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The analytic expressions of the Helmholtz free energy, the Gibbs thermodynamic potential the mean nearest neighbor distance between two atoms, the crystal parameters for bcc, fcc and hcp phases of defective and perfect substitutional alloys AB with interstitial atoms C and structural phase transition temperatures of these alloys at zero pressure and under pressure are derived by the statistical moment method.
HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2019-0031 Natural Sciences, 2019, Volume 64, Issue 6, pp 57-67 This paper is available online at http://stdb.hnue.edu.vn STUDY ON STRUCTURAL PHASE TRANSITIONS IN DEFECTIVE AND PERFECT SUBSTITUTIONAL ALLOYS AB WITH INTERSTITIAL ATOMS C UNDER PRESSURE Nguyen Quang Hoc1, Dinh Quang Vinh1, Le Hong Viet2, Ta Dinh Van1 and Pham Thanh Phong1 Faculty of Physics, Hanoi National University of Education Tran Quoc Tuan University, Co Dong, Son Tay, Hanoi Abstract The analytic expressions of the Helmholtz free energy, the Gibbs thermodynamic potential the mean nearest neighbor distance between two atoms, the crystal parameters for bcc, fcc and hcp phases of defective and perfect substitutional alloys AB with interstitial atoms C and structural phase transition temperatures of these alloys at zero pressure and under pressure are derived by the statistical moment method The structural phase transition temperatures of the main metal A, the substitutional alloy AB and the interstitial alloy AC are special cases of ones of the substitutional alloy AB with interstitial atoms C Keywords: Statistical moment method, Helmholtz free energy, Gibbs thermodynamic potential, structural phase transition temperature Introduction Structural phase transitions of crystals in general and metals and interstitial alloys in particular are specially interested by many theoretical and experimental researchers [1-7] In [8], the body centered cubic (bcc) - face centered cubic (fcc) phase transition temperature determined in solid nitrogen and carbon monoxide on the basis of the selfconsistent field approximation In [9], this phase transition temperature in solid nitrogen is calculated by the statistical moment method (SMM) The ( , bcc, fcc, hexagonal close packed (hcp)) phase transition temperature for rare-earth metals and substitutional alloys is also derived from the SMM [10] In this paper, we build the theory of , bcc, fcc, hcp) structural phase transition for defective and perfect substitutional alloys AB with interstitial atoms C at zero pressure and under pressure by the SMM [11-13] Received April 30, 2019 Revised June 15, 2019 Accepted July 22, 2019 Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 57 Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong Content In the case of perfect interstitial alloy AC with bcc structure (where the main atom A1 stays in body center, the main atom A2 stays in peaks and the interstitial atom C stays in face centers of cubic unit cell), the cohesive energy and the alloy’s parameters for atoms C, A1 and A2 in the approximation