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Bộ CA kiểm tra việc đảm bảo an ninh-trật tự Tây Nguyên - Bee - Khoa học & Đời sống Online

Bộ CA kiểm tra việc đảm bảo an ninh-trật tự Tây Nguyên - Bee - Khoa học & Đời sống Online

Chả lễ đến "Chính phủ VN" mà cũng không biết huyện KrongBuk là đang thuộc tỉnh Đăk Lăk, hay đang có dự định phân chia lại địa giới hành chính Tây Nguyên?!

80/13/6

Phu

Chuot

Abstract: In this paper we have investigated the 'native vacancy' and interstitial site for gas solubility in amorphous models SiO2, Al2O3 and FexB100-x using molecular dynamic and statistic relaxation techniques. We found a large number of native vacancies in models FexB100-x, the dominant contribution of them could interpret the correlation between pre- exponential factor and diffusion enthalpy for amorphous alloys. For models SiO 2 and Al2O3 the calculated number of accessible sites for He, Ne and Ar is of correct order of magnitude compared to experimental data . Furthermore, the position of the "first sharp diffraction peak" (FSDP) of silica glass determined on base of Elliott model is in good agreement with experiment , but the FSDP of Al2O3 system doesn't follow th at model.

 

Keywords: computer simulation, amorphous alloy, voids , vacancy, gas solubility

PACS numbers: 78.55.Qr, 61.43.Bn

The 'native vacancy' and interstitial site for gas solubility in amorphous solid

P.K.Hung*, L.T.Vinh**, P.H.Kien*
* Department of Computational Physics, Hanoi University of Technology, 1 Dai Co Viet, Hanoi Viet Nam, E -mail: pkhung@fpt.vn
** Vinh Technical Teachers Training University, Hung Dung block, Nghe An city, Viet
Nam

Abstract: In this paper we have investigated t he 'native vacancy' and interstitial si te for gas solubility in amorphous models SiO2, Al2O3 and FexB100-x using molecular dynamic and statistic relaxation techniques. We found a large number of native vacancies in models FexB100-x, the dominant contribution of them could interpret the correlation between pre- exponential factor and diffusion enthalpy for amorphous alloys. For models SiO 2 and Al2O3 the calculated number of accessible sites for He, Ne and Ar is of correct order of magnitude compared to experimental data . Furthermore, the position of the "first sharp diffraction peak" (FSDP) of silica glass determined on base of Elliott model is in good agreement with experiment , but the FSDP of Al2O3 system doesn't follow th at model.

Keywords: computer simulation, amorphous alloy, voids , vacancy, gas solubility
PACS numbers: 78.55.Qr, 61.43.Bn

Introduction
Void (or free volume) in amorphou s solids (AS) such as SiO2, Al2O3 and FexB100-x continues to be intensively studied due to several reasons. These materials are of great interest from both technological and scientific points of view [1-10]. Plenty of
experimental data accumulates that voids are involved in many physical problems such as the atomic transport, thermal stability and inert gas solubility [17-21]. For example, J.F. Shackelford found that the noble gas solubility in SiO2 glass is proportional to number of interstitial sites [18]. The study of effect of high hydrostatic pressure on diffusion in AS reveals the activation volume nearly equal to atomic volume [16] indicating the usual vacancy mechanism via large void. Therefore, the knowledge about void as well as free volume distribution has enabled the interpreting specific properties of AS. On the other hand, due to absence of crystalline lattice the exact definition of v acancy became difficult. Commonly, for AS the vacancy is termed a void with atomic size. Hence, the number of such voids is an important factor for diffusion in amorphous materials. As in the case of crystal, the therma l activated vacancies may occur such that their concentration depends on the temperature and the distribution of vacancy formation energy. Nevertheless, due to high structural disorder there are also a number of 'native vacancies' which weakly correlates with temperature. The 'native vacancy' is noted in some early simulations
[22,30,31]. However, previously, no one have reported the concentration of native vacancies in AS.
The experimental technique including the positron annihilation, small -angle x-ray scattering and electron microscopy [11-15] could provide the void radius distribution, but more detail, in particular, the exact number of specific voids can be obtained only from computer simulation [22-29]. To calculate the voids, authors in [22-27] consider the constituent atoms as a rigid sphere and they subdivide the simulation model into a mesh of points. At each mesh point , they calculate the distance to the nearest neighbor atoms and the radius of corresponding void is assigned. The accuracy of this method , of course depends on mesh resolution and it is computationally very expensive . Recently, various



