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Doorway States and the Super-Radiant Mechanism in Nuclear Reactions
Nuclear Physics A 834 (2010) 167c–171c www.elsevier.com/locate/nuclphysa
Doorway States and the Super-Radiant Mechanism in Nuclear Reactions N. Auerbacha a
School of Physics and Astronomy, Tel Aviv University, Tel Aviv , 69978, Israel, and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
In the early 1950s the possibility of forming a super-radiant (SR) state in a gas of atoms confined to a volume of a size smaller than the wave length of radiation was suggested by Dicke. The atoms are coupled through their common radiation field. This indirect interaction through the continuum leads to a redistribution of lifetimes among unstable intrinsic states. A rapidly decaying SR state is created at the expense of the rest of the states of the system that are robbed of their decay probability and become narrow. This mechanism is general and analogous phenomena should appear in many quantum systems when quasi-bound states are strongly coupled through common decay channels. Recently this mechanism was considered for generating doorway states in low-energy nuclear reactions. A discussion of the conditions for appearance of doorways in nuclear physics processes will is presented.
1. Introduction In the early 1950s the possibility of forming a ”super-radiant” (SR) state in a gas of atoms confined to a volume of a size smaller than the wave length of radiation was suggested by Dicke [1]. In the absence of a direct interaction, the atoms are coupled through their common radiation field. This indirect interaction through the continuum leads to a redistribution of lifetimes among unstable intrinsic states. A rapidly decaying SR state is created at the expense of the rest of the states of the system that have their decay probability greatly reduced and become narrow. This mechanism is of quite general origin, and analogous phenomena should appear in many quantum systems when quasi-bound states are strongly coupled through common decay channels. The SR approach has been used in many different fields. For example, it was applied in chemistry [2,3], atomic physics [4], condensed matter physics [5,6], intermediate energy nuclear physics [7,8], in particle physics [9], in the theory of nuclear reactions [6,10,11]. In the next section we will derive the SR formalism for some simple situations with the purpose to arrive in a heuristic fashion to the idea of a super-radiant state and its connection to the concept of a doorway. 0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.12.030
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N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
2. The effective Hamiltonian Following the projection operator formalism of Feshbach [7,12], let us divide the Hilbert space of nuclear states into two parts, the {Q}-subspace involving very complicated manybody states |q, and the subspace {P } of open channels |c. We use the notations Q and P for the corresponding projection operators onto the above subspaces. The total wave function of the system, |Ψ = Q|Ψ + P |Ψ,
(1)
satisfies the stationary Schr¨odinger equation H|Ψ = E|Ψ
(2)
that can be written as a set of coupled equations, (E − HQQ )Q|Ψ = HQP P |Ψ
(3)
and (E − HP P )P |Ψ = HP Q Q|Ψ,
(4)
where we use the notations HAB = AHB. Eliminating the part P |Ψ, we come to the equation in the Q-space,
˜ QQ Q|Ψ = 0 E−H
with the effective Hamiltonian 1 ˜ QQ = HQQ + HQP HP Q . H E (+) − HP P
(5)
(6)
Here E (+) = E + i0 contains the infinitesimal imaginary term ensuring right asymptotic conditions for the continuum wave functions. The second term of the effective Hamiltonian [6] contains a real and imaginary part of the propagator G(+) (E) =
E (+)
1 − HP P
(7)
emerging from the principal value and the delta-function δ(E − HP P ) (on-shell contributions from channels c open at energy E), respectively. The imaginary part, −(i/2)W , of the effective Hamiltonian is given by W = 2π
HQP |c c|HP Q .
