|Fig. 1: Schematic of a semi-circular focusing spectrometer.|
While the discovery of β-rays is attributed to Rutherford in his 1899 report, Becquerel, two years earlier, had discovered that the radiation from the radioactive nuclei is composed of two different particles. In his experiment with excited nuclei of Uranium, Becquerel used layers of Aluminum to investigate the radiation properties of the emitted particles. While taking his measurements, he discovered that absorption of the emitted rays remains constant for a couple of mm and then changes to a new value and remains constant for another hundred mm. The first absorbing material was called α, and the second, β. The discovery of Radium by Pierre and Marie Curie revolutionized the field of radioactivity. The signals from the previously used radioactive nuclei samples were too weak to fully investigate. In late 1899, researchers exposed the combination of the rays to a magnetic field, and they learned that only β rays bent in the field and not the αs. Using the magnet, the charge to the mass ratio of the β rays was determined and it appeared to be the same as the charge to the mass ratio of the electrons; that it was concluded that the β ray is in fact composed of electrons.
In 1900 the magnet experiment could not bend α rays; hence it was concluded that α particles do not have charge. During her studies, Marie Curie concluded that α rays are particles as well, and it was speculated that they are also charged particles. However, α particles had not been bent in the previous experiments as they were heavy. In the following year, Rutherford made another setup with a stronger magnetic field, and one year later he reported his results of deflection of α rays both in the electric and magnetic fields.
When atoms get excited beyond their ground state, the excess of their energy is released either as radiation or as a particle. α particles are the heaviest component emitted from an atom. The speed of these particles is approximately 1/10th that of the speed of light. Due to their heavy weight, they cannot penetrate far and can easily be absorbed by materials. β particles on the other hand are lighter. As they are smaller compared to the α particles, they can penetrate better into materials. γ rays, the third possible component of a radioactive emission, are light photons with high energy and large penetration depth that might be harmful for living cells and tissues.
Each of these three emitted particles can be used for the investigation of a radioactive nucleus. This article will describe the idea behind a β-ray spectrometer and its capabilities in characterizing materials.
By 1900, judging from their deflection by a magnetic field, it was clear that β-rays are negatively charged particles. In that year, Becquerel measured the charge-mass ratio of the β-rays and recognized that it is close to that of cathode rays, suggesting that β-rays are electrons, as well. However, β-rays are much faster than cathode rays and their velocity can reach 95% of the speed of light; hence they are relativistic particles. In a typical β-decay, the emitted particles are accompanied by neutrinos (anti-neutrino), which share the energy with β-rays. The presence of other particles in the reaction leads to a continuum distribution in the energy of the β-rays.
Like any other spectrometer this instrument also gives the distribution of particles as a function of their energy (momentum). Among various spectroscopic techniques the β-ray spectroscopy has one of the most versatile configurations; one can modify it via different shapes of the electric and magnetic fields to study different nuclei. Collecting the scattered radiation and increasing the resolution as much as possible were the two main challenges of the method. Typically, the spectrometers can be divided into electric and magnetic ones. Depending on the trajectory of the particles and the force lines, one can have either flat or helical spectrometers in the magnetic category. There, if the field lines are perpendicular to the rays, the particles travel a circular path, while a helical path would be traveled if the field lines are parallel to the rays.
Among very different designs there are semi-circular designs which use a constant magnetic field, and through a homogeneous field deflection one can measure the distribution of the energy of the emitted particles from the source. This method has an advantage over other typical methods in being cheap, easy to set up, and capable of measuring accurately the intensity of the field. Due to the diversity in the designs, in what follows we focus primarily on the semi-circular design of the spectrometer where the deflection and the points show the distribution of the β-ray energies in the radioactive atoms.
The first experiment on β-particles energy was done in 1910 by Baeyer and Hahn. They used the deflection of the emitted beam in a magnetic field to perform the measurement.  In their experiment the emitted rays from a radioactive source, after passing through a slit and a certain distance, were detected by a photographic surface. They realized that there are definite energy lines in the spectrum of the emitted β-rays. 
If the electrons are moving in a magnetic field normal to their plane, then the centrifugal force exerted on the particle will move them on a circular path. Assume that an electron with velocity v is moving in a flat homogeneous magnetic field of B. Then, according to the Lorentz force, one can write the following equation for its path as:
where R is the radius of the circle traveled by electrons in the plane. From the above equation, one can derive the following equation for the momentum:
In β-spectroscopy p is usually defined as a function of BR product. Note that in electrostatic spectroscopy one might relate the energy to the strength of the field; hence the voltage is used. But as the electrostatic cases are rare we stick to the case of magnetic fields and BR value. A magnetic spectrometer determines the distribution of particles as a function of momentum. In other words, the output of a magnetic β-spectrometer is N(p) dp.
If the radius of the motion and the magnetic field are known, the momentum of the electron can be easily determined. In a real measurement setup, one can restrict the direction of exiting electrons from a radioactive source by passing them through a narrow slit. The emitted electrons from the source are then directed in a semi-circular path as they are exposed to a constant magnetic field. For better control, a semi-circular tube guides the electron flow. At the exit end of the tube, the electrons will hit a detector which transforms them into photons. These photons are amplified with a photo-multiplier to increase the signal to noise ratio for further processing.
With a radius curvature of R and after half a revolution, the electron beams reach a minimum at the other side of the circle. However, due to the dispersion of the electron momenta, the electrons reach to different points on the detector. As shown in Fig. 1, only particles with momentum p = eBR will hit the detector at focus; those with momentum less than or greater than this value will miss it. By varying the magnetic field B, the intensity of particles with different momentum can be measured. However there is not any physical aperture that can narrow down the electron beam to a single emission angle. For an aperture with opening angle of 2φ the electrons reach to different points on the detector as shown in Fig. 1. In analogy with a typical imaging system this finite distribution on the detector plane is called aberration and its value is given by the equation
where s is the source width. This finite distribution limits the resolving power of the spectrometer. To overcome this problem, many design schemes have been proposed in literature to focus the electron beam at exit. For an elaborated discussion one might refer to some books as Siegbahn. 
In this work we presented a brief history characterizing the types of particles emitted from an excited nucleus. Later, we described how β-ray spectroscopy works and how it can be used to characterize an unknown atom via the momentum of its radiated particles.
© Hadiseh Alaeian. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
 O. von Baeyer and O. Hahn, "Magnetische Linienspektren von β-Strahlen," Phys. Z. 11, 488, (1910).
 O. van Baeyer, O. Hahn, and L. Meitner, "Über die β-Strahlen des aktiven Niederschlags des Thoriums," Phys. Z. 12, 273, (1911).
 K. Siegbahn, Beta- and Gamma-Ray Spectroscopy (North Holland, 1955).