General Theory I - Principle 1/ 1-Field Spectrometer 2/ Performance II - Analysis 1/ Particle identification - Time Labeling 2/ Examples III - Further resources

I - Principle

1/ 1-Field Spectrometer

In this section we explain the principle of 3D momentum imaging technique using a simple 1-Field spectrometers by looking at the motion of a charged particle originating from a source point (set as the origin). We define by:

• d: the length of the extraction region
• l: the length of the drift tube

The space focusing condition along the spectrometer axis (z), so-called Wiley McLaren conditions, are satisfied when l=2d. In the following we will only consider spectrometers which verify this criteria. The general expression for the time of flight is given by :
with When the kinetic energy (Ek) is much smaller than the work of the electrostatic force (qEd), i.e. Ek<<qEd, the expression of the time of flight can be linearized using a Taylor expansion. At the first order it becomes simply:
and The last quantity represents the nominal time of flight for a charged particle (q) of mass (m). This expression has a sens only when the sign of qE is positive, i.e that the field orientation has to be changed when detecting electron or ion.
Since the system is invariant under rotation around the spectrometer axis, the dimension of the problem can be reduced to 2D. The position of the charged particle on the detector is then expressed only by its radius with respect to the origin. Using uniform fields, there is no force acting on the particle, and the position is simply related to: where pr is the transverse momentum.

2/ Performance

• (pz) momentum along the spectrometer axis

To understand the resolution that one can achieve, it is useful to define the so-called "turn-around" time, i.e. time difference when the electron is emitted forward or backward (px=py=0). It can be expressed as: This time difference is directly connected to the maximum momentum transfer along the spectrometer axis. Since the measure of time requires a time-to-digital converter, there is an instrumental limitation related to the sampling rate. Indeed, for a certain value of the kinetic energy of the electron, the turn-around time is fixed, and the number of points to define (pz) distribution is limited by the instrumentation "speed". Nowadays fast electronics can reach "on the shelves" 4GHz (250 ps). In the figure, the corresponding sampling point that one can achieve is given as the function of the kinetic energy and the E field.

• Maximum kinetic energy

As a good approximation, particles which have been emitted perpendicular to spectrometer axis have a maximum radius on the detector. The condition for such a particle to reach the detector is : In the figure, this limit is presented for a given detector size as a function of the total length of the spectrometer (3d). Shorter is the spectrometer higher is the kinetic energy, but at the same time the condition Ek<<qEd is less and less justify. The momentum pz becomes thus non linear as a function of the time of flight (see Gisselbrecht et al for an exact solution of pz).

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II - Analysis

1/ Particle identification - Time Labeling

In this section we present a set of information related to the data analysis of a 3D momentum experiment. The particle identification does not have conceptual problems as long as the particles have different nature : electron, ion of type A, ion of type B… However in the case of identical particles, the acquisition system will detect particles one after the others. The "hardware' labels the particles with respect to their arrival time on the detector: the first particle is labelled 1, then the second 2 and so on… Unfortunately, there are physical situations where this labeling introduce confusion. Imagine the situation where two electrons are emitted one with a kinetic energy (Ek=20 eV) and the other with a kinetic energy (Ek'=5 eV). When the electron with Ek is emitted toward the detector while the other with Ek' in the opposite direction, the label will be :

• electron 1 : electron with Ek
• electron 2 : electron with Ek'

In the other case, when the electron with Ek' is emitted toward the detector while the other in the opposite direction, the label will be :

• electron 1: electron with Ek'
• electron 2 : electron with Ek

Thus the analysis has to consider such situation applying permutations on all identical particles.

In the following we consider that inputs data are given in this specific format where (E, thet, phi) represent the energy and the angles of emission of the particle with respect to the laboratory frame, e.g. polarisation of the light (see picture). ("e1", "e2", "i1"...) for the name of the particles, i.e. electron and ion, which are labeled ("1","2"...) by the acquisition system.

Event number	particle 1	particle 2	particle 3 ...
1	E_e1, thet_e1, phi_e1	E_e2, thet_e2, phi_e2	E_i1, thet_i1, phi_i1...
2	...	...	...
...

We assumed that the momentum of each particle can be determined by the 3D momentum imaging spectrometer. The label of each particle is then given by its arrival time on the detector. It is connected to the "id" of the particle (named "a" or "b"), by taking carefully into account the orientation of particles with respect to the polarization axis and the invariance per rotation around it.

2/ Examples

a - Polar limit


The transformation from particle "a" to particle "b" only occurs when theta is close to 0 [pi] (see figure). A simple test on the opening angle, tells whether or not one should consider the pa or pb (thus 1 or 2).
Concerning the polar angle, it is used to sort event which occur with the polarisation vector pointing the x axis (epsilon+), or with the polarisation vector pointing the opposite direction (epsilon -). These events are equivalent, as long as there is no symmetry breaking, and can be added together. However, one should take special care to keep the symmetry of the system invariant by the reflection through the vertical plan.

b - Azimutal limit
Let us consider phi_c, the azimutal angle for which the particle "a" switch from particle 1 to particle 2. It is defined as the azimutal angle for which the momenta along z axis between particle 1 and 2 are equal. As you can see in the figure on the right hand side, for a particle "a" emitted in the upper plan two cases ought to be considered:

• pa is 1
• pa is 2

The same hold for a partilce "a" emitted in the lower plan:

• pa is 2
• pa is 1

Each event can be unambiguously identified to one of these four cases testing conditions on the following azimutal angles: phi1, phi2, or phi12 (=phi2-phi1)
One should point that phi_c is up to 45° when the two particles have the same momenta in the plan (yOz). It strongly depends of the ratio of the momenta of the particles and the azimutal difference phi12.

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III - Further resources

Images

Last edited by Erik at Oct 23, 2009 11:21 AM - Edit content - View history - View source