- Email: [email protected]
On inelastic scattering of heavy particles with excitation of the collective states of nuclei
Volume 15, number 2
PHYSICS
but since the maximum of this peak is reached at about 50 MeV, extens~,on to still lower~energtes will probably not yield a higher L f l / L i ° ratio. This is already apparent in the 44 MeV value column 4). The variation of this ratio without Be 7 contribution does show a slight i n c r e a s e at 44 MeV and this is expected to r i s e as the energy is further d e c r e a s e d due to the difference in thresholds of the LiLTand Li 6 formation in C 12. That this may compensate for the d e c r e a s e of Be 7 is very improbable due to the rapid fall of that c r o s s - s e c tion near its threshold. In the c a s e of oxygen, the work of Albouy et al. [13] shows clearly that o(Be ~) does not exhibit a v e a k in the low energy region uo that the LiT/Li 6 ratio will essentially depend on the threshold effect which one may expect to be less pronounced than that of C 12. It is interesting to note that many of the Li 6 and Li 7 abundances measured recently in various T. Tauri type s t a r s [14] fall within the limits of column 4. If 9ne considers that the higher va~lues of the L f l / L i ° ratios observed a r e due to Li e destruction in the more central region of these s t a r s , the lowest values observed by astronomers may be signfficative of the production process in outer regions and may then be compared with the r e s u l t s given above. A Monte-Carlo cascade calculation followed by break-up of the residual excited nuclei has been
LETTERS
15 March 1965
performed in o r d e r to try to account for thel~ve2 results in the case of 155 MeV protons on C . The values obtained [15] a r e in excellent agreement with the above experimental data. References
I. R. A, Alpher and R. C. Herman, Rev. Mod. Phys. 22 (1950) 153. 2. E.M.Burbidge, G.R.Burbidge, W.A.Fowler F.Hoyle, Rev.Mod.Phys. 29 (1957) 547. 3. S.Bashkin andD.C.Pesslee, AstrophyB.J. 134 (1961) 981. 4. W.A.Fowler, J.L.Greenstein and F.Hoyle, Geophye.J, 6 (1962) 148. 5. F.Yiou, R.Klspisch, E.Gradsztajn, M.Epberre and R. Bernas, Advances in mass-spectrometry, vol.m (Pergamon press, London) in press. 6. R.Bernas, E.GradsztaJn, Nguyen LongDen, R. Klapisch and F.Yiou, Nucle~tr Instr. and Me~ods (to tm pubhshed). 7. J.B.Cumming, Ann.Rev.Nucl.Sc. 13 (1963) 261. 8. C.Bren, M.Lefort andX.Tarrago, J.Phys.Rad. 23 (1962) 37X. 9. E.Grad~ztaJn, M.Epherre etR.Bernem, Physics Letters 4 (1963) 257. 10. E. GradsztaJn, J. Phys. Rad. 21 (1960) 761. 11. G. Frtedlander, J. Hudis, R. Wolfgang, Phys. Rev. 99 (1955) 263. 12. R. Klapiseh ~mdR. Bernas (to be published). 13. G. Albouy et al., Physics Letters 2 (1962) 306. 14. G.M. Herbig, Ap. J. 140 (1964) 792; G.Wallerstein, Bull.Am.Phys.Soc, series II, 9 (1964) 711 and private communication. 15. E. Gradarj~jn, R. Klspisch, F.Yiou and R. Bernas, Phys. Rev. Letters (in press).
