Disentangling the kaonic-hydrogen K
F b-complex with DEAR
C. GUARALDO(1), M. BRAGADIREANU(1)(2), M. ILIESCU(1)(2)V. LUCHERINI(1), C. PETRASCU(1)(2) and T. PONTA(2)
(1) INFN, Laboratori Nazionali di Frascati - C.P. 13, I-00044 Frascati (Roma), Italy
(2) IFIN-HH - Bucharest-Magurele, P.O. Box MG6, R-76900 Romania
(ricevuto il 12 Settembre 1997; approvato l’8 Ottobre 1997)
Summary. — The possibility of disentangling for the first time the KF b-complex in kaonic hydrogen is shown to be realistic with the DEAR experiment. A precise identification of the pattern of lines in kaonic hydrogen and in kaonic deuterium can allow to obtain the first experimental determination of the KN sigma terms. PACS 13.75.Jz – Kaon-baryon interactions.
PACS 36.10 – Exotic atoms and molecules (containing mesons, muons, and other unusual particles).
1. – Kaonic-hydrogen formation and energy levels
An “exotic atom” is formed whenever an electron of an outer orbit of an atom is replaced by a heavier charged particle, such as a muon (m2), or hadrons like a pion (p2), a kaon (K2), an antiproton (p) or a sigma hyperon (S2).
Among the exotic atoms, the hydrogen-like systems are of particular importance, because they have the simplest structure and are free from any screening effect due to bound electrons. By studying them, one can, for instance, directly probe the hadron-nucleon interaction at zero energy.
The case of kaonic hydrogen.
The mechanism of formation of a kaonic hydrogen atom is the following: a negative kaon enters into a hydrogen target; it loses its kinetic energy by ionization and excitation of the hydrogen molecules until it is captured into an atomic orbit around the proton (by replacing the electron). The capture orbit has the principal quantum number
ncaptCkm/meC 25 (m and meare the reduced mass of the K2p system and the electron mass, respectively), the Bohr radius of the kaonic system corresponding to that of the hydrogen K-shell electrons. Then the kaon cascades down through the series of atomic levels. The processes co-participating in the de-excitation of kaonic hydrogen are:
1) Molecular dissociation: ( K2p )
i1 H2K ( K2p )f1 H 1 H, which occurs when D Eif4 Ei2 EfD 4.7 eV (dissociation energy of H2molecule).
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2) External Auger transition: (K2p )
i1 H K ( K2p )f1 H11 e2, which occurs when DEifD 15.6 eV (ionization energy of the H atom).
3) Radiative transition: (K2p )
iK ( K2p )f1 g.
4) Nuclear absorption, which is due to strong interaction and has a rate going as 1 /n3.
5) Stark mixing: the strong electric field of the protons causes a mixing between degenerate states with the same principal quantum number and different angular momenta, the consequence being a nuclear absorption from high n states, preventing the kaon from reaching low-lying states. The effect is relevant in liquid hydrogen, where the high density gives high electrical fields.
6) Weak decay: the lifetime of the kaon plays a role for a dilute gas target. The X-rays emitted in transitions to the 1 s state constitute the K-series (Ka: 2 p K
1 s; Kb: 3 p K1s; Kg: 4 p K1s; etc.), while the X-rays associated with transitions to the
2 p state give the L-series (La: 3 d K2p; Lb: 4 d K2p; etc.).
The energy levels of kaonic hydrogen can be calculated by solving the Klein-Gordon TABLE I. – Calculated energy levels of kaonic hydrogen. The solutions of the Klein-Gordon
equation are given. The error on the kaon mass reflects in an uncertainty, given in parenthesis, of the 1s energy level. The uncertainty due to the unknown kaon charge distribution affects the evaluation of the finite size effect and is also given in parenthesis. As a consequence, the transition energies corresponding to the K-series lines are affected by an uncertainty of 1 eV.
