Excellent question Craig! To first answer your question: yes, the rate of isotope decay can be changed.

Radioactive decay happens when an unstable atomic nucleus spontaneously changes to a lower energy state and shoots out a tiny amount of radiation. This process changes the original atom to a different element (changing the number of protons) or a different isotope of the same element (equal number of protons but different number of neutrons, so the mass changes but not the element). Since radioactivity is spontaneous, you can think of the decay rate as fixed, or constant, through time. However, this is not entirely true, there are events that can influence the decay process.

Because radioactive decay is spontaneous, or random, it is not possible to predict exactly when an individual atom will decay. The decay rate better describes the average time required for half of a large amount of atoms to have decayed. This means there are two main ways that decay can be altered. First, time could be manipulated. If the atom can be moved at near the speed of light, time slows down relative to an outside observer. Scientists use an apparatus called a particle accelerator, to shoot atoms or subatomic particles at incredibly high velocities by using electromagnetic fields. Slowing down relative time for an atom could allow processes like radioactivity to happen slow enough for a human to observe. There are more than 30,000 particle accelerators in the world being used for a variety of research questions.

I think the second way to alter decay applies most directly to your question. This involves changing the electron state surrounding the atom's nucleus. One type of radioactive decay is electron capture. In electron capture, an electron from the atom's inner shell is absorbed by, or drawn into, the atom's nucleus where it combines it with a proton to create a neutron and neutrino. The neutrino is ejected from the atom's nucleus, and the atom becomes a new element since it loses a proton while gaining a neutron.

An example of electron capture can represented by a carbon atom (with 6 protons and 5 neutrons) capturing an electron from its inner shell (typically from the K-shell), resulting in a boron atom with 5 protons and 6 neutrons. The more the atom's electrons overlap with the nucleus, the more likely the nucleus is to capture an electron. Thus, by exciting or deforming the atom's electrons into states that overlap more or less with the nucleus, the decay rate of the atom can be changed.

An excited-state atom is an atom in which the total energy of the electrons could be lowered by transferring one or more electrons to different orbitals; that is, in an excited-state atom, not all the electrons are in the lowest possible energy levels. The state of an atom's electrons could be altered by changing the types of atoms surrounding it, due to the energy required for chemical bonding between atoms. Another approach would be the removal electrons from the atom. If a radioactive element requires electron capture in order to decay, that atom could be prevented from decaying if all the electrons were ripped away. The final approach is by bombarding the atom with high-energy radiation.

Since radioactive decay is a nuclear reaction, inducing other nuclear reactions at the same time can interfere with the decay rate. This means that theoretically, large amounts of solar radiation from the sun could alter decay rates of some atoms. However, this would no longer be an independent decay event with a change to the decay rate, it would be more like decay soup, with competing reactions. Despite these three methods being able to affect decay rates, those fluctuations would be quite small.

Further, the decay rates defined for each radioactive element represent the rate at which the atoms will decay when at rest or in a particular chemical bonding configuration. Most changes to the decay of a particular material is very small, and minor fluctuations increasing or decreasing the decay rate will mostly be unnoticeable when considering that the decay rate is an average over time. So any isolated fluctuation may not alter the long term decay average. Any large changes to the decay rate would require elaborate, expensive, and high-energy instruments (such as particle accelerators mentioned earlier, as well as nuclear reactors and ion traps). In 2009, a research team at Purdue University did discuss some unexplained annual fluctuations in long-term measurements of decay rates of silicon-32 and chlorine-36 at the Brookhaven National Laboratory (BNL) in New York and for radium-226 at the Physikalisch-Technische Bundesanstalt (PTB) in Germany (Jenkins, et al., 2009, Astroparticle Physics, Evidence of correlations between nuclear decay rates and Earth–Sun distance. link here ). Many groups tried to relate these fluctuations to solar radiation from the sun. However, my understanding is that no clear evidence has yet tied those observations to solar radiation. In fact, in 2014, scientists using a more sensitive radioactivity detector were unable to replicate the 2009 study results for the amplitude of fluctuations measured, nor in the seasonal pattern of fluctuations (Kossert and Nähle, 2014, Astroparticle Physics, Long-term measurements of 36Cl to investigate potential solar influence on the decay rate. link here ).

To address your carbon-14 example. Carbon-14 decays to nitrogen-14 through beta decay, which is not directly impacted by solar radiation. Further, dates produced by carbon-14 dating are extensively calibrated with naturally occurring materials that exhibit annual growth phenomena such as tree rings, varves (seasonal/annual sediment layers deposited in lakes, ocean basins, and glacier deposits), and cave deposits, as well as with other materials of known ages based on archeological records, to test the accuracy of the results. Solar radiation can affect the formation of carbon-14, however. The majority of carbon-14 is produced in the upper troposphere and stratosphere (altitudes of ~30,000 to 49,000 feet) by cosmic ray spallation, or the formation of new elements from the impact of cosmic rays on an atom, of a nitrogen-14 atom absorbing a free neutron. Free neutrons are formed when cosmic rays enter Earth's atmosphere. The majority of cosmic rays arrive from outside our solar system, and production rates in Earth's atmosphere change depending on fluctuations in Earth's magnetic field and fluctuations in the Sun's magnetic field and solar wind (or the stream of charged particles released from the upper atmosphere of the Sun). Spikes in solar radiation have been recorded in Earth's history, can cause large increases in the production of carbon-14, and the impacts are difficult to fully understand.

