I'm wondering how long it takes the LHC to accelerate particles from rest to their top speed at 6.5 TeV. And related, how long a hypothetical linear accelerator would have to be to accelerate particles to the same energies.
The times in accelerators depend on the specific way of constructing them. Linear accelerators have different elements in their building up of the beams.
Here is a source for the LHC design.
Its length is 31kilometers.
To compact the 5 GeV electron and positron bunches to a sufficiently small size to be usefully collided, they will circulate for 0.1–0.2 seconds in a pair of damping rings, 3.24 km in circumference, in which they will be reduced in size to 6 mm in length and a vertical and horizontal emittance of 2 pm and 0.6 nm, respectively.
From the damping rings the particle bunches will be sent to the superconducting radio frequency main linacs, each 11 km long, where they will be accelerated to 250 GeV. At this energy each beam will have an average power of about 5.3 megawatts. Five bunch trains will be produced and accelerated per second.
This gives you the timing for this particular design.
Accelerators and colliders have individual characteristics.
Here is an official description from the LHC division: https://indico.cern.ch/event/434129/contributions/1917195/attachments/1205096/1765722/Nominal_cycle.pdf This is basically valid at least for LHC Run 2 (2013-2018) but mostly also before that, and probably also in the future. I have been myself in LHC control rooms a lot during runs (run 1 and 2).
In short, one "fill" of LHC has the following steps:
- "injection" (about 1 h), here individual bunches (very short bulks of particles) are injected from SPS pre-accelerators into LHC at an energy of 450 GeV.
- "prepare ramp" (a few minutes)
- "ramp" (about 30 mins). Beam is accelerated from 450 GeV to 6.5 TeV
- "flattop" (a few minutes)
- "squeeze" (a few minutes). Beams are adjusted in transverse size.
- "adjust" (a few minutes). Beams are finally brought to collision in the interaction points.
- "stable beams" (many hours, up to maybe almost one day). Experiments take data with the same beam. The number of particles decreases with time.
For a linear accelerator you can easily assume particles start at zero velocity on one side, and end up with end-energy on the other side. Particles like electrons (but also protons) will reach relativistic velocities after a few meters in the beginning. Thus, for a hypothetical planned linear collider with a length of 50 km the particles will always move just under the speed of light while they are continuously accelerated. This would take just 50 km/c = 0.00017 s.
Thus, a linear accelerator can be about 100 million times faster in accelerating particles. Not that this is of so much relevance...
I don't know about the LHC, but I remember visiting CERN in the SPS days of 1990's and there were display screens in various places that allowed you to see the strength of the magnetic field in the SPS ring as it was ramped up during the acceleration process. It took a couple of seconds to rise to full strength, and then drop quickly back to zero. I imagine that LHC has a similar duty cycle of a few seconds.
Edit: It look like the LHC is much slower. The ramp up to fullpower is 25min, and the beam is then stored for 10Hr. These details are in the following link"
The LHC has eight accelerating structures (radio frequency (RF) cavities) per beam grouped in four cryomodules. The acceleration from 450 GeV to 6500 GeV takes about 30 minutes. In this time the beam travels around the ring for approximately 20 million times.
Here is a picture of one of LHC RF cavities cryomodule without the external shielding, where you can see the four cavities:
Now, do you need to put 40 millions of those in a line to have a linac with similar energies? The answer is... come on, we can be smarter than that!!!
The key concept here is called beam loading. That is how much of the energy stored in the cavity under the form of electromagnetic field, gets transferred to the beam as it travels through it. With a higher beam loading, you need less accelerating structures (or less passages through them) but you need a more powerful RF production and transport system.
In a synchrotron, such as the LHC, the beam loading is kept to minimal values, typically few percents. The limitation actually comes from how fast you can ramp the current into the magnets, but even if that was not there, limiting factors from RF and beam stability would not be too far away.
A linac does not have any of these limitations so you can run it with a beam loading factor close to (if not exactly of) 100%. This means that all the energy stored into the RF cavities gets transferred to the beam. You will have to wait some time between each pulse to refill the cavities, but that may be ok. You can also optimize the cavities to operate at higher frequencies, so that the maximum accelerating field increases and, at the same time, they get smaller, so it is easier to pack more of them.
These two actions, combined, will give you about a factor 1000 in efficiency, so that the 40 million cavities now become 40 thousands. A high frequency cavity can be less than meter long (but let's say a meter including some additional instruments magnets here and there). So you need a linac which is about 40 km long to reach an energy comparable to the one of the LHC. Still a huge machine, but technically reasonable and potentially economically feasible in the context of a large international collaboration such as the ILC and CLIC.