Appendix 1 A.1 DERIVATION OF THE X-RAY FORM FACTOR An X-ray beam is incident in the direction defined by the unit vector s0 . We shall consider radiation scattered in the direction defined by the unit vector s [7] The path difference between the rays is: − = − OR QP S.r S.r 0 … (A.1-1) π Phase difference = 2 × path difference λ Where λ is the wavelength of the incident radiation Then 2π v v v Phase difference = s.(r − s ) … (A.1-2) λ 0 Hence the scattered amplitude from an element dr of electron density ρ(r) is given by: 2πi vvv − )ss.(r Scattered amplitude = ρ e)r( λ 0 dr … (A.1-3) v v = ρ e)r( r.dik dr … (A.1-4) 2π v v v Where k = , d = s − s λ 0 Thus, the total scattered amplitude is obtained by integration over the whole electron density distribution as: v v Total scattered amplitude ∝ ∫ ρ e)r( r.dik dr … (A.1-5) 60 Appendix 1 Let us write the atomic scattering factor )s(f from an electron distribution of density ρ( r )as s = kd For closed shell atom ρ )r( is spherically symmetrical. The angular integrations can then be performed by spherical polar coordinates θ ,,r φ . Then the volume element becomes r 2 dr sin θdθdφ , and since s.r = sr cos θ we can integrate over φ from 0 to 2π to obtain. ∞ π )s(f = 2π ∫ r 2 ρ dr)r( ∫ e cosisr θ sin θdθ … (A.1-6) 0 0 The integration over θ can be carried out, with the result ∞ sin sr )s(f = ∫ ρ )r( 4πr 2 dr … (A.1-7) 0 sr But ρ )r( 4πr2 = D )r( … (A.1-8) Then ∞ sin sr )s(f = ∫ )r(D dr … (A.1-9) 0 sr Q P ρ dr)r( r S0 S O R Fig.(A.1-1) x-ray scattering 61 Appendix 1 The solution of equation (A.1-9) by mathematical analysis is: For K-shell − ζ +ζ + = ( i j ).r ( ni n j ) D( r ) ∑∑ N i N j e r … (A.1-10) i j ( sr )3 ( sr )5 … (A.1-11) sin( sr ) = sr − + − ...... 3! 5! By substituting equation (A.1-11) and (A.1-10) in equation (A.1-9) 3 5 − ( sr ) + ( sr ) − ∞ sr ...... − ζ+ ζ + = ( i j ).r ( ni n j ) 3! 5! )s(f ∫ ∑∑ ci c j N i N j e r dr 0 i j sr … (A.1-12) = x ∑∑ci c j N i N j … (A.1-13) i j = ζ + ζ m i j … (A.1-14) = + … (A.1-15) p ni n j ∞ 2 4 − ( sr ) ( sr ) )s(f = x∫ r p e m.r (1 − + − ......) dr 0 3! 5! ∞ 2 4 p −m.r s p+2 −m.r s p+4 −m.r = x∫ r e − r e + r e dr 0 3! 5! 2 + 4 + = !p − s ( p 2 )! + s ( p 4 )! x p+1 p+3 p+5 m 3! m 5! m 62 Appendix 1 ( p + 2 k( ))! s 2( k ) = − k )s(f x∑( 1) p+2 ( k )+1 … (A.1-16) k =0 ( 2( k ) + 1)! m (n + n + 2(k)) s! 2()k = ()− k i j )s(f ∑∑c c N N ∑ 1 + + ()+ i j i j ni n j 2 k 1 i j k =0 + ()ζ + ζ ( 2( k ) 1)! i j … (A.1-17) The total scattering form factor is given by: For Li-like ions ∞ sin( sr ) = 1 … (A.1-18) )s(f total 3∫ DT ( r1 ) dr 1 0 sr 1 = 1 [ + + ] D r( ) D α β r( ) D α α r( ) D β α r( ) T 1 3 K K 1 KL 1 KL 1 (n + n + 2(p)) s! 2()p = ()− p i j )s(f 2∑∑c c N N ∑ 1 + + ()+ i j i j ni n j 2 p 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j (n + n + 2( p )) !s 2( p ) + ()− p i j ∑∑d d N N ∑ 1 + + + i j i j ni n j 2( p ) 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j … (A.1-19) For Be-like ions ∞ sin( sr ) = 1 )s(f total 4∫ DT r( 1 ) dr 1 … (A.1-20) 0 sr 1 63 Appendix 1 1 D )r( = [D r( )+ D )r( + D )r( + D r( )+ D )r( + D r( )] T 1 6 KKα β 1 KLα α 1 KLβ α 1 KLα β 1 KLβ β 1 LLα β 1 (n + n + 2(p)) s! 2()p = ()− p i j )s(f 3∑∑c c N N ∑ 1 + + ()+ i j i j ni n j 2 p 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j ()n + n + 2()p s! 2()p + ()− p i j 3∑∑d d N N ∑ 1 + + ()+ i j i j ni n j 2 p 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j … (A.1-21) 64 Appendix 2 A.2 Γ DERIVATIONS THE TWO-PARTICLE DENSITY HF (,1 2 ) FOR KαKβ , K αLα , K βLα , K αLβ , K βLβ and, L αLβ Γ The two-particle density function HF (,1 2 )for the individual shells for Li- Like ions can be written as [51]: 3 Γ = 1 ∗ ij (1,2 ) ∑ Aij Aij 2 i< j And for Be-like ions can be written as: 6 Γ = 1 ∗ ij (1,2 ) ∑ Aij Aij 2 i< j Γ i j ij (,1 2 ) Shell Γ 1 2 12 (,1 2 ) K α K β Γ 1 3 13 (,1 2 ) Kα Lα Γ 2 3 23 (,1 2 ) K β Lα Γ 1 4 14 (,)1 2 K α Lβ Γ 2 4 24 (,)1 2 K β Lβ Γ 3 4 34 (,)1 2 Lα Lβ = φ φ − φ … φ … (A.2.1) Aij i (1) j ( 2 ) j (1) i ( 2 ) • ≡ In this work Aij Aij due to the S-state symmetry. 65 Appendix 2 (1) Kα Kβ Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) −φ (1)φ ( 2 ) …(A.2.2) 12 2 1 2 2 1 1 2 Γ = [ϕ α ϕ β −φ β φ α ] …(A.2.4) 12 1s (1) (1) 1s ( 2 ) ( 2 ) 1s (1) (1) 1s ( 2 ) ( 2 ) 2 Since : ϕ ( 1 ) = R ( 1 )Y ( 1 ) s1 s1 s1 ϕ = …(A.2.5) s1 ( 2 ) R s1 ( 2 Y) s1 ( 2 ) After integrated over all spins in equation (A.2.4) and substituted equation (A.2.5) into (A.2.4) we get: Γ ′ = [ ] 2 ….(A.2.6) 12 ( spinless ) R1s (1 Y) 1s (1)R1s ( 2 Y) 1s ( 2 ) now we can integrate over all angular part to get: Γ = 2 2 '12 (1,2 ) R1s (1 R) 1s ( 2 ) …(A.2.7) (2) Kα Lα Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) 13 2 1 3 3 1 …(A.2.8) Γ = 1 [ϕ α ϕ α − ϕ α ϕ α ]2 13 (1,2 ) 1s (1) (1) 2s ( 2 ) ( 2 ) 2s (1) (1) 1s ( 2 ) ( 2 ) 2 ...(A.2.9) 66 Appendix 2 ϕ (1)ϕ ( 2 )−ϕ (1)ϕ ( 2 ) 2 Γ = []α α 2 1s 2s 2s 1s 13 ,( 21 ) (1) ( 2 ) …(A.2.10) 2 since ϕ = 1 s ( 1 ) R 1 s ( 1 Y) 1 s ( 1 ) ϕ ( 2 ) = R ( 2 Y) ( 2 ) 1 s 1 s 1 s …(A.2.11) ϕ (1) = R (1 Y) (1) 2 s 2 s 2 s ϕ = 2s ( 2 ) R2s ( 2 Y) 2s ( 2 ) And ∫α(1)β(1 d) σ = 0 ∫α(1)α(1 d) σ = 1 ∫ β(1)α(1 d) σ = 0 …(A.2.12) ∫ β(1)β(1 d) σ = 1 by substituted the equation (A.2.11) into (A.2.10) and integrated over all spins using equation (A.2.12) we get: 1 2 Γ ′( spinless ) = [R (1 Y) (1 R) ( 2 Y) ( 2 ) − R (1 Y) (1 R) ( 2 Y) ( 2 )] 13 2 1s 1 s 2s 2s 2s 2s 1s 1s ….