Chapter 5
Protonium and antiprotonic deuterium

In this Chapter we describe the measurement of X - rays   from antiprotonic hydrogen and deuterium atoms and discuss how the strong-interaction parameters are deduced from the data. Many interesting phenomena are associated with this search, a large number having been observed in the Asterix experiment. So, instead of introducing these phenomena as abstract concepts, we discuss them as they show up when presenting results from this experiment.

5.1 PS171: The Asterix experiment

The Asterix experiment was carried out by stopping antiprotons in gaseous H-
s2  at room temperature and pressure. A short description of the X-ray detector can be found in Sec. 2.4, a report on the performance of the full spectrometer in  [150].

5.1.1 X-ray spectra with pp   annihilation into charged particles

Figure 5.1 shows the X - ray   spectrum observed in coincidence with two charged particles after stopping 200MeV /c  antiprotons. A double-peak structure is seen at low energies while the high-energy spectrum contains much fewer entries. The low-energy peak at 1.74keV  is due to the Balmer-a  line, the smaller peak at 3.1keV  to a convolution of Balmer lines close to the series limit, denoted as Ls oo  . Part of the 3.1keV  peak is due to argon fluorescence: charged particles (or high-energy X-rays) kick an electron out of the K shell of an Ar atom. The K shell is then re-populated by emission of a 3 keV  photon. In a segmented detector the photon may escape the local detector segment and convert elsewhere. This contribution can be estimated from the X - ray   spectrum observed when there are only neutral particles in the final state (see Fig. 5.3a below) for which the argon fluorescence contribution is much smaller.

The large peak in Fig. 5.1 due to the Balmer-a  line is obviously not accompanied by a Lyman-a  line of similar strength. The spectrum evidences at the first glance the importance of annihilation from the 2P levels of protonium atoms: pp   atoms in 2P states annihilate instead of radiating Lyman-a  X-rays. A fit to the data  [152] shows some (weak) evidence for Lyman radiation with a fractional intensity of the Ksa  line of (1.0  0.4) 10-3  , compared to a total yield of the Balmer series of (12 2)  %. These data are used - together with Eq. (3.73) - to determine an average 2P strong interaction width of (45 18)  meV.

Figure 5.1: Energy spectrum of X - rays   emitted by protonium atoms annihilation into final states with two charged particles. The line represents a fit taking into account the Balmer series of pp    atoms, a (small) constant background, inner bremsstrahlung (solid line), and contributions from the Lyman series (dashed curve). The data on the right are multiplied by 20.

The continuous background at higher energies originates from inner bremsstrahlung which is emitted in the spontaneous acceleration of charged particles in the annihilation process. The emission of bremsstrahlung quanta can be calculated in the soft-photon approximation  [346]. For a final state characterized by a partial width dG0   , the charges Qi  of the outgoing mesons and their four-momenta ~qi  , the differential width for emission of a bremsstrahlung photon (after summation over all polarizations) can be written as

      (    )
dGB =  -a-  dWBw - 1dwdG0,


          sum   sum 
dWB  = -       QiQj --~qi.q~j---w2d_O_k,
          i  j      (~qi.~k)(~qj.~k)

where ~k  is the four-momentum vector (w,k)  of the photon, and d_O_k  the solid angle element into which the photon is emitted. The photon energy distribution (5.1) exhibits the expected w-1  behaviour, and Eq. (5.2) gives the angular correlation between bremsstrahlung and atomic X-rays.

Photons emitted radiatively as part of the atomic cascade lead to a small alignment of the pp
 angular momentum states; thus the direction of a primarily produced meson resonance is also correlated with the photon direction  [347]. But in the subsequent decay of the resonance, this weak correlation is washed out; angular correlations between, e.g., Balmer X - rays   and charged particles can be neglected.

Equations (5.1,5.2) can be evaluated in the rest system of the two charged particles assuming that they are pions; contamination by kaons is known to be small. The measured angular distributions depends on experimental cuts taken into account in the Monte Carlo simulations. In particular, to identify the X-ray, a minimum angular spacing is required between the direction of a X - ray   and that of the nearest charged particle.

Figure 5.2: Angular distributions of X - rays   originating from the atomic cascade of antiprotonic hydrogen atoms (E < 5keV )  and from inner bremsstrahlung (E > 5keV )  . Histogram: data; solid line: Monte Carlo simulation.

Figure 5.2 (upper panel) shows the angular distributions for X -rays   with energies below 5 keV  (mostly originating from the atomic cascade). The angle h  is defined with respect to the direction of the higher-momentum particle; the angle f   is the polar angle, f = p   for X - rays   which are found in the direction of the total laboratory momentum of the two charged particles. The solid lines represent an isotropic distribution in the laboratory system. Figure 5.2 (lower panel) shows the angular distributions for X - rays   above 5 keV  (originating mostly from bremsstrahlung). It was calculated from the measured particle momenta using formula (5.2) to calculate the direction of the photon emission. The differential bremsstrahlung-emission width was calculated for each individual event. After summation over all events, a theoretical bremsstrahlung spectrum is obtained.

The two processes - radiative transitions to lower energy levels with subsequent annihilation and annihilation with emission of inner bremsstrahlung - lead to the same final state, and interference effects may occur. They could be particularly disturbing for the Lyman series since in this case the amplitudes for radiative transitions and for bremsstrahlung are of comparable magnitude. The phase of the amplitude changes by 180o while crossing the resonance position while the phase of the hadronic transition amplitude remains, at least approximately, constant. Wrong results for the line centre and width could be obtained when constructive and destructive interferences on the two sides of the Ksa  -line are neglected. This effect could be particularly dangerous in low-statistics experiments in which the line shape cannot be unambiguously determined from data. In inclusive protonium experiments, many hadronic final states contribute likely having different, statistically distributed, hadronic phases. Hence there is a good chance that the superposition of all lines should result in an undistorted line shape. Only when exclusive final states are selected, interferences are likely to play a role.