ON I N E L A S T I C S C A T T E R I N G OF H E A V Y P A R T I C L E S W I T H E X C I T A T I O N OF T H E C O L L E C T I V E S T A T E S OF N U C L E I V. K. LUKYANOV and I. ~.. PETKOV Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR Received 18 January 1965
In the last years, great attention has been paid to the process of inelastic scattering of heavy particles. This is mainly due to the fact that the above process can be used to obtain information on the structure of low-lying nuclear levels. The Coulomb excitation of nuclei [1] is of especial importance because, according to the well-known laws of electromagnetic interaction, one succeeds in singling out in the f i r s t o r d e r of perturbation theory the part of the cross section deperding only on nu,:leus structure Eowever in some cases, especially in the scattering of heavy ions, the interaction may not be considered weak ~md the first o r d e r of pertur)ation theory yields an i n c o r r e c t result. Experimentally, this results in the observation of some strong h-ansitions forbidden in the f i r s t order of perturbation theory. 149
Volume IS) number 2
PHYSICS
LETTERS
15 March 1965
The cMculaflons are then p e r f o r m e d in an adiabatic approximation with the f o r m u l a [2]
d~ = d~R"2g,1
~ [<1,~lexp (-:~ f® v c 7 dO I,I,,) [2 (1) . I MiMf =~ where d a R is the Rutherford s c a t t e r i n g c r o s s section, @i, f a r e the wa!.~ functions of the initial I and the final f sta-fi~s of t h e nucleus. S o m e t i m e s , in detecting weak transitions, one has to i n c r e a s e the e n e r g y of incident p a r t i c l e s up to the Coulomb b a r r i e r I E , ' ~ UB ~. In this c a s e both the Coulomb and n u c l e a r interactions need to be taken iato account, i.e. . Ckml ~ . ~ o l Coul cad q~ ~ n ~ v
(OlY~(60)e l ~
(2)
where )~@ are the coordinates of the relative motion, and {~} are the internal coordtvates of the nu= cleus. F o r E ~ UB one may s t i l l assume that the particles travel along the Coulomb trajectories. Ins e r U ~ (2) into (I) one can estimate the contribution of the nuclear interactions to the excitation cross section. The corresponding calculations in the f i r s t order in Umt have been made in ref. 3; f o r 0+-0 " transitloas the result was obtained in an analytic form. Some conclusions were drawn about the energy dependence of the excitation cross section near the b a r r i e r and also about its sensitivity to the choice of the parameters of the nuclear potential. In this paper, inelastic scattering of heavy particles above the b a r r i e r is considered. In this c~ ;e the t ~ e af classical angles has a l i m i t e0 ~ U B / E < 1 and the scattering amplitude can be given in the
form [4]
.o
27r
f(O) = -~--~'=f
p do f
0
0
d~0 exp {-l?J~p sin ~0 cos ~p - ~
inserting here V = ~O(r) + Vi=t(rO where gO(r) = %(r).+ Vo(r) = ' ~ ( r )
f V dz} -,o
(3)
+ rJ~ =~ (r) + Vo(r) is the
cenL~al-symmetric part of the interaction potential of an incident ion with the nucleus, V0 is its a b s o r p tive part and Uint is of the form (2); the inelastic s c a t t e r i n g amplitude in an adiabatic approximation can be written a s [5] (kR AE/E << 1, AE is the excitation energy) finelastic = The usual method of calculation of
(¢)fl/(0~) 14)i)
(5)
f(O~) is the finding of the f i r s t o r d e r of perturbation t h e o r y in
oO ~1[~,° The r e s t r i c t i o ~ to only the f i r s t t e r m , liJ~ear in [J int, ,n ~he expansion e~) (- ~ f ~.~i~, d~) holds -oo merely in the range of angles 0 < (ankR) "1 where a n is the p a r a m e t e r of the nuclear deformation. However this interval of angles is not always interesting. F o r example, for incident heavy )~ns with k ~ 10 for a n ~ 0.3 and R ~ 10 we have 0 < 01 ~ 5° which c o r r e s p o n d s to angles difficult to r e a c h e x p e r i m e n tally. T h e r e f o r e in what follows we shall not use the assumption about the s m a l l n e s s of Utnt . Thus, we write the initial expression: o0 2~ ik i dp drp F(p) . e r p {-i2kpsin ½0cos ~p ~-~ f Uo(r) dz}- exp (-i 2~ L~v(pt~). e tv~0) (6) .~) = _~ f / _oo )~P o o where oo kv = ~ g Utnt(r~) dz t~} .cO ao
F(o) -- o.e,,p (-
i ~-~f
Yo(r)
(s)
--00
In the case c o n s i d e r e d for 0 > 01 one has 2kp sir, ~ 0 ~ ,~ 2kR sin !20 >> 1, The amplitude (6) can t h e r e f o r e be calc~dated using the method of stationary p~ase. It is worthwhile to note that, as e s t i m a t i o n s show, the number of oscillations of the e x p r e s s i o n 0~ F(p) exp (-i ~ LX~ Is not large. This m e a n s that in finding the saddle points the last c o f a c t o r may be d i s r e g a r d e d . Finally one has: 150
Volume 15, number 2
PHYSICS L E T T E R S ,
,,
15 March 1966
,,
......