of three coordination spheres are determined by [11-13] k Cbcc 2 AC i ui2 bcc 8r ( 3) AC r1bcc bcc 25 5r (1) (1) AC r1bcc ( 4) AC r1bcc r1bcc eq 24 bcc 4 AC 48 i ui2 ui2 16 r (1) bcc 16 ( 2) (1) bcc AC r1bcc bcc AC r1 AC r1bcc , Cbcc 1bcc C 2C , bcc r 5 r 1 eq 4 ACF 48 i u i4 bcc 1C 2bcc C ni AC (ri ) AC (r1bcc ) 2 AC r1bcc 4 AC r1bcc , i 1 u 0bcc C 8r ( 2) AC r1bcc ( 3) AC r1bcc ( 2) AC r1bcc bcc ( 4) bcc (3) bcc AC r1 AC r1 , 150 125r1bcc ( 3) AC r1bcc bcc r1bcc eq 4r1 bcc 25 r 8r ( 2) ACF r1bcc bcc 8r (1) AC r1bcc bcc ( 2) r1bcc AC bcc 25 r (1) r1bcc AC ( 4) bcc AC r1 25 (1) AC r1bcc , (2) bcc bcc bcc bcc bcc u0bcc A1 u A AC r1 A1 , A1 A1 A1 , k Abcc k Abcc bcc 1bcc A 1A 4 AC 48 i u i4 2bccA 2bccA 1 2 AC i u i2 ( 2) (1) k Abcc AB r1bcc AC r1bcc A A1 , bcc r A1 eq r r bcc A1 ( 4) bcc 1bcc AC r1 A1 A 24 r1bcc eq r r bcc A1 4 AC 48 i u i2 u i2 ( 2) r1bccA AC 1 A1 bcc A1 8r (1) r1bccA , AC 3 ( 3) ( 2) (1) 2bcc AC r1bcc AC r1bcc AC r1bcc A A1 A1 A1 , bcc bcc bcc r r1 A1 r1 A1 A1 eq r r bcc A1 bcc bcc bcc bcc bcc u0bcc A2 u A AC r1 A2 , A2 A2 A2 , k Abcc k Abcc 58 1 2 AC i u i2 (1) bcc ( 2) k Abcc 2 AC r1bcc bcc AC r1 A2 , A r A eq r r bcc A2 (3) Study on structural phase transitions in defective and perfect substitutional alloys AB 4 AC 48 i u i4 bcc 1bcc A 1A bcc A2 8r 2bccA 2bccA ( 4) bcc ( 3) 1bcc AC r1 A2 bcc AC r1bcc A A2 24 r A eq r r bcc A2 ( 2) r1bccA AC 4 AC2 48 i u i u i bcc A2 8r bcc A2 8r (1) r1bccA , AC ( 4) bcc ( 3) 2bcc AC r1bcc A AC r1 A2 A2 bcc r A2 eq r r bcc A2 ( 2) r1bccA AC bcc A2 8r (1) r1bccA , AC (4) where AC is the interaction potential between the atom A and the atom C, ni is the number of atoms on the ith coordination sphere with the radius ri (i 1, 2,3), bcc bcc r1bcc r1bcc C r01C y A1 (T ) is the nearest neighbor distance between the interstitial atom C and the metallic atom A at temperature T, r01bccC is the nearest neighbor distance between the interstitial atom C and the metallic atom A at 0K and is determined from bcc the minimum condition of the cohesive energy u 0bcc C , y0 A1 (T ) is the displacement of the atom A1 (the atom A stays in the bcc unit cell) from equilibrium position at temperature T ( m) AB m AC (ri ) / ri m (m 1,2,3,4, , x, y.z, and ui is the displacement bcc of the ith atom in the direction r1bcc A1 r1C is the nearest neighbor distance between bcc bcc bcc the atom A1 and atoms in crystalline lattice r1bcc A2 r01A2 y 0C (T ), r01A2 is the nearest neighbor distance between atom A2 and atoms in crystalline lattice at 0K and is bcc determined from the minimum condition of the cohesive energy u 0bcc A2 , y 0C (T ) is the bcc bcc bcc displacement of the atom C at temperature T In Eqs (3) and (4), u0bcc A , k A , A , A are the coressponding quantities in clean bcc metal A in the approximation of two coordination sphere [11-13] The equation of state for bcc interstitial alloy AC at temperature T and pressure P is written in the form u 0bcc k bcc bcc r1bcc bcc bcc , v Pv r x cthx bcc 2k bcc r1bcc 3 r1 At 0K and pressure P, this equation has the form bcc bcc u 0bcc 0bcc k bcc Pv bcc r1bcc bcc bcc 4k r1bcc