Why is beta 11 m? The explanation!

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Tau
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Why is beta 11 m? The explanation!

Post by Tau » Thu Mar 04, 2010 7:06 pm

Thanks to interesting discussions with photino :wave: I managed to get a way to explain what beta and "squeezing" is all about, without using formulas.

The short version: if beta is low, the beam is narrower, "squeezed". If beta is high, the beam is wide and straight.
Beta is the distance from the focus point that the beam with is twice as wide as the focus point.

But why don't you just make the beam narrow everywhere?
First, you have to realize that "the beam" is not a well-behaved bunch of protons all moving in exactly the same direction. Due to all kinds of reasons, the protons all have a varying amount of sideways motion. Keeping this in check is handled by the quadrupoles around the ring, and by tuning (see PCatoms' explanation elsewhere on this forum). There's also longitudinal motion, but that's another story (RF and sextupoles come in the story there: don't ask).

Anyway, if you want to focus the beam, you have to live with the fact that you can never get all the particles exactly through one point, because this sideways movement cannot be suppressed.
The "width" of the beam in the LHC is normally a few mm. The "epsilon" parameter, also called "emittance" of the beam, can be defined as "the smallest opening you can squeeze the beam through, in theory". If you squeeze the beam through such a small hole, the beam will very quickly spread out through that hole, and there is no magnet that can get it together anymore. The emittance of the LHC is about 4 micrometers (um).
On the other hand, if you make the particles all move nicely in the same direction, the beam is wider, but will barely spread out further. That is what happens in the long arc magnets: beam width is there about 7 mm. Wide straight beams are great for achieving high energies.
The shape of the beam has to vary throughout the LHC for focusing reasons, but the "emittance" does not vary around the LHC, whatever the shape of the beam.

So, basically, you have a choice between a wide "well behaved" beam, that can be focused to a narrow "chaotic" beam.
The "beta" parameter tell you what the width of the beam is at a point.
Normally in the LHC the beta is about 10 meters, so that it spreads out a few mm wider inside an arc magnet, and then is focused by a quadrupole into the next magnet.

But in the experiments, the beam will be "squeezed" as much as possible, to increase the number of collisions, so just before some experiments are strong focusing magnets that squeeze beta down to 0,5 m, with the focus point precisely a defined location. From the definition of beta we know that half a meter away from that point, the beam is twice as wide already, so it becomes quite wide before it can be reshaped again to a higher beta to enter the tube again.
Towards the focus point, because the proton got in an almost exactly straigth path, the beam becomes narrower. So at a distance of beta before the focus point, the beam is also twice as wide (clarification suggested by Joe on IRC).

PS if you really want to see a formula: beam width = sqrt (emittance*beta).
Disclaimer: I am not a physicist, but a mathematician. Although I am pretty sure the math is right, there might be inaccuracies in the details.

PS in the meantime, I found an article by a real physicist, Lucie de Nooij, working at the ATLAS detector, who gave in her blog the same explanation. So now I am sure it is correct!
Last edited by Tau on Thu Jun 10, 2010 2:49 pm, edited 2 times in total.
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Re: Why is beta 11 m? The explanation!

Post by oxodoes » Thu Mar 04, 2010 10:54 pm

Tau wrote:Beta is the distance from the focus point that the beam with is twice as wide as the focus point..
Hmm thats new to me. Could you please give a reference.

Just to make it a bit easier to imagine, think of the particle beam as a light beam that spreads out when left alone. (The spreading of a partice beam can be easily explained by the beam particles all having the same electric charge, so they repel from each other.)

EDIT: Some further reading has shown that electromagnetic repulsion seems to be negligible. (http://cas.web.cern.ch/cas/ZEUTHEN/Zeut ... ioCAS1.pdf, page 11) The widening of the beam is rather due to the dipole magnets bending particles with slightly different speeds slightly different (F=vxB) and some tiny errors in the accelerator structure.

Now we can use lenses (quadrupole magnets act exaclty like lenses) to focus the beam. (Actually quadrupoles only focus the beam in one plane and defocus it in the other plane. But as the focusing effect gets stronger when you move away from the center of the magnet one can use a vertically and a horizontally focusing magnet (a so called FODO cell) together to achieve overall focusation.) Once the beam passes the focus point it gets spread out again. So one has to install many FODO cells to contol the beam on its way around the accelerator.
(As a reference one might take http://indico.cern.ch/getFile.py/access ... nfId=57408 from last years Cern summer school.)

Just as a little example. This is the beta-function of the CERN AD as computed with WinAgile (lattice from sample data):
Image
Last edited by oxodoes on Fri Mar 05, 2010 2:30 pm, edited 2 times in total.

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Re: Why is beta 11 m? The explanation!

Post by Tau » Fri Mar 05, 2010 9:38 am

The whole story on the beta is in this document:
https://espace.cern.ch/be-dep/Lists/AXE ... tics_2.pdf
On page 12, there is an ellipse graph that shows exactly what beta is: the width of the ellipse divided by the height. After a bit of calculation, I came to the definition of "distance of doubling of width from the focus point". Beta will only change in a quadrupole.
As far as I know, that is not equal to the "betatron amplitude": this is the width of the beam itself, which changes along the path of the beam.
But your pretty picture nicely shows the focusing of the beam, and you can see that the beta is about one quarter of the cell length, which matches the actual value of 11 m.
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Re: Why is beta 11 m? The explanation!

Post by oxodoes » Fri Mar 05, 2010 1:35 pm

Tau wrote:After a bit of calculation, I came to the definition of "distance of doubling of width from the focus point".
IRC wrote: <photino> tau's argument, slightly fixed: take a single particle at the beginning of a drift space. it can have x, x' anywhere on the ellipse. so x is bounded by sqrt(epsilon beta), and x' by sqrt(epsilon/beta)
<photino> take the maximal values and ask after what minimum distance might the beam be twice as wide
<photino> answer is beta
<photino> of course a very worst case rough estimate
This calculation seems reasonable, yet we were not able to find any reference dealing with this kind of argumentation. So in my opinion this is not to be taken as a fact.

Tau wrote:As far as I know, that is not equal to the "betatron amplitude": this is the width of the beam itself, which changes along the path of the beam.
Looking at the dimensions of the graph axis clearly shows that the betatron function can not be the actual beam size. This reference (http://indico.cern.ch/getFile.py/access ... nfId=34185 (slide 13)) indeed shows that beta and betatron function are synonyms.

If you are looking for some further reading material you are sure to find something here: http://indico.cern.ch/scripts/SSLPdispl ... &nbweeks=7
http://lhcportal.com/Forum/viewtopic.php?f=5&t=136

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Re: Why is beta 11 m? The explanation!

Post by Tau » Sun Mar 07, 2010 3:57 pm

Right, thanks for the clarification.
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Re: Why is beta 11 m? The explanation!

Post by oxodoes » Wed Jun 09, 2010 4:10 pm

Tau you are confirmed!! This is from the CERN bulletin.
http://cdsweb.cern.ch/journal/CERNBulletin/2010/23/News%20Articles/1269066?ln=en wrote:What is a β*?

β* describes the beam size at the interaction point. More mathematically, it is a measure of the distance from the interaction point at which the beam is twice the size of that at the interaction point. The lower the beta, the smaller the beam at the interaction point, therefore the better for the physics. Before beam squeezing, beta is typically 11 m at ATLAS and CMS. Today, we’re running with a beta of 2 m, and the ultimate goal is to reduce it to 0.55 m.

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