Annihilation of Low Energy Antiprotons in Hydrogen
S.Yu. Ovchinnikov∗ and J.H. Macek∗
∗
Department of Physics, The University of Tennessee, Knoxville, TN 37831 and Oak Ridge National Laboratory,
Oak Ridge, TN 37996
Abstract. The cross sections for annihilation of antiprotons in hydrogen are very important for designing the HighPerformance Antiproton Trap (HiPAT). When antiprotons are trapped they undergo atomic reactions with background gases
which remove them from the trap. First, antiprotons are captured into highly excited bound states by ejecting the bound
electrons, then they are radiationally deexcited and, finally, they annihilate by nuclear interaction. An understanding of these
process require reliable cross sections for low-energy collisions of antiprotons with atoms. We have developed a theoretical
technique for accurate calculations of these cross sections.
INTRODUCTION
When antiprotons are trapped they undergo atomic reactions with background gases which remove them from
the trap. For example they may capture into bound states
with subsequent annihilation by nuclear interactions. An
understanding of these processes requires reliable cross
sections for low-energy collisions of antiprotons with
atoms.
At present ionization cross sections have been reported
[1, 2, 3] that are based on the semiclassical method that
is essentially exact for energies above a few hundred eV
with projectiles whose mass is comparable to the proton mass. In this range, there are a wide variety of techniques that can be reliably employed to compute ionization cross sections. Important for designing the HighPerformance Antiproton Trap (HiPAT), however, are energies in the 0-50 eV range. In this range one typically
uses some sort of adiabatic representation for the electron motion and a wave treatment for the relative motion
of the nuclei. This approach becomes problematical for
antiprotons owing to protonium formation which is not
represented in standard adiabatic bases. Only the hyperspherical representation is readily adapted to this region,
however, the hyperspherical adiabatic method is impractical owing to the need for a large number of basis states
that are difficult to compute. For that reason, the only
protonium formation cross sections that are available at
the present time employ some variant of the classical
trajectory Monte Carlo method [3], since bound protonium orbits can evolve from collisions of antiprotons
with atoms where the much lighter electrons are initially
also in bound orbits.
We have used the advanced adiabatic theory to compute protonium formation and ionization for low en-
ergy impact of antiprotons on atomic hydrogen. The advanced adiabatic method derives from an exact Sturmian
representation, given by Ovchinnikov and Macek [4],
of the full wave function for three particles interacting
via electrostatic interactions. For that reason it must include all reaction channels, including protonium channels even though no protonium wave functions are included in the basis set. Even in the one-Sturmian approximation, which is crucial to the advanced adiabatic theory, all physical channels are included.
ADVANCED ADIABATIC THEORY
The conventional adiabatic electron energies for an electron in the field of p + p are known as the potential
curves of the finite dipole. In the separated atom limit
they are the Stark energy levels of the H-atom in the
field of the antiproton. At some finite distance, called the
Fermi-Teller radius RFT = 0.693..., the electron just becomes unbound in the finite dipole field of the p, p system. The lowest adiabatic potential curve of the finite
dipole is shown on Fig. 1 below E = 0. At the united
atom limit, where the antiproton coincides with the proton, the electron nuclei potentials cancel and the electron is completely free. The ground state potential curves
ε (R) therefore move into the continuum with decreasing internuclear separation and the bound states become
quasi-stationary states. An approximate expression for
ε (R) for R less than RFT is [5]
εn1 n2 m (R) = R42 λ0 (R) − 72 (1)
− i R42 (2n1 + m + 1) 8λ0 (R) − 49
4
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
199
Finite dipole
where the classical actions in the initial and final states
are
(L + 1/2)2 1
+
,
(4)
K(R) = 2µ E − ε (R) −
2µ R2
R
1
0.8
E (a.u.)
0.6
0.4
K f (R) =
0.2
0
−0.2
0.5
1
1.5
Re d (a.u.)
2
2.5
FIGURE 1. The lowest adiabatic potential curve of the finite
dipole with charges Z = 1 as a function of the dipole momentum d = ZR.
with
1
λ0 (R) = n2 + m +
2
2
1
+
2
R
n2 + m + 12
2
,
(2)
where n1 n2 m are the parabolic quantum numbers of a
hydrogen in a constant electric field.
1
1.0
Im d (a.u.)
0.8
0.2
0.1
10.0
0.6
0.4
0.01
100.0
0.2
0.001
E=0
0
0
0.2
0.4
0.6
Re d (a.u.)