5.1.2 X-ray spectra with -
pp   annihilation into neutral particles only

The contributions from bremsstrahlung and the residual background can be suppressed quantitatively by two further techniques:

The fraction of -
p   annihilation in liquid H-
s2  into all-neutral final states is given by the ratio of frequencies f   [348]

f(pp-- >  neutral particles)
--f(all annihilations)--= (3.6 0.4)%

In all-neutral final states (defined by the absence of detected charged particles), a small bremsstrahlung contribution survives due to events in which two charged particles were produced but escaped detection. The coincidence requirement (together with an energy cut) displays only those events in which the atomic cascade reached the 2P level. Only the Ksa   line is observed and thus the interpretation of the resulting X - ray   spectrum is facilitated.

Figure 5.3: X - ray   spectrum of protonium for events with neutral particles only in the final state. a) The spectrum shows the Balmer series but the residual background is still too large for an unambiguous identification of the Lyman series. b) X - ray   spectrum of protonium for all-neutral events for two coincident X - rays  . The energy of the more energetic X - ray   is shown. The peak at 8.67keV  is due to the Lyman-a   line. The line shape theory of Ericson and Hambro is used to fit the data.

Results using 300 MeV /c   antiprotons were reported in  [151]. Figure 5.3a shows the X- ray
 spectrum obtained by stopping 105MeV /c  antiprotons and requiring absence of any charged particle in the final state  [153]. The spectrum is dominated by the Balmer series and does not yet allow unambiguous conclusions on the presence of the Lyman series. The situation improves when coincidences between two X - rays   are required. In Fig. 5.3b only events are considered with two X - rays   observed in coincidence; the energy of the more energetic X -ray   is plotted. The low-energy peak at 1.74keV  originates from coincidences of a line belonging to the M series with a Lsa   line; note the absence of Ls oo  . The broad peak at about 8 keV  stems from coincidences of an L X- ray   with a Ksa   line. The apparent width of the peak (~ 4keV  ) is much broader than the experimental resolution of 2.4keV  indicating a sizable broadening due to strong interactions. When a cut is made to select events with one X- ray   in the (6.9 - 10.9)keV  energy interval, the energy distribution of the second X - ray   shows contributions from the full Balmer series.

The proper line-shape theory for broad lines (for which G  e  is not satisfied) is by no means trivial. To first order, the Ksa  line-shape should correspond to a Lorentzian distribution. The detector resolution has to be taken into account. The convolution of a Lorentzian line with a Gaussian resolution function is called Voigt function

         + oo -------A---------      (E'--E)2-   '
F(E) =  - oo  (E - E0)2 + (G/2)2 exp{-  2 s2  }dE

When the detection efficiency varies across the line, the Voigtian line shape has to be folded with it. The numerator A  in (5.4) is constant only when the transition matrix element for the radiative transition does not dependent on energy. This is, however, not the case. Electric dipole transitions prefer more energetic transitions suggesting that A  in (5.4) is proportional to E3   . However, this integral would diverge. The situation was analysed by Ericson and Hambro  [349] who developed a line-shape theory for broad lines. From the requirement of probability conservation, they deduced that the term A  contains an energy dependence

         ----(2--b)(1+-b)2------[   (             b---2)]2
IEH(E) =  b2(2 + b)7(4 -b)2(3- b)2  2F1  5,1- b,5- b;b + 2    ,

where b = (1/4+ E/12.49)-1/2  and s2  Fs1  is the hypergeometric function. In the proximity of the resonance energy E
  0   , the relation I   (E) = E
 EH  holds to a good approximation. A description of results from fitting the data of Fig. 5.3 with different line shape theories can be found in  [350]. Eq. (5.5) gives the best description of the data, with a central energy of the line at E  = 8.67 0.15keV
 1S  . The photon energy is shifted to a lower value than expected from QED (e
 1S  is negative), the 1S level is shifted to a higher energy (DE
   1S  is positive). Thus a final result of

e  = -DE   = - 0.74  0.15keV ,    G  = 1.60 0.40 keV ,
1S       1S                       1S

is obtained.

The yield of Balmer X - rays   in all-neutral events is (13 2)  %. This is surprisingly close to the value (12 2)  % obtained with charged particles in the final state. Annihilation from S-states into all-neutral events proceeds dominantly only via the 1S0   state (annihilation into any number of p0  and j  requires positive C  -conjugation) which has a statistical weight of 1/4. In P-wave, annihilation into all-neutral final states is allowed from spin-triplet states, with a statistical weight of 3/4. In spite of this statistical argument the probability of the protonium atom to annihilate into all-neutral events from P-states is only (13 2)/(12 2)  times larger from S-states. We deduce

fP(pp-- >--neutral particles)-= (3.9 1.0)% .
  fP(all annihilations)

The fraction of all-neutral events in which a Ksa  X -ray   is emitted after a Balmer X - ray
 is detected is rK = (0.79 0.20)  %. We may now assume that all-neutral events come only from positive-parity states. (This is true for the majority of all-neutral events but not for those which contain strange particles or w  -mesons decaying into p0g  .) And we assume that the protonium spin is conserved in radiative transitions, that there are no intercombination lines. Then we can deduce the strong interaction width of the 21P1

The total number of L X -ray   followed by annihilation into neutral particles is (13 2)%  (3.9 1.0)%  . The number of Ksa  X - ray   followed by all-neutral annihilation is rK  times smaller. This gives

G(21P1) = 51 18meV .

