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8.3 Electron spin

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By the end of this section, you will be able to:
  • Express the state of an electron in a hydrogen atom in terms of five quantum numbers
  • Use quantum numbers to calculate the magnitude and direction of the spin and magnetic moment of an electron
  • Explain the fine and hyperfine structure of the hydrogen spectrum in terms of magnetic interactions inside the hydrogen atom

In this section, we consider the effects of electron spin. Spin introduces two additional quantum numbers to our model of the hydrogen atom. Both were discovered by looking at the fine structure of atomic spectra. Spin is a fundamental characteristic of all particles, not just electrons, and is analogous to the intrinsic spin of extended bodies about their own axes, such as the daily rotation of Earth.

Spin is quantized in the same manner as orbital angular momentum. It has been found that the magnitude of the intrinsic spin angular momentum S of an electron is given by

S = s ( s + 1 ) ℏ ,

where s is defined to be the spin quantum number . This is similar to the quantization of L given in [link] , except that the only value allowed for s for an electron is s = 1 / 2 . The electron is said to be a “spin-half particle.” The spin projection quantum number m s is associated with the z -components of spin, expressed by

S z = m s ℏ .

In general, the allowed quantum numbers are

m s = − s , − s + 1 , … , 0 , … , + s − 1 , s .

For the special case of an electron ( s = 1 / 2 ),

m s = − 1 2 , 1 2 .

Directions of intrinsic spin are quantized, just as they were for orbital angular momentum. The m s = −1 / 2 state is called the “spin-down” state and has a z -component of spin, s z = −1 / 2 ; the m s = + 1 / 2 state is called the “spin-up” state and has a z -component of spin, s z = + 1 / 2 . These states are shown in [link] .

The two possible spin states of the electron are illustrated as vectors of equal length, one pointing up and right, representing vector S spin up, and the other pointing down and right, representing spin down. The two vectors are at the same angle to the horizontal. Spin up has a z component of plus h bar over two, and spin down has a z component of minus h bar over 2.
The two possible states of electron spin.

The intrinsic magnetic dipole moment of an electron μ e can also be expressed in terms of the spin quantum number. In analogy to the orbital angular momentum, the magnitude of the electron magnetic moment is

μ s = ( e 2 m e ) S .

According to the special theory of relativity, this value is low by a factor of 2. Thus, in vector form, the spin magnetic moment is

μ → = ( e m e ) S → .

The z -component of the magnetic moment is

μ z = − ( e m e ) S z = − ( e m e ) m s ℏ .

The spin projection quantum number has just two values ( m s = ± 1 / 2 ) , so the z- component of the magnetic moment also has just two values:

μ z = ± ( e 2 m e ) = ± μ B ℏ ,

where μ B is one Bohr magneton. An electron is magnetic, so we expect the electron to interact with other magnetic fields. We consider two special cases: the interaction of a free electron with an external (nonuniform) magnetic field, and an electron in a hydrogen atom with a magnetic field produced by the orbital angular momentum of the electron.

Electron spin and radiation

A hydrogen atom in the ground state is placed in an external uniform magnetic field ( B = 1.5 T ). Determine the frequency of radiation produced in a transition between the spin-up and spin-down states of the electron.

Strategy

The spin projection quantum number is m s = ± 1 / 2 , so the z- component of the magnetic moment is

μ z = ± ( e 2 m e ) = ± μ B ℏ .

The potential energy associated with the interaction between the electron magnetic moment and the external magnetic field is

U = − μ z B = ∓ μ B B .

The frequency of light emitted is proportional to the energy ( Δ E ) difference between these two states.

Solution

The energy difference between these states is Δ E = 2 μ B B , so the frequency of radiation produced is

f = Δ E h = 2 μ B B h = 2 ( 5.79 × 10 −5 eV T ) ( 1.5 T ) 4.136 × 10 −15 eV · s = 4.2 × 10 10 cycles s .

Significance

The electron magnetic moment couples with the external magnetic field. The energy of this system is different whether the electron is aligned or not with the proton. The frequency of radiation produced by a transition between these states is proportional to the energy difference. If we double the strength of the magnetic field, holding all other things constant, the frequency of the radiation doubles and its wavelength is cut in half.

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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