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On the damping of deep-lying hole states in spherical nuclei
Volume 167B, number 3
ON T H E D A M P I N G
PHYSICS LETTERS
13 February 1986
O F DEEP-LYING H O L E STATES IN SPHERICAL NUCLEI
B.B. MATVEEV, B.A. T U L U P O V Institute for Nuclear Research of the Academy of Sciences of the USSR, Moscow, USSR
S.E. MURAVIEV and M.G. URIN Moscow Physieal Engineering Institute, Moscow, USSR Received 8 January 1985; revised manuscript received 3 December 1985
A model for the description of deep-lying hole state strength functions in spherical nuclei is proposed, in which the coupling of one-quasiparticle states to non-collective many-particle configurations is described by means of a phenomenological optical model, while the coupling to the most collective states (phonons) is taken into account explicitly. The results of the calculations are compared with the available experimental data.
Recent experimental studies of neutron pick-up reactions with the excitations of deep-lying hole states (see, e.g. refs. [1,2]) stimulated the appearance of refs. [ 3 - 5 ] , devoted to the theoretical interpretation of the corresponding strength function energy dependence. In the present work for the solution of this problem the optical-phonon model (OPM) is suggested, which may be considered as an alternative approach. In this model the coupling of simple shellmodel excitations to non-collective many-particle configurations is taken into account in the average in the framework of the phenomenological optical model using the Green's function method [6], while the coupling to the most collective states (phonons) is taken explicitly into account by introducing the dynamic deformation parameters into the shell-model potential. Being applied to the description of the deep-lying hole state damping, the OPM proves to be the generalization of the well-known coupled channel approach [7]. In this approach the parameters of the OPM are those of the shell-model (single-particle) potential, the dynamic deformation parameters and the intensity of the optical potential imaginary part. Here the OPM is applied to the quantitative interpretation of the (lg91~)-i neutron configuration strength functions in 115,119Sn and 123,129Te isotopes. The deep-lying hole state strength function Sx(e ) may be connected with the imaginary part of the 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
averaged single-particle Green's function
g(rr'; e) = G(rr'; e)" ~=G(rr; e + il sign (e -/a)) , where/a is the chemical potential and I is an averaging interval. This connection follows from the spectral representation of G(rr'; e) [8] and is given in the pole approximation by
Sx(e ) = 1 Im f ~(r)g(rr'; e)~x(r' ) dr dr',
(1)
where e
G = G O+ G o Z G ,
(2)
where G O is the shell-model Green's function and Y, is the self-energy operator, which describes in the present case the coupling of single-particle configurations to the complex ones. Following the above described implications of the OPM one may represent the self-energy operator ~ as = Xst + ]~ph •
(3)
Here ]gst is the "statistical" part of ~, which may be considered as the smooth function of energy after averaging over an energy interval I, containing many 255
Volume 167B, number 3
PHYSICS LETTERS
compound levels; Y~phis that part of E which depends non-monotonously on energy after averaging due to the virtual excitations of phonons. One may treat the average "statistical" part Zst as the local quantity and parametrize it as follows [6] : ~st(rr'; e) = Y~st(rr'; e + iI sign (e - ~))
=
(r; e - u)
(r - r ' ) ,
(4)
where Ah(r; e -- la) -->--i I#(r; le --/.tl) sign(e --/.t) has the meaning of the phenomenological optical addition to the shell-model potential. According to (2) and (4) the appropriate averaged Green's function Gs t = gopt is the Green's function of the Schr6dinger equation with the optical model hamiltonian h = h 0 + a h [6]:
[h (r) - e]g°Pt(rr'; e) = - 6 ( r - r') .
(5)
In accordance with relation (3) one may rewrite the Dyson equation in the form G = Gst + GstZphG,
(6)
which is more convenient for further analysis. The self-energy part Zph is described by the graph of fig. 1 [8], where the thick line indicates the Green's function, satisfying eq. (6), the wavy line indicates the Green's function of the phonon with an energy coL and angular momentum L and the square denotes the field V(r), describing the coupling of quasiparticles to phonons. In the proposed phenomenological approach it appears due to a dependence of the shell-model potential on the nuclear dynamic deformation parameters. To the lowest order in these parameters the interaction V(r) may be written as follows:
V(r) = V(r) ~L~ aL#YLl~(n) -- ~VL(r)L' V(r) = r OU/3r,
(7)
where U(r) is the shell-model potential and the ~2 = Zu taL ~ 12 are the dynamic deformation parameters.