5.1.3 The cascade time

The Asterix collaboration determined the cascade time of p  p
 atoms  [154]. For 90 000 events with four tracks coming from a common vertex, the annihilation point was determined (with a resolution of  2  mm). The data were split into events with vertices in the entrance counter T2 and in the exit counter T4 and, along the Hs2  target, into 12 slices of 5 cm length (see Fig. 2.17). For each of these 14 positions, the mean time of the two inner proportional counters was determined.

Fig. 5.4 shows the mean time spent by antiprotons the scintillation counter at the entrance of the target and their annihilation. The time increases with the distance traveled by antiprotons before capture, but those stopping immediately after the scintillation counter T2 need

t = (5.1  0.7)ns

more time than those stopping in T2. The cascade time in solids is some ps only, hence the time difference t  corresponds to the time elapsed between capture in Hs2  and annihilation, at normal pressure and temperature,

The solid line in Fig. 5.4 is obtained by numerical integration of the Bethe-Bloch formula. The integration also yields the kinetic energy of antiprotons stopping in front of T4: when they entered the target they had Ekin = 3 MeV  .

The Obelix experiment  [351] performed similar measurements using Hs2  gas at four different pressures. They determined the time antiprotons need to reach the downstream end of the target. Antiprotons stopping in gas need a long time and produce the Gaussian time distribution in Fig. 5.5. Antiprotons reaching the exit wall may have some residual kinetic energy. These antiprotons traverse the target at a larger speed and the measured time difference is smaller. These antiprotons produce the steep rise at short times in Fig. 5.5. The time of flight of antiprotons stopping on the wall surface is obtained by linear extrapolation to the very latest antiproton annihilation on the wall. The difference between the Gaussian peak and extrapolated value gives the cascade time. A finite time resolution gives entries at late times, neglecting it favors cascade times which are too short. Table 5.1 lists the final results.

In liquid Hs2  , the cascade time is too short to be measured. However, for the S- p system, the time can be deduced which elapses from the moment where the S- has a residual velocity of 0.004 c (where the sum of the measured decay momenta does not yet vanish) to nuclear absorption  [352]. This time is an upper limit for the cascade time and, likely, a good estimate for it. It is included in Table 5.1. For convenience, we summarise the results in Fig. 5.6.

Table 5.1: Cascade time for antiprotons stopping in Hs2  as measured in the Asterix and Obelix experiments.

Pressure (mbar)
Cascade time (ns)

LHs2  S- p < 5 10-3   [352]
STP pp  5.1  0.7   [154]
150  1   -
pp  6.7  1.1   [351]
9.8 0.05  pp  34.3 2.4   [351]
5.8 0.05  pp  59.9 6.0   [351]
3.4 0.05   -
pp  84.1  10.3   [351]

Figure 5.4: The mean time at which annihilation occurs as a function of the annihilation vertex. Annihilation immediately after the entrance window is delayed by 5.1 ns compared to annihilation in the entrance window due to the time which elapsed from capture of an antiproton in gaseous Hs2  and annihilation.

Figure 5.5: Annihilation time distribution of events for which the vertex is reconstructed close to the exit wall. In-gas annihilation show a (late) Gaussian distribution. Antiprotons with higher velocities reach the end wall and annihilate early; the latest antiprotons (defined by the linear fit) annihilate on the wall surface.

Figure 5.6: Cascade time of antiprotons in Hs2   . Dots: Obelix; square: Asterix; upper limit ( |,  ): from S - stopping in liquid Ds2  .

5.2 PS174: The cold gas experiment

The cold gas experiment used a variable Hs2   gas density over the range from 10 to 1/8 times STP. The first results were obtained using antiprotons with momenta of 300MeV /c   and a Si(Li) detector  [168]. We discuss here the final results using Si(Li) detectors  [169] and two gas-scintillation proportional detectors (GSPD)  [167].

Figure 5.7: X - ray   spectrum of protonium and antiprotonic deuterium formed in gas at 0.25  -
rsSTP   as observed in the cold gas experiment by Baker et al. Individual peaks due the the Balmer series are clearly identified. The detection efficiency falls off rapidly at low energies: the L -
sa  line at 1.74keV  is the strongest line.

Figure 5.7 shows the low-energy part of the X - ray   spectrum of antiprotonic hydrogen and deuterium atoms. Contributions from individual lines of the Balmer series are clearly identified. Similar data were obtained for a wide range of Hs2   densities; they provide valuable information about the cascade processes which precede the emission of X - ray   lines. The intensities are listed in Tables 5.4 and 5.5.

The high-energy part of the X - ray   spectrum  [169] is shown in Fig. 5.8 for various Hs2
 densities. There are striking differences between the data sets: at the largest density, only one line is observed which can be identified with the limit of the Lyman series. At moderate densities a further line is seen, the Ksa  line, which becomes the strongest component at the lowest density. Below atmospheric pressure, background lines from antiprotonic oxygen and carbon show up due to inadequate gas tightness. In comparison to the Ksa  line, they have narrow widths. The data were fitted with a polynomial background and a complex of eight lines corresponding to Ksa  , Ksb  , ...  , Ks oo   transitions. Their relative strengths were taken from the results of cascade calculations (see Section 5.5). The published strong-interaction parameters were obtained with the Voigt function, the use of the Hambro-Ericson line shape not leading to any significant change  [353]. The strong interaction parameters determined from the five data sets are mutually consistent and give average values of

e1S = - 0.75 0.06keV ,    G1S = 0.90 0.18 keV ,  G2P = 45 10meV  .

Figure 5.8: X- ray   spectrum of protonium atoms for different gas H-
s2   gas densities. The line at 11.78 keV  is assigned to the Lyman series limit, the line at 8.66 keV  observed at low H -
s2   densities to the K-
sa  line.

In parallel to the data with the Si(Li) detectors, the GSPD was also used to detect the X - rays  . The fit to the results, shown in Fig. 5.9, gives  [170]

e1S = -0.73 0.05keV     G1S = 1.13 0.09 keV .

