NUCLEAR MAGNETIC RESONANCE 97 



mind, the one which agrees with the frequency of the Larmor Precession. 

 Think now of any two adjacent protons. The distance r between them 

 will fluctuate in a complicated w^ay as time goes on, but in this compli- 

 cated motion we distinguish, again in mind, the component which has 

 the frequency of the Larmor Precession. Well, according to earlier 

 theory it is this component of the vibrations — be they interpreted as 

 vibrations of the whole lattice or of two neighboring protons relative to 

 each other — which helps the protons to turn from down to up, or for 

 that matter from up to down. This is the channel by which energy 

 passes to and fro between the lattice and the spins. 



On submitting this idea to calculation it was found to give values of 

 Ti that are far too long. The next recourse was to take into account the 

 ''beat tones." Choose any frequency whatever in the elastic spectrum, 

 and then another frequency differing from the first one by the frequency 

 of the Larmor Precession. The sum of the two will present a beat fre- 

 quency equal to that of the precession. This is a mathematical state- 

 ment which seems as empty of physical meaning as — well, as seemed 

 in its turn the assertion that the alternating magnetic vector Hi directed 

 along the x-axis is the sum of two circularly-polarized vectors. But the 

 force between two neighboring protons is not linear (it is proportional 

 to the inverse third-power of the distance r), and this gives physical 

 meaning to the statement: the two frequencies conjointly act as if the 

 beat-frequency were present. When these channels of communication 

 between the spins and the lattice are added to the one first thought of, 

 the calculated relaxation times come down into the order of magnitude 

 of the real ones. More in the way of precise agreement can scarcely be 

 hoped for, because of the effect of impurities on Ti . 



Two other topics in the field of spin-lattice relaxation must at least 

 be mentioned. 



Some of the known values of Ti are too low to be measured by observ- 

 ing the rise and fall of the resonance-peak. To indicate how these are 

 measured, I recall that the energy of a wave-train is proportional to 

 the square of its amplitude. Hi in the present case. To speak of protons 

 for simplicity: if Ti were zero the number of protons in the "up" orienta- 

 tion would always be the same, and hence the height A of the peak 

 would be proportional to Hi . But since Ti is not zero the number of 

 protons capable of absorbing goes down as Hi goes up; and the curve 

 of A against Hi starts off tangent to the ideal straight line for Ti = 0, 

 but is concave-downward, drops away from the straight line and eventu- 

 ally will cease to rise. 



We may pursue the argument one step farther. Here is the equation 



