What is the Lens Addition Equation?

In summary: At level 1, which is intended for students, this equation would be sufficient. Higher level discussions would require the use of derivatives and integrals, and so on.
  • #1
Jrs580
20
4
TL;DR Summary
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.
 
Physics news on Phys.org
  • #2
Jrs580 said:
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.
 
  • #3
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.
 
  • #4
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.
 
  • Informative
  • Like
Likes vanhees71 and berkeman
  • #5
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.

IMG_20160418_105841772.jpgIMG_20160418_110136807.jpgIMG_20160418_105816780.jpgIMG_20160418_105755249.jpg
 
  • Informative
  • Like
Likes dlgoff and berkeman
  • #6
mfb said:
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.
 
  • #7
@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.
 
  • Like
Likes vanhees71 and mfb
  • #8
Vanadium 50 said:
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.
 
  • #9
Vanadium 50 said:
@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?
1628782280346.png
 
  • #10
Jrs580 said:
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:

1628785489780.png
 
  • #11
No, I meant the lens addition equation:

$$
\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}- \frac{d}{f_1f_2}
$$
 
  • Like
Likes Jrs580

FAQ: What is the Lens Addition Equation?

What is the purpose of focusing magnets at the LHC?

The focusing magnets at the LHC are used to steer and focus the beams of particles as they travel through the accelerator. This ensures that the particles collide at the correct energy and location within the detectors, allowing for accurate measurements and observations.

How do focusing magnets work?

Focusing magnets use strong magnetic fields to bend the path of charged particles, causing them to travel in a circular path. The strength and direction of the magnetic field can be adjusted to control the trajectory of the particles and keep them on the desired path.

What is the strength of the focusing magnets at the LHC?

The focusing magnets at the LHC have a strength of up to 8.3 tesla, which is about 100,000 times stronger than the Earth's magnetic field. This allows them to bend the path of particles traveling at nearly the speed of light.

Are there any risks associated with focusing magnets at the LHC?

There are potential risks associated with the high magnetic fields used in the LHC, but extensive safety measures are in place to protect both the equipment and personnel. The magnets are designed and tested to ensure they can withstand the forces and temperatures involved, and strict safety protocols are followed during operation.

What advancements have been made in focusing magnets at the LHC?

The LHC has continually improved and upgraded its focusing magnets since its initial operation in 2008. In 2015, a new type of focusing magnet called the "crab cavity" was successfully tested, allowing for even more precise control of particle beams and increasing the potential for new discoveries in particle physics.

Similar threads

Back
Top