Mass Density (a working hypothesis)

As demonstrated in Fig 1, matter density and gas-transition temperature are related. This is because they are both dependent upon the same inter-atomic forces (Fig 2; Fₑ & Fₘ).

Adjacent atomic elements are attracted to each other by the magnetic field force generated by their proton-electron pairs.

Adjacent atomic elements are forced apart by the electrical charge force in their nucleic protons, which varies linearly with temperature.

The neutronic ratio of each element determines its nucleic structural (lattice) arrangement, which is replicated in the structural arrangement of elemental atoms in matter, in both gaseous and viscous states.

The state of matter is defined thus (Fig 3);
Fₑ < Fₘ: matter will be viscous (liquid or solid),
Fₑ = Fₘ: viscous-gas-transition,
Fₑ > Fₘ: matter will be gaseous.

Whilst the density of viscous matter is [theoretically] constant, electron shells vary inversely with temperature, decreasing the attraction between adjacent atoms as temperature rises. When the combination of increasing repulsive electrical [charge] force, together with the reduction of electron sharing, exceeds the elemental magnetic force, matter will exist as a gas.

Inter-Atomic Forces

Subscripts:
'ₘ' = magnetic
'ₑ' = electrical or electron
'₁' = innermost shell; shell-1
'ₛ' = outermost shell

The magnetic field force (Fₘ) generated by an atom's proton-electron pairs holds adjacent atoms together. Its magnitude is constant and calculated thus:
Fₘ = μₒ . Iₛ² . (2π)² . ξₘ/Y . (ζ.Rₛ/d)³
Fₘ = hₑ².mₚ/Y . (ζ/d)³
Both of which give exactly the same value for any element at any temperature.
Whilst the density of all elements remains constant at any temperature, such a claim is theoretical. In actuality, density falls very gradually with increasing temperature and/or accompanying lattice changes.

The electrical force of repulsion between adjacent atoms varies linearly with temperature. Its magnitude is calculated thus:
Fₑ = p.d²
Fₑ = PE₁ / Y.d
Fₑ = μ₁.I₁².(2π)² . R₁/d / Y
Fₑ = k.e'₁² / (d.Rₙ.F_T)
All of which give exactly the same value for any element at any temperature.

Where:
Iₛ = electrical current in the proton-electron pair
Rₛ = electron orbital radius
d is the average inter-atomic spacing (³√[mₐ/ρ])
ζ = lattice factor
p = internal pressure (PVRT)
μₛ = mₑ.Rₛ/e²
R₁ = electron orbital radius
I₁ = electrical current
e'₁ = proton electrical charge
Ṯ₁ = proton-electrical pair temperature
Ṯₖ = temperature constant (Ṯₙ / Y.ξₘ = 19.4336001664707 K)
F_T = Ṯ₁/Ṯₖ

Lattice Structure

The lattice structure of elemental matter defines its density and its gas transition temperature. It is specified for each element by name, which represents a 3-D arrangement of its atoms. However, this same arrangement originates in an element's nucleus, and is replicated in elemental matter in both viscous and gaseous states.

Lattice factor 'ζ' applies @ Ṯ_g when Fₘ = Fₑ:
1 = Fₑ/Fₘ = (k.e'₁² / (d.Rₙ.F_T)) / (hₑ².mₚ / Y.d³) / ζ³
ζ = ³√[ (k.e'₁² / (d.Rₙ.F_T)) / (hₑ².mₚ / Y.d³) ]
ζ = ³√[ (Y.k.Ṯₖ) / (Rₙ.hₑ².mₚ) . (mₚ.RC.Ṯ_g/Ṯₙ)² . d²/Ṯ_g ]
ζ = ³√[ (d/Rₙ)² . Ṯ_g/Ṯₙ / ξₘ ]