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methods have been developed including :1/ Chan and Elliott method [28]; 2/ Voronoi– Delaunay method [29]; 3/ Rigid sphere [30]. The last approach is simple and enables the detecting all possible interstice s in AS. Therefore, we have employed the rigid sphere approach to study voids in two amorphous systems: amorphous metal -metalloid alloys and oxide glasses (SiO2 and Al2O3), which attain the random parking and random network structure respectively. The objectives of our simulation are (i) to estimate the concentration
of native vacancy and (ii) to give some insight into the correlation between void set and macroscopic properties of considered amorphous solids .

Calculation method
Usually, the calculation of void set in amorphous solid is performed on molecular dynamic
(MD) model. Such a simulation requests very long computing times to construct a large model. Therefore, we employ the statistic relaxation (SR) method; a computationally cheaper method to prepare a model FexB100-x of 2.104 atoms. Here x varies from 70 to 90. The initial conf iguration is generated by randomly placing all atoms in a cubic cube with periodic boundary condition . The Pak-Doyam-type potential [24-26] is applied and the density is adopted from real amorphous alloys. Then this sample is treated over 10 6 SR steps to reach the equilibrium state. The SR step length is equal to 0.01Å. After each running, the structural characteristics such as the pair radial distribution function (PRDF) and the coordination number are determined. To calculate the coordination number we us e
a cut-off distance chosen as the first minimum after first peak in PRDFs.
For glass model Al2O3 and SiO2, MD simulation is carried out using configuration containing 1998 atoms (666 Si and 1332 O) and 2000 atoms (800 Al and 1200 O) in a cubic cube with periodic boundary conditions. We use the BKS and Born -Mayer potentials
to construct SiO2 and Al2O3 system, respectively. These potentials correctly reproduce the structure and some properties of both liquid and glass. More detail about applied potentials can be seen in ref. 31,33. The starting configuration is melted at 6 000 K to remove possible memory effect and then it is subsequently cooling down to 4000, 2000 , 1000 and finally to
500 K with rate of 0.25 K/ps. The equation of motion is integrated using Verlet algorithm with time step of 0.46 fs. At temperatures of 4000 and 2000 K a relaxation with 2 .104 time steps has been accomplished and a 10 6 time step relaxation has been performe d to produce
a glass model at 500 K and at ambient pressure.

Table 1. The atomic radius
Fe B Si Al O
Radius, Å 1.28 0.78 1.46 1.23 0.73

Void is defined as a sphere that is in contact with four atomic spheres and not overlapped
with any atom (see Fig.1 A). The radius of atomic spheres is listed in Table 1 . To calculate the voids we consider all set of four atoms such that the distance between any two atoms of them is less than specified value. This condition decreases the numb er of sets of four atoms and thus saves computing times. Then a sphere (void) is inserted in con tact with those atoms. If the inserted void overlaps with any atom, it is removed from system. The algorithm employed here can be found elsewhere [31].

Result and discussion

Amorphous alloys Fe xB100-x
The structural characteristics of models Fe xB100-x are summarized in Table 2. They are very
close to simulation data in ref. 24-26 and reproduce well the diffraction experimental data



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in ref. [36]. A small discrepancy around first peak of PRDF between our model and that simulation relates to small size of t he models in ref. 24, 25 (2000 atoms). Fig.2 shows the RVD for models Fe 70B30 and Fe90B10. The form of RVD of both models are similar to each other and one can see two peaks located at 0.35 and 0.5 Å. For another model Fe xB100-x, the RVD attains the same form, but the position of main peak of RVD slightly shifts to bigger radius with increasing metalloid content .