f(Ol~) = . ~ + ) . e x p {-i~Xu Lxv(°c~) ( ' l ) v } + f(o) • exp {-I ~LAv(pe~)}
(9)
where f(~) = l i (1 * 1) J ~ e .
exp {tA(~ (oc)}.
" I) "½ Roc) (pc IA(~
(10)
QO
A(.)(O) = • ~
8in ½o- ~ f
dz
(11)
The saddle points ¢)c(0) (where ( = ± ) are determined from the equation d A ( ~ ) / ~ -- 0
(1~.)
Note that in the range o~ c l a s s i c a l angles O< O0 e a c h of the amplitudes f ( ~ is given by the contribution of the two r e a l saddle points. However, p r a c t i c a l l y the main contribution ~s given only by one of them. For 0 = 00 both points coincide, but h e r e A"O = 0; t h e r e f o r e it is n e c e s s a r y to take into account the contributlon of the third derivative in the expansion of the exponent. F o r 0 > 00, P( becomes complex and this leads to an exponent~l d e c r e a s e of the c r o s s sections with the observation angle [4]: Substituting (9) into (5) one gets the inelastic scattering amplitude in the form
~nelutic{ 0) = f(+)" (~'fl exp {-t ~/~v(pcg) (-l)v}l @l) + f(.). (@fi exp -i hv T, L~v(p~O I@i)
(13)
and the differential cross section is da =
1 i MZ~ I/lnei~tlJ0) 12 i-ii--~-iMf
(14)
The following conclusions may be drawn from the obtained expression. F i r s t l y , in a par/ttcular case i = f one gets the elastic s c a t t e r i n g amplitude. If Ulnt is due to the n u c l e a r deformation then this amplitude c o r r e s p o n d s to the elastic scattering in the d e f o r m e d potential field. Putting in (13) Utnt = 0 one obtains the elastic scattering amplitude in the c e n t r a ] - s y m m e t r i c field
f = f(+) + f(_)
05)
In the general case i ;, f the obtained expression (14) yields the excitation cross section for any n-phonon transition (multiple excitation) while the account of only the first term in the expansion of the exponential (13) leads to one-phonon transRlons [5]. The obtained result resembles in form the expression (1) which underlies the theory of multiple Coulomb excitation [2], moreover, in calculating the structure cofactor (@flexp (-I ~.~ L~,)I4, i) one can use the methods developed in this theory. The solution ~P for eq. (12) p~ = p((0) is necessary for specific calculation by the formulas (13) and (14). This problem has been considered in detail by the authors in the preprint of the JINR, P-1908. In conclusion we wish to acknowledge helpful discussions with Dr. S. I. Drozdov. ~CfC I'CPICC N
I. K.A. Ter-Marttrosyan, Zh. El~p. i Teor. Fiz. 22 (1952) 9.84. K.Alder, A.Bohr, T.Huus, B.Mottelson and A.Winther, Rev.Mod.Phys. 28 (1956) 432. 2. K. Alder ~ A. Winther, Kgl.Darmke Videru~kab.Selkab. Mat. Fys. Medd. 32 No. 8 (1960). 3. V.K.Lukyanov, Izv. Akad Nauk SSSR, ser.fys. 28 (1904) 2212. 4. L.D. Landau and E.M.Ltfshltz, quanCam meohanics (Fizmatgiz, 1963). 5. S.I.Drozdov, Zh. Eksp, t Teor. Fiz. 44 (1963) 335.
