[MAP] Correction (Re: Hamiltonian-gauge potentials

Fri Mar 18 01:48:46 EDT 2011

Rob,

To confirm that eigen-emittances are not constant in the nonlinear case, 
I've run the beam with the following parameters in g4beamline (MICE Step 
IV beam for wedge absorber simulations): p_ref=200 MeV/c, norm. 
longitudinal emittance 90 mm, norm. transverse emittance 6 mm, 
sigma_x=sigma_y=37 mm, sigma_px=sigma_py=17 MeV/c, sigma_pz=29 MeV/c, 
sigma_T=1.25 ns, no dispersion. The lattice is a simple drift of 3.3 m. 
I calculated eigen-emittance the way you suggested in your slides at the 
MAP meeting. Since the drift is nonlinear, and the beam occupies a 
rather large phase space, all three emittances grow in the similar way 
the rms emittances (=based on the second moment matrix) do.

Pavel

On 3/17/2011 21:38, Robert D Ryne wrote:
> What I wrote below is not correct. As Kirk just pointed out to me, and
> Alex wrote earlier, it is true only in the linear approximation.
> As Alex wrote earlier,
>> But the eigen emittances are invariant under linear symplectic
>> transformations, and therefore also under gauge transformations in the
>> linear approximation.
>
> Rob
>
>
> On Mar 17, 2011, at 4:29 PM, Robert D Ryne wrote:
>
>> Kirk,
>>
>>> On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:
>>> It remains that rms emittance (and eigen-emittance, I believe) is not
>>> gauge invariant. But we have to choose some gauge to proceed.
>>
>> The eigen-emittances /are/ gauge invariant.
>>
>> More precisely, consider a numerical distribution of particles in the
>> presence of electromagnetic fields. These can be defined, e.g., using
>> Cartesian coordinates and velocities at some time, or using transverse
>> coordinates and velocities along with energy and arrival time at some
>> location z, etc. Next, express the numerical distribution using
>> /canonical/ variables. Some elements of the 6-vector describing each
>> particle /will/ depend on your choice of gauge (e.g. p_{x,canonical}
>> and p_{y,canonical}). Next, compute the beam second moment matrix
>> \Sigma. Some of the matrix elements /will/ depend on your choice of
>> gauge. Next, compute the moduli of the eigenvalues of J \Sigma (these
>> are the eigen-emittances). Though the numerical distribution (the
>> collection of 6-vectors) and the \Sigma matrix will depend on the
>> gauge, the eigen-emittances will be /independent/ of the choice of gauge.
>>
>> Rob
>>
>>
>> On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:
>>
>>> Folks
>>> I believe that what Alex Dragt has been trying to tell us is that for
>>> applications involving Hamiltonian dynamics it is most favored to use
>>> the so-called Hamiltonian gauge, in which the scalar potential V is
>>> zero everywhere. Further, this ties in nicely with the use of
>>> coordinates (x,y,t) rather than (x,y,z).
>>> For a general expression for the vector potential A in this gauge,
>>> see sec. 8 of
>>> http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf
>>> For fields with time dependence e^{- i omega t), with nonzero wave
>>> number k = omega / c, then the vector potential is simply
>>> A = – i E / k. (Gaussian units).
>>> Note that A = 0 wherever E = 0, in which regions B can only be static.
>>> For a static magnetic field, the vector potential is the same as that
>>> in the Coulomb gauge (and in the Lorenz gauge). [In some static
>>> cases, such a toroidal magnets, the static vector potential will be
>>> nonzero in regions where B is zero; but in regions where a Fourier
>>> component B_omega is zero for nonzero omega, A_omega will be zero also.]
>>> A static electric field has the vector potential
>>> A = – c(t – t_0) E.
>>> This seems a bit odd, but will not bother us.
>>> An unusual feature of the Hamiltonian gauge vector potential is that
>>> is is not continuous at a perfectly conducting surface – which did
>>> bother me for quite a while.
>>> However, in using A to compute B, the (ill-defined) normal derivative
>>> at a perfectly conducting surface is not needed to deduce the
>>> tangential B.
>>> Of course, the Hamiltonian gauge vector potential makes the canonical
>>> momentum p = p_mech + q A / c
>>> discontinuous at the surface of the rf cavity.
>>> This is troublesome if we use coordinates (x,y,z) and have an
>>> accelerating cavity with E_z, such that p_z takes a step on entering
>>> or leaving the cavity.
>>> However, if we switch to coordinates (x,y,t) the t-canonical momentum is
>>> p_t = – E_mech – q V = – E_mech
>>> in the Hamiltonian gauge, which is continuous at the cavity wall.
>>> Also, canonical momenta p_x and p_y are continuous if the particles
>>> enter and leave the cavity through faces at constant z (of a good
>>> conducting material).
>>> All this is serendipitous for the cooling sections of a muon
>>> collider, where we are almost certain to use cavities with flat (Be)
>>> faces where the particles enter and exit.
>>> It remains that rms emittance (and eigen-emittance, I believe) is not
>>> gauge invariant. But we have to choose some gauge to proceed.
>>> It now looks like the Hamiltonian gauge is the one to use, along with
>>> coordinates (x,y,t) rather than (x,y,z).
>>> 1. Time-dependent A_omega follows immediately from knowledge of
>>> E_omega, and is zero where E_omega and B_omega are zero.
>>> 2. Static A is same as in the Coulomb gauge (which is what we almost
>>> always use now).
>>> 3. No steps in canonical momenta so long as all rf cavities have flat
>>> faces where particles enter/exit.
>>> 4. No scalar potential to worry about, so p_t = – E_mech, as in our
>>> present software. (It doesn’t hurt anything to define p_t = + E_mech,
>>> as we actually do.)
>>> ---------------------------------------------------------
>>> 5. Although phase volume is the same where or not we include the
>>> potentials in the momenta, this is not true for rms emittance (or
>>> eigen-emittance). So we should stop using ECAL9, and switch to
>>> emitcalc (with a better approximation to the vector potential) and/or
>>> equivalent programs developed by Tom Roberts, Chris Rogers, et al.
>>> --Kirk
>>> _______________________________________________
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>>> MAP-l at lists.bnl.gov <mailto:MAP-l at lists.bnl.gov>
>>> https://lists.bnl.gov/mailman/listinfo/map-l
>>
>
>
>
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