Luminance, luminosity, squeezing explained in English

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Tau
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Luminance, luminosity, squeezing explained in English

Post by Tau » Mon Apr 26, 2010 6:31 am

Hi Folks,
Having read lots of articles and questions about luminosity and luminance I will try to explain what I understood about it. Now that we have squeezed stable beams, this is becoming to be urgent.

The problem is that most of the articles use lots of math :shock: , and this scares away interested intelligent people who don't have enough math background :!: . This article tries to fill the gap, starting as simple as possible and ending by actually computing the luminosity for the first squeezed run!. 8-)

Remember the "work" the LHC produces, is collisions between protons. We measure that like this:
  • luminosity: how fast collisions are produced
  • luminance or integrated luminosity: how many collisions are produced
The luminosity is a measure of how many collisions are produced per second, and it depends on many variables.

You could compare this with rain: if you are waiting for the rain to give your plants enough water, one of these things will increase the amount of water for the plants:
  • If it rains very hard
  • If you collect the rain and give it to the plants directly
  • only give water at the roots of the plants
  • If you wait long enough
Each of these things are used in the LHC too. You can compare the protons to rain droplets, but instead of watering plants they are brought together to collide.
How much water your plants have got is called "luminance" or "integrated luminosity", and "luminosity" is how fast this is happening.
  • Higher intensity (more bunches in the beam, and more protons per bunch) correspond to more rain;
  • squeezing the beam makes it more narrow, corresponding to collecting the rain;
  • making the bunches shorter is equivalent to watering at the root of the plants;
  • waiting long enough is a matter of keeping the beam in for a long time (which is an art on itself
Luminosity is measured in a horrible unit called "per microbarn per second". Don't worry too much about this :? : just think of how fast the plants get wet; we'll worry about the details later :) .
One page 2 http://op-webtools.web.cern.ch/op-webto ... p?usr=LHC3 the luminance is given in that unit.
If you don't want to do any maths, is suffices to know that the luminosity number will have to be above say 10 or so for a few days on end to get any useful results for the experiments, and that the record to date is about one hundredth (0.01 or 1e-2), so it is still going really slow.

If you like maths you can read on from here, and than you are set to read some of the more advanced articles elsewhere.

About the unit:
The "barn" is a (very tiny) measure of surface area; it is the area of a square with sides of 10 femtometers. In SI units, 1 barn = 1e-28 m^2.
Luminosity is be measured in "per barn per second", equivalent to 1e28 per square meter per second, but this enormous number is still too small too use. Now instead of adding a scale factor to the luminosity proper, like "mega-per barn per second", they scale down the area, so that they get "per microbarn". (Like speaking about "meter per millihour" instead of "kilometer per hour": it's the same thing, only more confusing :) )

In order for the physicists to be happy, the LHC will have to produce several "inverse femtobarn" of data, written as fb^-1. One "inverse femtobarn", which is anoter way of saying "one per femtobarn" is equal to 1 billion per microbarn: for example, a thousand per microbarn per second, for a million seconds. Currently the LHC does about .011 per microbarn per second (only at the beginning of a run), which totals to is about 950 per microbarn per day (and in practice more like a quarter of that number). So, to get to an inverse femtobarn, we'll have to run a million days at least. But, in the future, with at several thousand times higher luminosity, we could get there.
For comparison:
  • the Tevatron produced about 8 inverse femtobarns of data to date (which is a lot, since Tevatron has high luminosity already).
  • All the initial runs up to this week at ATLAS gave 200 inverse microbarns.
  • The squeezed run gave 700 extra inverse microbarns! But still, were are only at .0000009 fb^-1, so there is still a long way to go.
The unit used on page 2 is "per microbarn per second", which is equal to 1e34 m^-2*s^-1, or, as stated in the log, 1e30 cm^-2*s^-1.

If you want to calculate the luminosity yourself, don't forget to take into account that the protons are not equally distributed in the tube, but are bunched together, increasing the probabilty of interaction (as the physicists would say: "concentrated in the longitudinal direction".
Combining all this, luminosity is "effective proton passages per area per second":
<number of protons>/<area of contact>*<rotations per second>/<bunch length per tunnel length>
  • Values during the first squeezed stable beams:
  • Diameter of beam at area of contact: 15 microns (15e-6)
  • Area of contact: pi/4*diameter squared = 1.77e-10
  • Number of protons: 20 billion, or 2e10 (there were more, but at any point, this is how many actually contributed).
  • Number of interactions per second: 11245 (rotations of the beam)
  • Length of a bunch: 30 cm, or 30e-2 m
  • Length of the tunnel: 27 km, or 27e6 m
Filling in the numbers for the first squeezed run, we get: 2e10/(1.77e-10)*11245/(30e-2/27e6) = 1.1e32 or 1.1e-2 inverse microbarns per second which nicely matches the luminostiy recorded.