Fig. 1. 256
13 February 1986
Using the standard rules of calculations we obtain the following relation for Zph(e
Zph(rr';e)
L
vL(r) V
(r')O(rr';e +coL).
(8)
Separating the angular variables in eqs. (1), (6) and (8) one has the formal OPM solution of the problem o f the description of deep-lying hole state strength functions: Sx(e) = l l m f x x ( r ) g i l ( r r ' ;
e)×x(r')drdr',
(9)
gfl(rr'; e) = g~/Pt(rr'; e) +
Li T
2 . .,, LOl,s t)fgj?o t erl; e) v(.l)
× gi,r(rlr2; e + coL) V(r2)gil(rlr';e) drl d r l - (10) Here the Xx ----Xe~jl are the shell-model radial wave functions of the hole states,g~/pt is the radial Green's function of the optical model Schr6dinger equation and the coupling parameters r 2 are equal to
(1/4n)132(21+ 1)(21'+ 1) (l 0 L OII'O)2W 2 ( ] 1 1 ' l ' ;~L). ' Eqs. (9), (10) have been used for the calculations of the (lg9/2) -1 neutron strength functions in the 115,119Sn and 123,129Te isotopes in the approximation of the surface interaction between quasi-particles and phonons (V(r) = RUo6(r - R), where U0 is the intensity of the neutron shell-model potential and R is the nuclear radius). The parameters of the realistic shell-model potential were chosen as in ref. [6], and the dynamic deformation parameters and phonon energies were taken from the data on the corresponding phonon excitations [9] in the neighbouring eveneven nuclei. The imaginary part of the optical potential was chosen in the shape of a surface absorption: W(r) = Wo4a dfws(r, R, a)/dr, where fws is the Woods-Saxon function and a = 0.63 fm, W0 = 0.9 MeV. This value of I~0 does not contradict the results of the coupled-channel approach analysis of the lowenergy neutron scattering data. For instance, in ref. [10] the value of W0 = 1 - 2 MeV, which may be somewhat overestimated due to the neglect of manyphonon configurations, was used to describe properly both the elastic and the inelastic neutron scattering data. The results are presented in fig. 2 in comparison with the (averaged) experimental data [2] and the
Volume 167B, number 3
PHYSICS LETTERS
1 15Sn
S~
13 February 1986
123Te
S~
I
I%
I I
0.4
i
0.4 t~
I-I I t
~' t
/:*.t ::~
,
/
,.o
,
: " ..
~
:
o.2
°: ;..° ~A/I/I~?'2.ZAI//A//I
3
4
5
6
7
8
9
s~
ll9sn 0.4
3
Ex(MeV)
4
5
6
7
8
iAi
9
S~
Ex(meV) 129Te
0.4
,-%
" o . 2 ~ i
02
3
4
5
6
7
8
9
Ex(MeV)
3
4
5
6
7
8
9
Ex(NIeV)
Fig. 2. Strength functions of lg9/2 neutron-hole states in 11s ,1t9Sn and 12a,lZgTe isotopes. The full curves are the results of the present work; the dotted curves show the results of the "ladder" approximation calculations; the dashed curves represent the quasipartiele-phonon model results [ 3 ] ; the dashed areas are the (averaged) experimental data [ 2 ]. corresponding results o f the quasiparticle-phonon model [3]. The details of these calculations, in which the coupling o f one-quasiparticle states to the lowlying 2 +- and 3--phonons was taken into account, are described in ref. [11 ]. In fig. 2 also the results of the strength function calculations in the "ladder" approximation are given. This approximation corresponds to the substitution G ~ G°P t in eq. (8) and implies that the coupling to one-phonon states is taken into account only. The qualitative and quantitative analysis of eq. (10) shows that the main contribution to the origin of the gross structure o f the strength function is induced by the coupling of the considered quasiparticle state to the low-lying 2+-phonon. As is seen from fig. 2 in the case o f strong coupling, which occurs in the 123,129Te isotopes, the role o f non-linear effects, in other words, the role of many-phonon configurations, proves to the very important for the interpretation o f the energy dependence o f the strength function. Apparently owing to this reason the present results sometimes differ essentially from the quasiparticlephonon model results.
