The Basics of Nuclear Chemistry and Radiochemistry: An Introduction to Nuclear Transformations and Radioactive Emissions

Abstract

Radiopharmaceutical chemistry and nuclear medicine make use of radioactive elements and compounds labeled with them. This chapter describes the fundamentals of radioactivity in the context of life sciences. It addresses principal questions such as: What is the composition of an atomic nucleus and what are the forces which hold nucleons bound within the nucleus? Even so, some nuclei are stable, and many others are not—why? The fate of unstable nuclei is transforming into more stable nucleon configurations—but what are the basic pathways to do so? What’s going on inside the nucleus? What are the energetics and velocities of these transformations? And finally, the various changes inside the nucleus and in part also in the electron shell of the unstable nucleus are accompanied by individual emissions of electromagnetic and particular radiations: beta-electrons and positrons, electron neutrinos and anti-electron neutrinos, alpha particles, gamma-photons and X-rays, conversion electrons and Auger/Coster-Kronig electrons, and bremsstrahlung. What are those emissions in detail?

Processes of Transformations: Overview

In order to understand the origin and character of individual radioactive emissions accompanying nuclear transformation processes, we first need to answer three questions:

• What is an unstable nucleus?

• What is its motivation to transform?

• What is the best way for it to transform?

It is important to note that the following discussion aims at describing phenomena relevant to radiopharmaceutical chemistry and nuclear medicine. See recently published texts for a comprehensive review on all aspects of nuclear chemistry related to radiopharmaceutical chemistry [1,2,3].

Composition and Mass of an Atomic Nucleus

The atom is composed-->of the nucleus and the shell. All nuclei of atoms (except for one of the isotopes of hydrogen, which contains one proton and no neutrons) are composed of two kinds of nucleons: protons and neutrons. The shell of the atom is populated by electrons. For an electrically neutral atom, the number of electrons in the shell is equal to the number of protons in the nucleus. Table 1 summarizes the characteristic parameters for these three subatomic particles. The classical properties of these particles (i.e. their absolute mass and charge) can be expressed in terms of real mass.

Table 1 Summary of the basic properties of the three basic constituents of atoms of chemical elements: the electron, proton, and neutron

The nomenclature of nuclear chemistry and physics presents the nucleus in the following way: the number of protons (Z) and the number of neutrons (N) are displayed as lower indices to the left and right of the symbol of the chemical element, while the overall mass number (A)—i.e. the sum of the number of protons and neutrons—is presented to the upper left of the symbol of the chemical element. Figure 1 illustrates this for the three most relevant nuclei of the chemical element hydrogen. The three nuclei all have the same number of protons, namely, one, and all have one electron in their shell, which makes the nucleus the chemical element hydrogen. The number of neutrons, however, differs, and so does the mass number. The individual nuclei are called “isotopes”, and in the case of hydrogen (and exclusively for that chemical element and no other element), the three isotopes have individual names: hydrogen, deuterium, and tritium--> (with deuterium and tritium reflecting the mass number).

Fig. 1

Notation of nuclides in nuclear and radiochemistry and examples for three isotopes of hydrogen

Mass and Mass Defect

We now may believe that the mass of the nucleus is the sum of the masses of the protons and neutrons located in it. Let’s use the known absolute masses of the neutron and the proton and simply sum up according to the mass number, A, to yield the absolute mass of the nucleus. However, the result we obtain differs from our expectation: the simple sum of the masses of the individual—i.e. non-bound—nucleons does not reflect the real mass of the nucleus containing exactly the same nucleons bound together. The nucleus is lighter than its individual components! This represents one of the most fundamental effects of our material world. The difference is expressed as the mass defect: Δm^defect = m^nucleus − m^{sum of individual, non-bound nucleons}.

Figure 2 illustrates the situation for the nucleus of the helium isotope ⁴He. Let’s calculate the masses. What we need are three values: the absolute mass of the nucleus as determined experimentally, the absolute mass of the proton, and the absolute mass of the neutron as given in Table 1 in terms of kg.

Fig. 2

Mass defect: The nucleus of an atom is lighter in mass (and, consequently, lower in energy) than the weight of the sum of its identical but non-bound components

But wait a moment! Those absolute masses are extremely low and not convenient to handle. Accordingly, two other expressions of mass are preferred in nuclear sciences. One is the equivalent of mass in terms of energy according to E = mc². This yields energy values with the electron volt (eV) unit; see Table 1.

The other version is to utilize a relative mass parameter: the atomic mass unit. It considers the experimentally very precisely known mass of a stable isotope of a prominent atom, divides this mass by the number of nucleons, and provides a value that describes the mass an average nucleon contributes to the mass of the whole atom. The reference is the carbon isotope of mass number 12, a nice nucleus: very abundant on earth, very symmetric with 6 protons and 6 neutrons, all nucleons existing as pairs. The experimentally determined absolute mass of one single carbon-12 atom is 19.92648^.10⁻²⁷ kg. It is divided by its mass number 12. The value resulting from 19.92648^.10⁻²⁷ kg/12 is 1.66054^.10⁻²⁷ kg which is called the “atomic mass unit”, u. With this parameter in hand, the absolute mass of every other isotope is easily estimated by just multiplying the mass number, A, of the given isotope by the atomic mass unit, u. Also for the subatomic particles -->such as the electron, proton, and neutron, masses can be expressed as parts of u; see Table 1. For a systematic presentation of the individual values of atomic mass and other parameters such as mean nucleon binding energy, see the AMDC—Atomic Mass Data Center—IAEA Nuclear Data Services [4] and Atomic Mass Evaluations [5, 6]. More data compilations for 2016 can be found in references [7, 8].

Let’s now turn to a real example. The nucleus ⁴He (which represents the α-particle) is composed of two protons and two neutrons. The mass of the ⁴He nucleus expected by summing 2m_p (u) + 2m_n (u) is 4.03188 (u). The experimental value for the mass of the He atom is 4.00260325415 u. The corresponding value for the He nucleus (obtained by subtracting mass and binding energy contribution of the two electrons) results in 4.00150 u. The total mass of the nucleus is thus smaller than the sum of the four individual nucleon masses not bound together: Δm is 4.00150 u – 4.03188 u = − 0.030377 u (see Fig. 2). See Wang et al. for a tabulated presentation of the mass defect values for all the stable nuclei [6].

Binding Energy

Where is that mass—“the mass defect , Δm”—going? Of course, mass cannot disappear: -->it is translated into energy according to ΔE = Δmc². What happens? Once nucleons approach a very small distance between each other (on the order of fm, i.e. the dimension of the atom nucleus), they are attracted to each other by the “strong force”—the strongest force known in our universe—and combine to form a nucleus. The energy all the nucleons save once bound together compared to their former non-bound state is called the “overall binding energy”. The equivalents of Δm and ΔE thus reflect the overall binding energy, E_B, of the nucleus. Nucleon binding energies correlate with mass defect values via E_B = ΔE = Δmc². Accordingly, the overall binding energy of a nucleus increases with increasing numbers of nucleons in it. Table 2 gives examples for four nuclei. However, a more interesting parameter is the “mean binding energy”, which is the average binding energy contributed by an individual nucleon: Ē_B = $\frac{E_B}{A}$.

Table 2 Experimental masses of atoms, mass excess, as well as overall and mean binding energies for ⁴He, ¹²C, ⁵⁶Fe, and ²³⁸U. The nucleus ¹²C may serve as a relative scale again and is involved in defining a mass excess value, indicating the relative difference in binding energy between the “reference” ¹²C nucleus and any other nucleus

Let’s calculate the mean binding--> energies of the 4 nucleons of the helium-4 nucleus as well as the 12 nucleons within the carbon-12 nucleus. The overall binding energy of the ⁴He nucleus is 0.03038 u = 0.05045^.10⁻²⁷ kg in terms of mass and 4.53^.10⁻¹² J or 28.295660 MeV in terms of energy. The mean binding energy per nucleon within the ⁴He nucleus is Ē_B = 28.295660 MeV / 4 = 7.073915 MeV. For ¹²C, it is 7.680 MeV. Compared to Ē_B(⁴He) = 7.074 MeV, the 12 nucleons of carbon-12 are bound more strongly together within the ¹²C nucleus. This mean binding energy increases further with the increasing mass number, reaching maximum values of ~8.8 MeV for mass numbers around 56–62 but then starting to diminish for very heavy nuclei. Table 2 lists the values of experimental atomic masses, overall and mean binding energies, and the mass excess for ⁴He (a light nucleus), ¹²C, ⁵⁶Fe (a medium mass number nucleus), and ²³⁸U (a very heavy nucleus).

The maximum values for mean nucleon binding energy are Ē_B = 8.790 MeV for ⁵⁶Fe, 8.792 MeV for ⁵⁸Fe, and 8.794 MeV for ⁶²Ni. However, mean binding energies are quite similar compared to the strongly varying mass numbers and atomic weights, at least for most of the nuclei of A > 10. In this broad range of 10 < A < 238 for stable nuclei, average values for Ē_B are 8.2 ± 0.6 MeV. Ē_B values for the ~250 stable and more than 3000 unstable nuclei are tabulated in reference [6].

Models

A key question in the nuclear sciences is understanding the correlation between the mass number A (i.e.the total number of nucleons in the nucleus) and Ē_B, the mean nucleon binding energy. There is a huge data set for the absolute masses of the ~250 stable nuclei known and their corresponding mean nucleon binding energies. The basic theory is the “liquid drop model”, which is accompanied by a complementary “shell model”. The “liquid drop model” (LDM) of the atomic nucleus postulates that all protons are identical, all neutrons are identical, and all nucleons are distributed homogeneously within the nucleus like H₂O molecules within a drop of liquid water.

The semiempiric mathematics quantifying these experimentally known dependencies is the so-called Weizsäcker equation. The equation may be divided into five (or more) parts for volume, surface, Coulomb forces, symmetry, and pairing. Each term of this equation has a physical rationale that describes the various ways the two different types of nucleons contribute to binding energy. For some terms, there is a dependency on mass number, A, exclusively. For others, the individual contributions caused by either protons or neutrons are reflected as well. Finally, each of the terms gets a coefficient, values that are just adjustments of a polynomial to the “experimental” values of mean nucleon binding energy. The equation itself is inserted into Fig. 3. The polynomial correlation obtained is also shown in Fig. 3.

Fig. 3

Correlation of mean nucleon binding energies , Ē_B, versus mass number. Black squares are experimental values for stable nuclei, the three uranium isotopes U-234, U-235, and U-238, and Th-232; the red line indicates the polynomial according to the Weizsäcker equation inserted

Overall, the result is excellent—with some exceptions! For some mass numbers, there are extreme deviations between real values and the ones predicted by the LDM. This begs for another approach, which is reflected by the “shell model”. Among the existing sets of A, Z, and N with the ~250 stable nuclei known, there is a surprising over-expression of stable nuclei that possess 2, 8, 20, 28, 50, and 82 protons or neutrons. Why? As long as the reason for that (over-expression) was not clear, those numbers were called “magic”. Figure 4 shows that “over-expression” for isotopes of magic number 20.

Fig. 4

The “over-expression” of stable isotopes (black boxes) of magic number 20. Excerpt of the Chart of Nuclei. There are four stable isotopes representing N = 20 but zero for N = 19 and just one for N = 21. There are six stable isotopes representing Z = 20 (the chemical element calcium) but only three for Z = 19 and just one for Z = 21. Lower numbers in the boxes indicate the natural abundance of individual stable isotopes of one element. For calcium, the most abundant isotope is ⁴⁰Ca, a “double-magic” nucleus (20 protons + 20 neutrons)

Nuclei expressing these numbers for protons or neutrons seem to be (and are) more stable than predicted by the LDM. Consequently, another theory accompanies the liquid drop model theory: the “shell model” (SM). Similar to the orbital theory of electrons, both protons and neutrons are supposed to exist at characteristic shell levels with individual quantum numbers. This model centres on two key postulates that are dramatic departures from LDM:

• (i) The nucleons are not distributed homogeneously but rather in specific “shells”.

• (ii) All the protons and all the neutrons are different from each other, i.e.having individual characteristics that make each nucleon in the nucleus unique.

