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Right now I am speaking of lead and tin only.
With out a uniform ratio the plastic region is extended over a broader 
range of temperature. This is where some material is melted and some is not 
on a microscopic scale.
As one materials science book put it, the atoms exchange electrons and 
"loosely" form molecules but in a way that you can call them compounds. 
When the ratio is exactly 3 and the alloy is homogenous the
plastic region is completely eliminated. It is more than a mixture of 
materials, it is an alloy. When you plate lead and tin on a PC board it is 
not an alloy it has to be fused, melted,
so that the materials can diffuse though each other to become a homogenous 
alloy. 
  I did not say that all integer ratios are eutectic, only 3.
At 04:15 PM 9/2/2012 +0000, you wrote:
 
Bill,
 
I recalled that the ratios of lead and tin atoms had to be an integer
to for the lowest melting point.
 
But that's not right either! I hate to be so overly teacherish in this 
thread, but facts remain facts! 
In all possible compositions of tin-lead alloys, only one single ratio is 
eutectic. Exact ratios of atoms such as 1:1, 2:1, 1:2, are NOT eutectic, 
and fully melt at a higher temperature than the eutectic mix. 
Have a look at these two web pages for more detailed information, or 
google for phase diagrams of tin lead solders or other alloys. 
http://www.ami.ac.uk/courses/topics/0244_tsm/index.html
http://spaceflight.esa.int/impress/text/education/Solidification/Phase_Diagrams.html
Note that the eutectic point of tin-lead is given as 61.9/38.1, rather 
than the conventional 63/37. That's why there is so much 60/40 solder 
around. The 63/37 solder is only marginally better. The reason to use 
63/37 rather than 61.9/38.1 is in the inability to make an absolutely 
homogeneous alloy, and the fact that the pasty phase is shorter if the mix 
deviates from the eutectic point toward the tin side, then if it does 
toward the lead side. So, if the tolerance of an alloy happens to be 2%, 
it's likely that a nominal 63/37 mix will statistically end up having a 
shorter pasty range than a nominal 61.9/38.1 mix. 
Manfred
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