What is an electron environment? What is the does you are referring to? Electrons do not give a dose. Ionizing radiation can be measured in dose. If you are referring to ionizing radiation then the only difference in does would be due to the absorption of whatever gas is inside the hollow sphere. If it is a vacuum then they will be identical.

I forgot to mention that small detectors at the centers of the spheres define dose. But this is an unnecessary complication so I will change my question. Instead of asking about dose I am now asking about the radiation environments at the centers of the spheres. Comparing the solid sphere to the spherical shell, both used as a shield against trapped electrons in Earth's radiation belts, which one has the most severe radiation as seen by an observer at the center?

How can you have an observer in the center of a solid sphere? In any case, like I said earlier the only difference would be due to the atmosphere inside the hollow sphere. You still need to change your question a bit. It is not the electrons that are passing through the sphere, it is the electro-magnetic radiation that they produce.

Let me explain this in more detail. When electronic parts are being selected for use in a spacecraft, a preliminary assessment of the environment is needed to determine if the part can survive the space radiation environment. The preliminary assessment, regarding ionizing dose, assumes simple geometries for mass shielding (if greater accuracy is needed to reach a conclusion, we then go on to more advanced methods that consider the actual mass distribution of the spacecraft). Several simple geometries are typically considered and a dose-depth curve is calculated for each. This is a plot of ionizing dose in some reference material (usually either Si or SiO2) versus shield thickness (usually aluminum). Simple geometries include spherical shells and solid spheres, with the reference material regarded as small enough to effectively be a point at the center of the shield. For protons or heavy-ions, the straight-ahead approximation is adequate so there is no significant difference between the shielding geometries. However, electrons (found in Earth's radiation belts) are more complicated. Monte Carlo calculations, that include bremsstrahlung and all other reactions, show significant differences between the two shield geometries. My question is, is there a simple analytical argument that roughly explains these differences?

I thought the biggest problem for shielding spacecraft is gamma radiation, which is highly penetrating. Shielding from charged particles should not be difficult, since their mean path length in most solid materials is extremely small. I don't believe there is an analytical solution to either, since there are variety of interactions that can occur which have different cross sections (a factor that gives the probability of a certain type of interaction both elastic and non-elastic). A Monte Carlo approach is normally used to solve these types of problems.

You are not trying to be helpful, you are just trying to find ways to complain about my question, so I will end this discussion. But one thing that you should learn, for your own education, is that electrons do contribute to ionizing dose.

I could get your question.  You wish to know whether hollow sphere or solid sphere is better for radiation shielding i.e, high energy electrons. 

If you use solid sphere, the electrons(assuming electrons travel till the centre since they may have enough energy) scatter and reach the centre. Possibly the electrons from all the sides would converge (after superimposition ) to the centre.  In case of hollow sphere of some thickness, electrons after scattering to some distance through the thickness of the shell, they would travel straight in the volume where you have kept the detector. So i guess less radiation will be felt by the detector in case of the hollow sphere as compared to the solid sphere.  I wish to draw comments from other experts..

The electron and proton density at the lower energy spectrum (

Thank you for the help but my question is not whether the spherical shell is the worst case. My question is whether there is a simple qualitative explanation as to why it is the worst case.