An electron has spin angular momentum and orbital angular...

An electron has spin angular momentum and orbital angular momentum. For the $3 d$ electron in scandium, what percent of its total orbital angular momentum is its spin angular momentum in the $z$ direction?

Key Concepts

Comparison of Angular Momentum Components

To compare the electron’s spin angular momentum in the z direction to its total orbital angular momentum, one considers the measured z?component of spin (±1/2?) against the magnitude of the orbital angular momentum (?6? for a d electron with l = 2). By forming the ratio (1/2)?/(?6?), you obtain a value that, when multiplied by 100, gives approximately 20%—indicating that the spin z?component is about 20% of the total orbital angular momentum magnitude in this context.

Spin Angular Momentum

Spin angular momentum is an intrinsic property of electrons, independent of their orbital motion. For an electron, the spin quantum number is 1/2, meaning the magnitude of the spin angular momentum is ?(1/2(1/2+1))? = ?(3/4)?, but its measurable z?component is quantized to ±1/2?. This fixed value plays an essential role in many atomic phenomena and in comparing the contributions of spin and orbital parts.

Angular Momentum Quantization

In quantum mechanics, both orbital and spin angular momenta are quantized, meaning they can only take on discrete values. The magnitudes and z?components of these angular momenta are described by quantum numbers, and their eigenvalues are determined by formulas like ?[l(l+1)]? for orbital angular momentum and fixed values for spin angular momentum. This quantization is a cornerstone of understanding electron behavior in atoms.

Orbital Angular Momentum

Orbital angular momentum is associated with the motion of an electron in its atomic orbital and is characterized by the orbital quantum number (l). For electrons in a d orbital, l is equal to 2, so the magnitude of the orbital angular momentum is given by ?(2×3)? = ?6?. The orbital angular momentum has components determined by the magnetic quantum number, but its full magnitude is best represented by this square root expression.

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