Approximation of the Thomas-Fermi-Dirac potential for neutral atoms

Physica A 183 (1992) 361-377 North-Holland

~

~L~

Approximation of the Thomas-Fermi-Dirac potential for neutral atoms Aleksander Jablonski Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warszawa, Poland Received 18 July 1991 The frequently used analytical expression of Bonham and Strand approximating the Thomas-Fermi-Dirac (TFD) potential is closely analyzed. This expression does not satisfy the boundary conditions of the TFD differential equation, in particular, does not comprise the finite radius of the TFD potential. A modification of the analytical expression is proposed to adjust it to the boundary conditions. A new fit is made on the basis of the variational formulation of the TFD problem. An attempt is also made in the present work to develop a new numerical procedure providing very accurate solutions of this problem. Such solutions form a reference to check the quality of analytical approximations. Exemplary calculations of the elastic scattering cross sections are made for different expressions approximating the TFD potential to visualize the influence of the inaccuracies of the fit. It seems that the elastic scattering calculations should be based on extensive tables with the accurate values of the TFD screening function rather than on fitted analytical expressions.

1. Introduction T h e electrostatic potential in neutral a t o m s can be well a p p r o x i m a t e d by the T h o m a s - F e r m i - D i r a c ( T F D ) potential. T h e simplicity o f the only available analytical expression describing the T F D a t o m s considerably facilitates any calculations involving a t o m i c potentials. This expression, published by B o n h a m and Strand [1], is frequently referred to in the literature until present time [2-9]. R e c e n t l y , it has b e e n very successfully used in the theoretical description o f elastic electron backscattering f r o m surfaces [8, 9]. H o w e v e r , it t u r n e d out that the calculated p a r a m e t e r s characterizing electron backscattering are sensitive to variations in the scattering potential [9]. This stimulated the present analysis o f a c c u r a c y o f the fitted function p r o p o s e d by B o n h a m and Strand [1]. T h e analytical expression o f B o n h a m and Strand was fitted to solutions of the T F D e q u a t i o n published in 1954 by T h o m a s [10]. H o w e v e r , the fitted f u n c t i o n does not satisfy the b o u n d a r y conditions of the T F D equation. O n the 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B,V. All rights reserved

362

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

other hand, these boundary conditions were accounted for in calculations of Thomas [10]. This discrepancy may obviously influence the accuracy of the fit. Furthermore, the expression of Bonham and Strand does not include the finite radius of the TFD potential. The analytical function of Bonham and Strand can be modified to satisfy the boundary conditions. The objective of the present paper is to approximate the TFD potential by the corresponding function and to analyze its accuracy. This objective requires an accurate solution of the TFD equation as a reference. An effort is made to develop a new procedure providing the TFD potentials with very high accuracy.

2. Theory The electrostatic potential of an atom may be described by the so-called screening function, ft. This function is defined as a ratio of the electrostatic potential at a given distance, r, from the nucleus to the electrostatic potential due to the bare nucleus at the same distance. The dimensionless T h o m a s Fermi-Dirac differential equation describing the screening function ~/, can be written in the following form [11, 12]: dx 2 - x

x

+/30

'

(1)

where

fl0 ( 32 ~1/3 ] =

3--~2/

Z 2/3

and Z is the atomic number. The variable x is related to the distance from the nucleus r via the formula x = r/tx

,

where 1(9"rr2~ 1/3 a o /x=~ \ 2 /

1/----3 Z

and a 0 is the Bohr radius. The screening function qJ(x) is described by the boundary conditions

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

/3~/16)x0,

363

for x = 0 ,

(2)

for x = x 0 .

(3)

The screening function ~0(x) is limited by the finite radius r 0 = tzx 0. Its value can be determined on imposing the additional condition d6

Z-N

x~-~O-

Z

forx=x0,

where N is the number of electrons in the atom. For the neutral atom this condition simplifies to the form Xog,'(Xo)

-

(4)

~O(Xo) = o .

