2.1.  Measurement geometry of the source and the ionization chamber

Most re-entrant ionization chamber designs for activity measurements show a cylindrical symmetry as in Figure 4-7: the radioactive source, a solution in a cylindrical ampoule, is positioned at the centre of a cylindrical vessel having cylindrical electrodes. The symmetry of these structures is only interrupted by the entrance hole of the well, the electrical connections and the mountings of the electrodes covering a few percent of the total solid angle of $4\mathrm{\pi }\mathrm{sr}$. Earlier constructions of spherical ionization chambers, as shown in Figure 2-1 [308] copied in [309] and [260], Figure 21 [403] and Figure 20 [53], or chambers without a well, but with a lead entrance window (Figure 5 [264], [265]), are no longer in use for activity measurements of photon emitting radionuclides.

In the ideal case the radioactivity is homogeneously distributed in an aqueous solution of a density of about $1\mathrm{g}{\mathrm{cm}}^{-3}$ forming a cylindrical volume. Any point inside such a geometry could represent the origin of a trajectory of radiation from a decaying nucleus. In a few cases such as reference sources of ²²⁶Ra or ¹³⁷Cs solid sources are measured in ionization chambers. The volume distribution of the radioactivity should be homogeneous to render theoretical considerations about attenuation effects possible. Thin point sources with low amounts of material are almost never used in calibrations of re-entrant ionization chambers.

To visualize the operating principle of an ionization chamber, as for example would be required in a Monte-Carlo calculation, the development of the interactions of the emitted radiations and their trajectories can be followed in the manner of a flow chart. Only photon radiation (x and $\mathrm{\gamma }$ rays) and the very small number of $\mathrm{\beta }$ particles with energies above a few $\mathrm{MeV}$ can reach the sensitive volume of the ionization chamber. However, high-energy $\mathrm{\beta }$ particles are detected with low probability by bremsstrahlung photons. All photons from the source and those from secondary effects transfer energy to the surrounding matter by three principal interaction processes: photoelectric effect, Compton effect and electron-positron pair production, the latter having an energy threshold of $1.022\mathrm{MeV}$. Detailed descriptions of these effects may be found in books on nuclear physics or detector theory, such as [357], [425], [143], [316], [99] and [294]. Before the photons reach the sensitive volume of the chamber, their intensities undergo considerable attenuation in the material of the source, the containers and the sample holders, and in the ionization chamber walls. The photons that contribute to the ionization current are those that have transferred energy to electrons by direct interactions in the gas of the chamber, or in the matter around the gas from which electrons still reach the sensitive gas volume (Figure 6). All these electrons produce positive ions and electrons in the counting gas, usually referred to as ion pairs, with an average energy of about $30\mathrm{eV}$ per ion pair. When an electric field is applied to the chamber electrodes, the charges created move, are collected from the gas and measured as the ionization current of the chamber. A simple model of the processes is shown for a plane-parallel ionization chamber placed in a homogeneous photon radiation field in Figure 7 [31]. This may be extended to the more complicated cylindrical geometry with a radioactive source in the form of a solution, as shown in Figure 4-7.

2.2.  Ionization process and charge collection

Let us now consider the interaction of “fast” electrons with gases. “Fast” in this context means the ability to create ion pairs that contribute to the ionization current. Fast-moving electrons in matter lose energy by Coulomb interaction and by radiative processes. The first leads to the formation of excited electronic states or to ionization in the atoms which were hit. The second produces bremsstrahlung or electromagnetic radiation. The knowledge we have of the processes involving fast electrons can be considered as reliable, and average data of the phenomena are available [29], [30], [160] in the Encyclopedia of Physics (Flügge S., Ed.), [341], [257], [473], [258] and many others]. However, in the slowing down of electrons by multiple scattering to the velocities of atomic electrons and the transport of slow electrons in an electric field, there are still “several points on which the picture is slightly hazy” [160]. A theoretical hypothesis or experimental generalities concerning averaged experimental data must be introduced to complete the picture of a charge collection, as described, for example, by Grosswendt and Waibel ([189]). This is understandable given the large numbers involved. For example, an incident $1\mathrm{MeV}$ $\mathrm{\beta }$ particle, fully stopped in pure argon creates nearly $4\cdot {10}^4$ ion pairs, the detailed path history of which cannot be followed individually. Our understanding of ionization chamber characteristics is therefore mainly based on empirical studies of averaged quantities or parameters.

During slow-down, energy straggling, range straggling and angular straggling occur. In addition, not all primary energy can be transferred to the final signal. Not only does the mean number of ion pairs formed vary from gas to gas, but fluctuations in their number occur in the same gas and for identical incident-particle energies. This imposes a fundamental limit on the accuracy of the detector response [147], [148], which applies not only to detectors with energy resolution, but also to integrating detectors among them current ionization chambers. In practice, these effects are superposed on stronger disturbing effects such as noise and fluctuations from the current-measuring electronics. All the available experimental information can be summarized in the essential fact that, for current ionization chambers, the ionization (number of ion pairs formed) is proportional to the energy of the electrons stopped in the gas, where the proportionality constant is the average energy lost by forming an ion pair: this constant is often referred to as the $W$ value (Table 1). Because of the effects already noted, the $W$ value is higher than the ionization energy, that is the energy needed to eject an electron from an individual atom.

Within less than a microsecond of the slowing-down process, an electron cloud is created along the track of ionization. In this time the energy of the electrons not influenced by other external forces is reduced to thermal velocity values with energies of order $0.04\mathrm{eV}$. Very few free electrons may be expected to appear later by the depletion of metastable atomic states created along the track. If electric fields are weak or non-existent, the movement of electrons and ions can be described by theories of diffusion as a tendency to move away from zones of higher charge density. Charge transfer collisions occur, and electrons may be captured by positive ions in a process called recombination, or by neutral atoms (or molecules), a process called attachment [160]. There are two kinds of recombination effect. The first is called columnar recombination and takes place in the original ionization channel immediately after the creation of the ion pairs as a result of their high density in the tracks. The second is volume recombination and results from encounters between ions and electrons after they have left the immediate vicinity of the track.

When an external electric field is applied to the gas volume in the detector, the drift velocities are superposed on the random thermal motion of the ions and free electrons. In moving along the electric field, the ions and electrons gain kinetic energy, but lose it again through energy-loss processes. In the resulting steady state the average drift velocity $v$ of the ions is given by

where

$\mu$ = mobility of ion,
$E$ = electric field strength,
$p$ = gas pressure.

Applying a voltage of $500\mathrm{V}$ to the electrodes, separated by $1\mathrm{cm}$ in a chamber with an argon pressure of $2\mathrm{MPa}$ results in ion velocities of about $20\mathrm{cm}/\mathrm{s}$, so charge collection from the ion component takes about $50\mathrm{ms}$. This is a rather long time and imposes a lower limit on the charge integration time for the ionization current measurement, usually chosen to be several seconds. Fortunately, the electron-drift velocity is typically about 1000 times that for ions. An expression analogous to the equation for ions can be defined only for low-energy electrons moving in pure noble gases like argon. Generally, however, the behavior of the electrons cannot be predicted in detail. In a typical ionization chamber gas, such as pure argon or nitrogen, the excitation level of the outer shell is about $15\mathrm{eV}$. The electrons can be accelerated up to this energy, limited only by elastic scattering. They then establish an average drift velocity which depends on the constituents of the gas in a very sensitive manner, and in some cases it does not even increase monotonically with the strength of the applied electric field [160], [341]. It has sometimes been reported that the current collected from an ionization chamber exposed to constant radiation intensity changes in magnitude when the polarity of the collecting potential is reversed [30]. This may be caused by

• contact, thermal or electrolytic potentials of materials,

• “selective operating” electrode geometries with respect to the direction of the photon flux,

• low-energy secondary electron emission in chamber walls,

• space charge effects or distortions of the electric field.

The various effects of electrode geometry on charge collection are discussed by Boag ([30], [31]) and Colmenares ([76]).

2.3.  Ionization current

In activity measurements the photon flux of a radioactive source inside a current ionization chamber remains reasonably constant during the time of one measurement. In fact, for measuring times that are not short in comparison with the half-life of the radionuclide, a correction factor for radioactive decay has to be applied, see Equation (14). If the number of ion pairs created in the gas volume remains constant under well-defined measuring conditions, the charge collection as a function of the external polarizing voltage across the electrodes can be measured (Figure 8-1-Figure 8-2). This current-voltage characteristic curve is used to select the optimum working point of the ionization chamber. The measured current is proportional to the number of ion pairs collected. At low voltages, the current in the current-voltage characteristic curve increases strongly with increasing polarizing voltage. After this first increase, the slope of the curve decreases continuously to reach an almost constant current value, the so-called region of “ion saturation” (Figure 8-2). It serves no purpose to increase the voltage of the chamber to a value higher than this saturation region, because at very high voltages strongly ionizing processes occur and these may damage insulators and supports inside the chamber. In the region of saturation all charges due to ionization in the active chamber volume are collected and measured by the external current-measuring electronics. The result is an ionization current for chambers of the type under discussion in the range from about 10^–13 A to 10^–8 A for sources with activities ranging from a few hundred $\mathrm{kBq}$ up to about 10 GBq and radionuclides like ²²⁶Ra or ⁶⁰Co, which have a high exposure rate constant because of their strong photon radiation components. The upper limit can be pushed to even higher activities if, in the construction of the ionization chamber, particular care is taken to avoid saturation losses.

For complete saturation, the maximum sustainable number of ion pairs in the chamber volume has been reached by irradiation with a source of very high activity. Apart from this, several factors may result in incomplete charge collection with the consequence of saturation loss [35]. The main reason for saturation loss is recombination (Section 5.3).

Recombination may be caused by:

• high local densities of ion pairs from strong local irradiation,

• displacement of charge into a reduced volume under the action of the collecting field (radial effect in a cylindrical volume),

• slow movement of ion pairs to the collection electrodes (time of transit comparable with that of recombination),

• long or complicated trajectories in the electric field.

For these reasons complicated structures at the end of the cylindrical electrodes and regions of weak electric field strength should be avoided.

The ion optics of the electric field between the cylindrical electrodes is important. The inner diameter should not be too small. The field should have good cylindrical symmetry near the centre and the sensitive region of the chamber should have a proper cutoff. This makes it possible to avoid high field gradients at sharp edges and the creation of “screened” zones from which an ion pair can be collected after following a long trajectory.

Also to be avoided are filling gases which form negative ions during ionization, as a result either of their electro-chemical properties or because of their high attachment coefficients (as in the case of oxygen). In most cases these show higher recombination effects, and hence should be avoided in ionization chambers used for activity measurements. The older open-air ionization chambers [29] are now rarely used for activity measurements as they require gas density corrections, as a function of temperature and air pressure, to relate measurement values to the calibration obtained under standard conditions (Section 3.2.1). A more detailed description of ionization chamber constructions and ionization-current measurement techniques is given in Chapter 3 and Chapter 4.

2.4.  Ionization chamber calibration for activity measurements

It may be helpful for the understanding of the following text to begin with some definitions and to discuss a few simple relations which are valid provided that all measuring conditions of the ionization-chamber measuring system, including source parameters, remain uniform or stable during the measurements. The desired measurand is the activity $A$ as defined in the Chapter 1 which decreases exponentially in time according to the well-known law of radioactive decay

The activity $A_{\text{o}}$ is always referred to a fixed reference time $t=t_{\text{o}}$ . For a source used as a radioactivity standard this value is certified. Using Equation (2), the activity $A\left(t\right)$ at the time of measurement can be calculated in terms of the time difference $\Delta t=t-t_{\text{o}}$ and the relation $\lambda =\text{ln}2/T_{1/2}$, where $T_{1/2}$ is the half-life of the radionuclide. The half-life can be found in appropriate tables [272], [316], [51], [394], [225], [324].

Since the studies of Rutherford and Soddy ([367], [368]), the half-life is taken to be a fundamental parameter of a radionuclide characterizing the decay. Its value may be influenced by chemical effects or interactions, for example in the case of electron capture nuclides (section 8.3). To convert time units in Equation (2) from years (a) to days (d), e.g. for $T_{1/2}$, a factor of 365.25 d/a is usually applied for time intervals of less than a few hundred years. For intervals of up to 100 a this correctly averages over leap years and therefore, follows the calendar scale in days. Precise time differences in days used for decay corrections are calculated from the corresponding Julian dates given, for example, in a computer algorithm by Fliegel and Flandern ([152]).

The ionization chamber current $I$, originating from a source with a single radionuclide N under the same measuring conditions, is proportional to its activity, i.e.

The proportionality factor is called the radionuclide efficiency ${\epsilon }_{\text{N}}$. The values for various radionuclides are expressed in units of ampere per becquerel ($\mathrm{A}/\mathrm{Bq}$). In the equation given above, the measured current Im must be corrected for background current $I_{\text{b}}$, with $I=I_{\text{m}}-I_{\text{b}}$, determined under identical measuring conditions. In what follows it is always assumed that the ionization current is already corrected for background.

With most of the current-measuring instruments it is possible to adjust the instrument reading $R$ by an internal (or external) instrument setting. In modern current-measuring electronics this is done by adjusting the instrument amplification. This property can be described by an instrument constant $g$ in the formula

An instrument can be adjusted to show the activity reading $R_{\text{N}}=A$ for a defined radionuclide $\text{N}$ directly in units of becquerel or its multiples ($\mathrm{kBq}$, $\mathrm{MBq}$, etc.). The adjustment condition is $g_{\cancel{\text{N}}}{\epsilon }_{\text{N}}=1$ and it follows that

This technique is commonly applied to instruments such as radionuclide calibrators for performing quantitative assays of radiopharmaceuticals (Section 3.2.4) [398]. These instruments take the form of an ionization chamber coupled to current-measuring electronics with adjustments for particular radionuclides by fixed instrument settings chosen in advance. The settings are determined with the help of activity standards or transferred from a calibrated reference instrument by the manufacturer.

2.5.  Relative activity measurements and reference sources

Ionization chambers working under reproducible measuring conditions are very effective for relative (or indirect) measurements by comparison with an activity standard. For many applications it is useful to compare current ratios, i.e.

where i is an index specifying a particular sample. In quality assurance for standard source production, comparisons are made between samples of the same radionuclide, the same geometry, and almost the same activity. Alternatively, dilution factors are varified using samples with different ranges of activity. In each case, the terms expressing the radionuclide efficiencies in the formula cancel, and the unknown activity values are calculated from that of an activity standard.

In other cases, geometry correction factors are determined for radioactive solutions of various masses $m_{\text{i}}$ in standard ampoules or in ampoules of different materials and forms. For this purpose the sources are dispensed from the same mother solution (index S, the activity being measured in a standard geometry), this having a constant activity concentration $a=a_{\text{S}}=a_{\text{i}}=A_{\text{i}}/m_{\text{i}}$, the masses being determined by careful weighing. Dispensing procedures are described in the literature by Merritt and Taylor ([307]), Eijk and Vaninbroukx ([126]) and Campion in a monograph of BIPM ([63]). The related ratios of the currents per mass (or instrument readings per mass) are deduced from the preceding equations:

These ratios are independent of time only for radioactive solutions containing a single radionuclide, because in a mixed solution the ratios of the compositions change following the decay of the various radionuclides. Because the radiation mixtures change, variable radiation attenuation and detection effects are also involved in the ratios of Equation (7). For a solution with a single radionuclide measured in a non-standard geometry, the measurand is multiplied by the ratio ${\epsilon }_{\text{S}}/{\epsilon }_{\text{i}}$ to obtain the corresponding quantity in the standard calibration geometry. Several authors have studied filling correction factors or correction factors for displacements from a standard source position in the ionization chamber [95], [469], [372], [373]. Examples of this are presented in Chapter 6.

Perhaps the most important of relative measurements made with an ionization chamber are calibrations relative to a long-lived reference source of the type in general use in most national standards laboratories. Here, the activity of the sample to be determined is taken to be proportional to the quotient of the response to the radionuclide sample to the response to the reference source. In this way instrument instabilities are eliminated so long as they affect the responses to both sources equally. The preferred reference source is an old ²²⁶Ra source. In such a source the radium should be in equilibrium with its daughter nuclides which have shorter half-lives. The half-life of ²²⁶Ra is ($\mathrm{1 600}\pm 7$) a [449], [85], [86] or ($\mathrm{584 400}\pm \mathrm{2 600}$) d, corresponding to a decay correction of 0.043% per year. The longest-living radium daughter, ²¹⁰Pb, has a half-life of ($22.3\pm 0.2$) a and the photon radiation from the decay consists mainly of bremsstrahlung from the ²¹⁰Bi daughter ($T_{1/2}=5.0\text{ d}$). The $\pm$ terms represent one standard deviation. Many of the nuclear properties, including the half-lives of the nuclides in the uranium-radium decay chain, were given by Weigel ([465]) in the handbook of Gmelin. For ionization chamber measurements changes in time of the current ratios for mother and daughter nuclides of the radium decay chain are in most cases negligible if the reference sources are older than about 50 a and are kept in a sealed container. Equilibrium conditions are referred to in the literature (see Section 2.6) and the influence of the reference source on relative measurements of the half-lives of other long-lived radionuclides are reported, e.g. by Martin and Taylor ([299]).

