Chapter 9

DATA ANALYSIS AND REPORTING

John M. Keller

Oak Ridge National Laboratory

INTRODUCTION

Many of the areas discussed in this chapter are the procedures,
guidelines, expectations and other processes that should be well defined
in a laboratory=s Quality Assurance Plan (QAP). A general laboratory QAP
should have the flexibility to both define a minimum level of data
quality as well as provide guidance for higher levels of QA to meet
regulatory and compliance requirements. There are some who think all
laboratory work should require high levels of QA but the problem with
this position is the high cost. Another fundamental problem that can be
routinely observed in almost all environmental laboratories involves the
application of high level QA requirements for processing the laboratory
samples, but little or no requirements for the field sampling process.
It does not matter how good the laboratory analysis is performed if
there is no effort to collect representative samples. There is a process
called Data Quality Objectives (DQO) that was developed to define the
levels of QA needed by the customer and this process is designed to
apply a graded approach. If the DQO process is used and properly applied
it can have a significant impact on overall cost for laboratory
analyses. Since the level of QA can have a significant impact on cost,
many laboratories will have multiple project specific QA plans to
satisfy different customer needs.

The raw counting data is rarely useful without some type of additional
data manipulation to convert the counting results to an activity or
concentration. The data manipulations can be as simple as dividing the
counts per second by the detector efficiency to more complex
least-squares fitting techniques to define peak shapes which are then
used to calculate peak areas and/or heights. Typically the counting data
must be corrected for branching decay modes, gamma energy yields,
efficiency, and sample dilutions. Another type of correction applied to
counting data includes a mass absorption correction which can be
especially important for alpha counting and to a lesser degree beta
counting. Mass absorption corrections are usually less important for
higher energy gamma counting but may be required for specialized
applications. Overall, there are many data manipulations performed to
reach a final result which also means there are many opportunities to
make mistakes. Any complex process, such as what is involved with
reporting radiological data, requires a good technical review to ensure
data quality.

CHAIN OF CUSTODY & LABORATORY SAMPLE TRACKING

Any good laboratory should have control over the access and security of
analytical samples received from customers. At a minimum, this sample
control should include documentation to verify when the sample was
received, who delivered the sample, the customer's name, sample
identification, and the requested analyses. A typical form to document
this transaction is referred to as a Chain-of-Custody (COC) form. A COC
form is usually signed and dated by both the person delivering the
sample and the individual receiving the sample at the time the sample is
delivered to the laboratory. The person receiving the sample should
inspect and document the sample condition *as received* before accepting
custody of the sample. In this day-and-time any analytical sample could
become the subject of litigation, especially if the sample is related to
environmental compliance, environmental monitoring, or waste disposal.
As part of the sample receipt the inspection should also include a
radiological screening to ensure level of activity does not exceed any
operational limits for the laboratory. This is also an opportunity to
segregate samples with higher levels of activity from lower levels
sample to help minimize cross contamination problems. Any other
information available about the sample history should be documented
because it may be important for the interpretation of the results.

QA/QC RECORDS

There are various types of documents and records generated during the
operation of a radiochemistry laboratory and many of these records may
be needed for various audits, quality assurance requirements, and other
documentation needs. Therefore, there needs to be a document control
program which is usually defined by the laboratory quality assurance
program (QAP). The types of records that need to be controlled include
operating procedures, analytical methods, calibration records, sample
data files, control charts, and many other types of technical documents
created during the operation of a laboratory. The records to be
controlled need to be identified and listed in the laboratory QA plan.