A key challenge to organize the protons and neutrons of a nucleus into shell structures was to identify a system of shell arrangements, in which the balance of the nucleons involved represents “full” (or “closed”) shell occupancies and reflect the “magic” numbers. This is similar to the full occupancies of the noble gases in the Periodic Table of the Elements, in which electrons are filled into all the existing vacancies of one period.

The SM also helps to understand the occurrence of both excited and ground states for a nucleus. Once there are defined shell occupancies for protons and neutrons, it is straightforward to accept the idea that a certain nucleon may (for a certain period of time) exist within a higher shell as an excited level and subsequently “de-excite” back to its ground-state level. This is analogous to the well-known behaviour of excited electrons, which of course “de-excite” to their ground-state electron shell accompanied by the emission of electromagnetic radiation. In fact, following nuclear transformations within unstable nuclei, the nucleons of the newly formed nucleus frequently do not reside within their ground-state shells but rather at higher energy shells, defining an “excited nucleus”. Only when the excited nucleon “falls” to its lower energy shell can the ground-state nuclear level be achieved. This is the essence of radioactive emissions such as γ-rays (see below).

From Stable to Unstable Nuclei

Both LDM and SM were developed based on parameters (experimentally precisely determined masses) of ~ 250 stable atoms. Those nuclei are characterized by a set of proton and neutron and mass numbers Z, N, and A, respectively, which represents nuclei of maximum mean nucleon binding energy, accordingly. One can conclude that the stability of an atomic nucleus of mass A is basically a question of the right mixture between protons and neutrons for a given value of A. If “right”, the nucleus owns the optimum value of the mean nucleon binding energy for that value, reflecting the correlation Ē_B = f(A). If that particular mixture of nucleons behind those stable nuclei deviates from the optimum value, Ē_B values are lower, and the nucleus of that value of A is not stable anymore. Being not stable does not mean “not existing”. A suboptimal mean nucleon binding energy does guarantee stability but allows the nucleus to exist for a certain period of time. The question is: If the nucleus exists but is not stable, what is it doing?

The answer: Such an unstable nucleus tries to stabilize! Its “private” motivation is to improve the mean nucleon binding energy by optimising the existing mixture of protons and neutrons into a better, more stable mixture. This is the essence of radioactive transformations. The old unstable nucleus will find a way to form a new, lower mass nucleus that is characterized by increased mean nucleon binding energy. Put another way, the unstable nucleus simply obeys one of the general laws in the universe: to improve its status in terms of energy and mass. Consequently, the process itself is exothermic and spontaneous. The velocity of this transformation (we will later define this in terms of “half-life”) is simply proportional to the gain in terms of +Ē_B and −m.

The only issue remaining is to understand how a given unstable nucleus manages this transformation. In fact, there are several pathways, and we will soon learn how clever a nucleus can be in selecting the best route.

Transformation, non “Decay”

In the literature, the behaviour of an unstable nucleus is typically expressed as if it “decays”. Let’s first agree on a definition. Does the unstable nucleus really “decay”? The philosophic answer is that nothing decays, it only transforms into something new. An unstable nucleus, K1, thus transforms into a more stable one by optimising its mean nucleon binding energy. The absolute mass of the transformation product nucleus, K2—which may be truly stable or simply “more stable” (but still “radioactive”) and in need of another step of transformation—is less than the absolute mass of the initial unstable nucleus. This transformation thus proceeds exothermically. The difference in mass is typically expressed in terms of energy, ΔE, and is referred to as the Q value of a transformation. However, there is a third component to consider. This is the “radiation”, which is released and accompanies the transformation processes. This kind of emission is generally associated with “radioactivity”. At this stage, it is called “x” and subsumes the various kinds of “radiation” to be discussed later in detail (Fig. 5).

Fig. 5

Simplified scheme of primary transformation of an unstable nuclide, K1, into a more stable nuclide, K2. This “x” typically is a particle, such as a ⁴He nucleus (the α-particle) or an electron (as in β-processes)

Primary Transformations Versus Secondary Transitions and Post-processes

The primary goal of an unstable nucleus is to optimize its nucleon composition. “Radioactivity”—i.e. all of the forms of radioactive emission we observe—simply is a phenomenon accompanying the individual processes an unstable nucleus undergoes to increase its mean nucleon binding energy! In the following, let’s define a hierarchy of these processes of transformation: primary processes, secondary processes, and post-processes (Fig. 6).

Fig. 6

Hierarchy of the primary transformation of unstable nuclei, the subsequent secondary transitions, and the parallel post-processes

It all begins with a “primary” transformation: the change in the nucleon composition of the unstable nucleus: K1(A₁,Z₁,N₁). This change results in the formation of a new nucleus: K2 (A₂,Z₂,N₂). The three subtypes of primary transformations are β-processes (where A remains constant with A₂ = A₁, only Z and N changes by one unit to Z₂ = Z₁±1, N₂ = N₁±1), α-emission (where A changes A₂ = A₁–4, Z₂ = Z₁–2 and N₂ = N₁–2), and spontaneous fission.

In some cases, the nucleons of the newly formed K2 do not directly appear at the ground-state nuclear shell levels but instead occupy higher-energy shells. This situation is termed the formation of an “excited state” nucleus, ^ʘK2, which must subsequently de-excite to create the ground-state nucleus. The excited and de-excited nuclear states all belong to the same nucleus of identical set of (A₂,Z₂,N₂). This process of de-excitation encompasses the “secondary” transitions described here.

Finally, two classes of post-processes—both of which produce their own types of radioactive emissions—must also be considered. These processes do not concern the nucleus itself; instead, they either occur within the electron shell of the transforming nucleus or outside the atom.

Mechanism of Primary Transformation Processes

Three subtypes of primary processes differ --> in terms of the way that unstable nuclides convert into stable ones by changing the absolute number of nucleons (changing A) or by modifying the ratio between protons and neutron (changing the Z:N ratio for constant A). In the latter cases, an “excess” neutron “just” converts into a proton (supposing the nucleus owns an excess of neutrons over protons) or vice versa. In other cases, a nucleus releases a number of nucleons, typically as a small cluster of two neutrons and two protons (the α-particle), in order to lower its mass number, A. For a limited number of very heavy nuclides, there is a third option: spontaneously splitting the large nucleus into (usually two) fractions in a process called “spontaneous fission” (sf). The latter pathway is not relevant to molecular imaging or therapy and thus will not be discussed further (Fig. 7).

Fig. 7

Distribution of radionuclides in the Chart of Nuclides undergoing β-processes, α-emission, and spontaneous fission. According to the Karlsruhe Chart of Nuclides, β-processes are indicated either by blue (β⁻) or red (β⁺ or ε) colour, α-emission in yellow, and spontaneous fission in green

Secondary Transitions: No Change in Nucleon Composition

In some cases, the rearrangement of nucleons in primary transformations directly yields the ground state of the new nucleus, K2. In many other cases, the proton and/or neutron shell occupancies of the newly formed nucleus are not identical to those of the ground state of that nucleus. Consequently, the newly formed nucleus exists—for shorter or longer periods of time—in an “excited” state. Those excited states subsequently de-excite to levels of lower energy according to the shell model of the nucleus. Secondary processes proceed within one and the same nucleus, i.e.at both ΔZ and ΔA = 0. Those “secondary” processes are better described as “transitions” than “transformations”. Again, there are three subtypes of secondary transformations: the emission of electromagnetic radiation, the formation of inner conversion electrons, and pair formation. (The first, known as γ-emission, represents the most relevant subtype for SPECT imaging).

Post-processes

Some of the --> primary transformation mechanisms (in particular the electron capture process ) as well as a secondary transition pathway (namely, inner conversion) leave a hole within the electron shell surrounding that nucleus. While the new nucleus, K2, is already formed, the vacancy in the electron shell of the atom must be filled. The two ways to organize this are the emission of X-rays and the emission of Auger and Coster-Kronig shell electrons. These processes are categorized as “post-processes I”. The most relevant emission produced by these processes is X-rays. Like γ-emission, X-rays are electromagnetic radiation. However, their origin is different: while γ-emission is created within the nucleus via the de-excitation of excited nuclear levels, X-rays are generated within the electron shell.

Independently, the particle emission “x” released in primary and secondary processes interacts with the many other, stable atoms surrounding the newly formed nuclide, K2. The effects induced by these interactions are discussed as “post-processes II”. Most relevant (at least in the context of nuclear medicine) are β⁺ particles—i.e.positrons, formed in the primary β process. Positrons interact with electrons to induce an annihilation phenomenon, which produces a pair of 511 keV γ-rays that form the basis of PET.

β-Transformations

Three Pathways: β^--Process, β⁺-Process, and Electron Capture (ε)

Let’s start with a neutron-rich unstable isotope. What should it do to stabilize itself? The elimination of a neutron seems to be a good idea. However, this would require sufficient energy to eliminate that nucleon from the nucleus, which is not necessarily available. (Remember, the average binding energy per nucleon is around 8 MeV!) On the other hand, it is helpful to think about an “excess” of neutrons as tantamount to a “deficit” of protons. In light of this approach, the clever unstable nucleus comes up with a brilliant idea: converting a neutron into a proton would solve the problem in an elegant way. The inverse applies to neutron-deficient (proton-rich) isotopes, which can gain stability by converting a proton into a neutron. Converting a nucleon in excess to a nucleon in deficit is the foundation of the β-process. In this manner, the mass number of the nucleus will remain constant throughout the transformation.

The conversion of a neutron into a proton results in the process _ZK1 → _{Z + 1}K2. This is accompanied by the emission of a negatively charged electron and is called a β^--process. The conversion of a proton into a neutron results in the opposite case: _ZK1 → _Z−1K2. While there is only one approach for the _ZK1 → _{Z + 1}K2 conversion, there are two options for the _ZK1 → _Z−1K2 process. The one accompanied by the emission of a positively charged electron is called the β⁺-process. Alternatively, or in parallel, neutron-deficient nuclides may transform by the capture of an electron from the K electron shell. This type of β-process is named “electron capture” (ε).

From Isotopes to Isobars

All β-transformations of unstable nuclides proceed at A = constant. Neutron-rich isotopes transform via the neutron → proton conversion. The new nuclide, K2, has a composition of (Z + 1, N−1) and arrives at a nuclide that is a heavier chemical element. Proton-rich nuclides utilize proton → neutron conversion and yield a new nuclide, K2, of (Z−1, N + 1) composition. K2 represents a chemical element of lower Z. This transformation may continue in a stepwise fashion—K1 → K2 → K3, etc.—until the Z to N ratio reaches that of a stable nuclide.

This is illustrated in more detail in Fig. 8. An isobar line is indicated at A = 18 with ¹⁸O as the stable nuclide. The β⁺- and electron capture processes approach ¹⁸O coming from the proton-rich nuclides ¹⁸Ne and ¹⁸F, while the β⁻-processes approach ¹⁸O via the ¹⁸B → ¹⁸C → ¹⁸N cascade.

