The Correlation of the Parameters of Interatomic ... - Springer Link

ISSN 0018-151X, High Temperature, 2008, Vol. 46, No. 4, pp. 484–494. © Pleiades Publishing, Ltd., 2008. Original Russian Text © M.N. Magomedov, 2008, published in Teplofizika Vysokikh Temperatur, Vol. 46, No. 4, 2008, pp. 533–544.

THERMOPHYSICAL PROPERTIES OF MATERIALS

The Correlation of the Parameters of Interatomic Interaction in Crystals with the Position of Atom in the Periodic Table M. N. Magomedov Institute of Problems of Geothermy, Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, Dagestan, 367003 Russian Federation Received July 25, 2006

Abstract—The previously obtained parameters of interatomic potential of the type of Mie-Lennard-Jones potential are used to analyze the correlation of these parameters with the position of atom in the periodic table for the purpose of estimating the parameters of interatomic potential of substances for which no experimental data are available that would enable one to calculate these parameters. The parameters of interatomic potential for promethium (Pm-61) are estimated for two versions of structure, and the parameters for lanthanides are corrected. The similarity in double periodicity for lanthanides and actinides is used for estimating the values of coordinate of the minimum of potential ro for elements ranging from curium (Cm-96) to dubnium (Db-105). The value of ro is estimated for roentgenium (Rg-111) proceeding from the fact that, in the case of atomic number Za > 60, the value of ro decreases with increasing Za in elements from the same subgroup. Correct values of the parameters of potential are predicted for elements ranging from francium (Fr-87) to dubnium (Db-105). The obtained parameters of interatomic potential are used to calculate the Debye temperature and Grueneisen parameter for neptunium (Np-93) and americium (Am-95). PACS numbers: 34.20Gf, 61.50.Lt DOI: 10.1134/S0018151X0804007X

1. INTRODUCTION

The parameters of potential (1) were calculated in [3] using the approximation of “interaction of only nearest neighbors” from the following four equations:

The question of the dependence of the characteristics of monatomic substance on the position of atoms in the periodic table has been studied for quite some time [1, 2]. This is associated with the fact that the periodic dependence enables one both to predict the properties of hard-of-access elements and to reveal the effect of the atomic mass and of the number of filled electron shells on various properties. One of the fundamental problems in this field is the variation of the parameters of interatomic interaction in monatomic substances with increasing mass of the element. The present paper deals with the investigation of this problem.

r o = c 00 ,

b = 6γ 00 [ 1 + ( 2F b ) ] – 2, –1

b

where k3 is the first coordination number; kb and
are the Boltzmann and Planck constants, respectively; NA is 2

the Avogadro number; KR =
2/kb r o m; Fb = (4D/kbξ3Θ00) – 1; Aw(1) = Θ00/(2ξ3Fb), m is the atomic mass; and ξ3 = 9/k3. Four experimentally obtained parameters were used for calculation, namely, c00, L00, Θ00, and γ00, i.e., the distance between the centers of nearest atoms, the sublimation energy, the Debye temperature, and the Grueneisen parameter at zero values of temperature and pressure T = 0 K and P = 0. In [3], Eqs. (2) are used to calculate the parameters of potential (1) for numerous monatomic crystals of elements of the periodic table, for which the values of c00, L00, Θ00, and γ00 were known. In [4], special features were studied of covalent bond in crystals of elements of carbon subgroup (IVa) with a diamond-like structure such as Cdiam, Si, Ge, and α-Sn. The parameters of interaction potential (1), which were obtained in

a

= [ D/ ( b – a ) ] { a [ r o / ( c + r' ) ] – b [ r o / ( c + r' ) ] }. b

a

(2)

a = b/ { 1 + [ 5K R k 3 b ( b + 1 )/144 A w ( 1 ) ] },

We will represent the pair interaction of atoms in an elementary (monatomic) crystal in the form of the Mie– Lennard–Jones potential [3, 4], ϕ ( r ) = [ D/ ( b – a ) ] [ a ( r o /r ) – b ( r o /r ) ]

D/k b = ( 2/k 3 ) [ ( L 00 /N A k b ) + ( 9Θ 00 /8 ) ],

(1)

Here, D and ro denote the depth and coordinate of the minimum of potential well, respectively; b and a are the parameters which characterize the rigidity and long-range action of the potential, respectively; and c is the distance between the centers of nearest atoms in the crystal. 484

