detail treatment on Larmor Precession and Rabi Resonance

http://nukephysik101.files.wordpress.com/2011/01/a-treatment-on-larmor-precession-and-rabi-resonance.pdf

is a pdf files, 15 pages. i think it covered most material on this subject. did i?

NMR

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. 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 are not isolate, when it interact with environment, they will go back and align with the static B-field. 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.

Larmor Procession

Magnetic moment (mu) :

this is a magnet by angular momentum of charge or spin. its value is:

mu = gamma S ( in case of spin ) = gamma L ( in case of angular momentum )

where gamma is (g-factor) x ( Bohr magneton ) / ( h_bar [ in case that the S or L is not using natural unit ] )

the g-factor is a dimensionless number, which reflect the environment of the spin, for orbital angular momentum, g = 1.

Bohr magneton = electron charge x h_bar / ( 2 mass), since different particle has different mass, their Bohr magneton value are different. electron is the lightest particle, so, it has largest value on Bohr magneton.

J-coupling ( i am not sure it is correct to say so)

since the spin and angular momentum are mathematical equal. they give the same effect when react with magnetic field, thus, we use a new term:

J = L + S

mu = gamma J

in this case, since the g-factor of L and S are different, and when we combine with different particles, the Bohr magnetons are different, in that case, special treatment is needed. but it does not change the principle.

Larmor frequency:

When applied a magnetic field on a magnetic moment, the field will cause the moment process around the axis of the field. the precession frequency is called Larmor frequency.

the procession can be understood in classical way or QM way.

Classical way:

the change of angular momentum is equal to the applied torque. and the torque is equal to the magnetic moment  cross product with the magnetic field.

d J / d t = mu x B = gamma J x B

solving gives the procession frequency is :

w = - gamma B

the minus sign is very important, it indicated that the J is precessing by left hand rule.

QM way:


The Schrödinger equation is :

i  d/dt |Psi > = H |Psi >

H is the Hamiltonian = - mu . B = - gamma J.B = - gamma B Jz = w Jz

Jz is an operator.

the solution is for |Psi> is

|Psi ( t ) > = Exp[ - i w Jz ] |Psi ( t = 0 ) >

for up state coefficient a1 and down state coefficient a2,

solving a1 and a2 gives:

a1 = Exp [ - i w t ]
a2 = Exp [ + i w  t ]

Thus, in QM point of view, the state does not change but only the phase.

For not using natural unit, we knew, w h_bar = Energy.

Natural unit

on Size and Unit, we know that the speed of light better be equal to 1, that simplify the equation of relativity.

now, we impose 1 more things, the Reduced Planck constant, h_bar also set to 1. that simplify all equations with angular momentum or spin.

the Angular momentum:

L^2 | l, m > =  h_bar l ( l+1) |l, m >
Lz |l, m > = h_bar m |l, m >

now becomes :

L^2 | l, m > =   l ( l+1) |l, m >

Lz |l, m > =  m |l, m >

this is same for the spin. 

by doing so, the magnetic moment will be

mu = g x muB x S or g x muB x L

since there is no h_bar in S or L anymore. 

Hall effect

It is a short review.

The hall probe is perpendicular to the B field( pointing up) and have a current I passing through ( going forward ).

Due to the Lorentz force. The positron is moving to right and accumulate. The accumulating charge creates a electric force to the left to against further positron accumulate. The magnetic force will be balanced by the electric force. Due to the electric force, there is associated voltage across the hall probe. This voltage is called hall voltage.

FB= e v B = Vh d

Where e is positron charge, v is speed of positron, B is the magnetic field, Vh is the Hall voltage and d is the distance across the hall probe.

The current I is

I = A n e v

Where A is the cross section area of the hall probe, n is density of the positron carrier, v is the positron velocity.

Thus,

Vh = I B / ( V n e )

Where V is the volume of the hall probe. But the V n is equal to the total number of positron N.