There are many Laboratory Information Management Systems (LIMS)
available that provide customized database management tools and
interfaces to help manage the large amounts of information generated by
a laboratory operation. A typical LIMS system includes tools to track
samples, collect data, and generate reports. In addition to managing
sample data, most LIMS systems are designed to handle quality control
(QC) data such as spike recoveries, blanks, calibration checks, and
instrument performance data. Once the QC data is collected and stored in
the data base it can be used to generate control charts to monitor the
performance of both methods and instruments. Quality control charts for
calibration check standards and QC samples can be grouped by analyst and
used to document continuing qualification of the laboratory staff for
performing various methods. Since any well managed laboratory performs
routine monitoring of instrument performance, backgrounds, and method
performance, this information needs to be stored in a format that is
easy to retrieve, recall, and monitor for long term trends that could
indicate problems. Most short term trends are detected with weekly and
daily checks performed by laboratory staff.

Although it is very useful to store information electronically, the
computer database systems do not eliminate the need to retain hardcopies
of information related to the sample analysis. The laboratory needs to
maintain a well organized filing system that has controlled access and
is protected from fire and water damage. The types of hardcopy
information typically stored in this filing system would include
chain-of-custody documents, sample preparation records, computer
printouts from instruments and other forms of raw data, instrument
calibration and performance data, and any other type of sample
information not suitable for electronic storage. For many laboratories
it is more convenient to collect certain types of information in
laboratory notebooks or logbooks. If logbooks are used there needs to be
a system in place to link or associate the information with both the
hardcopy files and the sample database stored electronically; usually
these associations can be done with a sample identification code and a
date. For example, if a laboratory keeps a calibration logbook for
pipets and a pipet is identified on a sample preparation worksheet, an
auditor should be able to go back with a date and a pipet identification
code to verify that the pipet was properly calibrated.

In a regulatory or compliance environment the laboratory must document
and be able to track all aspects of the laboratory operations including
sample management, sample preparation, instrument calibration, standard
preparation, and analyst qualifications. Also, it is a good management
practice (required for some forms of regulatory work) to participate
national performance demonstration programs where a nationally
recognized group prepares samples with a known composition for
distribution to laboratories participating in the program. The
participating laboratories analyze the samples as unknowns and report
the data. The data for all the participating laboratories is then
summarized in a report that each laboratory can compare their
performance with the group average and the known values.

DATA PRESENTATION

There is no standard or historically based method on how to present
radiochemical data although there have been attempts over the last 10-15
years to encourage most laboratories to use the *International System of
Units* (SI) units. The SI units for reporting radioactivity are the
internationally accepted units and will eventually be the standard in
the United States; however, this is not the current practice for
reporting most environmental radioactivity data. The Becquerel (Bq) is
the SI unit for activity and is defined as one disintegration per second
(dps); the historical unit for activity is a Curie (Ci) which
corresponds to the activity for one gram of ^226^Ra. The relationship
between a Bq and Ci is defined as follows:

$$1\ \text{Ci} = 3\text{.}7x\text{10}^{\text{10}}\text{Bq}\ (\text{dps})$$

The level of radioactivity in environmental samples is usually discussed
in units of curies (Ci) and concentrations are generally expressed in
terms of fractions of curies per unit mass or volume. Since a curie
represents a comparatively high activity, it is more common to see
concentrations for environmental samples expressed as pCi/L or pCi/Kg
(pCi = 10^-12^ Ci). A list of some other common units used to express
activity is given the table below.

  ------------------------------ ----------------------------------------
  **Symbol**                     **Name**

  Ci                             curie

  dps or Bq                      disintegration per second

  dpm                            disintegration per minute

  mCi                            millicurie (10^-3^ Ci)

  μCi                            microcurie (10^-6^ Ci)

  nCi                            nanocurie (10^-9^ Ci)
  ------------------------------ ----------------------------------------

The ultimate goal when reporting data is to provide the customer or data
user with the information in the most useful format possible to meet the
customer's needs. There can be a wide variety of people, with a broad
range of technical background, that use the radiochemical data reported.
The various types of data users typically include radiation protection
technicians, project/program managers, scientists, environmental
monitoring groups, regulatory/compliance organizations, and even
lawyers. Therefore, it is important to know the audience that will use
the information to ensure the correct interpretation of the data and to
avoid misconceptions. Even though it is well known that there are
significant sources of natural radioactivity (cosmic radiation, ^40^K,
^232^Th, ^235^U, ^238^U, etc.) in the environment, the presence of any
radioactivity in the ground water, surface water, and soil is a very
sensitive area when dealing with the public. It is not uncommon to see
various media outlets, in an attempt to sensationalize news, to report
radiological data with units that result is a large numbers. For
example, consider a commercial reactor has a release of 1 mCi of ^60^Co
into a stream that results in a concentration downstream of 0.0001
mCi/mL; this would likely be reported to the public by the media as
100,000,000 pCi/L.