r1 (5) (6) 59 Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong If we know the interaction potential i0 , Equation (6) permits us to determine the nearest neighbour distance r1bcc X ( P,0)(X A, A1 , A , C) at pressure P and temperature we can determine the parameters r1bcc X ( P,0) bcc bcc bcc bcc k X ( P,0), 1X ( P,0), X ( P,0), X ( P,0) at pressure P and 0K for each case of X Then, the displacement y0bcc X ( P, T ) of atom X from the equilibrium position at temperature T and pressure P is calculated as in [11] From that, we can calculate the nearest neighbour distance r1bcc X ( P, T ) at temperature T and pressure P as follows: 0K When we know bcc bcc bcc bcc bcc r1bcc B ( P, T ) r1B ( P,0) y A1 ( P, T ), r1 A ( P, T ) r1 A ( P,0) y A ( P, T ), bcc bcc bcc bcc r1bcc A1 ( P, T ) r1B ( P, T ), r1 A2 ( P, T ) r1 A2 ( P,0) y B ( P, T ) (7) The mean nearest neighbour distance between two atoms in bcc interstitial alloy AC has the form bcc bcc r1bcc ( P, T ), A ( P, T ) r1 A ( P,0) y bcc bcc bcc r1bcc 3r1bcc A ( P,0) 1 cC r1 A ( P,0) cC r1 A ( P,0), r1 A ( P,0) C ( P,0), bcc bcc bcc y bcc ( P, T ) 1 7cC y bcc A ( P, T ) cC y B ( P, T ) 2cC y A1 ( P, T ) 4cC y A2 ( P, T ), (8) where r1bcc A ( P, T ) is the mean nearest neighbor distance between atoms A in the interstitial alloy AC at pressure P and temperature T, r1bcc A ( P,0) is the mean nearest neighbor distance between atoms A in the interstitial alloy AC at pressure P and temperature 0K, r1bcc A ( P,0) is the nearest neighbor distance between atoms A in the deformed clean metal A at pressure P and temperature 0K, r1Abcc ( P,0) is the nearest neighbor distance between atoms A in the zone containing the interstitial atom C at pressure P and temperature 0K and cC is the concentration of interstitial atoms C In the case of fcc interstitial alloy AC (where the main atom A1 stay in face centers, the main atom A2 stay in peaks and the interstitial atom C stays in body center of cubic unit cell), the corresponding formulas are as follows [11-13]: u 0fcc C k fcc C 2 AC i u i2 60 4 AB 48 i ui4 (9) (1) fcc ( 2) fcc ( 2) fcc ( 2) AC r1 fcc fcc AC r1 AC r1 fcc AC r1 r1 9r1 eq ( 2) 4 AC r1 fcc 1fcc C ni AC (ri ) 3 AC (r1 fcc ) 4 AC r1 fcc 12 AC r1 fcc , i 1 (1) fcc fcc AC r1 , Cfcc 1fcc C 2C , fcc 5r1 ( 4) AC r1 fcc r1bcc eq 24 ( 2) r1 fcc AC bcc 4r (1) r1 fcc AC ( 4) fcc AC r1 54 Study on structural phase transitions in defective and perfect substitutional alloys AB (3) fcc AC r1 fcc 27r1 27 r1 fcc fcc 2C ( 2) AC r1 fcc ( 3) AC r1 fcc fcc 125r1 25 r1 fcc 4 AC 48 i ui2 ui2 ( 3) AC r1 fcc fcc r1 fcc eq 2r1 81 r1 fcc ( 4) fcc 26 (3) fcc AC r1 AC r1 25 125r1 fcc 25 r1bcc fcc ( 2) r1 fcc AC r1 fcc fcc 3 bcc 125 r (1) r1 fcc AC (1) AC r1 fcc 16 r1 ( 2) AC r1bcc (1) AC r1 fcc , 125 r1 17 ( 4) fcc AC r1 150 ( 2) AC r1 fcc (1) AC r1 fcc ( 2) AC r1 fcc ( 4) fcc (3) fcc AC r1 fcc AC r1 8r1 r1 fcc (1) AC r1bcc , (10) fcc u0fccA1 u0fccA AC r1Afcc1 , Afcc 1fcc A1 A1 , k Afcc k Afcc fcc 1fcc A 1A 2fccA 2fccA 1 2 AC i u i2 4 AC 48 i u i 4 AC 48 i u i2 u i2 ( 2) k Afcc AC r1 Afcc1 , eq r r fcc A1 ( 4) fcc 1fcc AC r1 A1 , A 24 eq r r fcc A1 1 ( 3) 2fccA fcc AC r1 Afcc1 fcc r r A1 eq r r fcc A1 ( 2) r1Afcc AC 1 A1 r1 Afcc1 (1) r1Afcc , AC (11) fcc u0fccA2 u0fccA AC r1Afcc2 , Afcc2 1fcc A2 A2 , k fcc A1 k fcc A fcc 1fcc A 1A 2 AC i u i2 ( 2) fcc 23 (1) fcc k Afcc AC r1 A2 fcc AC r1 A2 , 6r1 A2 eq r r fcc 4 AC 48 i u i4 ( 4) fcc ( 3) 1fcc AC r1 A2 fcc AC r1 Afcc2 A 54 r A2 eq r r fcc fcc A2 9r A2 A2 ( 2) r1Afcc AC 2 fcc A2 9r (1) r1Afcc , AC 61 Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong 2fccA 2fccA 14 fcc A2 27 r Pv fcc 4 AC 48 i u i2 u i2 r1 fcc ( 2) r1Afcc AC ( 4) fcc ( 3) 2fccA AC r1 A2 AC r1 Afcc2 fcc 81 27 r A2 eq r r fcc A2 14 fcc A2 27 r (1) r1Afcc , AC (12) u 0fcc k fcc fcc fcc x cthx r bcc 2k fcc r1 fcc fcc , v u 0fcc 0fcc k fcc Pv fcc r1 fcc fcc r 4k fcc r1 fcc r1 fcc , (13) , (14) r1Bfcc ( P, T ) r1Bfcc ( P,0) y Afcc ( P, T ), r1Afcc ( P, T ) r1Afcc ( P,0) y Afcc ( P, T ), r1Afcc1 ( P, T ) r1Bfcc ( P, T ), r1Afcc2 ( P, T ) r1Afcc2 ( P,0) y Bfcc ( P, T ) (15) r1Afcc ( P, T ) r1Afcc ( P,0) y fcc ( P, T ), r1ccA ( P,0) 1 cC r1Afcc ( P,0) cC r1Afcc ( P,0), r1Afcc ( P,0) 2r1Cfcc ( P,0), y fcc ( P, T ) 1 15cC y Afcc ( P, T ) cC y Bfcc ( P, T ) 6cC y Afcc ( P, T ) 8cC y Afcc2 ( P, T ), (16) The mean nearest neighbor distance between two atoms A in the perfect bcc substitutional alloy AB with interstitial atom C at pressure P and temperature T is bcc a bcc ABC c AC a AC bcc BTAC bccF T B c B a Bbcc bcc BTB bcc T B bcc bcc , BTbcc c AC BTAC c B BTB , c AC c A cC , bccABC bccAC a bcc ( P, T ), a bcc ( P, T ), a Bbcc r1bcc ABC r1 A AC r1 A B ( P, T ), 2P bcc BTAC bcc TAC bcc 3 2 AC bccF bcc 4a AC 3N a AC T a bcc AC 3 bccF a0 AC 2P bcc , BTB bcc TB 3 2 Bbcc 4a Bbcc 3N a Bbcc2 T a bcc 3 Bbcc a0 B , bcc 2 bcc 2 bcc 2 AC 2 bcc 2 bcc bcc2 1 7cC bccA2 cC bccC 2cC bccA12 4cC bccA22 , a a a A a A a A T AC T C T T T Xbcc 2 Xbcc u 0bcc X bcc2 bcc 3N a X T a bcc 4k X X k bcc k bcc bccX bcc Xbcc , X A, A1 , A , B, C 2k X a X a X (17) The Helmholtz free energy of perfect bcc substitutional alloy AB with interstitial atom C before deformation with the condition cC c B c A has the form bcc bcc ABC AC c B Bbcc Abcc TScbccAC TScbccABC , bcc AC 1 7cC Abcc cC Cbcc 2cC Abcc 4cC Abcc TScbccAC , 62 Study on structural phase transitions in defective and perfect substitutional alloys AB bcc Xbcc U 0bcc X X 3N bcc bcc YX 2X bcc kX 2 bcc bcc YXbcc 1bcc X YX 1 X bcc kX 2 x bcc 0bcc X 3N x X ln e bcc X ,Y YXbcc 2 1bcc X 1 YXbcc bcc YXbcc 2 1bcc X 2X , bcc x bcc X coth x X , bcc X (18) where Xbcc is the Helmholtz free energy, YXbcc is an atom X in clean metals A, B or interstitial alloy AC, S cbccAC is the configuration entropy of bcc interstitial alloy AC and S cbccABC is the configuration entropy of bcc alloy ABC In the case of fcc interstitial alloy AC, the corresponding formulas are as follows: fcc fcc fcc fcc BTAC fcc BTB fcc a ABC c AC a AC cB aB , BTbcc c AC BTAC c B BTBfcc , c AC c A cC , fcc fcc BT BT fcc fcc a ABC r1AfccABC ( P, T ), a AC r1AfccAC ( P, T ), a Bfcc r1Bfcc ( P, T ), 2P fcc BTAC fcc 2 AC a fcc AC fccF TAC fccF a AC fcc 2 AC fcc 3N a AC fcc a AC 3 fcc a0 