Two-proton radioactivity and three-body decay. III. Integral formulae
  for decay widths in a simplified semianalytical approach
Two-proton radioactivity and three-body decay. III. Integral formulae for decay
widths in a simplified semianalytical approach.
L. V. Grigorenko1,2, 3 and M. V. Zhukov4
Flerov Laboratory of Nuclear Reactions, JINR, RU-141980 Dubna, Russia
Gesellschaft für Schwerionenforschung mbH, Planckstrasse 1, D-64291, Darmstadt, Germany
RRC “The Kurchatov Institute”, Kurchatov sq. 1, 123182 Moscow, Russia
Fundamental Physics, Chalmers University of Technology, S-41296 Göteborg, Sweden
Three-body decays of resonant states are studied using integral formulae for decay widths. Theo-
retical approach with a simplified Hamiltonian allows semianalytical treatment of the problem. The
model is applied to decays of the first excited 3/2− state of 17Ne and the 3/2− ground state of 45Fe.
The convergence of three-body hyperspherical model calculations to the exact result for widths and
energy distributions are studied. The theoretical results for 17Ne and 45Fe decays are updated and
uncertainties of the derived values are discussed in detail. Correlations for the decay of 17Ne 3/2−
state are also studied.
PACS numbers: 21.60.Gx – Cluster models, 21.45.+v – Few-body systems, 23.50.+z – Decay by proton
emission, 21.10.Tg – Lifetimes
I. INTRODUCTION
The idea of the “true” two-proton radioactivity was
proposed about 50 years ago in a classical paper of
Goldansky [1]. The word “true” denotes here that we
are dealing not with a relatively simple emission of two
protons, which becomes possible in every nucleus above
two-proton decay threshold, but with a specific situa-
tion where one-proton emission is energetically (due to
the proton separation energy in the daughter system) or
dynamically (due to various reasons) prohibited. Only
simultaneous emission of two protons is possible in that
case (see Fig. 1, more details on the modes of the three-
body decays can be found in Ref. [2]). The dynamics of
such decays can not be reduced to a sequence of two-body
decays and from theoretical point of view we have to deal
with a three-body Coulomb problem in the continuum,
which is known to be very complicated.
Progress in this field was quite slow. Only recently a
consistent quantum mechanical theory of the process was
developed [2, 3, 4], which allows to study the two-proton
(three-body) decay phenomenon in a three-body cluster
model. It has been applied to a range of a light nuclear
systems (12O, 16Ne [5], 6Be, 8Li∗, 9Be∗ [6], 17Ne∗, 19Mg
[7]). Systematic exploratory studies of heavier prospec-
tive 2p emitters 30Ar, 34Ca, 45Fe, 48Ni, 54Zn, 58Ge, 62Se,
and 66Kr [4, 8]) have been performed providing predic-
tions of lifetime ranges and possible correlations among
fragments.
Experimental studies of the two-proton radioactivity is
presently an actively developing field. Since the first ex-
perimental identification of 2p radioactivity in 45Fe [9, 10]
it was also found in 54Zn [11]. Some fingerprints of the
48Ni 2p decay were observed and the 45Fe lifetime and de-
cay energy were measured with improved accuracy [12].
There was an intriguing discovery of the extreme en-
hancement of the 2p decay mode for the high-spin 21+
isomer of 94Ag, interpreted so far only in terms of the
hyperdeformation of this state [13]. New experiments,
(A-2) + 2N
(A-1) + N
E2rE3r
(A-2) + 2N
(A-1) + N
(a) (b)
Three-body 
    decay 
  window
thresholds
FIG. 1: Energy conditions for different modes of the two-
nucleon emission (three-body decay): true three-body decay
(a), sequential decay (b).
aimed at more detailed 2p decay studies (e.g. observa-
tion of correlations), are under way at GSI (19Mg), MSU
(45Fe), GANIL (45Fe), and Jyväskylä (94Ag).
Several other theoretical approaches were applied to
the problem in the recent years. We should mention
the “diproton” model [14, 15], “R-matrix” approach
[16, 17, 18, 19], continuum shell model [20], and adiabatic
hyperspherical approach of [21]. Some issues of a com-
patibility between different approaches will be addressed
in this work.
Another, possibly very important, field of applica-
tion of the two-proton decay studies was shown in Refs.
[22, 23]. It was demonstrated in [22] that the importance
of direct resonant two-proton radiative capture processes
was underestimated in earlier treatment of the rp-process
waiting points [24]. The scale of modification of the astro-
physical 2p capture rates can be as large as several orders
of magnitude in certain temperature ranges. In paper [23]
it has been found that nonresonant E1 contributions to
three-body (two-proton) capture rates can also be much
larger than was expected before. The updated 2p as-
trophysical capture rate for the 15O(2p,γ)17Ne reaction
appears to be competing with the standard 15O(α,γ)19Ne
breakout reaction for the hot CNO cycle. The improve-
ments of the 2p capture rates obtained in [22, 23] are
connected to consistent quantum mechanical treatment
http://arxiv.org/abs/0704.0920v1
of the three-body Coulomb continuum in contrast to the
essentially quasiclassical approach typically used in as-
trophysical calculations of three-body capture reactions
(e.g. [24, 25]).
The growing quality of the experimental studies of the
2p decays and the high precision required for certain as-
trophysical calculations inspired us to revisit the issues
connected with different uncertainties and technical diffi-
culties of our studies. In this work we make the following.
(i) Extend the two-body formalism of the integral for-
mulae for width to the three-body case. We perform the
relevant derivations for the two-body case to make the
relevant approximations and assumptions explicit. (ii)
Formulate a simplified three-body model which has many
dynamical features similar to the realistic case, but allows
the exact semianalytical treatment and thus makes pos-
sible a precise calibration of three-body calculations. It
is also possible to study in great detail several impor-
tant dependencies of three-body widths in the frame of
this model. (iii) Perform practical studies of some sys-
tems of interest and demonstrate a connection between
the simplified semianalytical formalism and the realistic
three-body calculations.
The unit system h̄ = c = 1 is used in the article.
II. INTEGRAL FORMULA FOR WIDTH
Integral formalisms of width calculations for narrow
two-body states are known for a long time, e.g. [26, 27].
The prime objective of those studies was α-decay widths.
An interesting overview of this field can be found in the
book [28]. This approach, to our opinion, did not pro-
duce novel results as the inherent uncertainties of the
method are essentially the same as those of the R-matrix
phenomenology, which is technically much simpler (see
e.g. a discussion in [29]). An important nontrivial appli-
cation of the integral formalism was calculation of widths
for proton emission off deformed states [30, 31]. There
were attempts to extend the integral formalism to the
three-body decays, using a formal generalization for the
hyperspherical space [2, 32]. These were shown to be
difficult with respect to technical realisation and to be
inferior to other methods developed in [2, 3].
Here we develop an integral formalism for the three-
body (two-proton) decay width in a different way. How-
ever, first we review the standard formalism to define
(clearer) the approximations used.
A. Width definition, complex energy WF
For decay studies we consider the wave function (WF)
with complex pole energy
Ẽr = k̃
r/(2M) = Er − iΓ/2 , k̃r ≈ kr − iΓ/(2vr) ,
where v =
2E/M . The pole solution for Hamiltonian
(H − Ẽr)Ψ
lm (r) = (T + V − Ẽr)Ψ
lm (r) = 0
provides the WF with outgoing asymptotic
lm (r) = r
l (kr)Ylm(r̂) . (1)
For single channel two-body problem the pole solution is
formed only for one selected value of angular momentum
l. In the asymptotic region
l (k̃rr)
l (k̃rr) = Gl(k̃rr) + iFl(k̃rr) . (2)
The above asymptotic is growing exponentially
l (k̃rr)
∼ exp[+ik̃rr] ≈ exp[+ikrr] exp[+Γr/(2vr)]
as a function of the radius at pole energy. This unphys-
ical growth is connected to the use of time-independent
formalism and could be reliably neglected for typical ra-
dioactivity time scale as it has a noticeable effect at very
large distances.
Applying Green’s procedure to complex energy WF
Ψ(+)†
(H − Ẽr)Ψ
(H − Ẽr)Ψ
Ψ(+) = 0
we get for the partial components at pole energy Ẽr
After radial integration from 0 to R (here and below R
denotes the radius sufficiently large that the nuclear in-
teraction disappears) we obtain
which corresponds to a definition of the width as a decay
probability (reciprocal of the lifetime):
N = N0 exp[−t/τ ] = N0 exp[−Γt] .
The width Γ is then equal to the outgoing flux jl through
the sphere of sufficiently large radius R, divided by num-
ber of particles Nl inside the sphere.
Using Eq. (2) the flux in the asymptotic region could
be rewritten for k̃r → kr in terms of a Wronskian
= (kr/M)W (Fl(krR), Gl(krR)) = vr , (4)
where the Wronskian for real energy functions Fl, Gl is
W (Fl, Gl) = GlF
lFl ≡ 1 .
The effect of the complex energy is easy to estimate (ac-
tually without loss of a generality) in a small energy ap-
proximation
Fl(kr)
∼ Cl(kr)
l+1, Gl(kr)
(kr)−l
(2l+ 1)Cl
, (5)
where Cl is a Coulomb coefficient (defined e.g. in Ref.
