2 [1] KEK Report L C R L C R [2] 2
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- しょうじ てらわ
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1 Fundamentals of RF Acceleration Revised Edition Koji TAKATA
2 2 [1] KEK Report L C R L C R [2] 2
3 3 Slater [3] Collin [4] [5] [6] [7] MIT Radiation Laboratory Series 8 Principles of Microwave Circuits [8] 40 [ ] KEK Report 1 TEX
4
5 π APS
6
7 7 1 AR (Accumulation Ring) [9] MHz (λ = 58.9 cm ) m APS (Alternating Periodic Structure) kw 1.1 MV/m (1 3 MV) (38.1 cm ) 7.5 (WR1500 ) TE 10 9
8 AR 1. APS kv Q
9 9 90 kv 20 A 1.2 MW ( 65%) 1 W 1 m 2 3
10
11 (pillbox cavity) z r θ b d
12 12 2 r=b Hq Ez 0 2b d arbitrary scale E z H q c 01 r/b 2.1 b d TM 010 E z H θ z 0 J 0 (χ 01 r/b) 1 J 1 (χ 01 r/b) z χ 01 = E z Transverse Magnetic Mode (TM E ) H z Transverse Electric Mode (TE H ) TM 010 E z H θ z TM E z θ 1 2 E z r 1 3 E z z 2.1 E z H θ 0 0 (r = 0) 0 1 J 0 J 1
13 E z = Êz cos (ω 010 t) H θ = Ĥθ sin (ω 010 t + π) = Ĥθ sin (ω 010 t) E r = E θ = H z = H r = 0 (2.1) E 0 Ê z = E 0 J 0 (χ 01 r/b) Ĥ θ = E 0 ζ 0 J 1 (χ 01 r/b) (2.2) ω 010 = χ 01c b (2.3) d χ 01 = J 0 ζ 0 = µ 0 /ɛ 0 = Ω c (= m/s) µ 0 = H/m ɛ 0 = F/m (2.2) r/b = MHz 2π b cm Q Q Q W ω P Q = ωw P (2.4) P wall Q Q Q 0 2 W = µ 0 2 V Ĥ 2 dv = ɛ 0 2 V Ê 2 dv (2.5)
14 14 2 P wall P wall = ζ m 2 S Ĥ 2 ds (2.6) ζ m σ µ ζ m = ωµ 2σ (2.7) σ = m 1 Ω 1 µ = µ 0 = H/m ( ) 500 MHz ζ m = Ω δ TM 010 Q δ = 1 σζ m (2.8) Q 0 = ζ 0 2ζ m χ01d d + b (2.9) (2.9) d Q d Q L C R R (r.m.s.) ( R R a ) 2 TM L C R Q 0 2
15 ω 010 = 1 LC (2.10) Q 0 = R ω 010 L (2.11) L C R 1 (2.1) (2.2) E 0 z r = 0 d V (t) = E 0 d cos (ω 010 t) V 0 cos (ω 010 t) (2.12) L C R (= R a /2) 2.2 (2.12) z = d/2 z = d/2 z t 0 z = v(t t 0 ) V a (t 0 ) = E 0 d/2 d/2 ( ω010 z cos v = V 0 T cos (ω 010 t 0 ) ) + ω 010 t 0 dz V a cos (ω 010 t 0 ) (2.13) T = sin ( ) ω 010d 2v) 1 (2.14) ( ω010 d 2v
16 16 2 V a (2.12) V 0 T (T (Transit Time Factor) ) V 0 2 R V a = V 0 T = E 0 T d R = V a 2 2P (2.15) * 1 (2.15) (2.1) (2.2) (2.6) R = R a 2 = 1 2 R a = 2R (2.16) ζ 0 2 ζ m d 2 πj 2 1 (χ 01) b(b + d) T 2 (2.17) R ( R a ) 2 L C (2.3) (2.9) (2.11) L = R ω 010 Q 0 C = Q2 0 Rω 010 (2.18) d (2.15) (2.17) *1 5 I 0 2 R a R a I 0
17 T 2 R/Rmax Q/Q max pd/l f 010 = ω 010 /2π = 500MHz σ = m 1 Ω 1 c E 0 V/m E 0 b m ω 1 d m ω 1 U E 0 J ω 3 P wall E 0 W ω 3/2 Q Q ω 1/2 R a Ω ω 1/2 T ω 0 d = 0.44λ *2 d Q T R a MHz *2 r a r a = R a d = (E 0T ) 2 P wall /d d/λ = /3 2π/3 [10] [11]
18 δv n ω n W n δω n ω n = δw n W n (2.19) [12] n δw n F n F n = 1 4 ( µ 0 Ĥn 2 ɛ0 Ên 2 ) n F n n (2.20) n δw n δw n = δv δv F n dv (2.21)
19 δω n ω n = 1 4 δv ( µ Ĥn 0 2 ) ɛ0 Ên 2 dv ɛ 02 Ên 2 (2.22) dv V W n,e W n,h δω n δw n,e δw n,h (2.23) 2.1 δω n (2.1 ) (2.2) 75% 56% d 0.6b C CV 2 /2 C V δw E T 2.4 nose cone
20 20 2 H q E z Pill-box Cavity Cavity with Noze Cones 2.4 : PF 500MHz 500MHz PF cm 22cm 18cm 4cm 10cm 13cm 500MHz SUPERFISH R a = 9.9MΩ Q 0 44, 000 [13] 2.1 R a T
21 r R10mm R234.69mm Ez (r=0) R50mm R91.375mm z 220mm R130mm 300mm 2.5 PF 500MHz a r a r = a z E z (z) r a TM 0 TM 0 ω e jωt z β g exp (jβ g ) (tilde) Ã (β g) β g