Conditions (2), (3) and (4) fully define the radius x 0 and the solution ~O(x) of the differential equation (1). The above problem requires special methods to derive the solution satisfying all the boundary conditions. Let us briefly describe the numerical approaches which were successfully applied by different authors. The first method was published by Jensen et al. already in 1938 [13]. Eq. (1) was integrated from the origin x = 0 assuming different initial values of the derivative, q/(0). Integration was continued to such a radius x 0 at which condition (4) was fulfilled. From different resulting values of x 0 corresponding to different values of ~O'(0) only one value was selected which additionally satisfied condition (3). This method provided the radius x 0 with relatively low accuracy since this parameter turned out to be very sensitive to the starting value of derivative ~O'(0). Jensen et al. [13] published the solution of eq. (1) for only three neutral atoms of noble gases, i.e. Ar, Kr and Xe. Thomas [10] proposed a different method of solving the TFD equation. This equation can be transformed to the form dew d/2 - ¼w + e x p ( - ~t) [exp(¼/) w 1/2 + y]3

(5)

with boundary conditions W 1/2

= 17

dw d--~- + ½w - - 0

for t = O,

(6)

for t = O,

w---~(~2)y3/ZZexp(-½t)

(7) for t----~~ ,

(8)

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

364

where

3'

floX20=(~--~2) ~r2°

(9)

•

ao

Eq. (5) and conditions ( 6 ) - ( 8 ) can be obtained from eqs. ( 1 ) - ( 4 ) on substitution of 7/2

~b

w = x0

1/2 ,

X

t= -ln(x/Xo) .

T h o m a s [10] published the values of the radius r 0 with an accuracy of 5 decomal places for practically all elements, i.e. for atomic numbers varying from 2 to 105. He also listed extensive tables of the function rth, which is related to the screening function ~b by rib = ~ b Z - r/32"tr 2 . These solutions were, however, obtained for non-integer values of atomic numbers covering the range 0.6 < Z < 111.8. The solutions of Thomas were fitted later by B o n h a m and Strand [1] for atomic numbers varying in the range 1 . 6 < Z < 111.8. The third approach to solve eq. (1) is based on a variational formulation of the T F D problem. One can prove [14] that the T F D differential equation describes function ~(x) minimizing the functional xo

f 0

2

4

2

i1/

1/2

-14[[~b\l/2 (10)

T h e functional l(~b) is minimized simultaneously with respect to the radius x 0 and the function 4,(x). Thus, the admissible functions ~b(x) for this functional have one fixed boundary (~/,(0)= 1) and one boundary movable along the function F(xo) = ½ floXo 2 . It can be proved that the radius x 0 and the function ~(x) minimizing the functional I(~) must satisfy condition (4) [14]. For this reason, eq. (10) is the most compact formulation of the T h o m a s - F e r m i - D i r a c problem. It also offers a very convenient method for determining the analytical solutions of the T F D equation. However, it was effectively used to provide a very approximate solution, with one fitted constant [14]. A new procedure leading to very accurate solutions of the T F D problem is proposed in the present work. Let us consider the following variables:

365

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

u -- t = 2 In (qjx3),

(11)

t = --ln(x/Xo).

(12)

On substitution of eqs. (11) and (12) into eq. (1) we obtain

dE//

1 [ (du]Z]+E{exp[_l~(7t_u)]+yexp[_l(5t+u)]} dt 2 = ~ 1 - \ d t / J

3

(13)

with y given by eq. (9). The boundary conditions (2)-(4) are now the following: u =41n(ly) du dt

for t = 0 ,

1

(14)

for t = 0 ,

(15)

u - t= 21n[(~z)1/zy3/2Z]

for t--~ oo.

(16)

Since the right hand side of condition (16) is a constant for a given Z the function u ( t ) becomes linear with the increase of the variable t. We have then du dt

1

for t--~ o0.

(17)

Let us denote exp[ ½(u - t)]

R(t) = ~/%3-~

•

(18)

On comparison with condition (16) we have lim= R ( t ) = Z .

(19)

Eq. (13) can be represented as a system of two first-order ordinary differential equations, and solved starting from the initial condition given by eqs. (14) and (15). Integration is performed until the difference u ( t ) - t becomes constant (condition (16)) or until the derivative d u / d t approaches unity (condition (17)). Simultaneously, according to eq. (19), the function R ( t ) reaches a certain, usually non-integer, value of Z.

