# Soup-to-Nuts: How wide is my proton?

Here is one way to use data from the TOTEM Experiment in the LHC to study wave properties.

The goal and why:

Students measure an upper limit to the diameter of the proton using proton elastic scattering data from the TOTEM Experiment. Because protons are quantum objects, they "interfere" with each other rather than simply bounce off as if there were marbles. We can use the interference pattern to determine an effective width, which we can think of as the closest approach the protons can make with each other and still collide elastically. Students use the same wave interference they see in optics!

What you'll need:

Set-up:

Create a histogram space on the wall or the black/whiteboard marked in 10's from -240 to +240 microrad.

There are 80 events, 2 per page, in the data file. Divide these among the students and then divide them into small groups.

What happens:

Each student should find the scattering angle of both protons (shown as red and green dots) in each "Angular Topology" display they are given.

To find the scattering angle, follow the blue double-circle up or down from the dot the shortest way to the x-axis of the plot. Read the scattering angle in microradians - these are very small angles! Each display should show a positive and a negative scattering angle, usually very close in magnitude to each other. Record both in the tally sheet, rounding to the nearest multiple of 10 and making a mark next to that value.

When students are done measuring, they  should put sticky notes the histogram using the same number of stickies as marks on their tally sheet for each angle.

When all of the data is in the histogram, what you get should look like an interference pattern with the middle cut out. The reason is that, in the LHC, this is where the proton beam is and the TOTEM detector cannot reach there without interference with the operation of the accelerator. Student should use the spacing of the remaining maxima they do see estimate the order (n=0,1,2,...) of one of them. Using that angle, they can use a simple interference realtion to estimate the width of the proton interaction area: d = nλ/sinθ.

To get wavelength λ, you can either:

• Use E = hc/λ with E= 4 TeV = 4 x 1012 eV (1 eV = 1.609 x 10-19 J), c = 3 x 108 m/s, and h = 6.63 x 10-34 Js to solve for the deBroglie wavelength of the proton, or
• Give the students the wavelength of 3.1 x 10-13 μm to use.

Also, you should note that sinθ = θ to an extremely good approximation for the very small angles used here. Then if you have figured out n and θ, you can simply use d = nλ/θ.

Expect the result to be greater than the accepted proton diameter.