State Total energy (keV) Non-relativistic energy (keV) Vacuum polarization (keV) Relativistic correction (keV) Finite-size effect (keV) Higher-order vacuum polarization (keV) 1s 8.6339 8.6128 1 0.0219 1 0.0006 2 0.0016 1 0.0002 (6 0.0003) (60.0006) 2 p 2.1542 2.1532 1 0.0008 1 0.0002 — — 3 p 0.9573 0.9570 1 0.0002 1 0.0001 — — 4 p 0.5384 0.5383 1 0.0001 — — — 5 p 0.3445 0.3445 — — — — 6 p 0.2393 0.2393 — — — — 7 p 0.1758 0.1758 — — — — 8 p 0.1346 0.1346 — — — —
Transition X-ray line Energy (keV)
2 p K1s Ka 6.480 6 0.001 3 p K1s Kb 7.677 6 0.001 4 p K1s Kg 8.096 6 0.001 5 p K1s Kd 8.289 6 0.001 6 p K1s Ke 8.395 6 0.001 7 p K1s Kz 8.458 6 0.001 8 p K1s Kh 8.499 6 0.001 Q K 1 s KQ 8.634 6 0.001
equation and taking into account: relativistic corrections, vacuum polarization and finite-size effects. They are tabulated in table I. The energy levels can be evaluated with the precision of the tenth of eV (see table). However, the error on the kaon mass and the unknown charge distribution of the kaon turn out in an overall numerical uncertainty on the energy of the 1 s state of the order of one eV and of about few tenths of eV for the excited states. Consequently, the electromagnetic transition energies have an intrinsical uncertainty of 1 eV, as reported in table I.
When the kaon reaches a state with a small angular momentum, state in which the overlap between the kaon and proton wavefunctions is large, the kaon is absorbed by the proton due to strong interaction. The strong interaction causes a shift of the low-lying levels from their pure electromagnetic positions, while the widths are increased. Shift and broadening are appreciable only in the 1 s state and negligible in all the other states, i.e. only for the lines of the K-series. In practice, since the energy spacings between two adjacent lines of the KF b complex are much smaller than that between the Ka and the Kb lines (see table I), the spectrum of the KF b complex will display overlaps, depending on the energy resolution of the detector. The Ka line is
therefore the most important line in determining the level shift and the width of the 1 s level.
The shift e and the width G of the 1s state of kaonic hydrogen are related in a fairly model-independent way to the real and imaginary part of the s-wave scattering length aK2p:
e1 i
2G42a 3m2a
K2p4412aK2p eV fm21, (1)
with m reduced mass and a fine-structure constant. This expression is known as the Deser-Trueman formula [1]. If one neglects the mass difference between the K2p and K0n systems and the Coulomb correction, a
K2p is simply the average of the scattering lengths a0(I 40) and a1(I 41):
aK2p 4 1
2(a01 a1) . (2)
2. – Experiments on kaonic hydrogen
Three measurements of kaonic hydrogen X-rays were carried out at CERN and at Rutherford in the late 70’s through the early 80’s [2-4]. Recently, a fourth measure-ment has been performed at KEK [5]. In the three old measuremeasure-ments a liquid hydrogen target and Si(Li) detectors were used. Davies et al.[2] observed a 2 s peak at 6.52 6 0.06 keV, which they attributed to the Ka line. Izycki et al. [3] observed a weak
pattern of 3 lines at 6.96 60.09, 7.9960.07 and 8.6460.10 keV, with significance 2s, 3s and 2 s, which they assigned to the 2p K, 3pK and 4pK1s transitions. Bird et al. [4] saw a similar series of three lines with significance s, 3 s and 0.5 s, respectively.
All the three spectra suffered from large background and low statistics: X-ray signals were very difficult to identify, being strongly attenuated due to the Stark effect present in the liquid hydrogen target. A common feature, notwithstanding the extremely scarce quality of the data, was however apparent: the sign of the shift, and, consequently, that of the real part of the K2p scattering length, was positive (attractive strong interaction). This was in striking contradiction with the results of the analyses of
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TABLEII. – The "kaonic hydrogen puzzle" and its solution.
