bei48482_FM

Spin down

E 0 + μpzB

∆E = 2 μpzB

E 0 – μpzB

Spin up

ms = +^12

ms = –^12

E 0

B = 0 B > 0

Figure 11.5The energy levels of a proton in a magnetic field are split into spin-up (Szparallel to B)

and spin-down (Szantiparallel to B) sublevels.

Nuclear Structure 395

A photon with this energy will be emitted when a proton in the upper state flips its

spin to fall to the lower state. A proton in the lower state can be raised to the upper

one by absorbing a photon of this energy. The photon frequency Lthat corresponds

to Eis

L (11.6)

This is equal to the frequency with which a magnetic dipole precesses around a mag-

netic field (Fig. 11.6). It is named for Joseph Larmor, who derived Lfrom classical

physics for an orbiting electron in a magnetic field; his result can be generalized to any

magnetic dipole.

Example 11.3

(a) Find the energy difference between the spin-up and spin-down states of a proton in a mag-

netic field of B1.000 T (which is quite strong). (b) What is the Larmor frequency of a proton

in this field?

Solution

(a) The energy difference is

E 2 pzB(2)(2.793)(3.153 10 ^8 eV/T)(1.000 T)1.761 10 ^7 eV

If an electron rather than a proton were involved, Ewould be considerably greater.

(b) The Larmor frequency of the proton in this field is

L4.258 107 Hz42.58 MHz

From Fig. 2.2 we see that em radiation of this frequency is in the lower end of the microwave

part of the spectrum.

Nuclear Magnetic Resonance

Suppose we put a sample of some substance that contains nuclei with spins of ^12 in a

magnetic field B. The spins of most of these nuclei will become aligned parallel to B

1.761 10 ^7 eV



4.136 10 ^15 eV s

E



h

2 pzB



h

E



h

Larmor frequency

for protons

B

μ

Figure 11.6A nuclear magnetic

moment precesses around an

external magnetic field Bwith a

frequency called the Larmor fre-

quency that is proportional to B.

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