Ab initio molecular dynamics calculations have been carried out to search for the ground state structure of Fe_(n)Ti_(13-n)clusters and measure the thermal expansion of Fe_(n)Ti_(13-n).The volume of Fe_(n)Ti_(13-n)clu...Ab initio molecular dynamics calculations have been carried out to search for the ground state structure of Fe_(n)Ti_(13-n)clusters and measure the thermal expansion of Fe_(n)Ti_(13-n).The volume of Fe_(n)Ti_(13-n)clusters during thermal expansion is jointly determined by anharmonic interaction and magneto-volume effect.It has been found that Fe_(6)Ti_(7),Fe_9Ti_(4),Fe_(11)Ti_(2),and Fe_(13)clusters can exhibit the remarkable magneto-volume effect with abnormal volume behaviors and magnetic moment behaviors during thermal expansion.A prerequisite for the magneto-volume effect of Fe_(n)Ti_(13-n)clusters during thermal expansion has been revealed and the magnitude of the magneto-volume is also approximately determined.Furthermore,the magneto-volume behaviors of Fe_(n)Ti_(13-n)clusters are qualitatively characterized by the energy contour map.Our results shed light on the mechanism of the magneto-volume effect in Fe_(n)Ti_(13-n)clusters during thermal expansion,which can guide the design of nanomaterials with zero expansion or even controllable expansion properties.展开更多
This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An...This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.展开更多
基金the support from the National Natural Science Foundation of China(Grant No.52171038)key R&D projects in Shandong Province(Grant No.2021SFGC1001)+1 种基金supported by the Special Funding in the Project of the Taishan Scholar Construction Engineering and the program of Jinan Science and Technology Bureau(Grant No.2020GXRC019)new material demonstration platform construction project from Ministry of Industry and Information Technology of China(Grant No.2020-370104-34-03-043952-01-11)。
文摘Ab initio molecular dynamics calculations have been carried out to search for the ground state structure of Fe_(n)Ti_(13-n)clusters and measure the thermal expansion of Fe_(n)Ti_(13-n).The volume of Fe_(n)Ti_(13-n)clusters during thermal expansion is jointly determined by anharmonic interaction and magneto-volume effect.It has been found that Fe_(6)Ti_(7),Fe_9Ti_(4),Fe_(11)Ti_(2),and Fe_(13)clusters can exhibit the remarkable magneto-volume effect with abnormal volume behaviors and magnetic moment behaviors during thermal expansion.A prerequisite for the magneto-volume effect of Fe_(n)Ti_(13-n)clusters during thermal expansion has been revealed and the magnitude of the magneto-volume is also approximately determined.Furthermore,the magneto-volume behaviors of Fe_(n)Ti_(13-n)clusters are qualitatively characterized by the energy contour map.Our results shed light on the mechanism of the magneto-volume effect in Fe_(n)Ti_(13-n)clusters during thermal expansion,which can guide the design of nanomaterials with zero expansion or even controllable expansion properties.
文摘This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.