(This post fixed; I confused femto and nano in earlier versions.)
Last edited by Tau on Tue Apr 27, 2010 6:29 am, edited 6 times in total.
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Re: Luminance, luminosity, squeezing explained in English

Post by mrgumby » Mon Apr 26, 2010 8:02 am

thanks for that Tau......I understood very little but I shall read it several times until it sinks into my bewildered brain. :)

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Re: Luminance, luminosity, squeezing explained in English

Post by Kasuha » Mon Apr 26, 2010 8:37 am

I got lost...

I have no problems with that simple math part all the way down to number of collisions per bunch crossing (and what affects it). I'm just being puzzled by units used in this.

What is the relation between luminosity and number of collisions (per second)? Their units don't seem to match to me...

Is "inverse femtobarn" collisions a different number for LHC and Tevatron? Why don't they just say "we'll collect ten billions of events"...?

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Re: Luminance, luminosity, squeezing explained in English

Post by CharmQuark » Mon Apr 26, 2010 3:30 pm

Tau :D

Thank you so much for posting this :thumbup: don't make me feel all stupid known what it means ;)
Anyone who doesn't take truth seriously in small matters cannot be trusted with large ones either by Albert Einstein.

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Re: Luminance, luminosity, squeezing explained in English

Post by Tau » Mon Apr 26, 2010 4:51 pm

The luminosity unit is not so much "collisions" as well "collision opportunities". If you put enough protons through a narrow opening, they will collide.
The probability of two protons colliding depends on physical properties (size, for example) beyond my grasp.
But I bet that there is a not too hard relationship between luminance and actual collisions: something like "so many per femtobarn".
And why they don't collect a given number of events... I honestly don't know.
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Re: Luminance, luminosity, squeezing explained in English

Post by photino » Mon Apr 26, 2010 6:19 pm

why they don't collect a given number of events...
Ultimately this is of course just a convention, but I think it is because one is usually interested in the number of collision events *of a particular type* (e.g., Beauty production). A calculation on the theory side will traditionally produce the cross-section of some process as its output (with units of area), rather than a probability (with no unit). The barn is also a unit of area. So if you multiply the cross-section of the process you are interested in with the integrated luminosity (in inverse femtobarns) you get a number - the expected number of events.

The use of cross-sections to express collision probabilities is standard for scattering processes. Here is a nice introduction without mathematics - but I really liked the example from this Stanford article:

A femtobarn, then, is 10^-39 square centimeters ­ an incomprehensibly small unit of area.

Imagine you throw enough tomatoes at a (real) barn to get an average of two tomato hits per square foot. If the barn door is 10 feet by 15 feet, then the cross section for tomato-barn door interactions is 150 square feet, and the number of tomatoes that splat on the door is given by:

150 square feet x 2 tomatoes per square foot = 300 tomato interactions.

In this case, what physicists call the integrated luminosity is 2 tomatoes per square foot (or, in physics jargon, 2 "inverse square feet").

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Re: Luminance, luminosity, squeezing explained in English

Post by pixelmasseuse » Mon Apr 26, 2010 7:01 pm

this article http://pdg2.lbl.gov/atlasblog/?p=339 explains cross-section as a probability that a sub-atomic event will occur:
What the LHC has provided so far [article written April 5, 2010] … is on the order of hundred inverse micro-barns. This means that processes charcterized by production cross-section of ~10 nano-barn (nb) start to become accessible
As I understand it, rare events have a smaller cross-section, so more luminance (in inverse cross-section) is needed to make it probable that the event will be observed.

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Re: Luminance, luminosity, squeezing explained in English

Post by Kasuha » Mon Apr 26, 2010 8:19 pm

pixelmasseuse wrote:As I understand it, rare events have a smaller cross-section, so more luminance (in inverse cross-section) is needed to make it probable that the event will be observed.
As far as I know, proton energy alone increases its cross-section. Certain types of collisions are also only possible starting with certain energy. But if the energy is fixed I believe the distribution of different event probabilities is fixed too and making the proton beam denser does not change this distribution, just overall chance for a collision to occur.

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Re: Luminance, luminosity, squeezing explained in English

Post by Tau » Tue Apr 27, 2010 8:47 am

From https://twiki.cern.ch/twiki/bin/view/At ... esults2010 I now know you get (at ATLAS, at least) about 60 000 events per inverse µbarn.
(Or, as physicist would say: the area of an event is 16 picobarn.)
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Re: Luminance, luminosity, squeezing explained in English

Post by chriwi » Tue Apr 27, 2010 8:51 am

leaves only the questin: how many picobarns has the interactionregion of Atlas?
bye

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Re: Luminance, luminosity, squeezing explained in English

Post by Kasuha » Tue Apr 27, 2010 9:41 am

The graph for total event count is almost equal to graph for integrated luminance. I guess it means they are counting luminosity for planned beam cross-section, not actual one - otherwise I'd expect significant disproportion in the graph once squeeze was applied.
If that's so it means number of collisions is simple multiply of luminance.