Method Author aK2p(fm)
K2p scattering Sakitt et al. (1965) (20.91 6 0.05) 1 i(0.48 6 0.03)
analyses Kim et al. (1967) (20.87 6 0.04) 1 i(0.69 6 0.03) von Hippel et al. (1968) (20.89 6 0.02) 1 i(0.62 6 0.02) Martin and Ross (1970) (20.89 6 0.03) 1 i(0.66 6 0.03) Martin et al. (1981) (20.66 6 0.05) 1 i(0.64 6 0.04) Kaonic-hydrogen Davies et al. [2] ( 0.10 6 0.14) 1 i(0.0010.28
20.00)
X-ray Izycki et al. [3] ( 0.65 6 0.19) 1 i(0.68 6 0.31) measurements Bird et al. [4] ( 0.47 6 0.14) 1 i(0.1010.27
20.10)
Iwasaki et al. [5] (20.79 6 0.15 6 0.03) 1 i(0.49 6 0.25 6 0.12)
the low-energy scattering data (K2p cross-sections for elastic and inelastic processes, branching ratios for K2p absorption at rest, pS invariant-mass distribution), extrapolated to threshold and below, which showed a negative real part of the scattering length (repulsive strong interaction). This discrepancy between scattering data and X-ray measurements, reported in table II, is often referred to as the “kaonic
hydrogen puzzle”. This “puzzle” has received considerable theoretical attention in
about 15 years of tentatives to reconcile the two sets of data. No satisfactory theoretical explanation was however found, as well as various attempts to find a parameter set for a phenomenological potential which fitted the scattering data together with the kaonic-hydrogen measurements were unsuccessful. It is now clear, after the KEK results, that nothing had to be reconciled, being the so-called "puzzle" created by the bad quality of the kaonic-hydrogen data.
In the recent KEK experiment [5], instead of a liquid hydrogen target, a cryogenic pressurized gas target was used. The background of soft X-rays coming from the e.m. cascades initiated by high-energy photons, the major source of background in the previous experiments, was suppressed by selecting only the channels K2p KS6pZ followed by S6
K n p6, i.e. only channels without p0in the final state. These channels (A50% ) were identified by tagging on two charged pions having momenta higher than 150 MeV/c emitted in the final state, one produced in the hyperon production and the other in the hyperon decay. Moreover, only kaons stopping in the target volume were selected, by requiring that the two-pion vertex is in the hydrogen volume and that they have an appropriate time of flight.
The overall collected statistics was not enough to perform a precision measurement. However, the result
e 42 327663(stat.) 6 11(syst.) eV ,
(3)
G1 s4 406 6 206(stat.) 6 65(syst.) eV (4)
solves the long-standing “kaonic hydrogen puzzle”. In agreement with the scattering
data analyses and in contradiction with the previous X-ray measurements, the 1 s
level shift turns out to be negative, and consequently the real part of the scattering length, indicating that the kaon-proton strong interaction is repulsive.
Table II shows what was called the “kaonic hydrogen puzzle”, together with its solution.
3. – The DEAR experiment
A new measurement on kaonic hydrogen and the first measurement on kaonic deuterium will be performed by the DEAR experiment on the DAFNE e1e2collider at Frascati [6]. DEAR (DAFNE Exotic Atom Research) has substantial improvements, with respect to all the previous measurements on kaonic hydrogen, concerning mainly the beam quality and the choice of the detector.
– The kaon beam. It is the high-purity, low-momentum, narrow-momentum bite, good intensity, “kaon beam” constituted by the kaons coming from f decay in the DAFNE machine [7]. Less than two mm of a degrader are enough in order that kaons stop inside the hydrogen target, which is located one cm above the beam pipe (see fig. 1). – The detector. The Charge-Coupled Device (CCD) [8] is the tool to detect with good resolution (G 150 eV) an X-ray with energy in the range 2–12 keV, with an
enormous (orders of magnitude) background rejection from any charged or neutral background particle.
– The target. DEAR will adopt a pressurized cryogenic target, as done at KEK, in order to reduce Stark mixing and therefore not to decrease the K-series lines yield. Taking advantage of the KEK results and of the substantially background-free nature of the signal, an increase of the yield is obtainable by increasing the hydrogen density from the KEK value A7rNTP to A40rNTP. This means r 43.6 Q 1023 g cm23, corresponding to a hydrogen pressure of 3 atm and a temperature of 25 K.
The simulation of the experiment was performed with a Monte Carlo program in the framework of the CERN package GEANT3, in the version 3.21. As is known, this code is reliable only down to 10 keV, whilst, in order to evaluate the performance of the
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experimental set-up, the behaviour of photons, electrons and positrons has to be known below the above energy value. GEANT routines for electron and positron bremsstrahlung, photoelectric effect and photon total cross-section were checked on the available experimental data down to 1 keV, and modified when necessary [6].
An accurate evaluation of the background on CCDs was performed. Two sources of background have to be distinguished: the “blindness” of a CCD, i.e. the maximum number of hits which can be tolerated by a CCD avoiding double hits and the physical background, i.e. soft X-rays (G 10 keV), generated by the e.m. cascades initiated by high-energy photons, electrons, positrons, and that cannot be eliminated.