THE CORRELATION OF THE PARAMETERS OF INTERATOMIC INTERACTION

2. THE DEPENDENCE OF PARAMETERS OF POTENTIAL ON ATOMIC MASS The study of the dependences of the parameters of potential (1) (ro, D, a, and b, obtained in [3, 4]) on the atomic mass m revealed periodicities corresponding to periods and subgroups of the periodic table. 1. Minima for the dependence ro(m) were observed for subgroup IVa (Cdiam, Si, Ge, α-Sn) and subgroup VIIIb (Fe, Ru, Os), and maxima – for subgroup Ia of alkali metals (Li, Na, K, Rb, Cs, Fr). Double periodicity was observed in f-transition metals, i.e., lanthanides, with maxima for Eu and Yb. This regularity was observed previously as well in studying the atomic mass dependence of the atomic and ionic radii of elements, as well as the specific density of elementary crystals, as was described in detail in the monograph of V.K. Grigorovich [1, p. 146]. 2. Minima for the dependence D(m) were observed for subgroup VIIIa of inert gases (Ne, Ar, Kr, Xe, and Rn), and maxima – first for subgroup IVa (Cdiam, Si, Ge, α-Sn) and then for elements from subgroups Va and Vb (V, As, Nb, Sb, Bi). In so doing, the value of potential well depth Ds/kb calculated from the atomization energy L00 was used for Cdiam, Si, Ge, and α-Sn [4]. Intermediate extrema are present for elements from subgroup IIb (Zn, Cd, Hg) and lanthanides. This periodicity of the energy parameter D(m) is associated with similar extrema of numerous physical characteristics of matter such as the modulus of elasticity, the hardness of crystal, the coefficient of thermal expansion, the electron work function, the melting and boiling temperatures, and the heats of melting and evaporation [1, p. 155; 2]. 3. Maxima for the dependence b(m) were observed for subgroup VIIIa of inert gases (Ne, Ar, Kr, Xe, and Rn), and minima – for subgroup IVa (Cdiam, Si, Ge, α-Sn). Intermediate minima are present for subgroup IIa (Ca, Sr, Ba) and maxima for Mn, Ga, and In. Double periodicity was observed in f-transition metals, i.e., lanthanides, with maxima for Gd and Tm and a minimum for Er. 4. Maxima for the dependence a(m) were observed for subgroup VIIIa of inert gases (Ne, Ar, Kr, Xe, and Rn), and minima – for subgroup Ia of alkali metals (Li, Na, K, Rb, Cs, Fr). HIGH TEMPERATURE

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54 56 5.5

58

60 62

Za 64

66

68

70

72 250 2

1

5.0

200 150 Eu

4.0

Θ00, K

4.5

r0, Å

[3, 4], are used for formulating the problem of revealing the correlation of these parameters with the position of atom in the periodic table. In determining this correlation, two objectives are pursued, namely, 1) the correction of the parameters of potential of substances the experimental data for which exhibit a certain spread of values and 2) the estimation of the parameters of potential of substances for which no experimental data are available that would enable one to calculate these parameters.

485

100

Yb

50

3.5

Lanthanide compression

130

140

150 160 m, amu

170

0 180

Fig. 1. The coordinate of the minimum of interatomic potential (1, left-hand scale) and Debye temperature at P = 0 and T = 0 K (2, right-hand scale) as functions of atomic mass m (lower scale) and atomic number Za (upper scale) for lanthanides.

Unfortunately, the other regularities for the functions b(m) and a(m) could not be reliably revealed because of wide scatter of experimental data for lanthanides and actinides. Therefore, the regularities obtained for lighter elements will be used in correcting the parameters of interatomic potential of lanthanides and in estimating the parameters of potential (1) of actinides and heavier elements. 3. CORRECTION OF PARAMETERS OF POTENTIAL FOR LANTHANIDES Note that it is the value of c00, i.e., the distance between the centers of nearest atoms in crystal at T = 0 K and P = 0, that is determined most exactly from experimental data. Therefore, the parameter ro exhibits a low percentage of spread of values. The values of L00 and Θ00 are determined less exactly, which results in a wider range of possible values of potential parameter D. The Grueneisen parameter at T = 0 K and P = 0 is least exactly determined, which results in a wider range of indeterminacy for the potential parameters a and b. Figures 1 and 2 give the dependences ro(m), Θ00(m), D(m), and L00(m) for lanthanides. Based on these dependences, one can estimate the parameters of potential for promethium (Pm, Za = 61), which has no stable isotopes and for which no experimental data are available. Two of isotopes of promethium are of most interest, namely, isotope with the mass number m = 145 exhibiting the longest half-life [15] and isotope with the mass number m = 147 which finds practical application. Therefore, we determined the parameters of promethium for these two values of atomic mass.

486

MAGOMEDOV Za 54 56 58 60 62 64 66 68 70 72

D/kb, K

10000

1 2

70000 60000

La

8000

Gd Er

6000

40000 30000

Dy

4000 2000 130

50000

Nd

20000

Sm Eu Yb

140

150 160 m, amu

L00/NAkb, K

12000

170

10000 180

Fig. 2. The depth of the minimum of interatomic potential (1, left-hand scale) and specific energy of atomization at P = 0 and T = 0 K (2, right-hand scale) as functions of atomic mass m (lower scale) and atomic number Za (upper scale) for lanthanides.

In Fig. 1, one can clearly see “lanthanide compression”, i.e., the decrease in ro with increasing m or Za. The dotted lines are plotted in the region of promethium Pm (m = 145–147 amu) for which no experimental data are available. Linear interpolation for Pm gives for m = 145 amu: ro ≅ 3.66 Å, Θ00 ≅ 163 K; for m = 147 amu: ro ≅ 3.63 Å, Θ00 ≅ 166 K. In Fig. 2, one can clearly see “double periodicity”, i.e., the presence of two minima, those for Sm-Eu and for Yb. The vertical dotted lines indicate the position of Pm (m = 145–147 amu) for which no experimental data are available. Linear interpolation for Pm gives for m = 145 amu: D/kb ≅ 6370 K, L00/NAkb ≅ 37 520 K, L00 ≅ 312.0 kJ/mol; for m = 147 amu: D/kb ≅ 5500 K, L00/NAkb ≅ 32 420 K, L00 ≅ 269.6 kJ/mol. The thin horizontal line for L00/NAkb = 39 857.7 K indicates the position of the estimate of sublimation energy for Pm given in the data base of [18]: L00 = 331.393 kJ/mol. The parameters of Pm (L00, Θ00, and ro) obtained by linear interpolation from Figs. 1 and 2 are given in Table 1. These estimates of ro (in Å), namely, ro = 3.63– 3.66, agree well with the predictions of 3.62–3.64 from the monograph of Grigorovich [1, 16]. Our estimate of Debye temperature Θ00 = 163–166 K agrees well with the estimate of 162 K by Lounasmaa [17]. Given in Fig. 2 is the position of the estimate of sublimation energy for Pm from the electronic data base of [18], namely, L00 = 331.393 kJ/mol, which is close to our estimate for m = 145 amu of L00 ≅ 312.0 kJ/mol. Note, however, that the values of sublimation enthalpy in

