Why, precisely, is argon used in neutrino experiments? Why is argon used in neutrino detectors?  Other than liquid argon being denser than water or oil, what are its advantages?
 A: I think we're getting confused. Using Chlorine to trigger transitions to Argon is a way of detecting neutrinos are present.
Using a volume of liquid Noble element to detect the passage of charged particles is a technique of calorimetry. In my old experiment (NA48) we used a liquid Krypton calorimeter. 


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*A denser medium gives more ionisation per unit length (hence greater resolution).

*Electrons liberated by ionisation are able to drift long distances in a noble element (this is how we detect the ionisation). 


So hundreds of litres of liquid Kr provides a very dense medium that is good for tracking ionising particles.
However, liquid krypton is expensive - six times more expensive than single malt whisky. Luckily, we had a Russian team on our collaboration. In the 90s, the Russians had no money, but they did have plenty of raw materials, so they supplied the LKr (it's a by-product of producing liquid O2 for the steel industry).
A: There are many different types of neutrino detectors, using various techniques to turn a neutrino interaction into an electrical signal. Detectors that use a large quantity of water or oil, like Super-Kamiokande in Japan and MiniBooNE at Fermilab, are Cherenkov detectors. Their basic principle of operation is as follows: when neutrinos pass through the liquid and interact with it, they produce leptons that are energetic enough to move faster than the speed of light in the medium. When this happens, the leptons emit a cone of Cherenkov radiation, which is detected by photomultiplier tubes surrounding the medium; later, the intensity and relative timing of the light pulses are transformed into a reconstruction of the track of the charged lepton. 
The neutrino detectors using liquid argon are called Liquid Argon Time Projection Chambers, or LArTPCs, and work in a different way. These detectors consist of a large volume of liquid argon, with a grid of high-voltage bare wires passing through the liquid and a set of scintillators and photomultiplier tubes surrounding the liquid. When a neutrino interacts with an argon atom, it, as before, creates a high-energy charged lepton. This charged lepton ionizes other argon atoms as it passes through the liquid, leaving a trail of free electrons and argon ions in its wake. The electrons are drawn into the wires by the strong electric field and register as a pulse of current in a particular wire. Since liquid argon's electronegativity is so low, it's essentially transparent to these slower-moving electrons. A free electron typically hits the wire that it's closest to, and the further away it is from the wire grid, the longer it takes to hit it. So by analyzing which wires received current pulses as a function of time, it is possible to reconstruct the track of the charged lepton. This is not possible with water- or oil-based detectors because both are too electronegative - the free electrons will react with the medium before reaching the wire.
The high-energy charged particle also causes the argon itself to act as a scintillator in the far ultraviolet (with a wavelength of around 128 nm). The argon is essentially transparent to the scintillation light, so it's received by the scintillators on the walls, which convert the ultraviolet radiation to visible light, which the photomultiplier tubes are far more efficient at amplifying. This extra information can also be used for reconstruction of the hgih-energy charged lepton track. Liquid argon can also be used as a Cherenkov detector, since it has a similar refractive index as water (1.24 versus water's 1.33); it's this triple redundancy in the signal which has made liquid argon TPCs so appealing.
There are several other advantages to using liquid argon as opposed to water or oil. Liquid argon is, as you remarked, denser than either water or oil, which leads to a higher interaction frequency for the same incident neutrino intensity, which speeds up the data-taking rate (given that neutrino experiments have a very slow data-taking rate, this is the main bottleneck that determines how long an experiment must be run to collect a statistically-useful dataset). Argon is also a noble element, which means that it doesn't tend to interact with either impurities or the container it's stored in, so it's also somewhat easier to purify and keep clean, reducing the background that analyses have to account for and making it easier to see a signal.
Given that all of the above advantages also apply to the heavier noble gases such as krypton and xenon, you may be wondering why argon is used instead of those. The answer is simple: argon is much cheaper than the heavier noble gases, which allows you to build a bigger detector for the same budget.
A: There is a very specific reaction between a chlorine nucleus and a neutrino which produces an argon nucleus.  By detecting the production of argon in a very large volume of chlorine, the presence of a neutrino can be deduced. This means that an extremely sensitive neutrino detector can be designed around a great volume of chlorine which is periodically swept through an argon detector. 