22 22 2 Ẽ z (r, z) = Ẽ r (r, z) = H θ (r, z) = à (β g ) ) 2π J 0 ( β 2 βg 2 r e jβgz dβ g à (β g ) 2π à (β g ) 2π jβ g β 2 βg 2 J 1 ( β 2 β 2 g r ) e jβ gz dβ g jɛ 0 ω ( ) J 1 β 2 β 2 β 2 βg 2 g r e jβgz dβ g Ẽ θ = H z = H r = 0 (2.24) β ω c (2.25) ω r z r = a E z z d/2 { 0 z d/2 E z = 0 z > d/2 (2.26) z d/2 E z d E z = { E 0 = V 0 /d z d/2 = 0 z > d/2 (2.27) à (β g ) à (β g ) = V 0 β g d 2 sin (β g d/2) ) (2.28) J 0 ( β 2 βg 2 a (r = const.) v z = 0 φ
23 exp j ( ωz v + φ ) (2.29) (2.24) z z ( ) φ Ṽ (φ) Ã (β g ) ) = 2π J 0 ( β 2 βg 2 a e j[(ω/v βg)+φ] dβ g dz ( ) = Ã (ω/v) J 0 β 2 (ω/v) 2 a e jφ ( ) ω = Ã (ω/v) I 0 1 (v/c) 2 r e jφ (2.30) v I 0 (x) = J 0 (jx) V acc (2.30) (2.25) (2.28) ] V acc = Re [Ṽ (φ) sin ( ) ωd 2v = V 0 ) ( ωd 2v ( ω I 0 v I 0 ( ω v ) 1 (v/c) 2 r ) cos φ 1 (v/c) 2 a V a cos φ (2.31) φ = 0 V a v = c V a = V 0 sin (βd/2) βd/2 (2.32) T r (2.30) v = c r 0 (2.27) (2.28) (2.24) [14] β g
24 24 2 β g = ±jγ n (n = 1, 2,...) (2.33) J 0 n χ 0n β = 2π/λ Γ n (χ 0n /a) 2 β 2 (2.34) z < d/2 z > d/2 z d/2 sin (β g d/2) exp(jβ g d/2) exp( jβ g d/2) E z (r, z) E 0 = J 0(βr) J 0 (βa) 2 n=1 χ 0n cosh (Γ n z) e Γ nd/2 Γ 2 n a2 J 1 (χ 0n r/a) J 1 (χ 0n ) (2.35) I = 2πaH θ (a, d/2) (2.36) (2.35) V 0 = E 0 d C C = I jωv 0 (2.37) (2.35) (2.36) C = C 0 f ( ) d a, βa C 0 a d C 0 = ɛ 0 πa 2 d (2.38) (2.39) f C 0 ( ) f(x, y) = 2 1 e x χ 2 0n y2 n=1 χ 2 0n y2 = J 1(y) yj 0 (y) 2 n=1 2.6 (2.40) e x χ 2 0n y2 χ 2 0n y2 (2.40)
25 f (d/a, βa) d/a = βa 2.6 f(d/a, βa)
26
27 27 3 coupler φ 0 φ π π φ = π 2π π π φ = 0 π 0 ω φ = 0
28 28 3 π ω φ 0 bi-periodic structure z = 0 = π normal mode 3.1 TM π 2.2 C C R C C (3.1) 3.3 ĩ 1 ĩ 2
29 cell - 1 cell - 2 cell - 1 cell - 2 E H 0 - mode π - mode π ( jωl + 1 ) jωc ( jωl + 1 jωc ĩ jωc (ĩ1 ĩ 2 ) = 0 ) ĩ jωc (ĩ2 ĩ 1 ) = 0 (3.2) ĩ 1 = ĩ 2 ( 0) 0 ĩ 1 = ĩ 2 π π 2 0 ĩ 1 = ĩ 2 ω = 1 LC ω 0 (3.3) π ĩ 1 = ĩ 2 ω = ω C C ω π ω 0 ( 1 + C C ) > ω 0 (3.4)
30 30 3 L L i 1 ~ C C' C i 2 ~ ω 0 < ω π * E = 0 H = 0 (3.5) E = 0 H = 0 (3.6) *1 C L π ω 0 > ω π
31 π 3.2 e h cell e = ω 0 c h h = ω 0 c e e 2 dv = h 2 dv = 1 (3.7) cell e = 0 h = 0 (3.8) e h cell e = ω π c h h = ω π c e e 2 dv = h 2 dv = 1 (3.9) cell e = 0 h = 0 (3.10) ω 0 ω π [15] (A B B A) dv V = (B A A B) nds (3.11) S A e B e V 3.2 n S S
32 32 3 (e e ) n = 0 (3.12) (3.7) [ ( ωπ c ) 2 ( ω0 ) ] 2 v 0π = ω 0 (e h) nds (3.13) c c iris *2 iris v 0π v 0π cell e e dv (3.14) (3.13) (3.13) k c (e h) nds (3.15) ω 0 iris [ ( ) 2 ωπ 1] ω 0 = k v π (3.16) v 0π 1 e e v 0π 1 TM 010 k a λ 3.2 E z z = 0 z ± E z ±e 0 ( E r 0 z = aξη r = a (1 + ξ 2 ) (1 + η 2 ) (3.17) *2 TM 010 (3.13)
33 Φ = 2a π e ( 0 ξ tan 1 ξ + 1 ) (3.18) [16] e r e r (iris) = Φ r = z=0 2r π a 2 r 2 e 0 (3.19) h e z e z e 0 (3.7) h θ h θ (iris) = ω 0r 2c e 0 (3.20) (3.19) (3.20) (3.15) k = 4 3 a3 e 2 0 (3.21) b d (3.7) ( χ01 r ) e z = e 0 J 0 b (3.22) e 0 = 1 J1 (χ 01 ) d (3.23) k = 4a 3 3πb 2 dj 2 1 (χ 01) 1.57 a3 b 2 d (3.24) k a 3.5 (3.24) %
34 C C n ĩ n ( jωl + 1 ) ĩ n + 2ĩ n ĩ n+1 ĩ n 1 = 0 (3.25) jωc jωc φ ĩ n+1 = ĩ n e jφ (3.26) ω = ω 0 [1 + k (1 cos φ)] 1/2 ω 0 [ 1 + k 2 (1 cos φ) ] (3.27) ω 0 1 LC, k 2 C C ( 1) (3.28) φ π 3.5 dispersion curve ω 0 ω 0 (1 + k) pass band ω ± φ ñ ĩ n = Ã+e jn φ + à e jn φ (3.29)
35 ~ i = i 0 e 2jφ L i 0 e jφ i 0 i 0 e jφ i 0 e 2jφ L L L L C' C' C' C' C C C C C n = ω ω π ω ω 0 π φ 0 φ π φ 2πd/λ g 3.5 ( ) d λ g i n = A + cos (ωt + n φ + ψ + ) + A cos (ωt n φ + ψ ) (3.30) ( A ) ± 0 ψ ± n φ = 0 φ = π φ Ã+ Ã 0 d λ g β g λ g 2π = 2πd β g φ (3.31)
