Computational Tools to Evaluate High Pressure Ionic Prolarizabilities of Complex Solids: a Maple Implementation

Transcription

1 Instituto de Ciências Matemáticas e de Computação ISSN Computational Tools to Evaluate High Pressure Ionic Prolarizabilities of Complex Solids: a Maple Implementation Maria Carolina Monard Alicia Batana Jorge Bruno N ō 191 RELATÓRIOS TÉCNICOS DO ICMC São Carlos Fevereiro/2003 ftp://ftp.icmc.sc.usp.br/pub/biblioteca/rel_tec/rt_191.pdf

2 Computational Tools to Evaluate High Pressure Ionic Prolarizabilities of Complex Solids: a Maple Implementation Maria Carolina Monard Universidade de São Paulo Instituto de Ciências Matemáticas de São Carlos Departamento de Ciências de Computação e Estatística Caixa Postal 668, São Carlos, SP, Brasil mcmonard@icmsc.sc.usp.br Alicia Batana Jorge Bruno Universidad de Buenos Aires Facultad de Ciencias Exactas y Naturales Departamento de Química Inorgánica Analítica y Química Física and Universidad Nacional de La Patagonia San Juan Bosco Facultad de Ciencias Naturales {batana,jbruno}@q1.fcen.uba.ar Abstract An empirical method based on two models, with parameters fitted to ab initio and experimental data, has been developed for the evaluation of the pressure dependence of ionic polarizabilities in ionic crystals. The importance of this method is its predictive nature, particularly useful in the study of more complex materials for which there are no experimental data at high pressures. Both models have been implemented as Maple procedures and experimental results using these procedures on different salts are shown. Fevereiro 2003 Work partially supported by National Research Council CAPES/FINEP, Brasil Work partially supported by CONICET and ANPCyT, Argentina

3 Contents 1 Introduction 1 2 Method 1 3 Maple Implementation Data Structure Model Model Experimental Results 11 5 Conclusions 18 6 Appendix - Experimental Data 19 List of Algorithms 1 Model Model List of Figures 1 α = α e a R 2 for salt LiF Model α = α 3 α = α 1 1+aR 2 for salt LiF Model 2 s = aR 2 for salt LiF Model 2 s = List of Tables 1 List s Elements Mapping Data for LiF in cgs units Data for CaCl 2 in cgs units Results from Model 1 with Data in cgs Units Results from Model 2 for s = 3 with Data in cgs Units Results from Model 2 for s = 4 with Data in cgs Units Results from Model 1 with Data in au Units i

4 8 Results from Model 2 for s = 3 with Data in au Units Results from Model 2 for s = 4 with Data in au Units Experimental Data at Low Temperatures Values of 1 r +, r +, 1 r and r at Low Temperatures in cgs Units ii

5 1 Introduction High pressure studies of ionic materials are of geophysical interest; further importance is that one may also obtain a critical study of the theoretical models involved. The concept of ionic properties appears in these studies, such as polarizabilities of ions in the theories of dielectric constants, refractive index and optical properties of ionic solids. In the study of more complex systems, in general, there are no high pressure data available, thus it is important to have theoretical methods to evaluate these magnitudes; these may be based either on data only at P = 0 or on high pressure data of other solids that have either a cation or an anion in common with the one of interest. In this work we propose an alternative way, based on two different models [8], for evaluating high pressure ionics polarizabilities in more complex solids than the ones we used in previous work [1, 2, 9, 10, 11, 12, 13, 14]. Both models have been implemented as Maple procedures and experimental results obtained using different salts are shown and discussed. This work is organized as follows: in Section 2 the method proposed is presented and in Section 3 the corresponding Maple implementation is described. Experimental results are shown in Section 4 and conclusions are presented in Section 5. 2 Method Ab initio calculations for complex materials at high pressures are time consuming and expensive. We therefore propose to extend to these systems the behavior obtained in previous ab initio calculations of simple systems [4, 5, 6, 8]. In these studies it is shown that the anion polarizability curves α vs R, can be represented by different models equations 1 and 2 α α = e a R 2 (1) where α α = (1 + ar s ) for s = 3 or s = 4 (2) R is the equilibrium anion-cation distance (pressure p = 0); α is the value of α at R a is a parameter for a given anion in a given salt. For a given salt we propose to evaluate the parameter a that appears in both models by knowledge of α at the equilibrium R and α that should be evaluated independently. R is therefore obtained by diferentiating at the equilibrium R. If we know the value of the compressibility χ T one may evaluate as = R ( χ T R 3 ) (3) 1