Figure 5.9: X - ray   spectrum of protonium atoms. The line at 11.78 keV  is assigned to the Lyman series limit, the line at 8.66keV  observed at low Hs2   densities to the Ksa

The Balmer series was not observed because of the need of rather thick windows. Thus no value was determined for G2P  . A search for the Lyman series of p  d   atoms was not successful and only upper limits (95% confidence level) for the yield of Ks> g  X-rays per stopped antiproton were given: 8 10-4   and 5 10-4   at 0.25 and 0.92 rSTP  , respectively  [170].

5.3 PS175 and PS207: The inverse cyclotron experiment

5.3.1 PS175

The high stopping-power of the inverse cyclotron experiment allowed the use of very thin targets or of very low Hs2  gas densities, where Stark mixing plays only a minor role. Early experiments had to use a beam with 300 MeV /c   momentum  [173]; we show only the final results  [174].

Figure 5.10 shows expanded views of a high-statistics run using a target pressure of 30 mbar Hs2  (top) and Ds2  (bottom). The comparison of the two spectra evidences a peak at above 8 keV, the Ksa  line and a small contribution from Ks oo  . No sign is seen from any pd

Figure 5.10: X -ray  spectrum of protonium (top) and of -
pd   atoms (bottom). The data are collected using a high-resolution guard-ring protected Si(Li) detector. The low-energy range is shown on the left, the high-energy range on the right.

The authors of Ref.  [174] tried to extract information on the energy splitting between the 3S1
 and 1S0   ground states. The data are compatible with the assumption that the observed energy-distribution (after background subtraction) is composed of two lines, but one line is sufficient to fit the data.

More data were recorded than shown here. The K-lines were searched for with two Si(Li) detectors and an X -ray   drift chamber. The three sets of data were combined into the final results

e1S = -0.727 0.023 keV ,   G1S = 1.160  0.078keV .

Data with lower statistical significance were recorded at pressures from 16 to 120 mbar. The intensities are listed in Tables 5.4 and 5.5.

5.3.2 PS207

In 1990, it was proposed to combine the cyclotron trap with a high-resolution crystal spectrometer to study the line splitting and broadening of the 2P hyperfine levels of pp
 and pd   atoms  [354]. We first discuss the results obtained using as X -ray   detectors three Charged Coupled Devices (CCD’s) and not yet the crystal spectrometer. With these detectors, the Balmer series and the Lyman-a  line of protonium  [175176] and of antiprotonic deuterium  [177] atoms were studied.

Figure 5.11 shows the energy spectrum observed when antiprotons were stopped in the cyclotron trap operated at 20 mbar of Hs2  (a) and Ds2  (b) gas. The low-energy range demonstrates remarkable achievements in technology: individual lines are clearly identified, the Lsa  being the strongest one. Even the M series limit is observed, in Hs2  as a shoulder, in Ds2  as a peak.


Figure 5.11: X- ray  spectrum of protonium (left) and p  d atoms (right), from experiment PS207.

The high-energy part of the spectrum is contaminated by lines from electronic fluorescence and from heavier antiprotonic atoms. In addition, there are at least two further sources of background. One source is present in pp   and pd   data and called ‘standard CCD background’. It contains inner bremsstrahlung (as discussed in Section 5.1.1) and contributions from other sources. The ‘standard CCD background’ was assumed in  [176] to be the same for Hs2  and Ds2  . The difference of the spectra for Hs2  and Ds2  can, however, not be assigned to the Lyman series. There is an additional source of background of unknown origin, present only in the Hs2  data. This additional background was assigned to coherent interference between K-lines and bremsstrahlung, and then subtracted incoherently by the requirement that all three CCD’s gave compatible results on X- ray   energies, widths and yields for the K line series. The correlation of background fit and final result was not taken into account when the final errors were evaluated.

In Fig. 5.12 we present the energy spectra after background subtraction. The observation of Ksa  lines from pp   is obvious. The data are fitted with a single Voigtian function. The results from this fit and two analogous fits to data from two further CCD’s are displayed in Table 5.2.

Figure 5.12: X - ray   spectrum of protonium atoms after background subtraction and a fit using a Voigtian function. The dashed and dotted lines describe possible contributions from the spin triplet and spin singlet component.

Table 5.2: Energy shift and width (in eV) of the 1S width of p  p   and p  d   in the three CCD’s and the final result.

epp  Gpp  epd  Gpd

CCD1-(642.6  61.3)  1109.4  211.0  - (1077  380)  1496 762
CCD2-(714.4  23.8)  1023.3   74.6  - (838  243)  1130 452
CCD3-(751.7  51.0)  1182.6  176.0  - (1358  98)  541 205

Final -(712.3  20.3)  1053.5   65.3  - (1050  250)  1100 750

The authors attempted to split the line into contributions from ortho- and para-protonium even though the data show no visible shoulder. A free fit with two energies, two widths (folded with experimental resolution) and two intensities does, indeed, not converge. Hence the authors decided to guide the fit by subsequentially freezing and releasing parameters. Clearly, the parameter space is not fully explored. With this warning we quote their result on the hyperfine splitting:

3S1 :    e1S = -0.785  0.035keV ,    G1S = 0.940 0.080 keV .      (5.13)
1S0 :    e1S = -0.440  0.075keV ,    G1S = 1.200 0.250 keV .      (5.14)
The intensity ratio of the two hyperfine lines was determined to be
Y(2P --> 3S )/Y (2P --> 1S ) = 2.75 0.06.
         1           0

The evidence for Ksa  lines from pd   atoms is much weaker (Fig. 5.13b). There are several contaminant lines from gas impurities and from the target vessel; in particular the presence of a p  O(7-6) line at the proposed Ksa  energy is very unfortunate.

pict pict

Figure 5.13: X - ray   spectrum of -
p  d atoms before (left) and after (right) background subtraction. The sum of contamination lines and of a polynomial function is used to subtract the background. right) X -ray   spectrum of -
p  d   atoms after

Nevertheless it is possible to subtract the background contribution in such a way that an excess of events is seen in the region where the Ksa  line is expected. The results of fits to the three difference spectra listed in Table 5.2 are not fully compatible. A systematic error is introduced to account for the correlation between the results of the fits and the background subtraction. The final result reads
e1S = - 1.05 0.25keV ,  G1S = 1.10  0.75keV ,    80 < G2P < 350meV  .

