Showing posts with label Basic. Show all posts
Showing posts with label Basic. Show all posts
Sunday, January 30, 2011
Saturday, January 29, 2011
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?
is a pdf files, 15 pages. i think it covered most material on this subject. did i?
Wednesday, January 26, 2011
Changing of frame
sometimes, i will confuse on changing frame, especially between rotating frame and lab frame.
Monday, January 24, 2011
Mass of particles and nucleus
in Nuclear physics, the particle we deal with are so small and so light, if we use standard unit, then there will be many zero and we will lost in the zeros. for example, the electron has mass:
Mass( electron ) = 9.11 × 10-31 kilograms
Mass( proton ) = 1.67 × 10-27 kilograms
see? as the special relativity give us a translation tool - E = m c^2, thus, we can use MeV to talk about mass.
Mass ( electron ) = 0.511 MeV
Mass ( proton ) = 938.3 MeV
thus, we can see, Proton is roughtly 2000 times heavier then electron ( 1000 : 0.5 ).
Mass( neutron ) = 939.6 MeV
neutron is just 1.3 MeV heavier then proton.
The nucleus is formed by proton and neutron. so, in simple thought, an nucleus with Z proton and ( A-Z ) neutron should have mass
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) = Mass ( A, Z )
where A is the atomic mass number, which is equal the number of nucleons in the nucleus, and Z is the proton number.
However, scientists found that it is not true.
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) > Mass ( A, Z )
Some of the mass is missing! But that is explained why nucleus will not break down automatically. since it need extra energy to break it down.
we called the mass different is Mass Deficit. or Blinding energy.
Mass Deficit = Mass( A, Z) - Mass ( proton + neutron )
some one may think that the blinding energy is the energy for holding the nucleus together. in order to hold the nucleus, some mass was converted into the energy to holding it. this is INCORRECT. the correct argument is, the blinding energy is th energy require to break it down.
think about a simple 2 bodies system, like sun and earth. at far far away, when both of them are at rest, the total energy is Mass( sun) + Mass ( earth ) + Potential energy.
when the earth moves toward to sun, the potential energy converted to the Kinetic energy, so the earth moving faster and faster. but, in order to stay in the orbit, some K.E. must be lost so that it does not have enough ( or the same) energy to run away. Thus, the total energy of the system is lesser then the total mass.
another analogy is electron orbit. when an electron was captured by an atom, it radiate energy in order to stay in some energy level. thus, the total energy of the system again less then the total mass.
any any case, the mass of the sun and earth and electron does not change, but the potential changes to negative, thus it makes to total energy lesser.
similar idea hold for nucleus, but the potential of it are great different, because there are a Coulomb Barrier. Thus, in order to make a nucleus. we have to put so many K.E. to again this barrier, then the resultant nucleus release the Mass Deficit energy and also the input K.E..
Mass( electron ) = 9.11 × 10-31 kilograms
Mass( proton ) = 1.67 × 10-27 kilograms
see? as the special relativity give us a translation tool - E = m c^2, thus, we can use MeV to talk about mass.
Mass ( electron ) = 0.511 MeV
Mass ( proton ) = 938.3 MeV
thus, we can see, Proton is roughtly 2000 times heavier then electron ( 1000 : 0.5 ).
Mass( neutron ) = 939.6 MeV
neutron is just 1.3 MeV heavier then proton.
The nucleus is formed by proton and neutron. so, in simple thought, an nucleus with Z proton and ( A-Z ) neutron should have mass
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) = Mass ( A, Z )
where A is the atomic mass number, which is equal the number of nucleons in the nucleus, and Z is the proton number.
However, scientists found that it is not true.
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) > Mass ( A, Z )
Some of the mass is missing! But that is explained why nucleus will not break down automatically. since it need extra energy to break it down.
we called the mass different is Mass Deficit. or Blinding energy.
Mass Deficit = Mass( A, Z) - Mass ( proton + neutron )
some one may think that the blinding energy is the energy for holding the nucleus together. in order to hold the nucleus, some mass was converted into the energy to holding it. this is INCORRECT. the correct argument is, the blinding energy is th energy require to break it down.
think about a simple 2 bodies system, like sun and earth. at far far away, when both of them are at rest, the total energy is Mass( sun) + Mass ( earth ) + Potential energy.
when the earth moves toward to sun, the potential energy converted to the Kinetic energy, so the earth moving faster and faster. but, in order to stay in the orbit, some K.E. must be lost so that it does not have enough ( or the same) energy to run away. Thus, the total energy of the system is lesser then the total mass.
another analogy is electron orbit. when an electron was captured by an atom, it radiate energy in order to stay in some energy level. thus, the total energy of the system again less then the total mass.
any any case, the mass of the sun and earth and electron does not change, but the potential changes to negative, thus it makes to total energy lesser.
similar idea hold for nucleus, but the potential of it are great different, because there are a Coulomb Barrier. Thus, in order to make a nucleus. we have to put so many K.E. to again this barrier, then the resultant nucleus release the Mass Deficit energy and also the input K.E..
 |
| a scratch on the nuclear potential. there are a Coulomb Barrie. ( by wolframalpha.com) |
When the nucleus is radioactive and undergoes decay. this mean, it Mass deficit is positive. thus, it will automatically break down to another nucleus until it mass deficit is negative again. during this process, the emitted particle carry K.E. which is from the potential. Not the mass for one nucleons.