(6)
0.0001
p + H → (p, p)n + e−
0.000001
0.8
1
1.2
FIGURE 2. Parametric plot of Red and Imd as a function
of E for the quasi-stationary state continued from the lowest
bound state
The advanced adiabatic theory of Solov’ev [6] employs the quantity R(ε ) inverse to ε (R). This quantity
is the solution of the equation ε (R) = ε are shown on
Figs. 1 and 2. Here the single adiabatic function is understood in the sense articulated by Demkov [7]. The advanced adiabatic theory gives the probability for ionization P(E, εk ) as
dK(RM ) dRM
P(E, εk ) = 2π 2εk C(εk )
dE
d εk
R
2
R
M
T
× exp i
K(R)dR + i
K f (R)dR , (3)
Ri
1µ
+ εk
2 n2
where −µ /(2n2 ) is the binding energy of the n th state
of protonium. The advanced adiabatic theory emerges
when the integral representation for Ψ is evaluated for
large r in the stationary phase approximation. For ionization processes, the coefficient of the product of outgoing
waves for the
√ electron and the p, p pair exp(ikr + iKR),
where K = 2µ E, is computed. When E − εk is negative this same coefficient becomes the amplitude for the
rearrangement process
0.4
4.0
(5)
and where εk is the energy of the ionized electron, Ri is
a large value of R on the initial branch of ε (R), RT is the
turning point in final channel, RM is the complex value
of R where ε (R) = εk , and C(εk ) is the coefficient of the
outgoing electron wave exp(ikr) in the asymptotic Sturmian wave function. The quantities ε (R) and C(εk ) are
computed for complex values of R using the program of
Ovchinnikov and Solov’ev [8]. These numerical values
are used to compute the matching radius RM and classical actions in Eq. (3).
If the quantity E − εk is negative, then the wave function in R satisfies acceptable asymptotic conditions only
when the action takes on half integral multiples of π , i. e.
the R motion is quantized according to
E − εk = −
Finite dipole
2.0
(L + 1/2)2 1
2µ E − εk −
+
,
2µ R2
R
RM
200
(7)
up to a normalization constant. In this way one extracts
the rearrangement amplitude even though no protonium
states are included in the basis set.
There is little difference theoretically between ionization with and without protonium formation as is apparent from the continuity of ionization cross sections
across the protonium threshold seen in classical trajectory Monte Carlo calculations [1, 3]. This allows a simple way to compute protonium formation in the advanced
adiabatic theory, namely, we compute ionization without
reference to the quantization of the protonium energies
and then identify the cross section for negative values
of E − εk with protonium formation. This is convenient
for computations of total protonium formation cross sections, but by using the dE
dn energy interval weighting, ndistributions are also we obtained.
Protonium Formatiom
−
+
−
H + P = (H P ) + e
−
20
2
cm )
15
−16
2α
p
protonium formation given by σ p = π
E , an estimate
that is thought to be exact for vanishingly small E.
range where the straight line approximation is expected
to apply. Only at the lowest energy point (200 eV) do
departures, of the order of 15%, between the theories
appear. This good agreement gives us confidence in the
application of the advanced adiabatic theory for the study
of antiproton interactions.
The advanced adiabatic theory should be more reliable
at the lower energies where the relative velocities of the
heavy particles become much smaller than the electron
velocities. In this region ionization occurs only by protonium formation. Figure 4 shows our protonium formation cross in the region below the protonium formation
threshold at 27.21 eV. Also shown is the “upper limit"
cross section σ p . Our computed cross sections are below
this limit, as they must be. Also shown are CTMC results
10
σ (10
The advanced adiabatic theory should be more reliable at the lower energies where the relative velocities
of the heavy particles become much smaller than the
electron velocities. In the region below the ionization
threshold at 27.21 eV ionization occurs only by protonium formation. This allows a simple way to compute
protonium formation in the advanced adiabatic theory,
namely, we compute ionization without reference to the
quantization of the protonium energies and then identify the cross section with protonium formation. This is
convenient for computations of total protonium formation cross sections, but by using the dE
dn energy interval
weighting, n-distributions can also we obtained.
At very low energies, antiprotons may be temporarily trapped in the combined polarization and centrifugal
α
potentials Veff = L(L+1)
− 2Rp4 , where α p = 92 is the po2µ R2
larizability of the hydrogen atom. The associated orbiting resonances can decay by electron emission thereby
leading to protonium formation via a process similar to
associative ionization in negative ion collisions. If one
assumes unit probability for decay via electron emission
then one obtains an upper bound cross
section σ p for
5
RESULTS
0
0
5
10
15
20
25
E (eV)
Our computed ionization cross sections are shown in
Fig. 3 for energies between 10 ev and 100 keV. Also
shown are the results of essentially exact solutions of the
time-dependent Schrödinger equation in the straight line
approximation [3].