AC 2P , BTBfcc u 0fccX Xfcc fcc 4k Xfcc T a X k fcc fccX fcc 2k X a X TBfcc a BfccF k Xfcc fcc a X 2 Bfcc 3N a Bfcc T , 2 Afcc2 8cC a Afcc 2 T , T a Bfcc 3 fcc a0 B 2 Afcc 6cC a Afcc T 2 Cfcc cC fcc T aC 2 Afcc 1 15cC fcc a A T 2 Xfcc 3N a Xfcc T , X A, A1 , A , B, C, (19) fcc fcc ABC AC cB Bfcc Afcc TScfccAC TScfccABC , fcc AB 1 15c B Afcc c B Bfcc 6c B Afcc 8c B Afcc TScfccAC , Xfcc U 0fccX 0fccX 3N 2 fcc fcc YXfcc X YX 1 k Xfcc k fcc fcc YX 2X fcc X 1fcc X 0fccX 3N x Xfcc ln e 2 x Xfcc ,Y 2 2 1fcc X YXfcc 1 YXfcc fcc 2 1fcc X 2X fcc X x Xfcc coth x Xfcc YXfcc , (20) For perfect hcp interstitial alloy AC and perfect hcp substitutional alloy AB with interstitial atoms C, we have the same formulas as for perfect fcc interstitial alloy AC and perfect fcc substitutional alloy AB with interstitial atoms C The numerical 63 Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong calculations of the cohesive energy u0 and the alloy parameters k , , , of fcc alloy and ones of the hcp alloys are different When the phase equilibrium happens between the phase and the phase of perfect substitutional alloy AB with interstitial atoms C at zero pressure, (21) ABC ABC , TABC TABC TABC , bcc fcc where we call TABC as the - phase transition temperature of substitutional alloy AB with interstitial atoms C According to the thermodynamic relation, E TS Therefore, the - phase transition temperature of substitutional alloy AB with interstitial atoms Cat zero pressure can be determined by the following formular TABC ( P 0) E ABC E E ABC ABC S ABC S ABC S ABC (22) For example, when bcc, fcc, bcc E ABC 1 7cC E Abcc cC ECbcc 2cC E Abcc 4cC E Abcc c B EBbcc E Abcc , bcc E Xbcc U 0bcc X E0 X 3N bcc bcc YX 2X k Xbcc 1bcc X bcc bcc E0bcc X NY X , Z X 2 Z 2 bcc X x bcc X , sinh x bcc X bcc bcc 2X X Y Z , bcc X bcc S ABC 1 7cC S Abcc cC S Bbcc 2cC S Abcc 4cC S Abcc cB S Bbcc S Abcc , S bcc X S bcc 0X 3Nk Bo 1bcc X YXbcc Z Xbcc bcc kX S bcc 0X 3Nk Bo Y bcc X 2 ln sinh x bcc X bcc bcc 2X X Y , Z , bcc X fcc E ABC 1 15cC E Afcc cC EBfcc 6cC E Afcc 8cC E Afcc c B EBfcc E Afcc , fcc E Xfcc U 0bcc X E0 X 3N fcc fcc YX 2X k Xfcc 1fcc X E0fccX 3NYXfcc , Z Xfcc 2 Z 2 fcc X x Xfcc , sinh x Xfcc fcc 2X 2 YXfcc Z Xfcc , fcc S ABC 1 15cC S Afcc cC S Bfcc 6cC S Afcc 8cC S Afcc c B S Bfcc S Afcc , S Xfcc S 0fcc X 3Nk Bo 1fcc X YXfcc Z Xfcc fcc kX 2 fcc S 0fcc ln sinh x Xccc , X 3Nk Bo YX fcc 2X 2 YXfcc Z Xfcc , (23) where kBo is the Boltzmann constant When the phase equilibrium happens between the phase and the phase of perfect substitutional alloy AB with interstitial atoms C at preesure P, 64 Study on structural phase transitions in defective and perfect substitutional alloys AB G ABC G ABC , TABC TABC TABC , PABC PABC P, (24) where G is the Gibbs thermodynamic potential According to the thermodynamic relation, G PV U TS PV Therefore, the - phase transition temperature of substitutional alloy AB with interstitial atoms C at pressure P can be determined by the following formular: TABC E ABC PV ABC E E AB P V AB V AB AB S ABC S AB S AB (25) For example, when bcc, fcc, we also have Eq (23) and