Fig. 8

Left: β-transformation of unstable nuclides along the A = 18 isobar line with Ē_B values in MeV. ¹⁸O is the nucleus of maximum mean nucleon binding energy of this isobar line; it is stable. The β⁺ and electron capture processes approach ¹⁸O coming from the proton-rich nuclides ¹⁸Ne and ¹⁸F; the β⁻-processes accumulate at ¹⁸O via ¹⁸B → ¹⁸C → ¹⁸N →. Right: A selected isobar may be turned into a parabola (1) using a coordinate system of type Ē_B = f(Z)^{A = constant}. Note the direction of changes in Ē_B. (2) The proton-rich unstable nuclei successively transform via β⁺ or ε on the right side. The β⁻ transformations are on the left side. With either step of the transformation, Ē_B increases by characteristic amounts of ΔĒ_B. Interestingly, the individual values of ΔĒ_B become smaller for each step (3). Finally, both arms of the parabola approach the vertex of the parabola (4), where the most stable nuclide (or two stable ones) is (are) found

From Isobars to Parabolas

The diagonal isobar line may be converted into a parabola and gives a correlation of the type ΔĒ_B = f(Z) at A = constant. The blue nuclides from Fig. 8 shift to the left side of the parabola, because they are of low Z compared to the red nuclides, which are of higher Z. Each primary transformation step increases Ē_B values. Typically, the value of ΔĒ_B = f(Z) increases exponentially. This is reflected by the exponential expression of a parabola. The maximum mean nucleon binding energy is located at the vertex of the parabola, representing the stable nuclide. This is true for a single mass number A. It holds true for the neighboured mass numbers as well. Each of the many isobar lines of the Chart of Nuclides thus owns a maximum of mean nucleon binding energy for a specific value of Z. As the various terms of the Weizsäcker equation all include a multiple of mass number A, the equation may be transformed for the value of Z which lies at the vertex of the parabola. The expression is Z_A = f(A^constant). Z_A is the proton number with optimum mean nucleon binding energy.

$$ {\mathrm{Z}}_{\mathrm{A}}=\frac{\mathrm{A}}{2.0+0.0154{\mathrm{A}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} $$

As is characteristic for the mathematics of a parabola, the two ascents scale exponentially and thus become sharper and sharper. The x-axis, however, scales linearly with respect to Z. This indicates that the differences in mean nucleon energy between successive transformations of K1 → K2 are large at both “ends” of the parabola and become less and less pronounced the closer the transformation step is to the vertex. Simply put, the sharper the ascent, the more unstable the nuclides are.

However, those parabolas need a second look, which refers to the fact whether the number of protons or neutrons is even or odd. Let’s consider the combination of protons and neutrons in the nucleus in terms of (Z, N). For (Z = even, N = odd) and (Z = odd, N = even) nuclides, the new nuclide is of the same category: (even, odd) turns into (odd, even) and vice versa. This is the case for all isobars of odd mass number A. In this case (A = odd), there is only one parabola, and this is exemplified in Fig. 9 for mass number A = 95. In contrast, an (Z = odd, N = odd) nuclide turns into an (Z = even, N = even) nuclide and an (Z = even, N = even) nuclide turns into an (Z = odd, N = odd) one. This yields two separate curves as indicated in Fig. 10 for mass number A = 96.

Fig. 9

β-parabola for mass number A = 95: Isobars of odd mass number A represent transformations of (even, odd) nuclides into (odd, even) nuclides and vice versa. (1) one single parabola; (2) successive transformations of type β⁺ or ε at the right side and of type β⁻ at the left side; (3) decreasing differences in mean nucleon binding energy when approaching the vertex of the parabola; (4) Z_A is 40.937; (5) the only one stable nuclide is ⁹⁵Mo (Z = 42) with the largest value of Ē_B

Fig. 10

β-parabolas for mass number A = 96: Isobars of even mass number A represent transformations of (even, even) nuclides into (odd, odd) nuclides and vice versa. The impact of the parity term of the Weizsäcker equation creates two parabolas with a shift of ± δ/A^¾: (1) two separate parabolas with the one for (even, even) nuclides “below”, which is at higher values of Ē_B; (2) successive transformations of type β⁺ or ε at the right side and of type β⁻ at the left side alternating from (even, even) to (odd, odd) nuclides and so on; (3) the shift in energy between the two parabolas is 2δ/A^¾; (4) Z_A = 41.328; (5) the most stable nuclide is ⁹⁶Mo (largest value of nucleon binding energy). The stability of the two (odd, odd) nuclides needs to be studied in detail. In this case, ⁹⁶Ru is also stable, while ⁹⁶Zr has a half-life of 3.9^.10¹⁹ years (!)

From Two-Dimensional Isobars and Parabolas to the Three-Dimension Valley of β-Stability

There are many isobar lines across the Chart of Nuclides [9], ranging from short ones (e.g. A = 3 with the two nuclides ³H and ³He) to very long ones (e.g. A = 100 including 15 nuclides). Arranging these two-dimensional parabolas into a successive series of many parabolas creates a three-dimensional plot (Fig. 11). Unstable nuclides are positioned along the hillsides, stable nuclides at the bottom of the valley. The latter is called the “valley of β-stability”. The direction of the valley does not correspond to a straight line (which would have been the isodiaphere of N = A) but makes a soft turn to the right side. All the stable nuclides depicted in Fig. 4 of the chart of nuclide diagram lie in that “valley”.

Fig. 11

β transformation processes along a coordinate system of mean nucleon binding energy Ē_B vs. Z at A = constant form a parabola. If many parabolas are arranged with increasing A, the illustration reflects a valley formed by two hillsides. This “valley of β-stability” is composed of all the stable nuclides known

Quarks: The Elementary Particles Behind the Nucleons

The essence of the β-processes is turning either a neutron into a proton or vice versa. Nucleon binding energies improve, which is best expressed by the isobar parabola of Ē_B = f(Z) along an isobar. However, how can one sort of nucleon simply convert into the other one? In order to get an idea of this kind of wonder, a look into the theory of elementary particles and quantum physics is needed.

Elementary Particles

While proton, neutron, and electron have been classified as “subatomic”, it does not necessarily mean that these particles are not further divisible. While this holds true for the electron—which therefore is classified as “elementary particle”—the proton and the neutron are composed of other sub-nucleon particles. According to the development of the “standard theory of particle physics”, elementary particles (i.e.those which really cannot be divided further) can be arranged according to spin and electric charge. The spin of the particle may be half-integer or integer. Fermions all have half-integer spin values, while bosons have integer spin values. Fermions can be further subdivided according to charge. Fermions with integer electric charges are called “leptons”, while fermions with non-integer electric charges are called “quarks”. Both leptons and quarks can be subdivided further! For example, the electron is a fermion and a lepton (spin 1/2 and charge −1). There are likewise several types of bosons. The photon, for example, is a boson: spin = 1. The mediators allowing for the interactions between elementary particles are also called “field quanta”.

The elementary particles are summarized in Table 3. There are quarks—defined by non-integer spin and non-integer electric charge (yielding either +  2/3 or −  1/3)—and leptons, defined by half-integer spin and integer electric charge (0 or −1). In contrast, mediators or field quanta are characterized by integer electric charge (0, +1, or −1) and integer spin. This group belongs to the class of bosons. Gluons are the field quanta mediating the strong interaction (strong in power, short in distance), attracting nucleons, and being responsible for the formation of nuclei of atoms. In contrast, the W and Z bosons are correlated with the weak interaction. The photon is the field quantum mediating electromagnetic interaction.

Table 3 Overview on the system of elementary particles showing electric charge and intrinsic spin

Quarks and leptons are structured into three families, and—among other factors—arranged according to their mass (or energy) (Fig. 12).

Fig. 12

Families of elementary particles and their field quanta. The first family includes the up quark and the down quark, the electron and the electron neutrino. The second family collects the charmed quark, strange quark, muon, muon neutrino, etc.

Now we understand the composition of a nucleon. A proton is composed of two up quarks (2 times the electric charge of +  2/3 makes a +  4/3 charge) and one down quark (electric charge −  1/3). The resulting total charge thus is +1. A neutron consists of one up quark and two down quarks, and their particular electric charges compensate to the overall charge of 0 (Fig. 13).

Fig. 13

Left: The metamorphosis of quarks (here one d-quark into a u-quark) explains the conversion of a neutron into a proton. The other down quark and the up quark of the neutron remain unchanged. Right: Diagram of the process, indicating the appearance of the other members of the first family responsible for the β⁻ transformation: the W⁻ boson, the electron antineutrino, the β⁻ particle; see further in the text

Antimatter

Each quark and lepton has a “twin” that is identical with regard to all parameters except charge. These “twins” are called antiparticles. The most prominent antiparticle in the context of nuclear medicine and radiochemistry is the positron. It owns exactly all the properties of the electron (mass, spin), but its charge is +1 instead of −1. Another relevant system of elementary particle/antiparticle is the electron neutrino and its anti-electron neutrino (see below).

The First Family in the Context of β-Transformations

To understand the basic features of β-transformation processes of unstable nuclides, only the first family of elementary particles is relevant: two quarks (down and up), two leptons (electron and electron neutrino), the antimatter version of these two leptons (positron and electron antineutrino), and two field quanta (photon and gauge bosons). The essence of β⁻-processes is now accessible by utilising the concept of quarks. In all cases, only one of the three quarks of each nucleon is involved (the “actor”). The two other quarks just watch the others and are-->called “spectators”.

β^- Process

The conversion of a neutron into a proton is the metamorphosis of one d-quark into one u-quark. The initial composition of 2 × d + 1 u (= 2 × −$\frac{1}{3}$ + 1 × + $\frac{2}{3}$ = 0) thus turns into 1 × d + 2 u (= 1 × −$\frac{1}{3}$ + 2 × + $\frac{2}{3}$ = +1). The mechanism is illustrated for the β⁻-process. Figure 13 illustrates the principal changes among the quarks involved (up quark, down quark). Yet, there is one more question: why should one sort of quark turn into the other one? There is a force needed to manage this fundamental process: the mediators. The mediators relevant in β⁻, β⁺, and electron capture transformation processes are the W⁻, W⁺, and Z^o bosons, respectively. Feynman has suggested graphical presentations of this process (and many other processes in elementary particle physics). Figure 13 (right) shows how the W⁻ boson mediates the metamorphosis of the d-quark.

β⁺- and EC Processes

During the conversion --> of a proton into a neutron, the opposite occurs. A u-quark turns into a d-quark. In this case, it is the W⁺ boson and the Z^o boson, respectively, mediating the metamorphosis, and the elementary particles created are the positron and the electron neutrino.

β-Transformation and Laws of Symmetry

Figure 13 (right) indicates the appearance of the particle -->essential to β-transformation: the β⁻ electron. In addition, there is an electron neutrino. Let’s understand the origin of both of these particles. The metamorphoses of one member of the first family of quarks into the other one perfectly explain the balance in quarks and perfectly explain the conversion of one sort of nucleon into the other one. However, it introduces several other questions.

The first: What about the balance in charge? For β⁻-processes, a neutral nucleon had changed into a +1 charged nucleon. For β⁺ and EC processes, a positively charged nucleon had changed into a neutral one. Where is the missing charge going (for β⁻- and β⁺-processes) or coming from (for the EC process)? The answer is another elementary particle of the first family—the electron—is needed to carry the charge. Note that in the present context, this electron is referred to as β-particle. It is the origin of the electron which is responsible for this terminology: the β-particle electron is an electron created during these nuclear processes.

β⁻-Process

The emission of a “normal” electron within the n → p conversion satisfies the balance of electric charge: it is 0 → (+1) + (−1).

β⁺-Process

The p → n conversion requires the emission of the antimatter kind of electron, the +1 charged positron. The balance of electric charge then is (+1) → (0) + (+1).

EC Process

The p → n conversion can occur through another pathway, the electron capture (ε). Here, the proton captures a “normal” electron. The balance of electric charge then is (+1) + (−1) → (0).

The second question: What about the balance in orbital momentum, the spin? The answer is that another elementary particle of the first family—the electron neutrino—is needed to carry the spin. Let’s consider the n → p conversion of a neutron. The neutron’s spin is 1/2, so the total spin of the left side of the transformation equation is non-integer. Among the transformation products discussed so far, the spin 1/2 of the proton and the spin 1/2 of the electron combine to an integer number. So here comes a problem: the overall spins of the starting particles and the product particles differ! As simply postulated (!) by Pauli, a third reaction product is needed to solve the problem. It should have no electric charge so as to not disturb the symmetry in electric charge and (almost) no mass, in order to not disrupt the balance in electric charge and mass achieved so far. However, it should carry a half-integer spin. The neutrino hypothesis perfectly fits with all three subtypes of the β-process (Fig. 14).