Table 2. Structural characteristics of models FexB100-x. Here rFeFe, rFeB and rBB are the position of first peak in PRDFs for pair Fe-Fe, Fe-B and B-B respectively; ZFeFe, ZFeB, ZBFe and ZBB are the averaged coordination number for pair Fe-Fe, Fe-B, B-Fe and B-B, respectively; * - experimental
data from ref. 36, 41, 42 Model Density
g/cm3 rFeFe, Å rFeB, Å rBB, Å ZFeFe ZFeB ZBFe ZBB










Fe83B17* - 2.58 2.1 - 12.2 1.9 9.4 -

Voids are not distributed separately, but they could group into a large cluster involving two
or more voids (see Fig.1 B). Hence, the number of non-overlapped voids can't be determined directly from RVD and we calculate the averaged volume of space per atom Vvoid(r) that is occupied by voids with radius bigger than r. To determine the Vvoid(r) we randomly generate N points (N=8.106) in simulation box and then calculate the number of points Nin, which locate only inside voids with radius bigger than r. The volume Vvoid(r) is given as Vvoid(r) = VSB.Nin/N/Natom. Here VSB is a volume of simulation box ; Natom is a number of atoms in the model . The volume of space occupied by atoms Fe or B is calculated following similar way. The obtained results are listed in Table 3 . Here we also present the void fraction, which is a ratio between volume of space occupied by voids and volume VSB. As shown from Table 3, the averaged volume of space occupied by an atom
Fe or B, VFe and VB are almost unchanged for different models . In addition, those quantities
are close to volume of corresponding atomic sphere 4rFe3/3 or 4r 3/3. This fact indicates that the sample number N is enough large to ensure a sufficient accuracy .

Table 3. The averaged volume occupied by atom s or voids. VFe, VB are the averaged volume occupied by an Fe and B, respectively; VV is the averaged volume occupied by voids per atom;  is
the parking fraction; , 1 are the void and atom fraction, respectively.
Fe70B30 Fe75B25 Fe80B20 Fe85B15 Fe90B10






  0.1198 0.1131 0.1073 0.1073 0.098

The parking fraction  for binary mixture of hard spheres FexB100-x is defined by [7]





3

4 n r

3  n r 3

  Fe Fe B B

(1)

3 VSB

where nFe and nB are a number of iron and boron atoms; rFe, rB are the radius of iron and
boron, respectively . Table 3 lists the parking fraction and atom fraction, which is a ratio
between volume occupied by all atoms and VSB . According to data in Table 4 the parameter  is very close to atom fraction indicating the small overlapping volume Voverlap
(the volume of space where two or several atoms intersect; see Fig.1 C). As the boron
concentration increases, the parking fraction  of most models reduces, whereas void fraction  and volume VV change in opposite way. The exception is a model Fe 85B15, where
it possesses the largest VV and its parking fraction less than one for model Fe 20B20 (see
Table 3). The reason of the deviation from observed trend may be related to the metall oid
composition with better gla ss forming ability, which is experimentally observed around
x=18.2.
Fig.3 displays the dependence of volume Vvoid(r) on void radius. In the interval from 0.7 to
1.3 Å , the Vvoid(r) can be described by formula




V ()rexp

 4 r 3 
 


(2)

void

 3 

 


where parameters α and  are listed in Table 4. Note that the equation (2) is similar to the
formula developed by Turnbull and Cohen to estimate the probability of a free volume in liquid (see ref. 7)

Table 4. The parameters α and .
Fe70B30 Fe75B25 Fe80B20 Fe85B15 Fe90B10 SiO2 Al2O3
α, Å3 1.5 1.4 1.6 1.7 1.6 9.5 6.9
 ,Å-3 3.0 3.1 3.8 3.8 3.9 0.15 0.22



Table 5. The number s Nr1 and Nr2 .

Radius,

Fe70B30 Fe75B25 Fe80B20 Fe85B15 Fe90B10

Å Nr1 Nr2 Nr1 Nr2 Nr1 Nr2 Nr1 Nr2 Nr1 Nr2
0.7 279 580.6 202 430.0 142 275.5 167 274.2 108 139.7
0.75 138 307.2 104 227.6 66 145.3 72 135.8 44 67.0
0.8 76 155.0 56 114.8 26 73.0 38 63.5 13 30.2
0.85 39 74.1 28 54.9 16 34.8 12 27.9 8 12.7
0.9 17 33.4 13 24.7 11 15.6 6 11.4 3 4.9
0.95 7 14.1 5 10.5 4 6.6 2 4.3 3 1.8
1 1 5.6 4 4.1 4 2.6 0 1.5 0 0.6
1.05 1 2.0 2 1.5 2 0.9 0 0.5 0 0.2
1.1 1 0.7 0 0.5 1 0.3 0 0.1 0 0.0
1.15 1 0.2 0 0.2 1 0.1 0 0.0 0 0.0
1.2 0 0.1 0 0.0 1 0.0 0 0.0 0 0.0
1.25 0 0.0 0 0.0 1 0.0 0 0.0 0 0.0
1.3 0 0.0 0 0.0 1 0.0 0 0.0 0 0.0


Since large void could be a vacancy or interstitial sit e for impurity, hence it is of interest to
determine a number of spheres with radius r inside constructed model , which aren't intersected with any atom . In addition, they also aren 't intersected with each other.



