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Section 5.6 Spin Magnetic Moment in a Magnetic Field

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We are now ready to apply the major ideas of this chapter to some interesting examples which have important applications to medical physics, chemistry, and astronomy.

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Both protons and electrons have intrinsic spin, with spin quantum number s=1/2 (both are fermions). The proton also has a charge q=+e. Classically speaking, the proton is viewed as a tiny spinning charged sphere that has current loops associated with the circulating charge. In the previous unit on electricity and magnetism, we learned that a magnetic moment μ→ can be associated with these circulating currents. When these concepts are applied to the proton as a classical spinning charged sphere, we find that the magnetic moment of the proton is proportional to the proton's intrinsic angular momentum S→. In fact, the magnetic moment for the proton is found to be

(5.26)ΞΌβ†’=2ΞΌpℏSβ†’

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where ΞΌp=1.41Γ—10βˆ’26 \mathrm{J/T}. Since measuring any component of Sβ†’ results in a value of ±ℏ2, equation (5.26) tells us that, for a proton, ΞΌβ†’ has a magnitude of ΞΌp and points in the direction of Sβ†’.

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We learned in the unit on electricity and magnetism that if a magnetic moment μ→ is placed in a constant magnetic field B→, there is a magnetic potential energy associated with the orientation of the magnetic moment relative to the magnetic field given as

(5.27)U=βˆ’ΞΌβ†’β‹…Bβ†’.

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This energy is greatest when μ→ and B→ are pointing in opposite directions and least when pointing in the same direction. If we take B→ to be pointing in the z-direction such that B→=B0z^, then Eqs. (5.26) and (5.27) can be written as

(5.28)U=βˆ’ΞΌβ†’β‹…Bβ†’=βˆ’2ΞΌpℏB0Sz=βˆ’2ΞΌpB0ms

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where we have used equation (5.12) for the z-component of the proton's spin. Since ms=Β±1/2 for a spin-1/2 particle, each of the orientations of the proton (spin-up or spin-down) corresponds to a different energy of the proton in the magnetic field

U+=βˆ’ΞΌpB0(ms=+12, spin-up)Uβˆ’=+ΞΌpB0(ms=βˆ’12, spin-down).

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Consequently, the proton has a different energy depending on whether it is in the spin-up (|+z⟩) state or the spin-down (|βˆ’z⟩) state. Therefore the states |+z⟩ and |βˆ’z⟩, in addition to being states of definite z-component of spin Sz, are also states of definite energy U+ and Uβˆ’, respectively. Thus when placed in a magnetic field, the state of the proton splits into two possible energy states, as shown in Figure 5.11 separated in energy by

(5.29)Ξ”E proton β‰‘|U+βˆ’Uβˆ’|=2ΞΌpB0.
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Figure 5.11. Splitting of energy levels of a proton in a magnetic field

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Figure 5.11 suggests that for a spin-up proton |+z⟩ placed in a magnetic field Bβ†’=B0z^, if the proton then absorbs a photon of energy E photon =Ξ”E proton =2ΞΌpB0, then the proton will make a transition to the spin-down state |βˆ’z⟩, which has higher energy. The reversal of spin component Sz is called spin-flipping.

Consider a proton that is placed in a magnetic field such that it is in the spin-up state |+z⟩. If the magnitude of the magnetic field is B0=24.8mT, determine the frequency of the photon that would be absorbed by the proton causing it to make a transition to the spin-down state?

Solution.

For the magnetic field strength given, the energy difference between the spin-up and spin-down states is

Ξ”E proton =(2ΞΌpB0=(2(1.41Γ—10βˆ’26J/T)(24.8Γ—10βˆ’3T)β‰ˆ6.99Γ—10βˆ’28J.

Therefore the frequency associated with a photon of this energy is

(5.30)f photon =E photon h=Ξ”E proton h=6.99Γ—10βˆ’28J6.63Γ—10βˆ’34Jβ‹…s=1.05Γ—106Hz=1.05MHz.

This frequency lies in the region of the electromagnetic spectrum associated with radio waves.

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Like the proton, an electron also has intrinsic spin with spin quantum number s=1/2. In contrast to the proton, the negative charge of an electron means its magnetic moment ΞΌβ†’ points opposite the direction of Sβ†’. The electron has ΞΌe=βˆ’9.28Γ—10βˆ’24J/T, where the negative sign denotes the opposing directions of ΞΌβ†’ and Sβ†’. Replacing ΞΌp with ΞΌe in equation (5.28), we find that the higher energy level for an electron occurs when the electron is spin-up and the lower energy level occurs when the electron is spin-down (for the case where Bβ†’ is pointing in the positive z-direction).

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In a hydrogen atom, the nucleus (a proton) and the electron both have spin magnetic moments. Thus, the energy levels of the hydrogen atom are slightly split by the interaction between the electron spin and the nuclear spin, a phenomenon known as hyperfine splitting. When the electron's spin flips, the atom transitions between two hyperfine energy levels and a photon can be emitted or absorbed in the process. Because so much of the gas composing galaxies is neutral hydrogen, this spin-flip transition allowed astronomers to map the spiral structure of the Milky Way for the first time. In your homework, you'll take a closer look at hyperfine splitting.