The basic equations for indirect calibration of an ionization chamber for a radionuclide N against a reference source (index r) are derived hereafter. The ratio of the instrument readings of the source to be measured $R$ to that of the reference source $R_{\text{r}}$ is related to the ratio of the corresponding activities $A$ and $A_{\text{r}}$ by

If the decay correction factor for the reference source is written explicitly, resolving the equation for the activity of the radionuclide gives

with

where $A_{\text{ro}}$ is the activity of the reference source at $t_{\text{ro}}$, the reference time. The term $A_{\text{e}}$ is referred to as the “equivalent activity.” It refers to an individual radionuclide under reference measuring conditions related to the individual reference source at a certain, but fixed, instrument setting $g$. By definition, a source of a radionuclide with an activity value equal to the equivalent activity ($A=A_{\text{e}}$) produces the same instrument reading as that of the reference source used for the calibration ($R=R_{\text{ro}}$). Because this is valid for each instrument setting when both sources are measured under these conditions, they must also produce the same current ($I=I_{\text{ro}}$).

The equivalent activity $A_{\text{e}}$ is also sometimes called the “relative $K$ factor” [294], and a quantity equal or proportional to it is called the “calibration factor” $k_{\text{N}}$, or less precisely the “calibration constant”, of the ionization chamber for that particular radionuclide (Section 7.1). More briefly, in laboratory jargon, this is called the “chamber constant”. The reciprocal of this quantity represents the relative efficiency of the ionization chamber for a particular radionuclide ($k_{\text{N}}=1/{\epsilon }_{\text{N,rel}}$, or $A_{\text{e}}\sim 1/{\epsilon }_{\text{N,rel}}$).

The given relations can be used to adjust an activity meter (radionuclide calibrator) using the reference source belonging to the instrument. This source is placed in the instrument under reference conditions and the amplification of the current-measuring electronics is adjusted, for example by a potentiometer, until the instrument reading shows the equivalent activity value. With a series of instruments of common construction, this procedure can be used to transfer the values of calibration factors from a reference instrument to a new instrument or to one being recalibrated after repair. The procedure is also useful for an instrument already calibrated since it is only necessary to adjust the current-measuring electronics, for example after changes of the electronics, if the properties of the ionization chamber itself have not changed.

A further possibility for relative measurements is to use a simulated standard, or mock standard, as a reference source for a particular radionuclide. This is discussed in the literature with some more details on calibration constants: Examples include the simulation of ⁹⁹Tc^m with ¹⁴¹Ce [303], [163] or with ⁵⁷Co [22]. The simulated standards should have radiation characteristics similar to those of the radionuclide to be measured and, clearly, care must be taken to apply a correct conversion factor related to the corresponding calibration factors (Section 7.1).

2.6.  ²²⁶Ra sources

From earliest radioactivity standardizations by Mme Marie Curie ([87]), who prepared the first radium standard, to the 1950s, calibrated ²²⁶Ra sources were used as activity standards and were compared with the primary activity standard at the BIPM or with the Hönigschmid standards [88], [365], [467], [97], [79], [283], [295], [296], [316]. Some of the last Hönigschmid standards, dating from 1934, are still available (Figure 9). They are accurate to about 1%. Nowadays ²²⁶Ra sources are mainly used by national standards laboratories as reference sources for ionization chamber measuring systems. The measurement conditions and equilibrium problems of ²²⁶Ra reference sources are similar to those experienced with the older measurements of radium standards using ionization chambers. These are described by Meyer and Schweidler ([309]), by several editions of Kohlrausch’s handbook of [261], [260] and [264], or by Davenport et al. ([97]), Loftus et al. ([283]) and Mann et al. ([296]), when solution standards of radium were prepared at the NBS. In 1982, under the auspices of the BIPM, the PTB and a producer of radioactive sources (Amersham Buchler, Braunschweig [6]) jointly produced a set of about twenty ²²⁶Ra reference sources made from old British radium prepared in 1912.

The small fraction of decay per year for a reference source of ²²⁶Ra, with a half-life of $T_{1/2}=\left(\mathrm{1 600}\pm 7\right)$ a [449], sets a lower limit to the uncertainty component from decay corrections for this reference source for measurement periods of up to several decades [376]. A ²²⁶Ra reference source consists of purified ²²⁶Ra, free from radioactive contaminations, such as ²²⁸Ra. It has the chemical form of crystalline salt as chloride or sulphate, which may be mixed with inactive Ba sulphate. The confinement is a Pt-Ir or stainless steel (V2A) tube carefully sealed. For older sources, glass was frequently used. The seals must be checked for tightness at regular intervals to detect any possible escape of radon gas ²²²Rn, one of the short-lived daughters, with a half-life of $T_{1/2}=\left(3.823\pm 0.003\right)\text{ d}$. The ²²⁶Ra should be in equilibrium with its daughters. This is reached in a few weeks for the short-lived daughters, but equilibrium with the long-lived ²¹⁰Pb ($T_{1/2}=22.3$ a) is reached only a long time after preparation. The nuclide ²¹⁰Pb emits L-x rays of about $12\mathrm{keV}$ and $\mathrm{\gamma }$ rays of $46.5\mathrm{keV}$ with an emission probability per decay of about 4%. The low-energy photons are strongly attenuated in the source and the confinement layers, but the bremsstrahlung photons from the $\mathrm{\beta }$ decay of the following daughter ²¹⁰Bi are detected with high efficiency. To estimate the contribution of ²¹⁰Pb and its daughters to the ionization current of a ²²⁶Ra source and its degree of disequilibrium, the exact date of the separation of the Pb from the Ra must be known.

According to [469], the contribution of ²¹⁰Pb and its daughters is of the order of 0.2% of the total ionization current of the decay chain of ²²⁶Ra, and this is also the maximum error made in the course of time if the increase of ²¹⁰Pb in a newly-prepared radium source is not taken into account. Similar values are given by Rytz ([373]). Although small, this error may still have a strong effect on half-life measurements. Christmas et al. ([70]) have shown that for a sealed ²²⁶Ra sample, observed with an ionization chamber for which the chamber response to ²¹⁰Bi is 0.25% of the ²²⁶Ra response, the apparent half-life would vary from 1900 a, for a recently purified sample, to 1620 a, for preparations made at the time when the radium was discovered. A discussion of the effect on the half-life measurement of the longlived ¹³⁷Cs is given by Martin and Taylor ([299]), who attributed an uncertainty component of 0.22% to the ²¹⁰Pb/²²⁶Ra disequilibrium. The formulae for the change of activity ratios in decay chains are described in textbooks on radioactivity, for example by Evans ([143]). Numerical expressions for the uranium-radium chain are given in detail by Weiss in Kohlrausch ([266]), vol. III (1986). Fränz, in Kohlrausch ([264]) gives a formula and a diagram describing the influence on activity measurements of the ²²²Rn/²²⁶Ra disequilibrium in freshly prepared samples. A summary on the measurements of ²²⁶Ra and related problems can be found in a section by Weigel ([465]) on Radium, suppl. vol. 2, in the handbook of Gmelin.

In standards laboratories a set of about five ²²⁶Ra reference sources is usually available. These contain different amounts of radioactive substance covering an activity range of the order of 100. They are of similar construction and have the same attenuation properties. They are carefully compared at regular intervals to check the stability and linearity of the chamber. This makes it possible to measure samples of different activities, by choosing an appropriate reference source, in such a way that the current ratios fall within a range of about five, an arrangement which improves the accuracy. Results of current measurements from five reference sources covering a range of about 100 in activity (with a maximum of $308\mathrm{\mu g}$ of radium element) were published by Rytz ([376]) showing relative standard deviations of about $2\cdot {10}^{–4}$ over a period of about 7 a (Figure 10-1-Figure 10-2). The repeatability for a single source for a series of 50 measurements was about 0.09% ($1\sigma$).

3.1.  $4\mathrm{\pi }\mathrm{\gamma }$ ionization chambers

The optimum design of a re-entrant (or $4\mathrm{\pi }$) ionization chamber for activity measurements of photon-emitting radionuclides is nowadays a pressurized cylindrical construction. The earliest pressurized ionization chambers for capturing high-energy events can be found in the field of cosmic-ray measurements [136], [237], [260], [310], [77], [78], [428]. A detailed study on argon as filling gas is given by Hopfield ([214]). A review article with references by Steinke ([426]) can be found in the Handbook of Physics ([167]).

Another branch in the development of pressurized devices came from the field of dosimetry measurements, in earlier times associated with the names of Gray ([182], [183]) and Sievert ([411], [412]) who studied fundamental properties of ionization by radiation in gases as functions of pressure and other parameters. A short review of dosimetry and activity measurements is given by Allen ([4]). Many other references and details, mainly applied to dosimetry, can also be found in the review article on ionization chambers by Boag ([30]) and in Kment and Kuhn ([257]). The first practical designs of pressurized devices for dosimetry and radioprotection are given by Carmichael ([64], [65]) and by his co-workers Smith et al. ([417]), Steljes ([427]), Peabody ([338]), see Figure 11, which led to the development of ionization chambers for activity measurements of the form used today. These publications explain many technical details of the manufacturing of electrodes, insulators and gas fillings, etc. for these chambers. The working principles of the cylindrical ionization chambers filled with pressurized gases are more generally described by Siri ([416]), Taylor and Sharpe ([442]) and Sharpe ([407]).

Wade ([458]) constructed an ionization chamber (T.P.A. Mk IV), filled with argon at a pressure of 1 atmosphere (Figure 12), in which he added an inner aluminium tube for the reentrant well. This chamber is optimized for sources with $\mathrm{\beta }$-particle emitting radionuclides.

In the same way, one of the first pressurized ionization chambers (T.P.A. Mk II) was constructed with modified electrodes, a steel well and filled with argon at a pressure of 20 atmospheres by Sharpe and Wade ([405], [406]), Figure 13. More details of the design of this reentrant ionization chamber for activity measurements and calibration figures for some radionuclides can be found in Sharpe and Wade ([406]). This model is still in use today and, with minor modifications, it is available on the market from Centronic ([69]). Since then, Centronic ionization chambers have been installed in many national standards laboratories where they are used as secondary standard measuring systems for activity [307], [372], [376], [437], [395], [58], [441], [177]. A schematic drawing of this chamber is shown in Figure 14.

At about the same time as the work by Sharpe and Wade ([405]), other aspects of the design of a “high precision”, high-pressure ionization chamber, among them the location and construction of the insulators, guard rings and guard-ring connectors were developed by Shonka and Stephenson ([408]), Figure 15, and published by O’Kelley ([326]). This chamber was filled with 40 atmospheres of dry argon. The high sensitivity of such a chamber necessitates enclosure in a lead housing with walls some $10\mathrm{cm}$ thick to reduce environmental background effects [316].

Examples of other designs may still exist in standards laboratories [187], Figure 16, [45], [190] [27], Figure 17-1-Figure 17-2, [286]. Several pressurized chambers for activity measurements of photon-emitting radionuclides in the field of applications, especially in nuclear medicine, have been designed by manufacturers, such as Capintec, Merlin-Gerin, Meßelektronik Dresden (former Robotron) [402], ORIS, Philips, Physikalisch-Technische Werkstätten (PTW) Freiburg, Picker, Radcal, Siel (Nuclear Data), Veenstra, NE Technology (former Vinten), Zinsser Analytic. These have been or are still on the market (Section 3.2.4). Examples are shown in Figure 23-1-Figure 23-4. Among these the Vinten chamber, now manufactured by Nuclear Enterprises Technology Ltd. [319], should be emphasized. This was developed and tested at the NPL [476], [482], [479]. This system is delivered with a complete set of calibration figures measured at the NPL and the chamber can be delivered with a certificate from the NPL. It is thus traceable to NPL primary activity standards. It is in use in several national standards laboratories.

All these ionization chamber designs take the form of an external vessel of steel or aluminium with a thimble or well for introducing the sample (Figure 14). The vessel must be carefully welded or sealed to maintain a pressure of about $2\mathrm{MPa}$ (in old units about 20 atmospheres) over many years. The thickness of the walls should be about $3\mathrm{mm}$ to $4\mathrm{mm}$ to prevent deformation by pressure. One of the critical points in the design is the connectors (or insulators), which must be screwed and gas tight. For more details, see [442], [29], [160], [257]. The fabrication of ionization chambers is today an area which depends on the special know-how and competence of the manufacturing firms. The inner electrodes are nearly always made of aluminium foil. The copper or brass found in older constructions is no longer in use because of their larger content of $\mathrm{\alpha }$-particle emitting impurities which increase the background current, and because they have other unfavourable electrochemical and mechanical properties. A special chamber-electrode design has been suggested by Kostyleva et al. ([269]): this uses materials which optimize the attenuation effects of photons and electrons. An electrode coating with high-Z materials has been studied by Kleeven and Wijnhoven ([256]) with a view to increasing the chamber response.

At present, the most common filling gases are purified argon or nitrogen. Argon has a higher sensitivity, at least at photon energies below $100\mathrm{keV}$, but the response of nitrogen is more nearly linear with photon energy (Chapter 7).

3.2.  Special types of ionization chambers and their applications

4.1.  Electrometer picoammeters

An electrometer picoammeter usually has an input resistance of more than $100\mathrm{T\Omega }$ ($\mathrm{1 014}\mathrm{\Omega }$) and / or, depending on the connected voltage, an instrument leakage current (including offset contribution) of less than a few times 10^–15 A. These characteristics define a device that can measure extremely low currents, charges or voltages, the choice of any particular one depending on the way in which the feedback at the high-resistance entry stage is connected. Operational amplifier theory can be applied to explain the behaviour of various feedback types. A short summary is given in the next section.

To measure such low input currents, special care must be taken in the choice of insulating materials and in the cleanliness of the surfaces, and the path lengths on insulators should be long [251], [250]. The best material for use around the input point is sapphire, the next choice being teflon of high quality. For the capacitors, the models made from styroflex give good results, for example from Siemens ([410]). The input stage must be electromagnetically shielded in a metallic Faraday box against electromagnetic stray fields, and an optimum grounding of the components must be performed by direct connections in the housing: ground loops must be avoided in the circuitry. Radiation shielding of the entrance stage is recommended because ionizing effects in insulating materials may result in an undesirable increase of the low conductance. For all these reasons high-quality instruments have the preamplifier entry stage in a specially designed independent housing.

Of the four classes of electrometers referred to, only the dynamic condenser electrometer or vibrating-reed electrometer (VRE), and the solid-state device electrometer are still of practical interest. Their working principles are quite different.

• Vibrating-reed electrometers have one plate of their entrance capacitor charged by the current to be measured. This plate (the reed) vibrates with a stable frequency, modulating the electrostatic field of the capacitor and producing an AC signal which is capacitively coupled to an indicator stage, usually a vacuum tube, and amplified (Figure 27). Sophisticated nullindication or feedback methods with phase shift and rectification are used to transform the AC signal into a measure of the charge at the entrance stage. Such systems and their parameters are described by Palevsky et al. ([330]), Friedlander and Kennedy ([154], [155]), Staub ([425]), Fassbender ([149]), Loevinger ([280]), Frieseke and Hoepfner ([159]), Cary ([66]), Gühne and Rodloff ([191]), Guiho et al. ([193]) and Böhm ([34]). Provided that the working conditions (frequency, amplification, etc.) are stable, the transformed signal, or a compensation signal, can be used and subsequently displayed by a voltage-measuring instrument [245]. The impedance of such a system lies in practice close to infinity and measurements are limited only by the resistivity of the construction materials used for the entry stage.

• Solid-state electrometers possess the high input resistance of highly purified material in the entry stage. Today, metal-oxide-silicon field effect transistor (MOSFET) stages give the best results. A particular solid-state electrometer is characterized by the design of its entry stage and the stage in which the function and range adjustment occurs, it may display in analogue or digital form. In modern low-current-measuring systems, microcomputer-controlled digital electrometers are mainly used [248], [249], [250]. The entry stage of such an instrument is a high-resistance preamplifier with a MOSFET, a main amplifier with the range and function switching connected to an analogue-digital converter which in turn, is connected to a display and a standard interface (Figure 28), for example an IEEE-488 bus. All the components are surveyed or controlled by a microprocessor or small computer which may also serve as the interface between instrument and user.

The advantages of the MOSFET entry stage are [251]:

• good high-input impedance operating as a voltage amplifier,

• good low-current characteristics, down to several 10^–17 A,

• good stability under normal working conditions,

• easy realisation of input protection circuitry,

• commercial availability.

Other entry stage circuits are realized with junction-field-effect transistors (JFET). A JFET generally has lower input resistance, lower voltage noise and offset, but higher input current offset and noise than a MOSFET. Recent electrometer-input circuits go down to offset currents as low as a few 10^–15 A. Commercial Keithley instruments are practical realizations of such electrometers. Other industrial models are connected to instrument systems with ionization chambers for radiation protection and nuclear medicine (Section 3.2.4). A portable, “intelligent” electrometer for applications with ionization chambers in the field of radioprotection is presented by Halbig and Caine ([195]). Another field of low-current measurements is that of dosimetry where many helpful suggestions relating to electrometer techniques can be found, for examples, see [34], [37] and [266]. A study of the quality of a current integrator for dosimetry measurements realized with a commercial Keithley instrument with digital filter techniques is described by Brose ([49]).

The varactor bridge [251] which operates at a frequency of several hundred kilohertz works on quite a different principle. One of the capacitors in the bridge is a varactor diode, the capacitance of which changes if a DC voltage is applied to the bridge. This unbalances the bridge and creates an AC output signal which is proportional to the DC input being measured. Varactor bridge circuits are rarely used in electrometers as they:

• require extensive associated circuitry,

• take an excessive time to recover from overload,

• have high susceptibility to high frequency interferences,

• show high sensitivity to environmental changes or working conditions.