6 we propose to use the method we described previ- To evaluate the cation contribution α + ously [2] equation 4 1 α + α + 1 α = 3( 1 r + r + 1 r r ) (4) where r + and r are the cation and anion radii respectively; r + and r are their pressure derivatives. The method has been implemented in Maple 6 [3], a comprehensive computer system for advanced mathematics. Maple 6 includes facilities for interactive algebra, calculus, discrete mathematics, graphics, numerical computation and many other areas of mathematics. It also provides a unique environment for rapid development of mathematical programs using its vast library of built-in functions and operations. 3 Maple Implementation Both models have been implemented as Maple procedures. In what follows we shall refer as Model 1 and Model 2 to the models described by equation 1 and equation 2 respectively. For Model 1 only the data related to the salt needs to be informed. For Model 2 the data related to the salt and the aditional information related to the value of s = 3 or s = 4 are needed. Furthermore, for each model we want to calculate using equation 3 as well as α + using equation 4. Thus, as we would like the procedures to execute interactively as well as in batch, we decided to keep together all data needed for the calculations related to each individual salt, as described in next section. 3.1 Data Structure Maple has a rich set of build-in data structures and we decided to use the list data structure to group each salt s data. A list is an ordered sequence of distinct expressions enclosed in square brackets. The elements of a list may be extracted via the selection operation. Thus, if L is a list then the ith element of L can be obtained by L[i]. In our implementation, a salt is described by a list of 11 elements where each element in the list is indexed only through the following defined Maple constants > alfaminus:=1: Requil:=2: xi:=3: alfaminusrinfinity:= 4: a:=5: SaltName:=6: alfaplus:=7: rplus:=8: rminus:=9: DerRPlusDp:=10: DerRMinusDp:=11: Table 1 shows the mapping between these Maple constants and the corresponding information held by these list s elements. 2

7 Constant Data alfaminus α Requil R xi χ T alfaminusrinfinity α a a SaltName Salt s name alfaplus α + rplus r + rminus r DerRPlusDp r + DerRMinusDp r Table 1: List s Elements Mapping LiF_1:=[0.89, 1.996, 1.43, , -100, "LiF_1", 0.03, 0.60, 1.36, , ]: Table 2: Data for LiF in cgs units This data structure allows to identify any salts s data through its name. For example, the data related to salt LiF has been defined (cgs units) as shown in Table 2 thus, the value of α for this salt is given by LiF_1[alfaMinus], the value of R at equilibrium by LiF_1[Requil], the value of χ T by LiF_1[xi] and so on Table 1. Observe that the 5th element of the list is negative (-100) corresponding to the value of a, this means that the value of a for this salt is unknown. Also, a 4th element negative value means that the value of α for that salt is unknown. The Maple implementation of both models is described next. 3.2 Model 1 The Model 1 algorithm is presented in Algorithm 1 Algorithm 1 for a given salt is implemented by Model_1/1 Maple procedure, where: <arg-1> = Salt > Model_1 := proc(salt::list) > local Calc_a, Calc_alfaMinusRinfinity, > todraw, DerAlfMinusDR, DerAlfMinusDp,DerAlfPlusDp; > print("model 1 for salt ", Salt[SaltName],Salt); 3