[18], similar to the initial handbook [6], are approximate because of the absence of experimental data for trans-uranium elements. Unfortunately, the structure of promethium at T = 0 K and P = 0 is not clear: both the hcp (exhibited by the neighbors in the lanthanide series Nd and Sm and by the neighbor in subgroup VIIc Er) and bcc structures are possible. Therefore, the parameters γ00 and a were calculated for these two structures. The exact data for ro(m) and the physical regularities of filling of 4f and 5d electron shells in the case of lanthanides enable one to correct the dependence b(m). Starting with La (5d16s2), the filling of 4f electron shell occurs with invariant external 6s2 configuration of electrons. In so doing, the cation charge in metal lattice from La to Sm remains invariant and equal to 3+. This causes a higher density of inner electron shells and a reduction of atomic volume (“lanthanide compression”) up to Sm (4f 66s2). In the case of Eu (4f 76s2), because of stability of the group of seven unpaired electrons, only the outer two 6s2 electrons turn out to be valence electrons. As a result, the cation charge in the metal lattice of Eu decreases from 3+ to 2+. This causes an abrupt increase in the atomic volume of Eu, which affects the other properties of europium as well. Then, from Gd (5d16s2) to Tm (4f136s2), the filling of 4f electron shell is continued, and the cation charge is again equal to 3+. An abrupt decrease in charge occurs in the case of Yb (4f 146s2) whose 4f shell is completely filled. It is easy to understand that “lanthanide compression” associated with the increase in electron density must result in increasing rigidity of interatomic potential (see Fig. 3). This enables one to correct the values of parameter b (see Table 1). This correction is significant only for Ba, Ce, Sm, Ho, Er, and Hf, which points to inaccuracies in the experimental determination of Grueneisen parameter γ00 for these crystals. Given the corrected value of b and experimentally obtained values of ro, L00, and Θ00, one can select from Eq. (2) a value of γ00, which corresponds to the given value of b, and determine the values of the potential parameter a. The thus corrected (as well as uncorrected) values of b, γ00, and a are given in Table 1; the dependence a(m) obtained both from uncorrected values of b and γ00 and from those corrected in the manner described above is given in Fig. 4. The calculations for promethium were performed for two values of atomic mass and for two structures which are possible at T = 0 K and P = 0, namely, hcp and fcc. It was observed that, if the dependence b(m) for lanthanides in Fig. 3 agrees with the dependences ro(m) and Θ00(m) from Fig. 1, the behavior of a(m) in Fig. 4 correlates with the dependence of energy parameters D(m) and L00(m) given in Fig. 2. One can see in Figs. 2 and 4 that the decrease in the depth of potential D(m) results in the increase in parameter a from La to Sm. The dependence D(m) then increases, and the function a(m) decreases from Sm to Tb. The behaviors of these HIGH TEMPERATURE

Vol. 46

No. 4

2008

55

56

57

58

Cs

Ba

α-La

α-Ce

Vol. 46

No. 4

59

60

61

62

63

64

65

α-Pr

α-Nd

Pm

α-Sm

Eu

α-Gd

α-Tb

Za

1)

HIGH TEMPERATURE

Element

2008

158.9254

157.25

151.96

150.36

(147)

(145)

144.24

140.9077

140.12

138.9055

137.33

132.905

m, amu

12*

12*

8

12*

(8)

(12*)

(8)

(12*)

12*

12*

12

12*

8

8

k3

390.95 [6]

400.45 [6]

179.76 [6]

206.23 [6]

269.6

312.0

327.86 [6]

357.08 [6]

423.21 [10]

430.34 [6]

182.72 [10]

78.40 [6]

47020.82

48163.42

21620.32

24803.95

32420

37520

39432.78

42947.17

50900.83

51758.38

21976.32

9429.42

165

170

127

169

166

163

163

146

146

142

110.5

40

L00 , kJ/mol L00 /NAkb , K Θ00 , K [5]

1.929

1.77 [12]

1.923

1.79 [12]

1.329–1.412

0.434

0.450

3.52 [9]

3.58 [9]

3.96 [9]

7867.74

8059.11

5440.80

7.22

9.41

9.33

9.31

9.60

8.64

9.56

8.76

6.0–6.5

10.04

0.373

4165.68

8153.1

5435.4

9427.2

6284.8

6602.69

9.18

9.30

23)

3.59 [9]

3.63

3.66

3.66 [9]

7185.24

9.54

0.864

0.425

0.637

0.354

0.531

0.505

3.63 [9]

