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DOORWAY STATES, THE SUPER- RADIANT MECHANISM AND NUCLEAR REACTIONS N.Auerbach, TAU and MSU
Coherence in Spontaneous Radiation Process R.H.Dicke, Phys.Rev. 93, 99) 1954) “In the usual treatment of spontaneous radiation by a
gas, the radiation process is calculated as though the separate molecules radiate independently of each other…..
It is clear that this model is incapable of describing a coherent spontaneous radiation process…This simplified picture overlooks the fact that all the molecules are interacting with a common radiation field and hence cannot be treated as independent.”...
”A gas that radiates strongly because of coherence will be called “super-radiant””.
Superradiance, collectivization by Superradiance, collectivization by decaydecay
Dicke coherent stateN identical two-level atomscoupled via common radiation
Analog in nucleiInteraction via continuum(Trapped states ) self-organization
The Effective Hamiltonian
AHBH
QHPHE
PHQHE
EH
PQ
AB
PQPP
QPQQ
isnotation thewhere
and
:equations coupled ofset a into decomposed becan that
equationr Schrodinge thesatisfies
, system theoffunction wave totalThe
Effective Hamiltonian (cont’d)
PP
PQPP
QPQQef
efQQ
HEE
iEE
HHE
HH
QHE
P
1G
propagator theofpart imaginary and real a
containsn Hamiltonia effective theof termsecond The
0 Here
1H
:nHamiltonia effective with the
0)
:obtain we gEliminatin
The Effective Hamiltonian (cont’d)
These originate from the principal value and delta function .
The imaginary part, / 2 is given by:
2
where are the open channels.
The effective Hamiltonian in -space is non-He
PP
QP PQc
E H
i W
W H c c H
c
Q
eff
rmitian
H2
where is a symmetric real matrix that includes, apart from the
original Hamiltonian of the -space , ,
the principal value contribution of the -coupling.
The cross sec
iH W
H H
Q H
QP
, ' '
tion for a reaction is determined by the square of
the scattering amplitude:
1' ,
1
The eigenvalues of , E- / 2 , are complex poles of the scattering
baQP QPeff
q q qq
eff
a b
T E a H q q H bE H
S iT
H i
matrix,
corresponding to the resonances in the cross section.
Single Channel
To demonstrate in a simple way the effect of the anti-Hermitian term we look at the case of a single channel. Then the matrix W has a completely separable form:
cHq
AAqWq
QP
cq
cq
cq
*'
A
where, 2'
Separable interaction
.
.
.......
....... 2
.......
2323 13
3222 12
31212
1
AAAAA
AAAAAi
IH
AAAAA
“Super-radiant” state
The rank of the factorized matrix is 1, so that all eigenvalues of W are zero, except one that has the value equal to the trace of this matrix:
)2(
2
2
2
0
cqq
q
cq
q
A
AqWq
Effective interaction in the Q-space
q
q’
P
“Super-radiant” state
The special unstable state is often referred to as the “super-radiant” (SR), in analogy to the Dicke coherent state of a set of two-level atoms coupled through a common radiation field. Here, the coherence is generated by the common decay channel. The stable states are trapped and decoupled from the continuum.
R.H.Dicke, Phys.Rev. 93, 99 (1954)
General case
The phenomenon of super-radiance survives in a general situation of N intrinsic states and Nc open channels provided Nc<< N, if the mean level spacing D of internal states and their characteristic decay widths satisfy the conditions
1)(
D
cqc
General case
In this regime of overlapping resonances their interaction through the common continuum channels leads to restructuring of the complex energy spectrum, similarly to the formation of Dicke’s coherent state. Since the rank of the factorized W matrix is Nc , it has only Nc non-zero positive eigenvalues.
The intrinsic space Q is now divided into the SR subspace of dimension Nc and the subspace of trapped states It> of dimension N-Nc .