366

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

3. Results

The Sarafyan embedding method involving the Butcher fifth order formula [15] was used in the present work for integration of eq. (13). Integration was p e r f o r m e d with a variable step to maintain an assumed accuracy (usually better than 8 decimal places). Embedding (simultaneous determination of coefficients for the fourth- and fifth-order integration formulas) facilitates fast control of accuracy of the solution during integration. Numerous tests have proved that the embedding method based on the Butcher formula should be highly r e c o m m e n d e d for integration of differential equations (ref. [15], p. 103). In fact, the embedding method was successfully used in calculations of the elastic scattering cross sections [8, 9]. This method turned out to be also very reliable in the present work. The solution of the T F D differential equation consists of two separate problems: (i) determination of the radius r0, and (ii) calculations of the values of the screening function ~b for a given distance x. Particular care was taken in the present work to assure high accuracy of the values of the radius r 0. The corresponding calculations for a certain atomic number Z were made according to the following algorithm: 1. Assume an initial value of the radius r 0 and calculate the parameter ~ (eq.

(9)). 2. Integrate eq. (13) until the difference u ( t ) - t becomes constant. The u p p e r limit of the variable t usually varied between 30 and 40. 3. Determine the resulting atomic number Z (eqs. (18) and (19)). 4. R e p e a t calculations starting from different initial values of r 0 until the above procedure provides the required integer atomic number Z. Behaviour of the functions u(t), du/dt, u(t) - t, and R(t) is illustrated in fig. 1. T h e corresponding calculations were made for the radius r 0 equal to 4 atomic units. In the considered case the function R(t) approaches the value of Z = 8.850828024. Extensive calculations of the radii r 0 were performed for all integer atomic numbers varying between 1 and 105. They are listed in table I. Proper care was taken to obtain these values with an accuracy of 8 decimal places. The values of r 0 are in agreement with the values published by Thomas [10] within the accuracy reached by this author (4-5 decimal places). In principle, calculations of the screened function for a given atomic number can also be based on eq. (13) with boundary conditions (14) and (15). H o w e v e r , it has been found that the integration of eq. (1) is much faster. From eqs. (3) and (4) we have

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

367

u

I01

-~

=

15'

rr

o

I

(b)

u-t

I

I

I

...................................

10 l

~F

s .2 ¢

0

0

I

I

2

4 t:tn

6

8

10

(Xo/X)

Fig. 1. Illustration of the exemplary solution of the transformed TFD differential equation (eq. (13)). Calculations are made assuming the radius r 0 equal to 4 atomic units. (a) The functions u and du/dt; (b) the functions u - t and R(t).

Table I Parameters Pl, P2, q~, q2 and q3 for the function t#rFD(X) (eqs. (21) and (22)) approximating the solutions of the T h o m a s - F e r m i - D i r a c equation, and the corresponding values of the radius of the T h o m a s - F e r m i - D i r a c potential. Z 1 2 3 4 5 6 7 8

Pl

P2

ql (a~')

q2 (a~ 1)

q3

(ao 1)

ro (ao)

0.012895 0.017001 0.020162 0.022715 0.024858 0.026702 0.028319 0.029758

0.20346 0.24548 0.27594 0.29871 0.31663 0.33124 0.34348 0.35395

26.907 28.292 28.987 29.532 30.009 30.447 30.859 31.247

3.8316 4.1495 4.3214 4.4535 4.5656 4.6653 4.7561 4.8401

1.1012 1.1737 1.2152 1.2466 1.2726 1.2953 1.3156 1.3342

2.9753639 3.3210030 3.5162820 3.6506623 3.7521818 3.8332350 3.9003714 3.9574616

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

368 Table I (cont.). Z 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

pj

P2

ql (ao 1)

q2 ( a ; 1)

q3 (a; l)

ro (ao)

