Dear Masoud

As you know, the concentration of vacancies in materials have an exponential relationship with actual temperature. So, if the activation energy for the formation of one mole vacancies is  known for a specific material, you can calculate the ratio of vacancies concentration at different temperature by the attached relationship.

It should be noted that in this equation, it was assumed that no vacancy relaxation happens during rapid solidification.

Dear Mahmmod

thank you for devoting your time to considering my question. therefore, theoretically  equilibrium or non-equilibrium conditions do not effect on vacancy concentration!!!

Dear Masoud

As you know. the mentioned relationship is used between 0 K to melting point of the materials. So, if 800 K is the rapid solidification ending temperature, I think you can used this relation to calculate the vacancy concentration with this assumption that no vacancy relaxation happens during cooling. In other words, this relation specify the upper limit of the vacancy concentration. Gunter Gottstein in his book "Physical Foundation of Materials Science" represented a method for measuring the activation energy of the vacancy formation. He stated that in this method, the material is cooled from different temperatures rapidly and the resistivilty of the material is measured. The plot of the resistivity vs Temperature by and Arrhenius plot can be used for calculating the activation energy. Because the resistivily is proportional to the vacancy concentration, So, they assume that the rapid solidification retains the vacancy from high temperatures.

Best regards

M. Sharifitabar

Dear Mahmood

thank you for your comprehensive answers.

regards

masoud