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1 1 1
2 Fe C TEM C TEM Fe TEM
3 ,1 multiwall nanotube 1993 singlewall nanotube ( 1,) sp 7.4eV atom 1 1,3 c ( ) O A C h (chiral vector) O B T A B OBB A OB AB.4nm 1nm 1µm ( ) a1 a ( n m) C h na + ma, = (1.1) 1 = n m d θ n m t d t = 3a c c n + nm + m π (1.) θ = tan 1 3m n + m θ 6 π (1.3) 3
4 a (.14nm) c c n=m( = /6) (n,n) m=( =) (n,) ( 1, ) (armchair) (zigzag) n m (chiral) 1,5 T T (1.1) {( m + n) a ( n + m) a } 1 T d R = (1.4) dr (m+n) (n+m) T ( )l T 3l = (1.5) d R l = Ch = 3 ac c n + m + nm (1.6) (n,m) sp 3 1 1,6 (1.1) q C h k = πq (1.7) k (1.4) T 4
5 E k, k x y 5 ( ) k 1,7 K ( b ) 3 1 b (1.7) k K (1.7) (1.1) k= ( b ) 3 n m = 3q 1 b (1.8) n 3 ( 1,8) ( ) ( ) 1/3 / ( 1,9) 1 F + V = E + (1.9) F = p + h + s (1.1)
6 V ( 7s + 6h + 5 p) = (1.11) 3 E ( 7s + 6h + 5 p) = (1.1) (1.1),(1.11),(1.1) (1.9) s p + = h s + + p 1 + ( 7s + 6h + 5p) ( 7s + 6h + 5 p) 3 = 6 + (1.13) = ( 1,1) / cm 3 (1kV) 5V
7
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9 (, ) 9
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11 1-1,11 mm 1991 ( ) (Fe Co) ( / ) / Fe Ni Co Ni 1.11 Co (Ru Rh Pd Os Ir Pt ) (a ) (15A ) / 1 1.3nm 11
12 11 (SiC) 17 Si Si(A AB B) 1
13 13
14
15 - - (.1).1 z z= z= x E x iwt = A e sin kz (.1) x z= z= E = sin k = nπ ncπ k =, ω = (.) n n k= / (.) / n 15 x (.) sinkz n ( z ) z= z= ( ) f z (.) (.1) z= z= E ( ) <z< f z f ( ) z = n= A n πnz sin + B n π nz cos (.3) z= z ( z) f ( z ) f + = (.4) sinkz coskz z= (.3) k=n / (.)
16 sine cosine (.4) z= z= E E E = z z ( z = ) = E( z = ), ( z = ) = ( z ) (.5) x,y,z 3 τ τ a x,y,z 3 xˆ, yˆ, zˆ k ϖ ϖ k = k x xˆ + k yˆ + k y z zˆ A ϖ ρ ( i ω t ik r ) 16 c a (.6) exp (.7) r ρ A r ϖ = xxˆ + yyˆ + zzˆ x,y,z + k ϖ - k ϖ k ϖ n, n, n x= y= z= (.) k x π π π = nx, k y = ny, k z nz (.8) x y z
17 k = k + k + k x y z ( n + n n ) k = π x y + z (.9) n, n, n kc ω = ( n + n + n ) c ω = π x y z (.1) x, n, n, n x,y,z x y z R = n x + n y + n z ω = πc y z (.11) n, n, n 1/8 x y z π ω ω 3 = (.1) 3 πc 3π c +d ω π c 3 dω (.13) 3 +d ( ω ) dω dω π ω m = (.14) c = / +d 8πν c ( ν ) dν dν m 3 = (.15) 17
18 .1. n, n, n R x y z 18
19 - WU W WU W WU W ω = (.16) η (spontaneous emission) 1 A W( ) p ( U ) = A B W ( ω) (.17) U + U (induced emission stimulated emission) p ( U ) = B W ( ω ) (.18) U 1 B U = B U (.19) gu g (.19) g B = g B (.) U U U 1 (.16) (.17) A U B A B U 19
20 A B N P abs ( ω ) N = ηωbw (.1) N P emi { A BW( ω )} N U = η ω + (.) N U ηω = N kbt exp (.3) ( ) ( ) W ω W ( ) = ( ω ) ω W W ( ) th A th P abs = P emi (.) (.1) N U th ω = (.4) B N N U (.3) W th ( ) A 1 = ω kbt B e ω η (.5) 1 1 n ( U ) = ( n + )A P 1 (.6) ( U ) na P = (.7) m( ) +d dω
21 ( ω) m( ω) nηω W = (.8) (.6) (.7) A P ( U ) = + m ω ηω P ( U ) ( ) ( ω ) W ( ω ) A (.9) A = W ( ω ) (.3) m ηω (.17) (.18) A B = (.31) m ω η ( ) ω (.14) A B 3 ηω = m( ω) η ω = 3 (.3) π c (.5) W th ( ω ) ηω π c 1 3 = 3 ηω k T e B 1 (.33).3 1 d m( ) dω A = m ( ω) Bηω ηω (.17) (.18) W( ) ηω ( k B T ) exp ηω A= (.4) (.5) = A B π ν π A µ, 3 U Bν = µ 3ε hc 3ε h = U (.34) W th 1
22 B W( ) W th (.1) (.) B ( ω) = Bg( ω) (.35) B ( ω ) dω B B = (.36) g( ) B( ) NU N P ( N N ) ηωb( ω) c P = (.37) U z d P dz ( z) ( N N ) B( ω ) P( z) U ηω = (.38) c (amplitude absorption constant) ( ) d dz P ( z) α ( w) P( z) = (.39)