Fig. 14

Balances in electric charge and momentum for nucleon conversion representing the three subtypes of primary β-transformation. Grey and orange circles represent the neutron and the proton, respectively. In all cases, it is an electron or β-particle, respectively, which handles the balance in charge, though in different ways for the β⁻-process, the β⁺-process, and the electron capture ε. For symmetry in momentum, in all cases an electron neutrino is emitted in the context of nucleon conversions. It guaranties the conservation of spin

The last issue of symmetry to consider is that between the matter and antimatter, another fundamental law in physics. It requires a balance in terms of particles and antiparticles. For example, the metamorphosis of a neutron into a proton creates a β⁻ electron, an elementary particle. This now requires the simultaneous creation of an antiparticle. In the present case, we observe the formation of an electron antineutrino, not the electron neutrino. For the β⁺-process, the opposite occurs. Changing a proton into a neutron needs the formation of a positively charged --> -->β-particle: the positron. The positron is an antimatter particle, so the electron neutrino needed for reasons of symmetry in spin must be the “real” electron neutrino.

Energetics of β-Transformations: Values of ΔE and Q

The Q Value

The three subtypes of β-transformation all are characterized by a balance of mass between the initial unstable nuclide, K1, and the transformation product nuclide, K2. The new nuclide must be of lesser mass in order to guarantee an exothermic transformation. In different words, differences between the masses of the new nuclide and the old one are always positive: +Δm, which is also +ΔE. The value of mass refers to the whole nuclide (M) rather than the mass (m) of nuclei alone. If atomic mass data (in u) are used as tabulated, the mass of the nucleus is obtained by subtracting the mass of the electrons from the whole atom mass. The value of ΔE is specified as the Q value of the process. The three subtypes of transformations thus own individual values: Q_β ⁻, Q_β ⁺, and Q_ε. Supposing a given unstable nuclide is able to undergo two or all three subtypes of the transformation, each branch will thus be characterized by its individual amount of energy. Among the many unstable nuclei undergoing β-transformation, the range of Q values is very large. There are small Q values such as 18.55 keV for tritium and large ones such as 14.1 MeV for ⁸B. This covers about three orders of -->magnitude.

Specific Effects for β⁺-Emission Versus Electron Capture

The way the Q value is calculated—i.e. the difference between the masses of the nuclide formed minus the mass of the initial nuclide—is in part modified according to the role of the β-particles emitted and the electron captured, respectively. The β⁺-subtype starts from K1 and creates two components, namely K2 and the positron. The electron capture subtype starts from nuclide K1 and collects one additional electron on top of the initial electron shell configuration of the corresponding atom and only next forms K2. The overall masses to consider are thus the mass M of the nuclides and the masses of the electrons involved. For K1, the latter includes the masses of the number of shell electrons equivalent to the number of its protons (Z), i.e. {M_K1 – Z^.m_e}, while for K2, this number of shell electrons is one fewer, i.e. [M_K2 – (Z−1)^.m_e]. The masses of the electron antineutrino electron neutrino can be neglected. The resulting balances in mass are the following:

$$ {\displaystyle \begin{array}{c}\varDelta M\left({\beta}^{+}\right)=\left\{\left({\mathrm{M}}_{\mathrm{K}2}-\left(\mathrm{Z}-1\right)\ {\mathrm{m}}_{\mathrm{e}}\right)+{1}^{.}{\mathrm{m}}_{\mathrm{e}}\right\}-\left\{{\mathrm{M}}_{\mathrm{K}1}-\mathrm{Z}\ {\mathrm{m}}_{\mathrm{e}}\right\}\\ {}=\left({\mathrm{M}}_{\mathrm{K}2}-{\mathrm{M}}_{\mathrm{K}1}\right)+2\ {\mathrm{m}}_{\mathrm{e}}\end{array}} $$

$$ {\displaystyle \begin{array}{c}\varDelta M\left(\varepsilon \right)=\left\{{\mathrm{M}}_{\mathrm{K}2}-\left(\mathrm{Z}-1\right)\ {\mathrm{m}}_{\mathrm{e}}\right\}-\left\{\left({\mathrm{M}}_{\mathrm{K}1}-\mathrm{Z}\ {\mathrm{m}}_{\mathrm{e}}\right)+1\ {\mathrm{m}}_{\mathrm{e}}\right\}\\ {}=\left({\mathrm{M}}_{\mathrm{K}2}-{\mathrm{M}}_{\mathrm{K}1}\right)\end{array}} $$

Accordingly, whatever the difference in mass of the two nuclides, the β⁺ transformation requires an excess of that ΔM plus 2^.m_e. The amount of energy which equals the mass of two electrons is 2^.m_e ^.c² = 2. 0.511 MeV = 1.022 MeV. In contrast, electron capture and β⁻-processes are -->energetically satisfied by “just” M_K2 < M_K1. This discriminates the pathways of proton-rich unstable nuclides, i.e.β⁺ and ε-transformation. For example, the positron emitter ¹⁸F transforms to stable ¹⁸O. Atomic masses are 18.000937 u and 17.999160 u, Δu = 0.001777 u, and in terms of energy (1 u = 938.272 MeV), it is 1.667 MeV, i.e. >1.022 MeV. It allows to utilize both pathways, positron emission and electron capture. (In reality, it prefers positron emission 96.7% of the time.) ⁷Be transforms into stable ⁷Li. Atomic masses are 7.016929 u and 7.016003 u, Δu = 0.000926 u = 0.869 MeV, i.e. <1.022 MeV. As a result, ⁷Be is unable to undergo positron emission, and electron capture is its only option.

Electron Capture?

How can a proton, located in the nucleus of an atom, “capture” an electron? Didn’t we learn that the electrons orbit in electron shells far away from the nucleus? This takes us to the quantum mechanics of atomic shell electron. Their orbital momentum as characterized by the set of quantum numbers defines individual spatial distributions within an atom with certain probabilities. Interestingly, for s-orbital electrons (because of their orbital momentum of l = 0 and the corresponding spherical distribution of probabilities of existence), there is a very low probability that the electron exists close to and even “inside” the nucleus! Relatively speaking, this probability is most pronounced for K-shell electrons rather than L or even M-shell electrons. The probability of electron capture increases with decreasing distance of the K-shell to the nucleus. The higher the element’s proton number Z is, the higher the probability of electron capture. The distance between nucleus and K-shell follows a function of 1/Z². This allows-->us to draw several conclusions:

• 1. Unstable proton-rich nuclides that preferentially utilize β⁺ transformation are among the elements of the second period of the periodic table of the elements [e.g. carbon (¹¹C, t _½ = 20.38 min), nitrogen (¹³N, t _½ = 9.96 min), oxygen (¹⁵O, t _½ = 2.03 min), and fluorine (¹⁸F, t _½ = 109.7 min)]. In these cases, the abundance of the β⁺-subtype is 99.76%, 99%, 99.9%, and 96.7% for ¹¹C, ¹³N, ¹⁵O, and ¹⁸F, respectively. These nuclides have become key nuclides for medically important molecular imaging and diagnosis via positron emission tomography (PET) and find extensive application in radiopharmaceutical chemistry. Nevertheless, there are also unstable nuclides of elements above Z = 20 emitting positrons at percentages, which are relevant for practical application. Yet in these cases, the percentage of positron emission drops: ⁶⁴Cu = 17.9%, ⁶⁸Ga = 88.0%, ⁷³Se = 65.0%, ⁸⁶Y = 34.0%, ⁸⁹Zr = 23.0%, ⁹⁰Nb = 51.1%, and ¹²⁴I = 24.0%, for example.

• 2. Electron capture consequently dominates in the case of the unstable proton-rich nuclides of heavy elements. Many of the key radionuclides used in SPECT diagnosis undergo electron capture as the primary transformation and continue with secondary transitions yielding γ-emissions via excited nuclear levels. Examples of these nuclides include ⁶⁷Ga, ¹¹¹In, and ¹²³I.

Kinetic Energetics of β-Transformation Products

Recoil

Let’s assume the β-particle --> is ejected from K2, i.e. the former K1. The impulse it takes causes a somehow opposite impulse to K2. This is referred to as the “recoil energy” of K2. It is linked with (a) the Q value of the transformation, (b) its own mass, and (c) the kinetic energy, E_β, of the emitted β-particle and the electron neutrino (or the electron neutrino exclusively in case of electron capture). In addition, it is influenced by the spatial arrangements the two elementary particles are emitted. K2 recoil energies thus lie between the theoretical maximum value and zero. The maximum kinetic energy ^RECOILE_K2 ^max the recoil nucleus may get is

$$ {}^{\mathrm{RECOIL}}{\mathrm{E}}_{\mathrm{K}2}^{\mathrm{m}\mathrm{ax}}=\left(\frac{{\mathrm{E}}_{\beta}^{\mathrm{m}\mathrm{ax}}}{2{\mathrm{c}}^2}+{\mathrm{m}}_{\beta^0}\right)\frac{{\mathrm{E}}_{\beta}^{\mathrm{m}\mathrm{ax}}}{{\mathrm{m}}_{\mathrm{K}2}} $$

For example, the β⁻ transformation of ¹⁴C into ¹⁴N yields ^RECOILE_K2 ^max = E^max (¹⁴N) = 6.9 eV. (m_K2 = 14 u, m_β ^o = 0.511 keV, E_β ^max = 0.156 MeV).

The recoil energies of K2 are higher when the kinetic energy of the β-particle is high and the mass number of K2 is low. For example, at mass numbers (A) around 100 and maximum kinetic energies of the β-particle of 1 MeV, values of ^RECOILE_K2 ^max are about 10 eV.

Distribution of Kinetic Energies: β-Particle and Electron Neutrino

As the recoil nucleus just gets a very low amount of the total kinetic energy, the dominant fraction is left for the small particles emitted. In electron capture, all the remaining kinetic energy goes to the electron neutrino. Consequently, the electron neutrino gets a kinetic energy of a discrete energy value. However, in β⁻ and β⁺ transformations this is different. β-particles and the electron neutrinos share their fraction of kinetic energy “statistically”. There are cases in which the β-particle gets all the kinetic energy (E_β ^max), and nothing is left for the electron neutrino—or vice versa.

In reality, there is a distribution between both the elementary particles, and consequently, kinetic energies observed for β-particles and for electron neutrinos show a continuous spectrum. The β-particle kinetic energies thus lie between the theoretical maximum value and zero. For example, the β^--particles emitted from ³H and ¹⁴C show maximum kinetic energies E_β ^max of 18.591 keV and 156.476 keV, respectively. Typical maximum energies for β⁻ and β⁺ particles range from about 20 keV to a few MeV. However, the fraction of β-particles that reaches this maximum energy is very low. Most of the β^--particles show energies (most abundant average or mean energies (E_β ^mean or Ē_β) typically are around $\frac{1}{3}$E_β ^max. The same applies to positrons emitted within the β⁺-subtype of β-transformation. Figure 15 shows profiles of the continuous spectra of the positrons emitted from four relevant nuclides used in medical diagnosis (PET). The values of E_β ^max depend on the Q value of the transformation.

Fig. 15

Continuous spectra of positrons emitted from ¹⁵O, ¹³N, ¹¹C, and ¹⁸F

Quantum Theory of β-Transformation Phenomena

The process of --> nucleon transformation inside the nucleus of an atom is explained by quantum physics theory. The basic terminology is called “Fermi’s golden rule”. It defines the probability (P _fi) of transition (per unit of time) between initial (i) and final (f) states from one energy eigenstate of a quantum system (here represented by the nuclide K1) into another one (the final nuclide K2). Figure 16 compares the phenomenological process and the quantum physical approach.

Fig. 16

Phenomenology of the β-transformation (left) and the principal concept in quantum theory (right) showing wave functions Ψ_i and Ψ_f

Several parameters are needed to quantitatively understand β-transformation, such as phase space volumes, densities of energy states, probabilities of transition, and the overlap of wave functions of the initial state and the possible final states. Each state is expressed by a density profile, i.e. the number n of states per unit of energy (dn/dE). With the negligible mass of the electron neutrino and very small recoil energy of K2, the densities of states are expressed in terms of overall energy, Q_β, of the transformation relative to the maximum kinetic energy of the β-particle emitted. The mathematics relates the probability (P _fi) of transition (transition rate = transitions per unit time) to phase spaces via a matrix element {M _fi}². This matrix element considers the overlapping wave functions of the final and initial states, Ψ _f and Ψ _i, and includes the Hamilton operator Ĥ of the weak interaction. If the overlap of the wave functions is large, the probability of transition is high. The most relevant equations and their relationship are illustrated in Fig. 17.

