# Geiger Müller Design Notes

**1. Proportional Counter Theory in a Nutshell**

Figure 1.1 shows proportional counter theory in a nutshell. The main component is a gas filled Geiger Müller tube with a conducting outer cathode wall and a long anode wire down the middle, held at a high potential. As an ionizing particle passes through the tube it creates an ion trail which is dissipated into the surrounding circuitry as an electrical pulse.

Figure 1.1: Proportional Counter Theory in a Nut Shell. (Figure adapted from Gedeon [1].) |

The electrical pulse can be pulled off either the anode, Figure 1.1, or the cathode, Figure 1.2, of the tube (for a detailed discussion about the pros and cons see Centr☢nic's Geiger Müller Tubes guide [2]).

Figure 1.2: Cathode Method of Signal Capture for a Geiger Müller Tube (left; adapted from Centr☢nic [2]) in Proportional Mode (right; adapted from Fernow [3]). |

If the voltage across the tube is set appropriately, then the output pulse-height from the tube will be proportional to the energy deposited by the particle traversing the tube (*AKA*, the *dE/dx* loss of the particle): *i.e.*, the magnitude of the induced voltage signal goes as,

(cf., Figure 1.2.), | (1.1) |

where *e* is the fundamental electric charge, *ΔN* is the number of created ion pairs, *M* is the multiplication factor, *C* is the intrinsic capacitance of the tube, and *L* is the length of the tube (see chapter 9 of Fernow [3] for a more detailed theoretical discussion). The multiplication factor is dependent on gas mixture, pressure, *P*, temperature, *T*, and applied voltage. For our specific application, the tube's operating temperature is expected to vary, and, as such, its properties will need to be characterized as a function of temperature (*cf.*, Figure 9.5 of Fernow).

In proportional mode the number of created ion pairs is directly proportional to the energy loss of an ionizing particle as it traverses the tube: *i.e.*, *ΔN ∝ ΔE=∫ _{tube}(dE/dx) dx* … and so we have the relation

, | (1.2) |

which completes our description of Figure 1.2. (… well, OK, *dE/dx* is also a function of particle species.) So our tube response is experimentally determined by fitting

, | (1.3) |

with slope, *m _{T,p}* and intercept,

*b*, for fixed temperature,

_{T,p}*T*, and particle species,

*p*[1, 3]. In addition, if the particle is stopped in the tube then

, | (1.4) |

*V*, and tube starting voltage,

_{tube}*V*[2], … and so we can do limited particle spectroscopy.

_{start}As *ΔE* is path dependent, there is a spread in the spectrum due to the geometry of the tube. Therefore, if we are trying to measure the energy of a particle, *E < (ΔE) _{max}*, we must take into account the path lengths, the

*Δx*'s, of the loci of rays passing through the intersection of the bisecting plane of the tube:

*i.e.*, in cylindrical coordinates,

*E = ΔE(φ,z)*That said, this problem can be remedied by adding bits of material around the tube, such that,

_{tube}≠ "*Constant*."*ΔE(φ,z)*Geiger Müller tubes with this property are called energy compensated tubes. (Note that such tubes also have an energy window,

_{material}+ ΔE(φ,z)_{tube}= "*Constant*."*(ΔE)*, due to the cutoff energy of the material,

_{min}< E < (ΔE)_{max}*(ΔE)*, below which the particles are stopped.) Figure 1.3 shows an example of a radiation pattern for an energy compensated tube.

_{min}Figure 1.3: Example of E(φ) for an Energy Compensated Geiger Müller Tube. (Figure adapted from Centr☢nic [2].) |

*References:*

[1] G. Gedeon, *Radiation Detectors for Industrial Facility Systems (Course No: D06-001)*, CEDEngineering.Com, Continuing Education and Development, Inc., 9 Greyridge Farm Court, Stony Point, NY 10980.

[2] Geiger Müller Tubes, Centr☢nic.

[3] R. Fernow, *Introduction to Experimental Particle Physics*, Cambridge University Press, 1986.

*Last Updated: 8:14 PM EST Oct. 30 ^{th}, 2014*