Calculate the magnetic moment of spin half particle
Answers
Answered by
0
Spin" is a non-classical property of elementary particles, since classically the "spin angular momentum" of a material object is really just the total orbital angular momentaof the object's constituents about the rotation axis. Elementary particles are conceived as concepts which have no axis to "spin" around
Answered by
0
We can calculate the observable spin magnetic moment, a vector, μ→S, for a sub-atomic particle with charge q, mass m, and spin angular momentum (also a vector), S→, via:[n 1]
{\displaystyle {\vec {\mu }}_{\text{S}}\ =\ g\ {\frac {q}{2m}}\ {\vec {S}}=\gamma {\vec {S}}}
(1)
where {\displaystyle \gamma \equiv g{\frac {q}{2m}}} is the gyromagnetic ratio, g is a dimensionless number, called the g-factor, q is the charge, and m is the mass. The g-factor depends on the particle: it is g = −2.0023 for the electron, g = 5.586 for the proton, and g = −3.826 for the neutron. The proton and neutron are composed of quarks, which have a non-zero charge and a spin of ħ/2, and this must be taken into account when calculating their g-factors. Even though the neutron has a charge q = 0, its quarks give it a magnetic moment. The proton and electron's spin magnetic moments can be calculated by setting q = +e and q = −e, respectively, where e is the elementary charge.
The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magneton μB because g ≈ −2 and the electron's spin is also ħ/2:
{\displaystyle \mu _{\text{S}}\approx -2{\frac {-e}{2m_{\text{e}}}}{\frac {\hbar }{2}}=\mu _{\text{B}}}
(2)
Equation (1) is therefore normally written as[n 2]
{\displaystyle {\vec {\mu }}_{\text{S}}=-{\frac {g\mu _{\text{B}}{\vec {\sigma }}}{2}}}
(3)
Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured. Equations (1), (2), (3) give the physical observable, that component of the magnetic moment measured along an axis, relative to or along the applied field direction. Assuming a Cartesian coordinate system, conventionally, the z-axis is chosen but the observable values of the component of spin angular momentum along all three axes are each ±ħ/2. However, in order to obtain the magnitude of the total spin angular momentum, S→ be replaced by its eigenvalue, √s(s + 1), where s is the spin quantum number. In turn, calculation of the magnitude of the total spin magnetic moment requires that (3) be replaced by:
{\displaystyle |{\vec {\mu }}_{\text{S}}|=g\mu _{\text{B}}{\sqrt {s(s+1)}}}
(4)
Thus, for a single electron, with spin quantum number s = 1/2, the component of the magnetic moment along the field direction is, from (3), |μ→S,z| = μB, while the (magnitude of the) total spin magnetic moment is, from (4), |μ→S| = √3 μB, or approximately 1.73 Bohr magnetons.
The analysis is readily extended to the spin-only magnetic moment of an atom. For example, the total spin magnetic moment (sometimes referred to as the effective magnetic moment when the orbital moment contribution to the total magnetic moment is neglected) of a transition metal ion with a single d shell electron outside of closed shells (e.g. Titanium Ti3+) is 1.73 μB since s = 1/2, while an atom with two unpaired electrons (e.g. Vanadium V3+) with S = 1 would have an effective magnetic moment of 2.83 μB.
{\displaystyle {\vec {\mu }}_{\text{S}}\ =\ g\ {\frac {q}{2m}}\ {\vec {S}}=\gamma {\vec {S}}}
(1)
where {\displaystyle \gamma \equiv g{\frac {q}{2m}}} is the gyromagnetic ratio, g is a dimensionless number, called the g-factor, q is the charge, and m is the mass. The g-factor depends on the particle: it is g = −2.0023 for the electron, g = 5.586 for the proton, and g = −3.826 for the neutron. The proton and neutron are composed of quarks, which have a non-zero charge and a spin of ħ/2, and this must be taken into account when calculating their g-factors. Even though the neutron has a charge q = 0, its quarks give it a magnetic moment. The proton and electron's spin magnetic moments can be calculated by setting q = +e and q = −e, respectively, where e is the elementary charge.
The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magneton μB because g ≈ −2 and the electron's spin is also ħ/2:
{\displaystyle \mu _{\text{S}}\approx -2{\frac {-e}{2m_{\text{e}}}}{\frac {\hbar }{2}}=\mu _{\text{B}}}
(2)
Equation (1) is therefore normally written as[n 2]
{\displaystyle {\vec {\mu }}_{\text{S}}=-{\frac {g\mu _{\text{B}}{\vec {\sigma }}}{2}}}
(3)
Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured. Equations (1), (2), (3) give the physical observable, that component of the magnetic moment measured along an axis, relative to or along the applied field direction. Assuming a Cartesian coordinate system, conventionally, the z-axis is chosen but the observable values of the component of spin angular momentum along all three axes are each ±ħ/2. However, in order to obtain the magnitude of the total spin angular momentum, S→ be replaced by its eigenvalue, √s(s + 1), where s is the spin quantum number. In turn, calculation of the magnitude of the total spin magnetic moment requires that (3) be replaced by:
{\displaystyle |{\vec {\mu }}_{\text{S}}|=g\mu _{\text{B}}{\sqrt {s(s+1)}}}
(4)
Thus, for a single electron, with spin quantum number s = 1/2, the component of the magnetic moment along the field direction is, from (3), |μ→S,z| = μB, while the (magnitude of the) total spin magnetic moment is, from (4), |μ→S| = √3 μB, or approximately 1.73 Bohr magnetons.
The analysis is readily extended to the spin-only magnetic moment of an atom. For example, the total spin magnetic moment (sometimes referred to as the effective magnetic moment when the orbital moment contribution to the total magnetic moment is neglected) of a transition metal ion with a single d shell electron outside of closed shells (e.g. Titanium Ti3+) is 1.73 μB since s = 1/2, while an atom with two unpaired electrons (e.g. Vanadium V3+) with S = 1 would have an effective magnetic moment of 2.83 μB.
sumaylamansuri:
girl
9891559182
call
Similar questions