In the OPM the strength function width depends bth on the value of W0 and on the one-quasiparticle strength spreading due to the interaction with manyphonon configurations. In the case of weak coupling the width is approximately proportional to W0 while in the case of strong coupling it depends weakly on W0 • The analysis performed shows that the proposed optical-phonon model allows to describe sufficiently well the strength functions o f deep-lying hole states and also to indicate the main reasons which cause their gross structure. The authors are grateful to V.I. Bondarenko who kindly represented some programs for numerical calculations and took part in the initial stage o f this work.
References [1] M. Sakai and K. Kubo, Nuel. Phys. A185 (1972) 217; M. Tanaka et al., Phys. Lett. 78B (1978) 221. [2] E. Gerlic et al., Phys. Rev. 21C (1980) 124; S. Gal~s et al., Nucl. Phys. A381 (1982) 173. 257
Volume 167B, number 3
PHYSICS LETTERS
[3] V.G. Soloviev,Ch. Stoyanov and A.I. Vdovin, Nucl. Phys. A342 (1980) 261. [4 ] T. Koeling and F. Iachello, Nuel. Phys. A295 (1978) 45. [5] S.P. Klevansky and R.H. Lemmer, Phys. Rev. 25C (1982) 3137. [6] M.G. Urin, Soy. J. Part. Nucl. 15 (1984) 109. [7] T. Tamum, Rev. Mod. Phys. 37 (1965) 679.
258
13 February 1986
[8] Ad3. Migdal, Theory of finite Fermi-systems and applications to atomic nuclei (Interscience, New York, 1967). [9] Nuel. Data Sheets B7 (1972) 419;9 (1973) 157; 10 (1973) 91; 26 (1979) 207; 32 (1981) 287; 35 (1982) 375. [10] E.S. Konobeevsky and V.I. Popov, Sov. J. Nucl. Phys. 33 (1981) 7. [ 11 ] B.B. Matveev et al., preprint IYaI 11-0353 (1984).
ON T H E D A M P I N G
PHYSICS LETTERS
13 February 1986
O F DEEP-LYING H O L E STATES IN SPHERICAL NUCLEI
B.B. MATVEEV, B.A. T U L U P O V Institute for Nuclear Research of the Academy of Sciences of the USSR, Moscow, USSR
S.E. MURAVIEV and M.G. URIN Moscow Physieal Engineering Institute, Moscow, USSR Received 8 January 1985; revised manuscript received 3 December 1985
A model for the description of deep-lying hole state strength functions in spherical nuclei is proposed, in which the coupling of one-quasiparticle states to non-collective many-particle configurations is described by means of a phenomenological optical model, while the coupling to the most collective states (phonons) is taken into account explicitly. The results of the calculations are compared with the available experimental data.
Recent experimental studies of neutron pick-up reactions with the excitations of deep-lying hole states (see, e.g. refs. [1,2]) stimulated the appearance of refs. [ 3 - 5 ] , devoted to the theoretical interpretation of the corresponding strength function energy dependence. In the present work for the solution of this problem the optical-phonon model (OPM) is suggested, which may be considered as an alternative approach. In this model the coupling of simple shellmodel excitations to non-collective many-particle configurations is taken into account in the average in the framework of the phenomenological optical model using the Green's function method [6], while the coupling to the most collective states (phonons) is taken explicitly into account by introducing the dynamic deformation parameters into the shell-model potential. Being applied to the description of the deep-lying hole state damping, the OPM proves to be the generalization of the well-known coupled channel approach [7]. In this approach the parameters of the OPM are those of the shell-model (single-particle) potential, the dynamic deformation parameters and the intensity of the optical potential imaginary part. Here the OPM is applied to the quantitative interpretation of the (lg91~)-i neutron configuration strength functions in 115,119Sn and 123,129Te isotopes. The deep-lying hole state strength function Sx(e ) may be connected with the imaginary part of the 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
averaged single-particle Green's function
g(rr'; e) = G(rr'; e)" ~=G(rr; e + il sign (e -/a)) , where/a is the chemical potential and I is an averaging interval. This connection follows from the spectral representation of G(rr'; e) [8] and is given in the pole approximation by
Sx(e ) = 1 Im f ~(r)g(rr'; e)~x(r' ) dr dr',
(1)
where e
