Farley’s paper on absorbers

1. Farley’s paper on absorbers Very rough notes, rather late at night.... not as pedagogic as I would wish. ‘Standard’ treatment of dE/dX cooling cools because p-perp and p-long are reduced by dE/dX; p-long is replaced by RF; CMS in absorber causes divergence to increase. These effects give only two terms in the familiar expression for emittance decrease, involving dE/dX and radiation length. In principle, CMS in absorber also causes rms size of beam and x-x’ covariance to increase; there should be four terms in the expression for d(emittance)/dX. Absorbers for muon cooling are assumed to be located in a strong focussing field, at a waist and if certain conditions (to be understood – by me) are satisfied, then it is legitimate to ignore the increase in rms size and x-x’. This leads to a F.O.M for a material of dE/dX x rad. len. (MeV) (?? Scale??) MICE VC

2. Farley gives a treatment of a beam focussed on/in an absorber and solves the evolution equations for the beam x –x’ moments matrix (covariance matrix). This seems to be rather standard stuff assuming linear optics; in fact he just asks how a beam propagates in a drift. He then adds CMS and arrives at an expression for the emittance of the beam after it has passed through an absorber (= degrader), roughly: E(out)**2 = E(in)**2 + sqrt(AC –B**2) + (A *Sxx – 2*B*Sxx’ + C*Sx’x’) (don’t trust that expression – it’s from memory!!) where A,B,C are elements of moments matrix (~Twiss parameters) for the absorber and depend on dE/dX and Xo. Note that A,B and C depend on t, t**2 and t**3 respectively (if I remember rightly). He then minimises E(out)**2 wrt Sxx, Sxx’, Sx’x’ and finds that the beam should be focussed at the centre of the absorber and that on exit, with no CMS, it should have a moments matrix proportional to the ABC matrix. MICE VC

3. The main result is that there is an optimum optical setting for the beam – to match the absorber -- and that for a fixed dp in the absorber, the figure of merit for a material is FOM = (dE/dX)**2 * Xo in MeV**2 – cm. I have gone through all that part of the paper and it seems correct and very plausible (though I am not expert in this stuff!) Note the units for his FOM. Not only do the atomic properties matter, but also the density. Diamond is better than graphite. He concludes diamond is best, boron carbide is next. So, what are the differences wrt to our received wisdom? A – perhaps the – difference is that his treatment allows for the increase in the rms size of the beam which the standard dE/dX treatment does not. A task is to understand under what conditions this can be ignored in the context of muon-cooling. Presumably that is documented somewhere. Dan Kaplan has pointed me to a useful reference which points to another reference.... MICE VC

4. He – Farley – also goes on to discuss ionisation cooling and the like. I have • not followed those sections in any detail, except to note that: • He assumes that his optimisation is always possible. It may be neither • possible nor, for other reasons, desirable. But I do not know either way. • In a table at the end of the paper he compares the cooling rates of 315 • MeV/c muons in optimised LH2 and Boron carbide absorbers. • At face value, boron carbide wins by a fairly large factor. In fact, • when normalised to the same fractional energy loss, the fractional decrease • in emittance is identical – all cases seem to be well above any equilibrium • emittance (which is not defined per se).The emittances (30 -- 1000 mm-mr) • he discusses are well below MICE emittances. • As usual, there’s more to understand, especially the thinking of the muon • cooling experts. MICE VC