# Proton Structure: De Broglie Wavelength & Resolution

• Federica
In summary, the conversation discusses the De Broglie wavelength and its relation to resolving the substructure of protons. The speaker mentions that the relevant momentum for this is the momentum transfer between the proton and electron, rather than their total momentums. The speaker also presents their reasoning for calculating the momentum transfer, but notes that it is incorrect because it neglects the recoil motion. The conversation ends with a discussion on the change in momentum of the electron before and after the collision.
Federica
Homework Statement
An accelerator produces an electron beam with energy E=20 GeV. The electrons diffused at θ=6° are detected. Neglecting their recoil motion, what is the minimum structure in the proton that can be resolved?
Relevant Equations
De Broglie wavelength, λ=h/p and particle energy in natural units $$E =\sqrt{m^2 + p^2}$$
Hi all. I'd personally consider the De Broglie wavelength λ=h/p, with p being the momentum of the electron beam. I get $$\lambda \simeq 0.6 \times 10^{-18} m$$ and since the radius of a quark is ## \leq 10^{-19} m ##, the proton structure can't be resolved. I'm quite sure there's something wrong with my reasoning, since I didn't use the information on θ.

The relevant momentum is not the total momentum of the electron, but the momentum transfer between electron and proton.

Quarks are probably elementary without a radius, but that's not what you want to resolve to look inside protons: If you see quarks you see a substructure already.

mfb said:
The relevant momentum is not the total momentum of the electron, but the momentum transfer between electron and proton.

Quarks are probably elementary without a radius, but that's not what you want to resolve to look inside protons: If you see quarks you see a substructure already.

thank you for your reply. But how can I find the momentum transfer between proton and electron if I have to neglect the recoil motion? I assume this means the energy of the proton is always equal to its mass (if c is equal to 1)

Think of the electron momentum before and after the collision.

I know how to do this as long as I can consider the recoil motion. Here's my reasoning: I call ##E## and ##E'## the energies of the incoming and outgoing electron respectively, while ##p## and ##p'## are their momenta. ##E_{r}## is the recoil motion, ##m_e## and ##m_p## are the masses of the electron and proton. I have:

$$(E+m_p)^2 - p^2 = (E' + E_r)^2 - (\overrightarrow{p}' + \overrightarrow{p_r})^2$$

which gives

$$EM = E'E_r -p'p_rcos\theta = E'(E + M - E') - pp'cos\theta - p^2$$

$$E' = \frac{E}{1+\frac{E}{M}(1-cos\theta)}$$

and by ## E -E' ## I can get the momentum transfer between electron and proton. But since I have to neglect the recoil motion, I get:

$$(E+M)^2 - p^2 = (M+E')^2 - p'^2$$

$$E = E'$$

which implies the momentum transfer is equal to zero. What's wrong with my reasoning?

If you assume that the proton doesn't get energy (but can get momentum) then the electron has the same energy before and after. From the change in flight direction you can calculate its change in momentum.

As the proton is heavy it won't get much energy from a change in momentum.

## 1. What is the De Broglie wavelength and how does it relate to proton structure?

The De Broglie wavelength is a concept in quantum mechanics that describes the wavelength of a particle, such as a proton, based on its momentum. It is given by the equation λ = h/mv, where h is Planck's constant, m is the mass of the particle, and v is its velocity. In the context of proton structure, the De Broglie wavelength is used to understand the wave-like behavior of protons and their interactions with other particles.

## 2. How is the De Broglie wavelength of a proton measured?

The De Broglie wavelength of a proton can be measured using a technique called electron microscopy. In this technique, a beam of electrons is directed at a sample containing protons, and the resulting diffraction pattern is analyzed to determine the De Broglie wavelength of the protons in the sample.

## 3. What is the resolution of a proton and how does it relate to its structure?

The resolution of a proton refers to the smallest distance at which two protons can be distinguished from each other. It is directly related to the De Broglie wavelength of the protons, as a smaller De Broglie wavelength corresponds to a higher resolution. This means that the smaller the De Broglie wavelength of a proton, the more detailed its internal structure can be observed.

## 4. Can the De Broglie wavelength of a proton be changed?

Yes, the De Broglie wavelength of a proton can be changed by altering its momentum. This can be achieved by accelerating or decelerating the proton, or by changing its mass. For example, in high-energy particle accelerators, protons are accelerated to very high speeds, resulting in a shorter De Broglie wavelength and allowing for the study of their internal structure.

## 5. What is the significance of understanding proton structure and its De Broglie wavelength?

Understanding proton structure and its De Broglie wavelength is important in many areas of physics, including particle physics, nuclear physics, and materials science. It allows us to better understand the fundamental building blocks of matter and how they interact with each other. It also has practical applications, such as in medical imaging technologies like MRI, which rely on the interactions between protons and magnetic fields.

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