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this just pushes back the mystery back one step for me: why are nuclei with even atomic number more stable?

There is no big mystery about this. There is a pairing interaction in nuclei. It's loosely analogous to the Cooper pairs in a superconductor.

It's easy to construct an argument as to why, if you're looking for an element with no stable isotopes, Tc is a good candidate.

Any even atomic number is guaranteed to have an even-even isotope that's stable against beta decay, because even-even isotopes are more stable than odd-odd ones, due to pairing.

In general we expect a richer variety of stable isotopes for elements whose atomic numbers are near a magic proton number, or for elements such that for that element, the line of stability comes close to a magic neutron number.

Based on these considerations, if we were hoping to find a light element with no stable isotopes, it would be one that was an odd $Z$, one whose $Z$ was far from any magic proton number, and one for which the line of stability is not close to a magic neutron number. Technetium fulfills these requirements.

To retrodict this theoretically, the simplest thing we could try would be to use the semi-empirical mass formula, which is essentially the classical energy of a charged liquid drop, with a couple of terms thrown in to approximate quantum-mechanical effects. As John Rennie's answer describes, this works surprisingly well, in the sense that although it's incredibly crude, it correctly predicts that, e.g., 97Tc is very, very close to the dividing line between stability and instability.

The next step up in sophistication would be the Strutinsky smearing technique (Strutinsky 1968; also described in Salamon 2010). This method involves taking a classical liquid-drop energy and adding on a correction for quantum effects. It correctly reproduces the loss or gain in binding energy due to the higher- or lower-than-average density of single particle levels. E.g., it correctly predicts that magic numbers are far more bound. The technique has the advantage of being computationally cheap, and of working for both nuclei near closed shells and mid-shell nuclei.

If you want to know whether the Strutinsky technique is good enough to retrodict the beta-instability of 97Tc and 99Tc, I think the answer is basically that the question is ambiguous. These models have a fairly big number of adjustable parameters, maybe 30 or so, that need to be fitted to the experimental data. There are also qualitative choices involved, such as the use of a Woods-Saxon potential as opposed to Nilsson. When you use these ~30 parameters to predict some very large number of experimental observables across the entire chart of the nuclei (tens of thousands of masses, electric quadrupole moments, ground-state spins, ...), you get points for honesty but you don't get the best precision. People who are interested in a certain region of the chart of the nuclei, e.g., superheavy elements, will fit their parameters to that small region and get much better precision. If you keep on narrowing your focus like this, and you're making retrodictions about well-studied regions of the chart of the nuclei, then eventually what you're doing is just a fancy exercise in interpolation. I'm sure that at this level, one could correctly calculate the very small difference in binding energy between 97Tc and 97Mo to the precision needed in order to show that 97Tc is beta-unstable, but all you'd really be doing would be interpolating.

Strutinsky, Nucl. Phys. A122 (1968) 1

P. Salamon, http://arxiv.org/abs/1004.0079

this just pushes back the mystery back one step for me: why are nuclei with even atomic number more stable?

There is no big mystery about this. There is a pairing interaction in nuclei. It's loosely analogous to the Cooper pairs in a superconductor.

It's easy to construct an argument as to why, if you're looking for an element with no stable isotopes, Tc is a good candidate.

Any even atomic number is guaranteed to have an even-even isotope that's stable against beta decay, because even-even isotopes are more stable than odd-odd ones, due to pairing.

In general we expect a richer variety of stable isotopes for elements whose atomic numbers are near a magic proton number, or for elements such that for that element, the line of stability comes close to a magic neutron number.

Based on these considerations, if we were hoping to find a light element with no stable isotopes, it would be one that was an odd $Z$, one whose $Z$ was far from any magic proton number, and one for which the line of stability is not close to a magic neutron number. Technetium fulfills these requirements.

To retrodict this theoretically, the simplest thing we could try would be to use the semi-empirical mass formula, which is essentially the classical energy of a charged liquid drop, with a couple of terms thrown in to approximate quantum-mechanical effects. As John Rennie's answer describes, this works surprisingly well, in the sense that although it's incredibly crude, it correctly predicts that, e.g., 97Tc is very, very close to the dividing line between stability and instability.

