Computing Elastic Constants 2

Manual 08 for MD++
Computing elastic constants at nonzero temperature

Keonwook Kang, Seunghwa Ryu and Wei Cai

Jul 08 , 2009

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In the previous manual, the elastic constants of bulk silicon of diamond cubic structure are computed at 0 K. In this manual, we compute the elastic constants at nonzero finite temperature using fluctuation-dissipation theorem.

${\begin{array}{rcl}\mathbf {P} &\equiv &\sum _{i}m_{i}\mathbf {v} _{i}=\mathrm {Const.} \\\mathbf {L} &\equiv &\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\mathrm {Const.} \end{array}},$

Table 1. Elastic constants (GPa) of DC Si. Potential Model : SW

T (K)	C11	C22	C33	C12	C13	C23	C44	C55	C66
300	149.41	149.36	149.45	76.26	76.18	76.24	62.45	75.00	54.93
625	146.35	146.29	146.35	75.80	75.77	75.71	53.72	50.03	46.98
888	142.74	142.86	142.84	75.07	75.17	74.99	60.78	53.54	56.76
1164	139.11	139.06	139.29	74.28	74.33	74.27	47.30	60.86	51.17
1477	135.02	134.84	134.69	73.84	73.85	73.80	44.96	41.41	47.66

Table 2. Elastic constants (GPa) of FCC Au. Potential Model : EAM Foiles

T (K)	C11	C22	C33	C12	C13	C23	C44	C55	C66
170	177.48	177.45	177.53	151.05	150.80	151.04	36.32	36.34	36.41
300	170.72	170.25	170.44	146.01	146.00	146.06	34.16	34.07	34.20
600	152.37	152.40	151.99	132.33	132.30	131.92	29.05	29.35	29.10
900	132.23	132.02	132.78	118.10	117.71	117.08	23.90	23.44	23.99
1200	107.14	105.48	106.68	98.54	99.04	98.22	17.82	17.70	17.46

Table 3. Elastic constants (GPa) of FCC Cu. Potential Model : EAM Mishin

T (K)	C11	C22	C33	C12	C13	C23	C44	C55	C66
315	164.06	164.05	163.99	119.99	119.93	119.90	71.60	71.69	71.69
600	153.01	153.18	153.07	113.20	113.26	113.38	65.57	65.70	65.80
800	142.68	143.24	142.90	106.62	107.05	107.18	60.21	60.81	60.40
1000	131.30	131.57	132.30	99.71	99.89	99.32	55.13	54.24	55.26
1200	120.46	118.93	119.56	92.33	92.55	92.21	49.08	48.20	49.05

Table 4. Elastic constants (GPa) of BCC W. Potential Model : FS

T (K)	C11	C22	C33	C12	C13	C23	C44	C55	C66
310	489.45	491.04	489.23	209.52	209.81	209.69	162.63	163.67	163.52
600	411.34	411.16	409.90	200.96	200.95	200.85	156.75	156.66	156.84
900	347.50	347.06	348.17	190.88	191.04	190.38	148.43	148.48	148.78
1200	300.39	299.80	300.20	181.07	181.56	181.61	140.65	141.70	141.74
1500	271.06	271.35	270.86	177.32	177.67	177.69	135.83	136.35	135.78

Figure 5 Melting temperature Tm is experimental value.
File:Si elastconst.jpg
(a)	(b)
(c)	(d)