1t 15 1n5truct1ve t0 n0te that the peak p051t10n determ1ne5 t0 a 1ar9e extent the ma9n1tude 0fthe A-peak. 7he 10w-ener9y w1n9 0fthe tr1t1um 5pectrum 1n f19.2 15 c0mm0n f0r a11 three curve5. 1t 15 determ1ned 6y the 510pe 0f 221
V01ume 226, num6er 3,4
PHY51C5 LE77ER5 8
10 Au9u5t 1989
1.0 /
1
f 1
//
0.8
0.6
/ / / /~
0.4
1111 1600
f~00
1800
1900
[M0v]
0,2 1
r/Rc
F19. 1. E1k0na1 fact0r5 f0r ~2C. Fu1111ne: 1ne1a5t1c e1k0na1 fact0r f0r the (3He, t) react10n. Da5hed 11ne: the e1k0na1 fact0r f0r e1a5t1c 5catter1n9 1n the c0nv01ut10n m0de1.
F19. 2. 7he cr055 5ect10n d20/dE, d~2 1n m 6 / 5 r / M e V f0r the L~C(3He, t) react10n. 7he 11ne5 are ca1cu1at10n5 6a5ed 0n ref5. [6,7] f0r the nuc1ear re5p0n5e funct10n. 5011d 11ne: the ca1cu1at10n w1th the 1ne1a5t1c e1k0na1 fact0r. Da5hed 11ne: the ca1cu1at10n w1th the e1a5t1c e1k0na1 fact0r. D0tted 11ne: 4ua51free pr0duct10n w1th0ut med1um effect5. 7he 5h0rt ran9e 1nteract10n c0n5tant5 are 92 =0.5, 9 ~ = -0.2.
the (3He, t) vertex f0rm fact0r. 7heref0re, 1fthe p051t10n 0fthe A-peak 1n the nuc1ear re5p0n5e funct10n 15 h19her than - 3 0 0 MeV 0f the exc1tat10n ener9y, a c0n51dera61e part 0f 1t w111 6e cut 1n the tr1t1um 5pectrum 6y the (3He, t) f0rm fact0r at 2 6 e V k1net1c ener9y 0f a 3He. 7he 5011d 11ne 1n f19. 2 5t111 m155e5 50me part 0f the 5h1ft 0f the A-peak. Neverthe1e55, th15 51tuat10n 15 5at15fact0ry due t0 the appr0x1mat10n5 u5ed 1n the ca1cu1at10n5 pre5ented. F1r5t 0f a11, 0n1y 1A-1 h and 1p-1 h f1na1 5tate5 were taken 1nt0 acc0unt - th1515 n0t true f0r the ta11 0fthe 4ua51e1a5t1c peak wh1ch 151mp0rtant t0 repr0duce the r19ht w1n9 0fthe A-peak; 5ec0nd1y, the react10n take5 p1ace at the nuc1ear 5urface where the tar9et den51ty var1e5 rap1d1y. 1n the 10ca1 den51ty appr0x1mat10n u5ed 1n the ca1cu1at10n pre5ented the den51ty 9rad1ent5 were n0t acc0unted f0r. 1n 5ummary, 1t ha5 6een 5h0wn that the e1k0na1 fact0r f0r 1ne1a5t1c 0ne-5tep react10n5 1n nuc1eu5-nuc1eu5 c0111510n5 d1ffer5 fr0m the 0ne f0r e1a5t1c nuc1eu5-nuc1eu5 5catter1n9. 1t 91ve5 6etter penetrat10n 0f the pr0ject11e 1n51de the tar9et thu5 revea11n9 the med1um effect5 re5p0n5161e f0r the A-peak p051t10n 1n the (3He, t) react10n. 7he auth0r ackn0w1ed9e5 the d15cu5510n5 0fthe 5u6ject w1th V.L. K0r0tk1ch and L.C. Max1m0n
Reference5 [ 1 ] 1.8er94u15t et a1., Nuc1. Phy5. A 469 ( 1987 ) 648. [2] C. 6aarde, Nuc1. Phy5. A 478 (1988) 475c. [3] V.6. A61eev et a1., 50v. Phy5. JE7P Lett. 40 (1984) 763. [4] P. C0ntard0 et a1., Phy5. Lett. 8 198 (1986) 331. [5] 6. Chanfray and M. Er1c50n, Phy5. Len. 8 141 (1984) 163. [6 ] V.F. Dm1tr1ev and 7. 5u2uk1, Nuc1. Phy5. A 348 ( 1985 ) 697. [7] V.F. Dm1tr1ev, Yad. F12.46 (1987) 770; N0v05161r5k prepr1nt 1NP 118-86 (1986). [ 8 ] H. E56en5en and 7.H.5. Lee, Phy5. Rev. C 32 ( 1985 ) 1966. [9] W. C2y2 and L.C. Max1m0n, Ann. Phy5.52 (1969) 59.
222