[33]). The flux is then
l (k̃
l (k̃rr) − k̃
l (k̃
l (k̃rr)
2l(l+ 1)
+ l × o[Γ3]
So, the equality (4) is always valid for l = 0 and for l 6= 0
we get
l(l + 1)
B. Two-body case, real energy WF
Now we need a WF as real energy E = k2/2M solution
of Schrödinger equation
(H − E)Ψk(r) = (T + V
nuc + V coul − E)Ψk(r) = 0 ,
Ψk(r) = 4π
il(kr)−1ψl(kr)
Y ∗lm(k̂)Ylm(r̂) ,
in S-matrix representation, which means that for r > R
ψl(kr) =
[(Gl(kr) − iFl(kr)) − Sl(Gl(kr) + iFl(kr))] .
At resonance energy Er
Sl(Er) = e
2iδl(Er) = e2iπ/2 = −1
and in asymptotic region, defined by the maximal size of
nuclear interaction R,
ψl(krr)
= i Gl(krr) .
At resonance energy we can define a “quasibound” WF
ψ̃l as matching the irregular solution Gl and normalized
to unity for the integration in the internal region limited
by radius R:
ψ̃l(krr) =
(−i)ψl(krr)
|ψl(krx)|
ψl(krr)
. (6)
Now we introduce an auxiliary Hamiltonian H̄ with
different short range nuclear interaction V̄ nuc,
(H̄ − E)Φk(r) = (T + V̄
nuc + V coul − E)Φk(r) = 0 ,
and also construct other WF in S-matrix representation
Φk(r) = 4π
il(kr)−1ϕl(kr)
Y ∗lm(k̂)Ylm(r̂) ,
ϕl(kr) =
(Gl(kr)− iFl(kr)) − S̄l(Gl(kr) + iFl(kr))
for r > R. Or in equivalent form:
ϕl(kr) = exp(iδ̄l)
Fl(kr) cos(δ̄l) +Gl(kr) sin(δ̄l)
. (7)
The Hamiltonian H̄ should provide the WF Φk(r) which
at energy Er is sufficiently far from being a resonance
WF and for this WF δ̄l(Er) ∼ 0.
For real energy WFs Ψk(r) and Φk(r) we can write:
Φk(r)
† [(H − E)Ψk(r)] −
(H̄ − E)Φk(r)
Ψk(r) = 0 ,
ϕ∗l (V − V̄ )ψl =
For WFs taken at resonance energy Er this expression
provides
ϕ∗l (V − V̄ )ψldr = 2MiNl
ϕ∗l (V − V̄ )ψ̃ldr
= exp(−iδ̄l) cos(δ̄l) kr W (Fl(krR), Gl(krR)) ,(9)
1/2 =
−i exp(−iδ̄l) cos(δ̄l) kr
ϕ∗l (V − V̄ )ψ̃ldr
From Eqs. (3), (4), (6) and the approximation ψ
l ≈ ψl
it follows that
vr cos2(δ̄l)
ϕ∗l (V − V̄ )ψ̃ldr
. (10)
So, the idea of the integral method is to define the in-
ternal normalizations for the WF with resonant boundary
conditions (this is equivalent to determination of the out-
going flux for normalized “quasibound” WF) by the help
of the eigenfunction of the auxiliary Hamiltonian, which
has the same long-range behaviour and differs only in the
compact region.
III. ALTERNATIVE DERIVATION
Let us reformulate the derivation of Eq. (10) in a more
general way, so that the detailed knowledge of the WF
structure for ψl and ψ
l is not required. It would
allow a straightforward extension of the formalism to
the three-body case. We start from Schrödinger equa-
tion in continuum with solution Ψ(+) at the pole energy
Ẽr = Er + iΓ/2:
H − Ẽr
Ψ(+) =
T + V − Ẽr
Ψ(+) = 0 . (11)
Then we rewrite it identically via the auxiliary Hamilto-
nian H̄ = T + V̄
H + V̄ − V − Ẽr
Ψ(+) =
V̄ − V
H̄ − Er
Ψ(+) =
V̄ − V + iΓ/2
Ψ(+). (12)
Thus we can use the real-energy Green’s function ḠEr of
auxiliary Hamiltonian H̄ to “regenerate” the WF with
outgoing asymptotic
Ψ̄(+) = Ḡ
V̄ − V + iΓ/2
Ψ(+) . (13)
At this point in Eq. (13) Ψ̄(+) ≡ Ψ(+) and the bar in the
notation for “corrected” WF Ψ̄(+) is introduced for later
use to distinguish it from the “initial” WF Ψ(+) [the one
before application of Eq. (13)]. Further assumptions we
should consider separately in two-body and three-body
cases.
A. Two-body case
To define the width Γ by Eq. (3) we need to know
the complex-energy solution Ψ(+) at pole energy. For
narrow states Γ ≪ Er this solution can be obtained in a
simplified way using the following approximations.
(i) For narrow states we can always choose the auxil-
iary Hamiltonian in such a way that Γ ≪ V̄ −V , and we
can assume Γ → 0 in the Eq. (13).
(ii) Instead of complex-energy solution Ψ(+) in the
right-hand side of (13) we can use the normalized real-
energy quasibound solution Ψ̃ defined for one real reso-
nant value of energy Er = k
dr r2
Ψ̃lm(r)
≡ 1 .
So, the Eq. (13) is used in the form
lm = Ḡ
V̄ − V
Ψ̃lm . (14)
The solution Ψ̄(+) is matched to function
l (kr) = Gl(kr) + iFl(kr) , (15)
while the solution Ψ̃ is matched to function Gl. For deep
subbarrier energies it is reasonable to expect that in the
internal region r ≤ R
Gl ≫ Fl →
Re[Ψ̃(+)]
Im[Ψ̃(+)]
In the single channel case it can be shown by direct cal-
culation that an approximate equality
Re[Ψ̃(+)]
Im[Ψ̃(+)]
holds in the internal region and thus for narrow states
Γ ≪ Er the approximation (13) → (14) should be very
reliable.
(c) N2N1
X , lx
Y , ly
CoreCore
Y , ly
X , lx
(b)N1
r1 , l1
r2 , l2
FIG. 2: Single particle coordinate systems: (a) “V” system
typical for a shell model. In the Jacobi “T” system (b),
“diproton” and core are explicitly in configurations with def-
inite angular momenta lx and ly . For a heavy core the Jacobi
“Y” system (c) is close to the single particle system (a).
To derive Eq. (10) the WF with outgoing asymptotic
is generated using the Green’s function of the auxiliary
Hamiltonian H̄ and the “transition potential” (V − V̄ ).
The standard two-body Green’s function is
k2/(2m)
(r, r′) =
ϕl(kr)h
l (kr
′), r ≤ r′
l (kr)ϕl(kr
′), r > r′
Ylm(r̂)Y
lm(r̂
′) , (16)
where the radial WFs h
l and ϕl of the auxiliary Hamil-
tonian are defined in (15) and (7).
lm (r) =
dr′ Ḡ
k2/(2m)
(r, r′)
V̄ − V
Ψ̃l′m′(r
For the asymptotic region r > R
lm (r) =
l (krr)Ylm(r̂)
dr′ ϕl(krr
V̄ − V
ψ̃l(kr, r
The outgoing flux is then calculated [see Eq. (4)]
2l+ 1
lm (r)∇Ψ̄
As far as function Ψ̃ is normalized by construction then
Γ ≡ jl =
dr ϕl(krr)
V̄ − V
ψ̃l(kr, r)
. (17)
Note, that this equation differs from Eq. (10) only by a
factor 1/(cos2[δ̄l]) which should be very close to unity for
sufficiently high barriers.
B. Simplified model for three-body case
In papers [2, 3] the widths for three-body decays were
defined by the following procedure. We solve numerically
the problem
(H − E3r) Ψ̃ = 0
with some box boundary conditions (e.g. zero or qua-
sibound in diagonal channels at large distances) getting
the WF Ψ̃ normalized in the finite domain and the value
of the real resonant energy E3r. Thereupon we search for
the outgoing solution Ψ(+) of the equation
(H − E3r) Ψ
(+) = −iΓ/2 Ψ̃
with approximate boundary conditions of three-body
Coulomb problem (see Ref. [2] for details) and arbitrary
Γ. The width is then defined as the flux through the
hypersphere of the large radius divided by normalization
within this radius:
dΩ5 Ψ
(+)∗ρ5/2 d
ρ5/2Ψ(+)
ρ=ρmax
∫ ρmax
∣Ψ(+)
The 3-body WF with outgoing asymptotic is
JM (ρ,Ω5) = ρ
Kγ (ρ)J
Kγ (Ω5) , (19)
where the definitions of the hyperspherical variables ρ,
Ω5 and hyperspherical harmonics J
Kγ can be found in
Ref. [4].
Here we formulate the simplified three-body model in
the way which, on one hand, keeps the important dy-
namical features of the three-body decays (typical sizes
of the nuclear potentials, typical energies in the subsys-
tems, correct ratios of masses, etc.), and, on the other
hand, allows a semianalytical treatment of the problem.
Two types of approximations are made here.