bcc ABC a bcc ABC fcc a ABC (26) V Nv N ,V Nv N 3 The Helmholtz free energy of defective (or real) substitutional alloy AB with interstitial atoms C has the form R ABC ABC ngvf ( ABC ) TScABC* , bcc ABC fcc ABC fcc ABC gvf ( ABC ) cA gvf ( A) cC gvf (C ) cA1 gvf A1 cA2 gvf A2 cB gvf ( B), (1) g vf ( X ) n1 XX XX B X 1 XX , B X U0X X (1) , N XX X(1) , N XX X , (27) where ABC is the Helmholtz free energy of perfect substitutional alloy AB with interstitial atoms C, g vf (ABC ) is the Gibbs thermodynamic potential change of substitutional alloy AB with interstitial atoms C when one vacancy is formulated, g vf (X ) isthe Gibbs thermodynamic potential change of an atom X when one vacancy is formulated, ScABC * is the configurational entropy of alloy atoms and vacancies, N is the total numeber of atoms in alloy, n1 is the number of atoms on the first coordination (1) sphere, XX is the Helmholtz free energy of an atom X on the first coordination sphere with vacancy as centre, cA cB 7cC , cA1 2cC , cA2 4cC for bcc alloy and cA cB 15cC , cA1 6cC , cA2 8cC for fcc alloy The concentration of equilibrium atom is determined from the minimum condition of the Helmholtz free energy c g f ( B) c B g vf (C ) nv nvA exp B v exp , k Bo T k Bo T cA gvf ( A) cA1 gvf ( A1 ) c A2 gvf ( A2 ) n exp kBoT A v (28) Approximately, the mean nearest neighbor distance between two atoms in defective alloy is equal to one in perfect alloy The - phase transition temperature of defective substitutional alloy AB with interstitial atoms C at zero pressure and at pressure P is determined by 65 Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong R ABC T R R R EABC E ABC E ABC ( P 0) R , R R S ABC S ABC S ABC (29) Rbcc E ABC 1 nv n1 nv BA 11 c B 7cC E A nv n1 1 cB 7cC E A(1) 1 nv n1 nv BB 1cB EB nv n1cB EB(1) nv n1 nv BA1 cC E A1 2nv n1cC E A(11) nv n1 nv BA2 cC E A2 4nv n1cC E A(12) 1 nv n1 nv BC 1cC EC nv n1cC EC(1) , Rfcc E ABC 1 nv n1 nv BA 11 cB 15cC E A nv n1 1 cB 15cC E A(1) 1 nv n1 nv BB 1cB EB nv n1cB EB(1) nv n1 nv BA1 cC E A1 6nv n1cC E A(11) nv n1 nv BA2 cC E A2 8nv n1cC E A(12) 1 nv n1 nv BC 1cC EC nv n1cC EC(1) , Rbcc S ABC 1 nv n1 nv BA 11 cB 7cC S A nv n1 1 cB 7cC S A(1) 1 nv n1 nv BB 1cB S B nv n1cB S B(1) nv n1 nv BA1 cC S A1 2nv n1cC S A(11) nv n1 nv BA2 cC S A2 4nv n1cC S A(12) 1 nv n1 nv BC 1cC S C nv n1cC S C(1) , Rfcc S ABC 1 nv n1 nv BA 11 c B 15cC S A nv n1 1 c B 15cC S A(1) 1 nv n1 nv BB 1c B S B nv n1c B S B(1) nv n1 nv BA1 cC S A1 6nv n1cC S A(11) nv n1 nv BA2 cC S A2 8nv n1cC S A(12) 1 nv n1 nv BC 1cC S C nv n1cC S C(1) , R TABC (30) R R R E ABC PV ABC E ABC E ABC P V ABC V ABC R R R S ABC S ABC S ABC (31) When the concentration of interstitial atoms C is equal to zero, the theory of structural phase transition of substitutional alloy AB with interstitial atoms C becomes that of substitutional alloy AB When the concentration of substitutional atoms B is equal to zero, the theory of structural phase transition of substitutional alloy AB with interstitial atoms C becomes that of interstitial alloy AC When the concentrations of substitutional and interstitial