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A semiclassical approximation for the Wigner distribution function. Application to planar channeling
V~ume 94A, numb~ 9
A ~M~LA~AL
PHYSICS LETTERS
A~ROX~A~ON
4 Ap~ 1983
FOR THE ~ G N E R ~ S T R ~ U T ~ N
FUNC~O~
APPL~A~ON TO PLANAR CHANNEL~G ~ c t o r G. NOVffZKY
~s~u~ ~ M e ~ ~ i c s , ~
Kie~ USSR
ReceNed 7 Janu~y 1983
WffNn the ~amework of ~e W~n~ approach to ~e quamum mechaN~ of a s~#e p~fide the ~ m i d ~ c M approfim~ fion %r the ~ f i b m ~ n ~ n ~ n ~ the c~e of pNn~ channe~ng g obtMne~ ba~d on Rera~on of the dasNcM ~ t ~ n . Application of ~ m~hod m o ~ oneMimen~onM ~Nution NoNems N NoFo~&
Nn~e-particle ~ o ~ t i o n N one of the ~ m N e ~ phys~N probbms. At the same time its e x i t quantum-mechan~N sNution ~ known oNy ~ a ~ w c ~ (harrnoNc ~ c N ~ m , e~.) so l h ~ even one~imenNonN peneUation of a w a v e ~ k e t through a b ~ r i ~ m u ~ be sN~ed numericN~ [ 1]. N tNs ~ t u ~ n one must ~ o k for some app r o ~ m a ~ anNytic sNutiom A promiNng poss~flit~ appearing with the ~ o d u c f i o n of the Wigner d~tr~ufiom ~nction approach [2], win be conNde~d here with the application to the continuum model of N a n ~ channeling [3]. This m o d ~ ~ads to the proNem of de~rmining the evNufion of the initiN flux ~ an exactly perio~c on~ ~ m e n ~ o n ~ effective po~nti~. Quantum-mechaNcNly tNs m o d d w ~ derived ~ ~ [4], and the quantummechaNc~ sNution for a ~N~tic modal po~ntiN w ~ Nven ~ ~ [5] for the c ~ e cf a m o n o c h r o m ~ inifiN flux of channebd protons. Later an exact d ~ c ~ sdution ~ r ~ N t r g y ~ r p l a n ~ potentiN and nonmonochr~ matic initiN flux was constructed ~ ~ ~ ] . AccorSng to re£ [5], the effect of proton channefing d e m o n ~ s egenfia~y d ~ c N ~ u ~ s . This means that the ~miclassicN reset, if R N obtNnaNe, win pro~de a good a ~ procreation to the exact quantum-mechan~N sNution. Such r ~ t may p ~ s ~ s ~ n t i ~ l y quantum-mecha~cN properti~ in the case of the channe~ng of hght p ~ f i ~ . ~ the p N ~ n t p a p ~ the ~miclassic~ cor~ction to the ~ N c N sNution w ~ be obtNned ~ terms of the Wigner ~ t i b u t i o n function [2,7], wNch estab~sh~ the most n~urN ~nk between ~as~cN and quantum mec h a ~ . A ~cent ~ e w of Rs properti~ may be ~ u n d ~ ~ £ [8-10]. The sinNe~article one-~mensionN Wigner function ~ defined by f ( ~ ~ 0 = (gh)-I hs ~ u f i o n
~
f @ ~p@N~)ff*~
~ ~m~
by the ~ e r
_N 05~
(1)
+Y, O .
e ~
+ (p/m)Sff~ = L ~ = 0 ~ 2 ~ 1 f f
~ ~ ~p[N~
- p~]
[V(x +y) - V ~ - y ~ f ~ ~ O.
(2)
In the fimit ~ = 0 t~s equation c ~ n d d ~ with the Houfine equation for the d ~ f i c ~ phase~pace d~tribution function of an en~mNe of n o ~ n t ~ t ~ g particles in an extern~ p o ~ n t i ~ V~):
bfc~t + (P/m)bfc/bX = L c f e = V ' ~ ) b f e ~ P .