Another important aspect that needs to be considered when reporting
radiological data (or any analytical result) is the relative error
associated with the result. No measurement performed in a laboratory
provides an absolute value for the result and no result should be
reported without an estimate of the overall error. The overall error
includes the standard deviation of the measurement, which is a measure
of the precision, along with all other sources of error. The combination
of all these sources of error is referred to as propagation of error and
this subject is discussed in more detail in Chapter XXX. The overall
measurement error is especially important for environmental or low level
samples because as the measured activity approaches method detection
limits the analytical error can become very large. For low level samples
the counting statistics will frequently dominate the overall error. The
standard error (*σ~G~*) for a gross number of counts (*N*) is,

$$\sigma_{G} = \sqrt{N}$$

In a similar fashion, the standard error (*σ~g~*) for a count rate
(*N/t*) would be,

$$\sigma_{g} = \sqrt{\frac{N}{t^{2}}}$$

If a radionuclide of interest is not detected there needs to be a means
to report the minimum activity level the method could have measured.
Different conventions and terminology have been used to estimate the
lower limit of detection (*LLD*) and the technique suggested by the
Department of Energy Environmental Measurement Laboratory (DOE-EML) will
be discussed here. The *LLD* is defined as the smallest quantity of
sample radioactivity that will yield a net count for which there is a
predetermined level of confidence that radioactivity is present. The
*LLD* may be approximated as follows:

$$\text{LLD} = (k_{\alpha} + k_{\beta})\sigma_{r} = (k_{\alpha} + k_{\beta})\sqrt{\frac{G}{t_{G^{2}}} + \frac{B}{t_{B^{2}}}}$$

Where, *k~α~* = risk for concluding falsely that activity is present
(α),

> *k~β~* = degree of confidence for detecting the activity (1-β),

*σ~r~* = standard error for count rate (*r*),

*G* = gross counts collected in time *t~G~*,

*B* = background counts collected in time *t~B~*.

If α and β are equal then *(k~α~ + k~β~)* can be represented as
*2k~α/β~*. If the sample and background counting rates and count times
*(t~G~=t~B~)* are similar, as would be expected near the *LLD*, then the
above equation can be simplified to,

$$\text{LLD} = 2\sqrt{2}k\frac{\sqrt{B}}{t_{B}}$$

Some common values of k are given below

+---------------+---------------+---------------+--------------------+
| α/β           | (1-β)         | k~α/β~        | 2√2 k              |
+---------------+---------------+---------------+--------------------+
| 0.01          | 0.99          | 2.327         | 6.58               |
|               |               |               |                    |
| 0.02          | 0.98          | 2.054         | 5.81               |
|               |               |               |                    |
| 0.05          | 0.95          | 1.645         | 4.65               |
|               |               |               |                    |
| 0.10          | 0.90          | 1.282         | 3.63               |
|               |               |               |                    |
| 0.20          | 0.80          | 0.842         | 2.38               |
|               |               |               |                    |
| 0.50          | 0.50          | 0.0           | 0.0                |
+---------------+---------------+---------------+--------------------+

For *α* and *β* equal to 0.05 and when the gross count time is equal to
the background count time, then the smallest amount of radioactivity
that has a 5% *(α)* probability of a false positive and a 95% *(1-β)*
probability of being detected is,

$$\text{LLD}_{\text{95}}(\text{cps}) = 4\text{.}\text{65}\sigma_{B} = 4\text{.}\text{65}\frac{\sqrt{B}}{t_{B}}$$

