Derechos constitucionales afectados

In order to determine a velocity of the antihydrogen atoms from the experimental data shown in Fig. 4.8, we have developed a simple numerical model. Consider that N(ρ, v, F)dρdvdF antihydrogen atoms are produced at z=0 and t=0 within a radius of ρ and ρ+dρ, with a velocity towards the detection well between v and v+dv, and in an atomic state which ionizes at a electric fields between F and F +dF. Without a time varying field, the antihydrogen atom experience FDC(ρ, z = vt) on their way to

the detection well. The number of antihydrogen atoms detected is then N0 = Z ∞ 0 dρ Z ∞ 0 dv Z Fmax(ρ) FDC(ρ) N(ρ, v, F)dF. (4.12)

A value for ρmax is given below and Fmax(ρ) is the maximum electric field present in

the detection well.

When the time varying component of the pre–stripping field is switched on, then the atoms which travel towards the detection well pass an electric field given by

F(ρ, z, ω/v,Φ) =FDC (ρ, z) +FACρ, z) cos

_ωz

v + Φ

. (4.13)

The number of detected antihydrogen atoms is then given by N = Z ∞ 0 dρ Z ∞ 0 dv Z 2π 0 1 2πdφ Z Fmax(ρ) F0(ρ,ω/v,φ) N(ρ, v, F)dF (4.14) The lower border of the last integral, F0(ρ, ω/v, φ) is the maximum field magnitude that the antihydrogen atoms encounter in the pre–stripping region.

In order to derive a numerical result, we assume that the antihydrogen atoms are produced uniformly out to a radius ofρmax = 3 mm, which is approximately the radius

of a positron plasma cloud with 400 000 positrons. We also assume that N _∝ F−3

which has been measured by the experiment presented in section 4.3.4. Additionally it is assumed that all antihydrogen atoms have a velocity of v0. We therefore obtain

N(ρ, v, F) = 2ρ ρ2

max

δ(v−v0)F−3. (4.15)

The integral given in Eq. 4.12 is then solved numerically by usage of the approximations forFDC(ρ) and Fmax(ρ) and by use of Eq. 4.15. The result is depicted in Fig. 4.8 by the

4.4 Measuring the Velocity of Antihydrogen Atoms -63-

Figure 4.8: The fraction of the antihydrogen atoms detected in the detection well rel- ative to the number detected in the normalization well is shown by the solid points as a function of the frequency ω/(2π) of the oscillating pre–stripping field [GSS+_{04]. The}

error bars are statistical due to the uncertainties in the determination of the number of antiprotons in the normalization and detection well. The open square shows a measure- ment with the oscillating field switched off. The measured points are compared to a simple model discussed in the text (solid curve), which reveals that the average kinetic energy of the antihydrogen atom is about 200 meV.

blue dashed curve. The scale in that graph is chosen so that the curve is at a detection probability of one. The integral given in Eq. 4.14 is also solved numerically by usage of the approximations for F0(ρ) and Fmax(ρ) and by use of Eq. 4.15. The solid curves

in Fig. 4.8 show the functional behavior of the integral for several velocities v0. The

velocities are given in the graph via the corresponding kinetic energy 1/2m_H¯v₀2. where m_H¯ is the antihydrogen mass. As can be seen a kinetic energy of 200 meV is a good

fit to the experimental data. This result is rather insensitive to the assumptions. For example, if we had assumed that all antihydrogen atoms are produced on the central axis, then best fit would correspond to a kinetic energy of 100 meV. If we had assumed that all antihydrogen is produced 4 mm off axis, then we would have concluded that the antihydrogen velocity is about 300 meV. 200 meV correspond to an antihydrogen temperature of about 2300 K, which is much higher than the 0.36 meV kinetic energy corresponding to a temperature of 4.2 K.

The measured average antihydrogen energy of 200 meV corresponds to an antihydrogen velocity of about 6200 m/s. It is up to now unknown if this relative high velocity is caused by the rf–signal used to drive the antiprotons over the psoitrons or whether such a high velocity is characteristic of this production method. An antiproton speed that seems important for antihydrogen formation is the one that equals the most probable speed of the positrons, because in this case a relative large fraction of the positrons move collinearly with the antiprotons when these pass the positron cloud and one would expect an increased antihydrogen production. The most probable positron velocity ve+

for a temperature of T = 4.2K is given by ve+ =

s

2kBT

me+ ≈

11 300 m/s, (4.16)

where kB is the Boltzmann constant and me+ is the mass of a positron. The most

probable positron velocity is about a factor of two larger than the measured velocity of the antihydrogen atoms.

Another important antiproton speed can be calculated from the average rate of deex- citation collisions [GO91] between an initially bound positron and another positron. The expected e+ _−e+ _{collision rate should be of order n}

e+(πb2)v_e+ [GO91], where ne+ = 1.6×107cm−3 is a typical positron density andv_e+ is the positron velocity cal-

culated above. The distance of closest approach b comes from equating the potential energy e2

4π0b2 between two positrons and the thermal energy kBT. This yields a colli-

sion rate of about 9×106_s−1 _{and hence a mean time between two collisions of about} t = 1.1×10−7_{s. Thus for a least one deexcitation collision to occur while passing a}

positron plasma of a typical size s _≈ 1 mm, the antiproton velocity has to be lower thanv0 =s/t ≈9100 m/s, which is about 1.5 times larger than what has been observed experimentally. From this simple model the conclusion can therefore be drawn that the weakest bound atoms of which we have measured the velocity are formed from only one deexcitation collision. Notice that the antiproton speed v0 scales as ne+sT−3/2. Hence