I can now understand the "inverse femtobarn" goal though - it is goal for certain number of collisions per collision area which gives good chance all "collision angles" have been probed sufficiently.

... plus it gives answer to my earlier question: inverse femtobarn is indeed different number of collisions for LHC than for Tevatron

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Re: Luminance, luminosity, squeezing explained in English

Post by serych » Mon May 17, 2010 7:24 am

Thanks for the nice explanation. But I found one confusing error in your calculation - one km is 1e3 m and not 1e6 m. How it is possible, that it gives the right result? Is there any other error that compensates this one?
Thanks for clarification.
Jakub

Combining all this, luminosity is "effective proton passages per area per second":
<number of protons>/<area of contact>*<rotations per second>/<bunch length per tunnel length>

Values during the first squeezed stable beams:
* Diameter of beam at area of contact: 15 microns (15e-6)
* Area of contact: pi/4*diameter squared = 1.77e-10
* Number of protons: 20 billion, or 2e10 (there were more, but at any point, this is how many actually contributed).
* Number of interactions per second: 11245 (rotations of the beam)
* Length of a bunch: 30 cm, or 30e-2 m
* Length of the tunnel: 27 km, or 27e6 m


Filling in the numbers for the first squeezed run, we get: 2e10/(1.77e-10)*11245/(30e-2/27e6) = 1.1e32 or 1.1e-2 inverse microbarns per second which nicely matches the luminostiy recorded.

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Re: Luminance, luminosity, squeezing explained in English

Post by zaim » Tue May 18, 2010 1:05 am

nice post Tau.
The luminosity can be used to calculate expected rates (R). You just need to know the cross section (sigma) of the process and multiply it with the luminosity (L), ie
R=sigma L.

In this blog post http://pdg2.lbl.gov/atlasblog/?p=339 the author has plotted the cross section for different processes as a function of energy. Looking at 7TeV we see that W-boson production has a cross section of sigma = ~9e1 nb, so with something like 7nb^-1 of data so far, Atlas should have recorded ~630 W-boson events (this is a mean values, the distribution is probably gaussian). Of cause the detector is not 100% effective, so the real number is lower.

Another example: the most recent luminosity of the Tevatron is 330 ub^-1 s^-1 (i.e 5500 times more than the LHC record of 6e-2 ub^-1 s^-1. The Tevatron record is btw. 400 ub^-1 s^-1 only a few weeks ago). At the ~2 TeV of the Tevatron, the cross section of W-production is sigma= ~1e1 nb which means that every second the Tevatron delivers ~3,3 W-events.

If the Higgs particle has a mass of 150 GeV (no one knows it's mass yet), the Tevatron with 8.5 fb^-1 integrated luminosity so far will have produced 1105 Higgs particles since 2002. The LHC would only have produced 0,047 so far, even though the cross section is ~50 times higher a 7 TeV than at 2TeV.

At its peak luminosity of ~ 10 nb^-1 s^-1 and at 14TeV the LHC will produce ~1800 W-bosons and ~0,27 Higgs-bosons every second. Since Higgs particles might be very happy to decay into W-bosons, and W-bosons are very happy to decay into muons, we have the answer why CMS is called the compact muon solenoid.

Cheers Anders Lund

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Re: Luminance, luminosity, squeezing explained in English

Post by Tau » Tue May 18, 2010 6:30 am

Oops...
serych wrote:But I found one confusing error in your calculation - one km is 1e3 m and not 1e6 m. How it is possible, that it gives the right result? Is there any other error that compensates this one?
And I thought I understood it. :?
I'm sorry, but I have to admit that I am stupefied myself. I obviously was to optimistic with my assumptions about understanding this stuff.
If anybody can help me out, please do! :shifty:
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Re: Luminance, luminosity, squeezing explained in English

Post by photino » Tue May 18, 2010 3:10 pm

I am not sure why you included the length of the tunnel and the length of a bunch in your calculation?

Also, the number of protons enters the luminosity quadratically (they must hit each other, not a brick wall!).

There is also a geometric factor due to the beams crossing at an angle, rather than head-on (which changes the effective area of one beam as seen from the other), and a relativistic factor.

To be really precise, you would have to start with a probability distribution for where the protons are in a bunch (Gaussian is easiest of course), then calculate the overlap as two bunches move through each other.

You can find the full formula here:
https://edms.cern.ch/file/445831/5/Vol_1_Chapter_3.pdf

Hope this helps...

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