As far as the counting rate is concerned, at the following working conditions: luminosity L 41032cm22s21; target density r 43.6 Q 1023g cm23; (2p-1s) yield 3%; CCD efficiency e 460% at 6.5 keV, the number of detected X-rays (Ka) turns out
to be B20 h21.
4. – Disentangling the kaonic hydrogen KF b-complex with DEAR
In what follows, the possibility of identifying with DEAR not only the Ka line of
kaonic hydrogen, but also the Kb line, thus disentangling the KF b-complex, is investigated.
As was stressed by Batty [9], the presence of 2 or 3 lines from the K-complex, with their electromagnetic energy spacings, would aid considerably in their identification as originating from K2p atoms and therefore in distinguishing them from the background of other pionic and kaonic atom X-ray lines.
In fig. 2, position and width of the lines of the K-complex, as obtained by the KEK experiment [5], are reported. Looking at the spectrum, it turns out:
– the Ka line is positioned at 6.2 keV and its width is about 560 eV, which
corresponds to an intrinsical width of about 400 eV (the experimental resolution being about 400 eV);
– the KF b-complex cannot be disentangled and appears as a bump at about 8.1 keV, about 760 eV wide, which corresponds to an intrinsical width of about 650 eV.
The results of an iterative fitting procedure [10], with the intensities constrained by the cascade code of Borie and Leon [11], are reported in table III. The cascade code parameters kstkand T were fixed at 1.8 and 1.0 eV, respectively, and only G2 pwas allowed to vary. The value obtained for G2 pfrom the fit was about 0.3 meV, i.e. within the range of the commonly used values. In the same time, the intensity of Ka, which was not
cascade-constrained, agreed with the value predicted by the cascade code within 10 %. From table III, it turns out that, at the KEK density (r A7 rNTP) the ratio between the number of Kaevents and the number of events in KF b-complex is about 1 : 3, whilst the Kbintensity is about one half with respect to that of Ka. A reasonable assumption is
that, at the DEAR density (about 40 rSTP), these ratios should not substantially change. Taking as input the above KEK results, “experimental-like” distributions (corrected for detector efficiency) DEAR-compatible were generated and the precision obtainable in defining the position of the Kaand Kblines was studied as a function of the percentage
of background and of statistics. The percentage of background is defined as the ratio between the number of background events (physical background) and the total number of events in the region of the Kapeak (defined as the region given by the convolution of
the Kaintrinsical width with the experimental resolution).
TABLEIII. – Results of the cascade-constrained iterative fit in the KEK-experiment [10]. The two
columns show the intensity of the K-series lines, as obtained performing the fit at the Kaenergy
(6.5 keV) and in the full energy range 3–10 keV.
Ka(6.5 keV fit) (3–10) keV fit
DOF 31 137 x2 P 24.453 162.5 x2PODOF 0.789 1.19 e1 s(eV) 2327 6 61 2327 (fixed) G1 s(eV) 394 6 198 394 (fixed) Ka(counts) 113.2 6 24.5 102.7 Kb(counts) 50.1 (fixed) 50.3 Kg(counts) 68.5 (fixed) 68.8 Kd(counts) 112.5 (fixed) 112.9 Ke(counts) 60.9 (fixed) 61.2 Kz(counts) 18.0 (fixed) 18.1 Kh(counts) 5.3 (fixed) 5.3 G2 p(meV) 0.29 6 0.11 0.29 6 0.11
T (eV) 1.0 (fixed) 1.0 (fixed)
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The generated distributions contained:
1) a Voigtian function describing the Ka line, namely V(E , EKa4 6.2 keV, GKa4
400 eV, Rexp4 150 eV);
2) a Voigtian function describing the Kb line, namely V(E , EKb4 7.4 keV, GKb4
400 eV, Rexp4 150 eV);
3) a Voigtian function describing the KF g-complex, namely V(E , Ec4 8.1 keV, Gc4 650 eV, Rexp4 150 eV);
4) a function (linear) to describe the background, its shape being derived from the KEK results.
For different background percentages and statistics, 50 distributions were generated, and the resulting spectra were then fitted with a 10-parameter function: (5) ffit(E) 4AKaQ V(E , EKa, GKa, Rexp) 1AKbQ V(E , EKb, GKb, Rexp) 1
1AKcQ V(E , EKc, GKc, Rexp) 1 (aQE1b) . The 10 free parameters are: AKa, EKa, GKa, AKb, EKb, AKb, EKc, GKc, a and b, whilst GKbwas
constrained to be equal to GKa.
