Study on structural phase transitions in defective and perfect substitutional alloys AB with interstitial atoms C under pressure

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, r1Abcc ( 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 r1Afcc ( P,0), r1Afcc ( 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  NY 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  3NYXfcc , 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  PV 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  11  c B  7cC E A  nv n1 1  cB  7cC E A(1)  1  nv n1  nv BB  1cB 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  1cC EC  nv n1cC EC(1) , Rfcc E ABC  1  nv n1  nv BA  11  cB  15cC E A  nv n1 1  cB  15cC E A(1)  1  nv n1  nv BB  1cB 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  1cC EC  nv n1cC EC(1) , Rbcc S ABC  1  nv n1  nv BA  11  cB  7cC S A  nv n1 1  cB  7cC S A(1)  1  nv n1  nv BB  1cB 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  1cC S C  nv n1cC S C(1) , Rfcc S ABC  1  nv n1  nv BA  11  c B  15cC S A  nv n1 1  c B  15cC S A(1)  1  nv n1  nv BB  1c 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  1cC S C  nv n1cC S C(1) , R   TABC   (30)  R R R   E ABC  PV 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 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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