G = G O+ G o Z G ,
(2)
where G O is the shell-model Green's function and Y, is the self-energy operator, which describes in the present case the coupling of single-particle configurations to the complex ones. Following the above described implications of the OPM one may represent the self-energy operator ~ as = Xst + ]~ph •
(3)
Here ]gst is the "statistical" part of ~, which may be considered as the smooth function of energy after averaging over an energy interval I, containing many 255
Volume 167B, number 3
PHYSICS LETTERS
compound levels; Y~phis that part of E which depends non-monotonously on energy after averaging due to the virtual excitations of phonons. One may treat the average "statistical" part Zst as the local quantity and parametrize it as follows [6] : ~st(rr'; e) = Y~st(rr'; e + iI sign (e - ~))
=
(r; e - u)
(r - r ' ) ,
(4)
where Ah(r; e -- la) -->--i I#(r; le --/.tl) sign(e --/.t) has the meaning of the phenomenological optical addition to the shell-model potential. According to (2) and (4) the appropriate averaged Green's function Gs t = gopt is the Green's function of the Schr6dinger equation with the optical model hamiltonian h = h 0 + a h [6]:
[h (r) - e]g°Pt(rr'; e) = - 6 ( r - r') .
(5)
In accordance with relation (3) one may rewrite the Dyson equation in the form G = Gst + GstZphG,
(6)
which is more convenient for further analysis. The self-energy part Zph is described by the graph of fig. 1 [8], where the thick line indicates the Green's function, satisfying eq. (6), the wavy line indicates the Green's function of the phonon with an energy coL and angular momentum L and the square denotes the field V(r), describing the coupling of quasiparticles to phonons. In the proposed phenomenological approach it appears due to a dependence of the shell-model potential on the nuclear dynamic deformation parameters. To the lowest order in these parameters the interaction V(r) may be written as follows:
V(r) = V(r) ~L~ aL#YLl~(n) -- ~VL(r)L' V(r) = r OU/3r,
(7)
where U(r) is the shell-model potential and the ~2 = Zu taL ~ 12 are the dynamic deformation parameters.
13 February 1986
Using the standard rules of calculations we obtain the following relation for Zph(e
Zph(rr';e)
L
vL(r) V
(r')O(rr';e +coL).
(8)
Separating the angular variables in eqs. (1), (6) and (8) one has the formal OPM solution of the problem o f the description of deep-lying hole state strength functions: Sx(e) = l l m f x x ( r ) g i l ( r r ' ;
e)×x(r')drdr',
(9)
gfl(rr'; e) = g~/Pt(rr'; e) +
Li T
2 . .,, LOl,s t)fgj?o t erl; e) v(.l)
× gi,r(rlr2; e + coL) V(r2)gil(rlr';e) drl d r l - (10) Here the Xx ----Xe~jl are the shell-model radial wave functions of the hole states,g~/pt is the radial Green's function of the optical model Schr6dinger equation and the coupling parameters r 2 are equal to
(1/4n)132(21+ 1)(21'+ 1) (l 0 L OII'O)2W 2 ( ] 1 1 ' l ' ;~L). ' Eqs. (9), (10) have been used for the calculations of the (lg9/2) -1 neutron strength functions in the 115,119Sn and 123,129Te isotopes in the approximation of the surface interaction between quasi-particles and phonons (V(r) = RUo6(r - R), where U0 is the intensity of the neutron shell-model potential and R is the nuclear radius). The parameters of the realistic shell-model potential were chosen as in ref. [6], and the dynamic deformation parameters and phonon energies were taken from the data on the corresponding phonon excitations [9] in the neighbouring eveneven nuclei. The imaginary part of the optical potential was chosen in the shape of a surface absorption: W(r) = Wo4a dfws(r, R, a)/dr, where fws is the Woods-Saxon function and a = 0.63 fm, W0 = 0.9 MeV. This value of I~0 does not contradict the results of the coupled-channel approach analysis of the lowenergy neutron scattering data. For instance, in ref. [10] the value of W0 = 1 - 2 MeV, which may be somewhat overestimated due to the neglect of manyphonon configurations, was used to describe properly both the elastic and the inelastic neutron scattering data. The results are presented in fig. 2 in comparison with the (averaged) experimental data [2] and the
Volume 167B, number 3
PHYSICS LETTERS
1 15Sn
S~
13 February 1986
123Te
S~
I
I%
I I
0.4
i
0.4 t~
I-I I t
~' t
/:*.t ::~
,
/
,.o
,
: " ..