We caution the reader that in our view the identification of the observed structure with the K -
sa  line from -
pd   atoms is not unambiguously established. Also the yield of               -3
(2.3 1.4)  10   at a target pressure of 0.02rSTP  seems rather high. Batty, using cascade calculations, estimates the yield to be lower by one order of magnitude  [355].

In parallel to the search for K X-rays, the line profile of the Balmer-a  radiation from antiprotonic hydrogen and deuterium atoms was measured with a crystal spectrometer. To combine highest energy resolution with a sufficient count rate, a Bragg spectrometer was set up, in Johann geometry, equipped with spherically bent crystals. Three two-dimensional position-sensitive pixel detectors (CCD’s) were used for X-ray recording.

Fig. 5.14 shows the line profile of 3D to 2P transitions for one of the three detectors. The expected splitting of the D levels is negligibly small so that only four lines are expected, corresponding to transitions to the 23
 P2  , 23
 P1  23
 P0  , and 21
 P1  levels. The data on  -
 exhibit a shoulder at the high energy side. Its relative intensity was determined by a two-component fit to be (9.5  0.9)%, in good agreement with the statistical population of 8.3% for the   3
2 P0  level. This level is expected to have a large strong-interaction shift. Hence, the high-energy tail is likely due to  3      3
3 D1 --> 2 P0  transitions. The main part of the line is attributed to the (unresolved) transitions to the three hyperfine levels, labeled ( 3    1    3
2 P2,2 P1,2P1   ).


Figure 5.14: Balmer series of protonium atoms. The fine structure components 23  -
 Ps2  , 21  -
 P s1  have common strong interaction shifts and widths, the fine structure component 23  -
 P s0  can be identified as individual contribution.

The mean energy of the D --> (23P2,21P1,23  Ps1  ) transitions and the energy of the individual D--> 23P0  line were determined from a two-component fit to the measured line shape. The energy profile of the group was constructed from the individual contributions with positions and widths calculated from QED (see Tab. 3.3) and from NN   potential models or using a single (broadened) Gaussian or Voigtian distribution. Both treatments lead to the same results on strong interaction parameters for the 23P0   line. The relative intensities were always fixed to the statistical values. Under these assumptions the authors derived

e(23P2,21P1,23P1) = +4.0 5.8 MeV ,   G(23P2,21P1,23P1) =  38  9MeV , (5.17)
          e(23P0) = +139 20 MeV ,            G(23P0) = 120 25MeV . (5.18)
The energy shifts quoted in (5.17,5.18) are of hadronic nature. The determination of the average hadronic width G(23P ,21P ,23P )
     2    1    1  relies on the (very reasonable) assumption that the small splittings within the multiplet are known with sufficient precision.

From the intensity ratio r'= K /L
     a   tot  (see Eq. 3.73) the authors deduce a spin-averaged value

G  = 44  8 meV

The results on r' and on the direct crystal spectrometer measurement were combined assuming that GX   G(23P2)  ~~  G(21P1)  ~~  G(23P1)  yielding:

         G(23P0)  =   120   25 meV                (5.20)
G(23P2,21P1,23P1)  =   30.5  2.0 meV                (5.21)
             G2P  =   38.0  2.8 meV .              (5.22)
The last value uses our knowledge on the different fine-structure levels and is thus more reliable than (5.19), and more precise.

Using the same set up, data were taken also with D-
s2  as target gas. The splittings within the D levels were again neglected; hence a line quintuplet is expected. In a first attempt, QED splittings as given in  [259] were used with a common resolution given by the -
p  20  Ne calibration line. The fit did not reproduce the data. Much better agreement was obtained when the electromagnetic hyperfine splittings from  [260] (Table 3.3) were used. These splittings are small enough to treat the whole multiplet as a single line which is fitted with one Voigt profile. Imposing the splittings from  [260] and the hadronic shifts as given by  [261] did not effect the final result. A common broadening of all substates was a free parameter in the fits. The relative intensities of the hyperfine transitions were frozen to represent a statistical population of the 2P sublevels. The three detectors gave consistent results; we quote the weighted average as final result for the spin-averaged hadronic shift (negative, i.e., repulsive) and broadening of the 2P levels of antiprotonic deuterium:

e2P  =  - 243  26 meV ,                     (5.23)
G2P  =    489  30 meV .                     (5.24)

5.4 Summary of results on p p and p d atoms

The results on the strong interaction shift and width given in (5.10,5.11,5.12) and the mean value from from Table 5.2 are fully compatible even though we believe the errors to be sometimes underestimated. We give the linear average of the four measurements with a conservative estimate of the error:

e1S = - 0.730 0.030keV ,  G1S = 1.060 0.080keV .

Using the Trueman formula (3.90), we can relate these values to the complex S-wave scattering length

asco = (0.88 0.04)- i(0.64  0.05)fm.

Similarly, we obtain the imaginary part of the P-wave scattering volume:

    sc                 3
Im a1 = -(0.77  0.06)fm  .

The ratio of the real to imaginary part of strong-interaction amplitude is read as

r(E = 0) = G1S = - 1.38 0.12 .