A comparison of various electrometer types with notes on the concepts required for the design of femtoampere circuits is given by Patstone ([336]), together with some suggestions for guard ring configurations and the necessary specifications for resistors and capacitors for low-current applications.

For other electrometer components, like main amplifier, analogue-digital converter, etc., integrated electronic circuits or components of high quality, taken from modern standard data acquisition systems, are used [444].

4.2.  Feedback circuits and current integrators

General amplifier theory distinguishes between current and voltage feedback [103], [291], [448] and shows that the choice depends on the quantity to which the feedback signal is proportional; in the latter case the feedback voltage may be connected in series or in parallel with the input signal. The parallel arrangement is commonly called an operational amplifier [103], but this term may be extended to any linear amplifier [448] used for metrological applications or in analogue computers [268]. Electrometer circuit design can easily be understood by applying basic amplifier theory [103], [280], [178], [448], and is relevant for the design of solid-state device electrometers.

In the general sense, an operational amplifier is an electronic circuit with a positive and a negative input having very high input resistances, low offset currents and a linear response. The output voltage is given by the difference between the input potential and earth, multiplied by an amplification factor which is typically of the order of 10⁵. To this circuit a feedback network is taken directly from the output, or from a voltage divider at the output, to one of the inputs (normally to that of negative polarity) via a resistor, a capacitor or a shunt (Figure 29-1-Figure 29-4 [491]). In the latter case the active signal is fed into the input of opposite polarity (series feedback). To derive the relations between the input and output signals, Kirchhoff’s laws are applied either to the sum of the currents in a connecting knot or to the voltages on a closed loop of the network.

Four such types of feedback circuit are described by Zsdanszky ([491]) (Figure 29-1-Figure 29-4), and Mann et al. ([294]). Praglin ([340]) compares a number of electrometer constructions using electrometer tubes, vibrating reed and FETs in terms of noise, offset current and drift. More recent reviews from Keithley et al. ([251]) and Keithley ([250]) distinguish between the various circuits in terms of their applications, such as electrometer voltmeters, picoammeters and coulombmeters, but reach with equivalent formulae and parameters: see also [266]. Böhm ([34], [37]) describes the dynamic behavior of an integrator circuit so as to include impedance and stray components. An equivalent-circuit diagram of the current integrator is shown in Figure 30. In modern applications with high accuracy requirements the feedback-type current integrator is used [280], [193], [38], [390]. The measurement system installed by Santry et al. ([390]) corresponds to the working principle of a feedback coulombmeter, (Figure 28 [249], [251]).

In the current integrator (or coulombmeter) a feedback capacitor $C_{\text{F}}$ is charged, and the charge is integrated during the measuring time $\Delta t$ [491]. As for any inverting amplifier, the current summing point is held at virtual ground level ($V_{\text{i}}=0$) by the very high gain of the amplifier and its feedback network. This produces an output voltage $V_{\text{o}}$, Kirchhoff’s law for the feedback loop leads to a relation for the input current $I_{\text{i}}$, i.e.

where the current $I_{\text{i}}$ should be constant. The sensitivity of such a device is determined by the capacitor $C_{\text{F}}$ which gives an input capacitance $\mathrm{\alpha }\cdot C_{\text{F}}$, where $\mathrm{\alpha }$ is an amplification factor of order 10⁵. In comparison with the capacitance $C_{\text{F}}$, the capacitances of the operational amplifier input and the connecting cables are negligible.

Sophisticated current-measuring systems register several output voltage values as a function of time to pick up the optimum working condition of the electronics. Assuming that the ionization current to be measured is nearly constant, this makes it possible to carry out a linearity check in each individual time interval during integration. An RC filter at the integrator input is recommended with a time constant of order ${10}^{–3}\mathrm{s}$ to reduce instabilities from noise or hum at the entrance stage.

Practical conditions [251] for running the operational amplifier in this way are that

• the insulation resistances around the input are very high,

• the leakage currents during the time $\Delta t$ are negligible,

• high quality capacitors are used, without residual charges retained by dielectric absorption during the time of shorting.

Such a current integrator (or coulombmeter) system may be controlled either by preset timing [307], [441] or by voltage level discrimination [491], [372], which can be chosen to suit different orders of ionization current values. Alternatively, range switching can be done by changing the capacitor $C_{\text{F}}$. Some authors [169], [491] describe current integrators with voltage discrimination where the discriminator generates a reset pulse that removes a fixed charge from the integrating capacitor: The number of pulses per time is proportional to the current to be measured. [169] report on currents of 10 pA measured by this method with an accuracy of 1%.

A similar method of charge sampling is described by Yair [489]. In this method the charge generated by a current source is collected on a capacitor during a fixed time interval in which the capacitor is isolated from the amplifier. Discharging the capacitor produces an AC pulse which is then amplified. Since a direct current is converted into an AC pulse, there is no requirement for DC feedback over the entire system and the need for high performance feedback resistors is eliminated. The output display is obtained either by a peak-value measurement of the amplified AC pulse, employing sample-and-hold techniques and direct meter indication, or by integration of the amplified output pulses connected to an analogue-digital converter with a digital indication. The method is claimed to give good noise reduction with AC currents of order 5 fA. The need for a special construction with a reed-relay to switch the input capacitor is a disadvantage. The system is considered to be reliable for photon measurements in exposure monitoring.

Similar systems are described by other authors [5], [385], but in most cases they are not particularly adapted for use in ionization current measurements for activity determinations with high accuracy.

4.3.  Measurements by voltage drop across a high value resistor

Measurements of ionization current in terms of the voltage drop across a high value resistor have their origin is the method of constant instrument deflection [48], [290], [309], [236] in which the voltage drop is measured directly by an electrometer. A technical difficulty is the realization of high performance, “high-megohm” resistors of order ${10}^{10}\mathrm{\Omega }$ or higher with minimal size and mass. In earlier times, air resistors with ionizing radiation from radioactivity were used [48]. Later on, these were replaced by liquid column resistors or thin layer resistors of platinum on amber [263]. Today, modern industrial constructions [457] of a carbon-coated glass rod with silver contacts are available, the resistors being vacuum-sealed in a glass envelope and surface-treated with a special silicon product to eliminate the effects of moisture. It is evident that any contamination, and especially fingerprints, must be avoided on such a device. Resistance values of ${10}^{12}\mathrm{\Omega }$ to ${10}^{14}\mathrm{\Omega }$ can be obtained with these techniques.

A technical realization of the voltage drop method with compensation was published by Jaeger ([235]) and applied to ionization-current measurements in the range 10^–9 A to 10^–12 A in x-ray dosimetry. As a current source for the compensation, a Bronson resistance [48] is used, with ionizing radiation from a U₃O₈ layer inside an air capacitor running at saturation current. A compensation current, ranging from almost zero to $5\cdot {10}^{–10}\mathrm{A}$, is controlled by a variable diaphragm. Similar uranium cells have been used in compensation methods since the beginning of radioactivity measurements: see also the Townsend induction balance (Section 4.4).

A more recent current-measuring system is described by Walz and Weiss ([459]) using the voltage-drop method with compensation on the other side of a resistor of ${10}^{10}\mathrm{\Omega }$ to ${10}^{12}\mathrm{\Omega }$ (Figure 31). The system is directly connected to the ionization chamber together with a null instrument of very high input resistance, for example a vibrating-reed electrometer (VRE); the other side of the resistor is connected to a variable, but highly stable, voltage source realized by an electronic compensator. The input voltage at the VRE is compensated up to a small residual voltage of less than 10 mV by means of a variable voltage divider of high accuracy (relative adjustment of 0.001%). This residual voltage shows fluctuations caused by the statistical processes of radioactive decay, current detection and electronic noise. To record and average the voltage values, a voltage-frequency converter is connected to the output of the VRE. The pulses generated are fed to a scaler and integrated over a preselected time (typically 100 s). The number of counts in the scaler divided by the integration time is a measure of the mean value of the residual voltage. The residual voltage, added to the voltage reading of the compensator, gives the total “voltage drop across the high-ohmic resistor”. The ionization current can be calculated from the known value of the calibrated resistor. With such a measuring system, however, the current equivalent of a ²²⁶Ra reference source can also be measured, and a calibration in terms of equivalent activity can be carried out (Section 2.5). This avoids uncertainty components from the resistor calibration.

4.4.  Townsend induction balances with compensation

A Townsend induction balance [450] uses a current-integrating capacitor similar to the coulombmeter with an external capacitor as described in Section 4.2. The current source, here the ionization chamber, is connected to one side of the capacitor and a compensation voltage source to the other side; the electric charges on the two sides are balanced. The control signal is taken from that side of the capacitor to which the ionization chamber is connected, using an instrument or electronic circuit of very high resistance or very low leakage current. Uncertainties from current losses must be minimized. The correct compensation voltage is adjusted by “feedback” from the control signal, either continuously or in steps, a process which is comparable with the feedback in a coulombmeter with an external capacitor. More generally, an induction balance can be defined as a device where the charges related to the measured current are balanced on both sides of a capacitor during the short time interval of the control or “feedback” step. This permits the time behaviour of the control or feedback circuit to be described and takes into account the fact that the charges are not exactly balanced all the time.

Several types of compensation circuits have been constructed starting with Townsend ([450]) (Figure 32), who studied the theory of “Genesis of Ions by the Motion of Positive Ions in a Gas”. Different components for the compensation (Figure 33-1-Figure 33-3) were used, among them a cylinder capacitor charged by manual control [44], variable capacitors (Figure 33-2 [200]), supplementary current sources (Figure 33-3 [235]), a second ionization chamber as current source [234], [252], [253] and a uranium source with a second collector electrode inside the chamber [428]. For further details of compensation-system components, see the articles in [263] and [30].

Many devices for ionization current measurements have used a stable capacitor in combination with different control loops [30]. Photocell compensators were developed by Jacobsen [234] using two coupled ionization chambers of opposite polarity, by Leo and Hübner ([276]) and by Hübner ([221], [222]) with an electrostatic electrometer (Figure 34), and by Rothe and Willuhn ([358]) with a quadrant electrometer. These compensators are rather sensitive to deviations from the adjustment and to mechanical shocks. Lea ([274]) obtained automatic balance by a direct coupling in which he charged up the highly insulated grid of the first valve of an amplifier, its output being fed to one plate of the condenser. A motor-driven potentiometer, controlled by a Lindemann-Ryerson electrometer, was designed by Garfinkel ([164]) using stepwise compensation (Figure 35-1). Similarly, a vibrating-reed electrometer connected to a servomotor-driven compensation device worked almost continuously (Figure 35-2 [175]). A null balancing circuit was described by Geiger and Campion ([167]). Cloos and Heigwer ([71]) proposed an automatic compensation instrument with an electronic control circuit for the induction balance.

In other designs developed in national standards institutes for activity measurements with ionization chambers, vibrating reed electrometers have usually been used to control the compensation. Vibrating-reed electrometers with compensation by relay switching at well-defined time intervals have been described by Merritt and Taylor ([307]) (Figure 36-1-Figure 36-2), by Tapp ([441]) and by Guiho et al. ([193]). A vibrating-reed electrometer with compensation at a very stable voltage level and counting the compensation steps was used by Rytz ([372], [376]) (Figure 37-2). A digital feedback circuit with a vibrating-reed electrometer, a voltage-frequency converter with a pulse counter, and a digital-analogue converter of good accuracy as a compensation voltage source was developed by Deike and Walz ([104]) (Figure 38-1-Figure 38-2). A feedback circuit with fast, programmable analogue-digital and digital-analogue converters was described by Schrader ([397]) (Figure 39-1-Figure 39-3); the information from the control signal at the capacitor input is digitized by an analogue-digital converter at well-defined time intervals. The digitized signal is introduced into a fast computer, permitting an optimum compensation voltage value to be calculated in less than a millisecond. This value is fed to the capacitor by a digital-analogue converter of high accuracy [444]. A timing diagram for the compensation process is shown in Figure 39-3.

Common to all the compensation circuits with a fixed capacity value $C$ is the measurement of the change in voltage $\Delta U_{\text{c}}$ across the integrating capacitor (including the capacitance of the ionization chamber itself, the cables, etc.) — or of a quantity proportional to $\Delta U_{\text{c}}$ — during a time interval $\Delta t_{\text{m}}$. This leads for the current to the formula

This equation looks very similar to that for a feedback-type current integrator at Equation (12).

All these compensation methods require stable capacitors, e.g. the styroflex capacitors manufactured by Siemens ([410]). Another possibility is to construct a mechanical parallel plate capacitor [358], [350], [353] as used in national standards laboratories for calibration of capacitances [263]. A recent example of such a capacitor by Reher et al. ([353]) is shown in Figure 40. The capacitors of a measuring system should be mounted in a special box to protect them from moisture and contamination (Figure 36-2), and like other preamplifier components, they also must be shielded against external electromagnetic fields and protected from radiation (Section 4.2). To connect the capacitors at the beginning and end of a measurement or to switch ranges reed relays mounted near the capacitor should be used [254], [422], [50], [441]. Precautions must be taken against stray fields and protection measures comparable with those for the entry stages of low-current devices are required.

An advantage of compensation methods, like the Townsend induction balance, is that the voltage level at the ionization chamber output is maintained near zero. This reduces leakage currents from the electronic components coupled to the ionization chamber output, so the offset and leakage currents of the control instrument remain small in comparison with the current to be measured. Furthermore, this method does not increase the voltage in the ionization chamber and so avoids changes in the field strength applied in the collection of charge.

5.1.  Fluctuations from ionization and charge collection

This section details particular features of charge collection studied by current measurements with re-entrant ionization chambers. Few publications are available on the subject, but measurements of a minimum standard deviation with an ionization chamber are reported by Garfinkel ([164]), Merritt and Taylor ([307]) and Weiss ([469]) (Figure 41-1-Figure 41-4).

The first study of fluctuations in activity measurements with ionization chambers as a function of source activity $A$ was reported by Garfinkel ([164]). He measured the ionization currents produced by four radium sources, the activities $A$ of which ranged from $3\mu \text{Ci}$ to $200\mu \text{Ci}$, repeating each measurement several times during individual time intervals of duration $t$. The standard deviation ${\sigma }_{\text{I}}$ of the measured individual current values was calculated and plotted versus ${\left(At\right)}^{-1/2}$ or, assuming a proportionality between $A$ and $I$, versus ${\left(It\right)}^{–1/2}=Q^{–1/2}$. This is analogous to plotting ${\sigma }_{\text{N}}$ versus $N^{–1/2}$ for a counting experiment. In the experiment of Garfinkel the curve (Figure 41-1) takes the form of a straight line which may be extrapolated to ${\left(At\right)}^{–1/2}=0$, i.e. to the condition for a source of very large activity, giving a limit in “ultimate precision” of about 0.02%. This value represents the uncertainty of the complete ionization-chamber measuring system. Garfinkel used an unsealed chamber which, therefore, was at atmospheric air pressure, and interpreted his value as due to fluctuations in the density of the air in the chamber.

A similar plot of measured relative standard deviation versus the total charge collected in a single measurement is reported by Merritt and Taylor ([307]) (Figure 41-3). Since the data can be fitted with a straight line passing through the origin with a smallest relative standard deviation of 0.016%, they interpreted, the value as arising predominantly from the statistical nature of radioactive decay rather than from instrumental effects. An argument was also given which relates the relative standard deviation to the photon energies of various radionuclides.

Weiss ([469]) made repeated measurements of sources having different activities, and plotted the relative standard deviations $s$ of the single measurements against $Q^{–1/2}$, where $Q$ is the charge collected during the time of a single measurement (Figure 41-2) and is proportional to the $At$ of Garfinkel. To a first approximation, $s$ was assumed to be the sum of the internal standard deviation $s_{\text{i}}$ of the measuring system and the standard deviation sR of the radiation. By linear extrapolation of the plot to $Q^{–1/2}=0$ a minimum value of $s_{\text{i}}$ was found: this was close to zero. Weiss ([469]) gives a value of 0.005%. Recent measurement results from the secondary standard measuring system of the PTB are given in Figure 41-4 with a minimum relative standard deviation near to 0.01%.

The total uncertainty of current measurements with ionization chambers is further discussed by Guiho et al. ([193]). The values reported on the curves shown range from about 0.07% at 10^–9 A to about 3% at 10^–15 A and were obtained with a Townsend induction balance system (Section 4.4). The uncertainties of the electronic parameters of the current-measuring system are also discussed.

5.2.  Variation of electronic parameters

Apart from the statistical fluctuations, systematic variations may be caused, for example, by drifts of temperature or long-term changes in the electronic components. To minimize these influences, cyclic measurements of background, reference source and unknown source should not take more than one hour. Data evaluation with calibration against a reference source is recommended (Section 2.5) because this eliminates the long-term instabilities which produce similar effects during measurements of the sample and the reference source, but may change over periods longer than one hour.

Other variations may be caused by unstable power supplies or by electromagnetic fields which disturb the entry stage of the current-measuring electronics. These are mostly short-term effects which may be detected by statistical analysis of a series of measurements, and if necessary, eliminated.