Ionization Cross Section
−
+
−
H+P =H +P +e
−
2
1
σ (10
−16
2
cm )
1.5
0.5
0
2
10
3
10
E (eV)
FIGURE 4. Cross sections for protonium formation. Short
dash curve is CTMC from Schultz et al [3], long dash curve
is upper limit orbiting cross section, solid curve is advanced
adiabatic.
of Schultz et al [3]. The agreement of these results with
the advanced adiabatic cross sections is very good for
5 eV < E < 25 eV. Both results are larger than would be
obtained by extrapolating the TDSE calculations into this
region, indicating that orbiting probably plays an important role even though both calculations are below the "upper limit" orbiting cross section σ p . The orbiting effect is
expected to become important as E → 0. This trend is
apparent in the advanced adiabatic calculations, but not
in the CTMC results.
Results of our calculations of n and l distributions are
shown in Figs. 5 and 6 for E = 27 eV and in Figs. 7 and 8
for E = 1 eV.
4
10
FIGURE 3. Cross sections for the ionization of hydrogen
atoms by antiproton impact. Solid circles are exact TDSE and
dashed curve is CTMC from Schultz et al [3], solid curve is
advanced adiabatic.
The agreement between the advanced adiabatic theory
and the "exact" results is very good over the energy
201
SUMMARY AND CONCLUSIONS
We have used the advanced adiabatic theory to compute ionization and protonium formation for low energy
impact of antiprotons on atomic hydrogen. Our results
agree with the TDSE calculations of ionization at ener-
Protonium Formatiom
Protonium Formatiom
−
+
−
−
−
+
−
−
H + P = (H P ) + e
H + P = (H P ) + e
0.3
0.01
E=1 eV
E=27 eV
0.008
0.2
2
cm )
σ (10
σ (10
−16
−16
2
cm )
0.006
0.004
0.1
0.002
0
0
40
80
120
N
160
200
FIGURE 5. Cross sections for protonium formation as a
function of n for the initial energy E = 27 eV.
25
+
−
Protonium Formatiom
−
−
+
−
H + P = (H P ) + e
H + P = (H P ) + e
−
0.03
0.0003
E=1 eV
E=27 eV
N=80
N=29
0.02
σ (10
−16
−16
2
2
cm )
cm )
0.0002
σ (10
35
FIGURE 7. Cross sections for protonium formation as a
function of n for the initial energy E = 1 eV.
Protonium Formatiom
−
30
N
0.0001
0
0
10
20
30
L
40
50
0.01
0
60
0
10
20
30
L
FIGURE 6. Cross sections for protonium formation in the
selective n, l-state for the initial energy E = 27 eV and n = 80.
FIGURE 8. Cross sections for protonium formation in the
selective n,l-state for the initial energy E = 1 eV and n = 28.
gies above 300 eV, and with CTMC calculations of protonium formation in the energy range 5 eV < E < 25 eV.
Below 5 eV our cross sections rise above the CTMC results and approach a value indicative of capture via orbiting resonances. Protonium formation ends by annihilation and cross sections for annihilation are very important for designing the High-Performance Antiproton
Trap (HiPAT). The protonium formation cross sections
presented here are identical to annihilation cross section.
REFERENCES
ACKNOWLEDGMENTS
This research has been supported by the Chemical Science, Geosciences and Biosciences Division, Office of
Basic Energy Science, Office of Science, U.S. Department of Energy under Grant No. DE-FG02-02ER15283
202
1. Cohen, James S., Phys. Rev. A, 56:3583-960, 1997.
2. Schiwietz, G., Wille, U., Muniños, R. Díez, Fainstein, P. D.,
and Grande, P. D., J. Phys. B:At. Mol. Opt. Phys, 29:307-21,
1996.
3. Schultz, D. R., Krstić, P. S., and Reinhold. C. O., Phys.
Rev. Lett., 76:2882-5, 1996.
4. Macek, J. H. and Ovchinnikov, S. Yu. Phys. Rev. Lett.,
80:2298-301, 1998.
5. Abramov, D. I., Ovchinnikov, S. Yu., and Solov’ev E. A.
PIs’ma Zh. Eksp. Teor. Fiz., 47:424-8, 1988, [Eng. Trans.
JETP Lett. 47:504-8 (1988)].
6. Solov’ev, E. A., Zh. Eksp. Teor. Fiz., 70:872-82, 1976,
[Eng. Trans. Sov. Phys. JETP, 43:453-8 1976].
7. Demkov Yu. N. Invited Talks of the Fifth ICPEAC,
Lenningrad, 1967. Joint Institute for Laboratory
Astrophysics, Boulder CO, 186, 1968.
8. Ovchinnikov, S. Yu. and Solov’ev E. A. Zh. Eksp. Teor. Fiz.,
90:921-32, 1986, [Eng. Trans. Sov. Phys. JETP 63:538-44
(1986)].