8 Algorithm 1: Model 1 Require: Salt 1: procedure Model 1(Salt) 2: if α is known then 3: a := R 2 (ln α ln α + ) 4: else 5: α := α e a R 2 6: end if 7: todraw(r) := α e a R 2 Model 1 equation 1 8: R α + todraw(r) := R = 9: R ( χ T R 3 ) {at the equilibrium R equation 3} 10: := α +( 1 α + 3( 1 r + r + 1 r r )) {at the equilibrium R equation 4} 11: print all values 12: plot todraw(r) and at the equilibrium R 13: end > if Salt[alfaMinusRinfinity] > 0 > then Calc_alfaMinusRinfinity := Salt[alfaMinusRinfinity]; > Calc_a := calculate_aorrequilmodel_1(salt); > print("dataalfaminusrinfinity=", Calc_alfaMinusRinfinity, > "Calculated a=", Calc_a) > else Calc_a := Salt[a]; > Calc_alfaMinusRinfinity := calculate_aorrequilmodel_1(salt); > print("calculatedalfaminusrinfinity=",calc_alfaminusrinfinity,"data > a=", Calc_a) fi; > todraw := alfminusmodel_1(calc_alfaminusrinfinity,calc_a,r); > print("todraw=",todraw); > DerAlfMinusDR := eval(diff(todraw,r),r=salt[requil]); > print("deralfminusdr=", DerAlfMinusDR); > DerAlfMinusDp := -DerAlfMinusDR*Salt[xi]*Salt[Requil]/3; > print("deralfminusdp=", DerAlfMinusDp); > DerAlfPlusDp:=CationPressureDerivative(Salt,DerAlfMinusDp); > print("deralfplusdp",deralfplusdp); > plot({[[salt[requil],0],[salt[requil],eval(todraw,r=salt[requil])]], > todraw },R=0.5..4*Salt[Requil],title=Salt[SaltName]); > end: Model_1/1 calls the following three Maple procedures 1. calculate_aorrequilmodel_1/1 2. alfminusmodel_1/2 4

9 3. CationPressureDerivative/2 which is also called by Model_2/2 Procedure calculate_aorrequilmodel_1/1 calculates, for a given salt, the value of a if the value of α is known else, the value of a is known and it calculates the value of α using, in both cases, equation 1 for Model 1. > calculate_aorrequilmodel_1:=proc(salt::list) > if Salt[alfaMinusRinfinity] > 0 > then -(Salt[Requil]^2)* > (ln(salt[alfaminus])-ln(salt[alfaminusrinfinity])) > else Salt[alfaMinus]*exp(Salt[a]/(Salt[Requil]^2)) fi end: Procedure alfminusmodel_1/2 calculates the value of α related to equation 1 for Model 1, where: <arg-1> α <arg-2> a <arg-3> free variable used for plotting > alfminusmodel_1:=proc(alfinf::float,a::float,r) > AlfInf*exp(-a/(R^2)) end: Procedure CationPressureDerivative/2 calculates the cation pressure derivative α + (equation 4) of a given salt, given the anion pressure derivative (equation 3) previously computed. This procedure is used for Model 1 and 2 procedures. <arg-1> salt <arg-2> > CationPressureDerivative := proc(salt::list,deralfminusdp::float) > eval(salt[alfaplus]*(3*(salt[derrplusdp]/salt[rplus]- > Salt[DerRMinusDp]/Salt[rMinus]) + DerAlfMinusDp/Salt[alfaMinus])) > end: In order to illustrate it follows an execution example of Model_1/1 Maple procedure for salt LiF Table 2 pg.3. The call > Model_1(LiF_1); 5

10 has the following output as well as the graph shown in Figure 1 Model 1 for salt, LiF 1, [.89, 1.996, 1.43, , 100, LiF 1,.03,.60, 1.36,.2784,.6709] Data alfaminusrinfinity=, , Calculated a=, todraw=, e ( R 2 ) DerAlfMinusDR=, DerAlfMinusDp=, DerAlfPlusDp, Figure 1: α = α e a R 2 for salt LiF Model 1 Furthermore, as Maple handles symbols, it is possible to execute Maple procedures although some data is not available. As an example, consider the case of salt CaCl 2 where the value of χ T is unknown. The data related to this salt has been defined (cgs units) as shown in Table 3. In this case, it is possible to execute Model 1 procedure obtaining some partial results. CaCl2_1:=[2.8740, , xiunknown, , -100, "CaCl2_1*",.4732,.99, 1.81, , ]; > Model_1(CaCl2_1); Table 3: Data for CaCl 2 in cgs units Model 1 for salt, CaCl2 1*, [2.8740, , xiunknown, , 100, CaCl2 1*,.4732,.99, 1.81,.2194, ] Data alfaminusrinfinity=, , Calculated a=, todraw=, e ( R 2 ) DerAlfMinusDR=,

