Structure and Stability of BCC Crystals Fe , V , Nb and Ta under Hydrostatic Loading

The structural stability and theoretical strength of BCC crystals Fe, V, Nb and Ta under hydrostatic loading have been investigated by using the modified analytical embedded atom method (MAEAM). For all the calculated BCC crystals, the failures occur while the relation >0 is violated in compression and >0 is violated in tension. It found that the stable regions are 0.9269~1.1495, 0.9270~1.1545, 0.9268~1.1449 and 0.9268 ~ 1.1427 in the lattice stretch or the corresponding -408.89 ~ 123.54, -186.96 ~ 131.43, -259.07 ~ 152.53 and -283.92 ~137.04eV/nm in the theoretical strength for Fe, V, Nb and Ta, respectively. The calculated maximum tensile stresses max of Fe, V, Nb and Ta are 123.57, 131.74, 154.45 and 137.85eV/nm and the corresponding lattice stretch max are 1.1527, 1.1617, 1.1661 and 1.1545. The calculated maximum tensile stress max and the corresponding lattice stretch max of Fe are consistent well with the results of Ab initio calculation.


Introduction
Triaxial tension occurs in solids in the vicinity of some types of defects in the microstructure of crystalline solids, e.g., cracks, pores, voids, and complex phase or grain boundaries (Kelly & Macmillan, 1986).To understand the structure and behavior of these defects, information on local elastic constants and theoretical (ideal) strength is necessary.For example, the theoretical strength of a perfect crystal plays an important role in determining the stress distribution near the tip of a crack and is important in determining whether a material will exhibit brittle or ductile behavior (Kelly, 1966).
The theoretical strength of a material is defined as the stress at which a homogeneously deformed perfect crystal becomes elastically unstable with respect to internal displacements (Clatterbuck, Chrzan, & Morris Jr, 2003).The stable region is sensitive to the behavior of the loading mechanism and the theoretical strength is strongly depends on the direction of tension or compression since it is anisotropic.For these interesting properties, the structural and theoretical strength of solids have been widely investigated with many experimental (Fuente et al., 2002) and theoretical (Kanchana, Vaitheeswaran, & Rajagopalan, 2003) methods.Experimentally, the whiskers or nanoindentation experiments were used to investigate the stability or theoretical strength of the crystals under pressure.Theoretically, many simulation methods, such as the linear muffin-tin orbital (LMTO) method, the pair-potential theory (Milstein & Huang, 1978), the embedded-atom method (EAM) (Milstein and Chantasiriwan, 1998), the molecular-dynamics (MD) simulation (Qi, Zhang, & Hu, 2004), Ab initio calculation (Saib & Bouarissa, 2007) etc., have been used.Among all the theoretical methods, the Ab initio calculation based on quantum mechanics is thought to be the most precise, however, the complex calculations and long computation time requires higher-performance computing.Some properties of body-centered cubic (BCC) crystal Fe under hydrostatic loading (Černý et al., 2003) have been investigated with Ab initio.The main purpose of this paper is to compute theoretical strength and find out the stable region of BCC crystal Fe, V, Nb, and Ta under hydrostatic loading by using the modified analytical embedded atom method (MAEAM).This method has been used successfully in our previous papers to analyze the stability of the face-centered cubic (FCC) crystal Ni (Zhang, Li, & Xu, 2007), Au (Zhang et al., 2008a) and the BCC crystal Fe (Zhang et al., 2008b) under uniaxial loading and BCC transition metals Cr, Mo and W (Li, Zhang, & Xu, 2008) under hydrostatic loading respectively and the other properties of the metals and alloys.

MAEAM
In MAEAM, the total energy of a system t E is expressed as where i E is the energy contribution from atom i to the total energy t E , ( ) i F  is the energy to embed an atom in site i with electron density i  which is given by a linear superposition of the spherical averaged atomic electron densities of other atoms ( ) where the subscript e indicates equilibrium state and 1e r is the first nearest neighbor distance at equilibrium.In this paper, the atomic election density at equilibrium e f is chosen as where is the atomic volume in BCC metals ( 0 a is the unstressed lattice constant), c E and 1 f E are cohesion energy and mono-vacancy formation energy, respectively.
The other parameters n, F 0 , α, k 0 , k 1 , k 2 and k 3 in Eqs. ( 4)-( 6) can be determined by cohesion energy E c , mono-vacancy formation energy E 1f , lattice constant a 0 , and elastic constants C 11 , C 12 , C 44 of the metals considered   represented by Eq. ( 5) can be used only for the separated distance between atoms is shorter than the second neighbor distance r 2e and should be substituted by following cubic spline function (termed as a cutoff potential) while the separated distance between atoms ranges from r 2e to r c .
Four parameters l 0 , l 1 , l 2 , l 3 , and cutoff radius r c are taken as .
By substituting physical parameter a 0 , E c , E 1f , C 11 , C 12 and C 44 (listed in Table 1) into Eqs.( 8)-( 15), we can obtained the model parameters for Fe, V, Nb, and Ta (see Table 2).