atoms are equal to zero, the theory of structural phase transition of substitutional alloy AB with interstitial atoms C becomes that of main metal A Conclusions The analytic expressions of the alloy parameters, the mean nearest neighbour distance between two atoms, the Helmholtz free energy, the Gibbs thermodynamic potential, the energy and entropy for bcc, fcc and hcp phases of substitutional alloy AB with interstitial atoms C and the structural phase transition temperatures of these alloys at zero pressure and under pressure are derived by the statistical moment method The structural phase transition temperature of substitutional alloy AB, interstitial alloy AC and main metal A are special cases of that of substitutional alloy AB with interstitial atoms C In next paper, we will carry out numertical calculations for some real ternary ABC 66 Study on structural phase transitions in defective and perfect substitutional alloys AB REFERENCES [1] B R Cuenya, M.Doi, S.Löbus, R.Courths and W.Keune, 2001 Observation of the fcc-to-bcc Bain transformation in epitaxial Fe ultrathin films on Cu3Au (001) Surface science 493(1-3), 338-360 [2] B.Wang and H.M.Urbassek, 2014 Atomistic dynamics of the bcc↔fcc phase transition in iron: Competition of homo-and heterogeneous phase growth Computational Materials Science 81C, 170-177 [3] X.Ou, 2017 Molecular dynamics simulations of fcc-to-bcc transformation in pure iron: a review Materials Science and Technology 33(7), 822-835 [4] V.V.Hung and D.T.Hai, 2013 Melting curve of metals with defect: Pressure dependence Computational Materials Science 79, 789-794 [5] M.N.Magomedov, 2017 Change in the lattice properties and melting temperature of a face-centered cubic iron under compression Technical Physics 62(4), 569-576 [6] H.M.Strong, R.E.Tuft and R.E.Hanneman, 1973 The iron fusion curve and γ-δ-l triple point Metallurgical Transactions 4(11), 2657-2661 [7] F.P.Bundy, 1965 Pressure-temperature phase diagram of iron to 200 kbar and 900oC Journal of Applied Physics 36(2), 616-620 [8] N.V.Kotenok, 1972 Vestnik MU, Ser Fiz Astrono No.1, p.40 [9] N.Q.Hoc, D.D.Thand and N.Tang, 1996 Communications in Physics 6, 4, pp.1-8 [10] D.T.Hai, 2015 PhD Thesis, HNUE [11] N Tang, V V Hung, 1989, 1990, 1990, 1990 Phys Stat Sol (b)149, 511; 161, 165; 162, 371; 162, 379 [12] V V Hung, 2009 Statistical moment method in studying thwermodynamic and elastic property of crystal HNUE Publishing House [13] N Q Hoc, D Q Vinh, B D Tinh, T T C Loan, N L Phuong, T T Hue, D T T Thuy, 2015 Journal of Science of HNUE, Math and Phys Sci Vol 60, Number 7, p 146 67 ... the condition cC c B c A has the form bcc bcc ABC AC c B Bbcc Abcc TScbccAC TScbccABC , bcc AC 1 7cC Abcc cC Cbcc 2cC Abcc 4cC Abcc TScbccAC , 62 Study. .. bcc a bcc ABC c AC a AC bcc BTAC bccF T B c B a Bbcc bcc BTB bcc T B bcc bcc , BTbcc c AC BTAC c B BTB , c AC c A cC , bccABC bccAC a bcc ( P, T ), a bcc ( P, T ), a Bbcc r1bcc ABC... ABC S ABC S ABC (22) For example, when bcc, fcc, bcc E ABC 1 7cC E Abcc cC ECbcc 2cC E Abcc 4cC E Abcc c B EBbcc E Abcc , bcc E Xbcc U 0bcc X E0 X 3N bcc