252 



EXPONENTIAL M CURVE IN NONSTANDARD PROPAGATION 



We now substitute equation (37) in the integrand 

 of equation (29) and obtain 



1. n ± m 



P nm (X) = j o F(z) e" Xz dz = J o e~ x ° 

 \dz 



dz X 





(2 2 + D m ° + D n °)F + \P 



+ %(DJ>-D£)*J F(x)dx\ 



/to 

 e" X2 f (2) dz 



I 



+ X /.W 



= 1 + / e" x * F(«) 



(?" m — O 



6 



Co— 1 ■ , Ci- 



lOCi - 2 (r m 4 rj 



! | (2« + Z>„° + Z>„°) F + ^F 



(22 + £>„» + DJ) F + ±F 



C 2 = 

 C 3 = 



c„ = 



(r m - O 2 

 14C 2 - 2d (t w + t«) 



(4n + 2) C„_i - 2 (r m + r J C„_ 2 - C„_ 4 



cfe 



2X2 + X(Z) ra ° + DJ) + J X 3 + ± (DJ - D n °)*\ dz 



2X 



(r« — O 2 



2. n = m 



p nm = 1 + Si X + B 2 X 2 + • • • , (47) 



9 4 



R, — — Do R„ — — D 2(0) 



c l — q ^™ ) ■D 2 — , r ■t-'m j 



1 s 



r, = _ _i — _ n 3(0 



3 14 ^ 105 " ' 



•(46) 



= 1 - 2X 



d$ nm (X) 



d\ 



B„ = 



(2n 



-^ [2DJ B n _r + I S n _ 3 l . 



(48) 



+ /3„ m (X) [\(DJ + D B o) + | 3 + JL (DJ - D n y~\ . 



(42) 



It follows that the exchange integral /3„ m (X) satisfies 

 the first order differential equation 



dX ~ 2X + ^ (X) 



I -^+|(- D ™°+^°)+j+i A i( 2) ™ -- D » ) 2 



The solution of equation (43) is 

 /3„ m (X) = 



•(43) 



For intermediate values of X one may either use 

 the integral in equation (44) or integrate numerically 

 the differential equation (43). The latter procedure 

 was advocated by Hartree. 



For large values of X an asymptotic expansion can 

 be obtained directly from equation (43) by writing 

 it in the form 



P,m (X) = 



dp nm (X) 



- 2 + 4X 



d\ 



1 



Wx 



- CD +D°) + — -—(,d2, -Z>0)2 

 g 2 l m + *V + 12 4X m " 



dx _f (D o +i ,o ?i + j.. J3 o_ 7) o ) 2 



- e 2 m n' 12 41"! «' 



V: 



VI 



■ (44) 



25.5 PROPERTIES OF jS nm (X) 



For small X the solution of the differential equation 

 (43) can be started with a power series in X. 



X 3 + 2X (D° m + Dl) - 2 + i (Z)° - Z)°) 2 



X 3 + 2X (Dl + Dl) - 2 + i (D° - D°) 2 



8 |~3X 3 + 2X (D° + DJ) - 1 (Z>° - £°„) 2 ~] 



Tx 3 + 2X (Dl + Z)°) - 2 + i (£° - Z)°) 2 1 3 . 



(49) 



+ 



