[Halld-tagger] Info for the IPR 2009 review

[Richard Jones]

Eugene,
> Is the multiple scattering for those particles which produce coherent Bermsstahlung the same as the scattering in amorphous materials of the same thickness? 
 No they are not the same in general.  Coherent scattering is taken into 
account in the calculation of Mott scattering in a similar fashion to 
the way it is done for bremsstrahlung.  In fact the two integrals become 
the same  in the limit of zero bremsstrahlung photon energy, as you may 
easily imagine.  This relationship makes it easy to see the conditions 
under which coherent effects in the crystal make a big contribution over 
the incoherent: whenever the coherent part of the bremsstrahlung 
spectrum is a significant fraction of the total at the k->0 end of the 
spectrum.  Everybody has heard about how channeling can be used to 
deflect a high-energy electron beam without affecting its energy 
signficantly.  This is just multiple-scattering of course, but dominated 
by coherent scattering from a single lattice vector because the crystal 
is oriented to align the beam direction very close to one of the crystal 
axes.  In coherent bremsstrahlung, this corresponds to moving the 
coherent peak down to k->0.  On the other hand, the way we orient the 
crystal for Gluex, we are careful to keep the low-energy region of the 
spectrum free of any coherent peaks, so we negligible coherent effects 
there.   So there is no significant coherent contribution to multiple 
scattering at the kinematics for Gluex.

This raises a follow-up question: is the incoherent contribution in 
diamond reduced relative to an amorphous radiator of the same thickness 
because part of the momentum integral has been replaced with a discrete 
sum? The answer is yes, there is some effect.  Typical values I have 
seen in the literature are at the level of 10% when the peak is around 
0.5 E0.  It will be smaller for Gluex with the peak at 0.75 E0, roughly 
5%.  In the early days when I was first working out the properties of 
this source, I side-stepped this question by describing the radiator 
thickness in "radiation lengths".  This term requires a precise 
definition for a crystal target, so I defined it as the coefficient of 
the (1/k) factor that describes the bremsstrahlung spectrum in the 
low-energy region (assuming it is free of significant coherent 
components down there).  At this point we are now specifying the 
radiator thickness in microns, but we have not specified it at a level 
of precision that requires us to care about a 5% correction.

All of this to say, yes there is a shift, it reduces the multiple 
scattering somewhat, and we have not calculated the precise shift but 
based on values reported in the literature, we expect that MS in the 
diamond oriented for Gluex running will be lower than an amorphous 
radiator of the same g/cm^2 by about 5%, which is within the stated 
tolerance of +/1 micron on the thickness of the radiator.

> Another question - how long is the coherent path responsible for this Bremsstahlung? In other words, does the ratio of the coherent to regular radiation depend on the crystal thickness?
>   

I describe this in chapter 4 section 4.2.5 of the design report, with 
equation 4.3 showing the essential result.  It depends on the crystal 
orientation and which photon peak you are looking at, but for the Gluex 
orientation and photons at 9 GeV, the coherence length of the radiation 
process in the crystal is about 6 nm, or 20 lattice constants.  It is 
too small to be of any concern in the choice of radiator thickness.

Richard Jones

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