The next step up in sophistication would be the Strutinsky smearing technique (Strutinsky 1968; also described in Salamon 2010). This method involves taking a classical liquid-drop energy and adding on a correction for quantum effects. It correctly reproduces the loss or gain in binding energy due to the higher- or lower-than-average density of single particle levels. E.g., it correctly predicts that magic numbers are far more bound. The technique has the advantage of being computationally cheap, and of working for both nuclei near closed shells and mid-shell nuclei.

If you want to know whether the Strutinsky technique is good enough to retrodict the beta-instability of 97Tc and 99Tc, I think the answer is basically that the question is ambiguous. These models have a fairly big number of adjustable parameters, maybe 30 or so, that need to be fitted to the experimental data. There are also qualitative choices involved, such as the use of a Woods-Saxon potential as opposed to Nilsson. When you use these ~30 parameters to predict some very large number of experimental observables across the entire chart of the nuclei (tens of thousands of masses, electric quadrupole moments, ground-state spins, ...), you get points for honesty but you don't get the best precision. People who are interested in a certain region of the chart of the nuclei, e.g., superheavy elements, will fit their parameters to that small region and get much better precision. If you keep on narrowing your focus like this, and you're making retrodictions about well-studied regions of the chart of the nuclei, then eventually what you're doing is just a fancy exercise in interpolation. I'm sure that at this level, one could correctly calculate the very small difference in binding energy between 97Tc and 97Mo to the precision needed in order to show that 97Tc is beta-unstable, but all you'd really be doing would be interpolating.

Strutinsky, Nucl. Phys. A122 (1968) 1

P. Salamon, http://arxiv.org/abs/1004.0079

this just pushes back the mystery back one step for me: why are nuclei with even atomic number more stable?

There is no big mystery about this. There is a pairing interaction in nuclei. It's loosely analogous to the Cooper pairs in a superconductor.

It's easy to construct an argument as to why, if you're looking for an element with no stable isotopes, Tc is a good candidate.

Any even atomic number is guaranteed to have an even-even isotope that's stable against beta decay, because even-even isotopes are more stable than odd-odd ones, due to pairing.

In general we expect a richer variety of stable isotopes for elements whose atomic numbers are near a magic proton number, or for elements such that for that element, the line of stability comes close to a magic neutron number.

Based on these considerations, if we were hoping to find a light element with no stable isotopes, it would be one that was an odd $Z$, one whose $Z$ was far from any magic proton number, and one for which the line of stability is not close to a magic neutron number. Technetium fulfills these requirements.

To retrodict this theoretically, the simplest thing we could try would be to use the semi-empirical mass formula, which is essentially the classical energy of a charged liquid drop, with a couple of terms thrown in to approximate quantum-mechanical effects. As John Rennie's answer describes, this works surprisingly well, i.e., although it's incredibly crude, it correctly predicts that 97Tc is very, very close to the dividing line between stability and instability.

The next step up in sophistication would be the Strutinsky smearing technique (Strutinsky 1968; also described in Salamon 2010). This method involves taking a classical liquid-drop energy and adding on a correction for quantum effects. It correctly reproduces the loss or gain in binding energy due to the higher- or lower-than-average density of single particle levels. E.g., it correctly predicts that magic numbers are far more bound. The technique has the advantage of being computationally cheap, and of working for both nuclei near closed shells and mid-shell nuclei.

If you want to know whether the Strutinsky technique is good enough to retrodict the beta-instability of 97Tc and 99Tc, I think the answer is basically that the question is ambiguous. These models have a fairly big number of adjustable parameters, maybe 30 or so, that need to be fitted to the experimental data. There are also qualitative choices involved, such as the use of a Woods-Saxon potential as opposed to Nilsson. When you use these ~30 parameters to predict some very large number of experimental observables across the entire chart of the nuclei (tens of thousands of masses, electric quadrupole moments, ground-state spins, ...), you get points for honesty but you don't get the best precision. People who are interested in a certain region of the chart of the nuclei, e.g., superheavy elements, will fit their parameters to that small region and get much better precision. If you keep on narrowing your focus like this, and you're making retrodictions about well-studied regions of the chart of the nuclei, then eventually what you're doing is just a fancy exercise in interpolation. I'm sure that at this level, one could correctly calculate the very small difference in binding energy between 97Tc and 97Mo to the precision needed in order to show that 97Tc is beta-unstable, but all you'd really be doing would be interpolating.

