There is a convention about how to characterize dipole magnets. The convention is that the BACT value is expressed in length integrated magnetic field strength ("the integral of B dot dL"). Units are kG*m.
To determine the amount of angular deflection a beam would be kicked by such a magnet, there has been introduced a scaling coefficient called the "Magnetic Rigidity":
"Magnetic Rigidity" Cb = 33.35640952 kG*m/GeV
such that the angular deflection, theta, that an (electron) beam would see is:
theta (in radians) = BDES / (Cb * Energy)
If you are interested in how the parameters "KMOD", "LEFF", and "bend radius" are used for dipoles, check my original version.
There is convention about how to characterize quadrupole magnet strength. The convention is that BACT is "length integrated transverse field gradient". Units would be "kG*m/m" but people just say "kG".
If you are interested in how the parameters "KMOD", "LEFF", "bend radius", "transverse gradient", and "focal length" are used for quadrupoles, check my original version.
As a footnote, let me mention the convention for QTRMs, which is a little bizzare. BACT for a QTRM is in units of "equivalent main magnet string current". This becomes messy --- introducing issues of main/trim coil turn ratios, IVB for the main string, etc.
Here the convention is that BACT is "length integrated transverse field curvature" or "length integrated transverse quadrupole field gradient". Units would be kG*m/(m*2) where the numerator's m comes from length integration, and the two m's in the denominator come from the second derivative of transverse field strength. People just say kG/m.
If you think about this long enough, this explains why sextupole magnet BDES values tend to be so large (typically in the thousands).
I have never thought about how "KMOD", "LEFF", etc. are handled for sextupoles
Often there is an intermediate control system "thingy" in between you and the magnet itself (each bend and quadrupole magnet has it's own database called "BNDS" or "QUAS" meaning "bend station" or "quad station"). The intermediary LGPS can be confusing. Often BDES for LGPS dipoles is setup to reflect the beam energy. The problem is most typical where LGPS are used to power a string of many magnets, each of whom might have a different IVB.
Don't panic too much about the LGPS problem. You can always figure it out by translating the LGPS BACT to current (using LGPS IVB), then taking that current and shoving it into the BNDS or QUAS IVB. If you do that, you should get BACT values for which the above conventions hold.
I've started cataloging ways in which this information is useful. Here are a couple of ways you can take advantage of this stuff.
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The time honored technique to measure beta function at a quad by spinning the quad strength and looking at the tune change dnu/dB: Assimilating Sand's Chapter 2 about the guide field, I use EQ. 2.102 together with the notions of KMOD, BACT for quad, and Cb above to convince myself that:
Beta = 4*pi*Cb*Energy * ( dnu/dB)
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I recently struggled with the formulation of how to determine quad alignment by looking at the slope of a downstream position monitor vs. quad strength. Let's say we are interested in measuring the vertical offset in a quadrupole. Scan the quadrupole and sample the BACT and some BPM downstream. Fit the result to a line (Y axis as vertical position monitor readout, X axis as quadrupole strength). The information is in the fitted slope. Here's the formula:
R34 is the RMATRIX value between
slope; dY R34*Yoffs quad angle and BPM position.
----- = --------------
dBACT Energy * Cb Yoffs is the beam position in
the quad which we try to measure.
dY/dBACT we get from the fit
described above.
BE CAREFUL ABOUT THE UNITS!
Energy is beam energy.
Cb is "magnetic rigidity" as per
above.
If you want the proof, I wrote it up here
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Let's pretend we want to do beam based sextupole alignment in an electron storage ring. According to me; by changing the sextupole strength and looking for resulting tune shift, one should be able to infer the beam offset. Let's work it out:
Start with the beta function measurement for quadrupoles:
Beta = 4*pi*Cb*Energy*(dnu/dBq)where Bq is the quadrupole field. To generate confusion, I'll call Bq the quad strength and Bs the sextupole strength. From the explainations above it is clear that the quadrupole component from a sextupole at offset X is seen to be:
Bq(X) = X * Bswith a derivative:
dBq(X) = X * dBsThe first expression above can be re-arrainged to be:
Beta * dBq
dnu = ______________
4*pi*Cb*Energy
If we substitue the value of dBq as coming from the changing sextupole
field from above we get:
Beta * X * dBs
dnu = _________________
4*pi*Cb*Energy
Rearrainging terms yields:
4*pi*Cb*Energy
X = _______________ * (dnu/dBs)
Beta
Let's work out a typical example of this effect. Consider LER IR2 sextupole SCX3R1 in the horizontal plane. Presume there is a 2mm horizontal beam offset, and we vary the sextupole strength by 10%. Nominal BDES is 74 kG/m, the horizontal beta function is about 25m, energy is 3.1 GeV. Substituting these values into the above relation yields an expected horizontal tune shift of .0003 ... which is certainly measurable.
Note: Use caution when employing this technique! If the magnet in question has a large orbit offset, large beta function, or large field change, there may be secondary effects (such as orbit distortion i.e. dipole kick) which causes tune shift from other sources! Check that the global orbit does not change when you change the sextupole, and correct for the global orbit shift as necessary to eliminate misleading tune shift sources.
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