11 DerAlfMinusDp=, xiunknown Error, (in Model_1) CationPressureDerivative expects its 2nd argument, DerAlfMinusDp, to be of type float, but received *xiunknown 3.3 Model 2 The Model 2 algorithm is presented in Algorithm 2 Require: Salt, s 1: procedure Model 2(Salt,s) 2: if α is known then Algorithm 2: Model 2 3: a := α α α R s 4: else 5: α := α (1 + R 2 ) 6: end if 7: todraw(r) := α 1+aR {Model 2 equation 2} s α 8: todraw(r) R := R α 9: = R ( χ T R 3 ) {at the equilibrium R equation 3} α 10: + := α +( 1 α + 3( 1 r + r + 1 r r )) {at the equilibrium R equation 4} 11: print all values 12: plot todraw(r) and at the equilibrium R 13: end Algorithm 2 for a given salt and s = 3 or s = 4 is implemented by Model_2/1 Maple procedure, where: <arg-1> = Salt <arg-2> = s value for Model 2. Should be set as 3 or 4 otherwise the procedure informs the erro. > Model_2 := proc(salt::list, ModelSet::integer) > local Calc_a, Calc_alfaMinusRinfinity, > todraw, DerAlfMinusDR, DerAlfMinusDp,DerAlfPlusDp; > if ModelSet=3 then print("model 2 for salt ", Salt[SaltName],Salt) > elif ModelSet=4 then print("model 3 for salt ", > Salt[SaltName],Salt); print("s=",modelset) > else print("error ModelSet should be 3 or 4!!!!") fi; 7

12 > if Salt[alfaMinusRinfinity] > 0 > then Calc_alfaMinusRinfinity := Salt[alfaMinusRinfinity]; > Calc_a := calculate_aorrequilmodel_2(salt,modelset); > print("data alfaminusrinfinity=", > Calc_alfaMinusRinfinity, "Calculated a=",calc_a) > else Calc_a :=Salt[a]; > Calc_alfaMinusRinfinity := calculate_aorrequilmodel_2(salt,modelset); > print("calculated alfaminusrinfinity=",calc_alfaminusrinfinity,"data > a=", Calc_a) fi; > todraw := alfminusmodel_2(calc_alfaminusrinfinity, > Calc_a,R,ModelSet); > print("todraw=",todraw); > DerAlfMinusDR := eval(diff(todraw,r),r=salt[requil]); > print("deralfminusdr=", DerAlfMinusDR); > DerAlfMinusDp := -DerAlfMinusDR*Salt[xi]*Salt[Requil]/3; > print("deralfminusdp=", DerAlfMinusDp); > DerAlfPlusDp := CationPressureDerivative(Salt,DerAlfMinusDp); > print("deralfplusdp",deralfplusdp); > plot({[[salt[requil],0],[salt[requil],eval(todraw, > R=Salt[Requil])]],todraw },R=0.5..4*Salt[Requil],title=Salt[SaltName]); > end: Model_2/2 calls the following three Maple procedures 1. calculate_aorrequilmodel_2/2 2. alfminusmodel_2/2 3. CationPressureDerivative/2 which is also called by Model_1/1 and has been described in Section 3.2 pg. 3 Procedure calculate_aorrequilmodel_2/2 calculates, for a given salt and s the value of a if the value of α is known else it calculates the value of α using Model 2 equation 2 for s = 3 or s = 4. > calculate_aorrequilmodel_2 > :=proc(salt::list, ModelSet::integer) > if Salt[alfaMinusRinfinity] > 0 > then (Salt[alfaMinusRinfinity]-Salt[alfaMinus])/ > ((Salt[Requil]^(-ModelSet))*Salt[alfaMinus]) > else Salt[alfaMinus]*(1+Salt[a]*Salt[Requil]^(-ModelSet)) fi end: 8

