[MUSIC] Then let's talk about what happens when protons in the magnetic field. When protons are placed in a strong external magnetic field, the two aspects of the magnetic moment can be observed. And one is the magnetic moment tend to align parallel or anti-parallel to the external magnetic field. And then the other is, the magnetic moment is induced to rotate around the axis of the magnetic field, which is called precession. And we will talk about this two in detail. The protons are in the steady magnetic field now, which was randomly distributed without, when there is no external market field. And then when there is external market field, which it typically denoted as a B0, and then they are tend to align along the direction of parallel or anti-parallel to the field. So the B0 unit is typically in the unit of Tesla. So 1 Tesla is 10,000 Gauss, which is about 20,000 times stronger than the Earth's magnetic field. So the proton magnetic moments tend to align with B0, it's energy dependent. The parallel to the field has a lower energy state, and the anti-parallel to the field has higher energy state. Number of protons parallel to and anti-parallel to B0 are very similar. There are huge number of protons that are aligned along parallel or anti-parallel to the field. And most of them, they cancel each other. But there are very small number of spins a little bit higher for those spins, parallel to the field is a little bit higher than the number for the anti-parallel to the field. So this is called spin excess. So this is number of protons parallel to the B0 is slightly higher than the number of protons anti-parallel to the field. Summation is not 0 anymore. And there is some very small number net magnitation is not 0, and there is persistent amount of net magnetation, and we will talk about that later. So the number spins parallel to the field and the number of spin anti-parallel to the field. A subtraction between those two generates a non-zero component, and that is divided by the number of total protons. And that number is very small, so each about tenth minus five order in the room temperature at three Tesla, and which is given in this equation. So only 10 out of 1 million total protons contribute to the spin excess, and this spin excess is used to generate MR signals. So, as shown here, so there are two states, anti-parallel to the field and parallel to the field. So this figure is a little bit exaggerated. There are bunch of spins parallel to the field and anti-parallel to the field. Majority of them cancel each other, and there are very small number of spin excess to in this figure on this three. But anyway, so this spin excess will be used for MR imaging. So the other phenomenon is precession. And its analogy of precession to proton is a gyroscope. So a vertically oriented gyroscope rotate without any wobbling. But it's once tipped off axis, and then gravity causes the spinning gyroscope to wobble. So which is very similar to the protons precessed in the existence of an extra magnetic field. And the wobbling has its own frequency, so wobbling frequency as shown in this field, and which is considered independent of the spinning frequency. So there are two motions for each proton spin in their external magnetic field. One is, Spinning its own axis, which generate spin angular momentum and also generate magnetic moment. And the other is precession as shown in this figure, it's a wobbling motion. Okay, these two frequency rotation, these two motions has its own frequency, and they are completely independent. Let's talk about net magnetization. So each protons magnetization can be considered as a vector. Net magnetization of the precessing proton spins can be found using vector summation. For the case of longitudinal component, majority of parallel and anti-parallel spins cancel each other as I mentioned in the previous slide. So some additional parallel spins called spin excess remain and can be used to generate MR signals. So that is along the longitudinal direction. And for the transverse component, the phases of the photon spins for the spin excess are random. So the vector summation of these proton spins will be zero. So there will be no transport component. So overall, the net magnetization is in the longitudinal or z-direction. This is the same direction as the external magnetic field, B0. So when our body is placed inside MRI machine, so there is strong external magnetic field, and that generate longitudinal net magnetization, okay. So which is zero if there's no external magnetic field? But the net magnetization can be generated when our body is placed inside MRI machine. Shown in this figure, there are bunch of spins parallel to the field and anti-parallel to the field. They cancel each other, so there is only small number of spin excess exist. And then they generate net magnetization along z-direction, which is the same direction as the external magnetic field. And transverse component wise, the phases of these spins become summation of these transverse component becomes zero. Okay, let's talk about the precession phenomena in detail. And similar to the spin angular momentum and magnetic moment, the precession angular speed omega 0 is proportional to the external magnetic field B0 with proportionality constant gyromagnetic ratio gamma. So omega 0 is the rotation precession frequency and B0 is the external magnetic field. So this relationship is also linear and the proportional constant is gyromagnetic ratio gamma. So which is the same as the relationship between the angular momentum phi and magnetic momentum mu. So that the proportional constant was gamma, and this relationship is also same. So external precision frequency omega 0 equals gamma B0. So here B0 is external magnetic field strength. So the stronger the external magnetic field, the faster the precision speed. And typically both side of this equation can be divided by 2 pi, which make this angular frequency to be just a regular frequency in the unit over hertz. So f0 = gamma over 2 pi multiplied by B0. So in 1 Tesla MRI, this f0 = 42.98 megahertz. So you may want to remember this number. And it is typically assumed that the Larmor frequency is constant for a given spin system, for example, a proton. And the magnetic field is actually not constant due to magnetic field inhomogeneities caused by factors, such as hardware imperfection, and magnetic susceptibility, and the chemical shift. So you may want to remember this equation. And this equation is the most important equation for MR imaging. And you can explain all the MR phenomenon, and also it can explain magnetic resonance. And also it explain the imaging part, like a gradient one, what gradient it's doing. So the precession frequency is proportional to the magnetic field strength which can be used. And also which can be used to generate the imaging in this spacial domain, and we can modulate this frequency too for the imaging. And we will talk about that much later in detail. So this is common gyromagnetic ratio. And the proton is typically has the highest gyromagnetic ratio. And there is some other molecules, which has a little bit lower gyromagnetic ratio than protons. Which means the legend frequency at the same field strength is lower than the proton for other molecules. What you may want to remember, this gyromagnetic ratio for the proton is the most important. And so 42.58, so this is gyromagnetic ratio at 1 Tesla, which gives this frequency. So you may want to remember this number.