36 36 3 v p v p = ± ω β g = ± ωd φ (3.32) 3.5 (0, 0) (± φ, ω) v p /d r r ωl (3.33) (3.25) jωl jωl + r ( jωl + r + 1 ) ĩ n + 2ĩ n ĩ n+1 ĩ n 1 jωc jωc = 0 (3.34) r L r L C' C' C C C 3.6 ω Q exp ( ωt/q) Q Q = ωl r = 1 ωcr 3.2 R (3.35) Q = R = ωrc (3.36) ωl
37 ω (3.26) ĩ n+1 = ĩ n e jφ α (3.37) α (3.34) α φ (3.33) φ = 0 φ = π α ωcr k sin φ = 1 kq sin φ (3.38) α 1 (3.27) φ = 0 ω φ Cr α 2 ln (2/k) (3.39) φ = π ω r L (π φ) α 2 ln (π φ) (3.40) Bevensee 2 φ d Ẽ (x, y, z + d) = e jφ Ẽ (x, y, z) H (x, y, z + d) = e jφ H (x, y, z) (3.41) [4] φ 0 π R.
38 α k = 0.01 Q = φ (rad) ω/ω φ (rad) Q = 1000 k = 0.01 ω α 0.01 φ 0.99π φ ω 0 = 1/ LC M. Bevensee [15] [17] TM z = nd ( n ) (n) cell (n) (0) φ = 0 φ = π φ (0) (Ẽ, H) (0) (Ẽ, H) (E, H) (3.11) A e B E (0) (3.13) { [ω ] 2 (φ) 1} = ω (0) [ ] c + (E h) nds (3.42) ω (0) A (φ) right iris left iris
39 mode (f = 0) n = z z = 0 p - mode (f = p) n = z z = π A (φ) cell(0) E edv (3.43) (3.42) E e e e 0 E A e A = cell(0) E e dv (3.44) A A
40 40 3 E (cell(0), right iris) = A 2 e (cell(0), right iris) + A 2 e jφ e (cell(1), left iris) = A 2 ( 1 e jφ ) e (cell(0), right iris) (3.45) E (cell(0), left iris) = A 2 e (cell(0), left iris) + A 2 ejφ e (cell( 1), right iris) = A 2 ( 1 e jφ ) e (cell(0), left iris) (3.46) e n (3.42) (3.15) (3.44) (3.45) (3.46) { [ω ] 2 (φ) 1} A c (1 cos φ) (e h) nds ω (0) A ω(0) right iris = A k (1 cos φ) (3.47) A A /A 1 (3.27) 3.3 π π Floquet z d Ẽ z (x, y, z) = n= = e jβ gz Ẽ n,z e j(β g+ 2nπ d )z n= 2nπ j Ẽ n,z e d z (3.48)
41 3.3 π 41 e jβ gz d β g z z z ω 0 ω π ω a ω = ω a φ a 3.9 v p v p = ω a φ a d (3.49) v b 3.9 TW in +z direction TW in z direction ω v phase = ω/β g = v beam φ = β g /d 2π π φ a 0 φa π 2π φ a 2π 2π φ a 2π+φ a 3.9 +z z φ a φ a π π φ a π mπ m
42 42 3 m = 1 d ω v phase = v beam π 0 π 2π 3π φ = β g /d 3.10 φ a π π ω φ = 0 0 π (3.40) PEP [18] PETRA [19] 3.4 (N ) 3.4 N ( 3.11) 0
43 Rees Rees [20] ~ i 1 ~ i 2 ~ i 3 ~ i N 2 ~ i N 1 ~ i N L t C' L C' L L L C' C' L t C t C C C C C t 3.11 N L C C ω 0 1 LC, k 2C C (3.50) L t C t ω t 1 Lt C t, k t 2C t C (3.51) L L t C C t k k t 1 k 1 ( 1 + k ) ωt 2 2 ĩ1 kω2 0 2 ĩ2 = ω 2 ĩ 1 (1 + k) ω0ĩ2 2 kω2 0 2 ĩ1 kω2 0 2 ĩ3 = ω 2 ĩ 2. (1 + k) ω0ĩn 1 2 kω2 0 2 ĩn 2 kω2 0 2 ĩn = ω 2 ĩ N 1 ( 1 + k ) ωt 2 2 ĩn kω2 0 2 ĩn 1 = ω 2 ĩ N (3.52)
44 44 3 π ĩ 1 = ĩ 2 = ĩ 3 = = ( 1) N 1 ĩ N (3.53) (3.52) ω 2 t ω 2 0 = 1 + k (3.54) π ω π ω 2 π ω 2 0 = 1 + 2k (3.55) N Hĩ = ( ω ω π ) 2 ĩ (3.56) 1 k 2 k k 2 1 k k k 2 1 k k H = k 1 k k k 2 1 k k k 1 k 2 2 (3.57) ĩ = ĩ 1 ĩ 2 ĩ 3... ĩ N 2 ĩ N 1 ĩ N (3.58)
45 m (ω m /ω π ) 2 ω ( m mπ ) = 1 2k cos ω 2 π 2N ( mπ ) 1 k cos 2 (m = 1, 2,..., N) (3.59) 2N ĩm = {ĩ m,n }( n = 1, 2,..., N) ĩ 2 m = 1 [ ] 2 (2n 1) mπ ĩ m,n = (1 + δ mn ) N sin 2N (3.60) n δ mn m = N π ω N = ω π N = k N=5 k = r 3.14 (3.52)
46 46 3 mode number m = m = m = m = m = 5 amplitude cell number n ( 1 + k ) 2 + jωc tr ωt 2 ĩ1 kω2 0 2 ĩ2 = ω 2 ĩ 1 (1 + k + jωcr) ω0ĩ2 2 kω2 0 2 ĩ1 kω2 0 2 ĩ3 = ω 2 ĩ 2. (1+k+jωCr) ω0ĩn 1 2 kω2 0 2 ĩn 2 kω2 0 2 ĩn = ω 2 ĩ N 1 ( 1 + k ) 2 + jωc tr ωt 2 ĩn kω2 0 2 ĩn 1 = ω 2 ĩ N (3.61)