ζ³ = (d/Rₙ)² . Ṯ_g/Ṯₙ / ξₘ
d² = ζ³.Rₙ².Ṯₙ.ξₘ / Ṯ_g
d = Rₙ. √[ Ṯₙ.ξₘ . ζ³/Ṯ_g ]

ρ = mₐ / [ (ζ³ . hₑ².mₚ.Ṯ_g.Rₙ) / (Y.k.e'₁².Ṯₖ) ]¹˙⁵
ρ = mₐ / [ (ζ³ . hₑ².mₚ.Ṯ_g.Rₙ) / (Y.k.(Ṯ₁ . ξₘ.e/Ṯₙ)².Ṯₖ) ]¹˙⁵
ρ = mₐ / [ (ζ³.Ṯ_g . hₑ².mₚ.Rₙ.Ṯₙ²) / (Ṯ_g² . Y.k.ξₘ².e².Ṯₖ) ]¹˙⁵

Optimise the constants in the last formula
Constant Λ = (hₑ².mₚ.Rₙ.Ṯₙ²) / (Y.k.ξₘ².e².Ṯₖ) = ξₘ.Rₙ².Ṯₙ = 9.0882370009374E-18 {m².K}

The relationship between matter density and gas transition temperature is therefore:
ρ = mₐ / [ Λ . ζ³/Ṯ_g ]¹˙⁵
Ṯ_g = (ρ/mₐ)^⅔ . Λ.ζ³

Whilst there is an unmistakeable relationship between the documented lattice structures (some of which vary with source) as is evidenced in Fig 3, the actual (calculated) values reveal that they are simplistic; every lattice structure (ζ) is different.

The numerical values for each documented lattice structure, which define the elemental atoms are listed below:
bcc; 2.5
cubic; 3.25
cubic+2; 3.5
fcc; 2.45
hcp; 2.4
hcp+3; 2.5
monoclinic; 2.75
orthogonal; 2.5
rhombic; 2.75
rhombic+2; 3
tetra; 2.75

Conclusion

The average lattice factor is 2.58190719940102, and the range is; 0.35804463637539 (helium) < ζ < 3.6138801028218 (thorium)
which means that ???

As can be seen in Figs 1 to 3, there is a definite pattern to mass density, however,
you will notice in Fig 3 that the rare earth (57 to 71) densities are capped and those that follow have shifted two atomic numbers to the left. You will also notice from Fig 4 that Radon (86) is the only noble gas the Gamma value for which is greater than the integer and is not divisible by 18.
In fact, the Gamma value which is based upon the integer of the neutronic ratio is not only valid, the integer factor must also be correct; as evidenced in Fig 2, the pattern changes with atomic numbers that are divisible by the value ‘9’.
Therefore, something changes after Lutetium (71); but what?
You will also notice that the lattice value for every element is different. Therefore, it would appear that our categorisation; bcc, fcc, tetra', cph, etc. is incorrect.

But the variables in our calculation for matter density have no units (if you discount RAM {kg/mol}).
That said, there are a number of coincidents in this calculation technique:
1) the pattern is genuine,
2) the average mismatch over all the elements is only 1.5,
3) all the variables in the calculation are valid and appropriate,
4) these densities are for perfect (pure) crystals.
We must not forget that whilst empirical (documented) values for density should be more reliable than those for gas-transition, they are still subject to experimental error. So, we cannot automatically accept documented values as correct (elemental data).

The greatest change in density occurs in elemental matter at gas transition temperature, when the electrical (charge) repulsion between adjacent atoms exceeds the magnetic (field) attraction. At lower temperatures, all matter is viscous and the distance between adjacent atoms remains pretty much the same#. Albeit the force of attraction between them does weaken as temperature rises.

Whilst the distance between adjacent atoms varies with any given direction, once we have a correct value for density, we can establish the average distance between them like this; d = ³√[mₐ/ρ]
which is the same method we use to define the relationships between the coefficients of linear and volumetric expansivity.

Further Reading

You will find further reading on this subject in reference publications^{(69, 70, 71 & 73)}