13 Procedure alfminusmodel_2/2 calculates the value of α related to equation 2 for Model 2, where: <arg-1> α <arg-2> a <arg-3> free variable used for plotting <arg-4> value of s for Model 2. Should be 3 or 4 > alfminusmodel_2:=proc(alfinf::float,a::float, > R, ModelSet::integer) > AlfInf/(1+a*(R^(-ModelSet))) end: In order to illustrate it follows an execution example of Model_2/2 Maple procedure for salt LiF Table 2 pg.3 with s = 3 and s = 4. The call > Model_2(LiF_1,3); has the following output as well as the graph shown in Figure 2 Model 2 for salt, LiF 1, [.89, 1.996, 1.43, , 100, LiF 1,.03,.60, 1.36,.2784,.6709] s=, 3 Data alfaminusrinfinity=, , Calculated a=, todraw=, R 3 DerAlfMinusDR=, DerAlfMinusDp=, DerAlfPlusDp,

14 Figure 2: α = α 1 for salt LiF Model 2 s = 3 1+aR 2 10

15 The call > Model_2(LiF_1,4); has the following output as well as the graph shown in Figure 3 Model 3 for salt, LiF 1, [.89, 1.996, 1.43, , 100, LiF 1,.03,.60, 1.36,.2784,.6709] s=, 4 Data alfaminusrinfinity=, , Calculated a=, todraw=, R 4 DerAlfMinusDR=, DerAlfMinusDp=, DerAlfPlusDp, Figure 3: α = α 1 1+aR 2 for salt LiF Model 2 s = 4 4 Experimental Results In this section the results obtained for different salts using data in cgs and au units are listed. 11

16 R α + Salt a α LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI AgBr NaCN KCN NaClO NaBrO NH 4 F NH 4 Cl NH 4 Br NH 4 I CaCl SrCl SrBr BaCl BaBr Results Using Interpolated α Values NaCN KCN NaClO NaBrO Table 4: Results from Model 1 with Data in cgs Units 12

17 R α + Salt a α LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI AgBr NaCN KCN NaClO NaBrO NH 4 F NH 4 Cl NH 4 Br NH 4 I CaCl SrCl SrBr BaCl BaBr Results Using Interpolated α Values NaCN KCN NaClO NaBrO Table 5: Results from Model 2 for s = 3 with Data in cgs Units 13

18 R α + Salt a α LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI AgBr NaCN KCN NaClO NaBrO NH 4 F NH 4 Cl NH 4 Br NH 4 I CaCl SrCl SrBr BaCl BaBr Results Using Interpolated α Values NaCN KCN NaClO NaBrO Table 6: Results from Model 2 for s = 4 with Data in cgs Units 14

19 R α + Salt a α LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI AgBr NaCN KCN NaClO NaBrO NH 4 F NH 4 Cl NH 4 Br NH 4 I CaCl SrCl SrBr BaCl BaBr Results Using Interpolated α Values NaCN KCN NaClO NaBrO Table 7: Results from Model 1 with Data in au Units 15

20 R α + Salt a α LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI AgBr NaCN KCN NaClO NaBrO NH 4 F NH 4 Cl NH 4 Br NH 4 I CaCl SrCl SrBr BaCl BaBr Results Using Interpolated α Values NaCN KCN NaClO NaBrO Table 8: Results from Model 2 for s = 3 with Data in au Units 16

21 R α + Salt a α LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI AgBr NaCN KCN NaClO NaBrO NH 4 F NH 4 Cl NH 4 Br NH 4 I CaCl SrCl SrBr BaCl BaBr Results Using Interpolated α Values NaCN KCN NaClO NaBrO Table 9: Results from Model 2 for s = 4 with Data in au Units 17

22 5 Conclusions Salts for which there are experimental data at high pressures were studied in order to establish the validity of the model. Results of α obtained with the method developed in this report and implemented as Maple procedures are, in absolute value, lower than those calculated via experimental data; on the other hand, the empirical method we proposed previously [13] gave values higher than the experimental ones. We may therefore predict upper and lower bounds of the experimental data. Results for more complex materials are given in this report showing the same behaviour as described previously when compared to results obtained with other methods. 18

