2005.7.21 KEK
G(z 0 + "z) = G(z 0 ) + "z dg(z) dz z! # d" λ "G = G(z 0 ) + #cos dg(z) ( & dz ) * nv #., - d+ - - r 2 sin cosds e / r # ddr 4.r 2 #cos! "G = G(z 0 ) + #cos dg(z) ( & dz ) * nv 2+ + ds - d, - sin cosd 4+ 2. 0 0 0 0 0 "G = # 1 dg( nv * ds ds 3 & dz )
n(z + "z) = n(z 0 ) + "z (z) dz z d" λ "n = n(z 0 ) + #cos (z) ( d+ & dz ) * v cosds 4,! # #cos! J J = " #v 3 dz = "D dz D:
J = "nu " D dz # "nu λ λ J = "D * dz = " #* v 3 dz # U " n & ( dz θ
! i ds i cos " ik cos " ki ds k #r ik 2 n i ds k " i r ik! ik! ki n k ds i z = 0 Q = 0 # ds i # ds k i=!" " k= 0 ( i! k ) cos ik cos ki 2 &r ik! i ds i =! kds k cos " ki cos " ik ds i k #r ki 2! i ds i = 1 4 n i v ds i
Q = 4v 3 L! 0 1 H A 2 "p dl!* = " # = 4A H H A n v! = HL " 1 4 nv! = AL " nv L n ds Q =! v 8 dz I =! v 8 dz " H ds 1 2 # 2 cos d /2 " =! /2! "
Q =! v 8 dz I =! v 8 dz " H ds" /2 =! /2 1 2 # 2 cos d I = 16 3!a 3 ) I = 2a 2 b+ ln! + 1 +! 2 * + ( ) +! ln 1 + 1 +! 2 " #! ( ) 3/2 & ( 1 +! 2 3! + 1 +! 3 3!,. -. * I! 2a 2 16 b, + 3 " & ( ) ln 4# + 3 3/ 2 4 # 1 - ) ( /. " I = 4a 3 # ln 1 + 2 ( ) + 1! 2 3 & I = 3 4 a 3 ln 3 I = 32 3 a2 # /2 d! b 0 k 2 = 1! a 2 1 " k 2 sin 2! b 2 * I! 2"a 2 8 b, + 3 ln 19 / 4 & ( ) ln 4# + 3 4 # 1 - ) ( /.
Q =! v 8 dz I =! v 8 dz " H ds" /2 =! /2 1 2 # 2 cos d p = nkt Q v = QkT Q v = C( p 1 " p 2 ) p 1 L p 2 Q v = "AD * p 2 " p 1 L C = 2" 3L a3 v C = v 4 "a2 # 8a 3L
C Q p p 1 p 2 L x p(x) p(x+dx) A H # dp(x + dx) Q(x + dx) " Q(x) = "AD * dx " dp(x) dx & ( Q(x) q(x) Q(x+dx) Q(x + dx) " Q(x) = Hq(x)dx x x+dx d 2 p(x) dx 2 = " Hq(x) AD * = " Hq(x) CL
p( x) =! HLq 2C " # x L& 2 ( + ) HLq * 2C! p 0 { ( )! p(l) } + " x, + p(0) -# L& p(x) p(0) p(l) x C Q(0) = " HLq 2 + C p 0 Q( L) = HLq 2 + C p 0 { ( ) " p( L) } { ( ) " p( L) } Q(0) = "p( 0)S 0 Q( L) = p( L)S L
Q ij i = " G ij 2 + C ij( p i " p j ) Q ij j = G ij 2 + C ij( p i " p j ) pi!"# "# pj " Q ij i # S i p i = 0 j " # S 11 S n1... S ij... S 1n S nn " & # p 1 p j p n " = & # G * 1 G * i G * n & G * i = " h G ih 2 p = S "1 G
G S C S C S G # C + S 0 "C &# ( 0 C + S "C ( "C "C 2C + S ( p 1 p 2 p 3 # G& & 2 ( ( G( ( = ( 2 ( ( ( G ( G G S C S C S p p max p min L p min = G S " p max = 1 8C + 1 G # S& p max p min = S 8C +1
Q V dp dt = C p atm " p ( ) + Q " Sp C(p atm - p) p Sp V 1.E+05 1.E+04 1.E+03 sm all leakage pressure [Pa] 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-05 1.E-03 1.E-01 1.E+01 1.E+0 3 1.E+05 1.E+07 t [sec]
p(x) p(x+dx) A H " 2 p D * "x = "p 2 "t Q(x) x x+dx Q(x+dx) A"x #p #t = Q(x + dx) Q(x) = dp(x + dx) AD* & dx dp(x) dx ( ) * D * = "* v 3 D * = "* v 3 = " *2 3 " /v ( ) # " *2 3 1 (" /v) + s
D * = "* v 3 " 2 p D * "x = "p 2 "t p(x,t) = # exp " x 2 & 4D * ( t 2)D * t t = 3 " 6 6D * L 2
D * = "* v 3 = " *2 3 " /v ( ) # " *2 3 1 (" /v) + s " 2 p D * "x = "p 2 "t ( ) / p 0 = x L + 2 " p x,t * (#1) n 1 n exp # Dn 2 " 2 + & t) sin n"x & ) L 2 ( L ( n= o
D * = "* v 3 = " *2 3 " /v ( ) # " *2 3 1 (" /v) + s 1.E+10 1.E+09 N (Monte-Carlo) N (diffusion model)! 1.E+08 1.E+07 1.E+06 d hitting number N t = N! 1.E+05 L L 1.E+04 1.E+03 1.E+04 1.E+05 1.E+06 L/d