Instrument Geometry and Detection Efficiency: 2? vs. 4?

Posted on February 5, 2019

2Pi vs. 4Pi




The terms “2π" and "4π" are frequently used in describing several aspects of radiation detection and quantification. The same terms are also used to discuss the calibration certification of radioactive standards. It seems, however, that the actual meaning behind these terms is uncertain for many people. Furthermore, the terms "2π" and " " as used in radiation protection can imply additional information which may not be correctly inferred by the uninitiated. Improper or incomplete underselling of all the implications and uses of these terms could potentially lead to very large errors in calculation of surface (or airborne) contamination. This article, then, seeks to explain the use of these terms and their implications in a variety of uses as they apply to detection of alpha and beta contamination with hand-held instruments.

Geometry Basics

The circumference of any circle (i.e., the length of the line which defines the circle) can be calculated by multiplying the radius of that circle (i.e., the distance from the center to the edge) by twice the value of π (c = 2πr). This is also the basis of radian measure, in which multiples of π are used in place of degrees. In radian measure, 2π = 360°. In other words, in radian measure, 2π represents a complete, two-dimensional circle.

The three-dimensional analogue to the circumference of a circle is the surface area of a sphere. Surface area of any sphere can be calculated using the equation A = 4π2. This is used in three- dimensional angle measurement, in which the steradian is used. The term steradian is derived from "stereo radian” and is defined as a solid angle "subtended at the center of a sphere of unit radius by unit area on its surface.” Thus, "4π" (leaving the “r2” unsaid) represents an entire sphere. It can also be seen easily enough that "2π” represents exactly one-half of a sphere. Simply, then, when we refer to something as “4π,” we mean that something is analogous to a sphere. Similarly, "2π” refers to a half sphere, or hemisphere. How does this apply to radioactive sources and radiation detectors?


First, a word about detector efficiency.  For the purposes of this article, efficiency is defined as the ratio of the background-corrected counts observed by the detector for each radioactive disintegration occurring in the source

(counts/disintegration or c/d).

Efficiency can also be multiplied by 100 to express it as a percent for example, if my calibration source is listed as being 10,000 dpm (disintegrations per minute) and the detector reads 1,100 cpm (counts per minute) in a background of 50 cpm, the efficiency will be:

The efficiency in c/d is exactly the same as that which would be determined for cpm/dpm or cps/dps.

When the efficiency of a particular isotope is known, the background corrected rate of a sample can be used to determine the activity of a surface or a sample by dividing that background-corrected count rate by the efficiency. Using the efficiency determined above, if I get 250 cpm on a wipe with a 50 cpm background, the activity on that wipe is:

A 2π detector is one which can only see a single side of a radioactive source; all hand-held contamination detectors are inherently 2π. In other words, if a source is emitting radiation in all directions (i.e., in a 4π steradian), the detector will never be able to see more than half of that radiation, or have 50% efficiency. In actuality, the “theoretical maximum" efficiency will always be somewhat lower than 50%, for several reasons. First of all, for an actual 2π geometry, the center of the source must intersect with the flat plane of the detection hemisphere to achieve a full 50% coverage; the source would have to physically touch the detector face, where in reality the source is always held at least a small distance from the detector window (see Figure 1). The other factors all involve the loss of detection due to distance and shielding (e.g., the detector entry window).

While "4π" laboratory counters do exist, they are not particularly common. They generally consist of a spherical gas proportional counter in which the sample is suspended in the center. It is true that gamma spectrum analyzers in which the sample being counted is contained in a Marinelli beaker that surrounds the detector (as shown in Figure 2) are nearly 4π, those are not relevant to this pellicular article. Similarly, it could be argued that some exposure rate instruments are 4π, but exposure rate instruments are not typically used for measuring surface contamination.