The three-body Coulomb interaction is
V coul =
Z1Z2α
Z1Z3α
Y + A2X
A1+A2
Z2Z3α
Y − A1X
A1+A2
, (20)
where α is the fine structure constant. By convention, see
e.g. Fig. 2, in the “T” Jacobi system the core is particle
number 3 and in “Y” system it is particle number 2. We
assume that the above potential can be approximated by
Coulomb terms which depend on Jacobi variables X and
Y only:
V coulx (X) =
, V couly (Y ) =
(in reality for the small X and Y values the Coulomb
formfactors of the homogeneously charged sphere with
radius rsph are always used). The effective charges Zx
and Zy could be considered in two ways.
1. We can neglect one of the Coulomb interactions.
This approximation is consistent with physical situ-
ation of heavy core and treatment of two final state
interactions. Such a situation presumes that Jacobi
“Y” system is preferable and there is a symmetry
in the treatment of the X and Y coordinates, which
are close to shell-model single particle coordinates.
Zx = Z1Zcore , Zy = Z2Zcore . (21)
Further we refer this approximation as “no p-
p Coulomb” case, as typically the proton-proton
Coulomb interaction is neglected compared to
Coulomb interaction of a proton with heavy core.
2. We can also consider two particles on the X coor-
dinate as one single particle. The Coulomb inter-
action in p-p channel is thus somehow taken into
account effectively via a modification of the Zy
charge:
Zx = Z1Zcore , Zy = Z2(Zcore + Z1) . (22)
Below we call this situation as “effective p-p
Coulomb” case.
For nuclear interactions we can assume that
1. There is only one nuclear pairwise interaction and
H = T + V3(ρ) + V
x (X)
+V nucx (X) + V
y (Y ) ,
∆V (X,Y ) = V nucy (Y )− V3(ρ) . (23)
This approximation is good for methodological pur-
poses as it allows to focus on one degree of freedom
and isolate it from the others. From physical point
of view it could be reasonable if only one FSI is
strong [42], or we have reasons to think that de-
cay mechanism associated with this particular FSI
is dominating. Potential V nucy (Y ) in the auxiliary
Hamiltonian (27) is “unphysical” in that case and
can be put zero [43]. We further refer this model
as “one final state interaction” (OFSI).
2. We can consider two final state interactions (TFSI).
Simple form of the Green’s function in that case can
be preserved only if the core mass is considered as
infinite (the X and Y coordinates in the Jacobi
“Y” system coincide with single-particle core-p co-
ordinates). In that case both pairwise interactions
V nucx (X) and V
y (Y ) are treated as “physical”,
that means that they are both present in the ini-
tial and in the auxiliary Hamiltonians. Thus only
three-body potential “survive” the V̄ − V subtrac-
tion:
H = T + V3(ρ) + V
x (X) + V
x (X)
+V couly (Y ) + V
y (Y ) ,
∆V (X,Y ) = −V3(ρ) . (24)
The three-body potential is used in this work in
Woods-Saxon form
V3(ρ) = V
3 (1 + exp [(ρ− ρ0)/aρ])
, (25)
with ρ0 = 5 fm for
17Ne, ρ0 = 6 fm for
45Fe [44], and
a small value of diffuseness parameter aρ = 0.4 fm. Use
of such three-body potential is an important difference
from our previous calculations, where it was utilized in
the form
V3(ρ) = V
1 + (ρ/ρ0)
, (26)
which provides the long-range behaviour ∼ ρ−3. Such
an asymptotic in ρ variable is produced by short-range
pairwise nuclear interactions and thus the interpretation
of three-body potential (26) is phenomenological taking
into account those components of pairwise interactions
which were omitted for some reasons in calculations. In
this work the aim of the potential V3 is different. On
one hand we would like to keep the three-body energy
fixed while the properties (and number) of pairwise in-
teractions are varied. On the other hand we do not want
to change the properties of the Coulomb barriers beyond
the typical nuclear distance (this is achieved by the small
diffuseness of the potential). Thus this potential is phe-
nomenological taking into account interactions that act
only when both valence nucleons are close to the core
(both move in the mean field of the nucleus).
The auxiliary Hamiltonian is taken in the form that
allows a separate treatment of X and Y variables
H̄ = T+V coulx (X)+V
x (X)+V
y (Y )+V
y (Y ) (27)
In this formulation of the model the Coulomb potentials
are fixed as shown above. The nuclear potential V nucx (X)
[V nucy (Y ) if present] defines the position of the state in the
X [Y ] subsystem. The three-body potential V3(ρ) defines
the position of the three-body state, which is found using
the three-body HH approach of [2, 4]. After that a new
WF with outgoing asymptotic is generated by means of
the three-body Green’s function which can be written for
(27) in a factorized form (without paying attention to the
angular coupling)
(XY,X′Y′) =
(X,X′)G
(Y,Y′),
where E3r = Ex + Ey (Ex, Ex are energies of subsys-
tems). The two-body Green’s functions in the expres-
sions above are defined as in (16) via eigenfunctions of
the subhamiltonians
H̄x − Ex = Tx + V
x (X) + V
x (X)− Ex
H̄y − Ey = Ty + V
y (Y ) + V
y (Y )− Ey
In the OFSI case the nuclear potential in the “Y” sub-
system should be put V nucy (Y ) ≡ 0. The “corrected”
continuum WF Ψ̄(+) is
Ψ̄(+)(X,Y) =
dX′dY′
(X,X′)
(Y,Y′) ∆V (X ′, Y ′) Ψ(+)(X′Y′)
The “initial” solution Ψ(+) of Eq. (19) rewritten in the
coordinates X and Y is
JM (X,Y) =
ϕLlxlyS(X,Y )
[ly ⊗ lx]L ⊗ S
0 20 40 60 80 100
17Ne, "Y"-system,  core-p interaction in s-wave
  HH + FR  Kmax  →  12
  HH + FR  Kmax  →  24
  CorrectedΓ
FIG. 3: Convergence of the 17Ne width in a simplified model
in the “Y” Jacobi system. One final state interaction model
with experimental position E2r = 0.535 KeV of the s-wave
two-body resonance. Diamonds show the results of dynamic
HH calculations. Solid curves correspond to calculations with
effective FR potentials.
The asymptotic form of the ”corrected” continuum
WF Ψ̄
JM is
JM (X,Y) =
vx(ε)vy(ε)
×eikx(ε)X+iky(ε)Y
[ly ⊗ lx]L ⊗ S
Ex = εE3r ; Ey = (1− ε)E3r ; vi(ε) =
2Ei/Mi
A(ε) =
dY ′ ϕlx(kx(ε)X
′) ϕly (ky(ε)Y
×∆V (X ′, Y ′) ϕLlxlyS(X
′, Y ′) .(29)
The “corrected” outgoing flux jc can be calculated on the
sphere of the large radius for any of two Jacobi variables.
E.g. for X coordinate we have [45]
jc(E3r) = Im
Ψ̄(+)∗
Ψ̄(+)
= E23r
A∗(ε)
kx(ε)
A(ε′)
2π δ(ky(ε
′)− ky(ε)) . (30)
Values v′i above denote vi(ε
′). The flux is obtained as
jc(E3r) =
vx(ε)vy(ε)
|A(ε)|
. (31)
In principle as we have seen above that the widths ob-
tained with both fluxes Eqs. (18) and (31) should be
equal
≡ Γc =
. (32)
10 20 30 40 50 60 70 80
17Ne, "Y"-system,  core-p interaction in s-wave
HH + FR
   no nucl. potential
   E2R = 1.0 MeV
   E2R = 0.535 MeV
   E2R = 0.36 MeV
FIG. 4: Convergence of widths in OFSI model for different
positions E2r of the two-body resonance in the core-p channel
(Jacobi “Y” system). For Kmax > 24 the value of Kmax de-
note the size of the basis for Feshbach reduction toKmax = 24.
This is the idea of calibration procedure for the simplified
three-body model. The convergence of the HH method
(for WF Ψ
JM ) is expected to be fast in the internal re-
gion and much slower in the distant subbarrier region.
This should be true for the width Γ calculated in the HH
method. However, the procedure for calculation of the
“corrected” width Γc is exact under the barrier and it is
sensitive only to HH convergence in the internal region,
which is achieved easily. Below we demonstrate this in
particular calculations.
IV. DECAYS OF THE
NE 3/2− AND 45FE 3/2−
STATES IN A SIMPLIFIED MODEL
In this Section when we refer widths of 17Ne and 45Fe
we always mean the 17Ne 3/2− state (E3r = 0.344 MeV)
and the 45Fe 3/2− ground state (E3r = 1.154 MeV) cal-
culated in a very simple models. We expect that impor-
tant regularities found for these models should be true
also in realistic calculations. However, particular values
obtained in realistic models may differ significantly, and
this issue is considered specially in the Section V.