Vh = I B / ( N e )

Which is to say, the hall voltage is proportional to the magnetic field.

decay time constant and line width

the spectrum of energy always has a peak and a line width.

the reason for the line width is, this is decay.

i give 2 explanations, once is from classical point of view and i skipped the explanation for the imaginary part. so, i am not fully understand. the 2nd explanation is look better, but it is from QM. however, there is one hide question for that explanation is, why the imaginary energy is negative?

the simplest understanding of the relation is using fourier transform. (i think)

fourier transform is changing the time-frame into the frequency frame. i.e, i have a wave, propagating with frequency w. we can see a wave shape when plot with time. and we only see a line, when we plot with frequency, since there is only 1 single frequency. however, for a general wave, it is composite of many different frequencies, using fourier transform can tell us which frequency are involved. And energy is proportional to frequency.

when the particle or state under decay. the function is like

Exp[- R t] Exp[ i wo t]

where the R is decay constant, and wo is the wave frequency.

after fourier transform, assume there is nothing for t < 0

1/ ( R + i ( wo - w) )

the real part is

R/ ( R^2 + (w - wo)^2 )

which is a Lorentzian shape and have Full-Width-Half-Maximum (FWHM) is 2R. it comes from the cosine part of the fourier transform. thus, the real part.

and the imaginary part is

(w - wo)/( R^2 + (w - wo)^2 )

the imaginary part is corresponding to the since part, so, we can neglect it. (how exactly why we can neglect it? )

Thus, we can see, if there is no decay, R -> 0, thus, there is no line width.

therefore, we can see the line width in atomic transition, say, 2p to 1s. but there are many other mechanism to the line width, like Doppler broadening, or power broadening. So, Decay will product line width, but not every line width is from decay.

**********************************
another view of this relation is from the quantum mechanics.

the solution of Schroedinger equation is

Psi [x, t ] = phi[x] Exp[- i E/hbar t]

so, the probability conserved with time, i.e.:

| Psi [x,t] |^2 = | Psi[x,0] | ^2

if we assume the energy has small imaginary part

E = E0 - i R hbar/2 ( why the imaginary energy is nagative?)

 | Psi [x,t] |^2 = | Psi[x,0] | ^2 Exp[- R t]

that make the wavefunction be :

Psi [x, t ] = phi[x] Exp[- i E/hbar t] Exp[ - R/2 t ]

what is the meaning of the imaginary energy?

the wave function is on time-domain, but what is "physical", or observable is in Energy -domain. so, we want Psi[x,E] rather then Psi[x,t], the way to do the transform is by fourier transform.

and after the transform, the probability of finding particle at energy E is given by

|Psi[x,E]|^2 = Constant / ( R^2 + ( wo - w)^2 )

which give out the line width in energy.

and the relation between the FWHM(line width) and the decay time is

mean life time = hbar / FWHM

which once again verify the uncertainty principle.

WKB approximation

I was scared by this term once before. 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 pi / k, when it expressed in Exp[ - kapper x ], wavelength = 1/kapper.

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

A[x] Exp[ i phase[x] ], A[x] and phase[x] are real function.

sub this into the Schrödinger equation ( Psi ''[x] = - { 2 m / hb (E - V[x])} Psi[x] ). and separate the imaginary part and real part.

The imaginary part is can be simplified as:

D[ A^2 phase' ] = 0 => A = Const. /Sqrt[ phase'[x] ]

The real part is

A''[x] = ( phase'[x]^2 -  2 m / hb (E - V[x]) ) A[x]

we use the approx. that A''[x] = 0 since it varies slowly.

Thus,

phase'[x] =  Sqrt[2 m / hb (E - V[x])]
=>
phase[x] = Integrate[ Sqrt[2 m / hb (E - V[x])], x]

and the solution is, ( if we use p[x] = Sqrt[2 m / hb (E - V[x])] )

Psi [x] = Const. /Sqrt[ p[x] ] Exp[ i Integrate[ p[x] ] ]

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[- 2 Sqrt[ 2m / hb] Sqrt[area between V[x] and Energy] ]. Therefore, when we have an ugly potential, we can approx it by a rectangular potential with same area to give the similar estimation.

alpha decay

different decay cause by deferent mechanism, we first start on alpha decay.

i assume we know what is alpha decay, which is a process that bring excited nucleus to lower energy state by emitting an alpha particle.

The force govern this process is the strong force, due to the force is so strong, the interaction time is very short, base on the uncertainty principle that large change in energy leads to short time interval. however, the observed alpha decay constant is about 1.3 × 10¹⁰ year, which is about the age of our universe. That's why we still able to find it at the beginning of nuclear physics : discovery of radioactive matter.