Doorways
Frequently 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} states due to the admixtures of these selected states . The special states coupled directly to the continuum are the doorways Id>. They form the doorway subspace {D}. The corresponding projection operator will be denoted as D.
The remaining states will be denoted as~
q
Doorways
d c
~
q
Doorways
)expression above in the no is e that ther(Note
q~ states with mixed d components with the
q states back the give willexpression in thispart first theingDiagonaliz
H
: wayfollowing in the decomposed becan n Hamiltonia full The
~
~~~~
QP
PDDPPP
QDDQDD
H
HHH
HHHH
Doorways (Corridors)
Single doorway
doorway. theof h decay widt simpy the is width thisnaturally
2
: widthsa with state broad single a finds oneagain and
channels, ofnumber theof veirrespecti separable,again iselement matrix This
'2'
Then,
state thedoorway to theof admixture theis dq where,
,assumptiondoorway Under the
'2'q
:bygiven now are space-Q in the
operator W effective theof elementsmatrix The .doorway important one Assume
22
2
1
d
cDP
qs
cDP
DPDP
PDDP
N
c
cHddq
cHdqddqqWq
qcHddqcHq
qHccHqqW
d
c
When is this picture valid?
d
The criterion of validity is that the average spacing between
levels in {Q} -space is smaller than the decay width of such
a state "before" the SR is set at work. Consider the spreading
width , ,
q
of the doorway state for the fragmentation
into compound states q . If N is the number of compound states
in the interval covered by the spreading width, their average energy
spacing is :
Be
d
q
DN
2
q
d
fore the SR mechanism is turned on, the average decay
width of a q is
2 that can be estimated:
, therefore : thus, 1
QPc
qs s dq
q q d d d
q H c
N D
Examples
Single-particle resonance
.resonancesneutron therepresent states narrow theofrest The
theory.sFeshbach'in potential optical theofframework in the resonances
s.p. of width with theidentified be should width This . 2
widths.p. large a with state a produce willmechanism This doorway. a as serves
Thus .component s.p. the viacontinuum the tocouple states compound These
states. compound dcomplicate theamong state thisspreadsn interactio The threshold.
decay theabove isenergy itsbut spaceQ the tobelongs that state s.p. aConsider
2
....
....
..
cH
q
QPpsps
psps
ps
Examples
Isobaric analog state (IAS).
The IAS, A is the result of action of the isospin lowering operator on the parent state ,
A
In the compound nucleus, the IAS is surrounded by many compound states
of lower isospin T
T
const T
q
1. The Coulomb interaction does not conserve
isospin fragmenting the strength of the IAS over many states that results
in the spreading width of the IAS.
If located above the threshold, the IAA
T
q
S can also decay into several continuum channels that
gives rise to the decay width . In heavy nuclei .
The SR mechanism is relevant to this case, providing an explanation why the IAS appeaA A A
2
rs
as a single resonance with a decay width given by that of ,
2IAS QP
A
A H P
Pb208 Example
1Certainly
.80 and 170 which of ,250about is IAR theof width total theIn
A
AA208
A
A keVkeVkeVBi
Mixing of the IAS with T-1 states
IAS
states 1-T
Before mixing
After mixing
Neutron Induced Fission
Fission
Double humped potential in fission (intermediate structure)
section. crossfission in the structure
teintermedia observe willone aresult, As well.second theof states theof
each for itself repeatssituation This h.decay widt particle-single
aan smaller thbut state compound typicala ofthan that
hsdecay widtly wider considerab a find willone ation,diagonalizAfter
''
:as written becan doorway theof
vicinityin the states compound for thematrix W The . channels
fissionopen theinto state compound a ofdecay for the doorways
are minimum second in the states theformalismpresent In the
2f
fVdqddqqWq
d
qf
d
Double humped potential in fission
An outstanding example are the fission isomers in several heavy, nuclei such as Pu.
