Thursday, December 23, 2010

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.

Wednesday, December 22, 2010

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".

LY[l,m] =const. Y[l,m+1]
LY[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

LY[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.

Sunday, December 19, 2010

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.

Friday, December 17, 2010

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.

Thursday, December 16, 2010

type of accelarator I (Linac)

The machine used in nuclear physics is call Accelerator. Because it accelerate, speed up the particle.

There are basically 2 types, one is called Linear Accelerator (Linac) , another type is called Circular Accelerator.

in fact, there are many other type of accelerator, as long as they can accelerate particle, by definition, it can be called as accelerator. for example, The vacuum tube in old day TV is an electron accelerator! However, most other type of accelerators can only speed up particle at low speed, not comparable to speed of light. So, in modern nuclear physics, we don`t use them. Nevertheless, the mechanism of them may be reviewed and other type of accelerator may be invented in future, who know!

so, far, all Accelerator can only handle charged particle.

Linac

The particle being accelerated in Linac is moving in a straight line. that is why the name Linear. Linac constitutes of  many sectors, each sector is a mini accelerator, which speed up the particle by adding energy into it.

The simplest type of Linac is 2 parallel plate with a hole at the middle, and has the electric potential different V. when an electron passed through it, it will gain eV of energy and then speed up. so, if there is n sector, and each sector are identical, the final energy is neV. This LINEAR behavior also address the name.

Working Principle

Now, imagine you have many plates, when the particle passed plate 1, a voltage applied on plate 2 to speed up the particle. after the particle passed plate 2, you have to turn off the voltage and apply the voltage on plate 3, and the process go on. thus, the voltage on each plate is oscillating, which is the working frequency of Linac. Moreover, when the particle speed up, the time for it with in each section will be smaller and smaller, thus, the working frequency has to be increase and matching the particle speed. the other way around it, building each sector in different length, but this method will set a definite frequency of the Linac and the output energy.

The advantage of Linac is that :
1) it can produced continuous beam
2) less energy lost during acceleration. where all charged particle will radiate energy while accelerating.

The disadvantage is :
1) limited length, so the Maximum energy is limited
2)  The working frequency has to be tuned so accurate. it is harder to operate a Linac.

Special Relativity I

i just state the formula and the usage of it.

the basic equation is

E2 = (p c)2 + (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]

Method

As we know the world of nuclear physics is so small. ordinary method is not applicable to "see" this world.

we have no choice but just bombard the nucleus with electrons, protons, neutrons, etc... if we are Alice, who become much bigger then our earth and touch the moon easily. How do we understand human world? we simply pick a human, hitting on each other, see what is going on, what is the result. If we want to know how elevator work, we put a human, let him ride on it and see the result. but sometime, we will accidentally, put a car in elevator.

so, Most nuclear experiment is SCATTERING EXPERIMENT.

and the machine to conduce this kind of experiment is called Accelerator.

there is a famous quote i forget where it come from:

nuclear (particle) physics is like we figure out how a watch work by broken it and see the fragments.

The picture is really like this. we shoot particles into nucleus, and see what was knot out. how the incident particle changed. we can extract the energy change, the scatter angle, the polarization. basically are these 3 things. and using these 3 data, we construct the world of nucleus. Is it amazing???

There are some fundamental limits of the scattering experiment. 

1) since the size of the nucleus is very small, the chance of hitting it is very small for 1 particle to hit another particle. thus, we use many particles hit many particles. but even doing so, only a tiny fraction of reaction takes place. most of them just pass by, say a hello. Thus, we have to create a high density particle beam, and target. 

2) the particle should be moving very fast, almost same as speed of light, in order to carry enough energy to go inside the nucleus. because there is a barrier form the forces. the nucleus is something like a fortress, walled by forces. (sound like a star wars movie) another reason is, the larger the energy, the shortest distance we can probe. According to De Broglie, every particle can be treated as wave with wavelength is inversely proportional to the momentum. Thus, a faster particle has larger momentum and shorter wavelength. so, can see a smaller world. [ the De Broglie's wavelength has some debt on weather it is a physical wave or probability amplitude, this was solved by Dirac and proved by experiment that, it means both. for more info, see discussion on "interpolation on quantum wave function" ] 

This is a general property of wave. a Radio wave can easy pass though us because we are small compare to the wavelength, which is about 5 to 10 meters long. but red light can "see" us, or we can block red light, because we are much bigger. the idea is, if we want to see the detail, you have to use a smaller ruler. 