Special care must be taken to use a stable power supply for the polarizing voltage of the chamber. In older measuring systems, large accumulators connected in series to give a potential of several hundred volts were often used. Most of these have been replaced by electronic power supplies which provide low hum and low noise for the polarizing voltage of the ionization chamber. They are decoupled from the supply for the current-measuring electronics. This avoids disturbance of the very sensitive entry stage of the current-measuring electronics.

5.3.  Saturation-loss effects

When the ionization-chamber current from a high-activity sample is measured, the effects of saturation losses caused by incomplete ion collection and recombination effects [214], [30], [31], [160], [76], [35] must be considered, see Figure 42 from Colmenares ([76]). Much more intense radiation fields may be applied to cylindrical re-entrant ionization chambers before saturation losses become important than to the smaller plane-parallel plate chambers used for dosimetry applications, however, volume (general) recombination, initial recombination and diffusion losses during charge collection appear with the same characteristics in both. According to Boag ([30]), Böhm ([35]) and others, only volume recombination varies with the radiation field. The charge-collection defect due to volume recombination, depends on $I/U^2$, where $I$ is the measured ionization current and $U$ the collecting potential and occupies a certain region of an ($I,U$) diagram. Charge-collection defects due to initial recombination and diffusion loss depend on the reciprocal value $1/U$ of the collection potential. Measurements of these effects are mainly carried out on chambers for dosimetry applications [160], [223], [423], [30], [31], [35], [314].

Measurements of saturation-loss effects in a re-entrant cylindrical ionization chamber for activity determinations are described by Weiss ([469]) (Figure 43-2). The saturation loss is obtained by measuring the ionization current $I$ at increasing collecting potentials $U$ for a source of several hundred $\mathrm{MBq}$. By plotting the measured currents against $1/U$ a nearly straight line is obtained which, extrapolated to $1/U=0$, yields the saturation current $I_{\infty }$. The saturation loss is defined as $\left(I_{\infty }-I\right)/I$, where $I$ is the current measured with the collecting voltage applied usually. The diagram of Weiss (Figure 43-2) shows saturation loss as a function of the ionization current $I$. The values for a chamber filled with air at atmospheric pressure and normally operated at a collecting voltage of $400\mathrm{V}$ were determined by this method which, it should be noted, can be applied only if the effective chamber volume does not change with the collecting voltage. An alternative diagram is shown by Merritt and Taylor ([307]) who plot the chamber response versus the operating voltage (Figure 43-1).

A similar method has been used at the PTB to observe the saturation effects in different ionization chambers: in this, the ionization current $I$ is measured as a function of the collecting voltage $U$ for several sources covering a range of at least fifty in units of activity (or current). The collecting voltage is varied by about a factor of three and a relative quantity, the ratio $I/I_{\text{o}}$ of the ionization current to the current at $500\mathrm{V}$ (at the commonly used voltage), is plotted as a function of the voltage $U$ (Figure 44-1-Figure 44-2). Necessarily, the curves for all sources cross at the normalization point ($U=500\mathrm{V}$). For a chamber with good saturation qualities, the diagram shows straight lines with nearly the same values at the measured points for all sources. The slope of the straight line is about 0.05% in relative current change for a potential change of $100\mathrm{V}$ (Figure 44-1), which is comparable with the result of Merritt and Taylor ([307]) (Figure 43-2). For a chamber of low quality (Figure 44-2), the steepest slope was obtained at low collecting voltages (from $300\mathrm{V}$ to $400\mathrm{V}$) about 0.5% relative current change for a potential change of $100\mathrm{V}$ using the strongest source (this had about $40\mathrm{MBq}$ of ²²⁶Ra equivalent activity). This value changed by about 0.05% per $100\mathrm{V}$ for the weakest source of about $1\mathrm{MBq}$ of ²²⁶Ra equivalent activity. The variation of the slope is used as an empirical criterion for the charge-collection quality of the chamber. Woods et al. ([482]) give values of the saturation-loss characteristics for the radionuclide calibrator (Figure 45), developed at the NPL, with important saturation losses for currents only above ${10}^{–}8\mathrm{A}$.

5.4.  Linearity in response to activity

Good linearity in the response of an ionization chamber as a function of the source activity is one of its most essential qualities since the required range of measured activity values may cover more than five orders of magnitude. The linearity of an ionization-chamber measuring system is influenced mainly by saturation effects and by the linearity of the current-measuring electronics. In the latter, one important non-linearity effect is range switching. Generally, all the disturbing effects discussed in the preceding chapters may cause non-linearity in the activity response, but some may be eliminated by measurements relative to a reference source producing about the same ionization current as the sample to be measured. In this case, the activity ratios of the various reference sources must be known with good accuracy (Figure 10-1-Figure 10-2 [376]).

Linearity checks are often discussed in the literature on radionuclide calibrator measurements for nuclear medicine applications [270], [238], [3], [390], [389], and in the related standards for radionuclide calibrators or in regulations [8], [10], [476], [110], [230], [231]. The procedures described for these linearity checks apply to any measuring system with an ionization chamber.

Instrument linearity should be checked by measuring the ionization current from the maximum activity likely to be used down to the minimum reading possible with the instrument, following the decay of a radioactive source. This defines the activity range for which the instrument is properly calibrated and allows correction factors to be deduced for the radionuclide under test. A suitable radionuclide for this purpose is ⁹⁹Tc^m. The measured raw data are corrected for background and decay, applying the relevant half-life; $T_{1/2}$(⁹⁹Tc^m) = 0.25025 (8) d [220]. Examples for two different radionuclide calibrators are given in Figure 46-1-Figure 46-2, the first shows a continuous deterioration of linearity as the activity (or current) increases, and the second a sudden breakdown of linearity at a certain limiting level at high activity. From Equation (2) the activity values can be calculated as a function of measurement time and compared with the measured activity values. If the cause of a deviation can be explained and continuous measurements show reproducible effects, a correction factor can be deduced. In the case of electronic switching effects with adjustable electronic components, direct adjustment of the relevant electronics is preferable. After each adjustment a linearity check or a check against the reference source is required. A more sophisticated form of evaluation is to perform a linear regression analysis of the logarithm of the measurand as a function of time. This allows the residuals of the individual measuring points to be studied in detail and helps to show up trends (Figure 47-1-Figure 47-2). For example, decade switching of the time-measuring electronics exhibits a step function characteristic at the corresponding measurement value (Figure 47-2).

For high-quality instruments residuals of the order of 0.1% can be expected over an activity range of at least three orders of magnitude. At the extreme upper end of the range, the current value usually shows a continuous decrease relative to the activity because of saturation-loss effects. An abrupt cutoff of the current above a certain level may indicate a failure of the instrument electronics caused by subjecting particular components to excessive amplitudes or frequencies of signal. At the low activity end of the range, the statistics of the current-measuring process (see preceding section) increase the differences with respect to a linear curve (Figure 48-1-Figure 48-2). This effect may be superimposed on contributions from radionuclidic impurities with half-lives which differ from that of the radionuclide used as a check source. In the case of ⁹⁹Tc^m, for example, contamination with ⁹⁹Mo ($T_{1/2}=2.748\text{ d}$) may increase the current causing deviations from the expected values of the measurand in a logarithmic scale (Figure 48-1).

Regulations for radionuclide calibrators require that relative deviations from the calculated values be smaller than 5%. Today’s instruments, if correctly adjusted, are quite capable of achieving this, as is confirmed by Bullen [55], Kowalsky et al. [270], Jain and Rehman [238], Woods et al. [482], Ahluwalia [3], NCRP [316], Santry et al. [390], Santry and Bowes [389].

An alternative to the decaying source method is a linearity check using a set of sources having known activity values covering the range of interest. The sources are prepared by dividing a single strong source into parts and measuring these successively, or by preparing different quantitative dilutions to give a set of sources with suitable activity values. For both procedures careful weighing is essential, and care must be taken to use the same measurement geometry (containers, volumes, etc.) for all measurements. These procedures introduce supplementary uncertainty components, which may be small, but are avoided when using the decaying source method.

6.1.  Background

The background in ionization-chamber measurements, as in many other radiation measurements, is composed of radiation from

• materials of the chamber itself,

• radioactive gases inside the chamber well and nearby, such as radon and its daughters,

• materials around the chamber, such as sample holders, supports, shielding, any of which may experience unintentional contamination,

• natural radioactivity in the materials of the environment, such as the walls of the building and soil,

• external sources, especially near accelerators or reactors,

• cosmic-ray events, which produce secondary interactions in the earth’s atmosphere.

Inside the ionization chambers the background is due mainly to ionizing events following photon emission, but a small fraction comes from $\mathrm{\alpha }$ or $\mathrm{\beta }$ particles emitted from the chamber materials in such a way that they reach the sensitive chamber volume. The fundamental characteristics of background and shielding are described in publications on the background of low-level radioactivity measurement laboratories [188], [267], low-background Ge detector systems [61], [207], [13], [208], [209] and studies on environmental radioactivity with ionization chambers [404], [151], [171] including those on the background from decay products of radon [325], [334]. A typical background spectrum from a Ge detector is shown in Figure 49. All this can also be applied to the shielding of ionization-chamber measuring systems. For a general review of background and detector shielding, see [258], which contains a list and classification of the relevant radionuclides, for a discussion of effects arising in the detectors themselves: see also [316].

The radioactivity from materials commonly used in and around the chamber is due to low concentrations of naturally radioactive elements contained as impurities. Its components are radiation from potassium (⁴⁰K) and from the thorium- and the uranium-radium decay series [151], [258]. Other contaminating nuclides in materials are ⁷Be and ²²Na, produced by cosmic-ray reactions, and fission products from nuclear weapons and reactor accident fall-out, among them the well known ¹³⁷Cs (or ⁹⁵Zr and ¹⁴⁴Ce, with shorter half-lives than ¹³⁷Cs) which may be picked up during the material production process. Recently produced steel may also contain impurities of ⁶⁰Co, ¹⁰³Ru and ¹⁰⁶Ru from the furnaces. These radionuclides were frequently used in the 1950s to check for erosion of furnace liners in steel production. On the other hand, steel is a better construction material for the chamber than copper or brass because it has a low level of natural radioactivity, its origin being the thorium and uranium decay series [61]. Special care must be taken when aluminium is used as a construction material inside the chamber, as it is highly contaminated with decay products from the uranium and thorium series, and so releases $\mathrm{\alpha }$ particles [16], [404].

If properly shielded, the background reading for a pressurized re-entrant ionization chamber is typically about $50\mathrm{fA}$ for a sensitive volume of order of $10\mathrm{l}$. This is equivalent to a ²²⁶Ra activity of about $1\mathrm{kBq}$ to $3\mathrm{kBq}$ for a modern pressurized ionization chamber. In the activity determination of a sample, the measured current $I_{\text{m}}$ must always be corrected for background current $I_{\text{b}}$, the pure sample current being $I=I_{\text{m}}-I_{\text{b}}$ (Section 2.4), both measured with the same instrument settings and under identical measuring conditions.

One of the most important background components arises from the presence of radon and its decay products inside the ionization chamber well, in the space around the chamber and from a deposite on the wall materials inside the shielding. This component shows seasonal fluctuations with maximum values, for the northern hemisphere, in the months June, July and August and a minimum in January and February. This follows the general fluctuation in the release of radon from the soil by diffusion. Short-term fluctuations are also possible. It is therefore recommended that background measurements be taken close to the time of the actual sample measurement. To reduce the radon background, the room containing the measuring system should be well ventilated, a consideration which is of particular importance for measuring rooms below ground or at basement level.

Proper shielding reduces the background from external sources by about one order of magnitude. The usual shielding consists of a “castle” or “house”, made of lead bricks of about $5\mathrm{cm}$ thickness, around the cylindrical volume of the chamber and at a distance of several centimeters from it (Figure 24-1-Figure 24-2). Also cylindric lead rings may be used. The optimum thickness of the lead shielding is from $5\mathrm{cm}$ to $10\mathrm{cm}$, the lower limit being defined by the need to attenuate photons with energies of about $2\mathrm{MeV}$, the upper limit by the increase of radioactive contamination and cosmic-ray reactions in the lead volume and the need to avoid unnecessary weight. To reduce radiation from contaminations, a good quality lead should be chosen, one, for example, having an activity concentration of less than $30\mathrm{mBq}$ of ²¹⁰Pb per gram of lead. The studies on lead for low-level measurement systems [267], [335] also apply to the shielding of ionization chambers.

For any shielding, the scattering of photons and x-ray fluorescence effects increase the observed current component above the value caused by the source in the well. Modification of the shielding may thus change the values of calibration factors; for a measurement system of good accuracy, the same shielding should remain permanently around the ionization chamber.

The electronic background from instrument noise, leakage and offset currents, and electromagnetic field disturbances is usually an order of magnitude lower than the background from unwanted radioactivity. The electronic background depends on the quality of the current-measuring electronics and the cable installations (Chapter 4).

6.2.  Decay correction

When a radioactive source is put in an ionization chamber, the measurand determined by the chamber, normally the activity (or, proportional to it, the ionization current), decreases with time following Equation (2): $A=A\left(t\right)=A_{\text{o}}{e}^{-\lambda \Delta t}$ , with $\Delta t=t-t_{\text{o}}$. The activity $A_{\text{o}}$ is the activity at the reference time $t_{\text{o}}$. The decay constant is $\lambda =\text{ln}2/T_{1/2}$. The half-life $T_{1/2}$ of the radionuclide can be found in data files or tables [272], [316], [51], [394], [324].

In this context, the duration of the measurement is regarded as a point on the time scale, which is not true in practice. The ionization-current measurement takes, for example, a time of 100 s which may cause a systematic bias of more than 1% for radionuclides with half-lives shorter than a few hours. To calculate the necessary correction term, the decreasing current is averaged in time over the duration of one measurement (here 100 s). The exponential decrease of the current is regarded as a function of time which is integrated over the interval and referred to a well-defined reference time, usually the beginning of the individual measurement interval. It is also possible to refer to the midpoint of the measurement interval, but this has the effect that, for times of measurement which are not constant because of the particular current-measurement technique used, the interval between successive times on the scale is variable. For this reason the beginning of the measurement interval is preferred. This leads to a correction term $C\left(t_{\text{d}},T_{1/2}\right)$ which is the ratio of the charge collected with constant current to the charge collected with an exponential decrease of current in the same measurement interval $t_{\text{d}}$ and to the equation

hence

For example, for ⁹⁹Tc^m ($T_{1/2}=\mathrm{0.250 25}\text{ d}$) and a measurement interval of $300\mathrm{s}$, thus for $t_{\text{d}}/T_{1/2}=\mathrm{0.013 9}$, the correction factor is 1.00563, it necessarily refers to the start of the measurement, as defined. It is 1.000347 for a ratio $t_{\text{d}}/T_{\text{1/2}}=0.001$. For smaller ratios, the correction is negligible.

6.3.  Variations of sample dimensions and materials

Solution standards, prepared and measured in national standards laboratories, consist of a radioactive solution of given mass defined by careful weighing, dispensed in a glass ampoule of standard geometry and flame sealed. Various types of ampoules are used for calibration purposes [91], [472], [58], NIST SP 250. For international comparisons the BIPM ampoule [372] or the NBS (NIST) ampoule [60], [409] filled with $3.6\mathrm{g}$ of radioactive solution, is used (Figure 50-1). These ampoules are similar, the BIPM ampoule being standard for measurements with the International Reference System (SIR) at the BIPM (Chapter 10). The characteristic dimensions of this ampoule are given in Table 2, together with the results of an investigation on the influence of variations in these dimensions.

It is important that laboratories work with ampoules which are as uniform as possible in geometry and material. For this reason, ampoules are normally made in batches of several thousand and are checked for possible variations in the dimensions. For the BIPM ampoule, the following variations have been observed: 1% in the side-wall thickness of the cylinder, a few percent in the bottom thickness and 0.7% in the inner diameter of the cylinder [372]. In other laboratories similar standard ampoules are in use with filling masses of $1\mathrm{g}$ to $5\mathrm{g}$. When very small amounts of solution mass are used, the relative uncertainty of the activity measurement is rather large, due to the spread in weighing and the related filling correction applied to an ionization chamber reading. When large solution masses are used, some radionuclides may present problems of cost and availability. As a compromise, at the PTB, a standard ampoule of $15.2\mathrm{mm}$ outer cylinder diameter, $0.47\mathrm{mm}$ side wall thickness and a mass of radioactive solution of $2\mathrm{g}$ has been adopted.

For practical applications, such as the distribution of radiopharmaceuticals, ampoules and vials of many shapes, dimensions and filling volumes are available (Figure 51). These are usually made of glass, but plastic materials are also used. In principle, the vials used for radioactivity measurements conform to the standard IEC 583 ([228]), with a first supplement IEC 583A ([229]), but the dimensions chosen are too large for ionization chamber measurements. A more recent standard for glass ampoules (Figure 50-2) is given in ISO 9187-1, and in the corresponding ISO standard with the same number and title, and this seems suitable for metrological purposes. Plastic vials in particular, but also glass vials with wall thicknesses exceeding $1\mathrm{mm}$ or having large variations of the thickness, are not suitable for radioactivity measurements of metrological quality. If an ampoule from a standards laboratory cannot be used for a measurement or calibration, a cylindrical injection vial (also called a dose or penicillin vial) can be used, for example the vial P6 (Figure 51) from the Amersham ([6]) catalogue. This ampoule is made of glass, has a cylindrical shape with an outer diameter of ($20.0\pm 0.25$) $\mathrm{mm}$, a height of ($54.5\pm 0.5$) $\mathrm{mm}$ and $10\mathrm{ml}$ nominal filling volume; it is a good choice for routine measurements in the quality control of radiopharmaceuticals or for measurements with radionuclide calibrators. For the production and distribution of radiopharmaceuticals, standardized ampoules are recommended.