This equation for *LLD~95~* is used by many laboratories; however, one
must use care that the given conditions apply. If the gross and
background count times are different and/or different probabilities are
used the first equation must be used to calculate *LLD*. To obtain the
*LLD~95~* in units of activity one must correct for detection efficiency
*(E)*, decay branching ratio *(β)*, and sample quantity *(Q)* which
would include any dilutions.

$$\text{LLD}_{\text{95}}(\frac{\text{Bq}}{\text{mL}}) = \frac{\text{LLD}_{\text{95}}(\text{cps})}{\text{EQ}\beta}$$

Continue this discussion to include reporting negative results vs.
detection limits.

DATA REVIEW

There are different types of reviews that are routinely performed on
analytical data, including the radiochemical results. One type of review
that is commonly used on regulatory data packages is referred to a data
validation. Data validation is more of a clerical review of the data to
ensure that a data package includes all the information and raw data
needed to reconstruct and defend the final results. These extensive data
packages are required by the customer to meet regulatory or compliance
requirements and typically include documents such as the raw data,
instrument calibration information, certificates for standards, sample
preparation information, chain-of-custody, along with the final results.
Whereas a data validation may include some verification of calculations,
this type of validation typically does not include an extensive
scientific review of the final results. An acceptable technical review
requires much more experience and technical background than normally
available for data validation. Some suggested areas to look at for a
technical review will be the topic for the rest of this section.

A good technical review of radiochemical data usually requires more than
just a laboratory background and should draw from a broad range of
experiences including nuclear chemistry, nuclear engineering, reactor
operations, waste operations, and even some basic background in nuclear
weapon testing is useful when reviewing environmental data because of
the global fallout from above ground testing. A technical reviewer
should have a good understanding of the nuclear processes that produce
the various radionuclides. For most laboratories the radionuclides of
interest would involve natural products (i.e., ^40^K, ^232^Th, ^235^U,
^238^U, etc.), fission products (e.g., ^90^Sr, ^137^Cs ^154^Eu, etc.),
activation products (e.g., ^60^Co, ^63^Ni, ^65^Zn, etc.), and decay
daughters (^90^Y, ^208^Tl, ^210^Po, ^228^Th, etc.). However, do not
forget the sample history because some unusual radionuclides could be
found around accelerator sites, medical isotope production facilities,
university and government nuclear operations, and even hospitals with
nuclear medicine treatment facilities.

To begin the review process it is always good to do a *reality check* to
make sure the data makes sense. For example, one would not expect to see
short lived fission products (i.e., ^103^Ru, ^131^I, ^144^Ce, etc.) in
an old waste tank that historically collected liquid waste from a spent
fuel processing facility but had not been used for ten years. Due to the
relative fission yields and the half-lives for fission products, the
activity in aged (5-10 years decay) fission product waste would be
dominated by ^90^Sr (t~1/2~ = 29.1 y) and ^137^Cs (t~1/2~ = 30.2 y)
followed by ^154^Eu (t~1/2~ = 8.59 y) and ^155^Eu (t~1/2~ = 4.71 y). If
short lived fission products were reported on this ten year old waste
the data would be questionable. Either there was an error in the
identification of the radionuclides or there was much more recent use of
the waste tank than originally indicated. These types of *reality
checks* on the data are not always so obvious and it may take a
experienced reviewer to recognize basic problems with the data that do
not make sense physically or chemically. If there is some sample history
available, each radioactive species identified in the sample should have
reasonably defined pathway to explain why each radionuclide would be
present in the sample matrix. Any historical information about the site
where samples are collected is useful for the technical review and
validation of radiochemical data. The types of site information should
include information on the various production operations such as waste
processing, topsoil characteristics in the area sampled, surface water
locations, annual rain fall, and sometimes even the average daytime
temperatures may be useful in explaining sample results. Historical
information might also include radiological data where past samples were
collected at or near the same site. If historical data is available and
there is some knowledge of activities performed at the site during the
period between the sample collections times, there may be enough
information to estimate some expected outcomes such as types of
radionuclides present and even activity levels after decay corrections.