Fig. 17

Fermi’s golden rule: The number of states for β-particle and electron neutrino per volume segments combines to densities of states for both elementary particles. Probabilities of transition are described by the transition matrix element, M _fi, while several numerical parameters are combined to a constant C = V² / (2p ³ c³ ħ⁷). This equation includes an additional factor, the Fermi correction term. It reflects the fraction of kinetic energy an electron loses after its emission from the nucleus due to Coulomb attraction and the positron gains due to Coulomb repulsion

Velocities of β-Transformations

Correlations Between Q Value and ΔĒ_B with Half-Life

Q values --> correlate with the half-life of the transformation. For larger Q_β-values, the transformation steps proceed quickly. This perfectly fits with the β-transformation parabolas shown in Figs. 9 and 10, for example. The further the nuclides are from the vertex of the parabola, the steeper the sides of the parabola become. While the unit of the x-axis is Z±1 and is thus linear, the y-axis representing the mean nucleon binding energy is exponential. Figure 18 compares the “win” in mean nucleon binding energy, ΔĒ_B, with the corresponding half-life of this transformation for all the unstable nuclides covered by both Figs. 9 and 10, i.e. for all β-transformations along the isobars of mass numbers 95 and 96.

Fig. 18

Half-lives of β-transformations along parabolas for mass numbers A = 95 and 96 correlate with the successive gain in ΔĒ_B per transformation towards the vertex of the parabola. For values of ΔĒ_B > 0.1 MeV, the half-lives are seconds or less. For values of ΔĒ_B < 0.1 MeV, half-lives approach hours, days, and even years

Similarly, the Q values are (in general) inversely proportional to the half-life or directly proportional to the transformation constant. The larger the value of ΔĒ_B, the larger the value of E_β ^max (or Q_β) and the faster the transformations. Figure 19 illustrates the correlation between E_β ^max and the half-life (t_½) and transformation constant (λ; t_½ = ln2 / λ) for the same nuclides as shown in Fig. 18. Clearly, small changes in energy (Q_β or E_β ^max) have an impressive impact on the half-life of the transformation.

Fig. 19

Correlation of maximum kinetic energy of the emitted β-particles and the transformation constant (left) and half-life (right) of the β transformation for nuclides of mass numbers A = 95 and 96. For changes in E_β ^max of one order of magnitude (e.g. from 1 to 10 MeV), the transformation constant/the half-life changes by seven orders of magnitude

Logft Values

The correlation between nuclear transformation energetics and velocities is also addressed by quantum mechanics as introduced via the Fermi equation. The equation introduced in Fig. 17 can be modified towards a version expressing the transformation constant, λ (Fig. 20). It separates two parts and defines the integral on the left as velocity (λ), while the integral on the right is subsumed as the f-value. If velocity is expressed as half-life t_½ = t, a product ft is derived. It is typically given on a logarithmic scale. The relevant message here is that low values of logft reflect high probabilities of nuclear transformation and short half-lives. The larger a logft value becomes, the lower the probability of transformation and the longer the half-life.

Fig. 20

Deriving a ft-parameter from the Fermi equation

Selection Rules

The logft concept overlaps with other systematics in nuclear transitions: selection rules. In this regard, the two relevant nuclear properties are the overall spin of a nuclear level and its parity.

Overall Nuclear Spin J

Each nucleon in a--> nucleus owns its characteristic individual orbital spin. The sum of all individual spins creates the overall spin, J, of a given nuclear state. Overall spin values thus may be different between the initial state of the unstable nuclide transforming, K1, and the ground state of the new nuclide, K2. In addition, the new nucleus formed may be the ground state of K2 or an intermediate excited nuclear state, ^ʘK2. Those different nuclear states of the same nucleus may differ in J. An excited nuclear level is characterized by individual nucleons populating higher-energy shell positions of quantum numbers different to the corresponding ground state of the same nucleus. Accordingly, overall nuclear spin J numbers may differ between excited and ground-state levels of K2.

Parity Π

In quantum physics, parity refers to changes of physical quantities under spatial inversion within a polar coordinate system. Mathematically, parity refers to how wave functions with corresponding eigenvalues and parity operators change in the course of spatial inversion. While the three coordinates change from, e.g. (+x,+y,+z) to (−x, −y, −z), the quantum parameters in terms of wave functions and eigenvalues may also change or not. Parity is thus indicated as + or −.

Overall, the spin and parity of a certain nuclear level are expressed as J ^Π. Now, the transformations must be discussed in terms of changes in overall spin and parity, i.e. ΔJ and ΔΠ: changes are ΔJ = 0, 1, 2, 3, 4, …., and either ΔΠ = + or –. The termini derived from selection rules are “allowed” and “forbidden” with internal gradations and reflect the dimension of the changes. Allowed transitions are either “superallowed” or just “allowed”. Superallowed refers to the absence of changes in overall spin and parity, i.e.ΔJ = 0, and ΔΠ = +. They overlap with “allowed” transitions, which still remain ΔΠ = + but may accept the lowest change in overall spin: ΔJ = 1.

“Forbidden”: The more changes there are in J, the more the transitions become forbidden. Forbidden nuclear transitions are of much lower probability compared to less forbidden or allowed transitions: the more forbidden a transition is, the lower its velocity.

Excited States in β-Transformations

Primary transformation processes of unstable nuclides, K1, do not necessarily directly yield the ground state of the newly formed nuclide, K2. Instead, the energetically excited levels (^ʘiK2) of the new nuclide may be populated. Excited nuclear levels of a certain nucleus differ from the ground state of that nucleus simply because one or more nucleons of the nucleus exist—for a certain period of time: typically 10⁻¹² s and in other cases for seconds, minutes, and years—in a higher-energy nucleon shell. (This is introduced here in the context of β-transformations but also holds true for the α-emission pathway as well.) These energetically different states all belong to the new nuclide in terms of mass number A, proton number Z, and neutron number N, but a nucleon may occupy a higher-energy nucleon shell. Accordingly, the nucleon of an excited nuclear level owns a quantum number different from the one it belongs to in its ground state. For the whole nucleus, the “overall nuclear spin” may be different compared to its ground state. Consequently, every nuclear state is defined by its characteristic set of overall spin and parity.

Figure 21 illustrates various excited levels for ⁹⁰Y, a β^--emitting radionuclide that is medically relevant due to its role in endoradiation therapy. The ground state of ⁹⁰Y is 2⁻, and β⁻ transformation starts from that level. The transformation product nuclide is stable ⁹⁰Zr. There are two relevant individual excited nuclear states to discuss. Its highest-energy excited state (^ʘ2K2) is of 2⁺. The energetically lower excited level of ^ʘ1K2 is of 0⁺. Next, there is a ground state, ^ʘK2, which is of 0⁺ again. Theoretically, there are three principle primary transformations, namely, K1 → ²K2, K1 → ^ʘ1K2, and K1 → ^gK2, with the corresponding logft values. The most probable transformations are those in which the changes in overall spin and parity are lowest. The dominating transformation is K1 → ^gK2, with no change in J and a change in Π. This set is true also for K1 → ^ʘ2K2, but the two routes differ in their logft values: 8.1 vs. 9.4 in favour of K1 → ^gK2. Accordingly, the experimentally observed relative probabilities of the three possible transformations for K1 towards ^gK2, ^ʘ1K2, and ^ʘ2K2 are 99.982%, 0.017%, and < 10⁻⁶%, respectively.

Fig. 21

β⁻ transformation pathways of ⁹⁰Y towards ground-state and excited nuclear levels of stable ⁹⁰Zr

α-Emission

From β-Transformation to α-Emission

For all mass numbers (A) from 1 to 209, β-processes yield one definite stable nuclide or two, depending on (even, even) or (even, odd) nucleon composition (see Figs. 9 and 10). This paradigm does not continue when A > 209. As an example, let’s consider the mechanism of β-transformation of the A = 226 isobar. The radium isotope 226 represents the most stable nucleon composition along this isobar. ²²⁶Ra by far shows the longest half-life of this isobar at 1600 years. The neighbours at Z + i are of much lower stability, and their half-lives are in the range of hours (29 h for ²²⁶Ac) and minutes (31 min and 1.8 min for ²²⁶Th and ²²⁶Pa, respectively) and decrease further down to milliseconds (280 ms for ²²⁶U and 31 ms for ²²⁶Np). For the Z-i arm of the parabola, ²²⁶Fr and ²²⁶Rn show half-lives of 48 s and 7.4 min, respectively. In the present case, the nuclide at the vertex of the isobar parabola of Ē_B = f(Z) is ²²⁶Ra, yet it is not stable (Fig. 22).

Fig. 22

β-transformation processes along the isobar A = 226. The most stable (but not really stable) and longest-lived nuclide is ²²⁶Ra. This unstable nuclide of optimum mean nucleon binding energy along the isobar transforms through α-emission to ²²²Rn, thereby switching to a new, lower isobar. In the Karlsruhe Chart of Nuclides [9], α-emitting radionuclides are indicated by yellow colour

Thus, β-transformation has done its best to build the most stable nuclide of the A = 226 isobar, but it has not been able to create a stable nucleon configuration. Consequently, ²²⁶Ra must transform to a more stable nucleon configuration by a mode other than β-transformation. This is the moment that the unstable nuclides cannot continue following the A = constant strategy for stabilization. So, what should this unstable nuclide do? The answer lies in two classes of transformations of ^AK2 < ^AK1: cluster emission (the most relevant version is the emission of α particles) and spontaneous fission.

The emission of an α-particle--> immediately reduces the mass of the unstable nuclide, K1, and changes both its proton and neutron numbers: it is a primary transformation. The reason the α-particle is preferred lies in its very high “internal” stability. The mean nucleon binding energy of ⁴He nucleus is 7.074 MeV, and the nucleus is further stabilized due to a double-magic nucleon shell configuration (Z = 2, N = 2). The α-transformation thus balances mass between the initial unstable nuclide, K1, and the transformation product nuclide, K2 -->, in a clear way: the mass number of the new nuclide is reduced by 4. Those changes in ΔA (4) and in ΔZ (2) result in increased mean nucleon binding energies according to the LDM (Fig. 23), at least for heavy elements with A > ca. 130.

Fig. 23

Each α-emission from K1 creates a new nucleus, K2, with mass A-4, i.e. the new nucleus is located left to the initial one. For high mass number A (approximately above A ca. 130), this “automatically” gives a gain in mean nucleon binding energy. The corresponding values are calculated via the Weizsäcker equation, and the parameters changing between K1 and K2 are those for A, Z, and N in terms of A-4, Z-2, and N-2

The emission of one α-particle-->may generate a stable nucleon mixture but also may result in a nuclide that is “more stable” but not actually stable. This effect can be explained following the example given in Fig. 22. ²²⁶Ra starts to transform by α-emission; it follows an isodiaphere line forming a product of Z-2 and N-2 composition: the transformation product is ²²²Rn and wins mean nucleon binding energy: Ē_B = 7.695 MeV for ²²²Rn vs. 7.662 MeV for ²²⁶Ra. Yet, this new nuclide is not stable. The transformation may continue via another α-emission. This is exactly the case for ²²⁶Ra as illustrated by the natural chain of transformations of the 4n + 2 series: ²²⁶Ra originates from ²³⁰Th by α-emission, and ²²⁶Ra itself continues to form daughters by successive α-emission as ²²⁶Ra → α → ²²²Rn → α → ²¹⁸Po → α → ²¹⁴Pb (Fig. 24).

From α-Transformations to β-Processes

With each individual α-emission process, the nucleus increases the ratio between the number of its neutrons and protons. It is 138/88 = 1.568 for ²²⁶Ra, 136/86 = 1.581 for ²²²Rn, 134/84 = 1.595 for ²¹⁸Po, and 132/82 = 1.610 for ²¹⁴Pb. The excess of neutrons is reaching a dramatic level, and β-transformation is energetically favoured. Now, here comes the teamwork of α- and β-transformations: for ²¹⁴Pb, the β-process becomes the only pathway to further stabilize the nucleus. It happens along the neutron-rich arm of the isobar is parabola at A = 214 = constant until a new, local maximum of the mean nucleon binding energy for this particular isobar is reached. This new maximum of Ē_B could represent a stable nuclide, but this is not possible for A = 214; there is no stable nuclide. If not, a situation occurs like that explained in the beginning for transformations along the isobar A = 226, and another α-emission follows (Fig. 24).

