# AIDASoft/DD4hep

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# Fix multipole fields #322

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merged 4 commits into from Mar 7, 2018

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### danieljeans commented Feb 28, 2018

 I believe there are bugs in the definition of the multipole fields (both in the code and the documentation). I've compared to the definitions given in https://dd4hep.web.cern.ch/dd4hep/reference/classdd4hep_1_1MultipoleField.html BEGINRELEASENOTES correct coded expressions for quadrupole, sextupole and octopole fields. correct expression for octopole field in documentation ENDRELEASENOTES

### petricm reviewed Feb 28, 2018

 bx += coefficents[3] * (3.0*x2*y - y2*y) + skews[3]*(x2*x - 3.0*x*y2); [[fallthrough]]; case 3: // Sextupole momentum: by += -coefficents[2] * (x2 - y2) + skews[2] * 2.0 * xy; by += coefficents[2] * (x2 - y2) - skews[2] * 2.0 * xy; bx += coefficents[2] * 2.0 * xy + skews[2] * (x2 - y2);

#### petricm Feb 28, 2018

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Shouldn't this be

```by +=  0.5 * coefficents[2] * (x2 - y2) - skews[2] * xy;
bx +=  coefficents[2] * xy + 0.5 * skews[2] * (x2 - y2);```

to have the coefficient[1] vs. coefficient[2] properly normalised?

 @@ -99,16 +99,16 @@ void MultipoleField::fieldComponents(const double* pos, double* field) { double y2 = y*y; switch(coefficents.size()) { case 4: // Ocupole momentum by += coefficents[3] * (x2*x - 3.0*x*y2) + skews[3]*(y2*y - 3.0*x*y2);

#### petricm Feb 28, 2018

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I think this fix is correct, but the same comment as below, shouldn't there be a 1/6 in front for normalization?

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### danieljeans commented Mar 1, 2018

 I believe that what I implemented in the code is consistent with the definition in the documentation in the header file, which is in turn consistent with the references to which it links: for the nth multipole, By + i Bx = ( b + i a ) ( x + i y ) ^ (n-1). [b is the "coefficient", a is the "skew".] (eg see p10 of http://cas.web.cern.ch/sites/cas.web.cern.ch/files/lectures/bruges-2009/wolski-1.pdf) I'm sure that definitions with different normalisations are possible, but this one seems standard at least in the accelerator community.
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### petricm commented Mar 1, 2018 • edited

 I cannot agree with your statement this one seems standard at least in the accelerator community. If you look at this lecture from another accelerator school you find an explicit definition for the terms (page 23) which give you different normalization (the form of the expression is however the same).
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### danieljeans commented Mar 1, 2018

 @petricm thanks for pointing this out to me. I have no preference as to which convention to use, as long as it's consistent and well documented.
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### petricm commented Mar 1, 2018

 I'll ask someone in the magnets department for clarification and let's see what they say...
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### petricm commented Mar 1, 2018

 I still did not receive a reply but for book reference you can look at this book with the definition the same as the USPAS slides.
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### petricm commented Mar 5, 2018 • edited

 I have received a reply that we should consider Field Computation for Accelerator Magnets by Stephan Russenschuck as the reference for all matters accelerator magnets. If you wish you can download the book inside the CERN network here. The conclusion is that a sextupole is parameterised as ``````Bx(x,y)= 1/r_0^2 ( A3(r_0) (x^2-y^2) + 2B3(r_0)xy) `````` where r_0 is an arbitrary reference radius, so the boundary conditions defines you normalization. If you however do a multipole expansion as our code does, so add a dipole field to a quadropole, then there is a 1/(n-1)! term. Chapter 6 discussion from (6.80) to (6.83) and the definition of normalized gradient. So I would like to kindly ask you if you can fix the code to include this.
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### danieljeans commented Mar 6, 2018

 thanks for the info. It's easy to include the (n-1)! scaling, I'll do it right away. if you want the ability to explicitly include the r_0 normalisation (ie to be able to specify the coefficients and skews at any particular reference radius), we would need an extra parameter. In my opinion, we can explicitly say that r_0 = 1 and avoid this complication: do you agree?

### danieljeans added some commits Feb 28, 2018

``` correct expressions for multipole fields ```
``` 2a911d6 ```
``` correct expression for octopole field in documentation ```
``` b717b14 ```
``` update documentation: links to external references ```
``` c861c6d ```
``` FieldTypes: add factorial function to definition of multipole coeffic… ```
`…ients/skews`
``` e0b9630 ```

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### danieljeans commented Mar 6, 2018

 by the way, I could not get my hands on the book you referenced: the KEK copy is out on loan, and I cannot get into the cern network: so please check that what I implemented looks as expected.

### petricm merged commit `91a695f` into AIDASoft:master Mar 7, 2018 1 check passed

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### petricm commented Mar 7, 2018

 I went through the equations again and they are correct.
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