2.2 The Reaction Microscope
2.2.2 Computation of the momenta
In addition to the measured spatial information and the time-of-flight, the precise dimen-sions of the spectrometer, the voltage applied to the spectrometer and for the electrons the strength of the magnetic field are needed to reconstruct the momenta of the fragments.
The formulas needed for a spectrometer without a drift zone are presented in this sec-tion. They are separated in formulas for ions and electrons and for longitudinal (along the spectrometer axis) and transversal (perpendicular to the spectrometer axis) momentum components. For a spectrometer with a drift zone the reader may refer to [116].
2.2.2.1 Ions longitudinal
An ion which is produced in the interaction point gains a momentum pk and therefore an energy Ek = p2/2m by the reaction. The electric field is applied in direction of the time-of-flight and leads to an acceleration in the longitudinal direction. When the coordinate system is chosen as in illustration 2.6 the spectrometer axis is along thez axis and the ion is born at the origin. The acceleration is:
¨ z = qU
ma, (2.10)
where a is the length of the acceleration and U stands for the potential between the interaction point and the end of the acceleration. Integrating twice with respect to time results in position z(t). If nowz(t) is replaced with the end of the acceleration, it follows from the above for the time-of-flight of an ion from the interaction point to the detector:
t+/− Ek
=f ·√
m· 2a
pEk+qU ±p
Ek with f = 791.9ns cm
r eV
amu, (2.11) where the ”+” stands for ions that are flying after the initial reaction towards the detector and the ”-” correspondingly for ions that are flying away. The constant factorf originates from the transformation of the units and it is chosen so the mass m of the ion is inserted in atomic mass units (1 amu = 121m12C). Accordingly the energies Ek and qU in eV and the acceleration lengtha have to be given in cm.
It is a good approximation that the energy that is gained in the electric fieldqU is much larger than the primary energy Ek, because of this, the time-of-flight can be developed in
a Taylor series around Ek. This way, the time difference to ions with Ek = 0 can be approximated as:
∆t =t Ek
−t Ek = 0
≈
"
dt Ek
dEk
· dEk dpk
#
pk=0
·pk. (2.12)
With the help of equation 2.11 for the longitudinal momentum in atomic units it is un-folding (see [115]):
pk =
8.04·10−3a.u.·cm eV ·ns
· qU
a ·∆t. (2.13)
By means of this equation the advantage of the previous analysis is clear. For the cal-culation of the longitudinal momentum pk only the time difference between ions carrying momentum and ions with longitudinal momentum pk = 0 is needed, which can be eas-ily obtained from the mean value of the temporal distribution for the corresponding ion species (for example see fig. 2.8 for the example O+ ions from O2). The low accuracy in the absolute measurement of the time does not affect the achievable resolution. The low accuracy comes from the fact that the run time of the signals and their delay in the electronic modules are not exactly known.
Figure 2.8: Left: Time of flight spectrum of O+ ions. Illustrated is the evaluation of the time difference ∆t which is needed for the determination of the momentum along the time-of-flight axis. The time-of-flight axis (3500-4100 ns) displayed does not correspond to the absolute time-of-flight but to the time relative to the last trigger from the FEL bunch.
Right: spatial picture of the O+ ions, where the plotted parameters R and ϕ correspond to the computation of the momenta.
2.2.2.2 Ions transversal
In the transversal direction there is no acceleration. Therefore, the traveled distance in this direction with respect to the reaction point is depending on the original energy in transversal direction E⊥ and the time-of-flight to the detector. Due to symmetry reasons the image of the reaction point coincides with the center of mass of the distribution on the detector. The distanceR to the center of the distribution in the detector plane results from:
R= 1 f ·
√E⊥
M ·ttof. (2.14)
If in equation 2.14 the absolute time-of-flight is replaced by the mean time-of-flight t(Ek) from equation 2.11, the momentum in the transversal direction becomes:
p⊥ =
11.6 a.u.
amu eV · R
2a ·p
qU ·m. (2.15)
The substitution of t(Ek) by the mean time-of-flight follows the same reasons as in the last section. Additionally, the azimuthally direction angle ϕ in the detector plane can be determined using the two independent x and y coordinates on the detector. With this direction angle and the absolute value of the momentump⊥ the momentum in the detector plane can be displayed in polar coordinates.