The fission cross section in the neutron induced reaction shows structures with larger widths spaced several hundred eV apart , in addition to the usual compound resonances which are spaced a few eV. This phenomenon is interpreted as a direct consequence of a double humped potential in the fission process of the compound nucleus. The energy spacing between the few excited (usually collective-rotational or vibrational) states in the second shallower well are larger than between the compound states in the deeper well. When the compound states in the deeper well are in the vicinity of a state in the second well, they couple forming mixed states which, couple to the fission channel via the admixture of the state from the second well.
Intermediate Structure in Fission
Example; Giant Resonances (GR)
One describes the giant resonances in nuclei in terms of 1p-1h configurations. The residual interaction forms collective states out of these configurations. However, usually the GRs are located in the particle continuum. The 1p-1h are surrounded by a vast spectrum of 2p-2h excitations which will mix with the GR.
1that condition under the , 2
:hdecay widt thehave willstate SR theand onerank of ismatrix W Again the
''
:bygiven are elementsmatrix W The doorway. a as serves GR The
states).2h -2p theinto GR the(of bG admixture the viacontinuum theinto
decay willstates2h -2p The .Gby resonancegiant theand , b as states theseDenote
2
G
G
cG
c
cVG
GVccVGbGGbbWb
Giant Resonances (CRPA)
The super-radiant mechanism applied to intermediate energy nuclear physics; examples.
The SR mechanism is applied to several intermediate (and high) energy phenomena.
N-Nbar excitations
Multi-quark states
The SR mechanism is universal and can take place in very distinctively different systems.It is possible that in the sector of quark physics there are situations in which preconditions exist for the appearance SR states followed by very narrow trapped resonances. Some examples are taken from the multi-quark systems. In some cases it is claimed that uncharacteristically narrow resonances are observed. For example the 1545 MeV pentaquark, the X(3872) tetraquark and other tetraquarks.
Superradiance in resonant spectra
2 5 0 3 0 0 3 5 0
0
5
1 0
1 5
2 0
Co
un
ts
W - m1 2 C
[ M e V ]
2 5 0 3 0 0 3 5 0
0
5
1 0
1 5
2 0
Co
un
ts
W - m1 2 C
[ M e V ]
2 5 0 3 0 0 3 5 0
0
5
1 0
1 5
2 0
Co
un
ts
W - m1 2 C
[ M e V ]
2 5 0 3 0 0 3 5 0
0
5
1 0
1 5
2 0
Co
un
ts
W - m1 2 C
[ M e V ]
M(nK+) [ GeV/c2 ]
Event
s
0
5
10
15
20
25
30
35
1.5 1.6 1.7 1.8 1.9
Narrow resonances and broad superradiant state
in 12C
Pentaquark as a possible candidate for
superradianceStepanyan et.al. hep-ex/0307018
Bartsch et.al. Eur. Phys. J. A 4, 209 (1999)
+ pentaquark as a two-state interference
Effective Hamiltonian
2,1 , 2 iii
Two-level interaction via the continuum
Kn scattering crossectionE
ve
nts
0
5
1 0
1 5
2 0
2 5
3 0
3 5
1 .5 1 .6 1 .7 1 .8 1 .9
M(nK ) [GeV/c ]+ 2
Sensible parameters under requirement•Resonant energy Er=1540 MeV•Kn threshold energy•Width of broad peak
1535 MeV1(Er) =120 MeV2 =1560 MeV2(Er) =60 MeVv=1 MeV(green), 500 (red) MeV
Ghosts
New claims of very narrow quark states arround 4 GeV mass. Are these the ghosts of the pentaquark?
Or maybe real narrow states?
Other examples
One could envisage other situations in the field ofintermediate energy when the SR mechanism
mightproduce narrow states in addition to a very broad
state.Narrow resonances in deeply bound hadronic atoms(pionic, anti-nucleonic), in deeply bound anti-kaons,
insigma hypernuclei, etc
Collaborator
V.Zelevinsky, MSU