23 (.38) (.39) α ηω = U (.4) c ( ω ) ( N N ) B( ω ) (.35) (.34) A h = πη ν = ω π (.3) B = π ε η µ α ( ω) ( N N ) µ g( ω) 3 3 U πω = U U (.41) 6ε ηc (orentzian) Gaussian g ( ω) 1 = π ω g ( ω ω ) ( ) (.4) + ω ω ω = ± ω g ω g ω =1 π ω ( ) ( ) ω (half width at half maximum HWHM) ω ln 1 ω ω g G ( ω ) = exp ln (.43) ω ω ω ω ω = ln 1.47 g G ( ω ) = = (.44) π ω ω g ( ω ) π ln = g ( ω) g ( ω ).4 ω G (.19) ( ) NU N ( ω) = ( N N ) σ ( ω) α (.45) U (.45) (.41) (.4)
24 σ ηω ( ω) = µ g( ω) = B( ω) 3ε πω ηc U c (.46) NU N (.3) N U ηω kbt ( e ) N N = 1 (.47) ηω T N ηω k B N N U (.48) k T B ( ) ηω T k B N N N (.49) U ω x F dx dt dx F + γ + ω x = (.5) xt m ( ) t iω m -e E ω e (.5) d x + dt dx γ + ω dt e x = E m iωt ( ω) e (.51) x x iω t ( ω ) e = x (.5) (.51) ( ω) e E x ( ω ) = (.53) m ω iγω ω 4
25 x ( ω) e = mω E ω ω ( ω) iγ (.54) N N ( ) t iω P ω e P ( ) = ex( ω )( ) ω (.55) N N U χ ω = χ ω iχ ω ( ω ) ε χ ( ω ) E( ω ) P (complex susceptibility) ( ) ( ) ( ) = (.56) (.54) (.55) U χ ( ω) = ( N N ) ε U mω e ω 1 ω iγ (.57) ( ) ( ) ( ) ( ) ( ) χ ω χ ω χ ω = χ ω iχ ω χ χ ( ω ) ( ω ) = = ( N N ) U e ω ω ε mω γ ( N N ) ( ω ω ) + U e γ ε mω γ ( ω ω ) + (.58) (.59).5 ( ) χ ω ( ω) = ε { 1 χ( ω) } ε + (.6) µ = µ η η ( ω) ε η = η κ = = 1+ χ ε ( ω) i (.61) ( ) z exp iωt ikz η iκ 5
26 k ω ( η iκ ) c = (.6) ( iωt ikz ) = exp z exp iωt iη z κω c exp (.63) e αz ω α = κ (.64) c χ ( ω) κ χ ω c 1 (.59) α ( N N ) U e γ 4ε mc γ ( ω ω ) + (.65) e m ω 3 η (.41) g( ) (.4) 6 µ U χ χ µ U ω ω N N U (.66) 3ε η ( ω ω ) + γ ( ω) = ( ) ( ω ) = ( ) µ U γ 3ε γ N N U (.67) ( ω ω ) + f (.66) (.67) e f m ω µ 3η U = (.68) mω f µ 3e η U = (.69) f (oscillator Strength) NU N (.67) χ N N χ U
27 .3.4 g ω g ω ( ) G ( ).5 ( ) 7
28 -3 N U N N N U N N U (inverted population) N NU (pumping) -4 3 (three-level laser) 3 1,,3 W 1, W, W3 N 1, N, N3.6 W1 W W 3 N1 N N (relaxation) (radiative process) (non-radiativve Process) (relaxation rate) (relaxation constant) (fluorescence lifetime) 8
29 W W U γ U WU W γ N U U WU W γ = = U N γ U, NU N exp (.7) kbt γ γ U U WU W = exp kbt (.71) NU N 3 (rate equation) dn dt dn dt dn dt 1 ( Γ + γ 1 + γ 13 ) N1 + γ 1N + γ 31N3 = (.7) ( γ 1 + γ 3 ) N γ 3 3 γ N 1N1 + 3 = (.73) ( Γ + γ 13) N1γ 3N ( γ 31 + γ 3 ) N3 = (.74) N 1 + N + N 3 = const = N 3 (.7) (.74) k B T (.71) γ 1 γ 1, γ 13 γ 31, γ 3 γ 3 γ, γ 13 γ 3 (.7) (.74) 1 γ 1 ( γ 31 + γ 3 ) ( γ + γ ) + ( γ + γ ) N N 1 γ = (.75) N = γ 1 Γ ( γ + γ ) + ( γ + γ ) 31 γ Γ N (.76) 9
30 γ Γ > γ 3 γ (.77) N1 N (.77) γ γ 3 γ N = N N 1 (.75) (.76) N = γ 1 γ 3Γ γ 1( γ 31 + γ 3 ) ( γ + γ ) + ( γ + γ ) Γ N (.78).7 lim N Γ = γ γ 1 3 N + γ 3 = N γ 1+ γ 1 3 (.79) γ γ 1 3 NU N (.65) χ ( NU N ) χ e α z z ( ) - G z e α = e Gz z (gain) G (gain constant) G/ (amplification constant) N = N N 1 1 G ηω N B ω c ( ) = (.8) 3
31 G Nσ ( ω) = (.81) g 1, g g N g 1 N1 N g N = 1 ηω exp g k T 1 B (.8) g g 1 N N 1 N - N1 1 N 1, N N = N N 1 g N 1 N = (.83) g N g 1 g g 1 B B g = = (.84) B1 g1b1 1 (.8) B (.84) (.83) (.84) B (.8) (.83) (.35) (.34) B ν 3ε η U B = π µ 31
32 G g πω = N N1 µ 1 g( ω ) g (.85) 1 3 ε c η G g1 πω = N N1 µ 1 g( ω) g (.86) 3ε cη,1 1 P Q ( ) Q c ωw P = (.87) 1 dw κ = = W dt ω (.88) Q c (l= ) U P ωu = Q c (.89) N 3
