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Radii

Radii and radius vectors are used to express spatial concepts like extension and containment. For example, consider a particle P characterized by some repetitive chain of events

\Psi ^{\mathsf{P}} = \left(     \mathsf{\Omega}_{1} ,     \mathsf{\Omega}_{2} ,     \mathsf{\Omega}_{3}  \;   \ldots  \;       \right)

where each orbital cycle is a bundle of quarks

\mathsf{\Omega} = \left(    \mathsf{q}_{1},       \mathsf{q}_{2}   \;   \ldots   \; \mathsf{q}_{N}\right)

that are described by their quark coefficients  n and internal energies  U. Recall that  H_{chem} notes the enthalpy of the chemical quarks in P. And let a constant  k_{\mathsf{F}} have a positive value, with units of a force. These quantities are used to describe the shape and extent of P as follows. The chemical radius is

\rho_{chem}   \equiv  \dfrac{H_{chem}}{k_{\mathsf{F}}}

The magnetic radius of P is

\rho_{m} \equiv       \dfrac{ {\Delta}n^{  \mathsf{A}}  U^{  \mathsf{A}} - {\Delta}n^{ \mathsf{M}}   U^{  \mathsf{M}}    }{ k_{\mathsf{F}}  }

the electric radius is defined by

\rho_{e} \equiv       \dfrac{ {\Delta}n^{  \mathsf{G}}  U^{   \mathsf{G}} - {\Delta}n^{  \mathsf{E}}   U^{   \mathsf{E}}   }{ k_{\mathsf{F}}  }

and the polar radius of P is given as

\rho_{z} \equiv    \,  \rho_{chem}   +  \dfrac{{\Delta}n^{\mathsf{U}} U^{\mathsf{U}} - {\Delta}n^{\mathsf{D}}  U^{\mathsf{D}}}{k_{\mathsf{F}}}

An ordered set of these radii defines the radius vector of particle P as

\overline{\rho} \equiv  \left(   \rho_{m}, \rho_{e}, \rho_{z} \right)

The three components of  \overline{\rho} are conserved because they are defined from sums of quark coefficients, and quarks are conserved. So if some generic particles  \mathbb{X},  \mathbb{Y} and  \mathbb{Z} interact like \mathbb{ X } + \mathbb{ Y } \leftrightarrow \mathbb{ Z } then by the associative properties of addition, radii are related as

\overline{\rho}^{ \mathbb{X}} + \overline{\rho}^{\mathbb{Y}}  = \overline{\rho}^{\mathbb{Z}}

Also  \Delta n ^{\mathsf{Z}} \left( \mathsf{P} \right) = - \Delta n^{\mathsf{Z}} \left( \mathsf{\overline{P}} \right) and  H \left( \mathsf{P} \right) = - H \left( \mathsf{\overline{P}} \right). So particles and anti-particles have symmetrically opposed radius vectors

\overline{\rho}  \left( \mathsf{P} \right) = - \overline{\rho}  \left(  \mathsf{\overline{P}} \right)

Going forward, we use these radii and radius-vectors to develop a description of particle shape. As a starting example; if P has the same radius vector for every cycle, then we call it a rigid particle.

Radii and Range

Some particles are not very solid or exactly located. Then an analysis of shape might not provide a useful description. So instead we may use the following radii to describe their containment or range. The inner radius is defined as

\rho_{in}   \equiv   \dfrac{ \left| \Delta n^{\mathsf{D}} \right| }{8}       \sqrt{  \dfrac{hc}{2\pi k_{\mathsf{F}}}  \vphantom{ \dfrac{hc}{2\pi k_{\mathsf{F}}}  ^{X}} }

And the outer radius of P is

\rho_{out}   \equiv   \dfrac{N^{\mathsf{D}}}{8}       \sqrt{ \dfrac{hc}{2\pi k_{\mathsf{F}}}  \vphantom{ \dfrac{hc}{2\pi k_{\mathsf{F}}}    ^{X} } }

We say that a particle is free when  \left| \Delta n^{\mathsf{D}}  \right| \gtrsim 8 and  N^{\mathsf{D}}  \to \infty. That is, when the inner radius is small and the outer radius is big. But if  \left| \Delta n^{\mathsf{D}}  \right| < 8 then we say that P is a core particle.

Sensory Interpretation

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Here is a sensory explanation of the radius vector. In these formulae  \Delta n means that contributions from sensations on the right side are cancelled by sensations felt on the left. The radius vector depends on their net magnitude. Coefficients of baryonic quarks do not appear in these definitions, only dynamic quarks. So the radius does not depend on thermal sensation, only somatic and visual sensations. Also the null value for energy is referred to down-quarks which are objectified from black sensations. So overall, the radius vector is interpreted as a description of size for visual sensations, relative to black sensations, net right from left.

Next we look at how to use these radius vectors to make a mathematical vector space for quark models.

Quark models introduce spatial concepts like radii and containment, somewhat like this rattan basket from Borneo.
Ajat basket, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Hornbill motif. Photograph by D Dunlop.