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Thermal ellipsoid

Thermal ellipsoid is a engineering topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Thermal ellipsoid rather than just read about it. In short: In crystallography, thermal ellipsoids, more formally termed atomic displacement parameters or anisotropic displacement parameters, are ellipsoids used to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structures. Since the vibrations are usually anisotropic (different magnitudes in different directions in space), an ellipsoid is a convenient way of visualising the vibration and…

Thermal ellipsoid — main illustration
Thermal ellipsoid — illustration

Key takeaways

  • Thermal ellipsoid belongs to engineering; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Thermal ellipsoid to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Thermal ellipsoid from memory before moving on to harder problems.

Reference excerpt

In crystallography, thermal ellipsoids, more formally termed atomic displacement parameters or anisotropic displacement parameters, are ellipsoids used to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structures. Since the vibrations are usually anisotropic (different magnitudes in different directions in space), an ellipsoid is a convenient way of visualising the vibration and therefore the symmetry and time averaged position of an atom in a crystal. Their theoretical framework was introduced by D. W. J. Cruickshank in 1956 and the concept was popularized through the program ORTEP (Oak Ridge Thermal-Ellipsoid Plot Program), first released in 1965. Thermal ellipsoids can be defined by a tensor, a mathematical object which allows the definition of magnitude and orientation of vibration with respect to three mutually perpendicular axes. The three principal axes of the thermal vibration of an atom are denoted U 1 {\displaystyle U_{1}} , U 2 {\displaystyle U_{2}} , and U 3 {\displaystyle U_{3}} , and the corresponding thermal ellipsoid is based on these axes. The size of the ellipsoid is scaled so that it occupies the space in which there is a particular probability of finding the electron density of the atom. The particular probability is usually 50%.

See also Debye–Waller factor

References

Illustrations

Thermal ellipsoid: A thermal ellipsoid model of one stable conformation of the organic molecule, diphenyl ether, formulae .mw-parser-output .template-chem2-su{display:inline-block;font-size:80%;line-height:1;vertical-align:-0.35em}.mw-parser-output .template-chem2-su>span{display:block;text-align:left}.mw-parser-output sub.template-chem2-sub{font-size:80%;vertical-align:-0.35em}.mw-parser-output sup.template-chem2-sup{font-size:80%;vertical-align:0.65em}C12H10O or (C6H5)2O, abbreviated Ph2O. Carbons (C) are shown in black, hydrogens (H) in grey-white, and the oxygen (O) in red. The thermal ellipsoids are set at a 50% probability level, and the positions of atoms and the anisotropies of position reflected in the ellipsoids derive from a crystal structure of the molecule.[1]
A thermal ellipsoid model of one stable conformation of the organic molecule, diphenyl ether, formulae .mw-parser-output .template-chem2-su{display:inline-block;font-size:80%;line-height:1;vertical-align:-0.35em}.mw-parser-output .template-chem2-su>span{display:block;text-align:left}.mw-parser-output sub.template-chem2-sub{font-size:80%;vertical-align:-0.35em}.mw-parser-output sup.template-chem2-sup{font-size:80%;vertical-align:0.65em}C12H10O or (C6H5)2O, abbreviated Ph2O. Carbons (C) are shown in black, hydrogens (H) in grey-white, and the oxygen (O) in red. The thermal ellipsoids are set at a 50% probability level, and the positions of atoms and the anisotropies of position reflected in the ellipsoids derive from a crystal structure of the molecule.[1]
Thermal ellipsoid: A thermal ellipsoid model[2] of the coordination environment of the chlorine atom in ClO+2 (the chloryl cation), in crystalline chloryl hexafluoroantimonate, formula [ClO2][SbF6]. The chlorine atom (Cl) is in the +5 oxidation state, and is at the center in bright green; the two oxygens (O) are in red, and four fluoride anions from a hexafluoroantimonate (SbF−6) anion that coordinate to the electropositive chlorine atom are shown in yellowish-green at the periphery, at right (with light lines indicating the coordinating F-Cl interactions. This reactive compound is prepared by treatment of FClO2 with the perfluoro-Lewis acid, SbF5.[3]
A thermal ellipsoid model[2] of the coordination environment of the chlorine atom in ClO+2 (the chloryl cation), in crystalline chloryl hexafluoroantimonate, formula [ClO2][SbF6]. The chlorine atom (Cl) is in the +5 oxidation state, and is at the center in bright green; the two oxygens (O) are in red, and four fluoride anions from a hexafluoroantimonate (SbF−6) anion that coordinate to the electropositive chlorine atom are shown in yellowish-green at the periphery, at right (with light lines indicating the coordinating F-Cl interactions. This reactive compound is prepared by treatment of FClO2 with the perfluoro-Lewis acid, SbF5.[3]

Worked examples

Example 1 — a first encounter with Thermal ellipsoid

Start with the simplest possible case. Write down what Thermal ellipsoid claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Thermal ellipsoid before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Thermal ellipsoid ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Thermal ellipsoid

In research
Thermal ellipsoid appears in engineering research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Thermal ellipsoid in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Thermal ellipsoid is common in secondary-school and first-year university syllabi. It links to neighbouring topics Crystallography, Ellipsoids, so understanding it makes those chapters shorter.
In everyday life
Look for Thermal ellipsoid outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Thermal ellipsoid in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Thermal ellipsoid means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Thermal ellipsoid out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Thermal ellipsoid in simple terms?

In crystallography, thermal ellipsoids, more formally termed atomic displacement parameters or anisotropic displacement parameters, are ellipsoids used to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structures. Since the vibrations are usually anisotropic (di…

Why does Thermal ellipsoid matter?

Because it connects several engineering ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Thermal ellipsoid?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Thermal ellipsoid.

Tags

  • Crystallography
  • Ellipsoids

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