The equations below provide approximate scaling laws for relating the destructive radius of each effect with yield:

r_thermal = Y^0.41 * constant_th
r_blast = Y^0.33 * constant_bl
r_radiation = Y^0.19 * constant_rad

If Y is in multiples (or fractions) of 2.5 kt, then the result is in km (and all the constants equal one). This is based on thermal radiation just sufficient to cause 3rd degree burns (8 calories/cm^2); a 4.6 psi blast overpressure (and optimum burst height); and a 500 rem radiation dose.

The underlying principles behind these scaling laws are easy to explain. The fraction of a bomb's yield emitted as thermal radiation, blast, and ionizing radiation are essentially constant for all yields, but the way the different forms of energy interact with air and targets vary dramatically.

Air is essentially transparent to thermal radiation. The thermal radiation affects exposed surfaces, producing damage by rapid heating. A bomb that is 100 times larger can produce equal thermal radiation intensities over areas 100 times larger. The area of an (imaginary) sphere centered on the explosion increases with the square of the radius. Thus the destructive radius increases with the square root of the yield (this is the familiar inverse square law of electromagnetic radiation). Actually the rate of increase is somewhat less, partly due to the fact that larger bombs emit heat more slowly which reduces the damage produced by each calorie of heat. It is important to note that the area subjected to damage by thermal radiation increases almost linearly with yield.

Blast effect is a volume effect. The blast wave deposits energy in the material it passes through, including air. When the blast wave passes through solid material, the energy left behind causes damage. When it passes through air it simply grows weaker. The more matter the energy travels through, the smaller the effect. The amount of matter increases with the volume of the imaginary sphere centered on the explosion. Blast effects thus scale with the inverse cube law which relates radius to volume.

The intensity of nuclear radiation decreases with the inverse square law like thermal radiation. However nuclear radiation is also strongly absorbed by the air it travels through, which causes the intensity to drop off much more rapidly.

These scaling laws show that the effects of thermal radiation grow rapidly with yield (relative to blast), while those of radiation rapidly decline.

In the Hiroshima attack (bomb yield approx. 15 kt) casualties (including fatalities) were seen from all three causes. Burns (including those caused by the ensuing fire storm) were the most prevalent serious injury (two thirds of those who died the first day were burned), and occurred at the greatest range. Blast and burn injuries were both found in 60-70% of all survivors. People close enough to suffer significant radiation illness were well inside the lethal effects radius for blast and flash burns, as a result only 30% of injured survivors showed radiation illness. Many of these people were sheltered from burns and blast and thus escaped their main effects. Even so, most victims with radiation illness also had blast injuries or burns as well.

With yields in the range of hundreds of kilotons or greater (typical for strategic warheads) immediate radiation injury becomes insignificant. Dangerous radiation levels only exist so close to the explosion that surviving the blast is impossible. On the other hand, fatal burns can be inflicted well beyond the range of substantial blast damage. A 20 megaton bomb can cause potentially fatal third degree burns at a range of 40 km, where the blast can do little more than break windows and cause superficial cuts.

It should be noted that the atomic bombings of Hiroshima and Nagasaki caused fatality rates were ONE TO TWO ORDERS OF MAGNITUDE higher than the rates in conventional fire raids on other Japanese cities. Eventually on the order of 200,000 fatalities, which is about one-quarter of all Japanese bombing deaths, occurred in these two cities with a combined population of less than 500,000. This is due to the fact that the bombs inflicted damage on people and buildings virtually instantaneously and without warning, and did so with the combined effects of flash, blast, and radiation. Widespread fatal injuries were thus inflicted instantly, and the many more people were incapacitated and thus unable to escape the rapidly developing fires in the suddenly ruined cities. Fire raids in comparison, inflicted few immediate or direct casualties; and a couple of hours elapsed from the raid's beginning to the time when conflagrations became general, during which time the population could flee.

A convenient rule of thumb for estimating the short-term fatalities from all causes due to a nuclear attack is to count everyone inside the 5 psi blast overpressure contour around the hypocenter as a fatality. In reality, substantial numbers of people inside the contour will survive and substantial numbers outside the contour will die, but the assumption is that these two groups will be roughly equal in size and balance out. This completely ignores any possible fallout effects.

Information provided by: http://www.fas.org