Fig. 24

Continuation of the naturally occurring ²³⁸U transformation chain subsequent to the α-emissions from ²²⁶Ra. The direct chain of α-emission terminates at ²¹⁴Pb. The next primary transformations are two β⁻-processes. At ²¹⁴Bi, a branched chain starts with simultaneous transformation via β⁻ and α-emission. Those individual directions finally terminate together at the stable nuclide ²⁰⁶Pb

Simultaneous β- and α-Emission

As indicated in Fig. 22, α- and --> β-transformation not only may alternate from one transformation step to the next, they may appear simultaneously for one and the same nuclide! Obviously, ΔE values may be positive for different primary transformation options. In this case, each pathway gets its individual absolute value according to the different balances in mass, for which notations are Q_α and Q_β, respectively. Figure 24 shows the routes for parallel β- and α-emission (²¹⁴Bi and ²¹⁰Pb). Another example is ²¹³Bi (Fig. 25). In addition, α-emission and electron capture may occur simultaneously as, for example, in ²¹¹At. This is an example of an artificially produced radionuclide.

Fig. 25

Notation of parallel options of primary transformations for one and the same nuclide. ²¹³Bi: 2.1% α-emission (Q_α = 5.982 MeV, main α-energy 5.869 MeV) + 97.9% β⁻ emission (Q_β = 1.427 MeV, main maximum β⁻ energy 0.986 MeV), main γ-emission 440 keV. ²¹¹At: 41.8% α-emission (Q_α = 5.982 MeV, main α-energy 5.87 MeV) + 58.2% electron capture (ε) emission (Q_ε = 785 keV). See other studies for individual numbers [4,5,6,7,8]. The size of the colour-coded area qualitatively indicates the proportions between the different branches of transformation

Energetics of α-Emission

Absolute Values of Q_α

The α-transformation process occurs spontaneously and is nonreversible like all the other primary transformation pathways. The absolute value of Q_α basically depends on the masses of the two nuclides and their difference, accordingly, and involves the mass of the α-particle emitted. The range of Q_α values is rather small, approximately between 1 and 10 MeV.

Kinetic Energies of α-Particles and Recoil Nuclei

Similar to β-transformations, the impulses (p = m^.v) and kinetic --> energies (E = ½ m^.v²) refer to the two species formed in the primary nuclear transition. This results in a balance for the α-particle emitted and K2 recoiled as m_α ^.E_α = m_K2 ^.E_K2. The overall energy, Q_α, is allocated to the α-particle emitted and the recoil nucleus ^RECOILK2 according to the following equations.

$$ {p}_{\mathrm{K}2}={p}_a $$

$$ {\mathrm{Q}}_{\propto } =^{\mathrm{RECOIL}}{\mathrm{E}}_{\mathrm{K}2}+{\mathrm{E}}_{\propto } $$

Kinetic energies are distributed between the α-particle and K2 directly and depend only on the mass number of K2 (the mass of the α-particle is always the same). Consequently, the kinetic energy of the α-particle is discrete. Its value is nuclide-specific and representative, like a fingerprint. Absolute values of kinetic energies of the α-particle are higher in the case --> in which the Q_α-value is high and the mass number of the nuclide is low.

$$ {\mathrm{E}}_a=\frac{{\mathrm{Q}}_a}{1+\frac{{\mathrm{m}}_a}{{\mathrm{m}}_{\mathrm{K}2}}} $$

Table 4 gives the corresponding numbers for the α-emission of ²³⁸U.

Table 4 Values of Q_α and kinetic and recoil energies of the transformation products of the ²³⁸U α-emission process. Mass excess data are used to determine Q_α. The kinetic energies of the α-particles are obtained by simply using mass numbers

Velocities of Transformation

Q_α values correlate with the half-life of the radionuclide. This exactly has the same --> tendency as discussed for β-transformations. The larger the Q_α-value, the larger the gain in energy a nuclide “wins” in terms of mean nucleon binding energy, ΔĒ_B, when transforming (and the faster the transformation proceeds in terms of short half-life t_½ or large transformation constant, λ). Similar to β-transformation, α-transformations cover half-lives of milliseconds to billions of years. Figure 26 shows an experimental α-spectrum generated by the naturally occuring transformation chain of ²²⁸Th → ²²⁴Ra → ²²⁰Rn → ²¹⁶Po. It reflects the relationship between the energy of the α-particle and the half-life of the transformation: the higher the energy of the α-particle, the shorter the half-life.

Fig. 26

Experimental α-spectrum including the four α-emitting members successively formed in the ²³²Th chain. The higher the energy of the α-particle emitted and the Q_α values (given in MeV), the shorter the half-life. ²²⁸Th (Q_α = 5.520 MeV, E_α = 5.340 and 5.423 MeV t_½ = 1.913 a), ²²⁴Ra (Q_α = 5.789 MeV, E_α = 5.685 MeV, t_½ = 3.66 d), ²²⁰Rn (Q_α = 6.405 MeV, E_α = 6.288 MeV, t_½ = 55.6 s) and ²¹⁶Po (Q_α = 6.907 MeV, E_α = 6.778 MeV, t_½ = 0.15 s)

Quantum Mechanics of α-Transformation Phenomena

The Quantum Mechanical Phenomenon of “Tunnelling”

Let’s consider the nucleus ²³⁸U again. The -->protons inside induce a potential energy U^C due to --> coulomb forces of ≈ 28.5 MeV at a radius of 9.3 fm, the radius of the uranium nucleus. Consequently, one must expect that an α-particle leaving this nucleus should have at least 28.5 MeV energy. However, the kinetic energy of the emitted α-particle is 4.198 MeV (only!), as calculated in Table 4. It precisely corresponds to the experimentally measured kinetic energy of the α-particle as released from ²³⁸U. What’s wrong? Nothing! The key wording here is “tunnelling”, and this phenomenon may look like the schematic drawn in Fig. 27. The α-particle has left the potential well and has “tunnelled” through the potential wall. It becomes a “free” particle. The kinetic energy of the α-particle after tunnelling through the Coulomb barrier is much lower than the height of the barrier and corresponds to the energy at which tunnelling was successful.

Fig. 27

Concept of the tunnelling of an α-particle through a Coulomb barrier of a nucleus. E^C gives the amount of potential energy due to the Coulomb forces, and E_α the kinetic energy of the α-particle after tunnelling at a virtual radius r^C

Mathematics of the Tunnel Effect

As introduced for β-transformation, quantum mechanical models consider an initial and a final state of a transformation, with a corresponding probability, p _fi, of transition (per unit of time) from one energy of a quantum system (nuclide K1) into another one (nuclide K2). For α-emission, there are three particular aspects:

• 1. Prior to the emission, the α-particle must be preformed as such inside the homogeneous ensemble of the many individual nucleons within the large nucleus. This may happen with a given probability due to the special stability of the Z = 2 + N = 2 cluster of double-magic shell characteristics of the ⁴He nucleus, the α-particle.

• 2. Suggesting the cluster was formed anywhere within the nucleus, this cluster must be present close to the surface of the nucleus. This includes an anticipated sort of “transport” or “diffusion”.

• 3. Only following this imaginary formation and virtual transport, the tunnelling itself of the particle through the barrier is discussed.

Figure 28 illustrates this phenomenology. Although there is evidence for the processes of the preformation of an α-particle [1] and its diffusion to the surface of the nucleus [2], the mathematical model subsumes these two steps into the frequency factor, f (also called the “reduced transition probability”). This describes how often an α-cluster appears at the surface of the nucleus and “knocks on the door”. Once it has appeared at the surface, it gets a chance to leave the nucleus via the tunnelling effect. The probability for this step is defined by the penetrability factor , P (also the “transition factor” or “penetrability”).

Fig. 28

Phenomenology of the α-emission process in terms of frequency factor and penetrability factor: (1) the formation of an α-particle inside the nucleus, (2) the delivery of this nucleon cluster towards the surface of the nucleus, (3) the release of the α-particle and tunnelling the Coulomb barrier of the nucleus

The two components to handle this mathematically are the factors f and P. Both factors determine the overall efficacy, and the concept is to define the total velocity of an α-emission as a product, reflecting the velocity of the α-transformation in terms of the transformation constant λ = f ^.P. The individual expressions for the two factors are summarized in Fig. 29.

Fig. 29

Half-lives of α-emission correlated with quantum mechanical parameters: Key parameters are the frequency factor, f, and the penetrability factor, P. The exponent in P is called the Gamow factor, G

Most of the parameters involved in the equations in Fig. 28 reflect basic properties of the atomic nucleus: E^C = energy of the potential wall (typically around 28 MeV and 30 MeV); m´ = reduced mass (m´ = m_α m_K2/(m_α + m_K2), i.e.for heavy elements m´ ≈ m_K2); Z_K2 = proton number of K2; Q_a = overall energy of the α-transformation; and r_K2 = radius of K2. G is the Gamow factor and its values are on the order of G = 30–60. With this dimension it becomes clear that the probability of penetrating (tunnelling) a potential well is extremely low. The same is true for the frequency factor. Depending on the proton number, it is about Z_K2 ^–4/3, which makes it 1.9^.10⁻²⁰ s⁻¹ for Z_K2 between 58 and 98.

Excited States in α-Transformations

As in β-transformation, α-transformations do not necessarily yield the ground state of the nuclide K2 directly but may populate energetically enriched levels of the newly formed nuclide. Figure 30 shows two α-emitting nuclides, with the α-emission producing the ground state of K2 directly and exclusively (²¹²Po) or a transformation cascade through several excited states (²²⁶Ra.)

Fig. 30

Left: α-emission as direct transformation into the ground state of K2 (²¹²Po → ²⁰⁸Pb). Overall spin and parity are indicated for K1 and ^gK2. There are no changes. Right: α-emissions of ²²⁶Ra populating several excited states of ²²²Rn. Symmetry parameters are indicated for K1 and ^gK2 and for four excited levels, ^ʘiK2, of ²²²Rn. Each arrow shows the logft value for the transformation, the corresponding abundance of the five individual α-emissions, and the corresponding energies of the α-particles emitted. Kinetic energy of the emitted α-particle is maximum for K1 → ^gK2 (4.871 MeV). The most probable transformation is K1 → ^gK2 (94.04%) because of overall spin and parity values that are identical for both nuclei

Spontaneous Fission

For unstable nuclides of increasing mass number of about A > 234, α-emission is a promising choice of transformation. However, there is another possibility, the third and final type of primary transformation process—spontaneous fission—which creates even larger differences in ΔA. This option appears at very large nuclei. Fission yields two fragments, K2 and K3, of the initial nuclide K1 with characteristic mass distributions. Spontaneous fission is not discussed further in this chapter, because it is not relevant to radiopharmaceutical chemistry and nuclear medicine. It must be mentioned, however, that induced fission, e.g. of ²³⁵U, is indeed relevant, if only farther up the chain: it is the main source of fission products such as ⁹⁹Mo and ⁹⁰Sr, which are of the upmost importance to nuclear medicine.

Secondary Transformations

From Primary to Secondary Transformations

Excited Nuclear States

As already indicated in the sections on the “Post-processes of Primary Transformations and Secondary Transitions” for β- and α-processes, a primary transformation does not necessarily lead to the ground state of the new nuclide K2 formed. Instead, individual excited states, ^ʘiK2, are populated. Excited nuclear levels appear when one or more individual protons or neutrons of the newly formed nucleus K2 do not immediately find themselves within the nucleon shells corresponding to the ground state of the nucleus. Instead, they occupy higher-energy shells.

Similar to excited electrons of an atomic shell, these nucleons “fall” towards lower-energy nuclear levels. The transitions from a higher-energy nuclear state may proceed to a lower-energy excited nuclear state or to the final ground state of the nucleus, ^gK2. In each case, the specific differences of ΔE between the two nuclear levels are carried away by “secondary” transitions (Fig. 31). It is the essence of secondary transformations that the numbers for Z, N, and A do not change as long as only the individual nuclear levels of the product nucleus K″ are involved. This is the reason the terminology “transition” is preferred instead of “transformation”.