An ideal radioactive point source emits radiation in all directions, and thus is inherently "4π." This Ideal source would be weightless and dimensionless, which would serve to make its emissions pure and unimpeded. A real source, such as a calibration standard, varies from the ideal in several respects, due primarily to self-absorption and backscattering. A source typically consists of metal disk between 0.5 and 2 inches in diameter, with the radioactive material either dried or electroplated onto the surface. The radioactive material may or may not be protected with a thin (Mylar, Kapton, or other) covering. depending on the particular makeup and intended use of the source. Practically speaking, we may still consider any source a "point source” so long as it is small relative to the size of the detector area.

Self Absorption

Self absorption is the "self-shielding~ of emissions from the source by the protective covering or even from the thickness of the deposited radioactive material itself (see Figure 3). While self absorption exists to some extent in all radioactive sources, it is most significant in sources of isotopes which emit alpha particles (such as 230Th) or lower energy beta articles (such as 14C). This is why source manufacturers make strong efforts to keep the layer of radioactive material thin on calibration standards. If a source is made with an exact quantity of radioactive material, later analysis will show that there is apparently less radioactive material there if self-absorption is significant.


Backscattering means that some fraction of particles emitted in directions other than toward the detector bounce off the material the source is mounted on and change direction, often toward the detector. Backscattering will increase the emissions from the source in the direction of the detector, showing an apparent increase in activity over the quantity it was manufactured with (see Figure 4). Backscattering is much more significant with beta particles than with alpha, and more significant with higher-energy beta particles, such as those from 90Y. One method used by source manufacturers to overcome or at least minimize the effect of backscatter is to use a so-called weightless mounting. In one of these mountings. the isotope is placed on a piece of very thin Mylar (or other material) which is suspended in a more substantial metal ring (see Figure 5). While these do minimize backscatter very well, these sources are also very fragile and lire not recommended for general use.

For clarification, self-absorption and backscattering have been described separately. In reality, they with both exist simultaneously in a source to a greater or lesser extent.

and 4π in Describing Sources and Instrument Efficiency

When sources are purchased for the calibration of instruments, it is desirable to know the activity of the isotope on that standard with a high degree of accuracy, often within :5%. This is usually accomplished by depositing or electroplating an exactly known amount of a radioactive standard onto a holder or backing. That quantity of material would then be reported on the Certificate of Calibration which accompanies the source. It would then be considered a "4π" source, and an instrument efficiency determined using this activity would be a 4π efficiency.

Unfortunately, self absorption and backscattering can alter the number of particles actually being emitted from the active side of the source, as described above. These effects are compensated for by placing the source in a highly accurate windowless gas proportional counter and determining the rate of particle emissions off of the source. Since in this type of detector the source is actually within the active detector volume, the emissions are not attenuated by distance and don't need to pass through a detector window. The detector efficiency in this type of instrument is effectively 100%; all of the alpha or beta particles which make it off of the surface of the source are detected. The particle emission rate determined in this manner can be considered a "2π" activity. An instrument efficiency determined using this emission rate would be 2π efficiency.

In nearly all instances, "2π" efficiency Is effectively useless; in quantifying contamination we are trying to determine the amount of radioactive material present, not just the correct number of emissions from the surface.

So, the dilemma is this: We want to determine a 4π efficiency for our detectors, but the activity listed on the source certificate will give us an erroneous value. If we use the 2π emission rate, we will have an efficiency which will be accurate, but will not give us useful information regarding the quantity of contamination present on the surface. For example, if I measure the count rate on a contaminated surface, correct it for background, and divide by the 2π efficiency, the calculated activity on that surface will be approximately half of what is actually there.

Probably the best solution is to derive an "effective 4π activity" for the calibration standard by multiplying the 2π emission rate by 2, since 2π x 2 "" 4π. For example, I have a 14C source with a certified 4π activity of 3.477 kBq (kilobecquerels, or thousands of disintegrations per second). The activity, in dpm works out as follows:

This is the actual amount of 14C on the source.

Self Absorption Example

My G-M pancake detector, at 1 cm from this source, reads 8,000 cpm in a background of 50 cpm. The efficiency is calculated as:

This "efficiency" does not account for self absorption and is less than the true efficiency.