The Stability Criteria
For a cubic crystal undergoing homogeneous deformation, the strains   of the crystal are expressed as: where a α0 , i.e. a 0 above, is the initial unstressed lattice constant, a α is the instantaneous stressed lattice parameter.Since only a hydrostatic stress is applied, both the shear strains ε α and shear stresses σ α ( 4,5,6) equal to zero.
The normal stress σ α acting on a face of the unit cell is defined by the first order derivative of internal energy i E with respect to the normal strain The elastic parameters C αβ which can be expressed in terms of the second derivatives of the internal energy with respect to the lattice strains ε α .
Based on the crystal symmetry of the crystal and using Eqs.( 1)-( 3) and ( 22)-( 25), we can get: where j r , i.e. ij r above, is the separation distance of atom j from atom i, the subscript i is neglected here since the site i is selected as the origin of the coordinate system.
Under hydrostatic loading, the applied pressure P equals to the negative normal stress on each plane, viz.
. Three principal edges of the cubic lattice cell a 1 , a 2 and a 3 remain equal to each other throughout the hydrostatic loading and their included angles a 4 , a 5 and a 6 will retain their initial values of / 2  (at least until failure occurs) due to no shear stresses.For a cubic system under hydrostatic loading, the stability criteria are expressed as follows where κ is bulk modulus, μ and μ′ are the shear moduli, P is the applied pressure at current strain.The above stability criteria are presented by Born and then modified by Milstein.The theoretical strength is the value of the stress at which anyone of the above three conditions is not satisfied.

Results and Discussions
The variation of the internal energy per atom i E , the normal stress 1  , the stability criteria as a function of the principal stretch λ ( 1 ) are shown in Figure 1-3.Only the stability criteria of Fe are given in Figure 3 since the similar curves are obtained for V, Nb and Ta.From Figure 1, it can be seen that, for BCC crystal Fe, V, Nb, and Ta under hydrostatic loading, whether compressive or tensile loading from initial state, the internal energy per atom i E are all increased.Such behavior is due to the work which is done by applied stresses and the initial stress-free state corresponds to the minimum energy.The result of the internal energy per atom i E for Fe is similar to that of the Ab initio calculations.From Figure 2, we can see that, with decreasing the lattice stretch λ from initial stress-free state, the compressive stress σ 1 increases monotonously and quickly for Fe.However, the compressive stress 1  increases quickly at first and then slightly for V, Nb and Ta.The increased sequence of the compressive stress with decreasing λ is for Fe, V, Nb and Ta.While with increasing the lattice stretch λ from initial state, the tensile stress increases till the maximum tensile stress max  is reached and then decreases slightly.The calculated maximum tensile stresses max  are 123.57, 131.74, 154.45 and 137.85ev/nm 3 , and the corresponding lattice stretch max  are 1.1527, 1.1617, 1.1661and 1.1545 for Fe, V, Nb, Ta, respectively (listed in   .So the stable region of Fe is 0.9269~1.1495 in the lattice stretch λ.The similar curves are obtained for V, Nb and Ta and the stable regions are determined to be 0.9270~1.1545,0.9268~1.1449and 0.9268~1.1427 in the lattice stretch λ, respectively.The corresponding compressive and tensile theoretical strengths are 408.89 and 123.54, 186.96 and 131.43, 259.07 and 152.53, 283.92 and 3 137.04eV/nmfor Fe, V, Nb, Ta respectively (listed also in Table 3 for convenient).
It is interesting to note that from Table 3, for each of the four crystals, the tensile strength is slightly lower than its maximum tensile stress.Second, the compressive strength is higher than tensile strength especially for Fe.Third, the stable regions are almost equal with respect to the lattice stretch λ, especially in compressive region.

Conclusions
The structural stability and theoretical strength of BCC crystals Fe, V, Nb and Ta under hydrostatic loading have been investigated by combining the MAEAM with Milstein modified Born stability criteria.The conclusions are summarized as follows: (1) For all four BCC crystals under hydrostatic loading, as expected whether compressive or tensile loading from initial state, the internal energy per atom i E are all increased.(2) The failures occur while the condition μ>0 is violated in compressive region and κ>0 is violated in tensile region.
(3) The stable regions are almost equal with respect to the lattice stretch λ, especially in compressive region.
(4) For each of the four crystals, the tensile strength is slightly lower than the maximum tensile stress, however, the compressive strength is higher than tensile strength especially for Fe.

r
is the separation distance of atom j from atom i, ( ) ij r  is the pair potential between atoms i and j, and ( ) i M P is the modified term which describes the energy change due to the deviation from the linear superposition.The embedding function ( ) i F  , pair potential ( ) Zhang et al, the pair-potential ( ) ij r  where r 2e and r 3e are the second and the third neighbor distance at equilibrium,

Figure 1 .
Figure 1.Energy per atom as a function of      

Figure 2 .
Figure 2. Normal stress as a function of  

Figure 3 .
Figure 3.The stable criteria as a function of  for Fe

Table 1 .
The input physical parameters for Fe, V, Nb and Ta, 0

Table 2 .
The calculated parameters for Fe, V, Nb and Ta, n and e

Table 3 .
The maximum tensile stress max  and the corresponding lattice stretch max  , the stable region represented in lattice stretch λ and the compressive and tensile strengths for BCC metals Fe, V, Nb and Ta max 