Hakeem, the simplest experimental way to find the (linear or volume) coefficients of thermal expansion for any solid material is TMA or Thermal Mechanical Analysis. Small (a few mm in size) single crystals are ideal samples for such TMA measurements. One needs (1) to put a sample onto a TMA instrument measuring platform, (2) lower the instrument pirex probe having flat tip onto the sample surface under a minimal applied static load (a few mN are sufficient); (3) to place the assembly into the TMA furnace; (4) to select the heating program and (5) to accomplish the heating run. The so obtained TMA-curve (usually – a line steadily raising with temperature respective the horizontal baseline) is all one needs for calculating the expansion coefficients α (T1-Tref.). The average expansion coefficient for the temperature interval Tref ...T1 is mathematically defined as: α (T1-Tref.) = [dL/L0 (T1) - dL/L0 (Tref.)] / (T1 – Tref.), where L0 is the sample length measured at ambient temperature; T1 = upper temperature limit; Tref. = lower temperature limit named as reference temperature; dL/L0 – is the sample length (or thickness) change relative to L0.

Excellent TMA instruments (with nanometers linear displacement resolution) are produced for many years by, e.g., Perkin-Elmer, TI Instruments, Netzsch and others. Nowadays the TMA makers software usually permits to extract a lot of useful info (physical alpha curve and values, technical alpha curve and values, etc.) from the single TMA run. Also, the TMA curve will register the phase transitions (if any) within the sample in the temp range chosen. Good Luck!

Powder XRD as a function of temperature is the best and most reliable method to determine the thermal expansion coefficient .

The best way to have thermal expansion is the X-ray diffraction method, But at 50 K, it is some how difficult and expensive. In this range this phenomena is not linear with temperature as it is for temperatures above mostly 150K. Then theoretical calculations may be is more suitable but still accurate results can not obtained for the reason I mentioned for this range of temperature. But you may get approximate value. For this purpose I suggest to read the article published by L. G. Van Uilert, J. Appl. Phys. 50(12) 8052(1979) which may give answers for problem

Prof. Dr. M S Omar

As a possibly inexpensive way of finding the information you are looking for, try the Chemistry Data book on NIST website http://webbook.nist.gov/chemistry/, or Knovel Database at Knovel.com.

Hakeem, the simplest experimental way to find the (linear or volume) coefficients of thermal expansion for any solid material is TMA or Thermal Mechanical Analysis. Small (a few mm in size) single crystals are ideal samples for such TMA measurements. One needs (1) to put a sample onto a TMA instrument measuring platform, (2) lower the instrument pirex probe having flat tip onto the sample surface under a minimal applied static load (a few mN are sufficient); (3) to place the assembly into the TMA furnace; (4) to select the heating program and (5) to accomplish the heating run. The so obtained TMA-curve (usually – a line steadily raising with temperature respective the horizontal baseline) is all one needs for calculating the expansion coefficients α (T1-Tref.). The average expansion coefficient for the temperature interval Tref ...T1 is mathematically defined as: α (T1-Tref.) = [dL/L0 (T1) - dL/L0 (Tref.)] / (T1 – Tref.), where L0 is the sample length measured at ambient temperature; T1 = upper temperature limit; Tref. = lower temperature limit named as reference temperature; dL/L0 – is the sample length (or thickness) change relative to L0.

Excellent TMA instruments (with nanometers linear displacement resolution) are produced for many years by, e.g., Perkin-Elmer, TI Instruments, Netzsch and others. Nowadays the TMA makers software usually permits to extract a lot of useful info (physical alpha curve and values, technical alpha curve and values, etc.) from the single TMA run. Also, the TMA curve will register the phase transitions (if any) within the sample in the temp range chosen. Good Luck!

You can get the total expansion coefficient by using Thermomechanical Analysis. Use the quartz piston cell attachment. Run a blank in finely divided alumina. Then, push in several reasonably large crystals into the alumina and run again. You will measure the sum of the a,b,c-axies expansion. The individual crystal axis expansions can be measured on a hot stage with a filar micrometer on a good polarizing microscope. Tedious but accurate. If you have access to an X-ray diffractometer with a heated cell the expansion coefficients can be obtained from changes in the unit cells. Crystals in the Cubic system are much easier than non Cubic as a=b=c. Thermomechanical Analysis is probably best for your purpose as you want the sum of a,b,c

Good morning

Experimentally it is a trick measurement. In order to give you a good answer I have to make you a question, In which temperaure range to you need the thermal expansion coeficient?. Do you need it over the melting temperature also?

I agree with the answer of Alexander Shchegolikhin, I do not think RXD could be better for this research, while you could see how all the parameters of the substance are changing while you are increasing the temperature.

It depends what you want. The nanoscopic scale or the microscopic scale. If you need the nanoscale behavior of your material (lattice parameter), you can use the XRD method. But if you need the micro or macroscopic scale of the thermal expansion the best way is the TMA. This latter will give you the linear thermal expansion. For the XRd method you can get the linear and the volume thermal dilatation also.