1.916

1.53 [12]

1.896

1.884

1.88 [12]

1.860

1.88 [12]

0.372

4.85

9.12

8.87

4.98

8.22

6.72

b

9.14

8510.85

8653.02

5525.16

2368.61

D/kb , K

1.854

3.6496 [10]

3.73 [9]

4.3171 [10]

5.235 [9]

ro , Å

8.22

0.314

0.292

0.278

0.085

Aw(1), K

1.70 [12]

1.14 [13]

1.85 [12]

1.807

1.16 [11]

1.72)

1.45 [7, 8]

γ00

2.82

2.91

2.94

3.02

3.67–3.74

3.75

4.31

4.20

3.61

3.23

3.12

2.46

2.45

2.18

2.28

2.45

2.12

3.34

3.18

1.91

2.06

a

Table 1. Experimental data and the parameters of Mie-Lennard-Jones interatomic potential (1), calculated by these data using Eq. (2) for f-transition metals – lanthanides

THE CORRELATION OF THE PARAMETERS OF INTERATOMIC INTERACTION 487

1)

HIGH TEMPERATURE

Vol. 46

180.948

178.49

174.967

173.04

168.9342

167.26

164.93

162.5

m, amu

8

12*

12*

12

12*

12*

12*

12*

k3

706.26 [10]

618.81 [10]

428.87 [6]

154.17 [6]

233.57 [6]

322.17 [10]

302.43 [6]

297.06 [10]

84944.17

74426.28

51581.58

18542.52

28092.22

38748.43

36374.23

35728.37

247

256

210

117

200

193

188

183

L00 , kJ/mol L00 /NAkb , K Θ00 , K [5]

2.19 [13]

1.50 [14]

1.913–1.995

1.04 [8]

1.81 [12]

1.417–1.495

1.79 [12]

1.21 [14]

1.98 [12]

1.977

1.54 [12]

1.964

1.57 [12]

0.359

0.660

0.641

0.554

1.068

0.721

0.729

2.8648 [10]

3.1569 [10]

3.43 [9]

3.88 [9]

3.54 [9]

3.503 [10]

3.49 [9]

21305.51

12452.38

8636.31

3112.36

4719.54

6494.26

6097.62

6.12

b

11.16

7.01

9.5–10.0

4.25

8.88

6.5–7.0

8.78

5.29

9.93

9.90

7.27

9.82

7.45

9.74

5989.04

D/kb , K

1.951

3.5458 [10]

ro , Å

9.31

0.703

Aw(1), K

1.88 [12]

1.35 [13]

γ00

2.52

2.90

3.50–3.46

3.07

3.79

3.86–3.92

3.98

3.61

5.04

4.00

3.99

3.97

3.99

3.94

3.96

3.79

a

Za—atomic number. Given in bold italic in lower lines for b are the parameters determined from Fig. 3. For Pm, the estimates of L00 , Θ00 , ro , and b are obtained from Figs. 1–3, and the calculations of γ00 and a are performed for hcp and bcc structures. The values of mass in brackets (m) denote the mass of the most stable isotope of promethium Pm. The data on k3 are borrowed from [1, 5, 18]. The values of k3 = 12 and k3 = 8 correspond to fcc and bcc structures, respectively. The asterisk at k3 = 12* indicates the hcp structure. 2) The value was determined in [3] by studying the graphic dependence of γ on ln(m). 00 3) The value is taken to be the average value of Grueneisen parameter for metals: 1 ≤ γ00 ≤ 3 [7].

1)

73

Ta

70

α-Yb

72

69

Tm

α-Hf

68

Er

71

67

Ho

Lu

66

Za

α-Dy

Element

Table 1. (Contd.)

488 MAGOMEDOV

No. 4

2008

THE CORRELATION OF THE PARAMETERS OF INTERATOMIC INTERACTION b 11

Ta

La

9

4.5

Pr Nd

Ce

8

Sm

Lu Dy

Cs

Dy

4.0

Ho Er

Sm

Er

Ba

3.5

Yb

Nd

Hf

3.0

6

Yb

Ba

Gd

2.5

Ta

Tb

Pr Cs

5

Hf

4

2.0

La

130 140

Lu

Eu

Pm

Gd Tb

7

130

1 2

Tm

Tm

Eu

10

a 5.0

489

150

160

170

Fig. 3. The parameter of rigidity of interatomic potential (1) as a function of atomic mass m for lanthanides. The vertical lines indicate the region of scatter of experimental estimates of this parameter. The continuous line indicates the corrected dependence b(m) obtained in view of the increase in electron density of inner shells in the case of “lanthanide compression”.

4. ESTIMATION OF PARAMETERS OF POTENTIAL FOR ACTINIDES Figure 5 gives ro as a function of atomic number Za for lanthanides and physically similar actinides. The atomic weight of an element is usually calculated by its isotopic composition. This method is justified if the isotopes are stable or if they are characterized (as in the case of uranium) by long decay periods. However, this approach is invalid for artificially produced elements, because the isotopic composition of such elements depends on the method of their synthesis. Therefore, the calculations in the case of such elements are performed for the most stable isotope. As a result, for actinides we will use the dependence of the atomic number of element in the periodic system Za rather than on the atomic weight. Because the dependence ro(Za) for actinides must (by virtue of the periodic law) agree with the dependence ro(Za) for lanthanides, prediction was made for the values of coordinate of the minimum of interatomic potential (1) for elements from curium (Cm, Za = 96) to dubnium (Db, Za = 105). Figure 6 gives the coordinate of the minimum of interatomic potential ro as a function of atomic number Za for elements of subgroups IVb Vol. 46

No. 4

150

160

170

2008

180 m, amu

Fig. 4. The parameter of long-range action of interatomic potential (1) as a function of atomic mass m for lanthanides. The vertical lines indicate the region of scatter of experimental estimates of this parameter: (1) uncorrected data, (2) dependence a(m) calculated from the corrected dependence b(m).