A typical example of an “experimental-like” distribution (randomized and convoluted with the experimental resolution), for a background level of 60% and a statistics of 10000
Fig. 3. – “Experimental-like” distribution for the K-series lines of kaonic hydrogen, as seen by the DEAR experiment, for 60% of background and 10000 events in the Kapeak.
TABLE IV. – Precision obtainable in the determination of the Ka and Kb positions with DEAR.
No. of Kaevents % of background Kaprecision (eV) Kbprecision (eV)
10 000 0 3.4 6 0.1 8.3 6 0.5 10 3.6 6 0.1 8.8 6 0.5 30 4.3 6 0.2 10.3 6 1.0 60 6.0 6 0.3 13.8 6 2.3 90 11.9 6 0.4 22.9 6 3.5 5 000 0 4.7 6 0.2 11.9 6 0.6 10 5.2 6 0.2 12.6 6 1.5 30 6.1 6 0.3 14.4 6 1.8 60 8.5 6 0.6 18.7 6 2.1 90 18.6 6 1.9 33.9 6 3.4 2 000 0 7.4 6 0.2 18.3 6 2.1 10 8.0 6 0.6 21.6 6 3.8 30 9.6 6 0.8 23.7 6 5.0 60 13.3 6 1.2 28.6 6 5.4 90 29.0 6 6.4 60.8 6 34.3
events in the Kapeak, is shown in fig. 3. A pronounced shoulder is visible at the position
of the Kbline.
As far as the precision in determining the position of the Kaand Kblines is concerned,
the results of the fit on the experimental-like distributions, done with function (5) (mediated over the 50 distributions), are presented in table IV, as a function of the background level and of the number of events in the Kapeak.
From table IV, it turns out, for example, that for 10 000 events under the Kapeak, the
complex KF bcan be disentangled with DEAR and the Kbline identified with a precision
of about 4 % for a background of 30%. For the same conditions of background and statistics, it turns out also that the position on the Kaline can be determined at the same
level of precision of systematics (CCDs energy scale calibration plus error on e.m. energy levels), i.e. few eV. Therefore, the shift of the 1 s level in kaonic hydrogen due to strong interaction (of the order of 300 eV) can be determined with the precision of percent. The measurement of the shift of the Ka line in kaonic deuterium at the same
level of precision allows to pursue the main goals of the DEAR scientific programme, described in the following section.
5. – The DEAR scientific programme
The objective of the DEAR experiment is to perform a precise determination of the isoscalar a1 4 ( 1 O4 )(a01 3 a1) (6) and isovector a2 4 ( 1 O4 )(a12 a0) (7)
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K N scattering lengths (a0 and a1 are the I 40, 1 scattering lengths) through a 1% measurement of the Ka line shifts of kaonic hydrogen and kaonic deuterium due to
strong interaction.
The aim of this determination is understanding chiral symmetry breaking in meson-hadron interaction.
The fundamental quantities associated with the chiral symmetry breaking are the KN sigma terms, which give the measure of degree of chiral symmetry breaking in the KN system.
The KN sigma terms have been determined mostly by extrapolating to the unphysical region of the vanishing kaon mass by the use of KN and K N dispersion relations. The extrapolation requires knowledge of the relevant KN and K N amplitudes. The standard procedures use the full spectra of the K1p and K2p amplitudes, that include the below threshold contribution from not only the I 40 S-wave L(1405) resonance, but also from the I 41 P-wave S(1385) resonance.
The aK1p scattering length is known rather well, according to Barnes and Swanson [12]:
aK1p4 2 0.31 1 0.01 fm . (8)
The S(1385) parameters are determined better than the L(1405) parameters [13]: S( 1385 ) mass 41387.2 6 0.2 MeV, width 4 35.8 6 0.8 MeV ;
L( 1405 ) mass 41406.5 6 4.0 MeV, width 4 50 6 2 MeV .
It follows that the K N scattering lengths have to be determined at a precision level of
few percent. One expects [14] that a 1% measurement of the K N scattering lengths will enable the determination of the KN sigma terms within an uncertainty of about 20%.