Another aspect of date review is to make sure that the chemistry makes
sense, for example, the presence of thorium in a caustic solution would
be suspect because of the low solubility of thorium at high pH.
Sometimes the chemistry can become very complex and hard to predict but
for most cases the chemistry of the radionuclides will follow expected
behavior. For example, most anionic species tend to be water soluble and
are mobile in the environment. Two good examples are salts of
pertechnetate (^99^TcO~4~^2-^) and radioactive iodide (^129^I^-^); both
of these examples are long-lived fission products that are very mobile
in the environment. For the most part the radionuclides within the group
of transition metals, the lanthanide and actinide series, and even some
of the Group I and II elements that are present as cations in the
environment are absorbed on soils or precipitated at high pH values.
Some metals readily form complexes with common organic acids found in
the environment to form water soluble species. However, it important to
realize that soil in various geological locations can vary in
composition and chemistry, which can have a significant impact on the
environmental mobility of radionuclides.

The soil in east Tennessee has a strong affinity for actinides and
cesium, but there is almost no absorption of strontium. Even in areas
around nuclear facilities that experienced releases of fission product
waste from underground storage tanks and associated piping many years
ago, there has been minimal movement of the ^137^Cs and actinides from
the immediate area. However, the many years of exposure to ground water
and rain moving through the soil has removed most of the ^90^Sr from
these areas. The ^90^Sr moves so freely through the soil in east
Tennessee that if outflows around these areas are sampled soon after a
rainfall and immediately analyzed almost none of the ^90^Y normally
found in equilibrium with the ^90^Sr is present. As water moves through
the soil the ^90^Y is continuously removed by the soil and only the
^90^Sr is carried by the ground water. The identification of beta
activity from ^90^Sr that was [not]{.underline} in secular equilibrium
with ^90^Y has led to the misidentification of environmental beta
activity on more than one occasion.

Sometimes the sample history can help identify some expected trends in
the radiological data; however, there are additional expected outcomes
based upon the decay of radionuclides that [must]{.underline} be true if
the nuclide identification is correct. Many radionuclides, both natural
and manmade, have radioactive daughters. This decay process can continue
past the first daughter and result in decay chains where the activity of
the progeny can be estimated from the original parent. Some of the more
complex decay chains are found in nature. The ^238^U decay chain, often
referred to as the uranium series is illustrated on the next page. The
other two decay chains found in nature are the actinium series which
starts with ^235^U and the thorium series which begins with ^232^Th. The
neptunium series which starts with ^237^Np used to be found in nature
millions of years ago, but all currently available ^237^Np is manmade.
In general, most of the manmade radionuclides do not have a decay chain
as long or complex as the decay series found in nature, but many of the
isotopes routinely identified in the laboratory have one or two
daughters that also will be present in the sample. These daughters can
be useful for confirming nuclide identification and may even provide a
second confirmation of the parent activity. To fully appreciate the
nuclear decay process the reviewer needs to understand the ingrowth of
radioactive decay products and the three different types of
parent-to-daughter equilibriums that are possible.

A more detailed discussion of the ingrowth of decay products can be
found in several of the references at the end of this chapter; *Nuclear
and Radiochemistry* by Friedlander, et. al., provides an excellent
discussion of this subject. A set of equations can be derived, which are
referred to as the Bateman equations, to calculate the number of atoms
or the activity of each daughter in a decay series. The equation for the
first daughter, *N~2~(t)*, as a function of the initial amount of the
parent, *N~1~(0)*, and time (t) is given below in terms of the number of
daughter atoms,

$$N_{2}(t) = \frac{\lambda_{1}}{\lambda_{2} - \lambda_{1}}N_{1}(0)\left( e^{- \lambda_{1}t} - e^{- \lambda_{2}t} \right)$$

Since *A=λN*, this equation can be restated in terms of activity,

$$A_{2}(t) = \frac{\lambda_{2}}{\lambda_{2} - \lambda_{1}}A_{1}(0)\left( e^{- \lambda_{1}t} - e^{- \lambda_{2}t} \right)$$

These equations can be expanded to include additional progeny in a decay
chain, but only the first daughter in a series will be discussed here.
There are three special cases that depend upon the relative half-lives
for the parent and daughter in a decay chain summarized in the table
below.