~
:
o.2
°: ;..° ~A/I/I~?'2.ZAI//A//I
3
4
5
6
7
8
9
s~
ll9sn 0.4
3
Ex(MeV)
4
5
6
7
8
iAi
9
S~
Ex(meV) 129Te
0.4
,-%
" o . 2 ~ i
02
3
4
5
6
7
8
9
Ex(MeV)
3
4
5
6
7
8
9
Ex(NIeV)
Fig. 2. Strength functions of lg9/2 neutron-hole states in 11s ,1t9Sn and 12a,lZgTe isotopes. The full curves are the results of the present work; the dotted curves show the results of the "ladder" approximation calculations; the dashed curves represent the quasipartiele-phonon model results [ 3 ] ; the dashed areas are the (averaged) experimental data [ 2 ]. corresponding results o f the quasiparticle-phonon model [3]. The details of these calculations, in which the coupling o f one-quasiparticle states to the lowlying 2 +- and 3--phonons was taken into account, are described in ref. [11 ]. In fig. 2 also the results of the strength function calculations in the "ladder" approximation are given. This approximation corresponds to the substitution G ~ G°P t in eq. (8) and implies that the coupling to one-phonon states is taken into account only. The qualitative and quantitative analysis of eq. (10) shows that the main contribution to the origin of the gross structure o f the strength function is induced by the coupling of the considered quasiparticle state to the low-lying 2+-phonon. As is seen from fig. 2 in the case o f strong coupling, which occurs in the 123,129Te isotopes, the role o f non-linear effects, in other words, the role of many-phonon configurations, proves to the very important for the interpretation o f the energy dependence o f the strength function. Apparently owing to this reason the present results sometimes differ essentially from the quasiparticlephonon model results.
In the OPM the strength function width depends bth on the value of W0 and on the one-quasiparticle strength spreading due to the interaction with manyphonon configurations. In the case of weak coupling the width is approximately proportional to W0 while in the case of strong coupling it depends weakly on W0 • The analysis performed shows that the proposed optical-phonon model allows to describe sufficiently well the strength functions o f deep-lying hole states and also to indicate the main reasons which cause their gross structure. The authors are grateful to V.I. Bondarenko who kindly represented some programs for numerical calculations and took part in the initial stage o f this work.
References [1] M. Sakai and K. Kubo, Nuel. Phys. A185 (1972) 217; M. Tanaka et al., Phys. Lett. 78B (1978) 221. [2] E. Gerlic et al., Phys. Rev. 21C (1980) 124; S. Gal~s et al., Nucl. Phys. A381 (1982) 173. 257
Volume 167B, number 3
PHYSICS LETTERS
[3] V.G. Soloviev,Ch. Stoyanov and A.I. Vdovin, Nucl. Phys. A342 (1980) 261. [4 ] T. Koeling and F. Iachello, Nuel. Phys. A295 (1978) 45. [5] S.P. Klevansky and R.H. Lemmer, Phys. Rev. 25C (1982) 3137. [6] M.G. Urin, Soy. J. Part. Nucl. 15 (1984) 109. [7] T. Tamum, Rev. Mod. Phys. 37 (1965) 679.
258
13 February 1986
[8] Ad3. Migdal, Theory of finite Fermi-systems and applications to atomic nuclei (Interscience, New York, 1967). [9] Nuel. Data Sheets B7 (1972) 419;9 (1973) 157; 10 (1973) 91; 26 (1979) 207; 32 (1981) 287; 35 (1982) 375. [10] E.S. Konobeevsky and V.I. Popov, Sov. J. Nucl. Phys. 33 (1981) 7. [ 11 ] B.B. Matveev et al., preprint IYaI 11-0353 (1984).