Nuclear magnetic resonance is possible due to the intrinsic magnetism of the nucleus; all magnetically active nuclei that have a spin quantum number I >0 posses angular momentum, and as a consequence of this they also have a magnetic moment µ. By analogy, it can be convenient to think of the nuclear spin as a magnetic dipole, and if we consider a case of where we have just one such dipole then the magnetic moment of this dipole is proportional to the angular momentum
where ~ is Planck’s constant h/2π. The constant of proportionality γ is called the gyromagnetic ratio (units rad−1T−1) and each nucleus has a uniquely defined γ. The
fact that each nucleus has a unique γ means that (in principle) every magnetically active isotope can easily be distinguished from all others. The highest value of γ for the stable nuclei is that of the proton (1H) and corresponds to a precession frequency ν0 =γ/2π = 42.6 MHz per Tesla of applied magnetic field (B0).
If we now consider an idealised sample consisting of a large ensemble of identical spinI = 12 nuclei or magnetic dipoles, there will exist a net magnetic moment equivalent to a bulk magnetisation M. This is just the sum of the magnetic moment vectors per unit volume
M=X i
µi (2.2)
However we find in the absence of an external magnetic field the individual magnetic moments will point in all directions in space, in effect cancelling each other out (see Figure 2.1a) and there will be no bulk magnetisation (i.e. M = 0). If this idealized ensemble is then placed in a static uniform magnetic field B0, then there is a slight
preference for the magnetic moments to align with this magnetic field creating a bulk magnetisation (see Figure 2.1b). The proportion of nuclei that align with the magnetic
Figure 2.1: (a) In the absence of an external magnetic field the individual magnetic moments will point in all directions in space so that the average bulk magnetism will be zero (b) If placed in an external static magnetic field there will be a slight preference for the magnetic moments to align with the field, with those of a lower energy being parallel and those in the higher state being anti-parallel. As there are slightly more in the lower state then there will be a bulk magnet moment in the direction of B
field is generally very small and is given by
M= N γ
2I(I+ 1)
~2
3kT B0 (2.3)
which is considered to be a Curie-like relationship, where N is the number nuclei, k
is the Boltzmann constant and T is the temperature. At thermal equilibrium there will be slightly higher portion of the nuclei in the lower energy state caused by the Zeeman splitting of the ground state (see Section 2.4 ), those in the lower state align with the field and those in the higher energy state will be anti-aligned thus determining the magnetic moment. It is also apparent that the more spins in the ensemble that occupy the lower energy state will result in a larger magnetic moment. It follows from this that magnetic momentMwill be related to the total angular momentum Lof the system by
M=γL (2.4)
The net magnetic moment will not actually align with magnetic field as it has angular momentum and the external field exerts a torque on the magnetic moment
T =M×B0 (2.5)
and
T= d
dtL (2.6)
Then combining Equations 2.4 to 2.6 we get an expression describing the motion of M in the static field B0
dM
dt =γM×B0 (2.7)
If the magnetic moment is inclined to the fieldB0 at an angle θ then the net magnetic
moment will precess in a cone (see Figure 2.2b) around the direction of the magnetic field at a constant angular frequency ω0, where
Figure 2.2: (a) In a static magnetic field nuclei with a spinI >0 will precess around the field with frequencyω0 (b) When many nuclei are present they contribute to a bulk
magnetism M which also precesses around B0 describing a cone with frequency ω0
.
B0 is the direction of the magnetic field and is usually defined to be in thez direction
in a Cartesian coordinate system,ω0 is called the Larmor frequency (however it is more
often presented as ν0 =ω0/2π). The negative sign beforeγ is related to the direction
of the precession, as γ can take positive or negative values. When looking along the direction of B0 then nuclei with a positive γ precess in an anti-clockwise direction and
those with a negative γ precess clockwise however the consequences of this are usually ignored.
Now that a precessing magnet moment in a static uniform field model is estab- lished, this system can be perturbed if a second smaller magnetic field is applied using an oscillating electromagnetic wave of frequency ωrf that is orthogonal to B0. This
time dependent field can be represented by two vectors that are rotating in opposite directions in the x-y plane and having frequencies±ω0. Only the component rotating
in the direction of the precessing magnetization is important which will be called B1;
the component rotating in the opposite direction will have little interaction with the magnetisation along B0.
To simplify the interaction between the oscillating field and the magnetisation along B0 it is useful to translate from the laboratory frame of reference x, y, z,to a rotating
frame x´,y´,z´, wherez =z´that rotates aroundz at frequencyωrf. In this rotating
start precessing around what is called the effective field, the effective field Bef f in the
rotating frame is given by
Bef f =B0+
ωrf
γ +B1 (2.9)
However if the radio frequency pulse (rf) is applied atωrf =ω0 then the contribution
of the B0 field is removed from the rotating frame and the magnetization appears
stationary (see Figure 2.3a). The magnetisation will then start to precess around the only remaining field which is B1. Furthermore, it can now be observed that the
component of B1 that was rotating the opposite direction to the magnetisation does
not interact in the rotating frame as it will have frequency 2ω0.
By implementing the correct intensity and duration of the B1 field it is possible
to rotate the magnetisation in the y´-z´ plane by any desired amount. The angle of rotation θ in radians during a time t that the B1 field is applied for is given by
θ =γB1t=ω1t (2.10)
Therefore, if the rf pulse is applied so that theB1 field lies along the direction of the
x´axis then by applying what is called a 90◦ orπ/2 pulse, then the magnetisation will be rotated onto the −y´axis (see Figure 2.3b). When the B1 field is turned off, the
Figure 2.3: (a) When transferred to rotating frame with frequency ω0 the bulk mag-
netism will appear static. (b) If a second oscillating magnetic field is applied along x´
at frequency ω0 so that it appears stationary in the rotating frame the magnetisation
will start to precess around it with frequency ω1. (c) If the second field is turned off
after the magnetisation has rotated 90◦ from z to lie on the y´ axis then the mag- netisation will again start to precess around z at ω0 however this time in the x - y
rotating frame is removed and the magnetisation will again start to precess around B0 at ω0 in the laboratory frame, however this will now occur in the x-y plane (see
Figure 2.3c). The magnetisation will not stay there permanently but will return to equilibrium along z with some finite time (see Section 2.3). This rotating magnetic moment in the x-y plane produces a circulating magnetic field, and this induces a current in an appropriately positioned receiver coil. This oscillating current is called the free induction decay (FID) or the NMR signal.