---
title: "Inside the Proton Interactive Lab | Sandbox Physics"
description: "What do a scattered electron’s energy and direction reveal about the proton? The scattered electron’s energy and angle determine how much momentum it transferred and the mass of the remaining hadronic system. Comparing events at the same momentum fraction shows the approximate scaling that helped reveal pointlike constituents inside the proton."
canonical_url: "https://sandboxphysics.com/labs/deep-inelastic-scattering/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/deep-inelastic-scattering/"
last_modified: "2026-10-24"
---

P003 · Looking inside the proton

# Deep Inelastic Scattering

**What do a scattered electron’s energy and direction reveal about the proton?** The scattered electron’s energy and angle determine how much momentum it transferred and the mass of the remaining hadronic system. Comparing events at the same momentum fraction shows the approximate scaling that helped reveal pointlike constituents inside the proton.

Swing the electron spectrometer around a proton target. Reconstruct the momentum transfer and the mass of the recoiling system, then explore a simple valence-quark structure function.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Drag the blue spectrometer. The target illustration is enlarged; the electron angle follows the relativistic event kinematics.

The curve is an illustrative leading-order valence model. The selected electron and the inclusive recoil system obey energy–momentum conservation. These are calculated examples, separate from historical SLAC data.

Model and scientific sources

A massless electron scatters from a stationary proton in the one-photon approximation. The illustrative valence densities integrate to two up quarks and one down quark. Their scale-independent shape gives a leading-order structure function; sea quarks, gluons, QCD evolution, radiative corrections and detector response are omitted. Deep-inelastic teaching cuts require momentum-transfer squared of at least one squared gigaelectronvolt and recoil mass of at least two gigaelectronvolts per light speed squared. The proton picture is schematic; the outgoing recoil is one inclusive system, not a prediction of individual hadron tracks.

$$
Q^2=4 E E'\sin^2\!\frac{\theta}{2},\quad x=\frac{Q^2}{2M(E-E')},\quad W^2=M^2+2M(E-E')-Q^2
$$

- [PDG 2024 · Structure functions](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-structure-functions.pdf)
- [Breidenbach et al. · Observed behavior (1969)](https://doi.org/10.1103/PhysRevLett.23.935)

Physics tutorial

## Reconstruct a collision from the electron

**Background** High-energy electrons can transfer enough momentum to a proton to break it into a heavier outgoing system. The electron itself is easy to follow into a spectrometer.

**Why it matters** The outgoing electron supplies both an energy and a direction. Conservation laws then reveal properties of the unobserved hadronic system.

Start with the essentials

- **Focus question**: Which features of a deep-inelastic event can be inferred from the escaping electron?
- **One-sentence intuition**: The scattered electron determines the momentum transfer and inclusive recoil mass.

### Core mathematical model

#### The electron fixes the transferred momentum

$$
Q^2=4EE'\sin^2\!\frac{\theta}{2},\qquad \nu=E-E'
$$

The incident and outgoing electron energies and their angle give the spacelike momentum transfer in the negligible-electron-mass approximation.

#### Infer the recoil system

$$
x=\frac{Q^2}{2M\nu},\qquad W^2=M^2+2M\nu-Q^2
$$

M is the target proton mass. W is the mass of the whole outgoing hadronic system. In the leading-order parton picture, x is the struck parton momentum fraction.

### Common difficulties

#### A structure function is an inclusive response

**Typical misconception** The curve is a photograph of where the three quarks sit.

**Better mental model** The structure function describes how often scattering occurs at different momentum fractions. It is not a map of positions inside one proton.

#### The deep-inelastic region

**Typical misconception** Every electron–proton collision resolves individual partons.

**Better mental model** Low momentum transfer or a small recoil mass needs a different physical description. The page marks events outside its teaching cuts.

### Run the experiment

1. 01

   #### Move the spectrometer

   Drag the blue detector along its arc.

   **What to observe:** The scattering angle, transferred momentum and inferred recoil mass change together.

2. 02

   #### Read the structure function

   Change the Bjorken momentum fraction.

   **What to observe:** The selected point moves along a varying curve, while the event geometry follows the same kinematics.

3. 03

   #### Leave the deep region

   Choose Leave the deep-inelastic region.

   **What to observe:** The page marks the low-transfer event and keeps the kinematic reconstruction visible.

Physical conclusion

An electron energy and angle reconstruct an inclusive collision; interpreting its rate requires a model of proton structure.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)