To keep only the most significant features of the sys-
tems we assume pure sd structure (lx = 0, ly = 2) for
17Ne and pure p2 structure (lx = 1, ly = 1) for
45Fe in
”Y” Jacobi system (see Fig. 2). Spin dependencies of
the interactions are neglected. The Gaussian formfactor
V nuci (r) = Vi0 exp[−(r/r0)
where i = {x, y}, is taken for 17Ne (see Table I), and a
standard Woods-Saxon formfactor is used for 45Fe (see
Table II),
V nuci (r) = Vi0 [1 + exp[(r − r0)/a]]
. (33)
The simplistic structure models can be expected to
overestimate the widths. There should be a considerable
−14 −12 −10 −8 −6 −4 −2 0
10-16
10-15
10-14
10-13
10-12
0.2 0.3 0.4 0.5 0.6 0.70.80.9 1 2 3
10-16
10-15
10-14
10-13
10-12 17
Three-body
    regime
E2r = E3r
E2r (exp.)
E2r  (MeV)
   Corrected
   Kmax = 24
   Kmax = 24 + FR
  Transition region 
E2r = (0.7−0.85) E3r
 Vx  (MeV)
FIG. 5: Width of the 17Ne 3/2− state as a function of two-
body resonance position E2r. Dashed, dotted and solid lines
show cases of pure HH calculations with Kmax = 24, the
same but with Feshbach reduction from Kmax = 100, and the
corrected width Γc. Inset shows the same, but as a function
of the potential depth parameter Vx0. Gray area shows the
transition region from three-body to two-body decay regime.
The gray curve shows simple analytical dependence of Eq.
(34).
weight of d2 component (lx = 2, ly = 2) in
17Ne and f2
component (lx = 3, ly = 3) in
45Fe. Also the spin-angular
coupling should lead to splitting of the single-particle
strength and corresponding reduction of the width es-
timates (e.g. we assume one s-wave state at 0.535 keV in
the “X” subsystem of 17Ne while in reality there are two
s-wave states in 16F: 0− at 0.535 MeV and 1− at 0.728
MeV). Thus the results of the simplified model should
most likely be regarded as upper limits for widths.
A. One final state interaction — core-p channel
First we take into account only the 0.535 MeV s-wave
two-body resonance in the 16F subsystem (this is the
experimental energy of the first state in 16F). Conver-
gence of the 17Ne width in a simplified model for Jacobi
“Y” system is shown in Fig. 3. The convergence of the
corrected width Γc as a function of Kmax is very fast:
Kmax > 8 for the width is stable within ∼ 1%. For maxi-
mal achieved in the fully dynamic calculation Kmax = 24
the three-body width Γ is calculated within 30% preci-
sion. Further increase of the effective basis size is possi-
ble within the adiabatic procedure based on the so called
Feschbach reduction (FR).
Feschbach reduction is a procedure, which eliminates
from the total WF Ψ = Ψp +Ψq an arbitrary subspace q
using the Green’s function of this subspace:
Hp = Tp + Vp + VpqGqVpq
In a certain adiabatic approximation we can assume that
the radial part of kinetic energy is small and constant un-
0.0 0.2 0.4 0.6 0.8 1.0
"Y"-system
 (p-core)-p
  Kmax = 6
  Kmax = 10
  Kmax = 14
  Kmax = 24
  Kmax = 100
  corrected
ε =Ex / E3r
FIG. 6: Convergence of energy distribution for 17Ne in the
“Y” Jacobi system.
der the centrifugal barrier in the channels with so high
centrifugal barrier that it is much higher than any other
interaction. In this approximation the reduction proce-
dure becomes trivial as it is reduced to construction of
effective three-body interactions V effKγ,K′γ′ by matrix op-
TABLE I: Parameters for 17Ne calculations. Potential pa-
rameters for 15O+p channel in s-wave (Vx0 in MeV, r0 = 3.53
fm) and 16F+p channel in d-wave (Vy0 in MeV). Radius of the
charged sphere is rsph = 3.904 fm. Widths Γi of the state in
the subsystem and experimental width values Γexp for really
existing at these energies states are given in keV. The cor-
rected three-body width Γc is given in the units 10
−14 MeV.
TFSI calculations with d-wave state at 1.2 MeV are made
with s-wave state at 0.728 MeV.
E2r lx (ly) Vx0 (Vy0) Γx (Γy) Γexp Γc
0.258 0 −14.4 0.221 144
0.275 0 −14.35 0.355 16.6
0.292 0 −14.3 0.544 7.75
0.360 0 −14.1 2.09 2.34
0.535 0 −13.55 17.9 25(5) [34] 0.545
0.728 0 −12.89 72.0 70(5) [34] 0.211
1.0 0 −12.0 252 0.093
2.0 0 −9.0 ∼ 1500 0.021
0.96 2 −87.06 3.5 6(3) [34] 4.73a
1.256 2 −85.98 12.2 < 15 [35] 2.0a
0.96 2 −66.46 3.6 6(3) [34] 1.37b
1.256 2 −65.4 13.7 < 15 [35] 0.584b
aThis is TFSI calculation with “no p-p” Coulomb, r0 = 2.75 fm.
bThis is TFSI calculation with “effective” Coulomb, r0 = 3.2 fm.
0.0 0.2 0.4 0.6 0.8 1.0
94.4%
68.0%
   no nucl. potential
   E2r = 1.0 MeV
   E2r = 0.535 MeV
   E2r = 0.360 MeV
   E2r = 0.284 MeV
   E2r = 0.266 MeV
   E2r = 0.249 MeV
ε =Ex / E3r
FIG. 7: Energy distributions for 17Ne in the “Y” Jacobi
system for different two-body resonance positions E2r. The
three-body decay energy is E3r = 0.344 MeV. The distri-
butions are normalized to have unity value on maximum of
three-body components. The values near the peaks show the
fraction of the total intensity concentrated within the peak.
Note the change of the scale at vertical axis.
erations
G−1Kγ,K′γ′ = (H − E)Kγ,K′γ′ = VKγ,K′γ′
Ef − E +
(K + 3/2)(K + 5/2)
δKγ,K′γ′ ,
V effKγ,K′γ′ = VKγ,K′γ′ +
VKγ,K̄γ̄GK̄γ̄,K̄′γ̄′VK̄′γ̄′,K′γ′ .
Summation over indexes with bar is made for eliminated
channels. No strong sensitivity to the exact value of the
“Feshbach energy” Ef is found and we take it as Ef ≡ E
in our calculations. More detailed account of the pro-
cedure applied within HH method can be found in Ref.
[36].
It can be seen in Fig. 3 (solid line) that Feschbach re-
duction procedure drastically improves the convergence.
However, the calculation converges to a width value,
which is somewhat smaller than the corrected width value
(that should be exact). The reason for this effect can
be understood if we make a reduction to a smaller “dy-
namic” basis size (Kmax = 12, gray line). The calculation
in this case also converges, but even to a smaller width
value. We can conclude that FR procedure allows any-
how to approach the real width value, but provides a
good result only for sufficiently large size of the dynamic
sector of the basis.
The next issue to be discussed is a convergence of the
width in calculations with different positions E2r of two-
body resonance in the core+p subsystem. It is demon-
strated for several energies E2r in Fig. 4. When the
resonance in the subsystem is absent (or located rela-
tively high) the convergence of the width value to the
exact result is very fast both in the pure three-body
and in the “corrected” calculation (in that case, how-
ever, much faster). Here even FR is not required as the
0 20 40 60 80 100 120 140
HH + FRHH
Gaussian potential in s-wave
   OFSI           TFSI   
Potential with repulsive core in s-wave
   OFSI           TFSI
17Ne, "Y"-system 
FIG. 8: Convergence of the 17Ne width in a simplified model.
Jacobi “Y” system. OFSI model with s-wave two-body reso-
nance at E2r = 0.535 MeV; Gaussian potential and potential
with repulsive core. TFSI model with d-wave two-body reso-
nance at E2r = 0.96 MeV.
convergent result is achieved in the HH calculations by
Kmax = 10 − 24. The closer two-body resonance ap-
proaches the decay window, the worse is convergence of
HH calculations. At energy E2r = 360 keV (which is al-
ready close to three-body decay window E3r = 344 keV)
even FR procedure provides a convergence to the width
value which is only about 65% of the exact value.
In Fig. 5 the calculations with different E2r values are
summarized. The width grows rapidly as the two-body
resonance moves closer to the decay window. The pen-
etrability enhancement provided by the two-body reso-
nance even before it moves into the three-body decay
window is very important. Difference of widths with no
core-p FSI and FSI providing the s-wave resonance to
be at it experimental position E2r = 0.535 MeV is more
than two orders of the magnitude. The convergence of
HH calculations also deteriorates as E2r moves closer to
the decay window. However, the disagreement between
the HH width and the exact value is within the order
of the magnitude, until the resonance achieves the range
E2r ∼ (0.7 − 0.85)E3r. Within this range a transition
from three-body to two-body regime happens (see also
discussion in [8]), which can be seen as a drastic change
of the width dependence on E2r. This means that a se-
quential decay via two-body resonanceE2r becomes more
efficient than the three-body decay. In that case the hy-
perspherical expansion can not treat the dynamics effi-
ciently any more and the disagreement with exact result
becomes as large as orders of the magnitude. The de-
cay dynamics in the transition region is also discussed in
details below.
It can be seen in Fig. 5 that in three-body regime the
dependence of the three-body width follows well the an-
alytical expression
Γ ∼ (E3r/2− E2r)
−2 (34)
The reasons of such a behaviour will be clarified in
0 20 40 60 80 100 120 140
  HH + FR  Kmax  →  24
  Corrected
17Ne, "T"-system,  p-p interaction only
FIG. 9: Convergence of the 17Ne width in a simplified model.