When I ordered that source, I also insisted that the manufacturer provide me with the 2π beta emission rate, as well. This listed as 78,891 beta emissions per minute (epm). Therefore, the effective 4π: activity Is:

Calculating efficiency using the same parameters listed above from the effective activity gives:

This efficiency is accurate, and is the one which should be used with the instrument. The efficiencies determined using certified activity and correcting for self-absorption using effective activity vary by almost 25%, and would cause an overestimate of the quantity of 14C on a surface if the improper efficiency were used.

Backscattering Example

On the other hand, failing to compensate for backscatter would result in overestimating the quantity of higher energy beta emitters such as 90Y or 32P. I have, for example, an 90Sr/Y source with a certified activity of 0.0946) µCi. This calculates to:

My G-M pancake detector, at 1 cm from this source in a 50 cpm background, reads 59,000 cpm. The efficiency is calculated as:

This source is mounted on a brass plate, and the manufacturer informed me that with backscatter, the 2π particle emission rate is 128,100 epm. The effective 4π activity is:

Calculating efficiency using the same parameters listed above from the effective activity gives:

In this example, the efficiencies determined from the certified activity and correcting for backscatter vary by around 0%. In this case, however, the error would underestimate the quantity of 90Sr/Y contamination (unless you were ensuring activity on a brass bench top). Occasionally, instrument manufacturers will list a nominal 2π efficiency for a given instrument instead of a 4π efficiency. I suspect that this is so that a person comparing two similar detectors will probably purchase the one with a higher efficiency. Wouldn't you rather get 10% efficiency for 14C than 5%? So remember to read the fine print and if the listed efficiency is 2π, simply divide this efficiency in half to determine the actual efficiency you can expect to obtain.

There remain, of course, other factors for calculating the correct efficiency for ensuring surface contamination levels with hand-held instruments which are not the subject of this article. These include compensating for the physical size of the source (or the area of contamination) and the type of material which is contaminated.


All radioactive sources are inherently 4π, in that they emit radiation in all directions. When the activity in a calibration standard is given in units of Ci, Bq, or dpm, that activity does not account for backscatter and self absorption.

On the other hand, all hand-held radiation detection Instruments are inherently 2π, in that they will never be able to detect more than half of the particle emissions from a radioactive source.

When determining the activity on a surface or on a sample, we want to know what the 4π activity is (i.e., the total activity present). Therefore, use only 4π efficiencies for hand-held instruments. This efficiency is best approximated by doubling the 2π emission rate. 


1. Require that alpha and beta calibration sources be supplied with both total activity (e.g., dpm, µCi, kBq) and the 2π mission rate (e.g., epm, αpm, ßpm).

2. Determine instrument counting efficiency by dividing the background- corrected count rate by twice the reported 2π particle emission rate. This is a "fail-safe” method, and “automatically” corrects for self-absorption and backscatter. It applies to all calibration sources--alpha, low-energy beta, and high-energy beta - whether they are mounted on Mylar, plastic, aluminum, or stainless steel.

3. If you have calibration sources whose calibration certificates do not list the 2π emission rate, contact the source manufacturer. Based on the isotope, quantity of radioactive material, and the type of backing (e.g., plastic, stainless steel), the manufacturer might be able to provide accurate corrections for backscatter and self-absorption.

The Author

Paul R. Steinmeyer is a health physicist at Radiation Safety Associates, Inc., and the co-author of Mathematics Review or Health Physics Technicians. He developed the Ludlum Measurements Model 44-110 Tritium Frisker and is an expert in hand-held radiation detector use, calibration, and repair. Paul has supervised numerous small and large-scale contamination and decommissioning projects. He is also Executive Director and Assistant RSO for RSA Laboratories, Inc.

Phone: 860/228-0487

Fox: 860/228-4402


Reprinted from RSO Magazine Ma


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