Remember, Mass( nucleus ) = Mass ( protons + neutrons ) + Potential
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.
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 :
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.
Wednesday, January 19, 2011
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.
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.
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.
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.
Tuesday, January 18, 2011
Detector
there are many types of detectors developed. now is mostly electronic and able to collect huge amount of data.
there are 3 main principles on particle detection.
here is a diagram from WIKI, which summarized
there are 3 main principles on particle detection.
- by the ionization between particle and gas
- by the excitation of energy level
- by the detection of photon
i will talk more about them when i feel i understand. :)
here is a diagram from WIKI, which summarized
Friday, January 14, 2011
Type of accelerator (Ring type)
Ring type accelerator solved the difficulties by linear type.
- the particle circulate inside the accelerator, so, it can be accelerate infinite time in principle.
- The space require is smaller compare with similar energy output Linac.
However, there is a draw back is, for charged particle running in a circular path, it will radiate energy by EM wave due to the centripetal acceleration, thus, even it is just running in constant speed, it will radiate and energy lost. this is called synchrotron radiation.
There are mainly 2 types of ring accelerators, 1) cyclotron, 2) synchrotron.
Cyclotron
cyclotron is the simplest type, it has 2 D shape cavities and the 2 D shape formed a circle. the 2 D shape cavities is under a magnetic field to blend the particle. and the 2 Ds have different electric potential. when a particle pass from 1 D to the others, due to the potential different, it will be accelerate.
as you can imagine, the potential of the Ds has to be oscillating so that the particle is accelerated when passing each gap between the 2Ds. That frequency is called cyclotron frequency. and it also reflects the particle circulating frequency. surprisingly, the cyclotron frequency only depends on the magnetic field strength, the charge and the mass of the particle.
f = (B q) / (2 pi m)
which means, no matter the particle position, it moves in same frequency. Thus, the outer particle move faster then the inner one.
now a day, cyclotron may not just contains 2 D cavities but any 2 pi/n cavities. where n is number of cavities. thus, 2Ds is also called pi - cavity.
The typical speed it can reach is about 10% of speed of light.
The only draw back is, the energy it can reach is limited, if using fixed B field or E field, due to relativistic effect. (i.e. the cyclotron frequency also depend on the speed ) the particle cannot match the frequency and accelerated, after it goes to relativistic speed.
another factor is the B field strength is limited, even using super conductive magnetic. and the limit of B field, limited the max output.
particle is released at the center of the B-field and go outward as it acquire speed. Thus, the limitation of radius also limited the max speed. and also, a large radius means a large B field area, which raise a problem on uniform on the B-field.
So, there are another type of cyclotron, which changing the B field or E field to cope with the changing frequency. such cyclotron is called Synchrocyclotron. but due to the velocity dependent of the frequency, only certain speed of particle can be accelerate, thus, the intensity of the beam is smaller then cyclotron.
Synchrotron
synchrotron can reach a great energy and accelerate particle very close to speed of light.
it uses a lot beam focusing devices and accelerating devices to accelerate the beam in a very large radius. each device is well tuned, and all devices are well synchronized for different particle. Thus it is a very delicates and sophisticated machine.
- particle can have every high energy
- high intensity of beam
- it can have some section only for linear motion with accelerate.
- it is not limited by the B field. since the narrow of the beam, a higher forcing B field can be applied.
The only factor reduced the power output is the synchrotron radiation.
particles identification
after a bombardment in accelerator, there are many fragments, different kind of particles produced. thus, the identification of particles (PID) is essential.
there are 3 main principles in PID.
there are 3 main principles in PID.
- decay mood
- time of flight
- energy loss to surrounding
Sunday, January 9, 2011
Informations we can extract
in scattering experiment, the raw informations we can know or observe are only 2 things:
- the number of particles counted at particular solid angle. ( when you have a unit sphere, the area on the surface is called solid angle)
- The polarization (spin)
- charge
- energy
- momentum (time of flight)
Since the number of particles counted is related to the intensity of the incident beam, the density of the target, the interaction and the differential cross section.
on the other hand, the number of particles counted should be related to intensity of incident beam, density of the target and interaction potential. Thus, the differential cross section is related to the interaction potential.
The polarization can be measured by 2nd scattering of known polarization target. or directly from a polarized primary target.
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.
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.
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.
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.
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.
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.
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.
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.
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 = m 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.
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 = m 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.
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 |
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