For non-standard ampoules, a sample-geometry correction factor must be determined by relative measurements with respect to the standard calibration geometry. With glass thicknesses exceeding $2\mathrm{mm}$ and radionuclides emitting photons with energies above $100\mathrm{keV}$, the corrections range from 2% to 5%. For low-energy photon emitters, like ¹²⁵I, ¹²³I or for radionuclides with high emission probabilities for the emission of $\mathrm{\gamma }$ or x rays with energies of around $30\mathrm{keV}$, relative corrections as large as 25% are not unusual [95], especially if the measurement is carried out using plastic vials or radiopharmaceutical syringes [477], [59], [60].

It is important that calibrations and measurements use solutions of the same density. When carrier solutions are used for a dilution with concentrations below $50\mathrm{\mu g}$ salt per gram of solution and have low acid concentration, the density of the radioactive solution remains close to $1\mathrm{g}/\mathrm{ml}$. This corresponds to standard calibration conditions so additional correction factors for attenuation of the photon radiation are not required.

Larger variations of the relative ionization-chamber response can be expected for $\mathrm{\beta }$-emitting radionuclides such as ³²P or ⁹⁰Sr. These depend on variations in ampoule-wall thickness or on the spread of other dimensions in the particular type of ampoule used [95], [478]. The difference in response observed with different ampoule-wall thicknesses and with different chamber-well materials must be explained in terms not only of a simple spread in absorption, but also of changes in the location of bremsstrahlung creation by the $\mathrm{\beta }$ particles in the well materials with respect to the sensitive detector gas. Studies of radiation absorption with sources of ³²P and ⁹⁰Sr in glass have been described by Dalmazzone and Guiho ([95]), and of ³²P and ²⁰⁴Tl in various materials such as Al, Cu, Sn and Pb by Dhaliwal et al. ([106]).

The measurement of noble gases like ⁴¹Ar, ⁸⁵Kr or ¹³³Xe in sealed glass ampoules presents a particular problem of geometry (see vial P1A in Figure 51). These ampoules have roughly the same shape and dimensions as flame-sealed ampoules for solutions, but the radioactive substance fills the total volume with a uniform activity concentration. Variations of dimensions such as ampoule height and, especially, the size of the sealed part, therefore result in changes in response. The calibration of ionization chambers for these radionuclides can be carried out by comparison with activity measurements using Ge detectors at larger source-to-detector distances. This minimizes geometric effects as shown by Merritt et al. ([306]) who carried out measurements using a thin-walled aluminium well chamber of the T.P.A. Mk IV type which has a dominant response peak at low photon energies.

Another extreme geometric correction factor is that which applies when solid sources are measured in an ionization chamber. An early calculation of attenuation effects in cylindrical solid sources of high-Z materials, like radium or uranium, was carried out by Evans and Evans ([142]). Other studies of correction factors can be found in the literature, among them, in the field of brachytherapy, factors for radionuclides like ¹²⁵I and ¹⁹²Ir measured in a radionuclide calibrator. When used in the form of seeds or wires both these radionuclide sources show strong photon-attenuation effects in the source material, either because the energies of the emitted photons are low or because the material has a high density and a high Z value. Some details of measurement techniques are given in the publications referred to below. In these either special ionization chambers were used [283], [170], [83] or radionuclide calibrators with specific calibration factors were applied [391], [488], [54], [453], [481].

6.4.  Variation of the sample position

A fundamental requirement for good relative activity measurements using ionization chambers is good reproducibility of the position of the samples to be compared. This reproducibility should be better than $0.1\mathrm{mm}$ for repeated measurements. The effect of variation in the source position and other sample-geometry effects have been studied by many authors [292], [75], [210], [469], [372], [440], [386], [482], [27]. Diagrams with deviations of response are shown in Figure 52-1-Figure 52-3. The effects may be understood by detailed study of the isodose zones inside the chamber well (Figure 17-2). In most chamber constructions the radial dependence of source displacement in the chamber well is one order of magnitude greater than the vertical dependence. The effects of displacement also depend on the photon energies. This can be understood qualitatively by noting that the photons may irradiate different material structures if the source is displaced. The strongest effects are observed if the source is placed near the bottom or top of the chamber well. Normally, the source to be measured should be positioned in the centre of the well and on its central axis in a source holder of thin plastic material. The holder should be symmetric in form, with its head fitting snugly into the upper part of the chamber well. A change of about 0.3% in chamber response is observed for a radial displacement of $3\mathrm{mm}$ from the centre of the well when using photon energies of about $1\mathrm{MeV}$ from radionuclides such as ⁵⁴Mn or ⁶⁰Co. Position dependence in a spherical $4\mathrm{\pi }\mathrm{\gamma }$ ionization chamber has been studied in detail by Bucina et al. ([53]) and formulae are given for this geometry.

A special two-detector method for comparisons of source strength (activity) was proposed by Valkonen and Kantele ([455]), and realized for two ionization chambers by Greupner and Groche ([186]). A relatively strong source is measured at a distance of about 1 m from two almost identical, horizontal, cylindrical ionization chambers. By combining the measurements of the two chambers, the errors caused by source displacements can be reduced considerably. The results obtained are in good agreement with theoretical calculations.

Displacement of the radioactive parts inside the source confinement must also be considered as an aspect of ampoule or source manipulation. First, some of the solution may stick in the neck of an ampoule as a result of uncontrolled movements, temperature changes and condensation, or surface tension in the narrow glass tube. Such displacements can be undone by centrifuging the ampoule after careful manipulation at a stable temperature. After this, the ampoule must be kept permanently in a vertical position. A check to verify the absence of plate out is also necessary [8], at least by observation. Critical chemical elements are those whose compounds have very low solubility, such as Ba and Ag. Finally, the volatility of the radioelement in particular chemical states or its passage through a gas phase by decay (noble gases like Kr, Xe, Rn) may lead to displacements within the source confinement and thus to changes in the measurement geometry.

6.5.  Filling correction for sources of varying solution volume

Radioactivity standards for ionization-chamber measurements are realized by a radioactive solution in a standard glass ampoule with a well-defined cylinder diameter: they thus have a given mass and a corresponding well-defined filling height. As explained in the preceding section, the chamber response depends on the distribution of radioactivity within the ampoule. In addition, the standard calibration geometry corresponds to a defined configuration of attenuating materials or attenuation lengths within the source. A correction factor must therefore be applied to the ionization current if the filling level differs from the adopted standard.

These effects have been studied by many authors [403], [95], [469], [372], [440], [482]. The filling correction curve is determined empirically either by stepwise addition of weighed amounts of an inactive carrier solution to a concentrated defined radioactive solution in an ampoule, or by carefully weighing and placing different amounts of solution, taken from a mother solution, in ampoules of the same type. In the first method, the activity in the ampoule is constant, but the measurements are performed stepwise and, as evaporation occurs when the ampoule is not sealed, this introduces some uncertainty. In the second method, all ampoules can be sealed immediately after filling and the measurements can be repeated in various independent cycles, but minor variations in the ampoule dimensions resulting from the manufacturing process must be considered. Experience shows that the second effect is negligible in high-quality standard ampoules with glass walls less than $0.5\mathrm{mm}$ thick if the ampoules come from a single manufacturing batch, as is done in national standards laboratories. In both methods the ratio of the ionization current to the measured mass is compared with that obtained for a standard mass. When plotted as a function of solution mass interpolation of these relative currents in the curve at a desired mass value allows a correction to be made (Figure 53-1-Figure 53-3).

The shape of the filling correction function depends on the energies of the photons emitted by the radionuclides in the solution to be measured. With high energy-photons, the relative current per mass decreases continuously with increasing mass. This behavior may change for photon emitters at energies below about $100\mathrm{keV}$, such as ¹⁰⁹Cd ($88\mathrm{keV}$), ²⁴¹Am ($59.5\mathrm{keV}$), ²¹⁰Pb ($46.5\mathrm{keV}$) or ¹²⁵I (about $30\mathrm{keV}$), see Figure 53-3 or in the publication of Dalmazzone and Guiho ([95]). The activity of these radionuclides is usually measured in ionization chambers with Al walls since these have a high response at these energies. The shape of the filling correction function depends strongly on mass for radionuclides which emit $\mathrm{\beta }$ particles with endpoint energies above about $1\mathrm{MeV}$, like ¹⁰⁶Ru/Rh, ¹⁴⁴Ce/Pr, ⁹⁰Sr/Y or ³²P [95], [478], which are accompanied by bremsstrahlung photons.

6.6.  Radionuclidic impurities

An ionization chamber with a current-measurement system for activity determination is an integrating device which, in principle, picks up all the ionizing radiation which enters its sensitive volume. One single measurement, therefore, does not suffice to distinguish the radiations from the different radionuclides present in the sample. In a radioactive source all radionuclides, except the nominal one and its daughter nuclides, are regarded as impurities. It is obviously very important to check for radionuclidic impurities in a sample and to correct for the contribution of impurity components to the ionization current. Well-known candidates for radionuclidic impurities are ⁵⁶Co and ⁵⁸Co in ⁵⁷Co, ⁹⁹Mo in ⁹⁹Tc^m, ¹²⁴I in ¹²³I, ¹²⁶I in ¹²⁵I, ¹⁵⁴Eu in ¹⁵²Eu or ²⁰⁰Tl and ²⁰²Tl in ²⁰¹Tl. For example, if a ⁵⁷Co source containing only 0.1% of ⁵⁶Co is measured in an ionization chamber and no correction applied, an error of about 1.5% in the nominal activity value of the ⁵⁷Co will result [240]. Fairly complete lists of probable or possible impurities in standard sources for about fifty radionuclides are given by Jedlovszky and Szörenyi ([240]), Rytz ([375]) and Dryak et al. ([119]), with the 2nd revised edition of Dryak et al. ([119]).

Radionuclidic impurities can be avoided, or at least minimized, during the production of a radionuclide source by choosing an optimal nuclear reaction with a pure target material. In spite of this, chemical purification is sometimes essential after the production. In any case, the contribution to the activity from the the radionuclidic impurities must be measured carefully. Any standard or certified source used in a metrology or quality assurance programme should be accompanied by certified values for the activities of its impurity components in addition to its nominal activity.

For photon-emitting radionuclides, the problem can be solved by supplementary measurements which yield the activity of each contributing radionuclide. These values are obtained using semiconductor detectors of high energy resolution, e.g. Ge detectors, to identify and to assess, in a quantitative manner, the radionuclidic impurities in terms of their photon energies and emission probabilities.

For pure $\mathrm{\beta }$ emitters, like ³²P with an admixture of ³³P, more sophisticated methods must be applied, such as discrimination in terms of half-life, which requires a series of measurements covering at least one half-life of the nominal radionuclide [399], or discriminator-ratio or detector-response ratio methods which depend on photon-energy filters (absorbers, liners). A practical example of the latter method is to check on ⁹⁹Mo breakthrough by an ionization-chamber measurement using a solution from a ⁹⁹Tc^m generator with and without a lead filter in the chamber. Various techniques to determine impurity corrections for applications to ionization-chamber measurements are described in the following sections.

Corrections for radionuclidic impurities always take the same form; only the known and unknown parameters vary. The activity of each radionuclide in the source contributes to the ionization current. The basic equation is derived by adding the current components $I_{\text{i}}={\epsilon }_{\text{i}}{A}_{\text{i}}$ (Section 2.4), that is, the components of the nominal radionuclide (index N) and the impurities (indices k). The radionuclide efficiencies ${\epsilon }_{\text{N}}$ and ${\epsilon }_{\text{k}}$ are the weighting factors of the activities $A_{\text{N}}$ and $A_{\text{k}}$. More generally, the current components $I_{\text{i}}$ can be replaced by any other quantity proportional to them and added in the sum. The measurand $R$ in the formula Equation (16) may be the instrument output, the ionization current itself or any other quantity proportional to it, such as the relative current related to the current of a ²²⁶Ra reference source or the direct reading of an adjusted instrument in terms of activity of a particular radionuclide. In any case, these quantities must be corrected for the corresponding background. If a more general definition than in Equation (3) is applied for the efficiencies ${\epsilon }_{\text{i}}$, which enter in $R_{\text{i}}={\epsilon }_{\text{i}}{A}_{\text{i}}$, the relevant values of these efficiencies must also be used in the expression for the impurity correction. In the case that the efficiencies ${\epsilon }_{\text{i}}$ are replaced by the reciprocal equivalent activities $1/A_{\text{ei}}$, the equivalent activities $A_{\text{ei}}$ defined in Section 2.5 should not be confused with the radionuclide activities $A_{\text{N}}$ or $A_{\text{k}}$. With the general definition of the efficiencies, the equation

is obtained. The activities $A_{\text{i}}$ (i = N, k) must be corrected for decay each with its own half-life every time $R$ is measured. To calculate $A_{\text{N}}$ from $R$, the values of $A_{\text{k}}$ (or $A_{\text{k}}/A_{\text{N}}$) and the corresponding ${\epsilon }_{\text{i}}$ (i = N, k) must be known. An efficiency ${\epsilon }_{\text{i}}$ or a quantity proportional to it, such as ${\epsilon }_{\text{N}}\sim 1/k_{\text{N}}$ (Section 7.1), is determined by direct calibration against an activity standard or by calculation from the energy-dependent photon-efficiency curve of the chamber (Section 7.6).

7.1.  Calibration factors for particular radionuclides

The term calibration refers to the relative (or indirect) measurement of a physical quantity [294], here the “activity”, in comparison with a standard that embodies this quantity, here an international or national “activity standard” or field standard, this last being traceable to the first ones. In our case, the comparison is made by a current measurement in an ionization chamber that represents, in a standards laboratory, a secondary standard measuring system to maintain the results of activity measurements from primary standardization. More generally, a re-entrant ionization chamber can be calibrated in terms of activity by comparison with an appropriate activity standard. For this type of measurement a calibration factor of the ionization chamber must be determined for the radionuclide of interest (see Section 2.4 and Section 2.5). The reciprocal of this calibration factor represents an efficiency of the ionization chamber for the particular radionuclide: $k_{\text{N}}=1/{\epsilon }_{\text{N}}$, with ${\epsilon }_{\text{N}}$ defined in Equation (3), or alternatively $k_{\text{N’}}=1/{\epsilon }_{\text{N’,rel}}$, for a quantity which is proportional to the ionization current.

In fact, calibration is the key to any activity determination using an ionization-chamber measuring system. Nearly all the preceding chapters deal with procedures, questions and problems related to calibration. The calibration factors, for a particular radionuclide in a well-defined measurement geometry, are the essential parameters of the measuring system. Fairly complete tables or files of calibration factors have been developed in national standards or calibration laboratories by Engelmann ([134]), Dale and Williams ([92]), Dalmazzone ([93]), Williams and Birdseye ([472]), Goodier et al. ([174]), Woods ([475]), Lewis et al. ([278]), Woods et al. ([483]), Woods and Lucas ([484]), Merritt and Gibson ([303]), Dias ([107]), Merritt and Gibson ([305]), Kobayashi and Ishikawa ([259]), Broda ([46]), Szörényi and Zsinka ([437]), Coursey et al. ([80]), Woods et al. ([482]), Szörényi et al. ([438]), NIST (NIST SP 250) and Vinten (now NE Technology ([319])). These values are frequently compared with the results of direct activity measurements, thereby maintaining the unit of activity over long time periods. After a measurement in a secondary-standard measuring system, such as a calibrated ionization chamber, the unit of activity is distributed to manufacturers and users of ionization-chamber measuring systems, for example, in the field of nuclear medicine. This can be done either by distributing suitable standards of activity for instrument calibrations, or by a rigorous quality-control programme for a particular type of instrument to which a file of published calibration factors, measured in a standards laboratory, is related. In both cases, the file of correctly determined calibration factors is the basis for all activity assays.

Sometimes a simulated standard, also called a mock standard, is used to estimate calibration figures. Such a standard is a radioactive source prepared from a long-lived radionuclide that approximates the radiation characteristics of the radionuclide to be simulated. A typical radionuclide simulation is ⁹⁹Tc^m by ⁵⁷Co or ¹³⁹Ce, but it is not recommended for realistic calibration of an ionization chamber [303], [22], [316]. Mock standards may serve as reference sources, but they should not be used for calibrations of other radionuclides, because the factors used do not usually take account of the energy-dependent photon-efficiency of the instrument and the emission probabilities per decay of the radionuclides under consideration.

Another way of determining the calibration factor of an ionization chamber for ⁹⁹Tc^m is described by Broda ([46]). He measured activity ratios in the mother-daughter decay of the nuclides ⁹⁹Mo and ⁹⁹Tc^m by spectrometric and direct counting methods and used decay parameters to calculate a calibration factor. The method is a variation of that in which calibration factors are computed from energy-dependent efficiency curves and emission probabilities per decay (Section 7.6). It uses the balance condition to estimate the ratios of activities in a decay chain.