151
PHYSICS
but since the maximum of this peak is reached at about 50 MeV, extens~,on to still lower~energtes will probably not yield a higher L f l / L i ° ratio. This is already apparent in the 44 MeV value column 4). The variation of this ratio without Be 7 contribution does show a slight i n c r e a s e at 44 MeV and this is expected to r i s e as the energy is further d e c r e a s e d due to the difference in thresholds of the LiLTand Li 6 formation in C 12. That this may compensate for the d e c r e a s e of Be 7 is very improbable due to the rapid fall of that c r o s s - s e c tion near its threshold. In the c a s e of oxygen, the work of Albouy et al. [13] shows clearly that o(Be ~) does not exhibit a v e a k in the low energy region uo that the LiT/Li 6 ratio will essentially depend on the threshold effect which one may expect to be less pronounced than that of C 12. It is interesting to note that many of the Li 6 and Li 7 abundances measured recently in various T. Tauri type s t a r s [14] fall within the limits of column 4. If 9ne considers that the higher va~lues of the L f l / L i ° ratios observed a r e due to Li e destruction in the more central region of these s t a r s , the lowest values observed by astronomers may be signfficative of the production process in outer regions and may then be compared with the r e s u l t s given above. A Monte-Carlo cascade calculation followed by break-up of the residual excited nuclei has been
LETTERS
15 March 1965
performed in o r d e r to try to account for thel~ve2 results in the case of 155 MeV protons on C . The values obtained [15] a r e in excellent agreement with the above experimental data. References
I. R. A, Alpher and R. C. Herman, Rev. Mod. Phys. 22 (1950) 153. 2. E.M.Burbidge, G.R.Burbidge, W.A.Fowler F.Hoyle, Rev.Mod.Phys. 29 (1957) 547. 3. S.Bashkin andD.C.Pesslee, AstrophyB.J. 134 (1961) 981. 4. W.A.Fowler, J.L.Greenstein and F.Hoyle, Geophye.J, 6 (1962) 148. 5. F.Yiou, R.Klspisch, E.Gradsztajn, M.Epberre and R. Bernas, Advances in mass-spectrometry, vol.m (Pergamon press, London) in press. 6. R.Bernas, E.GradsztaJn, Nguyen LongDen, R. Klapisch and F.Yiou, Nucle~tr Instr. and Me~ods (to tm pubhshed). 7. J.B.Cumming, Ann.Rev.Nucl.Sc. 13 (1963) 261. 8. C.Bren, M.Lefort andX.Tarrago, J.Phys.Rad. 23 (1962) 37X. 9. E.Grad~ztaJn, M.Epherre etR.Bernem, Physics Letters 4 (1963) 257. 10. E. GradsztaJn, J. Phys. Rad. 21 (1960) 761. 11. G. Frtedlander, J. Hudis, R. Wolfgang, Phys. Rev. 99 (1955) 263. 12. R. Klapiseh ~mdR. Bernas (to be published). 13. G. Albouy et al., Physics Letters 2 (1962) 306. 14. G.M. Herbig, Ap. J. 140 (1964) 792; G.Wallerstein, Bull.Am.Phys.Soc, series II, 9 (1964) 711 and private communication. 15. E. Gradarj~jn, R. Klspisch, F.Yiou and R. Bernas, Phys. Rev. Letters (in press).
