## NMR (nuclear magnetic resonance)

NMR is a technique to detect the state of nuclear spin. a similar technique on electron spin is call ESR ( electron spin resonance)

The principle of NMR is simple.

1. apply a B-field, and the spin will align with it due to interaction with surrounding and precessing along the B-field with Larmor frequency, and go to Boltzmann equilibrium. the time for the spin align with the field is call T1, longitudinal relaxation time.
2. Then, we send a pule perpendicular to the B-field, it usually a radio frequency pulse. the frequency is determined by the resonance frequency, which is same as the Larmor frequency. the function of this pulse is from the B-field of it and this perpendicular B-field with perturb the spin and flip it 90 degrees.
3. when the spin are rotate at 90 degrees with the static B-field, it will generate a strong enough signal around the coil. ( which is the same coil to generate the pule ) and this signal is called NMR signal.
4. since the spins will be affected by its environment, and experience a slightly different precession frequency. when the time goes, they will not aligned well, some precess faster, some slower. thus, the transverse magnetization will lost and look as if it decay. the time for this is called T2, transverse relaxation time.

by analyzing the T1 and T2 and also Larmor frequency, we can known the spin, the magnetization, the structure of the sample, the chemical element, the chemical formula, and alot many others thing by different kinds of techniques.

For nuclear physics, the use of NMR is for understand the nuclear spin. for example, the polarization of the spin.

## on the sum of 4 momentum and excited mass

when we have a decay process, there are many fragments, we can measure their momentum and energy and construct the 4-momentum

$\vec {P_i} = ( E_i , p_i )$

we use the c = 1 unit as usual.

to find out the mass before the decay, we can use

$\sum E_i^2 - \sum p_i^2 = m_{excite}^2$

the reason for the term “excited mass”, we can see by the following illustration.

consider a head on collision of 2 particles in C.M. frame, with momentum p and energy E1 and E2.

the mass for each one is given by

$m_1 = \sqrt {E_1^2 - p^2 }$

$m_2 =\sqrt {E_2^2 - p^2 }$

but if we use the sum of the 4 momentum and calculate the mass,

$\sqrt { (E_1 +E_2)^2 - (p - p)^2} = E_1+E_2$

which is not equal to

$\sqrt { E_1^2 - p^2} + \sqrt{E_2^2 - p^2 }$

in fact, it is larger.

the reason for its larger is, when using the sum of 4 momentum, we actually assumed the produce of collision is just 1 particles, and the collision is inelastic. Thus, if we think about the time-reverse process, which is a decay, thus, some of the mass will convert to K.E. for the decay product.

## WKB approximation

I was scared by this term once before. ( the approach an explanation from J.J. Sakurai’s book is not so good)  in fact, don’t panic, it is easy. Let me explain.

i just copy what written in Introduction to Quantum Mechanics by David Griffiths (1995) Chapter 8.

The approx. can be applied when the potential is varies slowly compare the wavelength of the wave function. when it expressed in $Exp( i k x)$, wavelength = 2 π / k, when it expressed in $Exp( - \kappa x )$, wavelength = 1/κ.

in general, the wavefunction can be expressed as amplitude and phase:

$\Psi(x) = A(x)Exp(i \phi(x))$

where $A(x)$ and $\phi(x)$ are real function

sub this into the time-independent Schrödinger equation (TISE)

$\Psi '' (x) = - \frac {2 m} {\hbar^2 } ( E - V(x) ) \Psi (x)$

$\Psi ''(x) = ( A''(x)- A(x) \phi'(x)^2 + 2 i A'(x) \phi'(x)+ i A(x)\phi''(x) ) Exp(i \phi (x) )$

and separate the imaginary part and real part.

The imaginary part is can be simplified as:

$2 A'(x) \phi '(x) + A(x) \phi ''(x) = 0 = \frac {d}{dx} ( A^2(x) \phi '(x)$

$A(x) = \frac {const.} {\sqrt {\phi '(x)}}$

The real part is

$A''(x) = \left ( \phi ''(x) - \frac {2m}{\hbar^2 } ( E - V(x) ) \right) A(x)$

we use the approx. that $A''(x) = 0$ ,  since it varies slowly.

Thus,

$\phi '(x) = \sqrt { \frac {2m}{\hbar^2} (E - V(x) ) }$

$\Rightarrow \phi(x) = \int \sqrt { \frac {2m}{\hbar ^2} ( E - V(x ) )} dx$

if we set,

$p(x) = \sqrt { \frac {2m}{ \hbar^2 } ( E - V(x) )}$

for clear display and $p(x)$ is the energy different between energy and the potential. the solution is :

$\Psi(x) = \frac {const.}{\sqrt {p(x)}} Exp \left( i \int p(x) dx \right)$

Simple! but one thing should keep in mind that, the WKB approx is not OK when Energy = potential.

This tell you, the phase part of the wave function is equal the square of the area of the different of Energy and the Potential.

when the energy is smaller then the potential, than, the wavefunction is under decay.

one direct application of WKB approxi is on the Tunneling effect.

if the potential is large enough, so, the transmittance is dominated by the decay, Thus, the probability of the tunneling is equal to

$Exp \left( - 2 \sqrt { \frac {2m}{\hbar ^2 } A_{area} ( V(x) - E )} \right)$

Therefore, when we have an ugly potential, we can approx it by a rectangular potential with same area to give the similar estimation.

## decay

the decay idea and mathematic is simple. so, i just state it.

Number of particle (time) = Initial # of particle × Exp( – time / T )

or in formula

$N(t) = N(0) Exp \left( - \frac {t} {T} \right )$

where T is time constant, which has a meaning that how long we should wait before it decay. T also has another name, “mean-lifetime“, coz when you find out the mean of their life by usually statistical method, integrate the whole area of the graph of decay time and make it equal to initial # of particle × “mean lifetime”. that is what you got. ( $\int_0^\infty Exp( - \frac {t}{T}) dt= T$ )

some people like to write the equation is other way:

$N(t) = N(0) Exp \left( - R t \right )$

where R is the chance of decay in unit time. which is just the “invert” meaning of T.

we also have “Half-Life$t_\frac {1}{2}$, which is the time that only half of the particle left. by the equation, we have:

$t_\frac {1}{2} = ln(2) T$

thus, a longer T, the particle live longer, as what is the T mean!

But above mathematics only tell us the statistic result of the decay, not about the mechanism, or physics of what cause the decay happen. why there is decay? why particles come out from nucleus? how many kind of decay ?

the easiest question is, there are 3 decay happen in nature and a lot more different decay happened in lab. the reason for only 3 decay is that, only these 3 live long enough to let us know. the other, they decay fast and all of them are done.

and the reason for nucleus decay is same as the reason for atomic decay. excited nucleus is unstable (why?) they will emit energy to become stable again.

and the physics behind decay, we will come back to it later.