7.2.  Efficiency as a function of photon energy

In an ionization chamber the current components from particular photons emitted in the decay of a radionuclide are detected and collected independently and their sum is the total ionization current measured. The nuclide efficiency ${\epsilon }_{\text{N}}$ can be expressed as the sum of these components (index i), i.e.

with $p_{\text{i}}\left(E_{\text{i}}\right)$ the emission probability per decay of photons of energy $E_{\text{i}}$, and ${\epsilon }_{\text{i}}\left(E_{\text{i}}\right)$ the energy-dependent photon efficiency of the ionization chamber. Such an expression also applies to quantities which are proportional to the photon efficiencies ${\epsilon }_{\text{i}}\left(E_{\text{i}}\right)$ and the related nuclide efficiency ${\epsilon }_{\text{N}}$, such as the energy-dependent reciprocal photon-equivalent activities $1/A_{\text{ei}}\left(E_{\text{i}}\right)$ and the reciprocal equivalent activity $1/A_{\text{e}}$. Rytz ([376]) preferred, for practical reasons, to write the equations for the SIR in terms of ${10}^6/A_{\text{e}}$.

Measurements undertaken with a view to establishing the energy-dependent efficiency curve should start by using radionuclides that emit practically monoenergetic photons in a region where the curve is nearly linear (Figure 57), for example at $835\mathrm{keV}$ (⁵⁴Mn) to $\mathrm{1 253}\mathrm{keV}$ (⁶⁰Co). The latter, emitting photons of two distinct energies ($\mathrm{1 173}\mathrm{keV}$ and $\mathrm{1 333}\mathrm{keV}$) in the linear region, can be considered as practically monoenergetic if the mean energy of the photons and the sum of their emission probabilities per decay are calculated. In further steps, radionuclides with more complex decay schemes can be used if the photons emitted have energies in the range of the efficiency curve already established with the exception of the one photon energy $E_{\text{k}}$ for which ${\epsilon }_{\text{k}}\left(E_{\text{k}}\right)$ should be calculated. Efficiency values for all the further photons can be interpolated, multiplied by the corresponding emission probability, and subtracted from the measured radionuclide efficiency ${\epsilon }_{\text{N}}$. The difference represents the unknown component $p_{\text{k}}\left(E_{\text{k}}\right){\epsilon }_{\text{k}}\left(E_{\text{k}}\right)$. A list of calibration nuclides for the establishment of the energy-dependent efficiency curve is given in Table 5.

Thin-wall aluminium (Al) chambers show a strong peak in the efficiency at photon energies around $50\mathrm{keV}$. This results from the rapid increase of the interaction cross section of the photoelectric effect in the sensitive chamber materials (mainly the filling gas) with decreasing energies, and the low energy cutoff with Al walls at about $20\mathrm{keV}$. For iron (Fe) walls a similar cutoff occurs at about $40\mathrm{keV}$. To establish the energy-dependent efficiency curve of an ionization chamber with aluminium walls, it is particularly useful to measure sources of relevant radionuclides both with and without a low-Z-material filter (liner). For an Fe liner, a thickness of $1\mathrm{mm}$ will suffice. The energy-dependent efficiency curve of an instrument fitted with such a liner is considerably displaced towards lower efficiency values at low photon energies. Current components from photons having energies below about $35\mathrm{keV}$ vanish almost completely while components from photons having energies higher than about $150\mathrm{keV}$ are only slightly altered by the liner. From the difference of the results of these two measurements several efficiency points in the low-energy region can be deduced. This method was desribed by Schrader and Weiss ([401]) in the case of a radionuclide calibrator with $3\mathrm{mm}$ thick aluminium walls and a $1\mathrm{mm}$ thick Fe liner (Figure 58-1-Figure 58-2). The fractional radionuclide calibrator response can also be derived using a copper filter and the properties of characteristic x rays. This is discussed by Dubuque et al. ([123]) for the radionuclides ¹²³I and ¹²⁹Cs and by Harris et al. ([198]) for ¹²³I.

The energy-dependent efficiency curve of an ionization chamber with Fe walls is similar in shape to that for a chamber with Al walls and an Fe liner with about the same wall thickness as the Fe wall chamber, see Figure 57 and Figure 59.

Several authors have published energy-dependent efficiency curves for different ionization chamber constructions; Mann and Seliger ([295]) (Figure 60-1-Figure 60-2), Weiss ([468]), Dale ([90]) (Figure 61-1-Figure 61-2), Kohlrausch ([265]), Oncescu and Rebigan ([327]), Bensch and Ledermann ([21]), Weiss ([469]), Dubuque et al. ([123]), Suzuki et al. ([431]) (Figure 62), Merritt et al. ([306]), Rytz ([376]) (Figure 63-1), Rytz and Müller ([380]), Dias ([108]), Schrader and Weiss ([401]) (Figure 58-1-Figure 58-2) and Blanchis ([27]). Apart from a proportionality factor in the definition of the calibration (see above), the main differences in the values of the efficiency curves quoted result from differences in the attenuating materials between the source and the sensitive chamber gas, among them variations in the density and geometry of the radioactive solutions, the materials of the ampoules or confinements, the chamber walls inside the well, and the chamber electrodes. Details are given in Section 7.5.

A further parameter influencing the efficiency is the sensitivity of the filling gas. The efficiency curve of a chamber filled with argon at a pressure of $2\mathrm{MPa}$ is steeper, as a function of energy, than that of a chamber filled with nitrogen at the same pressure. On the other hand, the latter is more nearly a linear function of energy.

That the efficiency is strongly affected by the choice of material for the wall of the well has already been noted (Figure 58-1-Figure 58-2). Another important consideration is the geometry of the electrodes. Constructions with a single cylindrical tube for each electrode (high-voltage and collector electrode) should be distinguished from designs with multiple tubes, one inside the other, with varying electric potentials. With this construction the photon flux caused by the increase of total volume of attenuating materials inside the chamber is reduced, but secondary-electron production in the sensitive chamber gas is increased by the materials, as is the quality of the electron collection if the geometry of the electric fields is optimized.

7.3.  Efficiency of detection of bremsstrahlung from $\mathrm{\beta }$-particle emitters

Many radionuclides of interest for ionization-chamber measurements decay by the emission of $\mathrm{\beta }$ particles (electrons or positrons). By such a decay, excited states of the daughter nucleus (with one nuclear charge less or more) are populated, and they de-excite by electromagnetic radiation, such as photon emission. However, some of the $\mathrm{\beta }$-particle emitters decay directly to the ground state of a nucleus without photon emission. The consequent deceleration of the $\mathrm{\beta }$ particles by Coulomb interactions in matter is accompanied by the emission of bremsstrahlung photons with a continuous energy distribution. A $\mathrm{\beta }$ spectrum has a distribution function (shape) below its end-point energy $E_{\text{o}}$ that depends on the selection rules applicable to the $\mathrm{\beta }$ decay from the parent to the daughter nuclear states described by the spins, parities, etc. A short description can be found in the book of Evans ([143]). An average $\mathrm{\beta }$-particle energy can be calculated from this distribution. The complete energy spectrum of the bremsstrahlung photons emitted consists of the individual photon distribution folded with the shape of the $\mathrm{\beta }$ spectrum, and is limited by the maximum $\mathrm{\beta }$ end-point energy. In the energy region of interest for ionization chamber measurements, the bremsstrahlung photon spectrum is roughly the same shape as the $\mathrm{\beta }$-particle energy distribution. The average $\mathrm{\beta }$-particle energy is therefore a good parameter with which to characterize the ionization-chamber response to bremsstrahlung photons from $\mathrm{\beta }$ decay. The order of magnitude of the $\mathrm{\beta }$-particle response may be characterized and estimated, because the probability of the Coulomb interaction creating the bremsstrahlung can be expected to be about three orders of magnitude smaller than the probability of the $\mathrm{\beta }$-particle emission, and the response to the bremsstrahlung is fixed, in turn, by the photon efficiency of the ionization chamber used (Section 7.2).

An ionization chamber can be calibrated using sufficiently strong activity standards of pure $\mathrm{\beta }$-particle emitting radionuclides like ⁹⁰Y (the daughter of ⁹⁰Sr) or ³²P [295], [134], [93], [384], [27], [478]. Other radionuclides, such as ¹⁰⁶Rh or ¹⁴⁴Pr (the daughters of ¹⁰⁶Ru or ¹⁴⁴Ce), decay by $\mathrm{\beta }$-particle emission, the spectrum having a very high $\mathrm{\beta }$-endpoint energy followed by the emission of photons from excited nuclear states. When the components of photon emission from the disintegration of excited states and the total radionuclide efficiency are known, a bremsstrahlung photon efficiency can be deduced by subtraction (Section 7.2). For these radionuclides a comparatively high bremsstrahlung-photon efficiency component results. In ionization chambers with an aluminium well only a few millimeters thick, the high $\mathrm{\beta }$-energy emitters ($E_{\text{o}}$ above $2\mathrm{MeV}$) may also produce small ionization current components from direct interactions of the $\mathrm{\beta }$ particles with the sensitive chamber gas and the wall materials.

In the way outlined above a list of calibration factors for $\mathrm{\beta }$ particle emitting radionuclides can be established in a range of average $\mathrm{\beta }$-particle energies from about $60\mathrm{keV}$ to $\mathrm{1 410}\mathrm{keV}$ with the following radionuclides (with daughters): ³⁵S, ¹⁴⁷Pm, ²⁰⁴Tl, ⁸⁵Kr, ²¹⁰Pb (²¹⁰Bi), ¹⁴⁰Ba (¹⁴⁰La), ⁸⁹Sr, ⁹¹Y, ³²P, ⁹⁰Sr (⁹⁰Y), ¹⁴⁴Ce (¹⁴⁴Pr) and ¹⁰⁶Ru (¹⁰⁶Rh). At low energies a cutoff of the response occurs, its position depending on the entrance window of the chamber. A typical example of efficiency values for an ionization chamber (Centronic-IG12) is given in Figure 56-1-Figure 56-2 [69]. Few $\mathrm{\beta }$-efficiency curves for $4\mathrm{\pi }\mathrm{\gamma }$-ionization chambers and few detailed calibration factors for $\mathrm{\beta }$-particle emitters have been published. It is evident that account must be taken of the bremsstrahlung component from a $\mathrm{\beta }$-particle emitter with a high average $\mathrm{\beta }$-particle energy in detailed calculations of the radionuclide efficiency for energy-dependent efficiency curves (Section 7.6).

7.4.  Representation and fitting of efficiency functions

In many measurement systems, the values of the photon efficiency, determined by experimental methods as described in Section 7.2, increase almost linearly with energy $E$. Efficiency curves, therefore, used to be plotted in a reduced form (Figure 63-1 [376], [380] and Figure 64-1-Figure 64-2 [470]). A linear function (${\epsilon }_{\text{o}}=a+bE$) which gives a good approximation to the experimental values in the range of interest is chosen and the ratio $f=\epsilon /{\epsilon }_{\text{o}}$ is plotted versus the photon energy $E$. The function of these plots was to allow easy interpolation of sufficient accuracy by reading directly from the diagram. Nowadays, these methods have been replaced by computer techniques which store the experimental values in a file and apply interpolation routines to calculate efficiency values at energies intermediate between the points of direct calibration. The difficulty of such calculations is to choose the best analytical function to be fitted to the experimental values. This is especially true in those energy regions where no or only few calibration values are available, for example in the energy range from $\mathrm{1 332}\mathrm{keV}$ (⁶⁰Co) to $\mathrm{1 836}\mathrm{keV}$ (⁸⁸Y).

One of the simplest ways to solve the problem is to plot the values on a large scale and to draw a smooth curve by eye through the calibration points [376], [380]. This method can be computerized using a polygon to represent the calibration points with linear interpolation between the points adding, if necessary, supplementary points, fitted by eye, to the polygon to improve the smooth interpolation of the curve (Figure 57).

A polynomial of the n^th order of the energy $E$ is another useful representation of an efficiency curve. Thus Bensch and Ledermann ([21]) proposed an equation with terms which were quadratic in the energy. Dryak and Dvorak ([122]) used a higher-order polynomial, and a cubic-spline function was tested by Janszen ([239]). Once determined, all these functions must be checked for consistency by efficiency calculations for radionuclides, such as ¹³⁴Cs or ¹⁵²Eu, which have several photon energies covering large regions of the curve, applying Equation (29) of Section 7.2.

Experimental efficiency measurements and consistency calculations have been published by Schrader and Weiss ([400], [395]), Rytz ([376]), Dryak and Dvorak ([122]) and Reher and Sibbens ([352]). Some of these results are given in Table 5 from Schrader and Weiss ([401]) and Table 6 from Dryak and Dvorak ([122]). The deviations of experimental calibration points from the curve of the International Reference System are plotted in Figure 63-1 [376].

The relative deviations from such curves are in the best cases a few parts per thousand and in critical cases as large as 3%. The origin of these deviations may be the form of the curve, but in some cases they may indicate an error in the direct calibration of a particular radionuclide, or at least a large uncertainty in the emission probabilities used in the calculations. Critical points to be checked are energy regions with two or several neighbouring photon energies from different radionuclides, such as ²²Na ($511\mathrm{keV}$) and ⁸⁵Sr ($514\mathrm{keV}$), ⁵⁸Co ($810\mathrm{keV}$) and ⁵⁴Mn ($835\mathrm{keV}$), and ⁵⁹Fe ($\mathrm{1 099}\mathrm{keV}$) and ⁴⁶Sc ($\mathrm{1 121}\mathrm{keV}$).

7.5.  Calculation of efficiency values from fundamental quantities of the interaction processes

In several publications by Droste ([118]), Bradt et al. ([43]), Sinclair ([413]), Staub ([425]), Dale ([90]), Oncescu and Rebigan ([327]), Kleeven ([255]) and Kleeven and Wijnhoven ([256]) attempts have been made to explain the shape of an efficiency curve for ionizing detectors (including gas counters) by calculations involving the fundamental quantities of the interaction processes, like the cross section for the photoelectric effect, the Compton effect and pair production (Figure 65-1-Figure 65-2).

In the early days of the development of ionization chambers and gas counters attempts were made to estimate the number of electrons collected in terms of their range, and in terms of the variation with photon energy, taking into account the interactions in the detector walls, such as photon absorption, Compton scattering and electron pair production [118], [43]. Bradt et al. ([43]) also studied the influence of various electrode materials, among them lead, aluminium and brass (Figure 65-2).

Another way of approaching the problem of the energy dependence of photon efficiencies and the calculation of radionuclide efficiencies from curves came from their use in dosimetry. An early study of the dependence of ionization currents on photon energy, using filtered x rays, was made by Behnken ([20]) who found that the response of an ionization chamber increased linearly with the accelerating potential of the x-ray tube. Further literature from that period can also be found in his publication. It was later noted by Gray ([183]), Dale ([90]), Oncescu and Rebigan ([327]), Bensch and Ledermann ([21]) that the variation in response is fairly well related to the “k-value” or specific $\mathrm{\gamma }$-ray emission constant of the radionuclide [316], at least in the ideal case of a chamber with walls of air-equivalent materials. Dale ([90]) expected practical limitations for commercially produced chambers and he gave plots for experimental calibration factors relative to the k-value for several commonly used radionuclides. He also described practical adjustments of the response curves as a function of photon energy, taking into account secondary-electron production by Compton scattering and the photoelectric effect in the well (or liner) material of the chamber. Calibration factors expressed in terms of ionization current per activity have been calculated for several radionuclides and compared with experimental values. The agreement is at the level of 1% to 5%.

Kleeven and Wijnhoven ([256]) proposed another way to explain the energy-dependent chamber response for different construction materials and included chambers in which the electrode was coated by a thin high-Z metal layer. The electrode coating considerably increases the chamber efficiency in the energy range from $100\mathrm{keV}$ to $500\mathrm{keV}$. This is useful for most of the radionuclides used in nuclear medicine applications since the required photon energies are in this region. For the calculations, the chamber efficiency is defined as the mean number of electron-ion pairs formed in the sensitive gas per photon entering the chamber. In order to calculate the mean amount of energy per photon transferred to the gas, the photoelectric, Compton and pair production interactions are considered separately. A geometric model for the electrodes in the form of cylindrical elements is constructed, and the efficiency components are computed by numerical integration in the individual zones of the model. Various gas fillings and electrode coatings (tantalum, platinum or lead, $100\mathrm{\mu m}$ thick, for instance) are studied. The calculations help to understand the processes and the influence of various materials and dimensions in terms of relative efficiency. However, the model is not suitable for precise calculation of the efficiency values of actual radionuclide calibrators.

Another way of calculating efficiency values for various source geometries, in terms of parameters for a standard geometry, is to apply Monte-Carlo codes. In a study by Dryak ([120]) correction coefficients for self-absorption were determined as the ratio of the ionization currents of a cylindrical source of aqueous ⁵⁷Co or ¹³⁷Cs solutions and a point source of very small mass. For this purpose, other modern Monte-Carlo codes also seem promising, among them the EGS4 code [318]: here EGS stands for electron-gamma shower. While this is a fairly complete system in computational physics, the authors of EGS4 consider the development of electron-photon Monte-Carlo transport techniques to be far from complete. As EGS4 is still not able to handle fluorescent x rays, Auger electrons, binding effects in Compton scattering or some types of multiple scattering, we cannot expect to evaluate calibration factors in absolute terms of current for such a complex geometry as an ionization chamber.

