SCIENCE

The Mathematics of Mass Spectrometry - An Even Closer Look

Mass spectrometers are analytical instruments that determine atomic and molecular masses with great accuracy. Low-pressure vapors of elements or molecules are hit by a beam of rapidly moving electrons. The collision knocks an electron off the sample atom or molecule, leaving it positively charged.

These newly-formed ions are accelerated out of the ionization chamber by an electric field. The speeds to which the ions can be accelerated by the electric field are determined by their masses. Lighter ions can go faster than heavier ones.

 Ion's path bent by external magnetic field Courtesy: McREL

The beam of positively-charged ions generates a slight magnetic field that interacts with an externally-applied magnetic field. The net result is that the trajectory of a charged particle is curved to an extent that depends on its speed (determined by its mass). When the beam of a mixture of isotopes of different masses falls on a photographic plate, the different isotopes converge at different points, corresponding to the different radii of their semicircular paths.

The mathematical equation that describes this phenomenon is: m/e = H2 r2 /2V, where m is the mass of the ion, e is the charge of the ion, H is the magnetic field strength, r is the radius of the semicircle, and V is the accelerating potential.

The radius of the semicircular path is proportional to the mass of the particle. A photographically-detected mass spectrum of natural germanium obtained under conditions of constant magnetic field and accelerating potential would look like the figure shown below:

 Atomic mass units

The relative abundances of the ions can be determined from the densities of the photographic images they produce.

On most spectrometer graphs the mass in amu's or atomic mass units is plotted on the x-axis and relative intensity of the ion stream is plotted on the y-axis. The relative intensity is a measure of how many ions of any given mass are detected.

With modern low-resolution instruments, one can expect to obtain an accuracy level of plus or minus 1 amu up to 1000 atomic mass units. High-resolution instruments can provide an accuracy of 2-3 parts per million (ppm) up to masses of 3000 amu.

Since H, V, and r can be controlled experimentally, the ratio m/e can be determined in one of several fashions. In a typical modern mass spectrometric experiment the strength of the external magnet is slowly varied, causing the paths of the various particles to sweep past an exit point where the mass spectrometer's ion detector is located. In other words, a signal is produced at the detector when the magnetic field is just strong enough to bend the pathway of charged particles of a given mass so that they arrive at the detector. The mass of the ion detected is then calculated from the accelerating voltage and the strength of the magnetic field required to produce the signal.

The mass spectrum is a graph of intensity of detector signal versus magnetic field strength. The positions of the peaks on the graph are used to calculate the masses of the ions, and the relative heights of the peaks indicate the relative proportions of the ions of various types that were in the sample. If the ion has a charge of one, then the mass is calculated easily. Mathematically, a double-positively-charged ion of mass 64 gives the same value of m/e as a single-positively-charged ion of mass 32. Under the usual conditions for operating a mass spectrometer most of the ions produced are singly charged.

A second technique would be to vary the accelerating voltage while keeping the magnetic field constant. This has the same result of sweeping the ion paths past the detector, leading once again to a spectrum that can be analyzed for ion masses and relative abundances.

A third type of mass spectrometry is time-of-flight measurement in which the ions are produced in spurts and allowed to diffuse toward the detector in a straight line. No magnetic field is used, so the ions continue forward in a single direction. The heavy ions move more slowly than the light ones, so they arrive at the detector later. The mass spectrum is a plot of intensity versus time of flight of the particle.

Interpreting the data from a mass spectrograph is only one of many examples of the uses of algebraic equation-solving in today's scientific world.

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