47 L 1 r L r L ~ i ~ ~ 1 i 2 i 3 C' C' C 1 C C 3.14 r 3.11 (3.56) H 1 (H + H 1 ) ĩ = ( ω ω π ) 2 ĩ (3.62) H 1 (3.36) j Q j 0 Q j 0 0 Q H j Q j Q j Q (3.63) H 1 1 (3.59) (3.60) 0 ( ) 2 ωm = ĩt m H 1ĩm (3.64) ĩm = N ω π p=1, m ĩ t p H 1ĩm ( ) 2 ( ) 2 ĩp (3.65) ωp ω π ω m ωπ t (3.64) (3.63) m ω m ω π ω m ω π ( 1 + j ) 2Q (3.66) H 1 (3.65) 0 0
48 n ω 0 ω 0 + δω 0 (3.67) H 1 n j Q j Q + 2δω 0 ω 0 (3.68) (3.65) 0 π ĩn (3.65) p = N Ṽ ejωt (3.62) 0. ( ) 2 ω (H + H 1 ) ĩ 0 ĩ = ω π jωcṽ 0. 0 (3.69) π Q π ω = ω π n = 1 jωcṽ Ĩ (3.70) (3.69) E N N
49 ~ Ve jωt r L r L C' C' cell number: n 1 C C C n n Ĩ (H + H 1 E) ĩ = 0. 0 (3.71) ĩ = N a m ĩ m (3.72) 1 (3.60) a m a N = j Q N Ĩ Q sin a m = j 1 jq m ( mπ 2N N 2 ) Ĩ (m N) (3.73) m ( ωm ω π ) 2 1 (3.74) π Q N a N π N π a N 1
50 50 3 (3.74) π (n = 1 N) ĩ 1 ĩ N (3.60) (3.73) ĩ 1 Ĩ j Q N ĩ Ñ I j Q N [ ] 2 sin (N 1)π 1 + 2N 1 Q N 1 ( 1)N sin [ (N 2)(N 1)π 2N 1 Q N 1 ] (3.75) Q 1000 ] ĩ N /ĩ 1 arg [( 1) N 1 ĩ N /ĩ N N 1 N 1 kπ2 2N 2 (3.76) N 2 π ~ ~ i (end cell) / i (first cell) N 3.16 Q N π Q = 1000 k = 0.05
51 3.5 APS 51 Arg{(-1) (N-1) ~ ~ i (end cell) / i (first cell)} 2 (rad) N 3.17 Q N π Q = 1000 k = APS φ a = π 0 0 π/ APS (alternating periodic structure) [21] [22] 1 π 0 2 π 0 2 φ = π 0 π 0 biperiodic structure π/2 coupling cell
![52 3 3.18 500 MHz APS 3.5.2 APS 3.19 SCS side coupled structure [23] ACS annular coupled structure [24] DAW disk-and-washer [25] 3.5.3 3.19 l s 1 φ ( jωl l + 1 + 2 jωc l jωc ( jωl s + 1 + 2 jωc s jωc ) ĩ l = ej φ 2 + e j φ 2 jωc ĩ s ) ĩ s = ej φ 2 + e j φ 2 jωc ĩ l (3.](/docs-images/91/105189328/images/52-0.jpg "77) φ/2 z ω l 1 Ll C l ω s 1 Ls C s (3.78) k l 2C l C k s 2C s C (3.79)")
52 MHz APS APS 3.19 SCS side coupled structure [23] ACS annular coupled structure [24] DAW disk-and-washer [25] l s 1 φ ( jωl l jωc l jωc ( jωl s jωc s jωc ) ĩ l = ej φ 2 + e j φ 2 jωc ĩ s ) ĩ s = ej φ 2 + e j φ 2 jωc ĩ l (3.77) φ/2 z ω l 1 Ll C l ω s 1 Ls C s (3.78) k l 2C l C k s 2C s C (3.79)
53 3.5 APS APS SCS (side coupled structure) ACS (annular coupled structure) DAW (disk-and-washer) ~ i = i l exp(-jf) i s exp(-jf/2) i i exp(jf/2) i exp(jf) l s l L l L s Ll L s L l C' C' C' C' C l C s C l Cs C l 3.20 (3.77) ĩ l ĩ s = k l ω 2 l cos φ 2 (1 + k l ) ω 2 l ω2 = (1 + k s) ω 2 s ω 2 k s ω 2 s cos φ 2 (3.80)
54 54 3 Ω l ω l 1 + kl Ω s ω s 1 + ks K ω l ω s kl k s (3.81) (3.80) ( ( ω 2 Ω 2 ) ( l ω 2 Ω 2 ) φ s = K 2 cos 2 2 ) (3.82) ω φ = 0 φ = π ω = Ω 1, Ω 2 (φ = 0) ω = Ω l, Ω s (φ = π) (3.83) Ω 2 1 Ω 2 l = + Ω2 s 2 ( ) Ω 2 ± K 2 + l Ω 2 2 s (3.84) 2 φ = π v g d ω φ d (3.82) K 2 sin φ v g = 4ω (2ω 2 Ω 2 l Ω2 s) d (3.85) Ω l Ω s φ = π v g = 0 Ω l = Ω s Ω confl (3.86) v g = ± K d (3.87) 4Ω confl Ω l = Ω s confluence (3.79) C l C s 3.18 TM 010 APS
55 3.5 APS 55 ω (Ω l Ω s ) 1/ Ω l /Ω s = φ 3.21 k s k l k l = 0.03 k s = 0.09 Ω l /Ω s l s ω = Ω l Ω s φ = π APS TM 010 APS φ = π φ = ω = Ω 1 ω = Ω 2 (3.80) π Ω 2 Ω s < Ω l l s
56 56 3 ω Ω l ω Ω s 3.22 φ = π APS 2 ω = Ω 2 (> Ω 1 ) ω = Ω φ = 0 APS 2 Ω 2 2 Ω 2 l = Ω 2 s Ω 2 1 ( Ω 2 = K 2 l Ω 2 s 2 ) 2 Ω2 l Ω2 s 2 (3.88) Ω 2 Ω l Ω l Ω s Ω 1 Ω s kl k s 2 (3.89)
57 3.5 APS 57 (k s, k l k) Ω 2 Ω 1 /Ω l k k s k l (3.81) (3.86) ω 2 l ω 2 s = 1 + k s 1 + k l > 1 (3.90) φ = (3.80) (ĩs ĩ l ) 2 = k s(1 + k l ) k l (1 + k s ) k s k l (3.91) k s k l C s C l (3.92) ĩ 2 s C s ĩ2 l C l (3.93)