0.031049 0.032221 0.033293 0.034275 0.035188 0.036033 0.036821 0.037556 0.038250 0.038904 0.039516 0.040100 0.040652 0.041172 0.041674 0.042149 0.042600 0.043034 0.043447 0.043848 0.044229 0.044599 0.044949 0.045288 0.045615 0.045929 0.046234 0.046526 0.046810 0.047083 0.047347 0.047608 0.047853 0.048095 0.048332 0.048558 0.048780 0.048994 0.049201 0.049404 0.049599 0.049792 0.049976 0.050159 0.050338 0.050510 0.050677 0.050845 0.051004

0.36304 0.37104 0.37816 0.38453 0.39031 0.39557 0.40037 0.40478 0.40888 0.41267 0.41618 0.41948 0.42256 0.42545 0.42818 0.43075 0.43317 0.43547 0.43765 0.43973 0.44170 0.44359 0.44538 0.44710 0.44874 0.45031 0.45181 0.45326 0.45465 0.45598 0.45727 0.45851 0.45970 0.46085 0.46197 0.46304 0.46409 0.46509 0.46606 0.46700 0.46791 0.46880 0.46965 0.47049 0.47130 0.47209 0.47284 0.47360 0.47432

31.620 31.977 32.320 32.654 32.975 33.286 33.590 33.887 34.173 34.453 34.729 34.996 35.259 35.518 35.769 36.016 36.261 36.499 36.735 36.964 37.192 37.414 37.636 37.852 38.067 38.278 38.486 38.692 38.895 39.095 39.294 39.488 39.683 39.874 40.061 40.248 40.432 40.615 40.797 40.976 41.154 41.329 41.504 41.675 41.845 42.014 42.183 42.347 42.512

4.9188 4.9930 5.0634 5.1307 5.1949 5.2566 5.3162 5.3738 5.4293 5.4831 5.5357 5.5866 5.6362 5.6849 5.7320 5.7782 5.8236 5.8678 5.9113 5.9537 5.9955 6.0363 6.0767 6.1163 6.1552 6.1936 6.2313 6.2685 6.3051 6.3412 6.3769 6.4118 6.4467 6.4809 6.5144 6.5478 6.5807 6.6133 6.6455 6.6773 6.7088 6.7399 6.7708 6.8012 6.8312 6.8611 6.8907 6.9197 6.9488

1.3514 1.3675 1.3826 1.3971 1.4108 1.4239 1.4366 1.4488 1.4606 1.4719 1.4830 1.4938 1.5042 1.5145 1.5244 1.5342 1.5437 1.5530 1.5622 1.5711 1.5799 1.5886 1.5971 1.6054 1.6137 1.6218 1.6297 1.6376 1.6454 1.6530 1.6606 1.6680 1.6753 1.6826 1.6897 1.6968 1.7038 1.7108 1.7176 1.7244 1.7311 1.7377 1.7443 1.7508 1.7572 1.7636 1.7699 1.7762 1.7824

4.0069778 4.0505920 4.0894854 4.1245228 4.1563552 4.1854844 4.2123051 4.2371332 4.2602250 4.2817916 4.3020084 4.3210230 4.3389603 4.3559275 4.3720165 4.3873075 4.4018700 4.4157653 4.4290474 4.4417640 4.4539578 4.4656669 4.4769254 4.4877641 4.4982107 4.5082906 4.5180264 4.5274392 5.4365481 4.5453703 4.5539222 4.5622184 4.5702726 4.5780974 4.5857045 4.5931045 4.6003077 4.6073234 4.6141603 4.6208266 4.6273298 4.6336772 4.6398753 4.6459305 4.6518487 4.6576355 4.6632960 4.6688352 4.6742578

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

369

Table I (cont.). Z

p~

P2

ql

q2

(ao 1)

q3 (ao 1)

ro (a0)

58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

0.051160 0.051312 0.051463 0.051609 0.051749 0.051890 0.052025 0.052158 0.052290 0.052416 0.052546 0.052664 0.052784 0.052902 0.053017 0.053130 0.053241 0.053352 0.053458 0.053561 0.053662 0.053764 0.053864 0.053958 0.054055 0.054149 0.054242 0.054331 0.054419 0.054506 0.054594 0.054674 0.054758 0.054840 0.054925 0.055000 0.055078 0.055156 0.055229 0.055304 0.055378 0.055448 0.055517 0.055585 0.055656 0.055724 0.055791 0.055857

