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InitialPhase

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Initial Phase in a transverse field experiment (e.g. GPS)

When a magnetic field is applied longitudinally to the muon beam - say, along z - the incoming muons travel along the lines of the stray magnetic field of the magnet. If the muon spin has initially a component transverse to z, as in when the Spin Rotator is active, this component starts to precess before implantation. The time of implantation is taken as zero for any subsequent analysis but in this case a sizeable, field dependent initial phase must be taken into account.

This extra initial phase is not present, of course, if the external field is zero and the spin precession is due to an internal field, e.g. if one has turned the Spin Rotator on, and the sample is single crystalline, with internal field along z.

Specifically, for the present magnet wiring of GPS (2005):

• The location of time t=0 is determined from the histogram prompt counts. These counts are given by transported particles hitting both the muon and the positron counters, an event which is normally vetoed by the counting logic, by it escapes the veto with some low probability, yielding a sizeable peak. This determination has a systematic error due to the time of flight over the distance between the muon and positron counter, which are few centimeters apart, hence the error amounts to fractions of ns.
• In a zero applied field experiment, with Spin Rotator on and muon asymmetry obtained from detectors 3 - 4 (UP - DOWN), the initial phase referred to time t=0 is zero for whatever frequency. It is rigorously zero if the prompt is exactly located at the center of a time bin. If the center of the prompt peak differs by dt from the center of the chosen time bin the error on the initial phase is:

{$ d\phi = 2 \pi \nu dt $}

in radiant for each frequency ν, or, equivalently

{$ d\phi = 360 \nu dt $}

in degrees. Hence the phase could be determined precisely from a calibration experiment with a suitable sample, providing a spectrum of few narrow frequencies, both low and high.

• In experiment with an applied field B at the sample, the initial phase due to the fringe field is given, in the same geometry, by

{$ d\phi = \frac{d\phi}{dB} B $}

and for GPS the derivative of φ with respect to B is equal to 0.0231 degree per Oersted within the same errorbar of Eq. (2). This initial phase is the same that would be acquired if the muons where let to precess in the field at the sample for a time

{$ \Delta t = \frac{d \phi}{dB} \frac {1}{360 \gamma} = 4.7355 \mbox{ns} $}

with {$\gamma= 0.000001355$} GHz/Oe and {$d\phi/dB$} in deg/Oe.

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