An attempt to use EGS4 in relative terms of current ratios has been made by Büermann et al. ([54]) in order to compute geometric correction factors for small cylindrical solid sources, for example ¹⁹²Ir brachytherapy sources. The attenuation-correction factor for the source was defined as the ratio of the efficiency of the radionuclide, in standard ampoule geometry, to the efficiency for the solid source, applying simple geometric models for the material layers and a homogeneous activity distribution in the solution and the solid radioactive material. Fluence spectra outside the sources were then calculated and the efficiency values for the different sources determined from the sum of photon components for the radionuclide, weighted with an experimental energy-dependent photon-efficiency curve in a point-source geometry. The results of calculation and experiment for several source assemblies, and also with various lead filters placed around the source, agree with each other to within 2%. In earlier studies the photon fluence from a solid ¹⁹²Ir source was measured with a radionuclide calibrator calibrated in terms of fluence [72].

Considerable effort will be required if Monte-Carlo calculations are to be used to penetrate to deeper layers of the chamber well and electrodes with a view to obtaining energy-dependent photon efficiencies. So far, no energy-dependent photon-efficiency curves for different chamber geometries and constructions have yet been calculated by Monte-Carlo methods and published. This is due to the complexity of the material distributions and geometric parameters in the practical realizations of re-entrant pressurized ionization chambers used for activity measurements.

7.6.  Calculation of radionuclide efficiencies from energy-dependent photon efficiencies and photon-emission probabilities per decay

Radionuclide efficiencies ${\epsilon }_{\text{N}}$ are calculated from energy-dependent photon efficiencies ${\epsilon }_{\text{i}}\left(E_{\text{i}}\right)$ (Figure 57) and photon-emission probabilities per decay $p_{\text{i}}\left(E_{\text{i}}\right)$ using Equation (29). An example can be found in Table 7 for Ir-192. The same form of equation is used for quantities proportional to the efficiencies ${\epsilon }_{\text{i}}\left(E_{\text{i}}\right)$ and ${\epsilon }_{\text{N}}$, such as the reciprocal equivalent activities $1/A_{\text{ei}}\left(E_{\text{i}}\right)$ and $1/A_{\text{e}}$. Methods for obtaining an analytical expression for, or a numerical representation of, the efficiency function are described in Section 7.2 together with the related interpolation procedures. An accurate calculation should include the bremsstrahlung components following high energy $\mathrm{\beta }$ decay in the sum of Equation (29), using an efficiency curve at average energy, as shown in Figure 56, and the corresponding $\mathrm{\beta }$-particle emission probabilities per decay.

The emission probabilities per decay $p_{\text{i}}$ entering into the calculation are taken from decay data compilations like [324], [272], [316], [51], [394]. Average $\mathrm{\beta }$-particle energies and related probabilities needed for a determination of the bremsstrahlung component may be found in [51] or [316]. Several authors have published calculated efficiency values of this kind, indicating deviations between the measured and the calculated values, for example Merritt and Gibson ([304]), Weiss ([470]), Schrader and Weiss ([401]) in Table 8, Dryak and Dvorak ([122]) in Table 6. The results typically show a deviation of a few per cent.

Heydorn ([211]) made critical comments on the use of these methods when low-energy x rays are neglected, for example in the case of ¹³⁷Cs. In contrast with Bensch and Ledermann ([21]) he pointed out that it is not the average energy of the radionuclide which should be used in these calculations. Instead, the calculation should be carried out applying energy-dependent efficiencies in a straightforward way. It is evident nowadays, that all the components contributing to the ionization current must be included in the calculation. An efficiency curve must be established individually for each particular measuring system.

It should be noted that the parameter system for efficiency calculations is strongly correlated. On one hand, the emission probabilities per decay are determined from the activity of a radioactive source and from relative measurements with a Ge detector. On the other hand, the corresponding radionuclide efficiency of the chamber is calculated from the emission probabilities per decay and an efficiency curve (from a calibration with other activity standards): this then allows the activity to be determined. This correlation must be taken into account in any discussion of uncertainties in the results (Chapter 8) and introduced, if necessary, in an algorithm with covariance matrices. This consideration applies to the calculated efficiency for an individual radionuclide for which a calibration standard is not available and for which the activity is to be determined [197]. It also applies to the development of photon-efficiency values for measurement systems which use activity standards to check calibration consistency. In this sense, a photon-efficiency curve of high accuracy, i.e. one having only small deviations of the calibration points from the curve, confirms the consistency of the calibration. At the same time, it confirms the quality of the activity standards used. Such a smooth curve has been given for the International Reference System (SIR) at the BIPM by Rytz ([376]), Rytz and Müller ([380]) (Figure 63-1), and for the ionization chamber of the PTB by Weiss ([470]) (Figure 57 and Figure 64-1-Figure 64-2). Figure 66 and Figure 67 show the relative deviations between the measured and the calculated efficiency values for the PTB curve and the BIPM curve, respectively, as a function of photon energy.

8.1.  Uncertainty components of an activity measurement

The conditions for obtaining high accuracy in activity measurements with ionization chambers have been discussed in the preceding chapters. In this section, these are summarized and the corresponding uncertainty components are listed. To do this, we take an equation similar to Equation (9) which is applied to the determination of an activity $A_{\text{o}}$, at a certain reference time $t_{\text{o}}$, with an ionization chamber measuring system. The activity is

The expression Equation (30) describes an ionization chamber measurement in which most parameters and quantities are subject to some uncertainty. The chamber is calibrated in terms of the equivalent activity $A_{\text{e}}\sim 1/{\epsilon }_{\text{N}}$ (or in terms proportional to it) for a radionuclide N with the half-life $T_{1/2}=\text{ln}2/{\lambda }_{\text{N}}$. The current caused by the sample, $I=I_{\text{m}}-I_{\text{b}}$, is measured at a time $t_{\text{m}}$ and that of the reference source $I_{\text{r}}$ at time $t_{\text{rm}}$. For the present discussion, a reference source of ²²⁶Ra is considered. The currents must be corrected for background (index b). The sample is measured in a particular geometry for which a correction factor $C_{\text{g}}$ provides the transformation to a measurement in a standard geometry, and the sample contains radionuclidic impurities (index i), for which corrections are made through the factor $C_{\text{i}}$.

The components ${\sigma }_{\text{j}}$ (written as $\sigma \left(Q_{\text{j}}\right)$ of the quantity $Q_{\text{j}}$) are discussed in the order in which they appear in Equation (30) from right to left. The first components originate from the current measurements. They contain the statistical uncertainty components ${\sigma }_{\text{stat}}\left(I\right)$ from fluctuations of the charge collection process (Section 2.3) similar to those of a counting experiment. The current measurements experience other systematic uncertainty components ${\sigma }_{\text{meas}}\left(I\right)$, arising from saturation-current losses (Section 2.3), leakage currents (Section 4.1) or instabilities in the current-measuring electronics (Section 5.1). To keep the values of these components as low as possible, good repeatability of the system is required, at least for times long enough to carry out several measurements in a series. Compensation for slow fluctuations or a slow shift is obtained by performing measurements relative to the reference source (Section 2.5), as both current values follow the same trend. Stability criteria are therefore required for the reference source seen as a source of stable current (Section 2.6). Other uncertainty components $\sigma \left(I_{\text{r}}\right)$ arise from the decay correction of the reference source (uncertainty of the half-life value), possible leaks of activity, deviations from equilibrium in the mother-daughter decay of ²²⁶Ra (Section 2.6) or any change in the measuring conditions of the reference source (e.g. nonreproducible geometry). The reference source activity (or the related current) chosen should remain near the equivalent activity (or current) of the source to be measured [376]. Other uncertainty components $\sigma \left(I_{\text{b}}\right)$, for example from fluctuations in the radiation level of the environment or changes of shielding conditions (Section 3.2.5 or Section 6.1), are related to the background measurements.

The uncertainty components $\sigma \left(C_{\text{i}}\right)$ describe the quantities connected with the radionuclidic impurity correction factor $C_{\text{i}}$ (Section 6.6), such as the activity ratio measured with a Ge detector, including uncertainties in its calibration, components from the decay corrections $\sigma \left(T_{1/2},\text{N}\right)$ and $\sigma \left(T_{1/2},\text{i}\right)$, and the ratios of the chamber calibration factors $\sigma \left(A_{\text{e}},\text{N}\right)$ and $\sigma \left(A_{\text{e}},\text{i}\right)$ of the contributing radionuclides N and i.

Good reproducibility is required for measurements made under standard measuring conditions, so changes in sample geometry (Section 6.3) which can be quantitatively determined, must be taken into account using a correction factor and a corresponding uncertainty component. Changes in the sample geometry, related to vials, ampoules or filling are described by a factor $C_{\text{g}}$ and an uncertainty $\sigma \left(C_{\text{g}}\right)$. In fact, the sample changes and the cyclic measurements (Section 3.2.5), involving reference source, background and sample, enter into such a component. Other components of this type result from the limited stability of the sample, among them leaks from the confinement and chemical instability of the solution (plate out of the source [8]) with components $\sigma \left(C_{\text{s}}\right)$.

Measurements of time, such as $t_{\text{m}}$ and $t_{\text{rm}}$ in Equation (30), are usually based on quartz-oscillator computer clocks, giving the time of registration (usually day, hour, minute and second) of the measured current value. The accuracy of these data is in most cases sufficient if radionuclides with half-lives greater than several hours are studied. Care must be taken with older clocks which may show a time shift of several minutes per month. These remarks also apply to the accuracy of the duration of a current measurement because, for most current-measurement systems, an accurate time interval is needed (Chapter 4). Small errors in the duration of the current measurement may produce substantial errors in the resulting current values. This problem is related to the adjustment error of compact electronics of activity meters in the case of time interval changes for activity range switching (Section 5.4).

The uncertainty components $\sigma \left(t\right)$ of the time measurements are closely related to the decay corrections necessary for all radionuclides involved in a measurement procedure. They thus include the sample, the radionuclidic impurities, the reference source and any external background sources and are described by uncertainty components $\sigma \left(T_{1/2}\right)$. A 0.1% relative error in the half-life produces a relative error in the corresponding activity value of 0.07% for each interval of one half-life between measurements, or 0.3% for four half-lives. Furthermore, the decay correction for the time elapsed during a current measurement (Section 6.2) has to be taken into account when dealing with very short half-lives and this also has an uncertainty component.

The last group of uncertainty components $\sigma \left(A_{\text{standard}}\right)$, in some cases the dominant ones, result from the procedure used to calibrate the activity value of the standard, that is from the direct (or absolute) activity measurement. The related uncertainty components $\sigma \left(A_{\text{standard}}\right)$ are transferred to the ionization chamber measuring system by the calibration, and the unit of activity is maintained there with the original quality of the direct measurement. With an optimized ionization chamber used as a secondary standard measuring system, a reproducibility of several 10^–4 is normally obtained for relative activity measurements with sources of one radionuclide. This is usually one order of magnitude better than the uncertainty for the corresponding direct activity measurement.

Indirect calibrations, where a calibration factor (or efficiency) is calculated from the energy-dependent efficiency curve and the emission probability per decay (Section 7.6), transfer the uncertainty components of the related quantities $\sigma \left({\epsilon }_{\text{i}}\left(E_{\text{i}}\right)\right)$ and $\sigma \left(p_{\text{i}}\left(E_{\text{i}}\right)\right)$ to the uncertainty of the activity value. Usually, relative uncertainties of this type amount to a few percent. Typical values for direct and indirect measurements are quoted by Dalmazzone and Guiho ([95]) and, for the earlier NPL ionization chamber, by Woods ([475]).

8.2.  Combined uncertainty from uncertainty components

To exemplify the relative magnitudes of the uncertainties discussed, values from optimized measuring systems used in national standards laboratories are listed in Table 9. Some values are included which are based on estimates obtained with activity meters under rather unfavorable conditions. The uncertainty values quoted are standard deviations derived from a large number of observations. A Gaussian distribution of the measurement values is assumed. Student factors, applicable when the number of events is small, are therefore not taken into account nor are other distribution shapes. The numerical values in the table should be regarded as a guide, because individual cases vary considerably depending on the chamber construction.

To calculate the overall uncertainty value of an activity measurement, the law of uncertainty propagation must be applied: The combined standard uncertainty $\sigma \left(A\right)$ is the positive square root of the sum of the squares of the individual uncertainty components $\sigma \left(Q_{\text{j}}\right)$, weighted with the partial derivatives of the function $A\left(Q_{\text{j}}\right)$ of Equation (30). Thus

The individual components depend on the particular procedure used to evaluate the $A\left(Q_{\text{j}}\right)$ for which Equation (30) represents only one example. Another example is published by Rytz ([376], [378]) for the efficiency curve of the SIR ionization chamber. The results are presented graphically in Figure 67. A similar presentation of uncertainties for various radionuclides is given in Figure 66 for an ionization chamber at the PTB. These pictures summarize the uncertainty in ionization chamber calibrations and show the individual contributions from various radionuclides.

If a calculated calibration factor (or efficiency) is used for an activity measurement, correlations are formed between parameters since they depend on the calibration points of the energy-dependent efficiency curve and the photon-emission probabilities, which are both related to other activity determinations. Correlations between the parameters must be taken into account by covariance terms in the sum for the combined uncertainty. The necessary theory of uncertainty propagation is beyond the scope of this review. Recommendations can be found in specialist papers and standards some of which include examples from metrological practice. Of particular interest is the “Guide to the Expression of Uncertainty in Measurement” ISO/IEC 98-3:1993.

9.1.  Quality control in national standards laboratories

Activity measurements with ionization chambers play an important role in the preparation of activity standards. The radioactive raw solution is quantitatively diluted by careful weighing, and the corresponding value of the activity concentration is checked after each dilution step by ionization-chamber measurements. Such dilution procedures have been described by Merritt and Taylor ([307]), Eijk and Vaninbroukx ([126]), in Campion’s monograph [23], in the report [316] and by Mann et al. ([294]). First, two or three ampoules of standard geometry are prepared gravimetrically from the raw solution and their activity concentration carefully compared. Other dilutions are then prepared gravimetrically and measured so as to produce a family of diluted solutions (Figure 68-1). The first stage may be the only one in the preparation of gravimetrically dripped solid point source standards, depending on the activity concentration needed to obtain an activity of several hundred kilobecquerels in the solid source, because the dispensed mass values are usually about $5\mathrm{mg}$ to $15\mathrm{mg}$ per drop. It is possible to dispense two or three drops onto the support and still obtain a fairly thin solid source with sufficiently low radiation attenuation. Another condition for diluting and preparing standards from a solution is a low salt concentration. For an inactive carrier solution, this is roughly $50\mathrm{\mu g}$ salt per gram of solution, a value which should not be exceeded for an aqueous radioactive solution. Many details of these procedures are given in Campion’s monograph [23]. In one of the first dilution steps it is often useful to prepare solid samples so as to measure the activity ratio of the radionuclidic impurities in the solution with a Ge spectrometer (Section 6.6). Obviously, the activity ratios of the radionuclidic components must not change during the dilution steps.

In each dilution by a single factor, which should not exceed about 100, large numbers of ampoules are prepared together along with other types of source, and the activity concentration of the ampoules is checked by ionization-chamber measurements (Figure 68-1). This process may be continued to produce concentrations of quite a low activity. Such dilution factor measurements, or quantitative checks of ampoules after refilling the solution into other ampoules, also serve to confirm the chemical stability of a solution.

All solutions in a dilution family are usually interconnected via the dilution factors to one of the first members with the strongest activity concentration, the so-called master solution. All subsequent calculations and comparisons of activity concentrations are based on this [23].

Uncertainties resulting from diluting and dispensing have been discussed by several authors [307], [42], [23] who show that, with reasonable precautions, these procedures should not introduce a relative uncertainty of more than 0.1% into the final result of an activity concentration measurement. Campion in BIPM ([23]) proposed an internal verification procedure using ⁶⁰Co and several dilutions combined with international comparisons of activity measurements, and suggested that participation should be compulsory for every standards laboratory.

In addition to measurements within a family of dilutions, direct standardization of a particular radionuclide is required from time to time, and a calibration factor for the ionization chamber measuring system is determined within the various solution families. Assuming stable conditions of measurement, the corresponding calibration factors from various measurements on a long-term scale are compared so that their evolution with time can be studied. If these calibration factors remain constant within a defined uncertainty this confirms that the unit of activity is accurately maintained by the ionization-chamber measurement system [395], [27], [218], [58], [451], NIST SP 250.

9.2.  Quality control for activity measurements and intercomparisons in nuclear medicine and industry

The main instrument for activity measurements in the field of nuclear medicine is the radionuclide calibrator, also called the activity meter. It is described in Section 3.2.4 where the technical details and calibration of the instrument are given. These instruments provide a direct reading in units of activity. Here, the aspects of quality control relevant to activity measurements of radiopharmaceuticals and to the corresponding regulations are described. The situation in countries practising nuclear medicine is regulated by laws, standards or recommendations from organizations which have official responsibilities in this field. These regulations apply to the manufacturers of radiopharmaceuticals and measuring instruments (radionuclide calibrators) [429], [430] and to users in hospitals or medical practices. The national standards laboratory, or a laboratory traceable to it, delivers appropriate activity standards for quality assurance of activity measurements for which the user is responsible. Modern regulations specify the accuracy required in measurements of activity and of the dose from radiopharmaceuticals administered to patients, and make it a legal requirement that records are kept. The final responsibility for these procedures in the sense of the radioprotection rests in the hands of the user of a radiopharmaceutical, usually the physician applying it. He is normally assisted by well-trained staff who carry out the necessary activity measurements and have experience in handling activity meters. Their duties include carrying out the checks necessary for daily routine measurements, for field calibration procedures and for the long-term peformance of the instrument. There is scientific interest not only in performing correct individual activity measurements for hospital work, but also in obtaining correct results for comparisons on a general research level, and hence to develop new procedures and methods in nuclear medicine.

