Applying A Magnetic Field
With the basic atomic structure fully described for the purpose of understanding MRI, the next step is to explain how the protons behave when placed in an external magnetic field. More rigorous mathematical descriptions are used in this chapter and a co-ordinate system introduced.
2.1 Alignment
When placed in a magnetic field, the randomly positioned protons – being like little magnets – align with the applied external field, much in the same way as a compass needle aligns with the Earths field. However there is an important difference ; the compass needle has only one way to align itself - facing north. Proton magnetisations on the other hand have two possible directions to point when placed in an external field. Figure 4 shows these two possible positions.
Fig 4 : Alignment
of protons in
external field
For obvious reasons the two different directions are known as the ‘up’ and ‘down’ states. It is also important to note that the magnetisation vectors of the protons do not lie parallel to the external field, but are at a slight angle to it. This occurs because of the complex rules of Quantum Mechanics and the explanation is well beyond the scope of this booklet.
The way in which a particular proton aligns itself is dependent upon its energy. That is, it is ‘easier’ for a proton to occupy the up state rather than oppose the external field and occupy the down state. Lower energy (cooler) protons therefore align up and higher energy (hotter) protons align down. Because of the distribution of thermal energies (Maxwell-Boltzmann distribution) this means there will always be slightly more protons in the up rather than in the down state for any given sample. In later sections we will see how this uneven balance leads to a net magnetic moment. That is, if both states were equally occupied the ‘up’ fields would cancel out the ‘down’ fields and there would be no overall field from the sample. With more protons in the up state however there will be a net field and we will see later that this points in the same direction as the applied field.
2.2 Precession : A Closer Look At Alignment
So far we have assumed that the protons just lay in the magnetic field, aligned in the up state or in the down state. However, the protons actually move around in the external field, again due to the rules of Quantum Mechanics. They rotate much in the same way as a wobbling spinning top and we call this type of movement precession. Figure 5 illustrates the movement of the proton together with its analogy.
Fig 5 : Precession
of proton in field [2]
For reasons we will see later, it is important to know how fast the protons precess in the external field. That is, we need to know the precession frequency – how many times the proton completes a full rotation in 1 second.
2.3 The Larmor Equation
The precession frequency of a proton is dependent upon the strength of the magnetic field it is placed in, with a stronger field leading to a faster rate of precession. This is just like the plucking of a guitar string ; the greater the tension in the string, the higher the frequency. Precise calculations of precession frequency are made using the Larmor Equation ;
w0 = Precession frequency / Larmor frequency [Hz]
B0 = Strength of external magnetic field [T]
g = Gyromagnetic ratio [Hz/T]
The gyromagnetic ratio is different for all materials, however for protons it is 42.5 MHzT-1.
2.4 Mathematical Analysis Of Alignment
Fig
6 : Coordinate
System [2]
We will now introduce a coordinate system to make future explanations more concise and ensure further diagrams are more simplistic. Also, from now on we will only illustrate the protons using their magnetisation vectors.
With this coordinate system in place we can calculate the
direction of the net magnetisation field of the precessing protons
mentioned in section 2.1. The following analysis uses only a
handful of protons ‘frozen in time’ to illustrate the
ideas but it is important to emphasise the vast quantities and
huge speeds involved ; millions of protons all rotating at around
40MHz (40 million rotations per second). 
To begin we can see how the magnetic moments of protons in different states (up or down) can ‘cancel’ each other out ;
Fig 7 : Cancellation
of precessing
proton fields [2]
With many millions of protons in the body it is easy to imagine that at any one time there will be a proton A’ which is in the exact opposite position of proton A. The field of A’ will therefore cancel out the field of A, much like we saw before. However, we have also seen there is an uneven distribution of protons so there will be some protons in the up state which are not cancelled out – four in the case of Figure 7. It is these ‘excess’ protons which make up the net magnetisation vector.
If we now look at the vector components of these excess up-protons we can see further cancellations of their individual fields.
Fig 8 : Cancellation
of excess
up–proton fields [2]
The y-component of A’ will cancel the y-component of A, as does the x-component of B’ with B. For the large numbers of protons in real life this process will occur right around the xy-plane, leaving only the z-components of each proton magnetisation vector. These components will thus add together and create a net field in the z-direction (shown as the thick yellow arrow in Figure 8).
2.5 Magnetic Field Of Patient
The magnetic field of the MRI machine is set up in such a way that the net magnetisation field of the protons in the body runs from the feet to the head. That is, the z-axis runs up the body, the x-axis points out from the chest and the y-axis is flat with the patients chest. This set-up allows data to be taken easily with the patient relaxed and lying on their back.
Move on to next section: 3. Resonance.