Jacobi “T” system. Final state interaction describes s-wave
p-p scattering.
the forthcoming paper [37]. The deviations from this de-
pendence can be found in the decay window (close to
“transition regime”) and at higher energies. This depen-
dence is quite universal; e.g. for 45Fe it is demonstrated
in Fig. 14, where it follows the calculation results even
with higher precision.
Another important issue is a convergence of energy dis-
tributions in the HH calculations, demonstrated in Fig.
6 for calculations with E2r = 535 keV. The distribution
is calculated in “Y” Jacobi subsystem, thus Ex is the en-
ergy between the core and one proton. The energy dis-
tribution convergence is fast: the distribution is stable at
Kmax = 10− 14 and does not change visibly with further
increase of the basis. There remain a visible disagree-
ment with exact (”corrected”) results, which give more
narrow energy distribution. We think that this effect was
understood in our work [4]. The three-body calculations
are typically done for ρmax ∼ 500−2000 fm (ρmax ∼ 1000
fm everywhere in this work). It was demonstrated in Ref.
[4] by construction of classical trajectories that we should
expect a complete stabilization of the energy distribution
in core+p subsystem at ρmax ∼ 30000−50000 fm and the
effect on the width of the energy distribution should be
comparable to one observed in Fig. 6.
The evolution of the energy distribution in core+p sub-
system with variation of E2r is shown in Fig. 7. When
we decrease the energy E2r the distribution is very stable
until the two-body resonance enters the three-body de-
cay energy window. After that the peak at about ε ∼ 0.5
first drifts to higher energy and then for E2r ∼ 0.85E3r
the noticeable second narrow peak for sequential decay
is formed. At E2r ∼ 0.7E3r the sequential peak becomes
so high that the three-body component of the spectrum
is practically disappeared in the background.
The result concerning the transition region obtained
in this model is consistent with conclusion of the paper
[8] (where much simpler model was used for estimates).
The three-body decay is a dominating decay mode, not
0.0 0.2 0.4 0.6 0.8 1.0
   Kmax = 10
   Kmax = 100
   corrected  (p-p FSI)
   corrected  (no p-p FSI)
ε =Ex / E3r
"T"-system   (p-p)-core
FIG. 10: Energy distributions for 17Ne in “T” Jacobi system
(between two protons).
only when the sequential decay is energy prohibited as
E2r > E3r. Also the three-body approach is valid when
the sequential decay is formally allowed (because E2r <
E3r) but is not taking place in reality due to Coulomb
suppression at E2r >∼ 0.8E3r.
Geometric characters of potentials can play an impor-
tant role in the width convergence. To test this aspect
of the convergence we have also made the calculations
for potential with repulsive core. This class of potentials
was employed in studies of 17Ne and 19Mg in Ref. [7]. A
comparison of the convergence of HH calculations with s-
wave 15O+p potential from [7] and Gaussian potential is
given in Fig. 8. The width convergence in the case of the
“complicated” potential with a repulsive core is drasti-
cally worse than in the “easy” case of Gaussian potential.
For typical dynamic calculations withKmax = 20−24 the
HH calculations provide only 20 − 25% of the width for
potential with a repulsive core. On the other hand the
calculations with both potentials provide practically the
same widths Γc [46] and FR provides practically the same
and very well converged result in both cases.
B. One final state interaction — p-p channel
As far as two-proton decay is often interpreted as
“diproton” decay we should also consider this case and
study how important this channel could be. For this cal-
culation we use a simple s-wave Gaussian p-p potential,
providing a good low-energy p-p phase shifts,
V (r) = −31 exp[−(r/1.8)2] . (35)
Calculations with this potential are shown in Fig. 9 (see
also Table V). First of all the penetrability enhancement
provided by p-p FSI is much less than the enhancement
provided by core-p FSI (the widths differs more than two
orders of the magnitude, see Fig. 3). This is the feature,
which has been already outlined in our works. The p-p in-
teraction may boost the penetrability strongly, but only
0 4 8 12 16 20
ρ0 /√2   (fm)
ρ0 = 6 fm
0 2 4 6 8
Ypeak  (fm)
FIG. 11: Comparison of the OFSI calculations (solid lines) for
45Fe in the “T” system with diproton model Eq. (36) (dashed
lines). Effective equivalent channel radius rch(dp) for “dipro-
ton emission” (a) as a function of radius ρ0 of the three-body
potential (25), the value ρ0/
2 should be comparable with
typical nuclear sizes. (b) as a function of the position of the
peak Ypeak in the three-body WF Ψ
(+) in Y coordinate. The
dashe lines are given to guide the eye.
in the situation, when protons occupy predominantly or-
bitals with high orbital momenta. In such a situation the
p-p interaction allows transitions to configurations with
smaller orbital momenta in the subbarrier region, which
provide a large increase of the penetrability. In our sim-
ple model for 17Ne 3/2− state, we have already assumed
the population of orbitals with minimal possible angular
momenta and thus no strong effect of the p-p interaction
is expected.
Also a very slow convergence of the decay width should
be noted in this case. For core-p interaction the Kmax ∼
10 − 40 were sufficient to obtain a reasonable result. In
the case of the p-p interaction theKmax ∼ 100 is required.
Energy distributions between two protons obtained in
this model are shown in Fig. 10. Important feature of
these distributions is a strong focusing of protons at small
p-p energies. This feature is connected, however, not
with attractive p-p FSI, but with dominating Coulomb
repulsion in the core-p channel. This is demonstrated by
the calculation with nuclear FSI turned off, which pro-
vides practically the same energy distributions. Similarly
to the case of the core-p FSI, very small Kmax > 10 is
sufficient to provide the converged energy distribution.
The converged HH distribution is very close to the exact
(”corrected”) one but it is, again, somewhat broader.
So far the diproton model has been treated by us as a
reliable upper limit for three-body width [8]. With some
technical improvements this model was used for the two-
proton widths calculations in Refs. [16, 17, 18, 19]. It
is important therefore to try to understand qualitatively
the reason of the small width values obtained in this form
of OFSI model, which evidently represents appropriately
formulated diproton model [47]. In Fig. 11 we compared
the results of the OFSI calculations for 45Fe in the “T”
0 20 40 60 80 100
17Ne, "Y"-system, core-p interactions 
                    in s- and d-waves
  HH + FR  Kmax  →  24
  Corrected
FIG. 12: Convergence of the 17Ne width for experimental
positions E2r = 0.535 MeV of the 0
+ two-body resonance in
the “X” subsystem and E2r = 0.96 MeV of the 2
+ two-body
resonance in the “Y” subsystem (TFSI model).
system with diproton width estimated by expression
Γdp =
Mredr
ch(dp)
Pl=0(0.95E3r, rch(dp), 2Zcore) ,
where Mred is the reduced mass for
43Cr-pp motion and
rch(dp) is channel radius for diproton emission. The en-
ergy for the relative 43Cr-ppmotion is taken 0.95E3r bas-
ing on the energy distribution in the p-p channel (see
Fig. 10 for example). In Fig. 11a we show the effective
equivalent channel radii for diproton emission obtained
by fulfilling condition Γdp ≡ Γc for OFSI model calcula-
tions with different radii ρ0 of the three-body potential
Eq. (25). It is easy to see that for realistic values of these
radii (ρ0 ∼ 6 fm for
45Fe) the equivalent diproton model
radii should be very small (∼ 1.5 fm). This happens pre-
sumably because the “diproton” is too large to be con-
sidered as emitted from nuclear surface of such small ρ0
radius. Technically it can be seen as the nonlinearity of
the rch(dp)-ρ0 dependence, with linear region achieved at
ρ0 ∼ 15−20 fm. Only at such unrealistically large ρ0 val-
ues the typical nuclear radius (when it becomes compa-
rable with the “size” of the diproton) can be reasonably
interpreted as the surface, off which the “diproton” is
emitted. It is interesting to note that in the nonlinearity
region for Fig. 11a there exists practically exact corre-
spondence between the Y coordinate of the WF peak in
the internal region and the channel radius for diproton
emission (Fig. 11b). This fact is reasonable to interpret
in such a way that the diproton is actually emitted not
from nuclear surface (as it is presumed by the existing
systematics of diproton calculations) but from the inte-
rior region, where the WF is mostly concentrated.
0 20 40 60 80
45Fe, "Y"-system, core-p interactions 
                           in p-waves
  HH + FR  Kmax  →  24
  Corrected
FIG. 13: Convergence of the 45Fe width for position of the 1−
two-body resonances in “X” and “Y” subsystems E2r = 1.48
C. Two final state interactions
As we have already mentioned the situation of one final
state interaction is comfortable for studies, but rarely re-
alized in practice. An exception is the case of the E1 tran-
sitions to continuum in the three-body systems, consid-
ered in our previous work [23]. For narrow states in typ-
ical nuclear system of the interest there are at least two
comparable final state interactions (in the core-p chan-
nel). For systems with heavy core this situation can be
treated reasonably well as the Y coordinate (in “Y” Ja-
cobi system) for such systems practically coincides with
the core-p coordinate. Below we treat in this way 17Ne
(for which this approximation could be not very consis-
tent) and 45Fe (for which this approximation should be
good). In the case of 17Ne we are thus interested in the
scale of the effect, rather in the precise width value.