Fig. 31

Excited nuclear states of the newly formed nucleus, K2, populated in primary transformations de-excite to lower-energy nuclear levels. Each excited and ground-state level of a particular nucleus owns its set of quantum numbers (such as overall spin and parity) and a characteristic energy. Theoretically, all possible transitions may occur, such as from each excited level directly into the ground state or from each excited level to the ones of lower energy. Transitions between two levels involved address a specific amount of energy, as represented by ΔE between the two levels. The question is how that specific amount of ΔE is managed by the nucleus. As shown in this illustration, there are three options

Metastable Nuclides/Nuclear Isomers

De-excitation between individual excited states or finally from one excited state to the ground state is very fast, typically lasting only 10⁻¹⁶ to 10⁻¹³ s. The overall secondary transformation is thus extremely fast, even when cascades of several transitions are involved. However, individual excited states, ^ʘiK2, may show half-lives much longer than the half-lives of the other excited levels of one and the same nuclide, ^ʘK2.

This is often observed for very small values of ^ʘiΔE and in the context of selection rules, i.e. whenever the differences in overall angular momentum are large (octa-, hexa-, or higher multipole orders) or the parity is violated (see below). This excited state is not really stable (this is true for the ground state of ^gK2, exclusively) but nevertheless remains “meta”-stable for a significant period of time and is referred to as ^mK2. Compared to the ground state, it reflects a nucleus of identical nucleon composition in terms of A, Z, and N and is therefore also referred to as “nuclear isomer”. There are many metastable isomers with half-lives in the range of minutes, hours, days, and years. The Chart of Nuclides involves more than 3000 stable and unstable nucleon configurations. In addition, about 700 of these nuclides show (at least one) metastable isomers. Some of these metastable radionuclides are of interest in fundamental research; others are relevant to important practical application. Such an example is depicted in Fig. 32: the ground-state and the metastable states of technetium-99. The metastable ^99mTc is the most relevant radionuclide in diagnostic nuclear medicine. Its half-life is 6.0 h. Its secondary transition in terms of γ-emission releases a 141 keV energy photon of high (89%) branching.

Fig. 32

Metastable nuclear isomer ^99mTc and ground state ^99gTc (2) of technetium-99 show individual half-lives. ^99mTc is populated (1) in high abundance from ⁹⁹Mo in the course of its many β⁻-transformations to individual excited nuclear levels of technetium-99, while there is no direct β⁻-transformation to the ground state ^99gTc (2). ^99mTc de-excites dominantly in a secondary transition accompanied by photon emission of 141 keV energy to the ground state ^99gTc (3). The ^99gTc continues the stabilization via β⁻ transformation to the ultimate stable nuclide of the A = 99 isobar: ⁹⁹Ru (4). In parallel, yet with a much lower probability, the metastable ^99mTc may transform independently via a direct β⁻-transformation to ⁹⁹Ru as well (5)

Options for Secondary Emissions

There are three principal routes to manage the difference in energy, ΔE, between different excited levels or one exited level and the ground state of a given nucleus.

The most frequently occurring (and for the detection of radioactive transformation very valuable) sort of secondary emission consists of the release of electromagnetic radiation as γ-quanta, i.e. photons, with ΔE = E_γ. The second option is the conversion of this particular amount of ΔE into the release of an already existing electron of that nuclide from its inner electron shell, creating a “conversion electron”. The third option is the transformation of ΔE directly into matter according to E = mc². It creates a pair of two particles, representing matter (electron) and antimatter (positron). Note that the three pathways may occur simultaneously for one and the same transition.

Photon Emission

According to the standard model, the photon is an elementary particle (see Table 3). It belongs to the field quanta and is the mediator of the electromagnetic force. Different from other mediators, its mass is zero. Still, it obeys the wave-particle duality of quantum mechanics. It is of no charge, and because its charge is zero, it has no antiparticle. Its spin is 1, and because of that, it is a boson. Its parity is −1 (Fig. 33).

Fig. 33

De-excitation between two excited nuclear states defined by quantum physical parameters J and Π for initial and final state. Both levels are characterized by the energy differences ΔE between the two states, the orbital quantum number (l), the magnetic orbital quantum number, the spin quantum number (s) and the resulting overall orbital momentum (L), the overall angular momentum (S), and the overall momentum (J) obtained from orbital-spin coupling. The characteristic difference in energy, ΔE, is released as γ-radiation. The transition via photon emission proceeds in specific values of ΔJ, and the photon must carry away this difference in overall momentum

The electromagnetic radiation created emitted from unstable nuclei can be divided into two subparts: γ-radiation and X-rays. Both represent photons, but the difference is not the absolute value of frequency, wavelength, and energy. Instead, it is their origin.

γ-Rays

Gamma radiation here is meant to originate from the nucleus of a radionuclide via the de-excitation of an excited level of the nucleus. In terms of wavelength, it is approximately <10⁻¹¹ m (which is <10 pm or < 0.1 A, i.e. shorter than the diameter of the nucleus of the atom); in terms of energy, it is >0.1 MeV.

X-Rays

In the --> framework of radioactive processes (but also in conventional X-ray spectroscopy), electromagnetic radiation emitted from the shell of the atom—i.e. not from the nucleus—is signified as X-radiation (the creation and properties of those X-rays is discussed later under post-processes I). Compared to γ-rays, their wavelengths are higher (ca. 0.01–10 nm), and their energies are lower (ca. 0.1 keV to ca. 0.1 MeV). Thus, the spectral domain of X-ray lies between γ-rays and UV light. In terms of energy, a further phenomenological notation relates to the penetration power of the radiation and discriminates between “soft” X-rays (energies <10 keV) and “hard” X-rays (~10–120 keV).

According to the shell model of the atomic nucleus, the quantum number (basically the spin-orbit quantum number) of a nucleon within a higher shell may differ from those within lower shells. Accordingly, the overall spin of the excited nuclear level of that nucleus may differ from the overall spin of its ground state. Those changes between individual levels have to be seen in the light of quantum physics, e.g. how do overall spin and parity change? Let’s take an arbitrary example to identify the problem and also to understand the creation of a γ-spectrum. Figure 34 shows a hypothetical cascade of secondary transitions for altogether three states of K2: two excited ones, ^ʘ2K2 and ^ʘ1K2, and the ground state, ^gK2. Starting from K1, there are many possible transitions. The question is whether these different options proceed with identical probabilities? Or are some of the transition steps preferred and others not? If the latter is true, then why?

Fig. 34

Hypothetical cascade of secondary transitions for altogether three excited and one ground-state states of K2. For simplicity, three transitions are selected in red, namely ^ʘK4 → ^gK2, ^ʘ2K2 → ^ʘ1K2, and ^ʘ2K2 → ^gK2. In terms of ΔEγ, the order is ^ʘK4 → ^gK2 > ^ʘ2K2 → ^ʘ1K2 > ^ʘ2K2 → ^gK2. This is how the three photon emissions are recorded in the G-spectrum as f(ΔEγ). What about the intensity of the three lines? It is given here as Iγ = ^ʘ2K2 → ^ʘ1K2 > ^ʘ2K2 → ^gK2 > ^ʘK4 → ^gK2. The question is, how can we explain the individual branching of the three selected emissions?

Indeed, all of the potential steps have an individual branching. The reason for this defined protocol takes us back to intrinsic quantum physical parameters of the nucleons and of a certain nuclear state: overall orbital momentum (spin) and parity. Similar to primary β- and α-transformations, secondary photon transitions are also defined by initial and final wave functions as well as a transition matrix element. The most relevant quantum physical parameters needed for each initial and final state are the overall momentum J, and the parity II.

Conservation of Orbital Momentum

Whenever a secondary -->transition occurs, accompanied by the emission of a photon (i.e. ^ʘiK2 → ^ʘ(i−1)K2 + γ), the process must conserve quantum physical parameters. For the orbital momentum, the balance of spin is J(^ʘiK2) = J(^ʘ(i−1)K2) + l _γ. The photon takes away a spin of l = 1. Consequently, such a transition is impossible in the case J(^ʘiK2) = 0 and J(^{ʘ (i−1)}K2) = 0, i.e. ΔJ must be >0 to allow for the emission of a photon.

Transition Probabilities for Multipoles of Different Orders

Because the nucleus represents electromagnetic multipoles of different orders (multipoles, dipoles, quadrupoles, etc. depending on momentum numbers), there are selection rules for the release of photons. The selection rules refer to the important impact coming from the orbital moments. Individual velocities, λ, are a measure of transition probabilities: large values of λ (short half-lives) would indicate high probabilities and vice versa. Typically, those half-lives are extremely short and cover nanoseconds and picoseconds.

Conversion Electrons

“Inner conversion” or “internal conversion” (IC) represents a situation in which the difference in energy, ^ʘiΔE, between two nuclear levels is directly transferred to an atomic shell electron. If that amount of energy is larger than the electron binding energy of that electron shell, E_B(e), an electron is ejected from the atom. All of the internal conversion electrons ejected are former s-orbital electrons, and most of these electrons originate from s-orbital as “close” to the nucleus as possible. The kinetic energy, E_IC, of the ejected electron is equivalent to the value of ^ʘiΔE minus the energy, which was needed to overcompensate for the electron binding energy (Fig. 35). Internal conversion electrons thus have a discrete energy, different from β⁻ electrons.

Fig. 35

Origin of internal conversion electrons from individual s-shell orbitals of individual main quantum number n, i.e. K-shell, L-shell, etc.

$$ {\mathrm{E}}_{\mathrm{IC}}=\kern0.5em {}^{\odot \mathrm{i}}\varDelta \mathrm{E}-{\mathrm{E}}_{\mathrm{B}\left(\mathrm{e}\right)} $$

Internal conversion is a domain for low-energy ^ʘiΔE transitions and in particular for monopole modes of 0⁺ → 0⁺, for which γ-emission is not possible.

Pair Formation

The third pathway of secondary transitions is “pair formation”. In this case, the equivalent of ^ʘiΔE is converted into real matter: an electron and its antiparticle, the positron. The masses of both particles are 0.511 MeV (or 0.00055 u). As a result, pair formation thus exclusively emerges in cases in which ^ʘiΔE > 1.022 MeV. Clearly, this only occurs for relatively high differences between the energy of the two nuclear levels involved.

Post-processes of Primary Transformations and Secondary Transitions

Finally, there are additional processes—termed post-processes—which inherently accompany some of the primary and secondary processes. In principle, there are two classes of post-processes. The first kind is caused by vacancies in the electron shell of the transforming atom and the pathway the atom chooses to refill those holes. The second is caused by electrons, which initially originated from primary transformations or secondary transitions, namely, beta particles (the positron or the electron) and IC electrons. It is of upmost importance to note that these two classes of post-effects in turn create new kinds of radioactive emissions, which are not yet seen in primary and secondary processes.

Vacancies of Shell Electrons

Remember that the primary electron capture transformation involves the capture of a shell electron to combine with a proton in the atoms nucleus. Remember also that the secondary inner conversion process is predicated on the release of a shell electron due to the de-excitation of excited nuclear levels (Fig. 36). Yet even when these two processes are done forming a new nucleus, the newly formed atom is still “not ready”. It lacks an electron in its shell, i.e. Z ≠ e.

Fig. 36

Post-effects motivated by a vacancy in an electron shell of the atom. This occurs after a primary electron capture transformation of a proton-rich nuclide or after a secondary internal conversion

Let’s have a look how the new-born atom handles this vacancy in one of its shell. Electron vacancies typically appear in electron shells close to the nucleus. The K-shell is preferred, the L-shell is less affected, etc. (Note that only s-electrons are also involved, no matter what shell is affected.) There are two different pathways, each of which handles this vacancy promptly and induces different radiative emissions (see below).

