I want to beam into solid concrete as a bomb. I have transporters and unlike Star Trek I'm not afraid to use them. Terrorists have taken the tech and have nasty plans for us.

The bomb is 1 kg sphere of solid concrete, and the target is also a 2m thick solid concrete wall. The ball will be beamed directly into the center of mass making the density of that volume exactly double. Both have equal density ρ = 2,400 kg/m³

Concrete composition
The concrete in question has quartz aggregate in the following ratios:

Giving the following final molecular composition:

• CaO = 4%

• SiO₂ = 89%

• Fe₂O₃ = 1%

• H₂O = 6%

Calculating number of atoms in the merged mass
I will use the composition above to invent a "concrete atom" with the average weight of these components. This cement is 64% Oxygen, 30% Silicon, 4% Hydrogen, and 2% Ca or Fe. The weighted average atomic weight of a "concrete atom" is therefore:

$$ \text {Weight of Concrete Atom} = w_\text c = 0.64(w_\text O)+0.3(w_\text {Si})+0.04(w_\text H)+0.02(w_\frac{(Ca+2Fe)}{3}) $$ $$w_\text c = 10.23+8.415+0.04+1.012 = 19.6968 \text{AMU}$$

Therefore, the initial 2kg concrete mass contains $ \frac{2\text{kg}}{19.6968 \text{AMU}} = 6.115\text E{25}$ representative "concrete atoms" in a volume $v_i = \frac{m}{\rho} = 833.3\mu\text m^3$.

Each "concrete atom" shares a volume of space = $ \frac {833.3\mu\text m^3}{6.115\text E{25}} = 1.3627\text E-23\mu\text m^3$ having a diameter:

$$ d = 2r = 2 \sqrt[3]\frac{3(1.3627\text E-23)}{4\pi} = 3.0\text E-8\mu \text m$$

This math has something wrong... the atoms are not 0.008pm apart in static pressure!

Assume the atoms will align perfectly in between each other.

Neglect the strong nuclear force because no baryonic intersections will occur. Also neglect electromagnetic bonding forces (ionic and covalent), but obviously the other various van der Waals forces apply.

My assumptions: the force needed to compress the combined 2kg mass of concrete to 50% of its volume will be the same force exploding the volume of space when its density instantly doubles (but I could be wrong).

Also acceptable is simply calculating the Lennard-Jones Potential energy for the atoms suddenly reducing their distance by 1/2.

The yield calculation should be KE=1/2â‹…mâ‹…v²

Where m is 2 kg, and v will be the integral of the velocities of each atom of concrete as the cumulative van der Waals force repels the atoms back toward their van der Waals contact distance.

I believe the final formula can be derived from W=ΔKE where Work (W) is the the work done moving the atoms back into their normal density with v_initial and v_final Calculated from the force acting on each atom.

I'm hoping for the equation so I can scale it for different yields and possibly different material densities.

What is the simplified equation for yield in gigajoules for this kind of bomb?

(I'm calling this a kinetic weapon because the damage is done by accelerating the mass radially outward via nuclear forces.J