Nuclear Spin is a Vector quantity, actually Angular Momentum, and it is not a Scalar quantity as Pauli had incorrectly assumed in 1930. As a Vector, both the Amplitude and Direction are important. A Vector Addition Diagram of the Nuclear Spin of a Neutron's beta-decay happens to be an equilateral triangle, which has a Neutron's Spin of 1/2 unit as one side of that triangle with a Proton's Spin of 1/2 unit and an Electron's Spin of 1/2 unit being the other two Vector sides (as shown in the graphic below). Wolfgang Pauli did not seem to know that and in incorrectly thinking along Scalar lines (only considering the amplitude with no direction), he dreamed up the need for Neutrinos (with another [Scalar] Spin of 1/2 unit). His (incorrect) Scalar thinking did not add up correctly as Scalar quantities, and so he thought he needed to invent a neutrino, for a single purpose, that of supplying an extra Scalar quantity of Nuclear Spin in order to comply with the Conservation of Angular Momentum. He was smart and he certainly should have known better than to try to apply Scalar thinking and logic to Vector quantities.
There is additional statistical scientific and precise mathematical evidence that Neutrinos do not exist inside atomic nuclei in the highly respected NIST data resource. Here are two related articles.
Nuclear Physics May be Fairly Simple
Nuclear Physics - Statistical Analysis of Isotope Masses
There is a curious simple but huge flaw in Nuclear Physics today regarding this incorrect reasoning of Pauli (for which he even later received the Nobel Prize in Physics in 1945.) Nuclear Spin is a Vector quantity, actually Angular Momentum, which every beginning Physics student knows! It is not a Scalar quantity like most of the things we encounter in normal life. The difference is that every Vector quantity has both an Amplitude and a Direction, while all Scalar quantities only have an amplitude and they do not have any defined direction. Examples are speed, which is a Scalar, and velocity, which is a Vector, the difference being that velocity tells you what direction the object is moving in, along with its speed.
Physicists (should) all know that any quantity like Nuclear Spin is necessarily a Vector, where both the rate of rotation is specified, as well as the orientation, which is defined by the direction of the spin axis. We say the Earth has a Spin Vector of "once a day" rotational speed and a direction of exactly due north. If the Earth's spin Vector was pointed south instead, that would tell us the Earth was spinning the other way around.
During the 1920s, Physicists discovered that all nuclear particles have Angular Momentum or Spin Vectors. They were mostly concerned with one specific object, the Neutron, which was known to spontaneously decay into a Proton and an Electron with a half-life of around 15 minutes. More confusing yet, the Proton and Electron could soon fuse back together to reform a Neutron.
Physics has long been based on several Conservation Laws, including one of Conservation of Angular Momentum.
By about 1929, it was firmly known that each Proton, each Electron and each Neutron all had the exact same amount of "Nuclear Spin", which is described as being 1/2 unit of Spin, using an actual scientific quantity of Angular Momentum defined as Planck's Constant divided by (2 * Pi), or h / 2π
Many Physicists repeated the same experiment where a Neutron spontaneously "beta-decayed" into a Proton and an Electron, and everyone was uncomfortable with the result. Pauli was one of the leading Physicists who was troubled by a Neutron starting out with a Spin of 1/2, and coming apart into two particles, a Proton with a Spin of 1/2 and an Electron with a Spin of 1/2. It was (incorrectly) thought that this violated the Conservation of Angular Momentum. And so Wolfgang Pauli dreamed up a new particle, which became known as a Neutrino (or, in this specific case, an Anti-Neutrino), which he said had no electrical charge, no mass, no size and only a single characteristic, its Spin of 1/2 unit.
In other words, the ONLY reason that the Neutrino was even speculated to exist was for the single purpose of trying to resolve a PROBLEM WHICH DID NOT ACTUALLY EVEN EXIST. There has never been any other reason suggested as to why Neutrinos should even exist!
Therefore, Pauli announced that he had the explanation for how Angular Momentum could be Conserved when a Neutron decayed. He explained (wrongly) that the Neutron broke apart into three particles and not just the obvious two, that is:
Neutron → Proton + Electron + (Anti)-Neutrino
He then explained that Nuclear Spin was conserved:
1/2 → 1/2 + 1/2 + -1/2
where the Neutrino involved was actually an Anti-Neutrino, meaning that it spins in the opposite direction so that the Spin Vector is in the opposite direction (and therefore the minus sign).
Everyone was thrilled that the brilliant Wolfgang Pauli had solved the biggest nuclear problem of the day! And, to this day, in 2017, everyone still absolutely accepts Pauli's explanation, as he was one of a handful of Physicists, with Einstein and Bohr, who was researching Nuclear Physics. Therefore, no one questions Pauli's announcement that there must be Neutrinos, in order to explain the Nuclear Spin Conservation issue of a Neutron's beta-decay.