O)
Fo~owing the m ~ h o d used ~ re~ [6], the Green function G c ~ r thg equation can be con~ructed ~ r an ~ b R ~ r y on~dimengonN p ~ e n t i ~ . Th~ e n a N ~ one to ~ansform the Cauchy proNem for eq. (2) with lhe i~tiN v~ue 0 031 ~ 163/83/0000-0000/$ 03.00 © 1983 North-Holland
437
Vo~me 94A, numb~ 9
PHYSICS LETTERS
4 April 1983
f ( ~ a o) = ~ ( ~ x)
~)
~ t o the following ~ N
equation
t
f=fc + f dtl Gc(t- tl)(Lq - L c ) f Q 1 ) ,
(5)
0 when fc( N ~ 0) = f 0 ( ~ x). A ~micl~NcN approximation ~ now obtNned by iterat~g eq. (5) OMs idea bdonN to SNmkov [9]) t ~ =fc + f l ,
f l = f dtlGcQ - t l ) ( t q - L c ) f c ~ l ) " 0
(6)
~ the case of Nan~ channeling V~) = Voo@),
f 0 ( ~ x) = / 0 ( P ) ,
o~) = ~
~ exp(~nx~),
where v(0) = Omin = 0, o ~ ) = Vm~ = 1, ~ = V_n, and 2a is the ~ r ~ a n ~
x ~,
p~m~)~p,
(7) spacing. In ~men~oN~s variables
t~/2~)~2at,
(8)
the das~cN ~ncaons are Nven by [6] ~
1
dq f dyGc( N q ; ~ y ; t ) ~ ) ,
~ ( N x, Q= f --~
O)
--1
Gc( p, q;x, y; t) = ~ {O(Eqy - 1)8(p - u)8(t - T(q, y, x) - nr) n
+O(1-~)[6(p-u)b(t-~x)-2nr)+6(p+u)b~+T~,-y,x)-(2n+
1)r)l)/lul,
(10)
where 0 g the stop ~ncfion, ~ y = q2 + o(y), u =u~G
x) = M Y - o ~ V 2
~gn~),
(11)
x
1
f eo%y y
(12, 13)
1 1
L(gat)
=fd~Sq+~-nrl)[O H -1
~ -
1)~O1)+0(1-~)0~-o~-l~u~ull.
(14)
Here u 1 = u ( ~ ~ ~, ~ = ~ ~ ~L rl = r ( ~ x) as defined ~ eqs. ( 1 1 ) ~ 1 3 ~ It shouN be noted ~ a eq. (10) and r~e O~ by wNch tNs Green ~ncfion can be ~ e d , are v ~ d oNy ~ r Ix l ~ 1 and ~ r Nn~ions pefio~c ~ coordinate ~ace (wMch g the case in the present d ~ u s ~ o n ~ ~ t h tMs in ~ n d and ~Mng ~ m account that ~ r the pommM (7) and variaNm ~ the owrator Lq (2) takes the ~ r m L q [ ( N x) = ( g / 2 ~ ~ ~ s i n ~ x ~ ( p n
438
- ng, x),
(15)
V~ume 94A, numb~ 9 ~e
g=( ~ m
PHYSICSLETTERS ~ ,
the ~m~hsfic~ correction to the d a t u m c o o ~ v s p ~ e t ~ 1 dtl 0
+ 0(1 - ~
~
4 April 1983
f
f dY [uIM ~m
-~
-1
denfi~ b ~ o m ~
{O~y-1)6Q-tl-T~x)-mr)
[~ Q - t 1 - T ~ ~ x ) - 2mr) + 6 Q - t 1 + T ~ , ~ , x ) - (2m + 1)~] }
- o~
X ~ ~ fin~)~n
~
tl)~ +nO~
~ tl)~q ] .
(16)
~ ~ncfion, ~ ~h ~ ~ ~ s the ~ d a ~ c ~ ~ ~ ~ ~ ~ ~ t ~g pm~om ~ to the ~ a n c e ~om ~ ~ sur~ce. It ~ ~so ~ n g m d ~ m ~ e the ~ ~ ~ n ~ y , whi~ ~ n be done by m~ng the ~ a c e ~ e [I1] ofeq. (16): d t P l ~ , 0 = s~ m 0 s S dt e- a P l ~ 0 .
