Stability of a nucleus ( Liquid Drop Model )

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when look at the table of the nuclear world, why there are some nucleus more stable then the other? which mean, why some will decay while some are not?

OK, this basically the ultimate question that nuclear physics want to answer.

so, the very fundamental reason, no one know.

but in the elementary level, or by experimental fact and some assumption. we have Binding Energy to estimate or predict the stability of a nucleus. when the Binding Energy is larger then Zero, it must be unstable and will decay under conservation laws. if it is less then zero, it may be stable or not, it depends on whether it reach the bottom of energy level.

Binding Energy can also be though as the energy required to break the nucleus.

In liquid drop model, we imagine the nucleus is like a liquid. and nucleons inside just like liquid molecules. experiments show that nucleus is a spherical object. and it density is a constant. and the interaction range of nuclear force is short, few fm. thus, it likes a incompressible liquid drop. the radius of it is related to the mass number:

R^3 = A

the Binding Energy ( = \Delta M(A,Z,N) = mass deficit) is given by theoretical assumption and experimental fact.

\Delta M(A,Z,N) = - a_1 A + a_2 A^{\frac {2}{3} } + a_3 Z^2 / A^{ \frac{1}{3}} + a_4 (N-Z)^2 /A \pm a_5 A^{- \frac{3}{4} }

the first 3 terms are theoretical assumption and the lat 2 terms are from experimental fact. All coefficients are given by experimental measurement.

The first term is the “volume energy” by the nuclear force, which is proportional to the number of nucleons.

the 2nd term is the “surface tension” from the “liquid”. we can see its dimension is area.  (why this term is + ? ) it  explained why smaller nucleus has less Binding energy.

the 3rd term is the Coulomb potential term.

the 4th term is the balance term.  if the number of neutron and proton is no balance,

the 5th term is the “Symmetry term“. for even-even of neutron and proton number, the nucleus is more stable, thus, we choose minus sign for it. for odd-odd combination, nucleus are more unstable, thus, plus sign for it. for other, like ood – even or even-odd combination, this term is zero.

the value of the coefficients are:

a_1 \simeq 15.6 MeV

a_2 \simeq 16.8 MeV

a_3 \simeq 0.72 MeV

a_4 \simeq 23.3 MeV

a_5 \simeq 34 MeV

The below plot is the Binding Energy per nucleon in  Z against N.

Lets use the liquid drop model and Binding Energy to look the β-decay. the β-decay conserved the mass number A. there are 2 β-decays.

\beta_- : n \rightarrow p + e^- + \bar{\nu_e}

\beta_+ : p \rightarrow n + e^+ + \nu_e

so, the β+ decay decrease the number of proton while β– decay increase the number of proton.

The below diagram show the β-decay for A = 22. we can see the 22Ne is stable, since no more β-decay can help to reach a lower energy level.

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2p-2p decay of 8C and isospin-allowed 2p decay of the isobaric-analog state in 8B

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DOI: 10.1103/PhysRevC.82.041304

this paper reports another 2 protons decay mode in 8C. They also discover an “enhancement” at small relative energy of 2 protons. They also reported that an isobaric analog state, 8C and 8B, have same 2-protons decay, which is not known before.

the 1st paragraph is a background and introduction. 2 protons decay is rare. lightest nucleus is 6Be and heaviest and well-studied is 45Fe. the decay time constant can be vary over 18 orders and the decay can be well treated by 3-body theory.

the 2nd paragraph describes the decay channel of 8C and 8B. it uses the Q-value to explain why the 2-protons decay is possible. it is because the 1-proton decay has negative binding energy, thus, it require external energy to make it decay. while 2-protons decay has positive binding energy, thus, the decay will automatic happen in order to bring the nucleus into lower energy state. it also consider the isospin, since the particle decay is govt by strong nuclear force, thus the isospin must be conserved. and this forbid of 1-proton decay.

it explains further on the concept of 2-protons decay and 2 1-protons decay. it argues that, in the 8C, the 2-protons decay is very short time, that is reflected on the large energy width, make the concept of 2 1-protons decay is a unmeasurable concept. however, for the 8Be, the life time is 7 zs (zepto-second 10^{-21} ), the 8Be moved 100 fm ( femto-meter $latex 10^{-15} ), and this length can be detected and separate the 4-protons emission in to 2 2-protons decay.

the 3rd paragraph explain the experiment apparatus – detector.

the 4th paragraph explains the excitation energy spectrum for 8C, 6Be.

the 5th and 6th paragraphs explain the excitation energy spectrum for the 6Be form 8C decay. since the 2 steps 2 – protons decay has 4 protons. the identification for the correct pair of the decay is important. they compare the energy spectrum for 8C , 6Be and 6Be from decay to do so.

the 7th paragraph tells that they anaylsis the system of 2-protons and the remaining daughter particle, by moving to center of mass frame ( actually is center of momentum frame ) and using Jacobi T coordinate system, to simplify the analysis. the Jacobi T coordinate is nothing but treating the 2-protons the 2 protons are on the arm of the T, and the daughter particle is on the foot of the T.

alpha decay

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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

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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.