0.47502 0.47570 0.47637 0.47702 0.47765 0.47827 0.47888 0.47946 0.48004 0.48060 0.48115 0.48168 0.48220 0.48271 0.48321 0.48370 0.48418 0.48465 0.48511 0.48556 0.48599 0.48643 0.48685 0.48725 0.48766 0.48806 0.48845 0.48883 0.48921 0.48957 0.48994 0.49028 0.49063 0.49098 0.49133 0.49164 0.49197 0.49229 0.49260 0.49291 0.49321 0.49351 0.49380 0.49408 0.49437 0.49465 0.49492 0.49519

42.675 42.837 42.997 43.156 43.315 43.471 43.627 43.781 43.933 44.085 44.233 44.385 44.533 44.680 44.826 44.971 45.115 45.257 45.400 45.541 45.682 45.821 45.959 46.097 46.233 46.368 46.502 46.637 46.771 46.903 47.033 47.167 47.296 47.425 47.551 47.680 47.807 47.932 48.059 48.183 48.307 48.432 48.555 48.678 48.798 48.919 49.039 49.158

6.9775 7.0060 7.0341 7.0619 7.0897 7.1171 7.1443 7.1712 7.1979 7.2245 7.2505 7.2768 7.3027 7.3283 7.3537 7.3790 7.4040 7.4288 7.4535 7.4781 7.5025 7.5266 7.5506 7.5746 7.5981 7.6216 7.6449 7.6682 7.6912 7.7142 7.7368 7.7596 7.7820 7.8043 7.8262 7.8484 7.8704 7.8920 7.9138 7.9352 7.9566 7.9780 7.9992 8.0203 8.0411 8.0619 8.0825 8.1030

1.7885 1.7946 1.8007 1.8066 1.8126 1.8185 1.8243 1.8301 1.8359 1.8416 1.8472 1.8528 1.8584 1.8640 1.8695 1.8749 1.8803 1.8857 1.8910 1.8963 1.9016 1.9068 1.9120 1.9172 1.9224 1.9275 1.9325 1.9376 1.9426 1.9476 1.9525 1.9574 1.9623 1.9672 1.9719 1.9768 1.9816 1.9863 1.9910 1.9957 2.0004 2.0051 2.0097 2.0143 2.0189 2.0234 2.0279 2.0324

4.6795681 4.6847704 4.6898686 4.6948664 4.6997673 4.7045748 4.7092920 4.7139220 4.7184676 4.7229317 4.7273168 4.7316256 4.7358602 4.7400231 4.7441165 4.7481423 4.7521027 4.7559995 4.7598346 4.7636097 4.7673265 4.7709867 4.7745917 4.7781432 4.7816423 4.7850909 4.7884898 4.7918405 4.7951443 4.7984023 4.8016157 4.8047854 4.8079128 4.8109987 4.8140441 4.8170499 4.8200173 4.8229469 4.8258397 4.8286964 4.8315180 4.8343052 4.8370587 4.8397793 4.8424677 4.8451245 4.8477506 4.8503463

(ao 1)

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

370 ,'(Xo)

=

~1 o . 2

(20)

When x 0 is known, eqs. (3) and (20) form an initial condition which makes possible the integration of eq. (1) towards decreasing values of the distance x. The function approximating the solution of the TFD equation should satisfy the boundary conditions (2)-(4). As already mentioned, the function proposed by Bonham and Strand [1] does not satisfy these conditions. This function is given by 3

qJBS(r) = ~ Pi e x p ( - - q i r ) i=1

or 3 ~Bs(X) = Z Pi e x p ( - q i ~ x ) , i=1

(21)

where pi and qi are constants depending on the atomic number. Actually, this function was used to approximate also other potentials, e.g. the ThomasFermi potential [16] and, more recently, the Dirac-Hartree-Fock-Slater potential [17]. A universal transformation has been proposed to adapt any monotonically decreasing function to the boundary conditions of the TFD equation [14]. A general form of this transformation applied to the function of Bonham and Strand is the following: ~TFD(X) = ~ B s ( x ) + A ( x / X o ) + B ( x / X o )

(22)

c .

