Incidencia jurídica de rectificaciones en los usuarios

ber of Positrons

We have performed experiments over several weeks with about 300 000 antiprotons and a varying number of positrons using the driving method described above. The data can be used to investigate the dependence of the number of antihydrogen atoms detected in the normalization well on the number of positrons. The number of antiprotons (= number of antihydrogen atoms) detected in the normalization well is depicted in Fig. 4.5 as a function of the number of positrons. In order to compare the individual experiments, the number of antiprotons has been normalized with respect to 250 000 antiprotons in the nested Penning trap. As can be seen the number of antiprotons depends linearly on the number of positrons with a slight deviation from linearity for small numbers of positrons (< 1 million). In order to understand the linear dependence, the shape and density of the positron plasma have to be taken into account. The positron plasma shape and density have been studied by use of an aperture method, and the method

and results are published in [OBP+_{04, Oxl03], so that only a brief account is given}

in the following. In these studies, the positron cloud is pulsed through the aperture presented by the ball valve electrode of the hbar1 Penning trap apparatus (its inner diameter is only 6 mm, while the inner diameter of all other electrodes is 12 mm). From the measured transmission efficiency and the measured number of positrons, the shape and density of the plasma can be calculated by a numerical code written by Spencer et al. [SRV93] and Parrott [Par] which solves the Poisson equation self–consistently for charged particles confined in a Penning trap. These studies reveal that as the positron number increases from 1 million to 4.5 million particles the positron plasma diameter remains at 1 cm and the density remains within a few percent of 2_×107_cm−3_{. The}

axial extent of the positron plasma however grows linearly from 1.5 mm to 3 mm. The antihydrogen production grows therefore linearly as the axial extent of the positron plasma. The observed deviation from linearity for less than one million positrons is due to the fact that the positron cloud still grows radially.

4.3.4

The State Distribution of the Formed Antihydrogen

For the experimental results presented in the previous section, only the antihydrogen atoms stripped in the normalization well have been taken into account. When carrying out these experiments the electric field in the analysis region has been additionally var- ied from one experiment to the other between 30 V/cm and 360 V/cm. An antihydrogen atom stripped by the pre–ionizing electric field in the analysis region is unable to deposit its antiproton in the detection well and therefore the number of detected antihydrogen atoms in the detection well reduces with increasing electric field in the analysis region. The number of detected antihydrogen atoms is shown in Fig. 4.6 as a function of the maximum electric field F present in the state analysis region. The ordinate to the left gives the number of antihydrogen atoms normalized to 250 000 antiprotons and five million positrons. Such a normalization is possible because in each trial about 300 000 antiprotons have been used and a linear dependence of the produced number of an- tihydrogen atoms on the number of positrons has been measured as presented in the previous section. The ordinate to the right shows the number of detected antihydrogen atoms normalized to the number of antiprotons used in the nested Penning trap. The number of antihydrogen atoms surviving an electric field F is proportional toF−2

orF−3/2 (see the dotted power laws in Fig. 4.6). This corresponds to state distributions dN/dF _∝F−3 _{ordN/dF ∝ F}−5/2_{. This is a very interesting and important result for}

the understanding of the production mechanism, since the current theory of the three- body formation of antihydrogen [GO91, Fed97] cannot explain the measured distribu- tion. A reason for the difficulties in the theoretical interpretation of the measured state distribution is that currently a numerical simulation of the antiproton–positron mixing process in a nested Penning trap seems unfeasible, because for a proper simulation the equations of motion during the mixing process have to be solved for each particle inde- pendently. In a particles’ equations of motion the interactions of the particle with all

4.3 State Distribution of Antihydrogen -57-

Figure 4.6: The number of antihydrogen atoms is shown as a function of the pre- stripping field F which the atoms have passed. The atoms can be characterized by their radial sizeρ, which is less than indicated by the horizontal axis at the top. Antihydrogen atoms with radii in the gray–shaded region are guiding center atoms (GCA), while atoms with radii in the white region are non–GCA. The error bars are statistical due to the uncertainty in the detection of the antiprotons released from the detection well.

other particles have to be taken into account. Thus, a simulation of typical experiments with about one million positrons and about 105 _{antiprotons requires due to the rela-}

tively large number of particles storage and calculation time that are currently beyond the possibilites of modern computer technology. Moreover the physical interpretation of numerical simulations are intractable. Another reason for the difficulties in the theoreti- cal interpretation of the measured state distribution is that so far theoretical treatments have only taken into account the relative simple model of guiding center atoms [GO91]. The interpretation of the measured state distribution requires probably the develop- ment of more advanced models. The results presented here have stimulated along with the results published by the ATHENA Collaboration new studies on the formation mechanism, see for example [Dri04, GBO+_{04, BD04, KO04a, KO04b, Rob04, RH04].}

However, as mentioned above, so far no satisfying explanation for the measured state distribution has been found, see in particular [Dri04, GBO+_04].

The electric fieldF by which an antihydrogen atom is ionized can be used to characterize the atoms. The most intuitive way to characterize an antihydrogen atom is by the

principal quantum state n to which the positron is excited. In the following a relation betweenF andn is derived. Consider an antihydrogen atom with the antiproton at the origin in the presence of an electric field in the z–direction. The potential experienced by the positron moving along the z–axis is given by the combined potential V(r) of the atomic core and the potential due the external electric field F

V(r) =−qF r− q

2

4π0r

, (4.7)

where q is the unit charge, 0 is the permittivity of the vacuum, and r is the classical

distance between the positron and antiproton. The maximum potential energy is ob- tained by solving V0(rmax) = 0 with respect to rmax and ensuring that V00(rmax) < 0.

This yields rmax =

q _q

2π0F and Vmax = V(rmax) = − 3q√qF

2√2π0. The energy of a positron

in an atomic state with principal quantum number n is given by E(n) = ₋Ryd_n2 , where Ryd ≈ 13.6 eV is the Rydberg energy. The antihydrogen atom is ionized when the energy of the positron is larger than the maximum energy of the combined potential. The minimum electric field by which an antihydrogen atom in a state with principal quantum number n is stripped is then given by solving E(n) > Vmax(F) for F which

yields

F ≥2.85×108V/cm× 1

n4. (4.8)

The quantum state which are ionized can be obtained by solving E(n) > Vmax(F) for

n which yields n _≥ 4

r

2.85×108_V/cm

F . (4.9)

As shown in Fig. 4.6, antihydrogen atoms that have passed a pre-stripping field between 30 V/cm and 360 V/cm are detected. Atoms that are not ionized by an electric field of 30 V/cm are according to Eq. 4.9 in an atomic state with n _≤55. Antihydrogen atoms which have passed a pre–stripping field of 360 V/cm are in a state withn≤29. A lower limit of the detected quantum states is given by n _≥ 24 set by the maximum electric field of the detection well, which is 860 V/cm.

The characterization of the detected highly excited antihydrogen atoms by the principal quantum number n is very intuitive and practical for any further estimates, as for example presented in chapter 5. However, due to the presence of the magnetic field of the Penning trap n is not a good quantum number for such highly excited states. The reason is that at a magnetic field strength of about [RGHW94]

Bcritical ≈8.3 30 n 3 T, (4.10)

the Lorentz force acting on the positron is of similar size as the Coulomb force caused by the atomic core on the positron. The Lorentz force cannot be treated as a perturbation