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Lesson #2 - Directions and Solid Angles

Reading Assignment: Intro. to Ch. 2, Section 2.1

This is one of the shortest readings in the course, but it is one of the hardest ones to master (so these notes will be more complete than usual).  The determination of solid angle draws on calculus that you may not have used in a while and requires a little thought.


At first glance, the direction part looks like it would be easy; we all know what "that way" means.  But the idea of direction depends on comparison with a set of reference directions. You are usually free to pick the reference directions to fit the problem at hand. 

The first choice of direction references that occurs to us is the 3 Cartesian directions - wpe3.gif (863 bytes),wpe5.gif (875 bytes), and wpe6.gif (887 bytes) -- which we will recall are unit length vectors in the directions of the x, y, and z axes, respectively:

wpeA.gif (2441 bytes)

Consider a vector wpeB.gif (897 bytes) that is also unit length and points in the 1st quadrant (i.e., +x,+y,+z):

wpeD.gif (2698 bytes)

The simplest way to characterize its direction is to "drop" perpendiculars to each of the three Cartesian axes and denote the direction from the lengths (u,v,w) of these three projections:

wpeE.gif (2999 bytes)

so that we have:

wpeF.gif (1189 bytes)

This 3-coordinate directional approach is intuitive, logical, and easy to understand.   Unfortunately, though, we seldom use it for two principal reasons:

    It has an extra variable that is not needed. It is wasteful because it uses 3 numbers for a 2-dimensional space.   Direction has only 2 degrees of freedom.  (A vector anchored at the center of the Earth has a direction completely specified by the latitude and longitude of the point where the ray with the same direction intersects the surface of the Earth.)  Mathematically, we know that we must have (for a unit length direction vector):
wpe10.gif (1076 bytes)

          so, if you know two of them, the third can be deduced from those two.

    More importantly, it is not very useful to nuclear engineers because radiation fields and cross section scattering distributions (the two places that direction is used) do not usually vary according to u or v or w, so it is not a useful way to describe direction.
A much more useful way of characterizing direction is to pick one of the axes (which we will call the polar axis) and begin by using the angle between this axis and wpeB.gif (897 bytes) as our first dimension, wpe1C.gif (880 bytes).   For the sake of discussion, we will use the z axis as the polar axis:

wpe1D.gif (2509 bytes)

This gives us one dimension, what about the other?  Well, in following our Earth analogy, that first angle gave us a latitude-like variable (although Earth latitude is measured from the Equator, not from the North Pole), so we follow with a longitude-like variable by projecting wpeB.gif (897 bytes) onto the x-y plane, call the new (flat) direction wpe15.gif (932 bytes), and use the angle between this projected vector and the (arbitrarily chosen) x axis as the second angle, which we will denote as wpe1E.gif (880 bytes):

wpe20.gif (2899 bytes)

This gives us a 2-dimensional representation of direction that is not only more concise than the (u,v,w) representation, but also turns out to be more useful (if the polar axis is properly chosen).

Solid Angle of a section of a sphere

This gives a representation of the direction of a vector, wpeB.gif (897 bytes),in terms of the angles wpe1C.gif (880 bytes) and wpe1E.gif (880 bytes), but since we are going to want to integrate over all directions, we must relate the differentials as well.  For this, we return to our graph and put in a differential area element, wpe21.gif (923 bytes), bounded by latitude and longitude lines:

wpe26.gif (3294 bytes)

From this figure, we see that the "north-to-south" lines that border the element have length wpe25.gif (887 bytes), but the "east-to-west" lines have a length equal to wpe27.gif (977 bytes)(since the distance you must travel "around the world" on a give latitude line gets shorter as you get closer to the North Pole).  Using these two differentials allows us to express the differential solid angle as:

wpe8.gif (1220 bytes)

This representation of wpe21.gif (923 bytes) is most useful for situations in which we want to determine the solid angle associated with a section on the surface of a sphere -- especially a section that is bordered by constant wpe1C.gif (880 bytes) and wpe1E.gif (880 bytes) lines.


The solid angle associated with a region on a sphere (not necessarily a unit sphere) bordered by wpe1.gif (896 bytes)wpe3.gif (902 bytes)wpe5.gif (900 bytes), and wpe6.gif (907 bytes) is given by:

wpe7.gif (2274 bytes)

Solid Angle of a Cartesian Surface Element

A second way of attacking solid angle that is equally valid, and better in some situations, is to recognize that solid angle subtended by a differential area (from a given point) is equal to the projection of the area (i.e., the area as seen from the point) divided by the square of the distance from the point to the differential area.  For example, in the following figure:

wpe2A.gif (3051 bytes)

the distance from Point P to the differential area is given by R and the projected area of dA from the point P is:

wpe2B.gif (1257 bytes),

the solid angle is the (slightly unwieldy):

wpe2D.gif (1787 bytes)

This representation is most useful for determining the solid angle of a rectangular surface, although the integrals tend to be difficult to work out. 


The solid angle subtended by a rectangular region of width W and length L, as seen from a point a distance z perpendicularly above the center is given by:

wpeA.gif (1837 bytes)

Homework problem 2.6 gives a solution for this in closed form.

Solid Angle of a Cylindrical Surface Element

Note that the same problem with the volume element expressed in cylindrical coordinates in the x-y plane would not be so messy:

wpe30.gif (3028 bytes)

In this case, the solid angle works out to be:

wpe33.gif (1998 bytes)


wpe35.gif (1124 bytes)

and z is a constant, we can differentiate both sides to get:

wpe36.gif (1162 bytes).

Substituting this gives us:

wpe37.gif (1574 bytes)

This representation is most useful for determining the solid angle of planar surfaces that are sections of disks. You may find this useful in doing Homework problem 2.1.

Reduction of more complicated shapes to one of the three

Now that we know how to attack solid angle determinations for three situations:
    Surfaces on spheres

    Flat rectangular surfaces

    Flat disk-section surfaces,

you need to recognize that you can do a lot more with these than you might think because the solid angle subtended by a solid object is the same as the solid angle subtended by the object's shadow, as cast by a light source at Point P.   Using this fact along with the fact that solid angles can be added and subtracted, gives us added flexibility.

Therefore, the solid angle of a given 2D or 3D object (as measured from a Point P) can be found by finding the solid angle of the object's shadow cast onto either a flat surface or an enclosing sphere, whichever is most convenient.

NOTE: The determination of the solid angle associated with a disk is more efficiently found by projecting the disk onto an enclosing sphere.  You may want to work homework problem 2.1 this way.

We will work some examples in class.

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