(8)
c
Thus, the effective Hamiltonian [7] in Q-space is non-Hermitian, ˜ QQ = H QQ − i W, H 2
(9)
where H QQ is a symmetric and real matrix that includes, apart from the original Hamiltonian in the Q-space, HQQ , the principal value contribution of the P Q-coupling. The
N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
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˜ E˜ = E − (i/2)Γ, are complex poles second part is anti-Hermitian. The eigenvalues of H, of the scattering matrix corresponding to the resonances in the cross sections. To demonstrate in a simple way the role of the anti-Hermitian term we assume that only one channel is open. Then the matrix W , Eq. [9], has a completely separable form, q|W |q = 2πAcq Ac∗ q ,
(10)
where Acq = q|HQP |c.
(11)
The rank of the matrix W is 1, so that all the eigenvalues of this matrix are zero except one that has the value equal to the trace of the matrix: Γs =
q|W |q = 2π
q
q
|Acq |2 ≡
q
Γ↑q
(12)
with Γ↑q denoting the escape width of the individual levels before the -matrix is diagonalized. The special unstable state with width Γs is often referred to as the super-radiant (SR), in analogy to the Dicke coherent state [1] of a set of two-level atoms coupled through the common radiation field. Here the coherence is generated by the common decay channel. The stable states are trapped and decoupled from the continuum. In the more general case of N intrinsic states and Nc open channels with Nc N the super-radiant mechanism survives if the mean level spacing D of internal states and their decay widths obey: κ=
Γ↑ > 1. D
(13)
3. Doorways Often only a subset of intrinsic states {Q} connects directly to the {P } space of channels. The rest of states in {Q} will connect to {P } only when they obtain admixtures of these selected states of the first type. The special states directly coupled to continuum are the doorways, |d. They form the doorway subspace {D} within {Q}, and the corresponding projection operator will be denoted here as D. The remaining states in {Q} ˜ will be denoted as |˜ q and the subspace as {Q}. The full Hamiltonian can be decomposed the following way:
H = HQ˜ Q˜ + HDD + HQD ˜ ˜ + HD Q
+ HP P + HDP + HP D .
(14)
Note that the terms HP Q˜ and HQP ˜ are missing because in accordance with the doorway hypothesis they are very small. Also note that diagonalizing the operator in the upper line of (14) would give back the states |q with the components of |d mixed with |˜ q states. The two last items in the above equation couple the doorway states, and therefore all the |q states to the open channels.
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N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
4. The case of a single doorway We discuss the case when there is only one important doorway state |d. The matrix elements of the effective operator W in the intrinsic space are now given by q|W |q = 2π
Nc
q|HDP |cc|HP D |q
(15)
c=1
with the doorway assumption: q|HDP |c = q|dd|HDP |c,
(16)
where q|d is the amplitude of the admixture of the doorway into the |q state. Eq. (16) becomes: q|W |q = 2π q|dd|q
|d|HDP |c|2 ;
(17)
c
again we have separable matrix elements of the matrix W , this time irrespective of the number of open channels Nc . As discussed above, the criterion of validity of the SR mechanism is that the average spacing between the levels in {Q} is smaller than the decay width of such a state ”before” the SR mechanism takes effect. This can be expressed , in the case of doorways, by considering the spreading width, Γ↓d , of the doorway state representing the fragmentation of |d into compound states |˜ q . It is easy to see [13] that the requirement for the SR doorway mechanism to be valid can be formulated as: Γ↑d > 1. Γ↓d
(18)
5. An Example; Isobaric Analog States The isobaric analog state (IAS), |A, is obtained from the parent state |π with certain isospin T [14] by acting with the isospin lowering operator T− that changes a neutron into a proton: |A = const · T− |π.
(19)
In a compound nucleus, the IAS is surrounded by many compound states |q of lower isospin T< = T − 1. The Coulomb interaction violates the isospin symmetry fragmenting the strength of the IAS over many states |q giving rise to the spreading width Γ↓A of the IAS. If located above thresholds, the IAS can also decay into several continuum channels that gives rise to the decay width Γ↑A . In medium and heavy mass nuclei the condition in Eq. (18) is satisfied. For example in 208 Pb the spreading width is about 80 keV while the escape width is about 160 keV [14], thus Γ↑A /Γ↓A ≈ 2. The SR mechanism is therefore relevant to this case providing a straightforward explanation why the IAS appears as a single resonance with the decay width given by that of |A: Γ↑A = 2π |A|HQP |P |2 . This work was done in collaboration with V. Zelevinsky.