The parameters A, B and C can be defined in different ways. In the present work the following formulas were used: A

=

~

20x0

-6

s(X0) -

B,

B = 6

s(X0) -

Xo (Xo),

C=2.

Eqs. (21) and (22) with the above parameters A, B and C have five fitted constants, i.e. two constants Pi (since E Pi ---- 1 ) , and three constants qi. These constants can be determined fitting the function ~bTFD(X) to the values ~0(x) calculated at selected points. However, the resulting constants pi and q~ depend on the number and values of the abscissae x and the criterion defining the best fit (minimum of the squared deviations, minimum of the absolute values of deviations, etc.). A unique criterion for the best fit can be based on the variational principle. The function ~0TFD(X), which approximates the solution qJ(x), most closely should realize the minimum of the functional corresponding to the TFD differential equation (eq. (10)). Thus, the variational representa-

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

371

tion of the TFD equation offers a possibility of determining the fitted function ~bTFD(X) by minimization of the functional I(~TFD) with respect to parameters defining the function ~TFD(X). Such an approach can also be used to determine the radius x 0 if the minimization is additionally performed with respect to this parameter [14]. However, it has been found in the present work that the functional I(qJTFD) is a very weak function of x 0, and the procedure for calculating x 0 outlined above is much more accurate. It has been decided to approximate the solution qJ(x) with the function given by eqs. (21) and (22) assuming the values of x 0 listed in table I. Eqs. (21) and (22) were introduced into functional (10) and the resulting integral was minimized with respect to five parameters, i.e. Pl, P2, ql, q2 and q3. Very extensive calculations were made for all atomic numbers in the range between 1 and 105. The fitted parameters calculated with an accuracy of five decimal places are compiled in table I. Figs. 2 and 3 compare the atomic number dependence of the fitted parameters obtained in the present work with the corresponding parameters calculated by Bonham and Strand [1]. Considerable differences are observed. They result from different criteria of the best fit used in both approaches, and also from modification of eq. (21) to satisfy the boundary conditions (2)-(4). Particularly dramatic deviations are observed for parameters qi in the range of low atomic numbers. The shape of atomic number dependences shown in figs. 2 and 3 is also different. The parameters qi calculated in the present work are the monotonically increasing functions of Z, while the corresponding depen-

0.15

g L 0.10' o

0.05 o 0...

0 0.6

~0.2 a_ i

,

,

,

,,,,,i

,

,

10

, , , , , , i

100

Atomic number Z

Fig. 2. Atomic number dependence of parameters Pl and P2 for analytical expression fitted to the TFD screening functions. Solid line: parameters calculated in the present work; dotted line: parameters of Bonham and Strand [1].

372

A . Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

8o! \

~, 6040-

20 9

E ~

53

2.2 ~ 1.8~

o

1.410

i

i

i

ILIII

L

10

I

I

'

Atomic n u m b e r

I t l l l

160 Z

Fig. 3. A t o m i c n u m b e r dependence of parameters ql, q2 and q3 for analytical expressions fitted to the T F D screening functions. Solid line: parameters calculated in the present work; dotted line: p a r a m e t e r s of B o n h a m and Strand [1].

dences published by Bonham and Strand [1] exhibit minima at Z equal to 10, 19 and 27. These authors drew attention to the fact that the minima occur in the vicinity of atomic numbers of inert gases (10, 18 and 36). However, in view of the present results it seems that this observation is meaningless.

4. Discussion

An important problem to consider is the accuracy of the fitted analytical solutions of the TFD equation. Results of calculations based on the approximated potential may be affected by the deviations from the accurate solution. This was demonstrated for different analytical functions approximating the Thomas-Fermi potential [18]. A similar effect can be expected for the Thomas-Fermi-Dirac potential. Figs. 4a-c compare the deviations of the function fitted in the present work from the accurate solution with the corresponding deviations for the function fitted by Bonham and Strand [1].

373

A . Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

10-

(a)

Z=l

/i I

~"

I1_

-5-

I

f

/ -10-

!