+---------------+------------------------------------------------------+
| Decay         | Type of Equilibrium                                  |
| Constant      |                                                      |
|               |                                                      |
| or            |                                                      |
|               |                                                      |
| Half-life     |                                                      |
+---------------+------------------------------------------------------+
| *λ~1~ \<      | [Transient equilibrium]{.underline}, where the ratio |
| λ~2~*         | of the parent and daughter decay rates become        |
|               | constant.                                            |
| *T~2~ \<      |                                                      |
| T~1~*         |                                                      |
+---------------+------------------------------------------------------+
| *λ~1~ \<\<    | [Secular equilibrium]{.underline}, where the         |
| λ~2~*         | activity for the daughter becomes approximately      |
|               | equal to the activity of the parent.                 |
| *T~2~ \<\<    |                                                      |
| T~1~*         |                                                      |
+---------------+------------------------------------------------------+
| *λ~1~ \>      | [No equilibrium]{.underline}.                        |
| λ~2~*         |                                                      |
|               |                                                      |
| *T~2~ \>      |                                                      |
| T~1~*         |                                                      |
+---------------+------------------------------------------------------+

The condition of secular equilibrium, where the half-life of the
daughter is much less than the parent, is the most useful case to the
radiochemist for the measurement of activity. Both ^90^Sr and ^137^Cs
are long-lived fission products routinely observed in environmental
samples that demonstrate secular equilibrium.

$$\text{Sr}\overrightarrow{\beta^{-}(\text{29}\text{.}1y)}Y\overrightarrow{\beta^{-}(2\text{.}\text{67}d)}\text{Zr}\ (\text{stable})$$

$$\text{Cs}\overrightarrow{\beta^{-}(\text{30}\text{.}2y)}\text{Ba}\overrightarrow{\text{IT}(2\text{.}\text{55}m)}\text{Ba}\ (\text{stable})$$

The time required for ingrowth of the daughter to approximately match
the parent activity is in the order of 10X the half-life of the daughter
radionuclide. This ingrowth time is illustrated for the ^90^Sr/^90^Y
secular equilibrium in the following curve.

![](media/image1.wmf){width="4.75in" height="3.059055118110236in"}

Although an understanding of daughter ingrowth and secular equilibrium
is very useful to the radiochemist doing measurements it also helps the
data reviewer validate and confirm results. There are a number of cases
where the identification of a radionuclide is based upon measurement of
a daughter in secular equilibrium; several examples are listed below,

$$\text{Ru}\overrightarrow{\beta^{-}(1\text{.}\text{02}y)}\text{Rh}\overrightarrow{\beta^{-}(\text{29}\text{.}9s)}\text{Pd}\ (\text{stable})$$

$$U\overrightarrow{\alpha(4\text{.}\text{47}x\text{10}^{9}y)}\text{Th}\overrightarrow{\beta^{-}(\text{24}\text{.}1d)}\text{Pa}\overrightarrow{\beta^{-}(1\text{.}\text{17}m)}U\text{.}\text{.}\text{.}\text{.}\text{.}$$

$$W\overrightarrow{\beta^{-}(\text{69}\text{.}4d)}\text{Re}\overrightarrow{\beta^{-}(\text{16}\text{.}9h)}\text{Os}(\text{stable})$$

For each of these examples listed, the daughter in secular equilibrium
is easier to measure by gamma spectrometry than the parent. This is
especially true for the ^106^Ru since this radionuclide represents a
"pure" beta emitter. Frequently, the 1001 KeV gamma from ^234m^Pa can be
used to quantify uranium in soil samples.