Strutinsky, Nucl. Phys. A122 (1968) 1

P. Salamon, http://arxiv.org/abs/1004.0079

this just pushes back the mystery back one step for me: why are nuclei with even atomic number more stable?

There is no big mystery about this. There is a pairing interaction in nuclei. It's loosely analogous to the Cooper pairs in a superconductor.

It's easy to construct an argument as to why, if you're looking for an element with no stable isotopes, Tc is a good candidate.

Any even atomic number is guaranteed to have an even-even isotope that's stable against beta decay, because even-even isotopes are more stable than odd-odd ones, due to pairing.

In general we expect a richer variety of stable isotopes for elements whose atomic numbers are near a magic proton number, or for elements such that for that element, the line of stability comes close to a magic neutron number.

Based on these considerations, if we were hoping to find a light element with no stable isotopes, it would be one that was an odd $Z$, one whose $Z$ was far from any magic proton number, and one for which the line of stability is not close to a magic neutron number. Technetium fulfills these requirements.

To retrodict this theoretically, the simplest thing we could try would be to use the semi-empirical mass formula, which is essentially the classical energy of a charged liquid drop, with a couple of terms thrown in to approximate quantum-mechanical effects. As John Rennie's answer describes, this works surprisingly well, in the sense that although it's incredibly crude, it correctly predicts that, e.g., 97Tc is very, very close to the dividing line between stability and instability.

The next step up in sophistication would be the Strutinsky smearing technique (Strutinsky 1968; also described in Salamon 2010). This method involves taking a classical liquid-drop energy and adding on a correction for quantum effects. It correctly reproduces the loss or gain in binding energy due to the higher- or lower-than-average density of single particle levels. E.g., it correctly predicts that magic numbers are far more bound. The technique has the advantage of being computationally cheap, and of working for both nuclei near closed shells and mid-shell nuclei.

If you want to know whether the Strutinsky technique is good enough to retrodict the beta-instability of 97Tc and 99Tc, I think the answer is basically that the question is ambiguous. These models have a fairly big number of adjustable parameters, maybe 30 or so, that need to be fitted to the experimental data. There are also qualitative choices involved, such as the use of a Woods-Saxon potential as opposed to Nilsson. When you use these ~30 parameters to predict some very large number of experimental observables across the entire chart of the nuclei (tens of thousands of masses, electric quadrupole moments, ground-state spins, ...), you get points for honesty but you don't get the best precision. People who are interested in a certain region of the chart of the nuclei, e.g., superheavy elements, will fit their parameters to that small region and get much better precision. If you keep on narrowing your focus like this, and you're making retrodictions about well-studied regions of the chart of the nuclei, then eventually what you're doing is just a fancy exercise in interpolation. I'm sure that at this level, one could correctly calculate the very small difference in binding energy between 97Tc and 97Mo to the precision needed in order to show that 97Tc is beta-unstable, but all you'd really be doing would be interpolating.

Strutinsky, Nucl. Phys. A122 (1968) 1

P. Salamon, http://arxiv.org/abs/1004.0079

this just pushes back the mystery back one step for me: why are nuclei with even atomic number more stable?

There is no big mystery about this. There is a pairing interaction in nuclei. It's loosely analogous to the Cooper pairs in a superconductor.

It's easy to construct an argument as to why, if you're looking for an element with no stable isotopes, Tc is a good candidate.

Any even atomic number is guaranteed to have an even-even isotope that's stable against beta decay, because even-even isotopes are more stable than odd-odd ones, due to pairing.

In general we expect a richer variety of stable isotopes for elements whose atomic numbers are near a magic proton number, or for elements such that for that element, the line of stability comes close to a magic neutron number.

Based on these considerations, if we were hoping to find a light element with no stable isotopes, it would be one that was an odd $Z$, one whose $Z$ was far from any magic proton number, and one for which the line of stability is not close to a magic neutron number. Technetium fulfills these requirements.

To retrodict this theoretically, the simplest thing we could try would be to use the semi-empirical mass formula, which is essentially the classical energy of a charged liquid drop, with a couple of terms thrown in to approximate quantum-mechanical effects. As John Rennie's answer describes, this works surprisingly well, i.e., although it's incredibly crude, it correctly predicts that 97Tc is very, very close to the dividing line between stability and instability.

