# THe-TRAP Project

## Physical basics

While a mass measurement in everyday life is often based on measuring a force, for example
personal or postal scales, this concept is not sufficient for our required accuracy
and is no longer applicable to atomic particles. To get an idea of the desired accuracy,
two comparisons with everyday life can be made. E.g. the mass of the currently largest
commercial aircraft the Airbus A380 with a maximum takeoff weight of 560 tons. With
an accuracy of 10^{-11} it can be measured with an uncertainty of 5.6 mg. This is approximately
the mass of one human hair. Applied to the Earth circumference of 40.000 km,
this would have to be measured with a precision of 0.4mm.

Instead of a force measurement, Penning trap mass spectrometry measures massdependent frequencies. In a uniform magnetic field the Lorentz force confines charged particle on a circular orbit. The frequency of the movement

ν_{c} = 1/2π ⋅ q/m ⋅ B_{0}(3.1)

is called free-space cyclotron frequency ν_{c}. Here *q* is the known charge, *m* the derived
mass of the particle and *B _{0}* the magnetic field. For a full spatial storage of the ion
perpendicular to the plane of the cyclotron motion an electric field must be applied,
which (see Figure 3.1) is generated by hyperbolic electrodes. This changes the motion
of the ion in the trap and the three resulting eigenmotions can be described by three
independent frequencies. These are

- the axial frequency

ν_{z} = 1/2π ⋅ √(q/m ⋅ U_{0}/d^{2})(3.2)

- the reduced cyclotron frequency

ν_{+} = ν_{c}/2 + √(ν^{2}_{c}/4 - ν^{2}_{z}/2)(3.3)

- and the magnetron frequency

ν_{-} = ν_{c}/2 - √(ν^{2}_{c}/4 - ν^{2}_{z}/2)(3.4)

Here *U _{0}* is the voltage applied to the electrode and

*d*is a characteristic trap size, which for hyperbolic traps results from the boundary conditions imposed by the electrodes. The invariance theorem states that the free-space cyclotron frequency can be calculated by summing up of the independent frequencies in quadrature:

ν^{2}_{c} = ν^{2}_{z} +
ν^{2}_{+} + ν^{2}_{-}(3.5)

In the experiment, ν_{c} will be determined for tritium and helium-3 and from this we get
the frequency ratio for these elements, which leads to a mass ratio, if the magnetic field
is stable enough.