/ /

% -15-20-

-25

,

,

,

, ,,,,i

0.01

i

0.1

i

~

i

i i i i

XJXo

5-

L

3.~ w

(b)

z :~

/

(c)

5-

/7

Z =t3

o

"\

%

%

/

\, / ,

5

Z=3 ,,"-~

,

)

,

,

,

, , , , t

51 j...._~<..."

,

,

i

,

, , j ,

Z=47

%

[ 5-

i i-,.'7

z=6

_

5-t

z=7g

0-

%

-5-t O.Ol

Ol ×/×,,

1

O.Ol

.......

bll

........

×/x,,

Fig. 4. Deviation of the approximating analytical expression, ~bFIT, from the accurate TFD screening function, ~b, as a function of the distance from nucleus, x. Solid line: analytical expression fitted in the present work (@r~r -= ~TFD); dotted line: analytical expression of Bonham and Strand [1] (~r~T------@as); (a) Z = 1; (b) Z = 2, 3 and 6; (c) Z = 13, 47 and 79.

374

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

One can see that the function proposed in the present work approximates much better the solution of the TFD equation than the function of Bonham and Strand in the range of low atomic numbers. This effect is particularly pronounced for hydrogen (fig. 4a) where the approximation of Bonham and Strand seems to fail. In the range of medium and high atomic numbers the quality of both approximating functions is comparable. It seems that the modified function, ~'rFD(X), somewhat better describes the screening function in the vicinity of the nucleus, while the function of Bonham and Strand, ~b~s(X), is closer to the screening function at distances approaching the radius r 0. Certainly, eqs. (21) and (22), together with parameters listed in table I, are recommended to use when calculations involving the TFD potential are made for low atomic number elements, or when the function ~(x) is expected to satisfy the boundary conditions. However, it is not easy to predict which approximating function, @Bs(X) or I~TFD(X), would provide more accurate results in the case of medium and high atomic number elements. There may be cases when application of the Bonham and Strand function is still preferable, e.g. when a given algorithm is sensitive to variations in the potential at large distances from the nucleus. Let us now consider possible errors in calculations involving the approximating functions. The TFD potential was used in a number of recent calculations of parameters describing elastic scattering [2, 3, 5-7]. For this reason, similar calculations based on different analytical approximations of the TFD potential were also made in the present work. Exemplary differential elastic scattering cross sections, dot/dO, as a function of the scattering angle, 0, were determined in the energy range from 50 eV to 1000 eV for gold, i.e. for an element for which the quality of the fit of both approximating functions is similar (fig. 4c). As a reference, the same calculations were repeated with accurate TFD potentials. The nonrelativistic partial wave expansion model (PWEM) proposed by Calogero [19] was used in the calculations. Details of the algorithm have been recently published [9]. This algorithm makes it possible to determine the phase shifts and the corresponding elastic scattering cross sections with high accuracy, which is crucial in the present case. Results of calculations are shown in figs. 5a-d. Comparison of cross sections related to both approximating expressions, qJBs(X) and I~tTFD(X), with the cross section corresponding to the accurate solution, ~O(x), leads to the following conclusions: 1. There are noticeable distortions of the differential elastic scattering cross section due to inaccuracies in the analytical functions approximating the TFD potential. 2. Deviations from the accurate cross section do-/dO are similar for both approximating analytical expressions. This is to be expected in view of fig. 4c.

375

A . Jablonski I T h o m a s - F e r m i - D i r a c potential f o r neutral atoms 4

[a)

11

20

50eV

b) I

~3

o

-o 1.5 o=

o

~

2

I

,jIl ,! A

200eV

o t0

i

b

o

i

:

I

al

I ~

0.5

-6

"

, \/

0

, N

60

120

0

180

60

Scattering eng{e 8 (deg)

2.0 (c)

120

180

Scattering angte 8 (deg)

400eV

1.0 (d)

"c

1.5

1.0

"~-~0.75

,~ 0 5 -

~ ,

°

,-

}

°

~,

,-

o 0.5

o~0.25

-5

-5

• r=

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o

g

0

0

~

I

60

120

Scattering angte 8 (deg)

°

180

OI

0

60 Scattering

120 angle

180

8 (deg)

Fig. 5. Angular dependence of the differential elastic scattering cross section calculated for Z = 79. Solid line: accurate solution of the T F D equation; dotted line: analytical expression fitted in the present work (qJTFD); dashed line: analytical expression of B o n h a m and Strand [1] (0ns).