For calculations with two FSI for 17Ne we used Gaus-
sian d-wave potential (see Table I), in addition to the
s-wave potential used in Section IVA. This potential
provides a d-wave state at 0.96 MeV (Γ = 13.5 keV),
which corresponds to the experimental position of the
first d-wave state in 16F. The convergence of the 17Ne
decay width is shown in Fig. 12. Comparing with Fig.
3 one can see that the absolute value of the width has
changed significantly (2−3 times) but not extremely and
the convergence is practically the same. Interesting new
feature is a kind of the convergence curve “staggering”
for odd and even values of K/2. Also the convergence
of the corrected calculations requires now a considerable
Kmax ∼ 12− 14.
The improved experimental data for 2p decay of 45Fe
is published recently in Ref. [12]: E3r = 1.154(16) MeV,
Γ2p = 2.85
+0.65
−0.68× 10
−19 MeV [T1/2(2p) = 1.6
−0.3 ms] for
two-proton branching ratio Br(2p) = 0.57. Below we use
the resonance energy from this work.
The convergence of the 45Fe width is shown in Fig. 13.
The character of this convergence is very similar to that
1 2 3
10-19
10-18
E2r   from Refs. [4,8]
E2r = E3r
Three-body
    regime
n    Corrected
   Kmax = 24
   Kmax = 24 + FR
E2r  (MeV)
FIG. 14: The 45Fe g.s. width as a function of the two-body
resonance position E2r. Dashed, dotted and solid lines show
cases of a pure HH calculation with Kmax = 24, the same but
with Feshbach reduction from Kmax = 100, and the corrected
width Γc. Gray area shows the transition region from three-
body to two-body decay regime. The gray curve shows simple
analytical dependence of Eq. (34).
in the 17Ne case, except the “staggering” feature is more
expressed.
The dependence of the 45Fe width on the two-body
resonance energy E2r is shown in Fig. 14. Potential
parameters for these 45Fe calculations are given in Ta-
ble II. The result calculated for E3r = 1.154 MeV and
E2r = 1.48 MeV in paper [4] for pure [p
2] configuration
is Γ = 2.85× 10−19 MeV. The value Kmax = 20 was used
in these calculations. If we take the HH width value from
Fig. 13 at Kmax = 20 it provides Γ = 2.62× 10
−19 MeV,
which is in a good agreement with a full HH three-body
model of Ref. [4]. However, from Fig. 13 we can conclude
that in the calculations of [4] the width was about 35%
underestimated. Thus the value of about Γ = 6.3×10−19
MeV should be expected in these calculations. On the
other hand much larger uncertainty could be inferred
from Fig. 14 due to uncertain energy of the 44Mn ground
state. If we assume a variation E2r = 1.1− 1.6 MeV the
TABLE II: Parameters for 45Fe calculations. Potential pa-
rameters for p-wave interactions (33) in 43Cr+p channel (Vx0
in MeV, r0 = 4.236 fm, rsph = 5.486 fm) and
44Mn+p (Vy0 in
MeV, r0 = 4.268 fm, rsph = 5.527 fm), a = 0.65 fm. Calcula-
tions are made with “effective Coulomb” of Eq. (22). Widths
Γx, Γy of the states in the subsystems are given in keV. Cor-
rected three-body widths are given in the units 10−19 MeV.
E2r Vx0 Γx Vy0 Γy Γc
1.0 −24.350 4.3× 10−3 −24.54 2.1 × 10−3 26.5
1.2 −24.03 0.032 −24.224 0.018 11.8
1.48 −23.58 0.26 −23.78 0.15 5.6
2.0 −22.7 3.6 −22.93 2.3 2.3
3.0 −20.93 58 −21.19 44 0.84
10 100
HH + FR
   TFSI
   Three-body realistic
17Ne 
FIG. 15: Interpolation of 17Ne decay width obtained in full
three-body calculations by means of TFSI convergence curves
(see Fig. 8). Upper curves correspond to TFSI case with
Gaussian potential in s-wave and compatible S1 case for full
three-body model. Lower curves correspond to TFSI case
with repulsive core potential in s-wave and compatible GMZ
case for full three-body model.
inferred from Fig. 14 uncertainty of the width would be
Γ = (4 − 16)× 10−19 MeV. On top of that we expect a
strong p2/f2 configuration mixing which could easily re-
duce the width within an order of the magnitude. Thus
we can conclude that a better knowledge about spectrum
of 44Mn and a reliable structure information about 45Fe
are still required to make sufficiently precise calculations
of the 45Fe width. More detailed account of these issues
is provided below.
V. THREE-BODY CALCULATIONS
Having in mind the experience of the convergence stud-
ies we have performed large-basis calculations for 45Fe
and 17Ne. They are made with dynamicalKmax = 16−18
(including Fechbach reduction from Kmax = 30− 40) for
17Ne and Kmax = 22 (FR from Kmax = 40) for
45Fe.
The calculated width values are extrapolated using the
convergence curves obtained in TFSI model (Figs. 15) for
17Ne and 13for 45Fe). We have no proof that the width
convergence in the realistic three-body case is asolutely
the same as in the TFSI case. However, the TFSI model
takes into account main dynamic features of the system
causing a slow convergence, and we are expecting that
the convergence should be nearly the same in both cases.
A. Widths and correlations in
The potentials used in the realistic calculations are the
same as used for 17Ne studies in Refs. [7, 38]. The GPT
potential [39] is used in the p-p channel. The core-p
potentials are referred in [38] as “GMZ” (potential in-
FIG. 16: Correlations for 17Ne decay in “T” and “Y” Jacobi systems. Three-body calculations with realistic (GMZ) potential.
troduced in [7]) and “high s” (with centroid of d-wave
states is shifted upward which is providing a higher con-
tent of s2 components in the 17Ne g.s. WF). Both poten-
tials provide correct low-lying spectrum of 16F and differ‘
only for d-wave continuum above 3 MeV (see Table III).
The core-p nuclear potentials, including central, ss and
ls terms, are taken as
V (r) =
V lc + (s1 · s2)V
1 + exp[(r − rl0)/a]
− (l · s)
2.0153V lls
× exp[(r − rl0)/a]
1 + exp[(r − rl0)/a]
,(37)
with parameters: a = 0.65 fm, r00 = 3.014 fm, r
2.94 fm, V 0c = −26.381 MeV, V
c = −9 MeV, V
−57.6 (−51.48) MeV, V 3c = −9 MeV, V
ss = 0.885 MeV,
V 2ss = 4.5 (12.66) MeV, Vls = 4.4 (13.5) MeV (the values
in brackets are for “high s” case). There are also repulsive
cores for s- and p-waves described by a = 0.4 fm, r00 =
0.89 fm, Vcore = 200 MeV. These potentials are used
together with Coulomb potential obtained for Gaussian
charge distribution reproducing the charge radius of 15O.
To have extra confidence in the results, the width of
the 17Ne 3/2− state is calculated in several models of
growing complexity (Tables IV-VI). One can see from
those Tables that improvements introduced on each step
provide quite smooth transition from the very simple to
the most sophisticated model.
In Table IV we demonstrate how the calculations in the
simplified model of Section IV are compared with calcu-
lations of the full three-body model with appropriately
truncated Hamiltonian. We can switch off correspond-
ing interactions in the full model to make it consistent
with approximations of the simplified model. To remind,
the differences of the full model and simplified model are
the following: (i) antisymmetrization between protons
is missing in the simplified model and (ii) Y coordinate
is only approximately equal to the coordinate between
core and second proton. Despite these approximations
the models demonstrate very close results: the worst dis-
agreement is not more than 30%.
In Table V we compare approximations of a different
kind: those connected with choice of the Jacobi coordi-
nate system in the simplified model. First we compare
the “pure Coulomb” case: all pairwise nuclear interac-
tions are off and the existence of the resonance is provided
solely by the three-body potential (25). This model pro-
vides some hint what should be the width of the system
without nuclear pairwise interactions. Then the models
are compared with the nuclear FSIs added. The addition
of nuclear FSI drastically increase width in all cases. It
is the most “efficient” (in the sense of width increase) in
the case of TFSI model in the “Y” system. Choice of
this model provides the largest widths and can be used
for the upper limit estimates.
In Table VI full three-body models are compared. The
simplistic S1 and S2 interactions correspond to calcula-
tions with simplified spectra of the 16F subsystem. For S1
case it includes one s-wave state at 0.535 MeV (Γ = 18.8
keV) and one d-wave state at 0.96 MeV (Γ = 3.5 keV).
These are two lower s- and d-wave states known experi-
mentally. In the S2 case we use instead the experimental
positions of the higher component of the s- and d-wave
doublets: s-wave at 0.72 MeV (Γ = 73.4 keV) and d-
TABLE III: Low-lying states of 16F obtained in the “GMZ”
and “high s” core-p potentials. The potential is diagonal in
the representation with definite total spin of core and proton
S, which is given in the third column.
Case GMZ high s Exp.
