What is the required thickness of concrete needed to entomb a leaking reactors emissions? What is the minimum thickness of an aggregate concrete membrane to prevent radioactive particles from escaping a broken reactor?
I am interested in calculating the amount of concrete required to entomb a reactor.
 A: Some amount of radiation always penetrates through a barrier of thickness $T$ - the proportion is given by $(1/2)^{(T/T_0)}$ where $T_0$ is the halving thickness. The decrease is exponential.
The halving thickness depends both on the material as well as the type of the radiation. Generally, gamma rays - heavier cousins of X-rays (photons) - are the most penetrating ones. See e.g.

http://en.wikipedia.org/wiki/Radiation_protection#Shielding_design

The halving thickness for lead and concrete is 1 and 6 centimeters, respectively. However, the gamma radiation obviously escapes in random directions and is only a concern in the very vicinity of the source.
Beta-radiation - electrons that fly nearly by the speed of light - is somewhat less penetrating than gamma-rays, so the halving thickness is a few millimeters for solids:

http://www.fas.harvard.edu/~scdiroff/lds/QuantumRelativity/PenetrationandShielding/PenetrationandShielding.html

The least penetrating ones are alpha particles and their heavier sisters, heavy radioactive nuclei, that are physically screened by dozens of micrometers of a solid; see the page above. So a minimal layer of a solid - with no holes - is always enough to protect a tomb against the release of some radioactive material.
When thinking about the tomb, it may be more important to guarantee that it will be mechanically stable against excessive pressure inside etc. that could lead to explosions etc. It's not quite trivial to decide how to lower the pressure in the hermetically isolated tomb because the stuff that creates the pressure is radioactive and harmful.
A: To prevent particles from escaping, a few centimeters of concrete is enough.
To prevent gamma radiation from escaping, one has to define what levels one considers acceptable outside the enclosure.
To prevent the hot air emanating from press weenies and the anti-nuclear lobby from escaping, several lightyears' thickness of lead might not be enough.