(hcp – Ti, Zr, Hf, Rf), data from [3, 10] and Fig. 5 (for Rf); Vb (bcc – V, Nb, Ta, Db), data from [3, 10] and Fig. 5 (for Db); and Ib (fcc – Cu, Ag, Au, Rg), data from [3, 10] and prediction for Rg. The data for actinides and roentgenium (Rg, Za = 111) are given in Table 2. It follows from Fig. 6 that, starting with Za > 60, the value of ro monotonically decreases with increasing atomic

86 88 5.5 5.0 4.5 r0, Å

dependences agree in what follows as well; in so doing, a plateau is present on both dependences in the Dy-HoEr interval. This is indicative of the fact that the deeper pair potential of interatomic interaction for lanthanides decreases with distance (at r > ro) less markedly, and the more “shallow” potential decreases more actively.

HIGH TEMPERATURE

140

180 m, amu

90

Za 92 94 96

98 100 102 104 106

1 2 3

4.0 3.5 3.0 2.5 54 56

58 60 62 64

66

68

70 72 74

Fig. 5. The coordinate of the minimum of interatomic potential for lanthanides (1, lower scale) and actinides (2, upper scale) as a function of atomic number Za; (3) predicted dependence ro(Za) for elements ranging from curium (Cm, Za = 96) to dubnium (Db, Za = 105) for which no experimental data are available.

490

MAGOMEDOV

r0, Å 3.2 3.1 3.0 2.9 2.8

IVb 4

6

2.7 2.6 2.5

r0, Å 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8

1 2 3

Ib

Vb 5

2.4 20

40

60

80

100

120 Za

Fig. 6. The coordinate of the minimum of interatomic potential as a function of atomic number Za for elements of subgroups (1) IVb (hcp—Ti-22, Zr-40, Hf-72, Rf-104), data from [10] and Fig. 5 (for Rf); (2) Vb (bcc—V-23, Nb-41, Ta-73, Db-105), data from [10] and Fig. 5 (for Db); and (3) Ib (fcc—Cu-29, Ag-47, Au-79, Rg-111), data from [10] and prediction for Rg. The lines indicate the quadratic approximation of dependence ro(Za): (4) for subgroup 2

IVb—ro = 2.45883 + 0.02745Za – 2.42243 × 10–4 Z a with the reliability factor Rcor = 0.98915; (5) for subgroup Vb— 2

ro = 2.07916 + 0.0289Za – 2.46434 × 10–4 Z a with Rcor = 0.99731; (6) for subgroup Ib—ro = 1.74341 + 0.03629Za – 2

2.74196 × 10–4 Z a with Rcor = 0.98861.

number Za for elements from the same subgroup. This must result in a monotonic increase in function b(Za) at Za > 60 for elements from the same subgroup. Figure 7 gives the coordinate of minimum ro and the parameter of rigidity b of interatomic potential as functions of atomic number Za for elements of subgroup VIIIc (the ro data from [3, 9], Table 1, and Fig. 5 (for Md)). Similar dependences are given in Fig. 8 for elements of subgroup VIc (the ro data from [9], Table 1, and Fig. 5 (for Es)). Based on these dependences, one can conclude that the dependence ro(Za) is inversely proportional to dependence b(Za) for elements from the same subgroup starting with Za > 60. Linear extrapolation of the dependences b(Za) in Figs. 7 and 8 was used to determine the values of the parameter of rigidity of potential (1): b(Md) = 10.36, b(α-U) = 11.7, and b(Es) = 12.2. Linear extrapolation of the dependences b(Za), which is given in Fig. 9, was used to determine the values of the parameter of rigidity of potential (1): b(Am) = 8.5, b(No) = 9, b(Cm) = 9.3, b(Lr) = 9.4, b(Db) = 14.7, b(Pa) = 11.7, b(Cf) = 12.2, b(Np) = 11.9, and b(Fm) = 12.5. The thus obtained values of b and the data from Table 2 were used to derive the dependence b(Za) for actinides, given in Fig. 10. Based on the simi-

1

b 10.4 10.3 10.2 10.1 10.0 9.9 9.8 9.7 9.6 9.5

2

VIIIc

60

70

80 Za

90

100

Fig. 7. The coordinate of the minimum and the parameter of rigidity of interatomic potential as functions of atomic number Za for elements of subgroup VIIIc: hcp—α-Sm-62, Tm-22-69, monc—α-Pu-94, and Md-101, ro data from [9], Table 1, and Fig. 5 (for Md): (1) quadratic approximation of 2

the form ro = 2.58808 + 0.03691Za – 3.34821 × 10–4 Z a with the reliability factor Rcor = 1; (2) linear extrapolation of the dependence b(Za).