In Table 5.3 we summarize the results on strong interaction parameters. They will be compared to theoretical predictions in Chapter 6.

The experiments gave X -ray   yields over a wide range of pressures. These are reproduced in Tables 5.4 and 5.5.

Table 5.3: Strong interaction shifts and widths of antiprotonic hydrogen and deuterium atoms.
Antiprotonic hydrogen atoms
Mean 2P level widths
Antiprotonic deuterium atoms

Energy shift
Energy width
e1S  = -730  30 eV  G1S  = 1060 80eV
e(23P2,21P1,23P1)  = + 4.0  5.8meV  G(23P2,21P1,23P1)  = 30.5  2.0meV
e(23P )
     0  = + 139  20meV  G 23P )
 (   0  = 120  25meV
G(2 P1)  = 51  18meV

using (3.72)
using (3.73)
G2P  = 38.0  2.8meV  G2P  = 44  8 meV

S-wave scattering length
P-wave scattering volume, imag. part
 - sc
aso  = (0.88 0.04) - i(0.64  0.05)fm
Im as1  = -              3
(0.77 0.06)fm

r  -parameter at threshold
r(E = 0)  = -1.38  0.12

Energy shift
Energy width
e1S  =-1.05  0.25  keV  G1S  =1.10 0.75  keV
e2P  = 243  26 meV  G2P  = 489  30 meV

Table 5.4: Intensities (in %) of L and K X-rays radiation from p   p atoms for different target densities.

L x-ray intensity
K x-ray intensity
rSTP  L-
sa  L-
sb  L-
stot  K-
sa  K -
sb  K-

0.016 51.9 11.0  9.2  2.5  70.6 11.6   [173]
0.016 53.2 9.3  7.7  2.4  71.9 10.0   [174]
0.03 40.7 8.5  3.7  1.3  52.3 8.9  0.62 0.17  0.91 0.19   [173]
0.03 40.2 5.0  6.2  1.3  55.1 5.6  0.81 0.15   [174]
0.06 34.7 7.5  3.5  1.5  47.7 8.0   [173]
0.06 31.9 3.8  4.8  0.9  44.3 4.1   [174]
0.120 26.5 3.3  5.2  1.0  38.9 3.7   [174]
0.125 17.2 6.5  8.4  1.9  35.7 7.0   [170]
0.25 10.3 2.2  5.2  0.6  24.0 2.4  0.28 0.08  0.03+-00..0503  0.52 0.12   [169]
0.25 0.37 0.05  0.09 0.04  0.78 0.08   [170]
0.30 9.5 2.6  1.6  1.1  17.8 3.6   [173]
0.92 3.4 0.8  2.4  0.3  11.2 1.0  0.10 0.04  0.01+-00..0401  0.36 0.07   [169]
0.92 3.4 0.8  2.4  0.3  11.2 1.0  0.18 0.04  < 0.02  0.53 0.06   [170]
1.0 13.0 2.0  0.26 0.14  0.65 0.32   [151]
1.0 5.5 1.5  12.0 2.0  0.10 0.05  0.14 0.06   [152]
2.0 3.4 0.9  1.5  0.2  8.1 1.0  0.04+-00.0.094  < 0.07  0.25 0.17   [169]
4.0 6.0 3.0  < 0.6   [134]
10 < 0.4  0.4  0.2  1.6 0.3  < 0.06  < 0.08  0.25 0.17   [169]

Table 5.5: Intensities (in %) of L X-ray radiation from p   d atoms for different target densities.

L X-ray intensity
 STP  Lsa  Lsb  Lstot

0.016 53.2 9.3  7.7  2.4  71.9 10.0   [174]
0.03 40.2 5.0  6.2  1.3  55.1  5.6   [174]
0.06 31.9 3.8  4.8  0.9  44.3  4.1   [174]
0.12 26.5 3.3  5.2  1.0  38.9  3.7   [174]
0.25 19.0 2.1  2.9  0.2  29.1  2.2   [169]
0.92 7.9 0.9  1.6  0.1  14.1  1.0   [169]
2.0 5.0 0.6  1.1  0.1  9.0 0.7   [169]
4.0 6.0 3.0   [146]
10 1.0 0.2  0.5  0.1  2.4 0.3   [169]

5.5 Cascade processes in p p and p d atoms

Most experiments on proton-antiproton annihilation at rest into exclusive final states were carried out by stopping antiprotons in a liquid hydrogen target. Annihilation at rest takes place from atomic orbits, when antiprotons with a kinetic energy of a few eV were captured by the Coulomb field of a proton or deuteron. The pp   system annihilates only from a small number of states with given quantum numbers which can be determined or at least restricted by using selection rules or by observing the X - rays   emitted in the course of the atomic cascade. The distribution of initial states can be changed by varying the target density. Hence we have a unique situation where annihilation processes can be studied with ab initio knowledge of the quantum numbers. In scattering experiments or in annihilation in flight, several partial-wave amplitudes contribute to the observables.

5.5.1 The capture process

Antiprotons stopping in Hs2  or Ds2  loose energy in collisions. Their energy loss per unit length is given by the Bethe-Bloch formula as long as their velocity is larger than ac  , corresponding to p   energies of ~ 25  keV. In Hs2  gas at STP, the range of, e.g., 3 MeV antiprotons is about 75cm; the antiprotons need 40 ns before they come to rest. Range and energy loss calculated with the Bethe-Bloch equation are in good agreement with data  [154] even though precision experiments reveal a small difference between energy-loss curves of protons and antiprotons  [356].