REPORTING ACTIVITY FROM MASS SPECTRAL DATA

Because of the improvements in sensitivity, reliability, and lower cost
mass spectrometers are becoming available for more radiochemistry
laboratories. Although Thermal Ionization Mass Spectrometers (TIMS) have
been around for many years for the measurement of isotopic ratios for
uranium and plutonium the Inductively Coupled Plasma Mass-Spectrometer
(ICP-MS) has recently become popular for a wider range of radionuclides,
including many of the long-lived fission products such as ^99^Tc,
^129^I, as ^135^Cs. Mass spectrometers provide a measurement of the
number of atoms for each isotope present which is directly proportion to
mass. Typically, results from a mass spectrometer are reported in units
of concentration in ng/g or ng/mL; most users of radiological data are
interested in units of activity such as Bq/mL or pCi/g. The conversion
factor between mass and activity is defined as the specific activity
(S~A~) which has units of activity per unit mass. The specific activity
for each radionuclide can be determined from the following:

$$S_{A}(\frac{\text{Bq}}{g}) = \frac{A_{t}(\text{Bq})}{m(g)} = \frac{\lambda\ N_{t}(\text{atom})}{m(g)}$$

Since,

$$\ \lambda\ (\text{sec}^{- 1}) = \frac{\text{ln}2}{T_{\frac{1}{2}}(\text{sec})}\ \text{and}\ N(\text{atom}) = \frac{m(g)\ N_{A}(\frac{\text{atom}}{\text{mol}})}{M_{A}(\frac{g}{\text{mol}})}$$

Then,

$$S_{A}(\frac{\text{Bq}}{g}) = \frac{\text{ln}2}{T_{\frac{1}{2}}(\text{sec})}\frac{N_{A}(\frac{\text{atom}}{\text{mol}})}{M_{A}(\frac{g}{\text{mol}})}$$

Where *T~1/2~* is the half-life in units of seconds, *N~A~* is
Avogadro's number (6.022 x 10^23^), and *M~A~* is the atomic mass of the
radionuclide of interest.

Some radionuclides can decay by more than one path such as ^227^Ac
(T~1/2~ = 21.77 y), which is a natural decay daughter of ^235^U.

$$(\text{98}\text{.}\text{62}\text{\%})\ \text{Th}\ \overrightarrow{\beta^{-}}\ \text{Ac}\ \overrightarrow{\alpha}\text{Fr}\ (1\text{.}\text{38}\text{\%})$$

The total decay constant *λ~T~* for ^227^Ac can be represented as the
sum of the partial decay constants *λ~α~* + *λ~β~*. Since decay
constants are a function of the half-life the concept of a partial
half-life can also be introduced. Since the decay constant is defined
as, *λ = ln 2 / T~1/2~*, the half-life for ^227^Ac can be represented
with partial half-lives as follows:

$$\frac{1}{T} = \frac{1}{T_{\alpha}} + \frac{1}{T_{\beta}}\ \text{or}\ T = \frac{T_{\alpha}T_{\beta}}{T_{\alpha} + T_{\beta}}$$

REFERENCES

Evans, R. D., *The Atomic Nucleus*, New York: McGraw-Hill, 1958.

Horrocks, D. L., *Applications of Liquid Scintillation Counting*, New
York: Academic, 1974.

Friedlander, G., Kennedy, J. W., Macias, E. S., Miller, J. M., Nuclear
and Radiochemistry, 3^rd^ ed. New York: Wiley-Interscience, 1981.

Tsoulfanidis, N., *Measurement and Detection of Radiation*, New York:
McGraw-Hill, 1983.

Knoll, G. F. *Radiation Detection and Measurement*, 2^nd^ ed. New York:
Wiley, 1989.

Ehmann, W. D. and Vance, D. E., *Radiochemistry and Nuclear Methods of
Analysis*, New York: Wiley-Interscience, 1991.