# PC 2014-11-11¶

Discussion on non-linear effects, 14:00 Wednesday 12th November 2014

## Notes¶

Plan to first study numerically:
1. Make a power series for the field, expand to N^th order, look at emittance growth
• Understand what are the dominant terms in the field expansion
• Track in homebrew code
2. Make a bunch of transfer maps, look order by order
• e.g. Marylie
• Isolate kinematic non-linearities

Other notes:

• Noted that MICE tracking is done using MAUS/Geant4, which makes (by default) 4th order RK with time as independent variable
• Noted that MAUS makes output using virtual planes in z; emittance is calculated by taking 4th root of determinant of the (x, px, y, py) covariance matrix (which should be conserved in solenoids in linear approximation)
Report in analysis meetings
• TBC Thursday 4th December 15:00 nb: collision with MAP CM
• TBC Thursday 18th December 15:00

Coil dimensions and maximum currents:
http://micewww.pp.rl.ac.uk/attachments/2476/As-built_coil_dimensions__positions_and_currents.pdf

Working assumptions for reasonable current densities (A/mm^2) EstimatedCurrentDensities.pdf

I also draw your attention to the following paper by Dragt, which is my best reference on non-linear effects in solenoids:

Numerical Third-Order Transfer Map for Solenoid, A. Dragt, Nuclear Instruments & Methods in Physics Research A, 1990, 298, 441-459

```
A method is described for computing numerically the Lie algebraic transfer map for a solenoid, including fringe-field effects,
through third order. The method can easily be extended to fifth and even higher order if desired. Examples are given for a
simple imaging system and a sextupole corrected spot forming system.
```

In particular he goes through the expansion of the Hamiltonian to 3rd order.

## Brief¶

Email to Rob Ryne:

Hi Rob, So you heard on the phone call yesterday that we have
some issues with non-linear effects in MICE. For the purposes of
MICE Demonstration of Ionisation Cooling, I think we will have to
come to a design decision with a relatively poor understanding of
the non-linearities.

However, we should seek to improve our understanding of the
effects as much as possible. I foresee two ways to investigate -
I think we need to develop a theoretical understanding of the
non-linear forces, for example using a beam envelope tool at
higher order; and we need to make some measurements at Step IV to
validate our theoretical understanding.

The issue with making measurements is that we will have only a
few week window some time in late summer next year to take data
without absorbers. You probably have heard the running has to be
very compressed for financial reasons. We are unlikely to get a
second chance after that.

So I would be very happy if someone could come up with a set of
run parameters and an analysis well in advance that could be used
to best demonstrate those non-linear effects. For example, * we
probably want to get a high beta function setting, what magnet
settings are required for that? * we probably want to use a high
emittance beam, what emittance is best? * do we have sufficient
resolution to make the measurement? * what statistics are
required? * how do we fight the systematics?

I wonder if you could do that? I think it should be a very fun
and interesting study.

We can arrange a phone call to discuss.