Jπ l S E2r (MeV) Γ (keV) E2r (MeV) Γ (keV) Γ (keV)
0− 0 0 0.535 18.8 0.535 18.8 25(5) [34]
1− 0 1 0.728 73.4 0.728 73.4 70(5) [34]
2− 2 0 0.96 3.5 0.96 3.5 6(3) [34]
3− 2 1 1.2 9.9 1.2 10.5 < 15 [35]
2− 2 1 3.2 430 7.6 ∼ 3000
1− 2 1 4.6 1350 ∼ 15 ∼ 6000
FIG. 17: Correlations for 17Ne decay in “T” and “Y” Jacobi systems. Three-body calculations with Coulomb FSIs only (all
nuclear pairwise potentials are turned off).
wave at 1.2 MeV (Γ = 10 keV). Parameters of the core-p
potentials can be found in Table I. Simple Gaussian p-
p potential (35) is used. The variation of the results
between these models is moderate (∼ 30%). The calcu-
lations with GMZ potential provide the width for 17Ne
3/2− state which comfortably rests in between the re-
sults obtained in the simplified S1 and S2 models. The
structure of the WF is also obtained quite close to these
calculations. The structure in the “high s” case is ob-
tained with a strong domination of the sd component.
The width in the “high s” case is obtained somewhat
larger (∼ 11%) than in GMZ case, but this increase is
consistent with the increase of the sd WF component,
(∼ 15%) which is expected to be more preferable for de-
cay than d2 component.
It is important for us that the results obtained in the
three-body models with considerably varying spectra of
the two-body subsystems and different convergence sys-
tematics appear to be quite close: Γ ∼ (5 − 8) × 10−15
MeV. Thus we have not found a factor which could lead
to a considerable variation of the three-body width, given
the ingredients of the model are reasonably realistic.
The decomposition of the 17Ne WF obtained with
GMZ potential is provided in Table VII in terms of par-
tial internal normalizations and partial widths. The cor-
respondence between the components with large weights
and large partial widths is typically good. However, there
are several components giving large contribution to the
width in spite of negligible presence in the interior.
Complete correlation information for three-body de-
cay of a resonant state can be described by two variables
(with omission of spin degrees of freedom). We use the
energy distribution parameter ε = Ex/E3r and the angle
cos(θk) = (kxky)/(kxky) between the Jacobi momenta.
The complete correlation information is provided in Fig.
16 for realistic 17Ne 3/2− decay calculations. We can see
that the profile of the energy distribution is characterized
by formation of the double-hump structure, expected so
far for p2 configurations (see, e.g. [4]). This structure
can be seen both in “T” system (in energy distribution)
and in “Y” system (in angular distribution). In the cal-
culations of ground states of the s-d shell nuclei we were
getting such distributions to be quite smooth. It can be
found that the profile of this distribution is defined by
the sd/d2 components ratio. For example in the calcula-
tions with “high s” potential the total domination of the
sd configuration leads to washing out of the double-hump
profile.
The correlations in the 17Ne (shown in Fig. 16) are
strongly influenced by the nuclear FSIs. Calculations for
only Coulomb pairwise FSIs left in the Hamiltonan are
TABLE IV: Comparison of widths for 17Ne (in 10−14 MeV
units) obtained in simplified model in “Y” Jacobi system
and in full three-body model with correspondingly truncated
Hamiltonian. Structure information is provided for the three-
body model. In the simplified model the weight of the [sd]
configuration is 100% by construction. “No p-p” column
shows the case where Coulomb interaction in p-p channel is
switched off (see, (21)). “Eff.” column corresponds to the ef-
fective treatment (see, (22)) of Coulomb interaction in the p-p
channel in the simplified model, but to the exact treatment
in full three-body model.
pure Coulomb OFSI TFSI
“no p-p” Eff. “no p-p” Eff. “no p-p” Eff.
Simpl. 0.017 0.0032 3.02 0.545 4.70 1.37
3-body 0.024a 0.0041a 3.22 0.555 3.91 0.445
[sd] 99.8 99.3 99.6 99.5 92.0 72.6
[p2] 0.2 0.6 0.3 0.4 0.1 0.2
[d2] 0 0 0 0 7.8 27.1
aSmall repulsion (∼ 0.5 MeV) was added in that case in the p-
wave core-p channel to split the states with sd and p2 structure
which appear practically degenerated and strongly mixed in this
model.
FIG. 18: Correlations for 17Ne decay calculated in simplified OFSI model in “T” (only p-p FSI) and in “Y” Jacobi systems
(only s-wave core-p FSI).
shown in Fig. 17. The strong peak at small p-p energy
is largely dissolved and the most prominent feature of
the correlation density in that case is a rise of the distri-
bution for cos(θk) → 1 in the “Y” Jacobi system. This
kinematical region corresponds to motion of protons in
the opposite directions from the core and is qualitatively
understandable feature of the three-body Coulomb in-
teraction (the p-p Coulomb interaction is minimal along
such a trajectory).
The distributions calculated in the simplified (OFSI)
model are shown in Fig. 18 on the same {ε, cos(θk)} plane
as in Figs. 16 and 17. It should be noted that here the
calculations in “T” and “Y” Jacobi systems represent
different calculations (with p-p FSI only and with core-p
FSI only). In Figs. 16 and 17 two panels show differ-
ent representations of the same result. Providing rea-
sonable (within factor 2 − 4) approximation to the full
three-body model in the sense of the decay width, the
simplified model is very deficient in the sense of correla-
TABLE V: Comparison of widths calculated for 17Ne (10−14
MeV units) and 45Fe (10−19 MeV units) with pure Coulomb
FSIs and for nuclear plus Coulomb FSIs. Simplified OFSI
model in “T”, TFSI in “Y” Jacobi systems (“effective”
Coulomb is used in both cases) and full three-body calcu-
lations.
pure Coulomb Nuclear+Coulomb
“T” “Y” 3-body “T” “Y” 3-body
17Ne 0.0011 0.0032 0.0041 0.0077 1.37 0.76a
[sd] 100 100 99.3 100 100 73.1
[p2] 0 0 0.6 0 0 1.8
[d2] 0 0 0 0 0 24.2
45Fe 0.0053 0.0167 0.26 0.034 4.94 6.3b
aThis is a calculation with S1 Hamiltonian.
bThis is a calculation providing pure p2 structure.
tions. The only feature of the realistic correlations which
is even qualitatively correctly described in the simplified
model is the energy distribution in the “Y” system. The
“diproton” model (OFSI model with p-p interaction) fails
especially strongly, which is certainly relevant to the very
small width provided by this calculation.
B. Width of
The calculation strategy is the same as in [4]. We start
with interactions in the core-p channel which give a res-
onance in p-wave at fixed energy E2r. Such a calculation
provides 45Fe with practically pure p2 structure. Then
we gradually increase the interaction in the f -wave, until
it replaces the p-wave resonance at fixed E2r and then
we gradually move the p-wave resonance to high energy.
Thus we generate a set of WFs with different p2/f2 mix-
ing ratios.
The results of the improved calculations with the same
settings as in [4] (the 44Mn g.s. is fixed to have E2r = 1.48
MeV) are shown in Fig. 19 (see also Table V) together
with updated experimental data [12]. The basis size used
in [4] was sufficient to provide stable correlation pictures
(as we have found in this work) and they are not updated.
TABLE VI: Width (in 10−14 MeV units) and structure of
17Ne 3/2− state calculated in a full three-body model with
different three-body Hamiltonians.
S1 S2 GMZ high s
Kmax = 18 0.35 0.27 0.14 0.16
Extrapolated 0.76 0.56 0.69 0.76
[sd] 73.1 71.7 80.2 95.1
[p2] 1.8 1.8 2.0 1.3
[d2] 24.2 25.7 16.8 3.1
The sensitivity of the obtained results to the experi-
mentally unknown energy of 44Mn can be easily studied
by means of Eq. (34). The results are shown in Fig. 20 in
terms of the regions consistent with experimental data
on the {E2r,W (p
2)} plane [W (p2) is the weight of p2
configuration in 45Fe WF]. It is evident from this plot
that our current experimental knowledge is not sufficient
to draw definite conclusions. However, it is also clear
that with increased precision of the lifetime and energy
measurements for 45Fe and the appearance of more de-
tailed information on 44Mn subsystem the restrictions on
the theoretical models should become strong enough to
provide the important structure information.
VI. DISCUSSION
General trends of the model calculations can be well
understood from Tables IV-VI. For the pure Coulomb
case the simplified model calculations (in the “Y” and
“T” systems) and three-body calculations provide rea-
sonably consistent results. The simplified calculations in
the “Y” system always give larger widths than those in
the “T” system. From decay dynamics point of view this
leads to understanding of the contradictory fact that the
sequential decay path is preferable even if no even virtual
sequential decay is possible (as the nuclear interactions
are totally absent in this case).
The calculations with attractive nuclear FSIs rather
TABLE VII: Partial widths ΓKγ of different components of
17Ne 3/2− WF calculated in “T” Jacobi systems. Partial
weights are given in “T” (valueN
) and in “Y” (valueN
Jacobi systems. Sx is the total spin of two protons.