larity of the dependences b(Za) for lanthanides and actinides, linear interpolation was used to estimate the values of the parameter of rigidity of potential: b(bccRa-88) = 9.8, b(fcc-Ac-89) = 10.5, b(fcc-α-Th-90) = 11, b(hcp-α-Bk-97) = 10.8, and b(hcp-α-Rf-104) = 12. The thus determined values of b are given in the bottom lines of Table 2. Given the value of b and the values of ro, L00, and Θ00 given in Table 2, one can select from Eq. (2) a value of γ00, which corresponds to the given value of b, and determine the value of the parameter a of long-range action of potential. The thus found values of b, γ00, and a are given in Table 2 for elements from Ra-88 to Np-93. Because the “zero-oscillation” energy for crystals of elements from Am-95 onwards is extremely low compared to the potential depth D (i.e., Fb  1 and Awξ3/Θ00 ≈ 0), the expression for the Grueneisen parameter, obtained in [22], transforms to the formula derived by Grueneisen himself (see [14, p. 14]), γ00 = [(b + 2)/6][1 + (Awξ3/Θ00)]–1 ≅ (b + 2)/6, and the expression for the depth of the minimum of potential well may be simplified to D/kb = (2/k3)[(L00/NAkb) + (9Θ00/8)] ≅ (2/k3)(L00/NAkb). The thus calculated values of γ00 and D/kb for elements from Am-95 and to Db-105 are given in Table 2. Because the dependence a(m) for lanthanides increases from La-57 to Sm-62 (see Fig. 4), a similar increase in a(m) is to be expected for physically similar actinides from Ac-89 to Pu-94 as well. However, for Np-93, the value of a = 1.11 drops out of the predicted HIGH TEMPERATURE

Vol. 46

No. 4

2008

THE CORRELATION OF THE PARAMETERS OF INTERATOMIC INTERACTION r0, Å 3.8

b 13.0

b 15

3.6

12.5

14

12.0

13

11.5

12

2

1

3.2

3

11.0

3.0

4

VIc

2.8

10.5 10.0

2.6 2.4

9.5 60

65

70

75

80 Za

85

90

95 100

2

b

approximation ro = 4.42888 – 0.00348Za – 1.5625 × 10–4 Z a (Rcor = 1); (3) linear extrapolation of the dependence b(Za); (4) scatter of data for the parameter b(α-U): 7–12.5 [3].

86 88 90 92 94 96 98 100 102 104 106 15 1 14 14 2 13 3 12 12 11 10 10 9 8 8 7 6 6 54 56 58 60 62 64 66 68 70 72 74 Za

Fig. 10. The parameter of rigidity of interatomic potential (1) as a function of atomic number Za for lanthanides (1, lower scale, data from Table 1 and Fig. 3) and physically similar actinides (2, upper scale, data from Table 1 and Figs. 7–9; (3) linear interpolations of dependence b(m) for actinides.

rise of function a(m). Linear interpolation of the dependence a(m) given in Fig. 11 was used to determine a(Np) = 3.34. In view of the fact that the atomic weight of Am-95 is smaller than that of Pu-94, linear interpolation of the dependence a(m) produced a(Am) = 3.52. The thus obtained potential parameters and the method described in [4, 22] were used to calculate the values of

HIGH TEMPERATURE

Vol. 46

No. 4

2008

10

IIIc

9

Vb IIc

8 7 6

Fig. 8. The coordinate of the minimum ro and the parameter of rigidity b of interatomic potential as functions of atomic number Za for elements of subgroup VIc: hcp—α-Nd-60 and Ho-67, orthr—α-U-92, and Es-99, ro data from [9], Table 1, and Fig. 5 (for Es): (1) linear approximation ro = 5.37451 + 0.02833Za (Rcor = 0.99926); (2) quadratic

Vc + VIIc

11

30

40

50

60

70

80

90 100 110 Za

Fig. 9. The parameter of rigidity of interatomic potential as a function of atomic number Za for elements of the following subgroups: IIc (bcc—Eu-63; hcp—α-Yb-70 and Am95; Ho-102), IIIc (hcp—α-Gd-64; Lu-71; α-Cm-96; Lr103), Vb (bcc—V-23; Nb-41; Ta-73; Db-105), Vc(hcp—αPu-59; α-Dy-66; bct—α-Pa-91; hcp—α-Cf-91), VIIc (Pm-61; hcp—α-Er-68; orthr—α-Np-93; Fm-100), data from [3] and Table 1. The lines indicate linear extrapolation of the dependence b(Za).

225

230

235

240

245 1 2

4.5 Ra

Sm

4.0 3.5

Np

Ba

U

a

3.4

491

3.0

Pu Am

Th

2.5 Pr La

2.0 1.5

Ac

135

Pa

140 145 m, amu

150

Fig. 11. The parameter of long-range action of interatomic potential (1) as a function of atomic mass m for lanthanides (1, lower scale) and actinides (2, upper scale) in the region of first minimum and maximum of double periodicity.