To summarize, yes, the rate of isotope decay can be changed. However, most naturally occurring decay is extremely close to constant. Solar radiation is unlikely to affect the majority of any decay reactions. Only cosmogenic isotopes, which are rare, are impacted by changes to solar radiation. However these cosmogenic isotopes require free neutrons from external cosmic rays to break down, and thus are not given decay rates. The product atoms of cosmogenic isotopes may be radioactive (such as carbon-14) or stable (such as helium-3). The radioactive product atoms will then follow the regular, natural decay rates as described above. It is a lot to tease out! But hopefully this helps answer your question.

This is discussed pretty thoroughly on this page and on this page. (Note that the second also contains information regarding changing apparent half-lives of large amounts of material, differs from the question in that the rate of decay of the isotopes.)

I can summarize here though.
In general, rates of nuclear decay are essentially fixed and insensitive to changes in their environment.

Beta decay may be changed slightly in some cases, but any changes which do occur are small. For decay by alpha emission and fission, the changes are below the level to which half lives are currently known. This constancy in beta decay is because this mechanism requires changes to the frequency at which electrons interact with particles in the nucleus. Since the inner electrons are more frequently "near" the nucleus, they are the most likely to be grabbed by the nucleus. However, these electrons, being the innermost, are also the least sensitive to external forces. The exceptions are those where the "inner" and "outer" electrons are essentially the same and when the energy produced from the decay is too small to liberate an inner electrons, meaning that only external (and therefore environment-sensitive) electrons are available for decay. The former is the case for beryllium, which has only 4 electrons. By surrounding radioactive Be-7 atoms with C-60 cages, the half-life was reduced by 0.8%, an effect attributed to chemical bonding affecting the wave functions of the electrons. One could also imagine preventing decay by electron capture by removing all of the electrons surrounding the atom. Obviously if there are no electrons to capture, then decay by this mechanism cannot occur. And if that particular element can only decay by electron capture, then decay could not occur at all.

Some years ago there was considerable excitement over the possibility that solar events could be affecting rates of nuclear decay, potentially due to enhanced neutrino emission. The findings were widely disputed though, and numerous - studies - since then have disproved any connection between neutrinos and radioactive decay. Instead, the effects are likely due to effects of environmental changes - on the detectors rather than on the decaying elements.

As to my knowledge, there are no events which would change the decay rate of an isotopic system. Isotope geochemists measure decay rates for different isotope systems (such as 87Rb decaying to 87Sr, or in the case of your question, 14C decaying to 14N). Usually these have some error associated with them, but are considered to be constant (within the analytical uncertainty of the instrument).

In the case of 14C, if a sample is over ~60,000 years old (roughly 6 half-lives of 14C), there is generally not enough of that isotope left to measure.

Many people think that radioactive half-lives are fixed, but that’s not completely true. Half-lives can be changed using time dilation effects.

According to the idea of relativity, everything that experiences time can be given a longer effective lifetime if time is dilated, or slowed down. The first way this can be done is through light speed. For example, if you shoot radioactive atoms through a tube at high speed in the lab, they will have their half-life lengthened relative to the lab. This has been done many times using particle accelerators.

The second way radioactive decay can be altered is by changing the state of the electrons that surround the nucleus. In a type of radioactive decay called “electron capture”, the nucleus absorbs a proton, combines it with a proton, and makes a neutron and a neutrino. By exciting or deforming the atom’s electrons into states that overlap less with the nucleus, the half-life can be reduced. Also, simply changing the neighboring atoms that are bonded to a radioactive isotope can change it’s half-life.

Interestingly, you can make the radioactive decay mode infinite if you theoretically strip all of the electrons off of a radioactive atom.

To answer your final question, the half-life of radioactive material can be changed by bombarding it with high-energy radiation. Theoretically, this “massive solar storm” could have altered the half-lives of radioactive atoms on Earth.

The rate at which time passes depends on the motion relative to the observer (as described by special relativity) and on the position in a gravitational potential well (as described by general relativity), both of which have been confirmed by experiment. Additionally, extreme pressure changes which isotopes are stable, but to get this pressure you need gravitational fields of the sort not found outside of neutron stars or black holes.

A massive solar flare would not change the decay rate of carbon-14. It would, however, create lots of carbon-14 when the high-energy radiation alters the isotopes of existing atoms, and the presence of a solar flare like this in the Sun's history could be detected that way.

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Forbes magazine had a different answer in 2011, which appears to be untrue. It's easy for scientists to get excited about new results that don't turn out to be correct, in the end.