---
title: "Nematic Defect Microscope Interactive Lab | Sandbox Physics"
description: "How can a direction turn through half a circle around a defect? A nematic direction has no arrowhead: turning a rod halfway around returns the same orientation."
canonical_url: "https://sandboxphysics.com/labs/nematic-defect-microscope/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/nematic-defect-microscope/"
last_modified: "2026-10-28"
---

FR22 · Read a field of directions

# Nematic Defect Microscope

**How can a direction turn through half a circle around a defect?** A nematic direction has no arrowhead: turning a rod halfway around returns the same orientation. Around a half defect, the direction changes by half a turn and joins smoothly to itself. Counting the signed turn distinguishes the two defect shapes.

Follow the rods around a bright core, then bring two opposite defects together. The microscope locates each defect from the sampled directions; a coarse grid can miss a close pair.

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

Model assumptions

The starting texture is an analytic, periodic alignment field with two positive and two negative half defects. This microscope samples a prescribed texture, with optional angular noise. It does not evolve a fluid or infer defects from an experimental image.

Positive half defect Negative half defect

Rod hue shows headless direction; marker locations are accurate to one sampling cell.

Positive half defects $2$

Negative half defects $2$

Total charge $0$

Mean nematic order $0.992$

### The direction turns as you circle a core

Amber and violet curves follow one positive and one negative detected core. Horizontal position is progress around a closed loop; vertical position is accumulated doubled-angle winding, in full turns.

Equations, approximations and sources

The analytic complex texture has four charge-balanced zeros on a torus. The detector wraps neighboring doubled-angle differences around each cell; close pairs and noise can change what the sampled field resolves.

The coarse browser model isolates coupling and sampling. It omits passive elastic stress and material calibration; its trajectories are numerical examples. Export includes raw tensor samples, velocity samples, seeds, parameters and tracked charges.

[Giomi (2015), active nematodynamics, equations 1–2 and Figure 1](https://arxiv.org/html/1409.1555v2)

[Continue through the active-matter course](https://sandboxphysics.com/stories/active-matter/)

[Read the active matter course: from persistent motion to defects and flow](https://sandboxphysics.com/stories/active-matter/)

Physics tutorial

## Nematic Defect Microscope

**Background** Follow the rods around a bright core, then bring two opposite defects together. The microscope locates each defect from the sampled directions; a coarse grid can miss a close pair.

**Why it matters** A field of headless rods can contain a winding that no smooth local rotation can erase.

Start with the essentials

- **Focus question**: How can a direction turn through half a circle around a defect?
- **One-sentence intuition**: Count the change of the doubled direction around a closed loop, then divide its winding by two.

### Core mathematical model

#### A direction without an arrowhead

$$
Q_{xx}=\frac{S}{2}\cos(2\theta),\qquad Q_{xy}=\frac{S}{2}\sin(2\theta)
$$

The tensor has the same value after a half turn. Its magnitude also falls near a defect core.

#### Charge from sampled directions

$$
s=\frac{1}{4\pi}\sum_{\square}\operatorname{wrap}\!\left(\Delta\arg(Q_{xx}+iQ_{xy})\right)
$$

Each phase difference is wrapped into the interval from minus pi to pi. A phase winding of one turn gives a half defect.

### Common difficulties

#### Two meanings of motion

**Typical misconception** A bright marker is a particle with a prescribed speed.

**Better mental model** Markers are found by winding the sampled tensor field around each cell; dragging prepares a new texture.

#### Model scale

**Typical misconception** The grid is a calibrated sample of a biological material.

**Better mental model** The starting texture is an analytic, periodic alignment field with two positive and two negative half defects. This microscope samples a prescribed texture, with optional angular noise. It does not evolve a fluid or infer defects from an experimental image.

### Run the experiment

1. 01

   #### Follow one core

   Choose Four half-turns. Trace the rods around an amber or violet marker.

   **What to observe:** The two colors have opposite signed winding. The four detected charges add to zero on the periodic square.

2. 02

   #### Bring the pair closer

   Decrease Defect separation parameter, then choose A coarse camera.

   **What to observe:** A pair closer than a sampling cell can cancel within that cell and escape detection. Resolve it with a finer grid.

3. 03

   #### Save the measured field

   Drag a highlighted core, then export the sampled record.

   **What to observe:** The export contains the sampled tensor values, detected charges and tracking history.

Physical conclusion

The starting texture is an analytic, periodic alignment field with two positive and two negative half defects. This microscope samples a prescribed texture, with optional angular noise. It does not evolve a fluid or infer defects from an experimental image.

## Related experiments and articles

Active matter · Reading path

### Active matter from swimming particles to moving defects

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a liquid-crystal defect can turn only halfway**](https://sandboxphysics.com/stories/active-matter/headless-order-and-defects/)
2. [Lab **LCD Pixel Microscope** Compare passive liquid-crystal optics](https://sandboxphysics.com/labs/lcd-pixel-microscope/)
3. [Article **Guide defects by choosing where rods prefer to point**](https://sandboxphysics.com/stories/active-matter/anchoring-and-confinement/)
4. [Lab **Active Nematic Flow** Follow defects and fluid together](https://sandboxphysics.com/labs/active-nematic-flow/)
5. [Article **Read active turbulence through a spectrum and a material path**](https://sandboxphysics.com/stories/active-matter/reading-active-turbulence/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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