(A.2.13) Since θ ϕ = θ ϕ …(A.2.14) Y1s ( , ) Y2s ( , ) 1 2 2 Γ ′(1,2 ) = [][]Y (1 Y) ( 2 ) R (1 R) ( 2 ) − R (1 R) ( 2 ) 13 2 1s 2s 1s 2s 2s 1s …(A.2.15) Now we can integrate over all the angular part so that we can get: 67 Appendix 2 R (1 R) (2)− R (1 R) (2) 2 Γ = 1s 2s 2s 1s ' 13 ( 1 , 2 ) …(A.2.16) 2 1 Γ )2,1(' = [R2 )1( R2 )2( + R2 )1( R2 )2( 13 2 1s 2s 2s 1s − ] 2R1s )1( R2s )1( R2s )2( R1s )2( ….(A.2.17) (3) Kβ Lα Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) …(A.2.18) 23 2 2 3 3 2 1 2 Γ (1,2 ) = [ϕ (1)β(1)ϕ ( 2 )α( 2 ) − ϕ (1)α(1)ϕ ( 2 )β( 2 )] 23 2 1s 2s 2s 1s ..(A.2.19) By substituted equation (A.2.11) into (A.2.19) and integrated over all spins using equation (A.2.12) we get: 1 Γ ′ ( spinless ) = [R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 ) + R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 )] 23 2 1 s 1 s 2 s 2 s 2 s 2s 1s 1s …(A.2.20) Using eq.(A.2.14) and integrate over all angular part to get: Γ / = 1 []2 2 + 2 2 23 ( 1 , 2 ) R 1 s ( 1 R) 2 s ( 2 ) R 2 s ( 1 R) 1 s ( 2 ) …(A.2.21) 2 (4) Kα Lβ shell 68 Appendix 2 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) 14 2 1 4 4 1 ...(A.2.22) 1 2 Γ (1,2 ) = [ϕ (1)α(1)ϕ ( 2 )β( 2 ) − ϕ (1)β(1)ϕ ( 2 )α( 2 )] 14 2 1s 2s 2s 1s ..(A.2.23) By substituting equation (A.2.11) into (A.2.23) and integrated over all spins using equation (A.2.12) we get: 1 Γ ′ ( spinless ) = [R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 ) + R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 )] 14 2 1s 1s 2s 2s 2s 2s 1s 1s ...(A.2.24) using equation (A.2.14) and integrate over all angular part to get: 1 Γ ' (1,2 ) = []R 2 (1 R) 2 ( 2 ) + R 2 (1 R) 2 ( 2 ) ...(A.2.25) 14 2 1s 2s 2s 1s (5) K β Lβ Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) ...(A.2.26) 24 2 2 4 4 2 1 2 Γ (1,2 ) = [ϕ (1)β(1)ϕ ( 2 )β( 2 ) − ϕ (1)β(1)ϕ ( 2 )β( 2 )] 24 2 1s 2s 2s 1s ...(A.2.27) 69 Appendix 2 By substituting equation (A.2.11) into (A.2.27) and integrated over all spins using equation (A.2.12) we get: 1 Γ ′ ( spinless ) = [R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 ) + R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 )] 24 2 1s 1s 2s 2s 2s 2s 1s 1s ...(A.2.28) using equation (A.2.14) and integrate over all angular part to get: 1 Γ ,(' 21 ) = [R 2 (1 R) 2 ( 2 ) + R 2 (1 R) 2 ( 2 ) 24 2 1s 2s 2s 1s − ] ...(A.2.29) 2R1s (1 R) 2s (1 R) 2s ( 2 R) 1s ( 2 ) (6) Lα Lβ shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) ...(A.2.30) 34 2 3 4 4 3 1 2 Γ (1,2 ) = [ϕ (1)α(1)ϕ ( 2 )β( 2 ) − φ (1)β(1)φ ( 2 )α( 2 )] 34 2 2 s 2 s 2s 2s ...(A.2.31) Since : ϕ (1)=R (1 Y) (1) 2s 2s 2s …(A.2.32) ϕ = 2s (2) R2s (2 Y) 2s (2) 70 Appendix 2 After integrated over all spins in equation (A.2.31) and substituted equation (A.2.32) into (A.2.31) we get: Γ ′ = [ ] 2 ….(A.2.33) 34 ( spinless ) R2s (1 Y) 2s (1)R2s ( 2 Y) 2s ( 2 ) now we can integrate over all angular part to get: Γ = 2 2 '34 (1,2 ) R2s (1 R) 2s ( 2 ) …(A.2.34) 71 References References 1.