For bizarre reasons that are beyond me, the brilliant Wolfgang Pauli and all Physicists since him seem to assume the Spin is a Scalar quantity, which would require the speculation of a neutrino's existence, with a Spin of 1/2, to Conserve Spin by a Scalar 1/2 = 1/2 + 1/2 - 1/2.
That is not remotely true! Pauli had made an enormous logical blunder in assuming that Nuclear Spin is a Scalar quantity! And so has every Physicist today still bought into that wrong assumption!
As an actual Vector quantity, it is clear that THERE IS NO NEED TO DREAM UP A NEUTRINO IN ORDER TO CONSERVE NUCLEAR SPIN.
Consider a (red) Spin vector of a Neutron which has an amplitude of "1/2 unit". (as shown in this graphic, to the upper right). The way Vector Addition works, is that two new Vectors can be formed from the original Vector, as long as they graphically add together, as the green (Proton Spin Vector) and blue (Electron Spin Vector) Vectors do in this graphic. In our example, the three Vectors form an equilateral triangle, meaning that all three Vectors shown have the same amplitudes.
This works the other way as well. Two Spin Vectors (with identical amplitude of 1/2 unit) (shown as green and blue arrows in our graphic) which happen to be oriented at a space angle of 120 degrees from each other, can also add, as Vectors, to become a new Spin Vector (shown as the red arrow in our graphic). The new Neutron Spin Vector has exactly the same amplitude (1/2 unit) but is now at an orientation of a third side of an equilateral triangle. There IS NO issue of any problem with Conservation of Angular Momentum regarding the Spin Vector of a Neutron and the consequent Spin Vectors of a Proton and Electron.
Vector Addition therefore works both ways, and Vectors can point in either direction, so a Neutron Spin can spawn two Vectors for a Proton and an Electron, or a Proton Spin and an Electron Spin can add to form a Neutron Spin Vector.
Note that the ONLY reason that Neutrinos supposedly need to exist is to Conserve Angular Momentum for beta decays such as when a Neutron decays into a Proton and Electron (which we know happens!) There has never been any other reason to speculate that Neutrinos even exist! And since Wolfgang Pauli had made such an enormous logical blunder in 1930 in his speculation, even that is not actually the case.
It is not as though neutrinos are really obvious objects. It was more than 25 years after Pauli had speculated the existence of neutrinos that the first experiment claimed to have detected any of them. Worse, such experiments, inside nuclear reactors, involved some rather creative speculations in making that claim.
How could any Physicist, then or now, believe that it was even needed to dream up a Neutrino, for the single purpose of Conserving Nuclear Spin? Yes, a really smart Physicist, Wolfgang Pauli, dreamed up the existence of Neutrinos to achieve this Scalar addition of the quantities which are actually Vectors, and apparently everyone just accepted that Pauli must be right! But he wasn't!
In fact, whether the Neutron is Free-Ranging across a room or within any atomic nucleus, the Vector nature of Nuclear Spin is such that the complete experimental explanation of the beta decay of a Neutron is simple and obvious! Just two objects are created, the Proton and the Electron! Much of the past 80 years of nuclear Physics has been centered on all the complexities Pauli had made necessary by his wrong explanation of a Neutron's decay and his wrong speculation of there therefore needing to be Neutrinos!
Even as a First Year Physics student in 1963, I was bothered by one other assumption that struck me as peculiar! IF this is all just Scalar additions, are all Protons and Neutrons and Electrons all lined up with their Spin axes neatly like trillions of soldiers? Otherwise, do physicists believe that a beta-decayed Neutron comes apart into two neatly lined up Proton and Electron objects? Because, 15 minutes later, it is assumed that the Proton and Electron re-combine to form a Neutron again! Do people think they stay lined up for all that time? On the other hand, do people just never think about such things? Apparently so!
As a Physics Major at the University of Chicago during the 1960s, I was told to always insist on absolute rigidity of my logic, but I was taught these very questionable speculations that seem to be rampant in the Physics community, then and now. Why don't other Physicists today see these same logical flaws which have troubled me during my whole career in Physics?
In any case, when I have examined such flaws as by Pauli (for which he was even given a Nobel Prize in Physics in 1945! Hideki got his in 1949), I see a far simpler explanation of nearly everything in nuclear physics. I spent several years studying the highly respected NIST data and I found many wonderful statistical patterns in various groupings of their Atomic Mass data for the thousands of Atomic Isotopes.
The implications are in many fields, and it took me several years (1996-2003) to analyze and digest. A rather comprehensive web-page on the patterns and results found is at Nuclear Physics - Statistical Analysis of Isotope Masses.