ON I N E L A S T I C S C A T T E R I N G OF H E A V Y P A R T I C L E S W I T H E X C I T A T I O N OF T H E C O L L E C T I V E S T A T E S OF N U C L E I V. K. LUKYANOV and I. ~.. PETKOV Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR Received 18 January 1965
In the last years, great attention has been paid to the process of inelastic scattering of heavy particles. This is mainly due to the fact that the above process can be used to obtain information on the structure of low-lying nuclear levels. The Coulomb excitation of nuclei [1] is of especial importance because, according to the well-known laws of electromagnetic interaction, one succeeds in singling out in the f i r s t o r d e r of perturbation theory the part of the cross section deperding only on nu,:leus structure Eowever in some cases, especially in the scattering of heavy ions, the interaction may not be considered weak ~md the first o r d e r of pertur)ation theory yields an i n c o r r e c t result. Experimentally, this results in the observation of some strong h-ansitions forbidden in the f i r s t order of perturbation theory. 149
Volume IS) number 2
PHYSICS
LETTERS
15 March 1965
The cMculaflons are then p e r f o r m e d in an adiabatic approximation with the f o r m u l a [2]
d~ = d~R"2g,1
~ [<1,~lexp (-:~ f® v c 7 dO I,I,,) [2 (1) . I MiMf =~ where d a R is the Rutherford s c a t t e r i n g c r o s s section, @i, f a r e the wa!.~ functions of the initial I and the final f sta-fi~s of t h e nucleus. S o m e t i m e s , in detecting weak transitions, one has to i n c r e a s e the e n e r g y of incident p a r t i c l e s up to the Coulomb b a r r i e r I E , ' ~ UB ~. In this c a s e both the Coulomb and n u c l e a r interactions need to be taken iato account, i.e. . Ckml ~ . ~ o l Coul cad q~ ~ n ~ v
(OlY~(60)e l ~
(2)
where )~@ are the coordinates of the relative motion, and {~} are the internal coordtvates of the nu= cleus. F o r E ~ UB one may s t i l l assume that the particles travel along the Coulomb trajectories. Ins e r U ~ (2) into (I) one can estimate the contribution of the nuclear interactions to the excitation cross section. The corresponding calculations in the f i r s t order in Umt have been made in ref. 3; f o r 0+-0 " transitloas the result was obtained in an analytic form. Some conclusions were drawn about the energy dependence of the excitation cross section near the b a r r i e r and also about its sensitivity to the choice of the parameters of the nuclear potential. In this paper, inelastic scattering of heavy particles above the b a r r i e r is considered. In this c~ ;e the t ~ e af classical angles has a l i m i t e0 ~ U B / E < 1 and the scattering amplitude can be given in the
form [4]
.o
27r
f(O) = -~--~'=f
p do f
0
0
d~0 exp {-l?J~p sin ~0 cos ~p - ~
inserting here V = ~O(r) + Vi=t(rO where gO(r) = %(r).+ Vo(r) = ' ~ ( r )
f V dz} -,o
(3)
+ rJ~ =~ (r) + Vo(r) is the
cenL~al-symmetric part of the interaction potential of an incident ion with the nucleus, V0 is its a b s o r p tive part and Uint is of the form (2); the inelastic s c a t t e r i n g amplitude in an adiabatic approximation can be written a s [5] (kR AE/E << 1, AE is the excitation energy) finelastic = The usual method of calculation of
(¢)fl/(0~) 14)i)
(5)
f(O~) is the finding of the f i r s t o r d e r of perturbation t h e o r y in
oO ~1[~,° The r e s t r i c t i o ~ to only the f i r s t t e r m , liJ~ear in [J int, ,n ~he expansion e~) (- ~ f ~.~i~, d~) holds -oo merely in the range of angles 0 < (ankR) "1 where a n is the p a r a m e t e r of the nuclear deformation. However this interval of angles is not always interesting. F o r example, for incident heavy )~ns with k ~ 10 for a n ~ 0.3 and R ~ 10 we have 0 < 01 ~ 5° which c o r r e s p o n d s to angles difficult to r e a c h e x p e r i m e n tally. T h e r e f o r e in what follows we shall not use the assumption about the s m a l l n e s s of Utnt . Thus, we write the initial expression: o0 2~ ik i dp drp F(p) . e r p {-i2kpsin ½0cos ~p ~-~ f Uo(r) dz}- exp (-i 2~ L~v(pt~). e tv~0) (6) .~) = _~ f / _oo )~P o o where oo kv = ~ g Utnt(r~) dz t~} .cO ao
F(o) -- o.e,,p (-
i ~-~f
Yo(r)
(s)
--00
In the case c o n s i d e r e d for 0 > 01 one has 2kp sir, ~ 0 ~ ,~ 2kR sin !20 >> 1, The amplitude (6) can t h e r e f o r e be calc~dated using the method of stationary p~ase. It is worthwhile to note that, as e s t i m a t i o n s show, the number of oscillations of the e x p r e s s i o n 0~ F(p) exp (-i ~ LX~ Is not large. This m e a n s that in finding the saddle points the last c o f a c t o r may be d i s r e g a r d e d . Finally one has: 150
Volume 15, number 2
PHYSICS L E T T E R S ,
,,
15 March 1966
,,
......