The next step up in sophistication would be the Strutinsky smearing technique (Strutinsky 1968; also described in Salamon 2010). This method involves taking a classical liquid-drop energy and adding on a correction for quantum effects.

If you want to know whether this technique is good enough to retrodict this fact about 97Tc, I think the answer is basically that the question is ambiguous. These models have a fairly big number of adjustable parameters, maybe 30 or so, that need to be fitted to the experimental data. There are also qualitative choices involved, such as the use of a Woods-Saxon potential as opposed to Nilsson. When you use these ~30 parameters to predict some very large number of experimental observables (tens of thousands of masses, electric quadrupole moments, ground-state spins, ...), you get points for honesty but you don't get the best precision. People who are interested in a certain region of the chart of the nuclei, e.g., superheavy elements, will fit their parameters to that small region and get much better precision. If you keep on narrowing your focus like this, and you're making retrodictions about well-studied regions of the chart of the nuclei, then eventually what you're doing is just a fancy exercise in interpolation. I'm sure that at this level, one could correctly calculate the very small difference in binding energy between 97Tc and 97Mo to the precision needed in order to show that 97Tc is beta-stable, but all you'd really be doing would be interpolating.

Strutinsky, Nucl. Phys. A122 (1968) 1

P. Salamon, http://arxiv.org/abs/1004.0079

this just pushes back the mystery back one step for me: why are nuclei with even atomic number more stable?

There is no big mystery about this. There is a pairing interaction in nuclei. It's loosely analogous to the Cooper pairs in a superconductor.

It's easy to construct an argument as to why, if you're looking for an element with no stable isotopes, Tc is a good candidate.

Any even atomic number is guaranteed to have an even-even isotope that's stable against beta decay, because even-even isotopes are more stable than odd-odd ones, due to pairing.

In general we expect a richer variety of stable isotopes for elements whose atomic numbers are near a magic proton number, or for elements such that for that element, the line of stability comes close to a magic neutron number.

Based on these considerations, if we were hoping to find a light element with no stable isotopes, it would be one that was an odd $Z$, one whose $Z$ was far from any magic proton number, and one for which the line of stability is not close to a magic neutron number. Technetium fulfills these requirements.

To retrodict this theoretically, the simplest thing we could try would be to use the semi-empirical mass formula, which is essentially the classical energy of a charged liquid drop, with a couple of terms thrown in to approximate quantum-mechanical effects. As John Rennie's answer describes, this works surprisingly well, i.e., although it's incredibly crude, it correctly predicts that 97Tc is very, very close to the dividing line between stability and instability.

The next step up in sophistication would be the Strutinsky smearing technique (Strutinsky 1968; also described in Salamon 2010). This method involves taking a classical liquid-drop energy and adding on a correction for quantum effects. It correctly reproduces the loss or gain in binding energy due to the higher- or lower-than-average density of single particle levels. E.g., it correctly predicts that magic numbers are far more bound. The technique has the advantage of being computationally cheap, and of working for both nuclei near closed shells and mid-shell nuclei.

If you want to know whether the Strutinsky technique is good enough to retrodict the beta-instability of 97Tc and 99Tc, I think the answer is basically that the question is ambiguous. These models have a fairly big number of adjustable parameters, maybe 30 or so, that need to be fitted to the experimental data. There are also qualitative choices involved, such as the use of a Woods-Saxon potential as opposed to Nilsson. When you use these ~30 parameters to predict some very large number of experimental observables across the entire chart of the nuclei (tens of thousands of masses, electric quadrupole moments, ground-state spins, ...), you get points for honesty but you don't get the best precision. People who are interested in a certain region of the chart of the nuclei, e.g., superheavy elements, will fit their parameters to that small region and get much better precision. If you keep on narrowing your focus like this, and you're making retrodictions about well-studied regions of the chart of the nuclei, then eventually what you're doing is just a fancy exercise in interpolation. I'm sure that at this level, one could correctly calculate the very small difference in binding energy between 97Tc and 97Mo to the precision needed in order to show that 97Tc is beta-unstable, but all you'd really be doing would be interpolating.

Strutinsky, Nucl. Phys. A122 (1968) 1

P. Salamon, http://arxiv.org/abs/1004.0079