One of the most powerful (and unexpected) results of that research is that there appears to be no statistical evidence that Neutrons even exist within atomic nuclei! The materials are certainly there, but they appear to exist as separate Protons and Electrons rather than as Neutrons! In fact, there appear to be specific patterns of motion of the Electrons inside atomic nuclei, where the negative charge of the Electrons provide wonderful sources of nuclear stability! Specifically, where it is certainly recognized that two Protons inside a specific atomic nucleus are so close to each other that they exert immense repulsive electrical force on each other, constantly trying to push each other out of the nucleus, I found that if a single (negatively-charged) Electron happened to be at a location exactly midway between those two Protons (at half the distsance to each), there is then an attractive force on both Protons that is four times as strong (due to the inverse square law of electrostatic force). The tiny Electron could not stay at that exact midpoint for very long without collapsing the nucleus! But I now believe that such intra-nuclear Electrons migrate between such midpoint locations, where each two Protons inside an atomic nucleus repel each other for around 3/4 of the time, while the brief presence of the centerpoint Electron enables the two Protons to attract each other four times as strongly, for the other 1/4 of the time. This results in the Protons vibrating with a specific frequency, which I believe is experimentally seen in some nuclei, and it therefore provides a meta-stability of the nuclei's structure, purely due to electrostatic forces of repulsion and attraction! No Strong Nuclear Force is necessary at all to provide nuclear stability!
The result of all this is that atomic nuclei seem to logically not contain Neutrinos at all, not to contain Neutrons at all, not to contain any Strong Nuclear Force, and not to contain massive numbers of Pions! I believe that the NIST data clearly shows that atomic nuclei only contain Protons and Migrating Electrons, and probably nothing else!
The full explanation of the NIST data analysis is rather complex, and probably not suitable for casual reading! But I believe it is strictly compliant with the Logic that Physics is supposed to demand.
The full analysis of the NIST data for all the activities within atomic nuclei, is at Nuclear Physics - Statistical Analysis of Isotope Masses which is a careful examination of the highly respected NIST data.
In the event that Neutrinos actually do exist, for many years I have been troubled by a variety of really bad assumptions that seem to have been made by my fellow Physicists regarding them!
In the 1820s, a brilliant scientist named Olbers realized a "Paradox" which has not yet been really resolved! He realized that with the astounding number of stars that we know exist in the Universe, then any precise direction we might ever look from Earth should eventually encounter the brilliant surface of some star! And so Olbers asked "Why is the sky dark at night?" The most acceptable answer has generally been that there is intervening gas which is somewhat opaque, which therefore blocks the light from many distant stars from ever getting to us. Opponents say that over billions of years, that should have resulted in heating of all that gas (due to absorbtion of the light and energy).
In any case, it is claimed that Neutrinos easily pass through the entire Earth, and so if Neutrinos are created in every fusion reaction in the core of every star in the Universe, then Neutrinos should likely get to us from every star in the Universe. Now, if the Sun generates all the countless billions of Neutrinos which my fellow Physicist experts claim constantly inundate us (in other words, from the specific direction of the Sun), then why are we not really inundated by 200,000 times as many Neutrinos which must be arriving from all those other directions and stars? But the "experts" totally ignore all this and apparently claim that only the Sun actually creates Neutrinos!
See anything lacking in such claims? Such as Logic?
See Neutrinos - Where Did they all Come From?
Also, there are Physicist "experts" who claim that the entire Universe is 20% or 80% or 90% made up of Neutrinos. I would ask "Where did they come from?" If each Neutrino is only created and emitted as a result of a nuclear beta decay of a Neutron, and they have at most a really tiny mass, how could anyone believe that even one one-millionth of the mass of each nucleon could become a Neutrino, then how could any more than one one-millionth of the Universe have ever become Neutrinos?
But an estimate can be made of the total number of galaxies that exist, and the total mass of each of them. The claims of really astounding amounts of (dark matter) Neutrinos mass is just totally silly!
See Neutrinos - Olbers Paradox Means Neutrinos from Everywhere How Many Neutrinos Could Exist?
Note that even separate from any concern about Pauli's logical flaw regarding the existence of Neutrinos, it turns out that the highly respected NIST data base provides another proof that Neutrinos cannot and do not exist! From around 1996 through 2003, I studied the precise Atomic Weights of every Isotope in the NIST data, and when I applied statistical analysis to those values, I found amazing new patterns in the data. There is no indication of any fifth-order or even third-order patterns, which indicates that there may not be any Strong Nuclear Force at all! But the patterns of same-mass Isotopes (from different elements) are even more amazing, where no energy seems to exist in any atomic nucleus to provide for 'energy for any Neutrino', none for any "energy which such a Neutrino might carry away after a beta-decay", no energy for any "Neutron self-binding energy", no energy for any "Pi-Mesons". In fact, the NIST data and a strict statistical analysis makes very clear that there is no energy inside any atomic nucleus other than the existence of Protons and the existence of Electrons, and a tiny amount of remaining energy which accounts for the kinetic energy of the movement of the Protons and Electrons inside the nucleus! A rather comprehensive web-page on the NIST patterns and results found in 1996-2003 is at Nuclear Physics - Statistical Analysis of Isotope Masses.
It really seems totally beyond any possible doubt that no Neutrinos actually exist anywhere! And so a Nobel Prize given in 2015 to scientists who claim to have found new Neutrino results, seems rather peculiar! They got a Nobel Prize for research on objects which do not even exist!
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