Chapter 5.4

Chapter 5 Quark Model

Chapter 55.1 Introduction5.2 Quark Model5.3 Meson and Baryon wave function5.4 Magnetic moment and masses of

baryons5.5 Interactive Exercise

5.4 Magnetic moment and masses of baryons

Dayalbagh Educational Institute 4

Quark Model Particle Physics

Magnetic moment and masses of baryonsMagnetic moment of Atom

For an atom, individual electron spins are added to get a total spin, and individual orbital angular momentum are added to get a total orbital angular momentum. These two then are added using angular momentum coupling to get a total angular momentum. The magnitude of the atomic dipole moment is then:

m ATOM = gJ μB √J(J + 1)

Where J is the total angular momentum quantum number, gJ is the Landé g-factor,, and μB is the Bohr magneton. The component of this magnetic moment along the direction of the magnetic field is then:

m Atom(z) = − mgJμB

Where m is called the magnetic quantum number or the equatorial quantum number, which can take on any of 2J + 1 values: − J, − (J − 1), , (J − 1), J

Magnetic moment of an electron

Electrons and many elementary particles also have intrinsic magnetic moments, an explanation of which requires a quantum mechanical treatment and relates to the intrinsic angular momentum of the particles.

Fig 1: Magnetic moment of electron



It is these intrinsic magnetic moments that give rise to the macroscopic effects of magnetism, and other phenomena, such as electron paramagnetic resonance. The magnetic moment of the electron is ms = -gs μB S/ћ

Where μB is the Bohr magneton, S is electron spin, and the electron g-factor is

gS = 2 in Dirac mechanics.

Again it is important to notice that m is a negative constant multiplied by the spin, so the magnetic moment of the electron is antiparallel to the spin.

This can be understood with the following classical picture: if we imagine that the spin angular momentum is created by the electron mass spinning around some axis, the electric current that this rotation creates circulates in the opposite direction, because of the negative charge of the electron; such current loops produce a magnetic moment which is antiparallel to the spin. Hence, for a positron (the anti-particle of the electron) the magnetic moment is parallel to its spin.

Magnetic moment of a nucleusThe nuclear system is a complex physical system consisting of nucleons, i.e., protons and neutrons. The quantum mechanical properties of the nucleons include the spin among others. Since the electromagnetic moments of the nucleus depend on the spin of the individual nucleons, one can look at these properties with measurements of nuclear moments, and more specifically the nuclear magnetic dipole moment.



Most common nuclei exist in their ground state, although nuclei of some isotopes have long-lived excited states.

Each energy state of a nucleus of a given isotope is characterized by a well-defined magnetic dipole moment, the magnitude of which is a fixed number, often measured experimentally to a great precision.

This number is very sensitive to the individual contributions from nucleons, and a measurement or prediction of its value can reveal important information about the content of the nuclear wave function.

The magnetic moments of baryons is calculated in unit of nuclear magneton. So, it is important to discuss nuclear magneton before calculating magnetic moments of baryons.

The nuclear magneton (symbol μN), is a physical constant of magnetic moment, defined by:

μN =

Where e is the elementary charge, ħ is the reduced Planck constant, mp is the proton rest mass.

In SI units, its value is approximately:

μN = 5.05078324(13)×10−27 J/T

The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei.



On the contrary, the dipole moment of the electron, which is much higher as a consequence of much higher charge-to-mass ratio, is usually expressed in corresponding units of the Bohr magneton.

In absence of orbital motion, the net magnetic moment is vector sum of the magnetic moments of three constituent quarks. If we have three quark with μ1, μ2 and μ3 as magnetic moment of 1st, 2nd and 3rd quark respectively. Then magnetic moment of particle is:

μ = μ1 + μ2 + μ3

Magnetic moment μ depends on quark’s flavors as three flavors carry different magnetic moments and on spin configuration as it determines the relative orientations. Magnetic moment of a spin- point particle of charge q and mass m is

μ =

Magnitude of magnetic moment for spin up s =

Magnetic moment formula for up quark (μu), down quark (μd) and strange quark (μs) is:

μu = μd = μs =

The Magnetic moment of baryon B is

μB = =

Where μB is the magnetic moment, B is baryon wave ↑function, μi is operator for magnetic moment of quark, SiZ is Pauli’s spin operator and i runs for up, down and strange quarks. By using this formula we can calculate the magnetic moment of baryon octet.



