X-ray Spot Temperature
The temperature of the X-ray spot on the target has a significant effect on the lifetime of an X-ray
The focal spot temperature increases with the input power and X-ray exposure time and decreases
with larger focal spot sizes and target diameters.
So why does the temperature on the target in an X-ray spot depend on the power divided by the spot
diameter and not the power divided by the area?
Here is an illustration on why the temperature has the dimensionality that it does.
Figure 1 shows a hemispherical section through an X-ray Target.
Figure 1: Section through Target near X-ray Spot
If we assume the spot power is deposited evenly on a small hemispherical surface as opposed to a
small flat surface, then the calculation of the temperature of that surface is rather straight forward.
From the definition of thermal conductivity through a thin layer of material:
Where P is the power in Watts, dt is some small thickness in meters through which the heat is conducted, λ is the thermal conductivity in Watts/m*K, A is the cross sectional area in square meters and dT is the temperature drop through the material in Kelvin.
In the situation in Figure 1 we have:
dt = dr, and A = 2πr2 Area of Hemisphere.
We may now integrate this expression from R1 to R2 and from T1 to T2:
If we take the limit as R2 goes to infinity, replace 2R1 with D (diameter of the spot) and replace T1 -T2
by T then we have:
We will therefore see that the temperature rise from the back of the target to the spot is given by an
expression that only depends on the ratio of the power to the spot diameter and not the area of the
The fact that the actual spot energy is deposited over a flat and not hemispherical area only changes
the expression by some constant of proportionality of order unity (~1).
The incremental temperature drops through the material decreases as the square of the distance from
the spot because the conduction area increases as the square of the distance. Adding all those drops
together total up so that the total temperature drop goes only as reciprocal of the distance, with the
smallest distance and thus the largest contribution occurring right at the spot.