7.2 Numerical Analysis
7.2.1 Description of the Algorithm
The time distribution of N antihydrogen annihilations is calculated by the algorithm described in the following. The basic concept is to calculate N times the time of flight of a Rydberg cesium atom from the excitation region to the location where the first charge–exchange occurs plus the time of flight of a positronium atom from the location of the first to the second charge–exchange plus the time of flight of an antihydrogen
7.2 Numerical Analysis -119-
atom from the location of the second charge exchange to the location of its annihilation at the electrodes taking into account different antihydrogen and cesium velocities as well as different antihydrogen and cesium paths from one repetition to the next. The sum of these three times of flight will be referred to in the following as total time of flight. The laser pulse length of about 20 ns is neglected for the calculation of the total time of flight because it is much smaller than the three times of flight and therefore considered to be of minor influence on the time distribution of the time annihilation spectrum.
In order to account for the effect caused by the temperature on the time distribution of antihydrogen annihilations, the velocities which are assigned per repetition to a Rydberg cesium atom and an antihydrogen atom are chosen so that the ensemble of all
N Rydberg cesium atoms and antihydrogen atoms obey a Maxwell–Boltzmann velocity distribution for a temperature TCs and TH¯, respectively. This is implemented in the
numerical code as follows. The temperature of the cesium atoms is taken to be TCs =
320 K, which is about the temperature of the cesium oven during an experiment as described in the previous chapter. Then, the most probable velocity of the cesium atoms given by vCs= r 2kBTCs mCs , (7.1)
wherekBis the Boltzmann constant andmCs is the atomic mass of cesium is calculated.
A random velocity between zero and 4·vCs is then assigned to the cesium atom by use
of the following method. Let r1 be a random number between zero and one. A value
for the velocity v is then calculated by v = 4·vCs·r1. The probability of this velocity
is given by the Maxwell–Boltzmann probability function according to [Vog95] f(v) = r 2 π m kBT 3/2 v2exp − mv 2 2kBT , (7.2)
with m = mCs, and T = TCs. The ratio q = f(v)/f(vCs) is compared with a second
random number r2. If r2 is larger than q then a new velocity v is calculated. If r2 is
less or equal than q thanv is assigned to the cesium atom.
Accordingly, the most probable antihydrogen velocity vH¯ for an antihydrogen temper-
ature TH¯ is calculated by
vH¯ =
s
2kBTH¯
mH¯ , (7.3)
wheremH¯ is the antihydrogen mass. Again, a random velocity between zero and 4·vH¯ is
assigned to an antihydrogen atom by the same method as described above for a cesium atom.
All positronium atoms are taken to be at the most probable velocity vP s = 10 000 m/s
according to the numerical result of Hessels et al. [HHC98]. A specific velocity distribu- tion for the positronium atoms is not taken into account because due to the high mean velocity the time of flight of a positronium atom is two orders of magnitude smaller than the times of flight of an antihydrogen atom or a cesium atom and is therefore considered to be of minor influence on the time distribution of the antihydrogen annihilations. With the particle velocities specified, what remains to be be done is to specify the path lengths that are covered by each particle species. Fig. 7.1 shows schematically the cesium excitation region by the green hatched triangle and the inner electrode surface of the lower electrode stack by the black rectangle.
For a cesium atom it is assumed that it is excited at a random spot within the 5.7 mm wide cesium excitation region and then travels the 14.8 mm long path to the center of the positron cloud, which is depicted by the bottom black dot in Fig. 7.1. The cesium path length is thus calculated bysCs= (14.8 + 5.7·x) mm, wherexis a random number
between zero and one, inclusively.
For a positronium atom it is assumed that it travels three millimeters, the distance between the centers of an antiproton and a positron cloud in a real experiment. The time of flight of all positronium atoms is then given by tP s =sP s/vP s = 300 ns.
The specifications of the cesium path and the positronium path imply that the radial and axial extensions of the antiproton and positron clouds are neglected in the numerical analysis. The reason is that the smearing of the time distribution due to the extensions of the clouds is considered to be negligible in view of the large uncertainties in the involved path lengths.
For an antihydrogen atom it is assumed that it leaves the antiprotons in a straight line and in a random direction within the white half plain inside the trap wall (because of the cylinder symmetry of the Penning trap electrodes, only this half plain needs to be considered in the simulation). The crossing point with the trap wall is determined numerically. The antihydrogen path is then obtained by the distance between the center of the antiproton cloud and the crossing point with the trap wall. For clarity, several arbitrarily chosen antihydrogen paths are indicated in Fig. 7.1 by the dashed red arrows. In order to obtain a time distribution of the N calculated annihilations, an array with 1024 elements called counts[n] (n = 0,1, ..,1023) is defined. All elements of the array are set to zero at the beginning of the algorithm. In each repetition, the element of counts[n] for which n is the closest integer number to the ratio between the calculated total time of flight and 0.64µs is increased by one.
As a summary of this section, the basic algorithm is summarized by the following list.
• For j = 1 toN
1. Determine a random velocity vCs,j for the cesium atom in the range between
7.2 Numerical Analysis -121-
12.7
14.8
12
3
positrons
antiprotons
antihydrogen path
Ps path
Cs path
optical fiber
Cs excitation region
120
30
trap wall
25.4°
5.7
Figure 7.1: The graph depicts schematically the cesium excitation region (green hatched triangle) and the electrode walls of the lower electrode stack (black rectangle). The upper horizontal electrode wall is assumed in the numerical code to be the ball valve, the lower horizontal line represents the degrader. All dimensions are given in millimeters. The width of the cesium excitation region along the path of the cesium beam is given by the diameter of the output cone (green hatched triangle) of the optical fiber along the cesium path. The specified value of 5.7 mm is obtained by the following consideration: The distance from the tip of the fiber to the cesium beam is as indicated in the graph 12.7 mm. According to section 2.4.3 the numerical aperture (NA) of the optical fiber is N A= 0.22. The relation between the numerical aperture and the half angleα/2 of the output cone is given by N A = sin(α/2). The relation yields 25.4◦ as indicated in the figure for the full angle of the output cone. From geometrical arguments follows then a width of 5.7 mm for the cesium excitation region.
TH¯ [K] tavg [µs] tstdev [µs] 4.2 159.6 1.7 5 154.7 1.7 7 146.2 1.6 10 138.5 1.4 25 124.2 1.2 50 117.0 1.1 250 107.4 1.1 500 105.1 1.1 1000 103.5 1.1 2000 102.3 1.1 2500 102.1 1.0 3000 101.8 1.1
Table 7.1: The first column gives the antihydrogen temperature specified in the numer- ical calculation. The second column gives the average time of flight and the last column gives the standard deviation from the average. tavg andtstdev have been calculated from
1000 time distributions of 5000 annihilation events.
2. Determine a random velocity vH,j¯ for the antihydrogen atom in the range
between zero and 4·vH¯.
3. Determine the random path length for the cesium atom via sCs,j = 14.8 +
2.8·xmm.
4. Cesium time of flight tCs,j =sCs,j/vCs,j.
5. Determine random antihydrogen path sH,j¯ .
6. Antihydrogen time of flight tH,j¯ =sH,j¯ /vH,j¯ .
7. Total time tj =tCs,j+tP s+tH,j¯ .
8. Increase the element of counts[n] for which n is the next closest integer of the valuetj/0.64µs by 1.
• End of loop