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Michael,

Why does KQUAD element like this one (without nonlinearities) generate some significant amplitude dependent tune shift?
KQUAD, N_KICKS=10, L=0.3, K1=+4.00

Are there any hidden nonlinearities (like fringe fields or something) that are taken into account?
Matrix version of this quadrupole of any order do not produce tune shift.

Alexey.

Alexei,

This is an interesting question. The KQUAD element does not have any fringe field modeling. Given that, I agree that the behavior of KQUAD is counterintuitive. QUAD displays no such behavior, even if I increase the ORDER to 3.

I also checked against the BMAP_XY element, which does integration through a 2D field map. Although it is a completely separate and very different implementation (non-symplectic), I'd expect very similar results to KQUAD if KQUAD is right. Indeed, that's what I get as you can see in the attached figure. The files I used are also attached. Based on this, I guess I need to look more carefully at the higher-order terms in the QUAD matrix.

However, I don't understand physically why the KQUAD and BMAPXY modeling gives these results. My original thought was that it was a result of the length of the quads, but some tests showed that wasn't the case. It also doesn't matter much if I use the 2nd or 4th order integrator.

By the way, using sddsnaff allows getting accurate frequencies with far fewer turns than needed for sddsfft.

--Michael

It looks like this tune shift in KQUAD comes from the v_z * H_y term in the Lorents force.
In qaudrupole vertical magnetic field is
H_y = G * x
Then horizontal Lorentz force is
F_x = (e/c) * v_z * G * x
The motion is linear as long as we assume v_z to be constant. This is only approximately true since
(v_z)^2 = v^2 - (v_x)^2 -(v_y)^2.