from the precisely measured lifetimes of the \( 6 p \, ^ { 2 } P _ { 1 / 2 } \) and \( 6 p ~ \AA ^ { 2 } \)\( ^ 2 P _ { 3 / 2 } \) states [35]. The other important E1 matrix elements are obtained using the RCC method in the sin-gles and doubles approximation (CCSD method) as de-scribed in [45, 46]. Contributions to \( \alpha _ { c } \) are determined using a relativistic random-phase approximation (RRPA) as described in [40]. It has been demonstrated that the RRPA method can give rise to very reliable results for the atomic systems having inert gas configurations [47].Smaller contributions from \( \alpha _ { c v } \) and from the high-lying excited states (tail contribution \( \alpha _ { t } \)) that are omitted in the above sum-over-states approach are estimated in the Dirac-Hartree-Fock (DHF) approximation.

It is not easy to get the dynamic electric permittivity of any real materials, so the convenient way of determin-ing these constants for simple metals such as gold is to use the Drude-Lorentz model as was done in Ref. [26]. In pursuance of obtaining more realistic values of these con-stants for different materials, we prefer to use the known real and imaginary parts of the refractive indices of a ma-terial at few real values of frequency \( \omega \). The imaginary parts of the dielectric permittivities of the materials can then be obtained using the relation

\[ I m \left[ \epsilon ( \omega ) \right] = 2 n ( \omega ) \kappa ( \omega ) , \eqno ( 9 ) \]

where \( n ( \omega ) \) and \( \kappa ( \omega ) \) are the respective real and imag-inary parts of the refractive index of a material at fre-quency \( \omega \). We use the optical data from the handbook by Palik [48] for the frequencies ranging from 0.1 eV to 10000 eV for Au metal to calculate \( I m [ \epsilon ( \omega ) ] \). Thereafter, the required real values of the dielectric constants at the imaginary frequencies \( ( \epsilon ( \iota \omega ) \)) are obtained by using the Kramers-Kronig formula. The available data, however, does not cover the whole frequency range to carry out the integration of Eq. (4). Thus, we extrapolate these val-ues for the lower frequencies to increase the domain over which the integrations are to be performed [49, 50]. For the frequencies below 0.1 eV, the classified values from[48] are extrapolated using the free electron Drude model in which the dielectric permittivity along the imaginary axis is represented as

\[ \epsilon ( \iota \omega ) = 1 - \frac { \omega _ { p } { } ^ { 2 } } { \omega ( \omega + \iota \gamma ) } , \eqno ( 1 0 ) \]

where \( \omega _ { p } = ( 2 \pi c / \lambda _ { p } ) \) is the plasma frequency and \( \gamma \) is the relaxation frequency. The optical data values for \( \omega _ { p } \) and \( \gamma \) available from various sources differ slightly, but we use these values as \( \omega _ { p } = 9 . 0 \) eV and \( \gamma \) = 0.035 eV as out-lined in [26, 49–51]. In case of Si (semiconductor),\( \mathrm { { S i O _ { 2 } } } \) (dielectric), Sapphire and YAG, the complex frequency-dependent dielectric permittivity are quoted for a wide range of energies in the handbook of Palik. Therefore, we use all these values for carrying out the integration and do not extrapolate any data. On the otherhand, experi-mental data of \( n ( \omega ) \) and \( k ( \omega ) \) for \( \mathrm { S i N _ { x } } \) are not available at all and we use Tauc-Lorentz model [33, 52] for esti-mating dielectric constants of this material.

TABLE III: Comparison of the calculated values of damping function \( f _ { 3 } ( a ) \) defined in Eq.(4) for different separation dis-tances a for the interaction of Cs atom with the gold surface.

a
(a. u.)	f3(a)
This work	Ref. [26]
11×10	1.00	0.99780
12×10	1.00	0.99408
15×10	0.98	0.97903
21×10	0.95	0.95169
22×10	0.89	0.90210
25×10	0.78	0.79521
31×10	0.66	0.68309
32×10	0.53	0.54485
35×10	0.33	0.33918

TABLE IV: Comparison of \( C _ { 3 } \) coefficients ratios of Li, Na, K and Rb atoms with the Cs atom for the perfect conductor, metal, semiconductor and dielectric surfaces.

iC \( C _ { 3 } ^ { \mathrm { L } } \) /Cs3	\( \gamma _ { 2 } ^ { \mathrm { N } } \) aC/Cs3	C \( \gamma _ { 2 } ^ { \mathrm { F } } \) /Cs3	Rb
C s \( \gamma _ { 3 } ^ { 0 } \) 3/
Perfect conductor	0.313	0.393	0.638	0.773
Metal: Au	0.416	0.464	0.715	0.811
Semiconductor: Si
Dielectric:	0.430	0.477	0.726	0.818
SiO2	0.409	0.458	0.708	0.807
SiNx	0.418	0.464	0.709	0.801

Again, the interactions between the ground states of atoms with an anisotropic surface have been studied be-fore [53, 54]. These studies demonstrate that for a uni-form birefringent dielectric surface, with the symmetry axis normal to the interface, the interaction potential de-scribed Eq. (3) is still applicable if \( \epsilon \) is replaced by an effective quantity \( \bar { \epsilon } \) defined as

\[ \bar { \epsilon } ( \iota \omega ) = \left[ \epsilon _ { | | } ( \iota \omega ) \epsilon _ { \bot } ( \iota \omega ) \right] ^ { \frac { 1 } { 2 } } , \eqno ( 1 1 ) \]

where \( \epsilon _ { | | } \) and \( \epsilon _ { \perp } \) are the respective dielectric permittiv-ities for the electric fields parallel and perpendicular to the interface between the atom and the dielectric surface.

# III. RESULTS AND DISCUSSION

The evaluation of \( C _ { 3 } \) coefficients requires the precise estimation of dynamic polarizabilities of the Cs atom. Table I presents the scalar polarizabilities of the Cs atom in its ground state along with the E1 matrix elements for different transitions that are used to estimate \( \alpha _ { v } \) and other contributions to the polarizabilities. The E1 ma-trix elements for the \( 6 S _ { 1 / 2 } { - } 6 P _ { 1 / 2 , 3 / 2 } \) transitions are taken from the experimentally measured data given in [35].Our calculated value of the scalar polarizability \( ( \alpha ( 0 ) ) \)