The reason for such a long decay time is due to the Coulomb barrier of the nuclear potential. since the proton carry positive charge, thus. it creates a positive potential wall in the nucleus. that potential not only repulse proton from outside but also the proton from inside which try to get out. thus, the inside protons are bounded back and forth inside the nucleus. due to the momentum carried by the protons, it has frequency 6  × 10²¹ per sec.

Due to the Quantum tunneling effect, the probability of tunneling is 4 × 10^-40. which is a very small chance. But , don't forget there are  6  × 10²¹ trails per sec. Thus, the chance per sec is 2.4 × 10^-18 . and the mean life time is inverse of the probability, thus it is approx 1.3 × 10¹⁰ year.

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

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

Sqrt[ Sum[E_i]^2 - Sum[p_i]^2 ] = excited mass.

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

m1=Sqrt[ E1^2 - p^2 ]
m2=Sqrt[ E2^2 - p^2 ]

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

Sqrt[ (E1+E2) ^2 - (p - p)^2 ] = E1 + E2

which is not equal to Sqrt[ E1^2 - p^2 ] + Sqrt[ E2^2 - 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.

Scattering phase shift

for a central potential, the angular momentum is a conserved quantity. Thus, we can expand the wave function by the angular momentum wave function:

Sum { a[ l ] Y[ l, m = 0] R[ l , k , r] }

the m=0 is because the spherical symmetry. the R is the radial part of the wave function. and a is a constant. k is the linear momentum and r is the radial distance.

for free particle, potential equal to zero,

R -> J[ l , k r]

which is reasonable when r is infinite and the nuclear potential is very short distance. when r goes to infinity,

J[ l, k r ] -> 1/ (k r ) Sin{ k r - 1/2 l pi}

for elastic scattering, the probability of the current density is conserved in each angular wave function, thus,
the effect of the nuclear potential can only change the phase inside the sin function:

1/ (k r ) Sin{ k r - 1/2 l pi + d[ l ]}

with further treatment, the total cross section is proportional to Sin{ d[ l ] }^2.

thus, by knowing the scattering phase shift, we can know the properties of the nuclear potential.

for more detail : check this website

magic number

we knew that for some atoms are more stable that others. like He, Ne, Ar, etc, which are belonged to noble gas. the reason for they are un-reactive is, there outer most electron shell is filled out.

similar things happened in nuclei. in the shell model of nuclei, protons and neutrons just like the electrons in atom. if the outer most shell of proton or neutron is filled out, the nuclei is very stable. and we called this number of proton or neutron be MAGIC NUMBER.

the first magic number is 2. this nuclei of 2 protons is very stable. If there are 2 protons and 2 neutrons, we called this double magic number, and this nuclei, which is He is very very stable. 

the list of magic number is 2, 8, 20, 28, 50, 82, 126 in theory prediction. 

however, when the nuclei become heavier and heavier, the stability of nuclei in the magic number lost. to understand this. we have to know that the magic number is come from the large spin-orbital coupling term in the Hamiltonian of the nuclei. and recent research suggest that, the spin-orbital coupling may change by the number of nucleons. 

Building of special relativity graphically.

Differential Cross Section

In nuclear physics, cross section is a raw data from experiment. Or more precisely differential cross section, which is some angle of the cross section, coz we cannot measure every scatter angle and the differential cross section gives us more detail on how the scattering going on.

The differential cross section (d.s.c.) is the square of the Form factor, which is the Fourier transform of the density.

d.s.c. = |F(θ)|^2 = Fourier[ ρ(r), Δp , r ]

Where the angle θ come from the momentum change. So, sometime we will see the graph is plotted against momentum change instead of angle.

On the other hand, F(θ) is the amplitude of the scatter spherical wave.

Therefore, by measuring the yield of different angle. Yield is the intensity of scattered particle. We can plot a graph of the Form factor, and then find out the density of the nuclear or particle.

However, the density is not in usual meaning, it depends on what kind of particle we are using as detector. For example, if we use electron, which is carry elected charge, than it can feel the coulomb potential by the proton and it reflected on the "density", so we can think it is kind of charge density.

Another cross section is the total cross section, which is sum over the d.s.c. in all angle. Thus, the plot always is against energy. This plot give us the spectrum of the particle, like excitation energy, different energy levels.