Below 25  keV the p   continues to loose its energy by ionisation until its energy is in the few eV range. Then it is captured by the Coulomb field of a proton by Auger emission of an electron.

p+ H2 --->  pp(nl) + e-+ H

The capture process can be followed numerically using the Classical Trajectory Monte Carlo (CTMC) method. It describes a three-body problem (antiproton, proton and electron) using a classical Hamiltonian to derive equations of motion, which are solved for a statistical choice of the so-called micro-canonical variables. Figure 5.15 shows the simulation of a capture process. The H atom is described by a classical p + e- system with a radius corresponding to the first Bohr orbit. Phases and eccentricity are chosen randomly. After ejection of the electron, antiproton and proton are bound in a flat ellipse, corresponding to a classical radius of 0.5  and to a principal quantum number n ~ 32  .

Figure 5.15: Simulation of the capture of a 5eV antiproton by a H atom using the Classical Trajectory Monte Carlo method. The plot shows the respective distances between proton and antiproton and electron (from  [357]).

The cross section for protonium formation rises steeply for very low energies and is 2-3 2   for p   energies below 13.6 eV  [358]. Most antiprotons are captured when their energy is below the H ionisation energy; the principal quantum number n  is most often between 30 and 50 (sometimes even larger than 100) and the average orbital angular momentum l  about 20  [358]. Qualitatively, the preference for protonium capture into high Rydberg states can be understood when the overlap of electronic and antiprotonic wave functions is considered. Capture will occur with high probability, when the classical radius of protonium atoms is matched to the size of ground-state hydrogen atoms. The expectation value of the atomic radius is related to n  and l  via

<rn,l> = a0 (3n2- l(l+ 1))

The “best” choice of the principal quantum number is then in the range

 V~ ----             V~ ---------
  2m-< nc < -3/2+    9+ 3 2m-,
  me                 4  2 me

or 32 < nc < 36  . For the angular momentum states after capture, a statistical population seems plausible. Calculations show that the preferred distribution in l  has its maximum at about n/2   [359].

5.5.2 Collisions between protonium atoms and Hs2  molecules

Once formed, protonium atoms collide with Hs2  molecules where they experience large electric fields inducing transitions from initial (n,l)  protonium states to other levels via dissociation of neighbouring molecules, Auger effect or Stark mixing. They are schematically represented in Fig. 5.16.

Figure 5.16: Level scheme and atomic cascade of antiprotonic hydrogen.

Chemical effects
In very high levels (for n > 20  ), pp   atoms de-excite by dissociation of the colliding Hs2  molecules:
(pp)ni,li + H2 ---> (pp)nf,lf + H + H ,
Gchem = N v p (rni)   for   dEni-->nf > D ,     li = lf .

The rate for this effect is assumed to be given by the classical “size” of the -
pp   atom and the collision frequency  [140]. D = 4.7  eV is the dissociation energy of H-
s2  molecules, N  is the density of hydrogen atoms, and v  is the protonium velocity.

Auger effect
For n ~ 20  the classical radius becomes too small to allow chemical effects to play a significant role. Yet one of the H atoms of a Hs2  molecule can be ionized and an Auger process can take place.
 -  ni,li          - nf,lf          -
(pp)   + H2 --->  (pp)     +H + p + e ,
        16p-N--( nf)2           -1/2
GAuger = 3  m2p- Rni  (2dE + 1.39)       for    dE + 1.39 > 15.2eV .

The Auger effect is induced by the electric field seen by the p-  p atom in the collision and is governed by the same matrix element (Rnnf)2
   i  as radiative de-excitation. But while radiative transitions prefer large transition energies, Auger transitions occur most frequently with a minimal change in the principal quantum number. The energy gain is then just sufficient to knock out an electron. As in radiative transitions, angular momentum changes according to lf = li 1

In collisions, peak electric field strengths of typically ~ V/ are experienced (for an impact parameter of 1.5 ) for about 20 fs. The electron density integrated over the collision time is ~ 0.1r0  fs where r0  is the electron density of H atoms at r = 0  . Protonium atoms are neutral; hence they move along straight lines if the is attraction between the two collision partners is neglected. The path can be calculated using, e.g., the CTMC method  [360]. For an impact parameter b  of 1.5 , the minimal distance shrinks with increasing principal quantum number of the protonium atom. For n = 20  , the minimal distance is by a factor 2 smaller than the impact parameter, for n = 5  this effect is negligible. The straight-line approximation therefore underestimates the effect of the electric field and in particular the Stark mixing probability.

Stark mixing
Stark mixing of states with different angular momenta is extremely important for the cascade of p-  p and p  d atoms, as first demonstrated by Day, Snow and Sucher  [139]. Many transitions between different nearly mass-degenerate angular momentum states occur in a single collision between a p  p or p  d atom and a Hs2  molecule. The electric field induces Stark mixing transitions between different orbital angular momentum states having the same principal quantum number n  . Since the direction of the electric field changes during the collision, not only transitions with Dm  = 0  occur but also transitions in which Dm =  1  . In principle, the theory involves n2   coupled Schrödinger equations with a time-dependent electric field. Leon and Bethe avoided this difficulty and use instead a shuffling model which takes into account the net effect of back and forth transitions between different l  . T
            2l+-1-                2
Gn,l-->n,l+1 = 2l- 1 Gn,l-->n,l- 1 = pN vr0

In a microscopic model, the n2  coupled differential equation are integrated numerically and transition rates from any initial state (n,l)  to the other states (n,l')  are determined.

5.5.3 The cascade

The microscopic cascade model of Reifenröther and Klempt  [361] begins with an initial population pn,l  of the protonium levels. All levels can radiate to lower levels or annihilate at any time. Collisions with different impact parameters may take place with their respective probabilities. Five different impact parameters are chosen in a way that the electric field strength in a collision reaches a maximum value of   -2  - 1    2
10  ,10  ,..,10 V  /. The impact parameters corresponding to these field values and hence the collision frequencies depend on n  . These collisions induce external Auger effect and Stark mixing. For the Figures presented here, the Auger effect was enhanced by a factor 2. This adjustment leads to a better agreement with data.

