Magnetic Moments

Outline

Here is a sketch for making some experimentally testable predictions.

Magnetic Susceptibility

The magnetic susceptibility is an indication of how much a particle is influenced by surrounding magnetic fields. It is a dimensionless constant noted by $\chi_{m}$ . In an extension of the assumption of conjugate symmetry we presume that ordinary-quarks and anti-quarks have the same susceptibilities. Then the quark index $\zeta$ can specify values of $\chi_{m}$ as shown in the adjacent table. These numbers are obtained by analyzing laboratory observations of nuclear particles.

Magnetic Susceptibility

𝜁	Z	$\chi_{m} \rule[-6px]{0.1pt}{0.1pt}$
1	U	2.449148
2	D	0.535786
3	E	2.093534
4	G	1.767340
5	M	-0.096763
6	A	0.229434
7	T	1.220846
8	B	0.360208
9	S	-1.953003
10	C	2.433428

Sensory interpretation: The magnetic-susceptibility describes some implied mixing between different classes of sensation. Remember that magnetic fields and muonic quarks are associated with redness. So ultimately, they are objectified from the sight of blood. And recall Ernst Mach’s remark that the perception of a sensation is connected to “dispositions of mind, feelings, and volitions”. So magnetic-susceptibility may be viewed as a mathematical description of how the sight of blood affects other perceptions. This is especially relevant for distinguishing between safe and dangerous conditions.

Induced Charge

The safety of a thermal sensation is represented by its baryonic quarks, and subsequently its charge. But magnetic-susceptibility is a more generalized concept, so we introduce a related quantity called the induced charge which is noted by $\mathtt{q} .$ Sensory imbalances are mathematically described by $\Delta n .$ And any imbalances are broadly associated with risk. So we account for the relationship between seeing blood and danger by defining the induced charge as

$\mathtt{q} \equiv e \chi_{m} \, \Delta n$

The number $e$ is a constant called the elementary charge. It is measured in Coulombs and abbreviated by (C).

Consider some particle P characterized by its period $\hat{\tau}$ . In a Cartesian descriptive framework P is rotating. Then the current $I ,$ due to the rotation of the induced charge $\mathtt{q}$ is given by

$I = \mathtt{q} / \hat{\tau}$

This current is measured in Coulombs per second, or Amperes, and abbreviated by (A). The magnetic moment due to the rotation of any $\zeta$ -type quarks may be defined from the current as

$\overline{\mu}^{\, \zeta} \equiv \mathscr{A} I^{\zeta} \; \widehat{z}$

where $\mathscr{A}$ is P’s cross-sectional area and $\widehat{z} \equiv (0, 0, 1)$ is P’s central axis. The norm of a moment is written without an overline as $\mu \equiv \left\| \, \overline{\mu} \, \right\|$ . By this definition the magnetic-moment is given by the product of a current and an area, so the measurement units used for $\mu$ are abbreviated as (A∙m²). The magnetic-moment of the whole particle P is defined by a sum over quark moments

$\displaystyle \overline{\mu}^{\, \mathsf{P}} \equiv \sum_{\zeta=1}^{10} \overline{\mu}^{\, \zeta}$

All quark moments are aligned with the central-axis, so by these definitions

$\displaystyle \mu^{\, \mathsf{P}} = \sum_{\zeta=1}^{10} \mu^{\, \zeta} = \mathscr{A} \sum_{\zeta=1}^{10} I^{\zeta} = \pi R^{2} \sum_{\zeta=1}^{10} I^{\zeta}$

where $R$ is the orbital radius of P. This radius is given by

$R \equiv \dfrac{hc}{2\pi} \dfrac{ \sqrt{\textsl{\textsf{J}} \; }}{E} = \dfrac{ h \sqrt{\textsl{\textsf{J}} \; }}{2\pi \, mc}$

where $\textsl{\textsf{J}}$ is P’s total angular-momentum quantum number, and $E$ is P’s mechanical energy. We assume that P is stationary or in slow motion so that its energy can be written as $E \! = \! m c^{2}$ where $m$ is P’s rest mass. We may also write the cross-sectional area in terms of these quantities as

$\mathscr{A} \equiv \pi R^{2} = \dfrac{h^{2} \textsl{\textsf{J}} }{4\pi \, m^{2}c^{2}}$

The foregoing equations may all be combined to state the magnetic-moment of P as

$\displaystyle \mu^{\mathsf{P}} = \dfrac{h^{2} \textsl{\textsf{J}} }{4\pi \, m^{2} c^{2}} \sum_{\zeta=1}^{10} I^{\zeta}$

We can also use Planck’s postulate to express the current as

$I^{\zeta} = \dfrac{ \; \mathtt{q}^{\zeta} \; }{\hat{\tau}} = \dfrac{\mathtt{q}^{\zeta} E}{h} = \dfrac{\mathtt{q}^{\zeta} mc^{2}}{h}$

Then the magnetic-moment of P may be written as

$\displaystyle \mu^{\mathsf{P}} = \dfrac{h \textsl{\textsf{J}} }{4\pi \, m} \sum_{\zeta=1}^{10} \mathtt{q}^{\zeta}$

Recall that the induced-charge $\mathtt{q}$ is related to the magnetic-susceptibility $\chi_{m}$ by the definition $\mathtt{q} \equiv e \chi_{m} \, \Delta n$ where $n$ notes P’s quark coefficients and $e$ is a constant called the elementary charge. Then finally we can express the magnetic-moment in terms of quark-coefficients as

$\displaystyle \mu^{\mathsf{P}} = \dfrac{eh}{4\pi} \dfrac{ \textsl{\textsf{J}} }{m} \sum_{\zeta=1}^{10} \chi_{m}^{\zeta} \hspace{0.8px} \Delta n^{\zeta}$

Experimental Comparison

The forgoing expression summarizes all thirteen known nuclear magnetic-moments to within experimental error.¹J. Beringer et al. (Particle Data Group), The Review of Particle Physics, Phys. Rev. D86, 010001, 2012.^, ²J. DiSciacca et al. (ATRAP Collaboration), One-Particle Measurement of the Antiproton Magnetic Moment Phys. Rev. Lett. 110, 130801, 2013.^, ³G. Lopez Castro, A. Mariano, Determination of the Delta++ Magnetic Dipole Moment arXiv:nucl-th/0006031, 2001. The representation uses ten adjustable parameters, i.e. the magnetic-susceptibilities of the ten different types of thermodynamic quarks.

Here is a graph comparing calculated versus observed magnetic-moments for baryons. The electron and muon are far off the scale of this image, but the moments of both particles are within experimental error as well.

A graph of calculated and observed magnetic moments for nuclear particles.

Using 10 parameters to represent 13 observations is a lackluster feat of data compression. But the quark-coefficients of other nuclear particles are already known from mass and lifetime experiments. So this pattern might be used to make predictions for particles that have not yet had their magnetic-moments measured.

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References
1	J. Beringer et al. (Particle Data Group), The Review of Particle Physics, Phys. Rev. D86, 010001, 2012.
2	J. DiSciacca et al. (ATRAP Collaboration), One-Particle Measurement of the Antiproton Magnetic Moment Phys. Rev. Lett. 110, 130801, 2013.
3	G. Lopez Castro, A. Mariano, Determination of the Delta++ Magnetic Dipole Moment arXiv:nucl-th/0006031, 2001.