e+r4e–e+

where Eγ is the photon energy, α is fine-structure constant,d is the electric dipole moment operator, and Ψf (Ef ) is thewavefunction of PsH(2,4 Pe ) continuum state corresponding to
the energy Ef = Ei − Eγ , normalized per unit energy.
For numerical calculation of dΓγ /dEγ , we introduce thecomplex rotated Hamiltonian Hθ where the distance r is scaled
by a complex factor of eiθ as r → reiθ . |Ψf (Ef )$#Ψf (Ef )| can be
expressed with the H(θ) and complex rotation operator R(θ) as"#111|Ψf (Ef )$#Ψf (Ef )| =R(−θ)R(θ) − R(θ)R(−θ)
2iπH(θ) − Ef
H(−θ) − Ef(2)e–
R5R4r5

ρ4

ρ5pe–

pe–

c=4

c=5

FIG. 2. Jacobian coordinate systems for PsH.

hottµ(1s)

X-raydµ(1s)ddµtµ(1s)
dtµ

dtµ*

ttµ

dtµ

Fusion
µHe

dtµ*

µCF0

1234

d-t distance (pm)562.0

hotdµ(1s)ttµ*Intensity

dtµ*Intensity (keV−1 ps−1)ddµ*

1.5tµ(n=2)

1.0dµ(n=2)

0.5tµ*

0.0dµ*

Energy (keV)µtive transitions have attracted astrophysical attentions in that
The eigenstates of Hθ , ψnθ , satisfies
they could contribute to stellar absorption spectra, and haveHθ ψnθ = Enθ ψnθ ,(3)0 been investigated theoretically as well as photoionization processes [22].
#ψ̄nθ |ψmθ $ = δnm ,(4)The radiative dissociation process has not been discussed indetail for PsH(2,4 S o ). In contrast to the H− (3 Pe ), PsH(2,4 S o )where #ψ̄nθ | denotes complex-conjugate of $ψnθ except for its
Ddtµ*
has a number of dissociationchannels.
AsshowninFig.
1,radial part. Utilizing the closure relation of n |ψmθ $#ψ̄nθ | = 1,3dσgPsH(2,4 S o ) results in 2,4 Pe continuum
whereH(n=2)+4fσu
the dΓγ /dEγ can be written in terms of the eigenfunctions ofHPs(nPs ≥ 1) and H(nH ≥ 1) + Ps(nPs = 2) dissociation threshH(θ) as
olds are located. Note that due to the even parity of Ltµ(n=2)= 1+ d dΓ"#
4 3 3 1 % #ψ̄nθ |d(θ)|Ψiθ $2γ
radiative dissociation into H(nH = 1) + Ps(nPs = 1) is for= α Eγ Im,(5)bidden. There could be a transition to the autoionizing statedEγ 3πEnθ − Efnembedded in the 2,4 Pe continuum. Since the PsH(2,4 S o ) is lowhere d(θ) = R(θ)dR(−θ) is the rotated electric dipole mocated just below the H(2p) + Ps(2p) dissociation threshold−1 energy, the structure of PsH(2,4 S o ) is expected to be a loosely
ment operator, and Ψiθ = R(θ)Ψi is the rotated wavefunctionof the bound state.
bound state of these excited atoms. As described by Mitroy
2,4 oIn the present work we evaluate dΓγ /dEγ both in the lengthand Bromley, the lifetime of PsH( S ) would be comparable−8−9and velocity gauges. In the length gauge, d(θ) is written as
to the lifetimes of H(2p) and Ps(2p), in the order of 10 -10s [19].
&'iθ−−+d(θ) = −e rp→e1 + re →e2 ,(6)In this paper, we calculate the radiation spectrum associ2,4 oated with the dissociative decay of PsH( S ) based on thep to one of the electrons, e−1 , andX-raymethod where rp→e−1 is a vector fromvariational method and complex coordinate rotation
re+ →e−2 is a vector from e+ to the other electron, e−2 . In the(CCRM) [23]. From the spectrum, we investigate the dissovelocity gauge,ciation channels of PsH(2,4 S o ) and its stability. The conver()gence of the120results is examined both in the length and veloce−iθ 11
Atomic units (a.u.; me = ! = e = 1)
−2 ity gauge calculations.
d(θ) = −∇p→e−1 +∇e+ →e−2 ,(7)100Eγ µpe−1µe+ e−2
are used throughout this paper, except where mentioned oth80erwise.
where µi j is the reduced mass between the particle i and j, and60∇i j is the differential operator associated with ri j . dΓγ /dEγ
40would give the same results in the length and velocity gauges
II. THEORY
20
if the Ψiθ and ψnθ were exact. However, as described above,0PsH(2,4 Pe ) continuum wavefunction in principle involves infidµ(1s)+t
X-ray energy
A. Radiative dissociation rates
1.7
1.8
2.0
nite1.6
number of dissociation
channels,
which1.9
makes it difficult
to construct the exact scattering wavefunction. In this work,
X−ray energy(keV)
tµ(1s)
Ψiθ +anddψnθ are calculated in variational method for energy
We consider the four-body Hamiltonian H consisting of theX in terms of the square integrable functions toand expanded
kinetic energy operators and all inter-particle Coulomb poten−3 tial operators. The numerical method to calculate the radiativesatisfy
2,4 odissociation rates of PsH( S ) follows the method utilizing#Ψ̄iθ |H(θ)|Ψiθ $ = Eiθ ,(8)the complex coordinate rotated wavefunction [23–26]. Noting