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Obviously, the number of such spheres satisfies the condition: Nr < Nr2 =
Vvoid(r)Natom/(4r3/3). The number of spheres Nr can be estimated by another way: firstly,
we determine all voids with radius bigger than r from the simulation just described. Let Nr0
is a number of finding voids. T hen we insert Nr0 spheres with radius r in the model. The
center of those spheres coincides with the center of void s. Secondly, we check the
overlapping between any two spheres of them. If two spheres are overlapped, then we remove one sphere from system. We obtain, thereby a number of sphere s Nr1 which separately locate and aren't overlapped with any atom. Obviously, Nr1


N  3 V void ()r Natom
r 4 r 3


(3)


here the parameter χ has a value in the interval from 1.0 to 3.0 .
Now we estimate the numbers Nr for spheres with radius rFe and rB. Obviously, they are the number of native vacancy for iron and boron. By taking the parameter χ equal to 2.0 and substituting it into formula (3) we get the approximate fraction of native vacancy. The native vacancy fraction for boron and iron is listed in Table 6.

Table 6. The native vacancy fraction.
Fe70B30 Fe75B25 Fe80B20 Fe85B15 Fe90B10
CNVacFe 9.4.10-8 6.9.10-8 4.1.10-8 9.3.10-9 2.2.10-9
CNVacB 3.9.10-3 2.9.10-3 1.8.10-3 1.6.10-3 7.5.10-4

In accordance to experimental data for variety of amorphous alloys (see the review in ref .
10) there is a correlation between the pre -exponential factor D0 and diffusion enthalpy H deduced from Arrhenius -type LnD -1/T curves. As a consequence of this correlation , the enthalpy H is too low in the system with low D0 (10-5-10-9 m2/s). This may indicate that the enthalpy value is equal to the migration enthal py, but not the sum of migration enthalpy and formation enthalpy of vacancy as in the case of crystal. This observation usually is interpreted by the collective diffusion mechanism involving a large group of atoms. Our finding shows a new explanation of that experimental observation: the concentration of native vacancies CNVac weakly depends on temperature; hence it results in the including CNVac in pre-exponential factor D0 and the absence of formation entha lpy in diffusion enthalpy H. In accordance to data in Table 9, the D0 value may be decreased over 4-9 orders for diffusio n of Fe or B via native vacancies. Thus, the diffusion mechanism with dominant contribution of native vacancy may lead to small value of both D0 and H for
diffusion in amorphous alloys.

Amorphous glasses SiO 2 and Al2O3 (AG)

Table 7, 8 shows the structural characteristics of AG models and they are close to experimental data from ref. 37-39 and to calculated da ta in ref. 24, 29. Compared to model FexB100-x, GA attains a greater void fraction and less parking fraction indicating their loose structure. In addition, the atom fraction is significantly less than parking fraction. This demonstrates the strong bond T-O in units TO4 of AG in comparison with Fe -B bond in system FexB100-x. Consequently, the overlapping volume in AG models is also significantly larger than one in FexB100-x system.




5

Table 7. Structural characteristics of AG models. Here rTT, rTO and rOO is the position of first peak
in PRDF (Å) for pair T-T, T-O and O-O; ZTO is the coordination number for pair T-O; θ is bond
angle (degree); T is Si or Al. * - experimental data from ref. 37, 38; ** - experimental data from ref.
39;
Model ZTO rTT rTO rOO θ (O-T-O) θ (T-O-T)




Al2O3** 4.1 3.20 1.80 2.80 - -



Table 8. The structure and void characteristics of AG model
Model VT, Å3 VO, Å3 VV, Å3   
1
SiO2 13.03 1.63 6.11 0.4375 0.3916 0.4928
Al2O3 8.18 1.63 4.87 0.4001 0.3904 0.4587