58
59 59 4 aperture loop matched coupling undercoupling overcoupling (well-padded) 4.1 S a
60 60 4 S a S S a 4.1 S m S a S m S a S m S (4.1) 4.1 ( 4.1 ) z ( ) k k S a n k k = n on S a Sa Sm n n = -k z V 4.1 S a (e n, h n ) (4.1) (e n, h n ) e jωt (4.2)
61 V e n, h n ) V ( 2 + ω2 n c 2 ) e n(r) = 0 h n (r) (n = 1, 2, 3,...) (4.3) S e n (r) = 0 h n (r) = 0 (4.4) n e n (r) = 0 n h n (r) = 0 (4.5) δ nn V V e n e n dv = δ nn h n h n dv = δ nn (4.6) e n (r) e n (r) m 3/2 e n (r) = c h n (r) ω n h n (r) = c e n (r) ω n (4.7) Ẽ H
62 62 4 Ẽ + jωµ H 0 = 0 H jωɛ 0 Ẽ = 0 Ẽ H Ẽ H (n A B) = (A B) n (4.8) (B A A B) dv V = [A ( B) B ( A)] n ds (4.9) S A = Ẽ B = e n A = H B = h n (4.5) (4.7) 2 ( ω 2 ωn 2 ) V ( ω 2 ωn 2 ) V ( ) Ẽ e n dv ω n c n Ẽ h n ds = ω n c S m H h n dv j ω µ 0 S m S a ( ) n Ẽ h n ds = j ω µ 0 S a ( ) n Ẽ h n ds ( ) n Ẽ h n ds (4.10) Ẽ Ẽ H 2 δ = µ 0 ωσ n Ẽ = 1 + j 2 µ 0ωδ H (4.11) Q n Q n 2 δ n S m h 2 n ds (4.12) δ n ω = ω n Q n n Q Q n
63 (4.10) 2 1/Q n Ẽ e n H h n j ɛ 0 /µ 0 V H h n dv j ) S a (n Ẽ µ 0 (ω jω n Q n h n ds ) (4.13) ω 2 n ω ω n ( ω n 1 1 ) ω n (4.14) 2Q n Q n (4.13) n (e n, h n ) 0 rotational irrotational *1 [26] H g m g m S g m (r) = 0 ( ) 2 + ω2 m g c 2 m (r) = 0 (4.15) g m n = 0 (4.16) (4.10) 2 g m H h n j ( ) S a n Ẽ h n ds ) + n µ 0 (ω [ j ( ) ] g jω n Q n ω 2 m n n µ ω m 0 ω Ẽ g m ds S a (4.17) ω n = 0 *1 Ẽ H
64 z k S a S a (4.17) h n S a Ẽ k +z (t) (z) (x, y) z A (x, y, z) e jωt = [A t (x, y) + A z (x, y)] e j(ωt β gz) (4.18) β g z TM (transverse magnetic) z TE (transverse electric) Z g H t,g (x, y) = ±k E t,g (x, y) (4.19) [27] + e jβgz +z e jβgz z Z g wave impedance ζ 0 = Ω β β g TM Z g 1 Y g = β g β ζ 0 (4.20) TE Z g 1 Y g = β β g ζ 0 (4.21) Y g
65 e g t (x, y) h g t (x, y) Ẽ g,t (x, y, z) = Ṽ (z)e g t (x, y) H g,t (x, y, z) = Ĩ(z)h g t (x, y) (4.22) e g t (x, y) h g t (x, y) S g eg t (x, y) 2 dxdy = hg t (x, y) 2 dxdy = 1 (4.23) S g S g (4.19) (4.23) e g t (x, y) = 1 (4.24) h g t (x, y) (4.22) +z + z (4.19) (4.24) Ṽ + (z) = Z g Ĩ + (z) = Ĩ+(z)/Y g e jβ gz Ṽ (z) = Z g Ĩ (z) = Ĩ (z)/y g e +jβ gz (4.25) z = 0 Z L z( 0) z = 0 Ṽ (z) = Ṽ+(z) + Ṽ (z) Ĩ(z) = Ĩ+(z) + Ĩ (z) (4.26) Ṽ (0) = Z L Ĩ(0) (4.27) (4.25) (4.26) (4.27) Ṽ (0) Ṽ + (0) = Z L Z g Z L + Z g R (4.28)
66 66 4 R z Z in = Z g 1 + Re 2jβ gz 1 Re 2jβ gz = Z g Z L jz g tan β g z Z g jz L tan β g z (4.29) S a Ẽ t (aperture) = Ṽge g t H t (aperture) = Ĩgh g t (4.30) z Ṽg Ĩg (4.30) (4.17) Ĩ g h g t n h n jṽg S a (n e g t ) h n ds ) + µ 0 (ω [ ] jṽg g jω n Q n ω 2 m (n e g t ) g m ds n µ ω m 0 ω S a (4.31) e g t (n A) B = n (A B) (4.23) (4.24) (e g t h g t ) kds = (e g t h g t ) nds = 1 (4.32) S a S a S a [ 2 (n e Sa g t) h n ds] [ ] 2 1 Ĩ g jṽg ) n µ 0 (ω jω n Q n ω 2 jṽg (n e g t ) g m ds (4.33) n µ ω m 0 ω S a [ ]
67 (4.33) ω n ω n (4.33) Ṽ g µ 0ω n [ ] 2 (n e g t ) h n ds r n (4.34) Q n S a Ĩ g r n r n C n 1 ω n r nq n (4.35) L n r nq n ω n (4.36) L 0 µ 0 [ ] 2 (4.37) n S (n e a g t) g m ds r C L (Ω) (Farad) (Henry) S a Ỹin = 1/ Z in Ỹ in = Ĩg Ṽ g = 1 jωl 0 + n 1 1 (4.38) jωc n + r n + jωl n (1) ω 0 (4.38) n ω Ỹc (ω 0) Ỹ in (ω 0) 1 jωl 0 + Ỹc (ω 0) (4.39)