33 P (.37) G P G Nηω B( ω)u = (.9) P P N G th 1 Q c = N th ( ) ηb ω (.91) R1 R /c ( 1 R )U R 1 c P ( 1 R1 R ) U = (.9) (.89) Q Q c = c ω ( 1 R R ) 1 (.93) (.9) (.93) ( R R ) c 1 1 ηωb( ω ) N th = (.94) z= +z E t( ωt kz) ( z t) = E e, (.95) k K G k k ε = ε { 1+ χ( ω )} G ω = k + i = 1 + χ c ( ω) ( ω ) k (.96) χ 33
34 k G ω 1 = 1 + χ ω c ( ) k (.97) ω = χ c ( ω ) G (.98) χ G (.95) (.96) z ( 1 ) ( t k ) ( t) E e G e i, E = (.99) Z, r 1, r (.99) z r -z z= r1 +z ( t k ) ( t) r r E e G e i ω, = E 1 (.1) (.95) z i t E e ω ik r r e G e 1 (.11) 1 = r1r r r R R e iθ 1 = 1 (.11) G R R e 1 (.1) 1 = n k = nπ +θ (.13) (.1) (.98) ω c 1 ( ) = ln R1 R χ ω (.14) ( ) R 1 R 1 ln R1R = R1R 1 Q (.93) (.14) 34
35 1 χ ( ω ) = (.15) Q c (.13) (.97) ω c { + χ ( ω )} = nπ + θ (.16) (.66) (.67) ( ω) = χ ( ω) ω ω (.17) χ γ (.15) χ ( ω) ω ω = γq c (.18) (.16) (.18) ω c ω ω + = π γ n Qc + θ (.19) ω (.16) χ = c ω = nπ π + c (.11) (.19) (.11) ω (.88) Q c c ω ω κ c ω + ω = γ (.111) ω κω + γω c = κ + γ (.11) ( 35
36 ) Q Q ω γ (.11) Qω + Qcω c ω = (.113) Q + Q c Q Q Q Q c > 1 (frequency pulling) 1 (.11) c / = Q
37 3-1 3,1 (Nd) 53 m 1Hz Q Q Q 1 ( ) 53nm mJ 5. 5 Torr
38 3.1 38
39 3- ( ) mm ( ) mm ( ) mm 5. mm 1 1.mm 3. Fe 3 1 Fe Fe 7 Fe Fe A A B TEM 39
40 3-3 SEM( ) TEM( ) SEM TEM TEM TEM ( ) SEM TEM 4
41 4 4-1 SEM 4,1 4,( ) 4,3( ) Cu.3 m 5nm 4,1 41
42 4, 4,3 4
43 4- Fe C TEM TEM 4.4 a B 5 c 1 4,4c 3 A B.71 C ( ) ( ) ( ) 43
44 44
45 45
46 4-3 C TEM TEM 4,5 1 A H A B C ( ) 46
47 47
48 4-4 Fe TEM TEM 4,6 1 (a) 5 (b) 4,6a ( ) 48
49 49
50 4,6c 3nm.g cm g cm 4.7 6% 3 ( ) 4.7 5
51 51
52 4-5 C %
53 5 ( 1983) ( ) Carbon Nanotubes and Related Structures edited by Peter J.F.Harris (Cambridge University Press 1999 The Science and Technology of Carbon Nanotubes edited by K.Tanaka, T.Yamabe and K.Fukui Elsevier (Elsevier Science 1999) 53
54 6 TEM SEM TEM 54
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
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. 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.........
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
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1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
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液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
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II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
![II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re](/thumbs/94/118770263.jpg "II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re")
II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
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() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
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I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
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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)
19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
19 σ = P/A o σ B Maximum tensile strength σ 0. 0.% 0.% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional limit ε p = 0.% ε e = σ 0. /E plastic strain ε = ε e
) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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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