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Does Earth's magnetic field block (or divert) only lower energy radiation while the hardest, high speed heavy ions, run through to LEO? Are the hardest rays dampened? Do the van Allen Belts, not far from LEO, actually generate electron rays that hit the ISS?
I'd like to know what (the potentially harmful part of) the radiation spectrum in the ISS looks like.
The ISS gets its radiation from galactic cosmic rays, solar cosmic rays, and the Van Allen belts. You are correct that the magnetic field does a pretty good job protecting the ISS from solar radiation, but the high energy and/or high mass galactic cosmic rays plow on through; fortunately, the flux of galactic cosmic rays that make it through is much less than that from the other sources.
The Van Allen belts do cause problems for the ISS. If the magnetic field was symmetric around the Earth, it wouldn't be too much of a problem because the ISS would fly below the belts, but the field is offset such that the belts dip down closer to the Earth over the South Atlantic causing the South Atlantic Anomaly, which the station does have to fly through. It is the inner belt protons and the outer belt electrons that are problematic. I read a National Academies report years ago that mentioned that on the Russian station Mir, about half of the radiation dose the cosmonauts received came from the couple percent of the time they spent in the SAA at the station flew through it.
As far as radiation, the biggest thing that NASA is worried about seems to be as follows:
Gamma rays
While fusion in the core does produce gamma rays, these high-energy photons are absorbed and re-emitted many, many times at lower frequencies in the solar plasma. However, gamma rays can enter the solar system from outside the system: these cosmic gamma rays do pose a hazard to the astronauts.
X-rays
The Sun produces this form of radiation during solar flares. They can pose hazard to the astronauts if they get too intense, but so far there's been no indication of a solar storm severe enough to warrant emergency measures (which would probably mean evacuation of the station and a return to Earth for all astronauts).
Protons and Neutrons
These forms of radiation can be problematic, and are monitored on the ISS. (Electron radiation--called beta radiation--cannot penetrate the walls of the ISS.)
How much radiation is there? Enough that the ISS monitors those radiation levels constantly, but not enough for radiation poisoning or anything of the like. Astronauts are at a higher risk for cancer, cataracts, and cellular damage, but usually they don't experience enough for acute radiation-related health issues.
All information from http://www.nasa.gov/mission_pages/station/research/experiments/1043.html and http://www.universetoday.com/60065/radiation-from-the-sun/.