These requirements for the quality control of activity assays in nuclear medicine can be fulfilled by calibrations made by direct comparisons with standards and by careful application of the regulations. To participate in comparisons of activity measurements organized by national authorities or organizations, for example by the national standards laboratory or a calibration service traceable to it, is strongly recommended [8], [1], [110], [230], [231]. Comparisons are one of the best tools in quality control for field applications, because not only the instrument calibration is checked but also the entire measuring chain, including the personnel operating it. One of the first pilot studies on the quality assurance of activity measurements using radionuclide calibrators was carried out by Garfinkel and Hine ([165]), measuring radionuclide sources in US hospitals with different types of calibrator. Since then, many other comparisons have been organized in countries practising nuclear medicine [201], [158], [172], [202], [476], [332], [387], [388], [96], [333].

In such a comparison samples of solution containing a known radionuclide, but with an activity unknown to the participants, are distributed. After careful measurement by the organizer, the participants measure the activity under the usual conditions and report their values to the organizer of the comparison. Later each individual participant is informed of the certified value of the activity of his sample. This procedure serves the purpose of providing a recalibration of the user’s instrument. All the results received are treated by statistical methods and are reported anonymously (Figure 69-1-Figure 69-4 [100], Figure 70-1-Figure 70-4 [477]).

In recent years many such comparisons have been organized by national standards laboratories and by calibration services providing traceability. The results have not always been satisfactory [277]. In earlier comparisons deviations by more than 30% from the nominal values were found, far beyond the limits imposed by the regulations [168], [197], [287], [337], [424], [356], [434], [435], [477], [436], [60]. As a result of the continuing attention given to quality assurance, the situation has improved in recent years [81], [60], [486], [351], [100], [162].

The critical radionuclides are those in which the solution to be measured contains radionuclidic impurities or where parts of the radiation are emitted within an energy range with which users are not familiar [275], [202], [331], [477]. This is well known in the metrology of radioactivity, see preceding chapters. A further source of error is that the measuring conditions for calibration and for practical measurement are not the same (e.g. a variation in measurement geometry). In some cases, a systematic trend in an assembly of instruments from a specific manufacturer has also been observed (Figure 69-4 or Figure 70-3-Figure 70-4).

The same concern for quality assurance is necessary in the production of radioactive sources for the nuclear industry and in applications involving radioprotection and environmental surveillance programmes [21], [281], [172], [181]. When, in these fields, ionization chambers are used to measure radioactivity [210], [302], the arguments valid for legal metrology or nuclear medicine obviously apply, and the same measurement conditions must be fulfilled for quality control.

9.3.  Half-life measurements

Direct decay measurements using an ionization chamber are the most obvious and least complicated means of determining the half-life of a radionuclide [480]. The equation for this method is given by the law of radioactive decay in Section 2.4, Equation (2). Basically, the method consists in repeated measurements of activity values $A\left(t\right)$ of the radionuclide under study as a function of time and a fit to the two parameters, namely the half-life $T_{1/2}$ (or decay constant $\lambda =\text{ln}2/T_{1/2}$) and an appropriate value for the initial activity $A_{\text{o}}=A\left(t_{\text{o}}\right)$. In the decay method, $A\left(t\right)=\text{d}N\left(t\right)/\text{d}t$ is defined as the probability of changing a defined nuclear state where $N$ is the number of nuclides in this state [226].

The half-life is assumed to be an innate parameter of a radionuclide and characteristic of its decay [367], [368]. Its value may be slightly influenced by chemical effects (or interactions) in the case of electron capture nuclides, for which the electron density in the shell plays an important role. These chemical effects are normally of order 10^–4 of the actual half-life value of the capturing radionuclide. A survey of experimental and theoretical perturbation data of the decay constant ($\Delta \lambda /\lambda$), including many references to historical publications of that problem, from about 1900, is given by Hahn et al. ([194]). Special techniques have been developed to measure these effects, for example differential experiments using two ionization chambers with samples of the same radionuclide in different physical or chemical states [132], [32], [301], [243], [33], [15], [321]. Kakiuchi et al. ([243]) also reported an application of this differential method for the measurement of a long half-life in a short period of time, e.g. with ⁶⁰Co in a 10 d run giving $T_{1/2}=\left(5.27\pm 0.07\right)\text{ a}$.

Other measuring methods compete with the decay methods and offer the possibility of identifying and counting the different nuclides directly, for example by mass separation of the material containing the decaying radionuclide [101], [102]. These methods are useful for radionuclides with long half-lives, such as ²³³U [456], or if identification of the radionuclide by radiation discrimination is difficult, as in the case of ²⁴¹Pu. For this, the half-life was measured by mass spectrometry of the decreasing ratio of ²⁴¹Pu / ²⁴²Pu [297]. The number of atoms of each nuclide $N_{\text{i}}$ in the sample is determined, and the relation $A\left(t\right)=\text{d}\mathrm{N}\left(t\right)/\text{d}t=-\lambda N\left(t\right)$ for the decaying radionuclide (²⁴¹Pu), or an equivalent relationship for the ratio of the nuclides (²⁴¹Pu / ²⁴²Pu) as a function of time, is used.

In other methods for long-lived radionuclides the decay heat is measured by microcalorimetry [292], for example to determine the half-life of ²¹⁰Pb [343]. At the other extreme, half-lives of less than a minute can be measured by sophisticated electronic methods which involve the time behaviour of scalers and related system properties such as multiscaling or multianalyzing [258], [311], selective sampling [312] and others [150]. Some of these alternative methods are summarized in an early review by Rowlands ([359]), which also gives many references to other articles of that period. The direct decay methods with ionization chambers do not give access to very long or very short half-lives.

One difficulty of the direct decay method is that of selecting only the decay from the nuclear state which is to be measured. This requires precise knowledge of all disturbing effects. In an integrating instrument, like an ionization chamber, the detection of radiation from other radionuclides must be avoided and an accurate measurement of the measurand as a function of time is essential. The second point implies the absence of any instrumental fluctuation over both short and long time periods or, more realistically, permanent monitoring of the stability of the instrument response and the application of a correction when necessary. Some of the effects, and their consequences for data fitting, are discussed by Tagesen and Winkler ([439]). This is particularly important for measurements of rather long-lived radionuclides like ⁹⁰Sr with $T_{1/2}=\left(28.7\pm 0.3\right)$ a [184], [490] and, more recently, $T_{1/2}=\left(28.91\pm 0.04\right)$ a [298], and ¹³⁷Cs with $T_{1/2}=\left(30.18\pm 0.15\right)$ a [480], [225], see Table 10. Considerable effort has been made in a number of standards laboratories to improve the accuracy of half-life values [220], [215], [272], [461], [396], [225], [452]. The determination of accurate half-life values using ionization chambers is discussed in the following sub-sections which describes the experimental techniques, the fitting procedures for measurement data and the evaluation of existing half-life values from the literature for use in decay corrections to measurement data.

10.1.  Intercomparisons of activity standards and traceability

When an international comparison of activity measurements is organized for a given radionuclide, a raw solution is carefully selected and checked for suitability (i.e. activity concentration, chemical stability and radionuclidic impurities) by the BIPM or another agreed laboratory. The radioactive solution is then distributed to participants in flame-sealed ampoules of BIPM/NIST standard geometry with a solution mass of about $3.6\mathrm{g}$ per sample (Figure 50-1). In recent years it has been usual to organize a trial comparison before the main comparison to pick up and resolve any measurement problem that might arise. First, a preliminary ionization-chamber measurement of the radioactivity is required for the original ampoule. Then, the standardization is performed by one or several direct methods, using recommended techniques for source preparation and measurement [23], [316], [294]. A questionnaire is circulated to all participants in the comparison. In this details of techniques, experimental parameters, possible correction factors, uncertainty components and the final activity concentration value at reference time are reported by the participant. The filled-in form is returned to the organizer (BIPM). The results are processed by statistical methods, illustrated in diagrams, analysed, and published in a report which is distributed to the participants.

International comparisons of this kind have been organized since the early 1960s by the BIPM under the auspices of the Comité Consultatif pour les Etalons de Mesure des Rayonnements Ionisants (CCEMRI), Section II, Mesure des Radionucléides. The first comparison was with ³²P in 1961, followed by ¹³¹I, ¹⁹⁸Au and ⁶⁰Co in 1961 and 1962 [62], [24] and many others, for example the ¹²⁵I trial comparison in 1987 [347], the ¹²⁵I main comparison in 1989 [348], [344], a ⁷⁵Se trial comparison in 1991 [349], [345], [346] and the ⁷⁵Se main comparison in 1992. The level of agreement in an international comparison is typically about 2% ($1\sigma$) for a single participant [24], with maximum deviations of about 5% for particular cases. In most of the comparisons since the 1960s the number of participants, mainly from national standards laboratories, has averaged around 20. The highest accuracy was obtained in the ¹³⁴Cs comparison in 1978, with a relative standard deviation of 0.17% (with 24 participants) and a maximum deviation of 0.68% [379], [374], see Figure 73. For some radionuclides, standardizations could not be performed by direct measuring methods like the $4\mathrm{\pi }\mathrm{\beta }-\mathrm{\gamma }$ coincidence technique. Special methods or technique were required: for ⁵⁵Fe, a tracer or calibrated-detector-response method [381], [419], for ⁹³Nb^m liquid-scintillation counting (LSC, see Section 10.4) or similarly, a calibrated-detector-response method [84], for ¹⁰⁹Cd, a calibrated $4\mathrm{\pi }$ detector, tracer or special coincidence methods [216], [433], for ¹²⁵I, photon-photon coincidence or sum-peak methods [216], [344], and for ¹³⁷Cs, coincidence methods with tracer [382], [383] and [377], see Figure 74. The SIR results for a period of five years are included in this diagram.

Consistent results in international comparisons is evidence for the quality of a national standards laboratory on an international level. They establish traceability between the distributors and users of activity standards from different countries (Figure 68-2). International comparisons of a particular radionuclide solution organized by the BIPM, or comparisons of samples measured by the SIR, represent the highest level in a “Traceability Tree” [294] describing the hierarchy of radioactivity-standardization laboratories and organizations [295], [187], [292], [355], [293], [67], [244], [246], [316], [294]. A complete chain of traceability is required linking quality control by users — hospitals, industrial groups [172], universities or government agencies — via regulatory bodies and national standards laboratories to international organisations like the BIPM, the IAEA or the ICRM. Guidelines for the international acceptance of radioactivity calibration sources have been formulated by Hoppes and Hutchinson ([217]) in response to a request from the International Committee for Radionuclide (ICRM). Laboratories participating in the traceability system must observe special quality assurance (or quality management) criteria. These include responsibilities for the training of staff, for installations and for measurement procedures. The criteria are set with the aim of proving that the laboratory has the ability to produce certified radioactivity standards or reference materials to agreed standards, so checks in the course of production and calibration must be both carried out and documented. The recommended quality assurance criteria provide a means of bringing the results of the international work on standardization into standards certification procedures and ultimately into user laboratories. They are the basis by which legal, safety and other requirements for the acceptance of standards will be met in an international frame [217].

10.2.  Measuring equipment and procedures of the SIR

The SIR is an ionization-chamber measuring system consisting of two re-entrant ionization chambers with current-measuring electronics and data storage [372], [376]. A photograph of this system can be seen in Figure 37-1 [376], [25]. Two identical chambers (20^th Century, IG11/N20 [69]), filled with nitrogen gas at a pressure of $2\mathrm{MPa}$, are set up inside a common lead shielding $5\mathrm{cm}$ thick. The two chambers are connected alternately to a current-measuring device, based on the Townsend-balance principle with stepwise compensation (Section 4.4), in which the time to charge the capacitor of the balance to a constant voltage level is measured a preset number of times. The background and leakage current of the system is typically about $33\mathrm{fA}$. The current of the samples is measured relative to that of a ²²⁶Ra reference source. Five reference sources, with a radium content from about $0.003\mathrm{mg}$ to $0.3\mathrm{mg}$, can be chosen to match the required activity range. The sources were manufactured by Union Minière in Brussels in 1973 from aged radium. They contain radioactive RaSO₄ with an admixture of inactive BaSO₄ in a Pt-Ir sealing. Several tests of this measuring system have been described by Rytz ([376]). The sources have been carefully compared, and the ratios are known to about $\pm 2\cdot {10}^{–4}$, at least for the four stronger ones. The absence of leaks is evidenced by the absence of trends in these measurements. To check that the radium daughters in the sources are close to equilibrium, two additional reference sources also containing aged radium were acquired in 1983 and included in the monthly test measurements. The repeatability and reproducibility of the two-chamber system have been tested with the reference sources and the data show that the current ratios are satisfactorily stable. Tests for ampoule geometry and source position dependence have also been carried out. Including an uncertainty component for the ampoule wall thickness as a function of energy, all uncertainty components add up to a combined relative uncertainty ($1\sigma$) of 0.1% for photon emitters of about $60\mathrm{keV}$ (²⁴¹Am), and of 0.05% for emitters of about $1.25\mathrm{MeV}$ (⁶⁰Co) [294].

The results of the measurements of SIR samples are given in terms of equivalent activity $A_{\text{e}}$ (Section 2.5), which is related to the current (about $330\mathrm{pA}$) produced by the strongest reference source ($0.308\mathrm{mg}$ Ra element) according to the formula [376]:

(These definitions are taken from the original publication and should not be confused with other definitions in this review using the same notation).

The measurements are performed in cycles consisting of a background run, ten readings with the reference source, ten with the sample and again ten with the reference source. The determination of $A_{\text{e}}$ is carried out by a computer programme taking into account the measurement parameters, corrections for decay and radionuclidic impurities. The final result of $A_{\text{e}}$, expressed in $\mathrm{kBq}$, with its uncertainty components and the sample parameters, is entered into a table. Lower limits of activity and impurity levels acceptable for the SIR are given in a report by Rytz ([375]).

10.3.  Results of measurements with the SIR

The radionuclide sheets which register equivalent activity values $A_{\text{e}}$, measured with the SIR, list the results obtained up to date. Until 1993, they gave details on about 570 ampoules containing about 50 radionuclides. Several of the most frequently used radionuclides, such as ⁵⁷Co, ⁶⁰Co or ¹³³Ba, have at least about 25 entries per nuclide. Each year about twenty new measurements for various radionuclides are performed and added to the file that is distributed annually to the participants in the SIR. An example of such a sheet for ¹⁹²Ir is given in Table 11. The standard deviation of the $A_{\text{e}}$ values of an individual radionuclide varies from 0.25% ($1\sigma$) for ⁶⁰Co, in the best case, with 43 measurements from about twenty laboratories, to about 1.1% ($1\sigma$), for the unfavourable case of ⁸⁸Y, with 25 measurements from twelve laboratories.

A graphical representation showing the consistency of the results by establishing the energy-dependent efficiency curve (Figure 63-1), see Section 7.2, has been published by Rytz and Müller ([380]), Rytz ([376]) and Rytz and Müller ([383]) with a supplement by Ratel ([346]) for the low-energy part from $30\mathrm{keV}$ to $250\mathrm{keV}$ (Figure 63-2). In Figure 63-1 the relative detector response values $f\left(E_{\text{n}}\right)$ are displayed, with the definitions from the publication of Rytz ([376]), namely

Here $E_{\text{n}}$ and $E_{\text{i}}$ are the photon energies, $p_{\text{n}}$ and $p_{\text{i}}$ the corresponding emission probabilities per decay of the radionuclide, and $k$ is 60, an arbitrarily chosen constant factor, if energies are expressed in mega electron volts ($\mathrm{MeV}$). In this equation, $f_{\text{i}}$ is the relative photon efficiency at energy $E_{\text{i}}$, as defined in Section 7.4; it should not be confused with the definition of $f$ in Equation (32). The value $f\left(E_{\text{n}}\right)$ of a radionuclide is shown at the photon energy $E_{\text{n}}$ for which the emission probability $p_{\text{n}}$ is largest and gives the major contribution to the ionization current. The value of $f\left(E_{\text{n}}\right)$ is calculated from the total reciprocal equivalent activity value ${10}^6/A_{\text{e}}$, proportional to the efficiency ${\epsilon }_{\text{N}}$ for the nuclide, by subtracting the weaker components, after interpolation and weighting with the corresponding emission probabilities $p_{\text{i}}$ (Section 7.2). Figure 63-1 shows remarkably large deviations of the individual values from a smooth curve in an energy range from $847\mathrm{keV}$ (⁵⁶Co) to $\mathrm{1 408}\mathrm{keV}$ (¹⁵²Eu), for example for ⁵⁶Co, ⁵⁹Fe, ¹⁸²Ta, ¹⁵⁴Eu and ¹⁵²Eu, but also quite large error bars. Radionuclides emitting many photons of different energies and covering a large energy range are also included in the representation, among them ¹³³Ba in the low-energy part, and ¹⁹²Ir, ¹³⁴Cs, ¹⁵²Eu and ⁵⁶Co, mainly with highenergy photons. A small deviation from the curve for these radionuclides illustrates the consistency of the calibration data.