# [ RadSafe ] formula and its derivation for rectangularextended radioactive sources exposure rate

George Chabot george_chabot at uml.edu
Fri Jul 12 16:47:40 CDT 2013

```I'm sorry the attachment for the rectangular source did not go through. I've
reproduced it below, but I had to transpose the Word equation into
computerese. I'm not sure it will be understandable, but a reference is
provided.

Regarding the exposure rate from a rectangular source, the solution cannot
be obtained in a closed algebraic form. If Sa is the activity per unit area
and G is the gamma ray exposure rate constant for a radionuclide of interest
and you set up the differential equation that shows the exposure rate from a
differential rectangular surface element dxdy, and the respective x and y
variables go from 0 to L and 0 to W, the integration to obtain exposure rate
from the entire source at a point on a perpendicular line at height H
directly above one corner of the rectangular source is

(1) X=Sa*G*(Integral from 0 to W of dy)*(Integral from 0 to L of
dx/(x^2+y^2+H^2))

Integrating with respect to x yields

(2) X=Sa*G*(Integral from 0 to W of
dy/(y^2+H^2)^0.5)*arctan(L/(y^2+H^2)^0.5).

This equation can be integrated by numerical techniques or other possible
methods. This equation, or essentially the same equation, may be found in a
text by Shultis and Faw (Radiation Shielding, Prentice Hall, 1996)

The reason for getting the result above one corner of the source is so that
the same equation may be used for determining exposure rate from any uniform
rectangular source regardless of where above the plane of the source the
dose point is located. For the simplest case when the dose point is directly
above the center of the rectangle, L and W, respectively, represent one half
of the rectangle length and one half of the rectangle width. The total
exposure rate would then be four times that obtained from equation 2. When
the dose point is at other locations above the source, but within the source
boundaries, the individual values for L or W would differ for each of the
four source elements, and the results from the four calculations using
equation 2 would be summed. If the dose point lies outside of the source
boundaries one can visualize drawing an extension or extensions of the
source so that the extended edge(s) lies directly below the dose point and
using the same equation 2 to calculate doses from the source plus
hypothetical extensions and subtracting out the contributions from the
individual source extension(s).

-----Original Message-----
From: radsafe-bounces at health.phys.iit.edu
[mailto:radsafe-bounces at health.phys.iit.edu] On Behalf Of Simpson, Daniel S.
(GSFC-3500)
Sent: Friday, July 12, 2013 2:35 PM
To: The International Radiation Protection (Health Physics) Mailing List
Subject: Re: [ RadSafe ] formula and its derivation for rectangularextended
radioactive sources exposure rate

George:

I don't see a WORD file attached.

Daniel S. Simpson
Deputy Chief, Code 350 Occupational Safety & Health Division
National Aeronautics and Space Administration
Goddard Space Flight Center
Greenbelt, MD  20771
(301) 286-2281
Fax: (301) 286-1745

-----Original Message-----
From: radsafe-bounces at health.phys.iit.edu
[mailto:radsafe-bounces at health.phys.iit.edu] On Behalf Of George Chabot
Sent: Friday, July 12, 2013 2:22 PM
To: 'The International Radiation Protection (Health Physics) MailingList'
Subject: Re: [ RadSafe ] formula and its derivation for rectangular extended
radioactive sources exposure rate

Regarding the rectangular source, attached is a Word document showing the
equation development. It is not possible to obtain a complete algebraic
solution to the required integration, but the resulting equation is easily
solved using numerical integration.

George Chabot

-----Original Message-----

From:
du>

[mailto:radsafe-bounces at health.phys.iit.edu] On Behalf Of Danak, Subhash M
CTR DOD CAPMED FBCH

Sent: Thursday, July 11, 2013 10:28 AM

Subject: [ RadSafe ] formula and its derivation for rectangular extended
radioactive sources exposure rate

I like to review formula for rectangular radioactive sources exposure rate.

I thank you.

Subhash

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