Hydrogen Atom (Bohr Model)

OK, here is a little off track. But that is what i were learning and learned. like to share in here. and understand the concept of hydrogen is very helpful to understand the nuclear, because many ideas in nuclear physics are borrow from it, like "shell".

The interesting thing is about the energy level of Hydrogen atom. the most simple atomic system. it only contains a proton at the center, um.. almost center, and an electron moving around. well, this is the "picture". the fact is, there is no "trajectory" or locus for the electron, so technically, it is hard to say it is moving!

why i suddenly do that is because, many text books said it is easy to calculate the energy level and spectrum for it. Moreover, many famous physicists said it is easy. like Feynman, Dirac, Landau, Pauli, etc... OK, lets check how easy it is.

anyway, we follow the usual way in every text book. we put the Coulomb potential in the Schrödinger equation, change the coordinate to spherical. that is better and easy for calculation because the coulomb potential is spherical symmetric. by that mean, the momentum operator (any one don't know what is OPERATOR, the simplest explanation is : it is a function of function.) automatically separated into 2 parts : radial and angular part. The angular part can be so simple that it is the Spherical harmonic.

Thus the solution of the "wave function" of the electron, which is also the probability distribution of  the electron location, contains 2 parts as well. the radial part is not so trivial, but the angular part is so easy. and it is just Y[l,m].

if we denote the angular momentum as L, and the z component of it is Lz, thus we have,

L²Y[l,m] = l (l+1) Y[l,m]
Lz Y[l,m] = m Y[l,m]

as every quadratic operator, there are "ladder" operator for "up" and "down".

L⁺ Y[l,m] =const. Y[l,m+1]
L^- Y[l,m] = const. Y[l,m-1]

which means, the UP operator is increase the z-component by 1, the constant there does not brother us.

it is truly easy to find out the exact form of the Y[l,m] by using the ladder operator. as we know, The z component of the a VECTOR must have some maximum. so, there exist an Y[l,max] such that

L⁺ Y[l,max] = 0

since there is no more higher z-component.

by solve this equation, we can find out the exact form of Y[l,max] and sub this in to L², we can know max = l. and apply the DOWN operator, we can fins out all Y[l,m], and the normalization constant is easy to find by the normalization condition in spherical coordinate, the normalization factor is sin[theta], instead of 1 in rectangular coordinate.

decay

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

Number of particle (time) = Initial # of particle X Exp( - time / T )

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 X "mean time". that is what you got. ( Integrate[ Exp[- t / T], {t,0, infinite}] = T )

some people like to write the equation is other way:

Number of particle (time) = Initial # of particle X Exp( - R time )

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

we also have "Half-Life", which is the time that only half of the particle left. by the equation, we have:

half-life = In (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.

Special Relativity II

We are going to talk about coordinate transform from "center of momentum frame" to "Laboratory frame".

At the center of momentum [C.M.] frame, the total momentum is zero.

Refer here on Google Docs. for the Mathematica 7 code and calculation steps.

anyone want to have the .nb file, feel free to ask.

i am just discuss on the result.

1) In the C.M. frame, energy of each particle reserved, there is no exchange in energy and momentum. after collision, they just change the moving direction.

2) the energy is C.M frame is always smaller then Lab frame, or other fame. which is also from the face that, at the CM frame, total momentum is zero and the corresponding energy is the Proper Energy.

3) In Lab frame, the scatter angle is always smaller than 90 degree for incident particle's mass > target particle.

4) In Lab frame, The larger the scatter angle, the smaller the momentum and larger the momentum transfer.

Special Relativity I

i just state the formula and the usage of it.

the basic equation is

E² = (p c)² + (m c²)²

where E is total energy, p is momentum

here we can see the advantage of using MeV as unit of mass. the equation is now further simplified by using MeV/c as momentum unit.

E² = p² + m²

which is Pythagorean theorem!

the speed of the particle is from the formula

 β = v / c = p / E

For example, proton mass is 940MeV/ c², if we say an proton is moving at 94MeV, or a 94 MeV proton. we mean, the KINETIC Energy (K.E.) of proton is 94MeV. The total energy is

Mass + K.E. = E

Thus, a 94 MeV proton is moving at 41.7% of light speed. by using a right-angle triangle of base 10, side 11, and the hight is Square-Root 21.

another way around is, a proton at 90% speed of light, how much K.E. it has? which is around 3000 MeV or 3 GeV [Giga eV]