What is the Lens Addition Equation?

[Jrs580]

TL;DR

How do colliders constrain charged particles in the transverse direction?

We all know how quadra-pole and octo-pole magnets are used to accelerate the particles in the longitudinal direction, but what keeps the similarly charged particles constrained in the transverse directions? Especially after the focusing magnets...during the collision you have 10^11 positively charged particles in a ~cubic micron volume.

 

[Vanadium 50]

We all know how quadra-pole and octo-pole magnets are used to accelerate the particles in the longitudinal direction

Oh, I hope not, since it's not true. Magnetic fields do no work, remember?

Quadrupoles are focusing magnets.

 

[SBoh]

First of all, high order magnets are used for transverse direction focus of particle beams.
It means that magnets are used to "constrain particles in transverse direction".
(The interesting point is that different setting of such magnets are required depending on particle beam energy due to the space charge effect. That is reason why multiple accelerators are required to provide particle beam to the LHC.)

rAcceleration, longitudinal emittance corection, compensation of energy losses cause by betatron radiations of particle beams are performed using ratio frequency (RF) electromagnetic standing waves in RF cavities. You can search for principle of synchrotrons.

 

[mfb]

Quadrupole magnets are the main focusing magnets. Quadrupole magnets always focus in one dimension and defocus in the other (both orthogonal to the beam axis). The common basic design here is a FODO structure: Focusing, some empty space, defocusing, more empty space. A repeated FODO is FODO in both dimensions just with an 180 degree offset.

https://arxiv.org/abs/1303.6514

Higher order magnets are used for smaller corrections.

 

[nsaspook]

Some reference pictures of small industrial ion accelerator Quadrupole magnets with the vacuum chamber beam-guide removed. It's easier to see the scale and physical properties with something smaller than the LHC.

 

[Jrs580]

Quadrupole magnets are the main focusing magnets. Quadrupole magnets always focus in one dimension and defocus in the other (both orthogonal to the beam axis). The common basic design here is a FODO structure: Focusing, some empty space, defocusing, more empty space. A repeated FODO is FODO in both dimensions just with an 180 degree offset.

https://arxiv.org/abs/1303.6514

Higher order magnets are used for smaller corrections.

Thank you for the input, my understanding was that quadrupole magnets focus in 1 dimension like you said and de-focus in another. So how have you alternate them in a way to have a net focus in all dimensions? And what about the z direction? We care very much about the length of the bunch I believe.

 

[Vanadium 50]

@Jrs580 do you really want this at A-level?

If I have a (equal strength) focusing and a defocusing lens in either order (what @mfb was talking about with the FODO lattice) the the lens addition equation tells me the net result is focusing.

 

[Jrs580]

Oh, I hope not, since it's not true. Magnetic fields do no work, remember?

Quadrupoles are focusing magnets.

You are correct, I misspoke. Magnetic fields do not do any work because the force is always orthogonal to the dr of the particle.

 

[Jrs580]

@Jrs580 do you really want this at A-level?

If I have a (equal strength) focusing and a defocusing lens in either order (what @mfb was talking about with the FODO lattice) the the lens addition equation tells me the net result is focusing.

Forgive me for I am new here. Can you direct me to where to find what the different levels are? Or just tell me, either way.
Is this the lens equation you are referring to?

 

[berkeman]

Forgive me for I am new here. Can you direct me to where to find what the different levels are? Or just tell me, either way.

When you start a thread, there is a pull-down menu where you select the prefix that signifies at what level you want the discussion held:

 

[Vanadium 50]

No, I meant the lens addition equation:

$$
\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}- \frac{d}{f_1f_2}
$$