Calculator inputs
Use the responsive three-column form on large screens, two columns on smaller screens, and one column on mobile.
Plotly graph
The chart below plots the potential curve across the selected range. When results exist, it also shows force on a secondary axis and highlights the chosen separation.
Example data table
This illustrative table uses ε = 0.997 kJ/mol and σ = 3.400 Å to show how the potential and force change with separation.
| Separation r (Å) | Potential U(r) (kJ/mol) | Force F(r) (kJ/mol/Å) | Behavior |
|---|---|---|---|
| 3.000 | 9.457136 | 54.730274 | Repulsive |
| 3.400 | 0.000000 | 7.037647 | Repulsive |
| 3.816 | -0.997000 | 0.001830 | Repulsive |
| 4.200 | -0.806490 | -0.700884 | Attractive |
| 5.100 | -0.319376 | -0.339575 | Attractive |
Formula used
Potential energy equation
U(r) = 4ε[(σ/r)12 − (σ/r)6]
Force equation
F(r) = (24ε/r)[2(σ/r)12 − (σ/r)6]
Important reference distances
Equilibrium distance: req = 21/6σ
Zero-potential distance: r = σ
- ε sets the depth of the attractive well.
- σ sets the length scale where the potential becomes zero.
- r is the particle separation used for each calculation.
- Positive force indicates repulsion, while negative force indicates attraction in this radial convention.
- The force unit is displayed as energy unit divided by length unit.
How to use this calculator
- Enter the interaction strength as epsilon ε.
- Enter sigma σ, the zero-potential distance.
- Provide the specific separation distance r for the main result.
- Set the graph start, graph end, and number of curve points.
- Add unit labels if you want polished on-screen and exported output.
- Press the calculate button to display the result above the form.
- Review the summary cards, result table, and Plotly graph.
- Use the CSV or PDF buttons to download the current report.
Frequently asked questions
1) What does the Lennard-Jones potential describe?
It models the balance between short-range repulsion and longer-range attraction between neutral particles. The equation is widely used in molecular physics, chemistry, and simulation work.
2) What is epsilon ε in this calculator?
Epsilon is the well depth. It tells you how strong the attractive interaction is at the minimum of the potential energy curve.
3) What is sigma σ?
Sigma is the distance where the potential energy becomes zero. It is a key size parameter in the Lennard-Jones model.
4) Why is the force sometimes positive and sometimes negative?
Positive force corresponds to repulsion in this radial form. Negative force corresponds to attraction. The sign changes around the equilibrium region.
5) What happens at the equilibrium distance?
The potential reaches its minimum value and the force becomes zero. That distance equals 21/6σ.
6) Which units should I use?
Use any consistent set of units. Common choices are Å or nm for length and kJ/mol or eV for energy. The force unit becomes energy divided by length.
7) Why does the curve rise sharply at small distances?
The repulsive term depends on (σ/r)12, so it grows very rapidly when particles are pushed too close together. That produces a steep upward wall.
8) What do the download buttons include?
The CSV export includes summary values and the full curve data. The PDF export creates a compact report with the main inputs and outputs.