Magnetic moment of proton

Table 1: Magnetic moment of proton



Magnetic moment of neutron

Table 2: Magnetic moment of neutron



Magnetic moment of cascade minus

Table 2: Magnetic moment of cascade minus particle



Magnetic moment of sigma plus

Table 3: Magnetic moment of cascade plus particle



Magnetic moment of cascade minus

Table 4: Magnetic moment of cascade minus particle



Magnetic moment of cascade zero

Table 5: Magnetic moment of cascade zero particle



Magnetic moment of sigma zero

Table 6: magnetic moment of sigma zero particle



Magnetic moment of lambda

Table 7: Magnetic moment of lambda particle

In above table magnetic moment of baryon octet is calculated by using magnetic moment formula.



Table 8: Numerical value of baron octets

By calculating the values for μu, μd and μs and putting in the magnetic moment formula, numerical value of magnetic moment can be calculated by comparing it with experimental value given the error can be calculated as follows:



Masses of Baryon: octet and decupletQuarks only exist inside hadrons because they are confined by the strong (or color charge) force fields. Therefore, we cannot measure their mass by isolating them.

Furthermore, the mass of hadrons gets contributions from quark kinetic energy and from potential energy due to strong interactions.

For hadrons made of the light quark types, the quark mass is a small contribution to the total hadrons mass.

For example, compare the mass of a proton (0.938 GeV/c2) to the sum of the masses of two up quarks and one down quark (total of 0.02 GeV/c2) . So the question is, what do we mean by the mass of a quark and how do we measure it.

The quantity we call quark mass is actually related to the min F = ma (force = mass x acceleration). This equation tells us how an object will behave when a force is applied.

The equations of particle physics include, for example, calculations of what happens to a quark when struck by a high energy photon.

The parameter we call quark mass controls its acceleration when a force is applied.

It is fixed to give the best match between theory and experiment both for the ratio of masses of various hadrons and for the behavior of quarks in high energy experiments.

http://www2.slac.stanford.edu/vvc/theory/colorchrg.html%22%20%5Cl%20%22Confinement



However, neither of these methods can precisely determine quark masses.

If SU (3) were a perfect symmetry, all particles in a given supermultiplets would have the same mass. But it is not so. First it was attributed to the fact that s quark is more massive than u and d quark.

But it can’t be the only reason otherwise Λ would have same mass as Σ’s, and the Δ’s mass would match proton. Evidently, there is a spin-spin contribution, which is directly proportional to the dot product of spins and inversely proportional to the dot product of masses

M (baryon) = m1 + m2 + m3 +

Here, A' is a constant.

The spin product is easiest when the three quark masses are equal, for

J2 = (S1 + S2 + S3)2 = + 2(S1. S2 + S1. S3 + S2. S3)

And hence

S1. S2 + S1. S3 + S2. S3 =

=

In case of the decuplet the spins are all parallel, so

(S1 + S2)2 = + 2 S1 . S2 =

Hence for decuplet S1 . S2 = S1. S3 = S2. S3 =

Mass of Σ and Λ particle can be done by taking that up and down quarks combine to make isospin 1 0r 0 respectively.



For the spin/ flavor wave function to be symmetric under interchange of u and d quark, the spin must therefore combine to give a total of 1 or 0, respectively.

For Σ’s, (Su + Sd)2 = + 2 Su. Sd =

So, Su. Sd =

For Λ, (Su + Sd)2 =0 or Su. Sd =

Mass of proton or neutron

Table 9: Mass of neutron or proton



Mass of delta particle, Δ++ and Δ-

Table 10: Mass of delta plus plus or delta minus particle



Mass of omega particle, Ω-

Table 11: Mass of omega minus particle



Mass of sigma star particle

Table 12: Mass of sigma star plus, sigma star minus and sigma star zero particle



Mass of cascade particle

Table 13: Mass of cascade star minus or cascade star zero particle



Mass of sigma particle

Table 14: Mass of sigma plus, sigma minus or sigma zero particle



Mass of lambda particle

Table 15: Mass of lambda particle



Mass of cascade particle

Table 16: Mass of cascade minus and cascade zero particle



Comparison of confined and constituent mass in MeV/c2

From this table it is concluded that error for constituent mass is less means constituent mass (mu,d= 363 MeV/c2 and ms = 538 MeV/c2) is more close to the experimentally calculated mass.

Error for confined mass (mu,d = 336 MeV/c2 and ms = 510 MeV/c2) is more which means confined mass is not much close to experimentally calculated mass.

Table 17: Comparison of constituent and confined mass

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Chapter 5.4