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Let's look at the construction of the tube. The Geiger-Müller (GM) tube is essentially a cylinder with a "wire" down the middle for an anode, and the inside of the cylindrical housing as the cathode. It's got some gas in it, and the type of gas will vary a bit from tube to tube and what is desired in the design. The "end" has a "window" in it made of thin glass or possibly mica. This window "lets in" the radiation while sealing the tube and providing a minimum amount of shielding that might "block" radiation, particularly alpha and beta radiation. Recall that alpha particles are helium-4 nuclei, and they can't penetrate a sheet of paper. Beta particles are high energy electrons, or possibly positrons, which can be stopped by a sheet of aluminum foil. You can't detect them if you block them. The way it works is as the radiation passes through the GM tube, it leaves an ionized "trail" in the gas behind it. This is primary ionization. The high voltage across the gas causes the ions to be accelerated toward the appropriately charged element. Positive charges move toward the cathode, and negative charges (electrons) move toward the anode. The movement of these charges ionizes other gas atoms, called secondary ionization, and the total effect is to create a current avalanche. With the movement of these charges and the accompanying current "spike" or "jolt" set off by the high voltage, we'll observe a "pulse" that the supporting circuitry in the detector can "see" and a "hit" or "count" is recorded by the Geiger counter. Gamma rays are penetrating types of (electromagnetic) radiation. They blast through the window and they ionize the heck out of the gas inside. It is basically the cumulative effect of all this ionization that creates sufficient ions to initiate the current avalanche that cause the counter to "pulse" electrically. The gamma rays have a field day zapping their way through the cylinder and creating lots ions to create a "click" or a count. In contrast, alpha and beta particles will not penetrate very far into the tube because of their limited ability to do so. This means that the current avalanche is more confined to the "front" of the tube. The superior penetrating power of the gamma rays means that their current avalanche includes a lot of volume deeper in the tube, or more "in the middle" than at the end of the tube, like the particulate radiation. Links can be found below for more information.

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Q: Why should gamma rays be detected at a point within the Geiger-Muller tube rather that at the end of it?
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