Debye temperature at T = 0 K and P = 0, which are given in Table 2. The found dependences point to the fundamental correlation between the parameters of interatomic interaction and the position of atom in the periodic table and enable one to predict these parameters of potential (1) for elements for which they are not known. In addition, the fact that the dependences of parameters of

90

91

92

α-Th

α-Pa

α-U

HIGH TEMPERATURE

Vol. 46

α-Cm

Am

96

95

94

89

Ac

α-Pu

88

Ra

93

87

Fr

α-Np

Za

Element

247.0703

243.0614

244.064

237.048

238.0289

231.036

232.038

227.0278

226.0254

223.0197

m, amu

12*

12*

11

387.647 [18]

284.047 [18]

348.242 [18]

348.48 [6]

464.834 [18]

7

monc

465.15 [6]

535.418 [18]

7

orthr

535.78 [6]

606.751 [18]

10 orthr

607.16 [6]

594.441 [18]

575.63 [10]

410.529 [18]

410.8 [6]

163.08 [6]

74.973 [18]

72.89 [6]

L00 , kJ/mol

bct

12

12

(8)

8

k3

46 623.56

34 163.25

41912.82

55945.09

64439.99

73025.09

69232.88

49408.24

19614.16

8766.72

L00 /NAkb , K

1282)

116 [21]

150 [19]

171 [5]

1.88

1.75

2.87 [21]

2.04 [8]

0.3642)

0.221

0.480

No. 4

3.1

3.5 [16]

3.5 [16]; 3.614 [2]

3.46–3.64 [1, p. 239]

3.61 [9]

3.15 [1, p. 328]

3.1 [2, 9]

2.59 [2]

2.62 [9] 3.1 [1, p. 238] 0.4342)

0.115

2352)

2.31

2.754 [2]

2.28

2.75 [9]

3.26 [1, p. 238]

3.21 [2, 9]

3.595 [1, p. 239; 2]

3.5898 [10]

3.755 [2]

4.1 [1, p. 152]

3.76 [9]

4.7 [1, p. 152]

4.6**

5.63**

ro , Å

2.77–3.12 [1, p. 238]

0.411

0.216

0.288

0.142

0.202

0.053

Aw(1), K

2.42 [20]

1.50 [8]

2.28

2.16

1.34 [8, 13]

2.08

1.98

1.85**

1.65**

γ00

188 [19]

121 [5]

210 [19]

246 [5]

159 [5]

100 [19]

163 [5]

96.8**

89**

30.5**

Θ00 , K

7770.59

5693.88

7644.24

7655.49

16023.21

18490.50

14640.79

11569.38

8252.86

4928.57

2200.26

D/kb , K

9.3

8.5

15.25

10.27

11.9

11.7

12.55

7.02

11.7

11.0

6.05

10.5

9.8

9.12

7.92

b

3.521)

1.55

3.53

3.331)

1.11

3.47

3.38

3.70

2.00

2.70

3.02

1.66

4.05

2.18

2.30

a

Table 2. Experimental data and the parameters of Mie-Lennard-Jones interatomic potential (1), calculated by these data using Eq. (2) for f-transition metals—actinides

492 MAGOMEDOV

2008

98

α-Cf

2008

103

104

105

111

Lr

α-Rf

Db

Rg

(272)

(262)

(261)

260.1054

259.1009

258.0986

257.0951

252.0828

251.0796

247.0703

m, amu

(12)

(8)

(12*)

(12*)

(12*)

12*

12*

k3

129.704 [18]3)

133.888 [18]3)

140.243 [18]

165.347 [18]

309.616 [18]3)

L00 , kJ/mol

19886.82

37238.52

L00 /NAkb , K

Θ00 , K

2.78

2.33

1.90

1.83

2.06

2.42

2.37

2.37

2.13

γ00 Aw(1), K

2.4

2.4

(2.8) [16]

2.7

(2.96) [16]

3.0

(3.26) [16]

3.4

(3.7) [16]

2.9

(3.72) [16]

2.5

(3.74) [16]

2.55

(3.75) [16]

2.7

3.35 [16]

2.85

3.4 [16]

ro , Å

3314.47

6206.42

D/kb , K

14.7

12.0

9.4

9.0

10.36

12.5

12.2

12.2

10.8

b

a

Given in bold italic in lower lines for ro and b are the parameters determined from Figs. 5–10. Underlined are the values of ro, L00 , and Θ00 used in constructing Figs 5 and 6 and in the calculations of a. The values of mass in brackets (m) denote the mass of the most stable isotope of the element [15]. The data on k3 are borrowed from [5, 6, 16, 18]. The letters over k3 indicate the following structures: monc—simple monoclinic (α-Pu—the densest of six phases of plutonium): k3 ≅ 11; bct—body-centered tetragonal (of the type of α-Hg, α-Pa): k3 ≅ 10; orthr—orthorhombic (of the type of α-Ga, α-U, α-Np): k3 ≅ 7; k3 = 12*—hcp structure. 1) Determined from Fig. 11. 2) Calculated by the parameters of potential. 3) Enthalpy of formation at 298 K. ** the values are determined by studying the graphic dependence on atomic mass (on ln(m) for ro and γ00 and on (1/m1/2) for Θ00) in [3]

102

α-No

No. 4

101

Vol. 46

Md

100

99

97

α-Bk

Fm

HIGH TEMPERATURE

Es

Za

Element

Table 2. (Contd.)