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N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
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REFERENCES 1. R.H. Dicke, Phys. Rev. 93, 99 (1954). 2. V.B. Pavlov-Verevkin, Phys. Lett. A 129, 168 (1988). 3. F. Remacle, M. Munster, V.B. Pavlov-Verevkin, and M. Desouter-Lecomte, Phys. Lett. A 145, 265 (1990). 4. V.V. Flambaum, A.A. Gribakina, and G.F. Gribakin, Phys. Rev. A 54, 2066 (1996). ¨ 5. K. Pichugin, H. Schanz, and P. Seba, Phys. Rev. E 64, 056227 2001). 6. A. Volya and V. Zelevinsky, in: V. Zelevinsky (Ed.), Nuclei and Mesoscopic Physics (AIP Conference Proceedings, vol. 777, 2005) p. 229. 7. N. Auerbach, Phys. Rev. C 50, 1606 (1994). 8. N. Auerbach and V. Zelevinsky, Phys. Rev. C 65, 034601 (2002). 9. N. Auerbach, V. Zelevinsky, and A. Volya, Phys. Lett. B 590 (2004) 45. 10. V.V. Sokolov and V.G. Zelevinsky, Nucl. Phys. A504, 562 (1989). 11. I. Rotter, Rep. Prog. Phys. 54, 635 (1991). 12. H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958); H. Feshbach, Ann. Phys. (N.Y.) 19, 287 (1962). 13. N.Auerbach and V.G. Zelevinsky, Nucl. Phys. A781, 67 (2007). 14. N. Auerbach, J. Hufner, A.K. Kerman, and C.M. Shakin, Rev. Mod. Phys. 44, 48 (1972).
Doorway States and the Super-Radiant Mechanism in Nuclear Reactions N. Auerbacha a
School of Physics and Astronomy, Tel Aviv University, Tel Aviv , 69978, Israel, and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
In the early 1950s the possibility of forming a super-radiant (SR) state in a gas of atoms confined to a volume of a size smaller than the wave length of radiation was suggested by Dicke. The atoms are coupled through their common radiation field. This indirect interaction through the continuum leads to a redistribution of lifetimes among unstable intrinsic states. A rapidly decaying SR state is created at the expense of the rest of the states of the system that are robbed of their decay probability and become narrow. This mechanism is general and analogous phenomena should appear in many quantum systems when quasi-bound states are strongly coupled through common decay channels. Recently this mechanism was considered for generating doorway states in low-energy nuclear reactions. A discussion of the conditions for appearance of doorways in nuclear physics processes will is presented.