In the SU(3) description, the KN sigma terms are strongly correlated with the strangeness content of the proton, as emphasized by Jaffe and Korpa [15]. In particular, their I 40 part [16]: s( 0 )KN 4 ms1 m ms2 m 1 1y 1 2y
k
aNN2 3 4H8( 0 ) NNbl
1 Og
md1 mu ms2 mh
, (9)is an extremely sensitive and direct measurement of the strange-sea unpolarized quark component in the nucleon at rest, measured by the renormalisation-scale invariant ratio:
y 4 aNNssNNb
aNN(1O2)(u u1d d)NNb . (10)
One can easily see that the above KN sigma-term is a more direct indication of the
value of y than the pN one [15, 16]: spN4 m ms2 m 1 1 2y[aNN23H8( 0 ) NNb] , (11)
where it is hard to separate the strong dependence on y from the equally strong one on the light-quark mass ratio ms/m.
According to Jaffe and Korpa [15], when the sigma terms are determined with the expected 20 % accuracy, the strangeness content of the proton can be determined at about 5 % precision level.
6. – Conclusions
Disentangling, for the first time, the KF b complex in kaonic hydrogen is demonstrated to be possible with the DEAR experiment. For a statistics and a background level which allow the determination of the Kaline with the precision of 1 %, it
becomes realistic to identify the Kbpeak with a precision lower than 5%. This represents
a direct check for their identification as originating from the K2p atom and, consequently, allows to distinguish them from the background of other pionic and kaonic atoms X-ray lines, which were a serious problem in all previous experiments.
A background-free determination of the Ka line of kaonic hydrogen, together with
the same precision measurement of Ka in kaonic deuterium (the DEAR programme),
allow to make a real breakthrough in the field of low-energy K N interaction. It would in fact be possible to make the first determination of the I 40 KN sigma term, which gives the degree of chiral symmetry breaking in hadron interaction and defines the content of strangeness of the nucleon.
R E F E R E N C E S
[1] DESERS. et al., Phys. Rev., 96 (1954) 774; TRUEMANT. L., Nucl. Phys., 26 (1961) 57; DELOFFA.,
Phys. Rev. C, 13 (1976) 730.
[2] DAVIESJ. D. et al., Phys. Lett. B, 83 (1979) 55. [3] IZYCKIM. et al., Z. Phys. A, 297 (1980) 11. [4] BIRDP. M. et al., Nucl. Phys. A, 404 (1983) 482. [5] IWASAKIM. et al., Phys. Rev. Lett., 78 (1997) 3067.
[6] DEAR COLLABORATION(BALDINIR., GUARALDOC. et al.), DAFNE Exotic Atoms Research (DEAR Proposal), LNF-95/055 (IR) (1995); DEAR COLLABORATION(BIANCOS., GUARALDOC.
et al.), The DEAR case, unpublished.
[7] DAFNE Machine Project, Contribution to ECFA 96, LNF-96/033(P) (1996).
[8] EGGERJ.-P., CHATELLARDD. and JEANNETE., Particle World, 3 (1993) 139; FIORUCCIG. et al.,
Nucl. Instrum. Methods A, 292 (1990) 141; VARIDELD. et al., Nucl. Instrum. Methods A, 292 (1990) 147.
[9] BATTYC. J., Intense Hadron Facilities and Antiproton Physics, edited by T. BRESSANI, F. IAZZIand G. PAULI(SIF, Bologna) 1990, p. 117.
[10] ITOT. M., Observation of Kaonic Hydrogen Atom X Rays, Ph.D. Thesis, University of Tokyo, UTPN-227, 1997.
[11] BORIEE. and LEONM., Phys. Rev. A, 21 (1980) 1460.
[12] BARNEST. D. and SWANSONE. S., Phys. Rev. C, 49 (1994) 1166.
[13] BARNETTR. M. et al., Review of Particle Physics, Phys. Rev. D, 54 (1996) 623, 641.
[14] SEKIR., 6th DEAR Meeting, Frascati 1997, DEAR Internal Note, unpublished; DICLAUDIO B., VIOLINIG. and RODRIGUEZ-VARGAS, Lett. Nuovo Cimento, 26 (1979) 555.
[15] JAFFER. L. and KORPAC. L., Comm. Nucl. Part. Phys., 17 (1987) 163.
[16] GENSINI P., Physics and Detectors for DAFNE, edited by G. PANCHERI (Frascati) 1991, p. 455.