2.2.2.3 Electrons transversal
Due to their lower mass, electrons are much more deflected by the magnetic fieldB than the ions. The magnetic field that is applied in the longitudinal direction, forces the electrons onto cyclotron orbits with the frequency ωc = e B/m in the transversal plane. The calculation of the momentum components is again in polar coordinates (p⊥, ϕ).
Starting from the reaction point the electrons start their cyclotron motion with the starting momentum p⊥ that has to be determined. The radius r of the cyclotron orbit is:
r= p⊥
eB. (2.16)
Unfortunately, the cyclotron radius cannot be observed in the experiment. Therefore it has to be identified indirectly from the distance R between the point of incidence and the image of the interaction point on the detector. This image coincides, due to symmetry reasons, with the center of mass in the detector picture. The point of incidence depends on the rotation angle traveled in the time-of-flight ttof:
α=ωct = eB
mttof. (2.17)
With the help of the formula that is known from geometry for the chord [119] in dependence of the radius r and the rotation angle α follows:
sinα
2
= R
2r ⇒ r= R
2
sin ω2ct
. (2.18)
By inserting equation 2.16 and 2.17 into the initial transversal momentum in atomic units unfolds:
p⊥ =
8.04·10−3 a.u.
mm Gauss
· R B 2
sin ω2ct
. (2.19)
Figure 2.9: Spatial and time-of-flight spectrum of electrons ionized from neon at a photon energy of 44 eV. The values drawn into the spatial spectrum (left) are related to the momentum calculation as described in the text. (Picture taken from [120].)
In order to determine the direction besides the absolute value of the transversal mo-mentum, the emission angle ϕ is needed in relation to an arbitrary reference axis in the detector plane. To this end, the angleθ, which is enclosed by the positionRon the detector and the reference axis (see fig. 2.9) is measured. From this angle θ the angle α, which is covered in the time-of-flight, is subtracted or added, depending on the direction of the magnetic field, and the result is:
ϕ=θ±α = arctany x
± ωct
2 , (2.20)
where the sign of the angleα is specified by the direction of rotation in the detector plane that is induced by the magnetic field. The xaxis of the detector is chosen as the reference axis for the angle θ.
It has to be kept in mind that electrons with a time-of-flight corresponding to a mul-tiple integer of the cyclotron period 2π/ωc are, independent of their initial momentum,
mapped onto the image of the interaction point on the detector. For these electrons all the transversal momentum resolution is lost.
For the absolute value as well as for the angle of the momentum of the electron the absolute time-of-flight is needed, which can be determined by the periodicity of the electron motion. These electrons with time-of-flights matching a multiple of the cyclotron period are mapped onto the image of the interaction point on the detector. If, for example, R is plotted versus the time-of-flight, the cyclotron period can be revealed by means of the points with minimal transversal latitude, so called ”wiggles”. Because an electron with zero time-of-flight has started at the interaction point, this point in time has to coincide with a ”wiggle”. So if the time-of-flight is known with the accuracy of one cyclotron period, then the absolute time-of-flight can be determined with a precision of the measurement of the cyclotron frequency. With typical cyclotron frequencies in the order of 30 ns this precondition can easily be fulfilled.
2.2.2.4 Electrons longitudinal
The magnetic field does not affect the longitudinal electron motion in the chosen geometry.
Hence, only the time-of-flight is required for the computation of the longitudinal momen-tum. The approximation that the energy gained in the electric field is much larger than the energy obtained in the reaction is not valid in this case. Therefore equation 2.11 is solved for Ek and for the momentum in atomic units unfolds:
pk =
0.457·a.u. ns mm
a t −
80.4·10−3a.u. mm ns eV
qU
a te, (2.21) wherea is again the acceleration length andte stands for the time-of-flight of the electron.