376

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

3. The P W E M algorithm seems to become less sensitive to inaccuracies in the analytical approximations with increasing electron energy. Obviously, a better fit to the T F D screening function can be obtained using a more involved analytical expression with a larger number of adjustable parameters. This, however, may cause considerable computational difficulties, in particular the amount of calculations due to the minimization procedure may be drastically increased. This problem can be circumvented using the accurate m e t h o d shown in the present work, i.e. integration of eq. (1) with the value of r 0 taken from table I. It takes a fraction of a second on a PC to calculate one value of the screening function for a given value of the argument, x, assuming the accuracy of 8 decimal places. Such an approach is feasible when a limited n u m b e r of values of the function is required. Actually, this method was used in the present work to calculate the accurate elastic scattering cross sections shown in figs. 5(a)-(d). More complex calculations involving the T F D potential can be based on extensive tables containing the values of the T F D screening function (e.g. several hundred values). These values can be interpolated for any value of the argument x with considerable accuracy. In fact, such an approach should be recommended in any calculations based on the T F D potential. The necessary tables with the values of the T F D screening function, on diskettes if necessary, can be obtained from the author on request. In summary, the expression of Bonham and Strand approximating the T F D screening function is found to deviate considerably from the accurate screening function in the range of low atomic numbers. This function, modified to satisfy the boundary conditions of the T F D differential equation, approximates the T F D potential for low atomic numbers much better. The quality of the fit of considered analytical expressions to the screening functions corresponding to medium and high atomic number elements is similar. Inaccuracies of the fit may result in noticeable errors, as shown by the example of the algorithm providing the elastic scattering cross sections. This effect may be considerable at low electron energies. Thus, the most universal and accurate method of determining the values of the T F D potential is direct integration of the T F D differential equation with the radius r 0 taken from the present work, or interpolation within extensive tables containing accurate values of the TFD screening function.

References [1] R. Bonham and T.G. Strand, J. Chem. Phys. 39 (1963) 2200. [2] S. Ichimura, M. Aratama and R. Shimizu, J. Appl. Phys. 51 (1980) 2853. [3] S. Ichimura and R. Shimizu, Surf. Sci. 112 (1981) 386.

A. Jablonski / Thomas-Fermi-Dirac potential for neutral atoms

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

377

A.W. Sfienz and H. l]berall, Phys. Rev. B 25 (1982) 4418. L. Reimer and B. L6dding, Scanning 6 (1984) 128. D. Ze-jun and R. Shimizu, Surf. Interface Anal. 10 (1987) 253. D. Liljequist, F. Salvat, R. Mayol and J.D. Martinez, J. Appl. Phys. 65 (1989) 2431. A. Jablonski, J. Gryko, J. Kraaer and S. Tougaard, Phys. Rev. B 39 (1989) 61. A. Jablonski, Phys. Rev. B 43 (1991) 7546. L.H. Thomas, J. Chem. Phys. 22 (1954) 1758. P. Gombas, Die statistische Theorie des Atoms und ihre Anwendungen (Springer, Wien, 1949) p. 83. P. Gombas, in: Encyclopedia of Physics, vol. 36, S. Fl/igge, ed. (Springer, Berlin, 1956) p. 109. H. Jensen, G. Meyer-Gossler and H. Rohde, Z. Phys. 110 (1938) 277. A. Jablonski, Physica A 134 (1985) 27. L. Lapidus and J.H. Seifeld, Numerical Solutions of Ordinary Differential Equations (Academic Press, New York, 1971) p. 72. G. Moli~re, Z. Naturforsch. 2a (1947) 133. F. Salvat, J.D. Martinez, R. Mayol and J. Parellada, Phys. Rev. A 36 (1987) 467. A. Jablonski, J. Phys. B 15 (1982) L623. F. Calogero, Variable Phase Approach to Potential Scattering (Academic Press, New York, 1967) chs. 3, 13.