23 6 Appendix - Experimental Data Experimental data used in this work has been taken from the experimental data at Low Temperature LT and Room Temperature RT described in [7]. It should be noted that some data was not available at LT, in those cases the missing data (data with an in Table 10) has been substituted by data at RT from the same source. Struct. Salt α + α R χ T cm cm cm cm 2 dyn 1 fcc Li C=2 LiF LiCl LiBr *4.20 LiI * Na NaF NaCl NaBr NaI K 0.81 KF KCl KBr KI Rb 1.35 RbF *3.81 RbCl *6.40 RbBr *7.69 RbI *9.48 Cs 2.34 CsF *4.25 bcc Cs 2.34 C= CsCl CsBr CsI AgBr NaCN KCN NaClO NaBrO NH 4 F NH 4 Cl NH 4 Br NH 4 I Ca CaCl Sr SrCl SrBr Ba BaCl BaBr Table 10: Experimental Data at Low Temperatures for different anions and cations at Low Temper- The following values of r +, r +, r and r ature have been used. 19

24 Cation r + r + Anion r r Li F Na Cl K Br Rb I Cs CN Ag ClO Ca BrO Sr Ba NH 4 F NH 4 Cl NH 4 Br NH 4 I Table 11: Values of 1 r +, r +, 1 r and r at Low Temperatures in cgs Units 20

25 References [1] Batana, A.; Bruno, J.; Munn, R.W. Anion Polarizability in Alkali Halide Crystals. Molecular Physics. 1997, 92(6), [2] Batana, A.; Bruno, J.; Monard. M.C. Ionic Polarizabilities in Cristals at High Pressures. Journal of Physics and Chemistry of Solids. 2001, 62, [3] Char, B.W.; Geddes, K.O.; Gonnet, G.O.; Leong, B.L.; Monagan, M.B.; Watt, S.M. Maple V Language Reference Manual. Springer-Verlag, 1991 [4] Fowler, P.W.; Madden, P.A. The In-crystal Polarizability of the Fluoride Ion. Molecular Physics. 1993, 49(4), [5] Fowler, P.W.; Madden, P.A. In-crystal Polarizabilities of Alkali and Halide Ions. Physical Review B. 1984, 29(2), [6] Fowler, P.W.; Pyper, N.C. In Crystal Ionic Polarizabilities Derived by Combining Experimental and ab initio Results. Proc. Roy. Soc. A , 377, [7] Fracchia, R.M. A Theoretical Study of Anharmonic Properties of Solids. PhD Thesis, (in spanish), Facultad de Ciencias Exactas y Naturales, University of Buenos Aires, Argentina, [8] Jemmer, P.; Fowler, P.W. Environmental Effects on Anion Polarizability: Variation with Lattice Parameters and Coordination Number. Journal of Physical Chemistry. 1998, 102, [9] Monard, M.C.; Bruno, J.; Batana, A. A Computational System for the Calculation of Polarizabilities and their Pressure Dependence. Tech. Rep. ICMC-USP, No. 51, ftp: //ftp.icmc.sc.usp.br/pub/biblioteca/rel_tec/rt_51.ps.zip [10] Monard, M.C.; Batana, A.; Bruno, J. Calculation of the Pressure Dependence of Anionn Polarizabilities in Crystals. Computers & Chemistry. 1999, 23, [11] Monard, M.C.; Batana, A.; Bruno, J. Calculation of Ionic Radii at High Pressure in Different Crystals. Tech. Rep. ICMC-USP, No. 112, ftp://ftp.icmc.sc.usp.br/pub/biblioteca/rel_tec/rt_112.ps.zip [12] Monard, M.C.; Batana, A.; Bruno, J. Computational System for the Calculation of the Pressure Dependence of Ionic Polarizabilities in Crystals. Tech. Rep. ICMC-USP, No. 113, ftp://ftp.icmc.sc.usp.br/pub/biblioteca/rel_tec/rt_113.ps.zip [13] Monard, M.C.; Bruno, J.; Faour, J.; Batana, A. Pressure Dependence of Ionic Polarizabilities in Crystals using MAPLE Procedures. Computers & Chemistry. 2001, Vol.25(5), [14] Monard, M.C.; Batana, A.; Bruno, J. A New Method to Evaluate Pressure Derivatives of Ionic Radii: Case Study. Tech. Rep. ICMC-USP, No. 171, ftp://ftp.icmc.sc.usp.br/pub/biblioteca/rel_tec/rt_171.pdf.zip 21