K L lx ly Sx N
2 2 0 2 0 23.88 33.87 44.93
2 2 2 0 0 24.97 16.52 13.29
2 2 1 1 1 0.28 7.39 3.59
2 2 1 1 0 1.54
2 2 0 2 1 3.68
2 2 2 0 1 3.68
4 2 0 2 0 8.97 20.04 3.19
4 2 2 0 0 8.68 13.57 5.57
4 2 2 2 0 15.49 0.32 18.80
4 2 1 3 1 0.03 2.18 0.95
4 2 3 1 1 0 1.89 0.63
4 1 2 2 1 1.02
4 2 0 2 1 1.99
4 2 2 0 1 2.07
6 2 2 4 0 0.14 0.77 3.57
6 2 4 2 0 0.14 0.77 0.78
6 2 0 2 0 0.50 0.09 0.69
8 2 4 4 0 0.02 0.003 1.58
1.0 1.1 1.2 1.3
10−20
10−19
10−18
 f  2  
26Fe 
E3r  (MeV)
4.56 10-21
   MeV
E2r = 1.48 MeV
FIG. 19: The lifetime of 45Fe as a function of the 2p decay
energy E3r. The plot is analogue of Fig. 6a from [4] with up-
dated experimental data [12] and improved theoretical results.
Solid curves shows the cases of practically pure p2 and f2
configurations, dashed curves stand for different mixed p2/f2
cases. The numerical labels on the curves show the weights
of the s2 and p2 configurations in percents.
expectedly provide larger widths than the corresponding
calculations with Coulomb interaction only. The core-
proton FSI is much more efficient for width enhancement
than p-p FSI. This fact is correlated with the observation
of the previous point and is a very simple and strong indi-
cation that the wide-spread perception of the two-proton
decay as “diproton” decay is to some extent misleading.
As it has already been mentioned the p-p FSI influences
the penetration strongly in the very special case when the
decay occurs from high-l orbitals (e.g. f2 in the case of
45Fe). Thus we should consider as not fully consistent the
attempts to explain two-proton decay results only by the
FSI in the p-p channel (e.g. Ref. [19]) as much stronger
decay mechanism is neglected in these studies.
From techical point of view the states considered in
this work belong to the most complicated cases. The
complication is due to the ratio between the decay energy
and the strength of the Coulomb interaction (it defines
the subbarrier penetration range to be considered dy-
namically). Thus the convergence effects demonstrated
in this work for 17Ne have the strongest character among
the systems studied in our previous works [4, 6, 7, 8].
Because of the relatively small Kmax = 12 used in the
previous works we have found an order of the magnitude
underestimation of the 17Ne(3/2−) width. For systems
like 48Ni — 66Kr the underestimation of widths in our
previous calculations is expected to be about factor of 2.
A much smaller effect is expected for lighter systems.
It was demonstrated in [22, 23] that the capture rate
for the 15O(2p,γ)17Ne reaction depends strongly on the
two-proton width of the first excited 3/2− state in 17Ne.
This width was calculated in Ref. [7] as 4.1× 10−16 MeV
(some confusion can be connected with misprint in Table
1.0 1.2 1.4 1.6 1.8 2.0
E2r  (MeV)
FIG. 20: Compatibility of the measured width of the 45Fe
with different assumptions about position E2r of the ground
state in the 44Mn subsystem and structure of 45Fe [weights
of the p2 configuration W (p2) are shown on the vertical
axis]. Central gray area corresponds to experimental width
uncertainty Γ = 2.85+0.65
−0.68 × 10
−19 MeV [12]. The light
gray area also takes into account the energy uncertainty
E3r = 1.154(16) MeV from [12]. The vertical dashed line
corresponds to E2r used in Fig. 19.
III of Ref. [7], see erratum). However, in the subsequent
work [21], providing very similar to [7] properties of the
17Ne WFs for the ground and the lowest excited states,
the width of the 3/2− state was found to be 3.6× 10−12
MeV. It was supposed in [21] that such a strong disagree-
ment is connected with poor subbarrier convergence of
the HH method in [7] compared to Adiabatic Faddeev
HH method of [21]. This point was further reiterated in
Ref. [41]. We can see now that this statement has a cer-
tain ground. However, the convergence problems of the
HH method are far insufficient to explain the huge dis-
agreement: the width increase found in this work is only
one order of magnitude. The most conservative upper
limit Γ ∼ 5× 10−14 MeV (see Table IV) was obtained in
a TFSI calculation neglecting p-p Coulomb interaction.
The other models systematically produce smaller values,
with realistic calculations confined to the narrow range
Γ ∼ (5 − 8) × 10−15 MeV (Table VI). Thus the value
Γ ∼ 4× 10−12 MeV obtained in paper [21] is very likely
to be erroneous. That result is possibly connected with a
simplistic quasiclassical procedure for width calculations
employed in this work.
VII. CONCLUSION.
In this work we derive the integral formula for the
widths of the resonances decaying into the three-body
channel for simplified Hamiltonians and discuss various
aspects of its practical application. The basic idea of the
derivation is not new, but for our specific purpose (pre-
cision solution of the multichannel problem) several im-
portant features of the scheme have not been discussed.
We can draw the following conclusions from our stud-
(i) We presume that HH convergence in realistic calcu-
lations should be largely the same as in the simplified
calculations as they imitate the most important dynamic
aspects of the realistic situation. The width values were
somewhat underestimated in our previous calculations.
The typical underestimation ranges from few percent to
tens of percent for “simple” potential and from tens of
percent to an order of magnitude in “complicated” cases
(potentials with repulsive core).
(ii) Convergence of the width calculations in the three-
body HH model can be drastically improved by a simple
adiabatic version of the Feshbach reduction procedure.
For a sufficiently large dynamic sector of the basis the
calculation with effective FR potential converges from
below and practically up to the exact value of the width.
For a small dynamic basis the FR calculation converges
towards a width value smaller than the exact value, but
still improves considerably the result.
(iii) The energy distributions obtained in the HH calcu-
lations are quite close to the exact ones. Convergence
with respect to basis size is achieved at relatively small
Kmax values. The disagreement with exact distributions
is not very significant and is likely to be connected not
with basis size convergence but, with radial extent of the
calculations [4].
(iv) Contributions of different decay mechanisms were
evaluated in the simplified models. We have found that
the “diproton” decay path is much less efficient than the
“sequential” decay path. This is true even in the model
calculations without nuclear FSIs (no specific dynamics),
which means that the “sequential” decay path is some-
how kinematically preferable.
(v) The value of the width for 17Ne 3/2− state was un-
derestimated in our previous works by around an order of
magnitude. A very conservative upper limit is obtained
in this work as Γ ∼ 5 × 10−14 MeV, while typical values
for realistic calculations are within the (5 − 8) × 10−15
MeV range. Thus the value Γ ∼ 4×10−12 MeV obtained
in papers [21, 41] is likely to be erroneous.
From this paper it is clear that the convergence issue
is sufficiently serious, and in some cases were underesti-
mated in our previous works. However, from practical
point of view, the convergence issue is not a principle
problem. For example the uncertain structure issues and
subsystem properties impose typically much larger uncer-
tainties for width values. For heavy two-proton emitters
(e.g. 45Fe) the positions of resonances in the subsystems
are experimentally quite uncertain. For a moment this is
the issue most limiting the precision of theoretical predic-
tions. We have demonstrated that with increased preci-
sion the experimental data impose strong restrictions on
theoretical calculations allowing to extract an important
structure information.
VIII. ACKNOWLEDGEMENTS
The authors are grateful to Prof. K. Langanke and
Prof. M. Ploszajczak for interesting discussions. The au-
thors acknowledge the financial support from the Royal
Swedish Academy of Science. LVG is supported INTAS
Grants 03-51-4496 and 05-1000008-8272, Russian RFBR
Grants Nos. 05-02-16404 and 05-02-17535 and Russian
Ministry of Industry and Science grant NS-8756.2006.2.
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[42] A realistic example of this situation is the case of “E1”
(coupled to the ground state by the E1 operator) con-
http://arxiv.org/abs/nucl-ex/0603020
tinuum considered in Ref. [23]. This case is relevant to
the low energy radiative capture reactions, important for
astrophysics, but deal with nonresonant continuum only.
[43] Interesting numerical stability test is a variation of
the “unphysical” (for OFSI approximation) potential
V nucy (Y ) in the auxiliary Hamiltonian (27). It can be used
for numerical tests of the procedure as it should not in-
fluence the width. Really, for variation of this potential
from weak attraction (we should not allow an unphysical
resonance into decay window) to strong repulsion (scale
of the variation is tens of MeV for potential with some
typical radius) the width is varied only within couple of
percents. This shows high numerical stability of the pro-
cedure.
[44] These values can be evaluated as typical nuclear ra-
dius for the system multiplied by
2: 3.53
2 ≈ 5 and
2 ≈ 6.
[45] The derivation of the flux here is given in a schematic
form. The complete proof is quite bulky to be provided
in the limited space. We would mention only that it is
easy to check directly that the derived expression for flux
preserves the continuum normalization.
[46] We demonstrate in paper [37] that a three-body width
should depend linearly on two-body widths of the sub-
systems and only very weakly on various geometrical fac-
tors. This is confirmed very well by direct calculations.
[47] The assumed nuclear structure is very simple, but the
diproton penetration process is treated exactly — with-
out assumptions about the emission of diproton from
some nuclear surface, which should be made in “R-
matrix” approach.