1. Introduction In the early 1950s the possibility of forming a ”super-radiant” (SR) state in a gas of atoms confined to a volume of a size smaller than the wave length of radiation was suggested by Dicke [1]. In the absence of a direct interaction, the atoms are coupled through their common radiation field. This indirect interaction through the continuum leads to a redistribution of lifetimes among unstable intrinsic states. A rapidly decaying SR state is created at the expense of the rest of the states of the system that have their decay probability greatly reduced and become narrow. This mechanism is of quite general origin, and analogous phenomena should appear in many quantum systems when quasi-bound states are strongly coupled through common decay channels. The SR approach has been used in many different fields. For example, it was applied in chemistry [2,3], atomic physics [4], condensed matter physics [5,6], intermediate energy nuclear physics [7,8], in particle physics [9], in the theory of nuclear reactions [6,10,11]. In the next section we will derive the SR formalism for some simple situations with the purpose to arrive in a heuristic fashion to the idea of a super-radiant state and its connection to the concept of a doorway. 0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.12.030
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N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
2. The effective Hamiltonian Following the projection operator formalism of Feshbach [7,12], let us divide the Hilbert space of nuclear states into two parts, the {Q}-subspace involving very complicated manybody states |q, and the subspace {P } of open channels |c. We use the notations Q and P for the corresponding projection operators onto the above subspaces. The total wave function of the system, |Ψ = Q|Ψ + P |Ψ,
(1)
satisfies the stationary Schr¨odinger equation H|Ψ = E|Ψ
(2)
that can be written as a set of coupled equations, (E − HQQ )Q|Ψ = HQP P |Ψ
(3)
and (E − HP P )P |Ψ = HP Q Q|Ψ,
(4)
where we use the notations HAB = AHB. Eliminating the part P |Ψ, we come to the equation in the Q-space,
˜ QQ Q|Ψ = 0 E−H
with the effective Hamiltonian 1 ˜ QQ = HQQ + HQP HP Q . H E (+) − HP P
(5)
(6)
Here E (+) = E + i0 contains the infinitesimal imaginary term ensuring right asymptotic conditions for the continuum wave functions. The second term of the effective Hamiltonian [6] contains a real and imaginary part of the propagator G(+) (E) =
E (+)
1 − HP P
(7)
emerging from the principal value and the delta-function δ(E − HP P ) (on-shell contributions from channels c open at energy E), respectively. The imaginary part, −(i/2)W , of the effective Hamiltonian is given by W = 2π
HQP |c c|HP Q .
(8)
c
Thus, the effective Hamiltonian [7] in Q-space is non-Hermitian, ˜ QQ = H QQ − i W, H 2
(9)
where H QQ is a symmetric and real matrix that includes, apart from the original Hamiltonian in the Q-space, HQQ , the principal value contribution of the P Q-coupling. The
N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
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˜ E˜ = E − (i/2)Γ, are complex poles second part is anti-Hermitian. The eigenvalues of H, of the scattering matrix corresponding to the resonances in the cross sections. To demonstrate in a simple way the role of the anti-Hermitian term we assume that only one channel is open. Then the matrix W , Eq. [9], has a completely separable form, q|W |q = 2πAcq Ac∗ q ,
(10)
where Acq = q|HQP |c.
(11)
The rank of the matrix W is 1, so that all the eigenvalues of this matrix are zero except one that has the value equal to the trace of the matrix: Γs =
q|W |q = 2π
q
q
|Acq |2 ≡
q
Γ↑q
(12)
with Γ↑q denoting the escape width of the individual levels before the -matrix is diagonalized. The special unstable state with width Γs is often referred to as the super-radiant (SR), in analogy to the Dicke coherent state [1] of a set of two-level atoms coupled through the common radiation field. Here the coherence is generated by the common decay channel. The stable states are trapped and decoupled from the continuum. In the more general case of N intrinsic states and Nc open channels with Nc N the super-radiant mechanism survives if the mean level spacing D of internal states and their decay widths obey: κ=
Γ↑ > 1. D
(13)
3. Doorways Often only a subset of intrinsic states {Q} connects directly to the {P } space of channels. The rest of states in {Q} will connect to {P } only when they obtain admixtures of these selected states of the first type. The special states directly coupled to continuum are the doorways, |d. They form the doorway subspace {D} within {Q}, and the corresponding projection operator will be denoted here as D. The remaining states in {Q} ˜ will be denoted as |˜ q and the subspace as {Q}. The full Hamiltonian can be decomposed the following way:
H = HQ˜ Q˜ + HDD + HQD ˜ ˜ + HD Q
+ HP P + HDP + HP D .
(14)
Note that the terms HP Q˜ and HQP ˜ are missing because in accordance with the doorway hypothesis they are very small. Also note that diagonalizing the operator in the upper line of (14) would give back the states |q with the components of |d mixed with |˜ q states. The two last items in the above equation couple the doorway states, and therefore all the |q states to the open channels.