Electron Vacancies Filled via Electromagnetic Radiation

The first approach to refilling an electron vacancy proceeds through the import of electrons located in the higher shells of that atom. Figure 37 illustrates the pathways for filling a vacancy in the K- or L-shells. Any electron hole is typically filled from an electron from the shell close to it. Typically, an L-shell electron fills a K-shell electron vacancy. In parallel, an electron from the N-shell may transit to the same K-shell electron vacancy, but this process is less common. Suppose the initial vacancy appeared in the L-shell, analogous electrons from the M- or N-shell may fill that hole. The transitions are named K_α or K_β for L→K or M→K and L_α or L_β for M→L or N→L, respectively. Here, K or L indicate the position of the hole to be refilled, and the Greek index indicates whether the arriving electron descended from the closest (α) or the next (β) main shell.

Fig. 37

Transitions of electrons from higher shells to a vacancy in a K-shell (left) or L-shell (right) and their notations. Red represents the K-shall the electron vacancy is created. The hole is filled via successive transitions of electrons from higher-energy shells (L, M, N, etc.) with decreasing probability. ΔE is the difference in electron binding energies between the initial shell where the transiting electron originates and the final shell where the vacancy is. R_H is the Rydberg constant, 13.6 eV. Furthermore, there are notations such as K_α1 and K_α2. This indicates electron transitions into the K-shell vacancy from two different energetic levels within the L-shell (due to the different quantum numbers l = 0 and 1). Their differences in ΔE are very low, and the relative abundances between values of K_α1 and K_α2 are about 2:1

Characteristic X-Rays

Electron energies are a function of the main quantum number: E_B(e)(n) = −R_H Z². 1/n², R_H is the Rydberg constant, 13.6 eV. The difference in binding energies of the electron between the initial state and the final state (the original hole) is obtained as indicated in Fig. 37. Consequently, there is a characteristic difference in energy, ΔE, which is released in terms of electromagnetic radiation, depending on Z and Δn. This energy is called an X-ray (rather than a γ-ray) because of its different origin. For a given nucleus (Z), the X-rays get characteristic values, depending on the shells involved.

Electron Vacancies Filled via Electron Emissions

Electron vacancies as created in Fig. 35 can also be refilled by a radiation-free process. The basic step remains the same: the transit of an electron from a nearby higher shell. The difference is that the amount of ΔE this time is not released as X-ray but is spent to release another electron from a higher shell. The process is referred to as a radiation-less “reorganization” due to a “direct” interaction of two electrons. This particular electron is immediately ejected from the atom. In this case, no electromagnetic radiation is emitted. If electrons are emitted in the course of processes between different main shells (interstate transitions), they are called Auger (A) electrons . If the pathway involves subshells of one and the same main shell, they are called intrastate transitions or Coster-Kronig (CK) electrons .

The overall number of electron ejected thus is larger than the number of all the shells (and subshells) of a given atom and thus is larger in the case of chemical elements of high main quantum number relative to those located in lower quantum number periods. Indeed, the overall number of A and CK electrons ejected increases with increasing Z but does not directly mirror the number of atomic electron shells in the corresponding elements. The final state after emission of all the shell electrons leaves a highly charged cation instead of a neutral atom. ¹²⁵I, for example, is a nuclide which primarily transforms through electron capture to an excited state of ¹²⁵Te, leaving a vacancy in the K-shell. In this case, the number of Auger and Coster-Kronig electrons emitted is about 25. This, of course, must cause dramatic changes in the chemical environment of that newly created atom.

The energy of the electrons emitted parallels the binding energies of the individual shells involved. Typically, A and CK electrons are within a range of approximately 20 to 500 eV. Though each electron thus gets an individual discrete energy, but the various electrons emitted within several shell cascades represent a mixture of several characteristic energies.

X-Ray Emission Versus Electron Emission

Once a vacancy in an s-shell electron is handled, the radiative and radiation-less post-effects proceed in parallel yet with individual ratios between the two pathways. The fluorescence yield, ω_X-ray (ω_K, ω_L ^, …), gives the percentage for the filling of an electron vacancy through radiative processes. The total process of addressing the hole created by primary or secondary transformations of an unstable nuclide than is ω_X-ray + ω_{(A) + (CK)} = 1. The ratio between X-ray and Auger and Coster-Kronig electron emission depends on the proton number of the nucleus and thus the absolute energies of the electrons and the relative differences between individual electron shell levels. Fluorescence dominates at higher Z.

Post-Effects Caused by Emitted Electrons

The second sort of post-effects is caused by electrons which initially originate from primary or secondary transitions, namely, beta particles (positrons or electrons) and IC electrons. Post-effects here are due to the way these particles interact with the surrounding condensed matter.

The Destiny of the Positron

Let’s start with the positron, which itself is an antimatter particle. It thus is supposed to annihilate with its matter counterpart, the electron (Fig. 38, left). This annihilation happens whenever a positron is created. Clearly, this occurs after the primary β⁺ transformation of a proton-rich nuclide. However, a positron can also be created (together with an electron) in the process of pair formation.

Fig. 38

Post-effects related to the appearance of a positron. Left: The positron is emitted from the transforming unstable nucleus, K1, and interacts with the surrounding condensed matter, in particular with the shell electrons of atoms and molecules. At kinetic energy close to zero, it combines with an electron. The masses of the two elementary particles convert into energy as E = mc². This energy is released as γ-radiation in terms of two photons of each 511 keV energy, which are emitted in opposite directions. Right: This electromagnetic emission is recorded in a γ-spectrum as intense 511 keV peak, as shown for emitter ¹⁸F when transforming to stable ¹⁸O via positron emission. Note that the γ-radiation is not created directly in the course of the transformation of the unstable nucleus

What happens with the positron? In a vacuum, the answer to this question is nothing! Of course, things change in (condensed) matter. In this case, the positron intensely interacts via inelastic and elastic scattering with the electrons located in the shells of atoms or molecules. These interactions reduce the kinetic energy of the moving positron until it is at almost zero kinetic energy: it has “thermalized”. Now, the interaction of the positron with a shell electron becomes elastic and finally the positron may combine with an electron. This represents the formation of a pair of matter + antimatter particles and results in the annihilation of the two particles.

What is annihilation? The mass of the two particles converts into energy according to E = m_ec². With the (rest) mass of the electron of m _e = 9.109383^.10⁻²⁸ g = 0.00054858 u and the energy equivalent of E _e = 510.9989 keV (see Table 1), the overall energy is 2 × 510.9989 keV = 1.0219978 MeV. This energy is emitted as electromagnetic radiation composed of two photons of 510.9989 keV each, emitted in opposite directions (see Fig. 38).

Inner Bremsstrahlung

Whenever an electron—either emitted in the course of β⁻ transformation or created during internal conversion—interacts with the nuclei of atoms or molecules representing surrounding condensed matter, it induces bremsstrahlung (from bremsen “to brake” and Strahlung “radiation”, i.e.“braking radiation” or “deceleration radiation”). Bremsstrahlung is induced whenever an electron transits with a given kinetic energy along an atomic nucleus (Fig. 39). The interaction leads to a loss of kinetic energy, and the energy lost (ΔE) is converted into electromagnetic energy, in this case photons. This time, the electromagnetic radiation is called internal or “inner bremsstrahlung”. This energy of these photons is lower than the energy of X-rays emitted for the same nuclide. The number of these bremsstrahlung photons per transformation is <10⁻³ (ΔE)². Obviously, its impact is greater for high Z and low E_γ.

Fig. 39

The origin of inner bremsstrahlung: An electron originating from nuclear transformation processes is attenuated within the nucleus of surrounding atoms. The kinetic energy the electron loses from its initial E_i to lower values, E_ii, is emitted as electromagnetic radiation

Summary

An unstable nucleus, K1, transforms in order to improve the mean nucleon binding energy of its nucleons in a product nucleus, K2. This is managed by three principal kinds of primary transformation. In many cases, a primary transformation process populates energetically excited nuclear levels of K2, which de-excite via three options of secondary transitions. In parallel, several kinds of post-processes are induced. For most of those processes, we do not observe the fate of the transforming nucleus directly. However, the basic signals we observe are related to the radioactive emissions, which accompany those nuclear transformations and transitions.

This chapter has tried to describe the individual radiations and their corresponding sources. The emissions originating from primary transformations are β⁻ electrons, β⁺ electrons, (i.e. positrons), and α-particles. All three are extremely relevant to radiopharmaceutical chemistry and nuclear medicine. (The electron neutrinos also emitted cannot be recorded easily, and thus they are not relevant to radiopharmaceutical chemistry and nuclear medicine.)

The emissions originating from secondary transitions are the single γ-photons (extremely relevant to radiopharmaceutical chemistry and nuclear medicine, in particular for SPECT), conversion electrons (discussed as potential particles for therapy), and the products of pair formation.

Those emissions originating from post-effects are the 511 keV annihilation photons (extremely relevant to radiopharmaceutical chemistry and nuclear medicine, in particular for PET), X-ray photons, and Auger-type electrons.

In reality, several of those radiations can be observed for the same unstable nuclide simultaneously. Figure 40 summarizes the different radiations in terms of their origin as well as their character, i.e.particulate or electromagnetic.

Fig. 40

The spectrum of different radiations in terms of their origin (left) as well as in terms of their character, i.e. particulate or electromagnetic (right). A + CK means Auger and Coster-Kronig electrons

This broad spectrum of radioactive emissions is a gift of nature, in particular to scientists and physicians working in the field of radiopharmaceutical chemistry and nuclear medicine. Some isotopes can be selected for diagnostic purposes simply because they offer photon radiations—i.e. electromagnetic emissions—which are not (or only weakly) absorbed when penetrating the human body. This allows for the quantitative determination of absolute activities, which is a feature of PET tracers. Furthermore, it avoids critical radiation doses to the patient. In contrast, if a high but locally focused radiation dose is preferred for therapeutic purposes, there are particle-emitting radionuclides available as well! The particles considered routinely these days—α-particles and β-electrons—are emitted from--> unstable nuclei in the course of primary α- and β-transformations, though electrons emitted during secondary and post-processes (e.g.IC or Auger electrons) can be therapeutically relevant as well.

It is important to remember that for a given radionuclide, the desired radioactive emission is often accompanied by other parallel emissions. Some of them can be ignored (e.g.electron neutrinos), while others are negligible if they appear in low abundance. For example, ¹⁸F is not a “pure” positron emitter. About 4 of 100 nuclei of ¹⁸F transform via electron capture. As a consequence, there are emissions caused by refilling the vacancy created in the K-shell of the atom: X-rays, Auger electrons, and Coster-Kronig electrons. However, these emissions are rare enough such that they do not matter in terms of radiation detection and radiation dose. But what about ⁶⁴Cu, a radionuclide typically used for PET imaging? Its positron branching is only 17.9% and is accompanied by 43.1% of EC and 39% of β-emission. In this case, these parallel emissions may require further consideration.

And what if the primary positron emission pathway involves excited nuclear states? High-energy photons of high abundance originating from the de-excitation of those excited nuclear levels may complicate the quantitative registration of the 511 keV annihilation radiation of that “positron emitter”. This is an issue in particular for the positron-emitter ⁸⁶Y [10]. On the other hand, primary electron capture makes ¹¹¹In a typical SPECT isotope because the excited nuclear levels produce low-energy photons via secondary processes. But what about the post-processes? The electrons released in the course of refilling the hole in the electron shell (as created by the initial EC process) may be (and indeed are) of “therapeutic” interest (see, e.g. the “therapeutic” studies of ¹¹¹In-labeled octreotide tracers [11]). And finally, one more example: even if there is a neutron-reach radionuclide defined as a 100% β⁻ emitter, the emitted electron will definitely induce bremsstrahlung. This may be considered a risk, as it causes additional radiation dose. However, in other cases cases—for example, the therapeutic use of ⁹⁰Y-labeled radiopharmaceuticals—the bremsstrahlung emissions actually represent a benefit, as they allow for imaging.

In conclusion, the primary interest of radiopharmaceutical chemists is often in the use of a given radionuclide's “ideal” radioactive emission. However, it is nonetheless essential to identify and to understand the fully transformation pathway of each candidate. Once these things have been considered and a “best” candidate for a certain application has been identified, another set of questions arises: How can this particular radionuclide be produced in high yield, ideal radionuclidic purity, and chemical identity? These questions will be addressed in the following chapters.