68 68 4 L 1 L 2 L 0 r 1 r 2 C 1 C (4.38) Q ω 0 (2) (ω ω n ) Ỹc (ω n ) Ỹ in (ω ω n ) 1 1 jωc n + r n + jωl n + Ỹc (ω n ) (4.40) (3) Y g (= 1/Z g ) (4.12) Q Q n Q Q ext, n Q ext, n (4.40) Ln C n Y g = r n Y g Q n (4.41) Ỹ in (ω ω n) ( j ω ω n 1 Q ext, n ) Y g + ω n ω + 1 Ỹc (ω n ) (4.42) Q n (4) S a 1/4 z = ± π 2β g ( Ỹ in 1/ Z ) in S a ( Ỹ in 1/ Z in ) Ỹ in = 1 Z in = Y g 2 = Z in Ỹ in Zg 2 (4.43)
69 (4.29) (4.39) (4.40) Z in = n 1 jωl n 1 + jωc n + 1 R n + 1 jωc 0 (4.44) C 0 L 0Y 2 g C n L n Y 2 g L n C n Z 2 g R n r 1 n Z2 g (4.45) 4.3 L' 1 L' 2 C' 0 C' 1 C' 2 R' 1 R' λ g /4 (5) (4.34) (4.37) [ ] ( ) F ( V1 I 1 ) ( ) V2 = F I 2 (4.46) F = ( L1 /M jω ( L 1 L 2 M 2) ) /M 1/(jωM) L 2 /M (4.47)
70 70 4 I 1 M I 2 L 1 - M L 2 - M V 1 V 2 M L 1 L 2 m : L 1 L M m = L 1 /L 2 k = M/ L 1 L 2 (4.48) k = 1 M F ideal = ( m 0 ) 0 1/m (4.49) Z 2 1 Z 1 m 2 Z 2 Z 1 = V 1 I 1 = m2 V 2 I 2 = m 2 Z 2 (4.50) Z m 2 Z m : (4.34) (4.37) [ ] e g t h n g n e g t h n g n
71 m 0 :1 m 1 :1 m 2 :1 r'' 1 r'' 2 L'' 0 L'' 1 L'' 2 C'' 1 C'' S a h n g n e g t m n = m 0 = c n S a (e g t h n ) nds c 0 [ ] (4.51) m S (e a g t h n ) nds c 0 c n (4.34) (4.35) (4.36) (4.37) C n = C n/m 2 n L n = m 2 n L n r n = m 2 n r n L 0 = m 2 n L 0 (4.52) m 2 n m (4.51) m 0 m n (4.41) Q ext, n m 2 n
72
73 73 5 * 1 *1 kv
74 74 5 (beam loading) V J(x, y, z, t) U = 1 (E D + B H) 2 [28] [29] J EdV = V V du dt dv + (E H) nds S S V n (E H) Poynting vector 5.1 J (t, x, y, z) E + µ 0 H t = 0 H ɛ 0 E t = J (5.1) Ẽ H S a 0 (4.7) E H V h n ( J ) dv (5.2)
75 h n = 0 ( h n J ) dv = J ( h n ) dv = ω n c V V V J e n dv (5.3) V ( h n J ) ) dv J ( h n ) dv = ( J hn dv V V V ) = ( J hn nds = 0 S ( ) 2 t 2 + ω2 n V ( ) Ẽ e n dv + ω n c n Ẽ h n ds = S ɛ 0 t V J e n dv (5.4) ( ) 2 t 2 + ω2 n V H h n dv + ( ) n µ 0 t Ẽ h n ds = ω n c S V J e n dv (5.5) (5.4) (4.13) V Ẽ e n dv = ɛ 0 ω n j J e V n dv ( ω ω n j Q n Ẽ (x, y, z) = e n (x, y, z) Ẽ e n dv n V = n j V e n (x, y, z) J e n dv ( ɛ 0 ω n ω ω n ω n ω ) (5.6) j Q n ω n ω ) (5.7) ( = ω) q
76 76 5 z (x = y = 0) c +z T = 2π/ω [ ( J (t, x, y, z) = ki 0 δ(x)δ(y) cos p ωt ωz ) ] (5.8) c k +z p δ(x) δ(y) x y p=1 I 0 I 0 = q T (5.9) p = 1 n = 1 ω ω 1 ( J (t, x, y, z) 2kI 0 δ(x)δ(y) cos ωt ωz ) c (5.10) 2 * 2 J (x, y, z) = 2kI 0 δ(x)δ(y)e jωz/c (5.11) (5.7) n = 1 Ẽ (x, y, z) e 1 (x, y, z) 2j k e1 (0, 0, z) e jωz/c ( ) dz I 0 ɛ 0 ω 1 j Q 1 ω 1 ω ω ω 1 = e 1 (x, y, z) 2j e1 z (z) e jωz/c ( dz ɛ 0 ω 1 ω ω 1 j Q 1 ω 1 ω ) I 0 (5.12) *2 z q(z) = q 0 δ(z) q(z) = q 0 e z 2 2σ z 2 2πσz σ z r.m.s. q 0 2I 0 2I 0 e ω 2 σ 2 z c 2
77 e 1 z e 1 z (z) n = 1 ω = ω 1 Ẽ (x, y, z) = e 1 (x, y, z) 2Q 1I 0 ɛ 0 ω 1 e 1 z (z) e jωz/c dz (5.13) e jωz/c E (t, x, y, z) = Re [Ẽ (x, y, z) e jωt] = e 1 (x, y, z) 2Q 1I 0 ɛ 0 ω 1 ( e 1 z (z) cos ωt ωz c ) dz (5.14) z z E (t, z) e 1 z (z) z E (t, z) = e 1 z (z) cos ωt 2Q 1I 0 ɛ 0 ω 1 e 1 z (z) cos ωz dz (5.15) c z = ct ( V br ) V br = = 2Q 1I 0 ɛ 0 ω 1 E (z/c, z) dz [ e 1 z (z) cos ωz ] 2 c dz (5.16) (ω = ω 1 ) (5.16) V e 1 e 1 dv = 1 [ e 1 z (z) cos ωz ] 2 c dz = [ ] 2 E 1 z (z) cos ωz c dz V E 1 E 1 dv (5.17) (2.4) Q Q 1 = ɛ 0ω 1 V E 1 E 1 dv (5.18) 2P wall (5.16) V br = V br 2 (5.19) I 0 P wall
78 78 5 R a V br = I 0 R a 1 = 2I 0 R 1 (5.20) (5.12) Ṽ b = 2I 0 jωc jωl R 1 (5.21) C 1 = Q 1 ω 1 R 1 L 1 = R 1 ω 1 Q 1 (5.22) (5.23) 5.1 (5.12) 5.2 2I 0 e jωt ~ L 1 C 1 R 1 (= R a /2)