THE CORRELATION OF THE PARAMETERS OF INTERATOMIC INTERACTION 493

494

MAGOMEDOV

potential on atomic weight (or atomic number) may be used for revealing the position of atom in the periodic table points to the correctness of the method of determining these parameters of the Mie–Lennard–Jones potential (1) as suggested in [3, 4]. CONCLUSIONS 1. The correlation has been revealed of four parameters of interatomic interaction of Mie–Lennard–Jones potential (1) with the position of atom in the periodic table. 2. The parameters of interatomic potential, the Debye temperature, and the Grueneisen parameter for promethium (Pm-61) have been estimated for two versions of structure at T = 0 K P = 0, namely, hcp and bcc. 3. Proceeding from the correlation between the parameters of interatomic potential ro and 1/b, correct values of b, γ00, and a have been predicted for lanthanides. 4. The similarity in double periodicity for lanthanides and actinides was used to estimate the values of coordinate of the minimum of interatomic potential ro for elements from curium (Cm-96) to dubnium (Db105). 5. Proceeding from the fact that, at Za > 60, the value of ro monotonically decreases with increasing atomic number Za for elements from the same subgroup, the value of ro for roentgenium (Rg-111) has been estimated. 6. Proceeding from the correlation between the parameters of interatomic potential ro and 1/b (at Za > 60), linear extrapolation of the dependence b(Za) for elements from the same subgroup was performed for predicting correct values of b, γ00, and a for elements from francium (Fr-87) to dubnium (Db-105). 7. The thus obtained values of parameters of interatomic potential were used to calculate the Debye temperature and the Grueneisen parameter for neptunium (Np-93) and americium (Am-95). ACKNOWLEDGMENTS I am grateful to A.D. Filenko, K.N. Magomedov, Z.M. Surkhaeva, and M.M. Gadzhieva for valuable discussions and kind assistance. REFERENCES 1. Grigorovich, V.K., Periodicheskii zakon Mendeleeva i elektronnoe stroyenie metallov (Mendeleev’s Periodic Law and the Electron Structure of Metals), Moscow: Nauka, 1966.

2. Regel’, A.R. and Glazov, V.M., Periodicheskii zakon i fizicheskie svoistva elektronnykh rasplavov (The Periodic Law and Physical Properties of Electron Melts), Moscow: Nauka, 1978. 3. Magomedov, M.N., Teplofiz. Vys. Temp., 2006, vol. 44, no. 4, p. 518 (High Temp. (Engl. transl.), vol. 44, no. 4). 4. Magomedov, M.N., Teplofiz. Vys. Temp., 2005, vol. 43, no. 2, p. 202 (High Temp. (Engl. transl.), vol. 43, no. 2, p. 192). 5. Zinov’ev, V.E., Teplofizicheskie svoistva metallov pri vysokikh temperaturakh. Spravochnik (The Thermal Properties of Metals at High Temperatures: A Reference Book), Moscow: Metallurgiya, 1989. 6. Termicheskie konstanty veshchestv. Spravochnik v 10-ti vypuskakh (Thermal Constants of Substances: A Reference Book in 10 issues), Glushko, V.P., Ed., Moscow: VINITI, 1965–1982. 7. Tsagareishvili, D.Sh., Metody raschota termicheskikh i uprugikh svoistv kristallicheskikh neorganicheskikh veshchestv (Methods for the Calculation of the Thermal and Elastic Properties of Crystal Inorganic Substances), Tbilisi: Metsniereba, 1977. 8. Ogorodnikov, V.V. and Rogovoi, Yu.I., Fiz. Tekh. Vys. Davlenii, 1992, vol. 2, no. 4, p. 54. 9. Kittel, C., Introduction to Solid State Physics, New York: Wiley, 1976. Translated under the title Vvedenie v fiziku tverdogo tela, Moscow: Nauka, 1978. 10. Zhen Shu. and Davies, G.J., Physica Status Solidi (a), 1983, vol. 78, no. 2, p. 595. 11. Anderson, M.S., Swenson, C.A., and Peterson, D.T., Phys. Rev. B, 1990, vol. 41, no. 6, p. 3329. 12. Parshukov, A.V., Fiz. Tverd. Tela, 1985, vol. 27, no. 4, p. 1228. 13. Rodionov, K.N., Fiz. Met. Metalloved., 1967, vol. 23, no. 6, p. 1008. 14. Novikova, S.I., Teplovoe rashirenie tverdykh tel (Thermal Expansion of Solids), Moscow: Nauka, 1974. 15. CRC Handbook of Chemistry and Physics, Lide, D.R., Ed., London: CRC, 1993–1994. 16. Grigorovich, V.K., Metallicheskaya svyaz’ i struktura metallov (Metallic Bond and the Structure of Metals), Moscow: Nauka, 1988. 17. Lounasmaa, O.V., Phys. Rev., 1964, vol. 133, no. 1A, p. 219. 18. Elektronnoe izdanie: “Baza dannykh: Termicheskie konstanty veshchestv” (Electronic Data Base on Thermal Constants of Substances) httr://www.chem.msu.su/cgibin/tkv2. 19. Ashcroft, N. and Mermin, N., Solid State Physics, New York: Holt, Rinehart, and Winston, 1976. Translated under the title Fizika tverdogo tela, Moscow: Mir, 1979. 20. Neal, T., J. Phys. Chem. Solids, 1977, vol. 38, no. 3, p. 225. 21. Wallace, D.C., Phys. Rev. B, 1998, vol. 58, no. 23, p. 15 433. 22. Magomedov, M.N., Teplofiz. Vys. Temp., 2002, vol. 40, no. 4, p. 586 (High Temp. (Engl. transl.), vol. 40, no. 4, p. 542).

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2008