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N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
4. The case of a single doorway We discuss the case when there is only one important doorway state |d. The matrix elements of the effective operator W in the intrinsic space are now given by q|W |q = 2π
Nc
q|HDP |cc|HP D |q
(15)
c=1
with the doorway assumption: q|HDP |c = q|dd|HDP |c,
(16)
where q|d is the amplitude of the admixture of the doorway into the |q state. Eq. (16) becomes: q|W |q = 2π q|dd|q
|d|HDP |c|2 ;
(17)
c
again we have separable matrix elements of the matrix W , this time irrespective of the number of open channels Nc . As discussed above, the criterion of validity of the SR mechanism is that the average spacing between the levels in {Q} is smaller than the decay width of such a state ”before” the SR mechanism takes effect. This can be expressed , in the case of doorways, by considering the spreading width, Γ↓d , of the doorway state representing the fragmentation of |d into compound states |˜ q . It is easy to see [13] that the requirement for the SR doorway mechanism to be valid can be formulated as: Γ↑d > 1. Γ↓d
(18)
5. An Example; Isobaric Analog States The isobaric analog state (IAS), |A, is obtained from the parent state |π with certain isospin T [14] by acting with the isospin lowering operator T− that changes a neutron into a proton: |A = const · T− |π.
(19)
In a compound nucleus, the IAS is surrounded by many compound states |q of lower isospin T< = T − 1. The Coulomb interaction violates the isospin symmetry fragmenting the strength of the IAS over many states |q giving rise to the spreading width Γ↓A of the IAS. If located above thresholds, the IAS can also decay into several continuum channels that gives rise to the decay width Γ↑A . In medium and heavy mass nuclei the condition in Eq. (18) is satisfied. For example in 208 Pb the spreading width is about 80 keV while the escape width is about 160 keV [14], thus Γ↑A /Γ↓A ≈ 2. The SR mechanism is therefore relevant to this case providing a straightforward explanation why the IAS appears as a single resonance with the decay width given by that of |A: Γ↑A = 2π |A|HQP |P |2 . This work was done in collaboration with V. Zelevinsky.
(20)
N. Auerbach / Nuclear Physics A 834 (2010) 167c–171c
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REFERENCES 1. R.H. Dicke, Phys. Rev. 93, 99 (1954). 2. V.B. Pavlov-Verevkin, Phys. Lett. A 129, 168 (1988). 3. F. Remacle, M. Munster, V.B. Pavlov-Verevkin, and M. Desouter-Lecomte, Phys. Lett. A 145, 265 (1990). 4. V.V. Flambaum, A.A. Gribakina, and G.F. Gribakin, Phys. Rev. A 54, 2066 (1996). ¨ 5. K. Pichugin, H. Schanz, and P. Seba, Phys. Rev. E 64, 056227 2001). 6. A. Volya and V. Zelevinsky, in: V. Zelevinsky (Ed.), Nuclei and Mesoscopic Physics (AIP Conference Proceedings, vol. 777, 2005) p. 229. 7. N. Auerbach, Phys. Rev. C 50, 1606 (1994). 8. N. Auerbach and V. Zelevinsky, Phys. Rev. C 65, 034601 (2002). 9. N. Auerbach, V. Zelevinsky, and A. Volya, Phys. Lett. B 590 (2004) 45. 10. V.V. Sokolov and V.G. Zelevinsky, Nucl. Phys. A504, 562 (1989). 11. I. Rotter, Rep. Prog. Phys. 54, 635 (1991). 12. H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958); H. Feshbach, Ann. Phys. (N.Y.) 19, 287 (1962). 13. N.Auerbach and V.G. Zelevinsky, Nucl. Phys. A781, 67 (2007). 14. N. Auerbach, J. Hufner, A.K. Kerman, and C.M. Shakin, Rev. Mod. Phys. 44, 48 (1972).