79 L 1 L 2 2I 0 ejωt C 1 C 2 ~ R 1 (= R 1a /2) R 2 (= R 2a /2) R 1 (= R a 1 /2) jb 1 = j Q ( 1 ω R 1 ω 1 ) ω 1 ω Y 0 (= 1/Z 0 ) ±z ṽ g± 2Ĩg Y 0 n 2I g e jωt v g+ 1 : n 2I 0 e jωt Y 0 v g 1/R 1 jb 1 klystron waveguide cavity Y 0 (= 1/Z 0 )
80 80 5 1/n n 2 Z 0 /n 2 β R 1 Z 0 /n 2 (5.24) 2I g e jωt V g+ 2I 0 e jωt V g β/r 1 1/R 1 jb 1 klystron waveguide beam β Ṽg± (= ṽ g± /n) 1/2 Ĩ g R 1 /β *3 Ṽ g+ = R 1 β Ĩg Ṽ g = R 1 β Ĩg (5.25) Ṽ g+ + Ṽg Ĩ g + Ĩg (5.26) I 0 = 0 Ṽg *3
81 I g = 0 Ṽb Ṽg (5.25) (5.26) 5.4 Ṽ g = Ṽg+ + Ṽg = R 1 β (Ĩg Ĩg ) Ĩg = Ĩg + Ĩg 1 R 1 + jb 1 (5.27) Ṽ g = 2R β + jb 1 R 1 Ĩ g (5.28) (tuning angle) ψ ω ω 1 ξ ψ tan 1 ξ (5.29) ξ B 1R β = Q ( 1 ω 1 + β ω 1 ) ω 1 ω (5.30) Ṽ g = V gr cos ψe jψ+jθ (5.31) V gr (ω = ω 1 ) V gr 2R 1 Ĩg 1 + β (5.32) θ * 4 P g P g = R 1 Ĩg 2 2β (5.33) V gr V gr = 2 β 2R1 P g = 2 β Ra 1 P g (5.34) 1 + β 1 + β *4 π
82 82 5 Im ω < ω ' 1 ω = ω' 1 ~ Vg ω > ω ' 1 V gr ψ 0 θ Re 5.5 Ṽ g ψ θ ω ω V gr ω 1 Ĩg = 0 Ṽb Ṽ b = 2R 1I β cos ψejψ = V br cos ψe jψ (5.35) V br 2R 1I β = R a 1I β (5.36) * 5 *5 (5.20) 1 + β β
83 Im ~ V g ~ V c ψ V gr e jθ ψ V br V b 0 φ θ ~ V a Re 5.6 P. B. Wilson V c V a Ṽc Ṽ c = Ṽg + Ṽb V c e jφ (5.37) V c φ [30] Perry Wilson 5.6 [31] Ṽc (5.37) V a V a = V c cos φ (5.38) P b = I 0 V a (5.39)
84 84 5 P b V c (= P b + P wall ) (θ = φ) β [31] [32] β = 1 + P b P wall (5.40) P wall = V c 2 (5.41) 2R [32] [33] (z > 0 ) z P g (z) dp g dz = 2αP g (5.42) α m 1 E g dz E g dz E g P g de g dz = αe g (5.43) z = 0 E 0 (5.43) E g (z) = E 0 e αz (5.44)
85 I 0 E b dz I 0 E b dz z z + dz P b dp b dz = I 0E b 2αP b (5.45) r(ω/m) (z, z + dz) E b dz 2αP b dz rdz 5.1 rdz = (E bdz) 2 2αP b dz (5.45) E b (5.46) de b dz = αri 0 αe b (5.47) E b = 0 E b (z) = ri 0 (1 e αz ) (5.48) E(z) = E 0 e αz ri 0 (1 e αz ) (5.49) φ E g E b E(z) = E 0 cos φ e αz ri 0 (1 e αz ) (5.50) z V a V a (z) = z 0 E(z)dz = z 0 [ E0 cos φ e αz ri 0 (1 e αz ) ] dz (5.51)
86 86 5 r z constant impedance structure constant gradient structure P v g w P = v g w (5.52) P v g w E 2 w z
87 5.4 87
88
89 89 [1] KEK Report , [2] (, 1954). [3] John C. Slater, Microwave Electronics (D. Van Nostrand Co., Inc., 1950). [4] Robert E. Collin, Foundations for Microwave Engineering, 2nd Ed. (MacGrawhill, 1992). [5] J. D. Jackson, Classical Electrodynamics, 3rd edition (John Wiley & Sons, 1999). [6] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison- Wesley, 1962). [7] J. A. Stratton, Electromagnetic Theory (MacGrawhill, 1941). [8] C. G. Montgomery et al ed., Priciples of Microwave Circuits, M. I. T. Radiation Laboratory Series (MacGrawhill, 1948) Vol. 8. [9] TRISTAN Project Group, TRISTAN Electron-positron Colliding Beam Project, KEK Report 86-14, March [10] E. L. Ginzton, Rev. Sci. Instr., (1948). [11] M. Chodorow et al, Rev. Sci. Instr., (1955). [12] L. D. Landau and E. M. Lifshitz, Mechanics, 3rd Ed. (Pergamon Press, 1976). [13] Y. Yamazaki et al, KEK Publication, KEK Report 80-8, [14], (1953). [15] R. M. Bevensee, Electromagnetic Slow Wave Systems (John Wiley and Sons, 1964). [16] G. Wendt, Handbuch der Physik, ed. S. Flügge (Springer, Berlin, 1958), Vol. 16, p. 140 (Springer, 1958). [17] R. M. Bevensee, Annals of Physics, (1961) (John Wiley and Sons, 1964). [18] M. A. Allen et al, Proc. of 1977 Particle Accelerator Conference, IEEE Trans. Nucl. Sci. NS (1977). [19] G.-A. Voss, Proc. of 1977 Particle Accelerator Conference, IEEE Trans. Nucl. Sci. NS (1977).
