# Thermal expansion and lattice constants

#### FrankD

Hi all,

I'm not a physicist, but I need to calculate the lattice parameter of a cubic crystal in dependence of the temperature.

The linear expansion coefficient in dependence of Temperature is given by the general formula: $$\displaystyle \alpha(T) = A+B*T+C*T^{2}+C*T^{3}+D*T^{4}$$ where A,B, C, D are constants.

I have to calculate the lattice constant for various Temperatures between 300 and 1000K. I tried with a big spreadsheet through calculating the values for very small Temperature steps but I know there must be a more elegant way (maybe integrating?) but I'm not good at that so i would be glad if somebody can help me out here.

#### Woody

A bit of clarification

Are you trying to determine the coefficients of the general formula based on experimental results?

Are the quadratic and cubic coefficients of your quartic polynomial really supposed to be the same?

Are you using an EXCEL spread-sheet

If the answers to those questions are yes, no and yes respectively,
Look into using the "LINEST" function,
its a bit awkward and takes some careful thought to understand the "help" instructions,
but it is extremely useful when analysing experimental work, if you can master it.

#### benit13

Hi all,

I'm not a physicist, but I need to calculate the lattice parameter of a cubic crystal in dependence of the temperature.

The linear expansion coefficient in dependence of Temperature is given by the general formula: $$\displaystyle \alpha(T) = A+B*T+C*T^{2}+C*T^{3}+D*T^{4}$$ where A,B, C, D are constants.

I have to calculate the lattice constant for various Temperatures between 300 and 1000K. I tried with a big spreadsheet through calculating the values for very small Temperature steps but I know there must be a more elegant way (maybe integrating?) but I'm not good at that so i would be glad if somebody can help me out here.
The method you want to investigate is called "Least Squares Regression". This is the technique usually employed for curve fitting. Depending on the software you are using, you should have available to you some kind of curve fitting tool.

It requires a plug-in, but it should be able to help achieve what you're looking for.

I usually use python and numpy for tasks like that, so give me a shout if you want to try that instead.