mathematics

Inglis–Teller equation

Inglis–Teller equation is a mathematics 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 Inglis–Teller equation rather than just read about it. In short: The Inglis–Teller equation represents an approximate relationship between the plasma density and the principal quantum number of the highest bound state of an atom. The equation was derived by David R.

Key takeaways

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

Reference excerpt

The Inglis–Teller equation represents an approximate relationship between the plasma density and the principal quantum number of the highest bound state of an atom. The equation was derived by David R. Inglis and Edward Teller in 1939. In a plasma, atomic levels are broadened and shifted due to the Stark effect, caused by electric microfields formed by the charged plasma particles (ions and electrons). The Stark broadening increases with the principal quantum number n {\displaystyle n} , while the energy separation between the nearby levels n {\displaystyle n} and ( n + 1 ) {\displaystyle (n+1)} decreases. Therefore, above a certain n {\displaystyle n} all levels become merged. Assuming a neutral atomic radiator in a plasma consisting of singly charged ions (and neglecting the electrons), the equation reads

N n 15 / 2 = 0.027 a 0 − 3 , {\displaystyle Nn^{15/2}=0.027a_{0}^{-3}\,,}

where N {\displaystyle N} is the ion particle density and a 0 {\displaystyle a_{0}} is the Bohr radius. The equation readily generalizes to cases of multiply charged plasma ions and/or charged radiator. Allowance for the effect of electrons is also possible, as was discussed already in the original study. Spectroscopically, this phenomenon appears as discrete spectral lines merging into continuous spectrum. Therefore, by using the (appropriately generalized) Inglis–Teller equation it is possible to infer the density of laboratory and astrophysical plasmas.

References

Worked examples

Example 1 — a first encounter with Inglis–Teller equation

Start with the simplest possible case. Write down what Inglis–Teller equation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Inglis–Teller equation 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 Inglis–Teller equation 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 Inglis–Teller equation

In research: Inglis–Teller equation appears in mathematics 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 Inglis–Teller equation 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: Inglis–Teller equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1939 in science, Edward Teller, Spectroscopy, so understanding it makes those chapters shorter.
In everyday life: Look for Inglis–Teller equation 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 Inglis–Teller equation in 20 minutes

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

Frequently asked questions

What is Inglis–Teller equation in simple terms?

Why does Inglis–Teller equation matter?

Because it connects several mathematics 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 Inglis–Teller equation?

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 Inglis–Teller equation.