3) since we are using the building block to hit another building block in investigation. they are similar size. imagine a scenario that you want to measure to speed of a car, and you use another car to hit on it, and see the bounded back car to find out the original speed. you can see, the cars hit each other and changed the original speed, and there is no way to accurate to measure to speed! same things happen in nuclear physics. the scatter particle will change to state of target, that create an uncertainty. This was formulated by Heisenberg and now called the Heisenberg's Uncertainty Principle. which state that 

change of position X change of momentum  >= Planck's constant


Size and Unit

As we know, atom is very small. Lets get some idea how small it is.

Let us transform ourself, like Alice in Wonderland, i guess Alice is just 100 times smaller. but this time, we goes much much smaller, become same same as an atom. Hydrogen atom, which is the smallest one, is about 1 meter hight, and Uranium is about 2 meters. and Alice, now becomes mush better then the earth and she can touch the moon easily.

how large is proton in this scale? it is just a diameter of our hair! and electron is still much smaller, so far, we believe that it is just a point, no size. so no matter how powerful your microscope, you never see it! which also means, electron has no internal structure. Thus, you may ask the structure of proton. um... we talk this later.

When someone want to talk about physics, he has to bring up ENERGY. that make his sound professional. and we are going to do the same. However, our unit of energy is different from day usage, like Calorie, or kilo-Watt-hour (kWh). we use  MAGA-ELECTRON-VOLT (MeV). before we get some idea about these units. we should understand kilo [k] = 1000, maga [M] = 1000 X 1000.

1 Calorie, which is the energy require the raise 1 degree of 1 kg water. in a cold winter, say 10 degree. if you drink tap water of 350ml, then you lose about 9 Calorie. if you drink 350ml water at 60 degree. then, you gain, 17.5 Calorie. according to this website : If your body mass is 64 kg and you walk at a speed of 5.63 km/hr then you will burn approximately 4.6 Calories/minute - if you walk for one hour you will burn 60 × 4.6 = 276 Calories. well, not much help for the hot water.

1 kWh is equal to 3.6 mage joule. so, what is joule? joule is another energy unit, 1 Calorie = 4.2 kilo joule. Thus, 1 kWh = 860 Calorie = about 3 hour walk. next time when you check your electric bill, you can imagine how far you have to walk to consume this energy.

as you may see, 1 Cal can raise 1 kg water up 1 degree. how many water molecules in 1 kg of water? It is 3 × 1025 !! the world population is just 6 × 109. if this number is money, then every one is a billion billionaire!! So, you can see, each molecule only share very very little among of energy. in order to save some zero in front of 1. we use a very small unit. maga-electron-volt.

according to Einstein Energy-Mass equation: E = c2, we found that a proton is about 940 MeV/ c2. which is to say, if we want to make a proton out of no where, we have to at least give 940MeV. um, pretty small compare to our daily life. But it is very big in nuclear physics.

from this, we can know electron mass is about 0.5 MeV/ c2.

the reason why we use "energy" unit on "mass", is not just because Einstein tell us we can do so, but it has a practical reason. We always deal with relativity and using MeV as a unit of mass bring huge convenient, both on calculation and imagination.

Objects of Interest

Nuclear Physics is a study on nuclear matter which is fundamental building block of the world.

electron, proton , neutron, deuteron, tritium, etc... those are objects in nuclear, we call them "particle". the most simple particle in here is electron, proton and neutron.

The different between nuclear and atom is:

Nuclear core (sit in the center) + Electrons (moving around) = Atom

the mass of atom is almost contributed by nuclear. This is because the mass of proton is about 1830 times bigger than electron, and neutron's mass is only heavier a bit then proton.

There are many properties contained in each particle. there are electric charge, mass, spin, kinetic energy, etc... and the objective of nuclear physics is understand all these properties and how these properties affect the inter-reaction among them. for example, how a proton and neutron form a nuclear core in deuteron? how they attract each other?

these properties, some may say, are ASSIGNED to the particles. Basically, we can only measure the effect or the result from each interaction. we think, there is a FORCE to make particles able to INTER-ACT with each others. simple to say, when an electron meets another electron, they affect each other by ELECTROMAGNETIC force. but when consider an electron meet a neutron, they don't interact by electromagnetic(EM) force. in order to distinguish these. we assign an electric CHARGE to electron, and no charge for neutron.

so, basically, Nuclear Physics is study the PROPERTIES of particles and the INTERACTION among them.

There are 3 major forces/interactions, Weak force, EM force and Strong force. Until this moment, we only know the weak and EM force and not fully understand the strong. We neglect the gravity in here, because it is very weak and do no observable effect.

Force
Strength
Range
Strong
10,000
10-15m
EM
1000
long
Weak
1
10-18m