Laboratory Report Materials Physics Laboratory
The determination of the curie temperature of the Barium Titanate Yufei Chang • Group H
Abstract The aim of this experiment is to measure the relative permittivity (εr) of a BaTiO₃ ceramic at a series of temperature. and using these data to obtain Curie temperature. Another aim is to observe the microstructure of BaTiO₃ which its microstructure has been printed out.
Introduction The Curie temperature is the temperature above which ferroelectric behaviour is lost.At the temperature below the Curie point the magnetic moments are partially aligned within magnetic domains in ferromagnetic materials.As the temperature is increased from below the Curie point, thermal fluctuations increasingly destroy this alignment, after it reached and above the critical Curie temperature, dielectric and ,consequently, ferroelectric behaviour are lost. And the materials is purely paramagnetic. In some materials , such like Barium Titanate, the Curie temperature corresponds to a change in crystal structure from the distorted tetragonal structure to a normal cubic perovskite unit cell. The group of dielectric materials called ferroelectrics exhibit spontaneous polarization .They are the dielectric analogue of ferromagnetic materials,which may display permanent magnetic behavior.There must exist in ferroelectrics materials permanent electric dipoles.In this experiment the dielectric material which we used (BaTiO₃) is the most common ferroelectrics. The spontaneous polarisation is a consequence of the positioning of the Ba²⁺,Ti⁴⁺ and O²⁻ ions within the unit cell.And this will be discussed later in the discussion part.
Theory for this part Capacitance of a parallel plate condenser is given by: C=Aε0εr /d Rearrange this equation we get: εr =Cd/ε0 A Where C is the capacitance, A is the area of the plate, d is the distance between the two plates, εr is the relative permittivity of BaTiO3, and ε0 is the permittivity of free space(8.542 x 10-12 Fm-1) If we now apply a voltage to this capacitor then the charge stored, Q, is given by:
Which C is the capacitance, Q is charge stored, and V is the voltage applied. Q/A=P=CV/A P=εrε0V/d Where P is the polarization term, it presenting dipole moment per unit volume of material. Above the transformation temperature the relative permittivity of the material
obeys a law known as the Curie-Weiss Law. This is stated in the following equation. εr=A/(T-Tc)c Where T is the temperature of the sample in Kelvin, and Tc is the Curie temperature. This relationship implies a singularity in the relative permittivity at T=Tc and that is should vary inversely with temperature in the paraelectric phase. Re-write this equation: 1/εr= (1/A) (T-Tc) This means by plotting 1/εr-1 we could obtain a straight line.
Experimental We record the diameter from the label on the equipment. Then there are blue and red wires for us to choose, which for measuring capacitance of Alumina and Barium Titantate respectively. This component analyzer allows the measurement of the capacitance at room temperature. We can read the capacitance from the indicator. Then rose temperature gradually. Each time we change temperature, we record its capacitance.After we collected enough data up to 170C° ,plot two graphs. First one is to plot relative permittivity vs. temperature. Second one is to plotεr -1 against temperature.
Results and Calculations εr=Cd /Aε0 Where A=cross section of BaTiO3 d= thickness of sample εr= permittivity of free space =8.8542*10-12 C=capacitance At room temperature=23oC: For BaTiO3: εr =2.1 x 10-9 x 1.81 x 10-3/ (8.8542 x 10-12 x (3.14 x (10.755/2) x 10-3)2) =5427Fm-1 For series of temperature: Taking a sample of 30oC, εr=2.46 x 10-9 x 1.835 x 10-3/ (8.8542 x 10-12 x (3.14 x (10.79/2) x 10-3)2) =5515Fm-11 Then by repeating the above calculation, the table is obtained as attached in appendix. Plot as follows
fig.1
fig.2
The second graph (figure 2) is to plot Îľr-1 versus temperature (K) in the Paraelectric phase (T>Curie temperature). Above the transformation temperature the relative permittivity of material obeys Curie-Weiss Law:
εr = A/ (T-Tc) where T = temperature of the sample Tc= Curie temperature Arranging the formula: 1/εr = (T-Tc)/A 1/εr versus T is plotted on graph 2 and by finding the gradient we can find constant A. 1/A=gradient =2 x 10-6, therefore A=500000 This is the result by finding the gradient using EXCEL, so there is a deviation from the result we got by plotting roughly in a graph paper. Which is A=2636536.
Discussion According to this experiment result, fig.1. We could simply find out the Curie temperature is a critical boundary.The capacitance keeps increasing before this point and goes down rapidly after that.Before temperature reached the Curie point. The Ba²⁺ ions are located at the corners of the unit cell, which is of tetragonal symmetry,the dipole moment results from the relative displacement of the Ti⁴⁺ and O²⁻ ions from their symmetrical positions.The O²⁻ ions are located near, but slightly below, the centers of each of the six faces. Thus , a permanent ionic dipole moment is associated with each unit cell. However, when barium titanate is heated above its ferroelectric curie temperature, the unit cell becomes cubic, and all ions assume symmetric positions within the cubic unit cell; the material now has a perovskite crystal structure, and the ferroelectric behavior ceases. At high temperature, too much energy in the mate4+ rial, Ti becomes disordered displacement. This result can also be observed in the figure 2, when above Curie temperature, it becomes a linear relationship between temperature and .εr -1. In this situation, the temperature difference between temperature applied and the Curie temperature could be considered as a driving force to disorder the material. A recently developed theory concerned with the size effect of the free energy related to the polarization, shows that small crystallites (mostly single domained ) have large depolarization energy.Because of that, the total free energy has a higher value. The multi-domain structure of the larger crystallites, on the other hand ,reduce the depolarization energy and thus the total free energy has lower value and the tetragonal structure is stable. Curie temperature obtained in lab is 118℃ which is lower than the theoretical value 130℃. This is due to the presence of defects and impurities. Since defect increase its free energy inside, less energy (∆G) is required to transform from tetrahedral structure (ferroelectric) to cubic structure (Paraelectric) by heating up the material. Obviously tetrahedral structure has lower free energy than cubic struc-
ture. The most unavoidable error is human error. The value of capacitance changes continuously as the temperature changes. Errors can be introduced in the reading process. Calculating and plotting data introduces errors as well, i.e. eliminating digit numbers, estimating best fitted line. In fact, a sharp Curie temperature can not be obtained as there is no defect free material. For a material, not all domains require the same amount of energy to become unpinned. Also, there are local chemistry variations which mean that there is a distribution of curie temperatures within the material. Therefore, the transition from tetragonal to cubic is not instantaneous throughout the entire material. In the microstructure of BaTiO3, there are voids (pores) and equal size grains. The presence of voids (pores) can decrease Curie temperature as mentioned before. The size of grain is also a determinant of Curie temperature.
conclusion Curie temperature of BaTiO3 is 118℃ which is lower than the expected value due to presence of defects and impurities. Grain size can affect the Curie temperature. As a result, dope could be used for control the Curie temperature, so that this kind of materials could be used for different applications by changing its Curie temperature.
References 1 http://en.wikipedia.org/wiki/Curie_point
2 http://www.xray.cz/ms/bul99-2/marinkovic.pdf
3 lab script 4 MSE 205B note 5 materials science and engineering an introduction William D.Callister