Fluctuation of the Ambient Magnetic Fields
In the real vacuum chambers, there are fluctuations of the ambient magnetic field. The multi-pole magnetic suspension design is naturally immune to uniform magnetic fields, but will be sensitive to higher order gradients in the field.
R. Schofield has made some measurements on the LIGO site long time ago. Here we attach his measurement for further references:
Schofield_magnetic_field_measurement_99.pdf
Schofield_presentation_magnetic_field_noise_spectrum.pdf
We know that there are gradients in the chambers with a ~several cm spatial scale. These gradients will couple to the magnetic suspension. The induced forces and torques cannot be estimated without better knowledge of the actual field gradients. This will certainly depend on the actual suspension cage and the amount of magnetic shielding employed.
Stiffness of the tilt and coupling among different degrees of freedom
For a good suspension system, a low-frequency translation motion is not enough to isolate seismic noise. If the tilt has a high stiffness, any small coupling between tilt and translational motion will significant degrade the performance of the suspension system. We will include this in our model.
Metallic Components and associated Eddy-current damping
In the numerical model, we haven't considered any metallic components. Their presence will modify the distributions of magnetic fields. In addition, when the magnets vibrate with respect to them, eddy current will be induced which creates damping.
In addition to the eddy currents induced in the surrounding metal, there will be some non-zero currents in the opposing magnets. We need to figure out what the real conductivity of the magnets are and whether or not they should be plated.
Eddy currents will, of course, short out the suspension effect at some level. Our goal will be to reduce the eddy current effect enough to achieve 80 dB of isolation before the 1/f shorting comes into play.
Barkhausen Noise
The Barkhausen effect is due to flips of magnetic domaisn when the magnet is placed in a varying magnetic field. The Barkhausen noise has been measured in the initial LIGO mirror actuators. Most importantly, it was shown that the Barkhausen effect is much smaller in SmCo magnets than in the NdFeB magnets. Our magnetic suspension will use SmCo magnets.
Even so, we need to estimate the fluctuating magnetic field which comes from our control forces and use this to estimate the Barkhausen noise in the GW band. We can then make a Barkahusen measuring rig to get better estimates.
Hysteresis Damping
A distinct feature of ferromagnetic materials is the hysteresis damping. This happens when the ambient field changes at a time scale faster than the magnetization process. In this case, the magnetization lags the field change, which induces damping. If we are able to control the suspend magnet well in position and the control field of the control coil only changes at low frequencies, this effect should be small. This, along with eddy-current damping, are the main damping mechanism in the magnetic suspension system. In the linear regime, the fluctuation-dissipation theorem tells us that it will also induce undesired fluctuations.
Temperature dependence of magnetization
As it is well known, when temperature is higher than the Curie temperature, the magnet will be completely demagnetized. Of course, Curie temperature for typical permanent magnets are much higher than room temperature, and there are only small changes in the magnetization when the temperature fluctuates. Typically, magnetization will change by 0.1 percent when the temperature changes by 1 degree. In the vacuum chamber, the temperature fluctuation will only have a very low-frequency component. Therefore, it will not significantly affect the suspension performance. We can measure such effect in a R&D experiment.
Slow demagnetization of the permanent suspended magnets
Permanent magnet is "permanent" only if the environment is fixed. In reality, the magnet will gradually demagnetize, either due to the thermal excitation and the fluctuations of ambient magnetic field. Of course, this demagnetization will happen in a time scale much longer than the time scale we are interested in.
Current fluctuations in the control coil
Since we will use a coil to actively control the suspended magnet, the fluctuations of the current in the coil will induce a fluctuating force on the suspended magnet. The seriousness of this will depend on the current -> force coefficient of the coils. We expect that we can achieve ~9 orders of magnitude in dynamic range (Amps_peak / (A/rHz)) without too much trouble.
Stress-induced change of the magnetization
For typical magnets, the magnetization also depends on the mechanical stress and strain on the magnet. Since in the case for advanced low-frequency GW detector, the suspension needs to be able to load a weight of the order of 50 kg. If the stress is not properly distributed on the suspended magnets, the magnitization will change in an unpredictable way and significantly degrade the performance. This effect can be modeled in the Comsol, as we will investigate.