I agree with the answer of my colleague Carlos Gracia-Fernández

"In which temperaure range to you need the thermal expansion coeficient?.

check also my answer to https://www.researchgate.net/post/How_to_measure_the_density_of_a_inorganic_salt_say_NaNO3_from_room_temperature_to_its_melting_point_at_intervals_say_every_10_K_experimentally2

The answer largely depends if you want the value for the material (intrinsic property) or for a given specimen. You can easily sort the suggestions you receive in one of the two categories: XRD give the intrinsic values, dilatometry the average ones.

I would however follow the suggestion of Norberto and simply check the literature as most of this stuff is already published

Thermal expansion on melting needs the information of the temperature dependence of the density of the solid phase as well as that of the liquid phase. TMA is suitable for solid, but not for liquid. For liquid, Archimedian method or Maximum bubble pressure method can be used to estimate the temperature dependence.

Electrical and thermal resistivities and thermal expansion coefficient of NaNO3 with standard four-probe, absolute steady-state, and quartz dilatometer techniques. Measurementsshould be performed at range of temperaturetes

Good answer and very complete for this problem of Hatem Maraqah

Dear Lazaro

Thank you very much for your comment on my answer please press the green button to upvote the answer

Thank you again Lazardo

XRD studies at different temperatures as long as it is a solid give us an idea about the change of cell parameters from which thermal expansion ca be found out.

You can use dilatometry in thermal analysis

perhaps using a densimeter and then by fitting v =f(T)

I agree with the answer of Matteo Leoni that : XRD give the intrinsic values, dilatometry the average ones.

x-ray or neutron powder diffraction from very low temperatures to twice the Debye

temperature can provide lattice parameters and interplanar angles (for lower symmetry materials). These parameters allow calculation of true thermal expansion coefficients of materials with symmetries as low as triclinic. The data obtained can be fit by a model utilizing several Einstein frequencies to get an approximate lattice vibration distribution. In the absence of appreciable defects higher temperature thermal expansion may be relatively simply calculated with simple lattice theory. There are many data compilations out there and many experimental errors in the data evaluated. Higher temperature measurements are often difficult to control so quality thermal expansion measurements at very high temperatures are available but are not the norm. With decomposing solids one must also control the fugacities of appropriate gases so that thermodynamic equilibrium is maintained during measurements.

Thermal expansion is described by a second rang tensor, that means six independent coefficients. Thermal expansion would be isotropic (1 single coefficient) if NaNO3 was cubic, but it is trigonal and this implies the change of this property with direction (anisotropy). For trigonal crystals the second rang tensor has two independent coefficients, and they could be determined from XRD measurements at different temperatures. Of course, the liquid phase will have only one coefficient as an isotropic medium.

Thanks a lot to all for the valuable inputs

@ Carlos Gracia-Fernández

I need in the solid region, melting mushy zone and liquid region also.

@ Joonho Lee

Thanks for the input about finding the value in the liquid region.

What is your working temperature intervall ?

Dear Engineer,

In your case of study, just TMA would give you much more accurate results while material phases are changing.

Matteo Leoni got it right! NDE XRD all the way from liquid He to above melting point of the inorganic compound with potentially Femtometer precision in real time. Observe the Nano structure in real time with XRD Microscopy! Refer to: http://www.flickr.com/photos/85210325@N04/10515372183/

@Belaribi Farid - Working Temperature Interval is 10 K

There are three approaches from the simple side of things:

1. By far the most cost-effective is going to be horizontal pushrod LVDT dilatometry. In this case, you take a sizable sample (typically a circular cylinder of 1-5cm in length) and measure the thermal expansion relative to the expansion of a known standard (typically sapphire, sometimes stainless steel). Benefits are cost, simplicity, and precision (resolution is comparable with TMA but much larger samples may be used allowing greater precision in thermal expansion calculation). Drawbacks include a relatively lengthy test time (5°C/min ramp rate is the maximum usable without introducing significant error caused by thermal gradients in the sample) and a relatively high applied force and pressure, which can artificially depress dilatometric softening point temperatures. C-Therm offers industry-leading precision in horizontal dilatometry.

 2. The best for measuring softening points and deformable samples is going to be thermomechanical analysis (TMA). TMA is a vertical push-rod dilatometry method which allows for extremely low contact force (essentially 0 force contact). This allows the most accurate determination of softening point and measurement of soft samples. SETARAM offers an excellent line of TMAs. Cost is roughly twice that of a horizontal dilatometer.

3. Top-of-the-line precision would be to use an optical horizontal push-rod system based on Michelson interferometry. Such a system has extremely fine resolution (on the order of 0.1 nm) but can take the large sample size of a traditional horizontal dilatometer, resulting in very good precision in the determination of the coefficient of thermal expansion. I do know a few groups working in this field, but it is an emerging technology. 

XRD as a powder photography is the best way to calculate lattice thermal expansion for crystaline solid inorganic materials. That's by taking XR powder films in different temperatures. The lattices are in this case is a temperature dependent and one can calculate lattice thermal expansion 

the simplest experimental way to find the (CTE) coefficients of thermal expansion for any solid material is TMA or Thermal Mechanical Analysis. Small (a few micro meter in size).