f(Ol~) = . ~ + ) . e x p {-i~Xu Lxv(°c~) ( ' l ) v } + f(o) • exp {-I ~LAv(pe~)}
(9)
where f(~) = l i (1 * 1) J ~ e .
exp {tA(~ (oc)}.
" I) "½ Roc) (pc IA(~
(10)
QO
A(.)(O) = • ~
8in ½o- ~ f
dz
(11)
The saddle points ¢)c(0) (where ( = ± ) are determined from the equation d A ( ~ ) / ~ -- 0
(1~.)
Note that in the range o~ c l a s s i c a l angles O< O0 e a c h of the amplitudes f ( ~ is given by the contribution of the two r e a l saddle points. However, p r a c t i c a l l y the main contribution ~s given only by one of them. For 0 = 00 both points coincide, but h e r e A"O = 0; t h e r e f o r e it is n e c e s s a r y to take into account the contributlon of the third derivative in the expansion of the exponent. F o r 0 > 00, P( becomes complex and this leads to an exponent~l d e c r e a s e of the c r o s s sections with the observation angle [4]: Substituting (9) into (5) one gets the inelastic scattering amplitude in the form
~nelutic{ 0) = f(+)" (~'fl exp {-t ~/~v(pcg) (-l)v}l @l) + f(.). (@fi exp -i hv T, L~v(p~O I@i)
(13)
and the differential cross section is da =
1 i MZ~ I/lnei~tlJ0) 12 i-ii--~-iMf
(14)
The following conclusions may be drawn from the obtained expression. F i r s t l y , in a par/ttcular case i = f one gets the elastic s c a t t e r i n g amplitude. If Ulnt is due to the n u c l e a r deformation then this amplitude c o r r e s p o n d s to the elastic scattering in the d e f o r m e d potential field. Putting in (13) Utnt = 0 one obtains the elastic scattering amplitude in the c e n t r a ] - s y m m e t r i c field
f = f(+) + f(_)
05)
In the general case i ;, f the obtained expression (14) yields the excitation cross section for any n-phonon transition (multiple excitation) while the account of only the first term in the expansion of the exponential (13) leads to one-phonon transRlons [5]. The obtained result resembles in form the expression (1) which underlies the theory of multiple Coulomb excitation [2], moreover, in calculating the structure cofactor (@flexp (-I ~.~ L~,)I4, i) one can use the methods developed in this theory. The solution ~P for eq. (12) p~ = p((0) is necessary for specific calculation by the formulas (13) and (14). This problem has been considered in detail by the authors in the preprint of the JINR, P-1908. In conclusion we wish to acknowledge helpful discussions with Dr. S. I. Drozdov. ~CfC I'CPICC N
I. K.A. Ter-Marttrosyan, Zh. El~p. i Teor. Fiz. 22 (1952) 9.84. K.Alder, A.Bohr, T.Huus, B.Mottelson and A.Winther, Rev.Mod.Phys. 28 (1956) 432. 2. K. Alder ~ A. Winther, Kgl.Darmke Videru~kab.Selkab. Mat. Fys. Medd. 32 No. 8 (1960). 3. V.K.Lukyanov, Izv. Akad Nauk SSSR, ser.fys. 28 (1904) 2212. 4. L.D. Landau and E.M.Ltfshltz, quanCam meohanics (Fizmatgiz, 1963). 5. S.I.Drozdov, Zh. Eksp, t Teor. Fiz. 44 (1963) 335.
151