90 90 [20] J. R. Rees, SLAC Publication, PEP Notes 255, [21] T. Nishikawa et al, Rev. Sci. Instrum (1966). [22] K. Akai et al, Proc. 13th Int. Conf. High Energy Accelerators, vol.2, p.303, [23] E. A. Knapp et al, Rev. Sci. Instr., (1968). [24] T. Kageyama et al, Particle Accelerators (1990). [25] V. G. Andreev, Sov. Phys. JETP (1969). [26] F. E. Borgnis and C. H. Papas, Handbuch der Physik, ed. S. Flügge (Springer, Berlin, 1958), Vol.16, p.415. [27] J. D. Jackson, ibid, p. 359 ff. [28] J. D. Jackson, ibid, p. 258 ff. [29] J. Schwinger et al, Classical Electrodynamics (Perseus Books, 1998), p. 21 ff. [30] T. Nishikawa, IEEE Trans. Nucl. Sci., NS (1965). [31] P. B. Wilson, Proc. 9th Int. Conf. High Energy Accelerators, p.57, [32] P. B. Wilson, SLAC Publication, SLAC-PUB-2884, [33] J. E. Leiss, Linear Accelerators, ed. P. M. Lapostolle and A. L. Septier (North-Holland, 1970), p.147.
Slater[] Collin[] [3] [4] AR 508.6MHz λ =58.9cm 4 9.7m APSAlternating Periodic Structure 50kW.M/m 3M cm 7.5 WR500 f c TE 0 f c = 393MHz 9 90k 0
![Slater[] Collin[] [3] [4] AR 508.6MHz λ =58.9cm 4 9.7m APSAlternating Periodic Structure 50kW.M/m 3M cm 7.5 WR500 f c TE 0 f c = 393MHz 9 90k 0](/thumbs/91/105189250.jpg "Slater[] Collin[] [3] [4] AR 508.6MHz λ =58.9cm 4 9.7m APSAlternating Periodic Structure 50kW.M/m 3M cm 7.5 WR500 f c TE 0 f c = 393MHz 9 90k 0")
00 3-00 3-00 9 30 OHO 97 4.... 4.. 8.3.... 9 3 3..... 3 3.... 5 3.3 π.... 9 3.4.... 0 3.5 APS... 4 4 7 4.... 8 4.... 30 5 35 A 40 L C R L C R Appendix Slater[] Collin[] [3] [4] AR 508.6MHz λ =58.9cm 4
OHO.dvi
1 Coil D-shaped electrodes ( [1] ) Vacuum chamber Ion source Oscillator 1.1 m e v B F = evb (1) r m v2 = evb r v = erb (2) m r T = 2πr v = 2πm (3) eb v
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds
127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
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2003/12/04 TASK PAF A. Fukuyama et al., Comp. Phys. Rep. 4(1986) 137 A. Fukuyama et al., Nucl. Fusion 26(1986) 151 TASK/WM MHD ψ θ ϕ ψ θ e 1 = ψ, e 2 = θ, e 3 = ϕ ϕ E = E 1 e 1 + E 2 e 2 + E 3 e 3 J :
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Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K
2 2 T c µ T c 1 1.1 1911 Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ 58 213 email:takada@issp.u-tokyo.ac.jp 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B =
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4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
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X X 1 1 2 Free Electron Laser: FEL 2.1 2 2 3 SACLA 4 SACLA [1]-[6] [7] 1: S N λ [9] XFEL OHO 13 X [8] 2 2.1 2(a) (c) z y y (a) S N 90 λ u 4 [10, 11] Halbach (b) 2: (a) (b) (c) (c) 1 2 [11] B y = n=1 B
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I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t)
Radiation from moving harges# Liénard-Wiehert potential Yuji Chinone Maxwell Maxwell MKS E x, t + B x, t = B x, t = B x, t E x, t = µ j x, t 3 E x, t = ε ρ x, t 4 ε µ ε µ = E B ρ j A x, t φ x, t A x, t