Parity

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It is Pa-ri-ty, not Par-ty.

( it needs to clean up)

Parity is just a reflection on every space dimension.

\begin {pmatrix} x \\ y \\ z \end {pmatrix} \rightarrow \begin{pmatrix} - x \\ - y \\-z \end {pmatrix}

General

This is just a mirror reflection, although mirror reflection only reflects on 1 dimension, the dimension that perpendicular to the mirror surface.

May be we start on 2-D space instead of 3-D, draw a F and flips it upside down, and left-side right. then, you have a F just rotated 180 degree, not a reflection. however, in 3-D, then there is something different.

The parity transform is taking everything reverted. For example, when you stand up, your arms place horizontal and you left arm points forward and your right arm points right. After a parity transform. You right arm point left. Your left arm point backward, and you are standing on the ceiling, upside down. The result is a mirror image of your self. If we rotate the reverted-self from the ceiling to the ground.

Thus, parity also related as mirror reflection. In physics, we like to call the right-hand system (RHS) or left-hand system (LHS).

A simple RHS and LHS are on your hands! Although our left hand and right hand has some minor different, in general, they are the mirror image of each other. And the great interesting thing is, your left hand cannot overlap the right hand. They are equal but not the same.

Another thing is spring, when a wire is rolled clockwise and going upward, it form a left-hand spring and vice aver. Thus 2 springs are not the same.

For those which keep function as before parity transform, we called it parity positive, for those who are not, we called it parity negative.

Be reminded that the chiral material that interact circularly polarized light different still the parity positive. For example, a material which only let right hand light passes through, but not let the left hand pass. After parity transform, it lets left hand light pass through but not right hand .Thus, the left hand and right hand are work equally well!

We also cannot say our left hand is more weak then our right hand, then we called it parity negative. It is because, if we reflected ourself, our left hand is as good and right hand and the right hand is as weak as left hand.

A more physical example is the polarization of light, there are lefthand rotating light and right hand rotating light, called circular polarization. And material which interact differenty with different circular polarization are called chiral material. We should stop talking about examples in here. Because in nature, there are so many things has chiral property. Never the less, potenient and drug also has chirality. One book I recommend on general science for the chirality is “right-hand, left-hand” by chris McManus.

Physics

Physics encounters parity is because we believe if the whole world is reverted, every thing just work fine and the same. For example, if our orgasms are all reflected, left goes to right, right go to left. We still alive. In fact, there are some real cases, that some peole do have reverted orgasm. Because there should be symmetric in the world.

In normal day, parity positive never break. It is seem impossible to break. How coome some thing work differently under parity transform?

For position, linear momentum, parity just make them change

However, in mathematics, there are many parity negative things. One example is the spherical harmonic. It is can be parity positive and negative depends on the parameter.

Lets take a imaginary example in parity negative. If we use photon to hit a target, all photons are going left. Now, we reflet the whole system. But now, the photons are still going left.

The first discovery of parity negative is on beta- decay from Co-60. Whe. Applied an external magnetic field from down to up, the beta particle come out at left. When we change the magnetic field, now is from up to down, the beta particle should come out at right, if parity is positive. But it is not, it still keep coming out left!

The reason of it is beyond my understanding… Sorry.

decay time constant and line width

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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

f(t) = Exp(-R t) Exp ( i \omega_0 t)

where the R is decay constant, and ω0 is the wave frequency.

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

F(t) = \frac {1} { R + i ( \omega_0 - \omega )}

the real part is

Re(F(t)) = \frac {R} { R^2 + ( \omega_0 - \omega )^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

Im(F(t)) = \frac {\omega_0 - \omega}{R^2 + ( \omega_0 -\omega )^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.

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another view of this relation is from the quantum mechanics.

the solution of Schroedinger equation is

\Psi (x,t) = \phi(x) Exp \left( - i \frac {E}{\hbar} t \right)

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 = E_0 - \frac {i} {2} R \hbar

( 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 \frac {E}{\hbar} t ) Exp( - \frac {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 = \frac {Const.}{R^2 +(\omega_0 - \omega )^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.

decay

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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-Lifet_\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.