The cascade model starts at n = 30  and assumes an initial distribution in l  and calculates the depopulation of these states until the residual population of 0.1% is reached. Each X-ray emission or annihilation from a S or P state is recorded. Thus the X-ray yields, the fraction of S and P state capture and the cascade time are determined. The X -ray
 intensities listed in Table 5.4 are shown in Fig. 5.17 and 5.18 and compared with the calculation.

pict pict

Figure 5.17: L X - ray   intensity of p  p atoms as a function of target density. Left: Lsa  and Lsb intensity; right: sum of intensities of L-line series. The solid line is from the Mainz cascade model, the dotted line from Batty  [362].

pict pict

Figure 5.18: K X - ray   intensity of p  p atoms as a function of target density. Left: Ksa
and Ksb  intensity; right: sum of intensities of K-line series. The solid line is from the Mainz cascade model, the dotted line from Batty  [362].

pict pict

Figure 5.19: L X - ray   intensity of p  d atoms as a function of target density. Left: Lsa  and Lsb intensity; right: sum of intensities of L-line series. The solid line is from the Mainz cascade model, the dotted line from Batty.

The cascade of p  d atoms is very similar to that of protonium. Cascade calculations concentrated on the role of S- wave and P-wave capture  [361]. The X -ray   yields are reproduced in Fig. 5.19.

5.5.4 S- versus P capture

Cascade models predict the density-dependent probability for a protonium atom to annihilate from an atomic S-state or from a P-state. This is an important issue since the dynamics of the annihilation process depends on the angular momentum state from which annihilation occurs.

The fraction of S- and P-state capture can be determined using selection rules. For instance, annihilation at rest into  0
Ks    0
K l   is allowed from the 3
 S1   state, into   0
K s   0
Ks   from the states 3
 P0   and 3
 P2  . The number of   0
K s   0
Kl   and   0
K s   0
Ks   events found in bubble chambers at BNL and CERN  [348]:

787 events       pp  -->   K0sK0l ,                   (5.35)
 4 events       pp  -->   K0sK0s ,
show a strong preference for the annihilation into K0s  K0l   and evidence the dominance of S-wave capture. It was therefore a great surprise when Devons et al.  [141] found an unexpectedly large branching ratio for the reaction -
p  p annihilation -->  p0p0  , forbidden from S-state orbitals. After a long history of conflicting results, the Crystal Barrel Collaboration found a rate for 2p0  production of (6.93  0.43)10-4    [363] for antiprotons stopping in liquid H-
s2  , fully compatible with the old findings of Devons et al.  [141]. When compared to the p+p - rate, a P-wave fraction of 45% to -
p  p annihilation at rest in liquid H -
s2  can be derived  [364].

This large discrepancy is derived from two rare channels, with frequencies of about 0.3%  for -
pp  -->  p+p- and about 0.1%  for -
pp  --> K0s  K0l  . It reflects the a large coupling to p  p   and a small coupling to K  --
K   from P states. With the measured rates for -
pp  --> p  p    [156] and to K  --
K    [157] from P-states, the P-state contribution reduces to ~ (30 15)  %, and there is no more conflict between the results derived from K  --
K   and from p  p

In the derivation of the new P-state fraction, the assumption is made that, at the moment of annihilation, the statistical distribution of the fine-structure levels 2s+1  P-
sJ  -states for large n  is the same as for n = 2  . This assumption is likely not true: in high-n  levels, Stark mixing of atomic states is very strong. The 3   P-
s0  level has a strong interaction width four times larger than the mean 2P width (compare (5.20) and (5.21)). High-n 3
s0  levels can be repopulated after annihilation via Stark mixing collisions and the 3

Ps0  levels have a larger chance to contribute to annihilation. There is practically no Stark mixing for n=2; pp   atoms in the 2 P fine-structure levels annihilate and the 3
Ps0  level is not refilled after annihilation. When this effect is taken into account, the fraction of P-state capture for antiprotons stopping in liquid hydrogen reduces from 28.8 3.5  % to 12  2  %  [363362]. This is a value compatible with most partial-wave analyses. Figure 5.20 shows the fraction of P-state annihilation as a function of Hs2  density.

Batty  [362] also determined the fractional contributions of individual hyperfine structure states to annihilation as a function of the hydrogen density. He found that for any selected channel, the contributions of individual hyperfine states change by an enhancement factor BHF S  . Figure 5.21 shows these factors as functions of the Hs2  density.

We notice a substantial increase of the contribution of the 3P0   state with increasing density. This increase is responsible for the large p0  p0   branching ratio. In turn, this large branching ratio is only compatible with other determinations of the P-state capture rate, when we assign the majority of pp   annihilations into 2p0    to the 3P0    and not to the 3P2    initial state.

The fraction of P-state annihilation in p  n annihilations is even more uncertain. From a comparison of p-  p annihilation into p0p0  in liquid Hs2  and Ds2  , the P-state capture fraction in Ds2  was estimated to (22 4)  % (after a cut on the proton momentum to ensure annihilation on a quasi-free nucleon)  [365]. Batty  [355] estimated the P-state annihilation frequency from p  d annihilations into p  p   and into K  K--   and derived a fraction (34 4)
% P-state capture. From cascade calculations he estimated this fraction to 40%. In summary, a P-state fraction of 30% for antiprotons stopping in liquid Hs2  seems realistic.

Figure 5.20: Fraction of P-state capture as a function of H-
s2   density. The lower line reproduces results from the Mainz cascade model; the upper line uses the Borie-Leon model. The “experimental” points are derived in  [362].

Figure 5.21: Change in the population of protonium levels as functions of H -
s2  density.