ATM-12

Spin–orbit interval

E_{SO} = (A/2) [J(J+1) − L(L+1) − S(S+1)]. Landé interval rule.

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StructureSpin–orbit

Governing equation

E=\dfrac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr]

$J \in \left[0, 5\right]$

where

A: Interval A (cm^{-1})
L: L (—)
S: S (—)
J: J (—)
E_{SO}: Fine-structure energy (cm^{-1})

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

E=\dfrac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr]

$J \in \left[0, 5\right]$

$J = 0$ $E_{SO} = -27.5$

Along the curve

②

Domain of the variable — blue

J_{\min} = 0,\quad J_{\max} = 5

$J \in \left[0, 5\right]$

$J = 0$ $J = 0$

Along the curve

③

Response curve — green

E=\dfrac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr]

$J \in \left[0, 5\right]$

$J = 0$ $E_{SO} = -27.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E_{SO}}{\mathrm{d}J}

$J \in \left[0, 5\right]$

$J = 0.031$ $E_{SO}^{\prime} = 10.63$

Along the curve

Lecture brief

Historical brief

Balmer, Rydberg, Bohr (1913), fine structure, Zeeman and Stern–Gerlach built the spectrum of one atom. The lab computes levels, selection and magnetic splitting. This sheet (ATM-12 — Spin–orbit interval) is the form associated with Spin–orbit. Working symbols: $A$ , $L$ , $S$ , $J$ $\rightarrow$ $E_{SO}$ . A ∝ Z⁴ α² Rydberg. Fine-structure splitting of alkali D lines is this plus core penetration.

Purpose

Purpose: compute

E_{SO}

from

A

L

S

J

in Atomic physics via

E=\dfrac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr]

E_{SO} = (A/2) [J(J+1) − L(L+1) − S(S+1)]. Landé interval rule. Use it when a real atomic physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

A = 20.000\,\mathrm{cm^-1}

L = 1.000\,\mathrm{—}

S = 0.500\,\mathrm{—}

J = 1.500\,\mathrm{—}

, the governing relation

E=\dfrac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr]

yields

E_{SO} = 10.000\,\mathrm{cm^-1}

. A term that splits into J levels. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Interval A A20 cm^{-1}L L1 —S S0.5 —J J1.5 —

Outputs

Fine-structure energy E_{SO}10.000 cm^{-1}

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Narration of this film

A term that splits into J levels.

A ∝ Z⁴ α² Rydberg. Fine-structure splitting of alkali D lines is this plus core penetration.

Reading speed

Watch on YouTube