Fig.4 shows the RVD for AG and together with data determined by


    

p()rexp 1   r rRVD  


(4)

 RVD 

 

 RVD

 


It demonstrates that the simulation result is well fitted to Gauss function (4) with following
parameters: RVD= 0.26 Å and rRVD =1.02 Å for SiO2 glass; RVD= 0.225 Å and rRVD =0.89 Å
for Al2O3 glass. Here RVD, rRVD are the mean radius and width of RVD respectively.
According to experimental study by Sh ackelford [18] and later simulation [29], large void
could be an accessible site for noble gas and the number of non -overlapped voids with radius bigger specified value is approximately equal to the number of sites for solute gas atom. They found that this radius is adopted to 1.28, 1.38 and 1.70 Å for He, Ne and Ar in silica glass, respectively [29]. In present simulation we consider the accessible site for solute gas as a large sphere which locates inside the model and isn't intersected with any atom. The radius of those spheres for He, Ne and Ar is taken as ones in [29]. To determine the number of such no n-overlapped spheres we apply the same methods as used in the case
of system FexB100-x. The number of voids Nr1 and Nr2 for AG models are shown in Fig. 5. Note that the volume Vvoid(r) for AG also follows the equation (2 ) (see Fig.6 and Table 4). Using the equation (3) , the number of solubility sites is approximately estimated and listed
in Table 9. It can be seen that the calculated data is larger than exp erimental value, but it gives the correct order of magnitude. Compared to silica the Al2O3 model attains smaller number of solubility sites. In particular, the ratio of solubility sites between SiO2 and Al2O3 glass increases from 2 to 5 with increasing the radius of solute atom (see Table 9). This results from that the void fraction  as well as the averaged void volume per atom VV of Al2O3 glass is significantly smaller than one of SiO 2 system (see Table 8).


Table 9. The interstitial solubility sites (1027sites/m3) in silica and alumina glass
exp. data for silica [18] SiO2 model Al2O3 model
He 2.3 7.78 3.06
Ne 1.3 4.45 1.49
Ar 0.11 0.56 0.10





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In accordance to Elliott model, the position of FSDP relates to the average void-cation distance D by formula [40]


Q  3 (1  / 2)
2d


(5)


Where D/d=1+ε; d is the nearest nei ghbor cation-cation separation. The calculated result
together with experimental data from ref. 34,35 is presented in Table 10. One can see that the best agreement with experiment is found for silica, but for alumina, in conversely, we obtain a large discrepancy between calculated and experimental data . The reason of this discrepancy relates to that the Elliott model reproduces the FSDP only for system with tetrahedral network structure. For alumina glass where a considerable proporti on of units AlO5 and AlO6 presents; therefore, its tetrahedral network structure strongly modifies and
the Elliott model can't reproduce the experimental data of Q.

Table 10. The parameters d, D and Q of AG model
d, Å D, Å Q, Å-1 Q1, Å-1 [34,35]
SiO2 3.14 2.48 1.66 1.52-1.69
Al2O3 3.08 2.10 1.77 2.1

Conclusions
In this paper, we have investigated the native vacancy and interstitial site for gas solubility
in amorphous alloy FexB100-x and AG by means of statistic relaxation and mo lecular dynamic simulation. Several conclusions can be made as given follow
i) We found that the models FexB100-x possess a large number of native vacancies. The native vacancy fraction vary from 10 -4 to 10-6 for boron depending on metalloid concentration. This data for iron changes from 10 -9 to 10-7. The small value of both pre- exponential factor and diffusion enthalpy (close to migration enthalpy) observed experimentally may be related to that the contribution of native vacancies to diffusion in amorphous alloys is essential.
ii) We found that compared to system FexB100-x, the AG model is characterized b y greater value of void fraction and averaged void volume per atom. The RVDs of AG models are fitted to Gauss form, but for system FexB100-x the RVD possesses two peaks located at 0.35 and 0.5 Å. The number of accessible sites for solute gas determined in model SiO2 is consistent with experimental data. Compared to silica glass we found a systematical reduce
of number of solubility sites in alumina glass .
iii) The simulation confirms the validity of Elliot model for FSDP of silica glass, but the deviation of this model from experiment is observed for Al2O3 system because its tetrahedral network structure strongly modifies.

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