0
(17)
0
(~e c~n~g d~fic~ ~ncfion is ~ven ~ re~ [6].) E ~ n g necessary now to de~rm~e a ~ ~rm S=s
Sdt e-~t ft dtl 0
0
~
6 Q - tl _
T1 _
the ~
dependence from eq. (16), R is
mZl)6Q1 _ T2 - n z 2 ) ,
m,~
where I~1 ~ ~ ~ = 1,2~ ~ r ~ r m ~ g the ~ e g r ~ , S =s H ~ p ( ~ + n ~ + n ~ -i=1,2 n
~ obt~n the pmdu~ ~ ~ o ~ o m e ~ c ~ series:
=s H ~ p ( ~ ( ~ + ~ x p ~ u ) - l ] - l } . i=1,2
For s ~ 0 this ~ncfion b e h ~ l~e S = ~-1 +~rl f i ~ ( ~ ) _
~ +½r2 f i ~ ( ~ ) _
~ +O~1~
"
(18)
The ~ # ~ term ~ n g ~ m v ~ ~ f i c ~ y after ~ e ~ t ~egr~s over ~ and y. ~Mng ~ o accoum the ~ m m e t ~ p m ~ of the ~egrand and the demema~ rdafion 0~) + 0 ( ~ ) = 1, we obtain ~om eqs. ( 1 6 ~ 1 ~ the finM reset ~ r the ~ c M c ~ t ~ u ~ n to the d e c l a i m e d d e n n y ~
1
=
-
--~
X~
sin~)~(p
oe
O
y
-
-
÷
-
--1
- q +ng)~ 4 ~
- p)]~(u2)~2Zl~
[q ,
(19)
n
where u 1 = u(q, y, x ~ u 2 = u ( p , y, ~), 71 = r(q, y ~ r 2 = r ( p , y ~ T 1 = T(q, y, x ~ T 2 = T(p, y, ~). The term with the 6-functions in eq. (19) ori#n~es from the operator Lq - L c. ~ w~ ~ff unintegrated for convenience. Owing to complexity of the prob~m (periodic potenti~), the qu~itafive an~y~s of the semiclas~c~ solutions (16) and (19) ~ a difficult task. Th~ task may be fimp~fied in the d~cus~on of problems with a fine,humped p~ tenfi~ [1,12]. In condufion ff mum be emphafized that the proposed method can be applied to any on~dimension~ prob~m, or to any problem for which the dasfic~ sdufion ~ ava~able. The numeric~ ~sul~ concerning the channeling eft ~ct will be #yen hte~ 439
V o ~ m e 94A, n u m b ~ 9 R [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
440
PHYSICS LETTERS
~ R.A. Sacks and J.E. R~hardson, Phy~ Rev. B21 (1980) 1449. E. Wigner, Phys. Rev. 40 (1932) 749. J. Lindhard, K. Dan. Vid. Sdsk. Mat. Fys. Medd. 34, no. 14 (1965). Yu. Kagan and Yu.V. Konone~, Zh. Eksp. Teor. F~. 58 (1970) 226. Yu. Kagan, Yu.V. Kononets and A.A. Mamonto~ Phys. LetL 72A (1979) 247. V.G. Nofi~ky, Phys. Lett. 85A (1981) 38. J.E. Moy~, Pro~ Cambridge Philos. Soc. 45 (1949) 99. C.R. de Groot and L.G. Suttorp, Foundations of electrodynamics (North-Holland, Amsmrdam, 1972). Yu.M. Shirokov, Part. Nucl. 10 (1979) 5. G.J. Ia~ate, H.L. Grubin and D.K. Ferry, Phys. Lett. 87A (1982) 145. W. Kohn and J.M. Lutfinger, Phys. Rev. 108 (1957) 590. A. Har~ein, Z.A. Wdnberg and D.J. DiMaria, Phys. Rev. B25 (1982) 7174.
4 Apdl 1983
A ~M~LA~AL
PHYSICS LETTERS
A~ROX~A~ON
4 Ap~ 1983
FOR THE ~ G N E R ~ S T R ~ U T ~ N
FUNC~O~
APPL~A~ON TO PLANAR CHANNEL~G ~ c t o r G. NOVffZKY
~s~u~ ~ M e ~ ~ i c s , ~
Kie~ USSR
ReceNed 7 Janu~y 1983
WffNn the ~amework of ~e W~n~ approach to ~e quamum mechaN~ of a s~#e p~fide the ~ m i d ~ c M approfim~ fion %r the ~ f i b m ~ n ~ n ~ n ~ the c~e of pNn~ channe~ng g obtMne~ ba~d on Rera~on of the dasNcM ~ t ~ n . Application of ~ m~hod m o ~ oneMimen~onM ~Nution NoNems N NoFo~&
Nn~e-particle ~ o ~ t i o n N one of the ~ m N e ~ phys~N probbms. At the same time its e x i t quantum-mechan~N sNution ~ known oNy ~ a ~ w c ~ (harrnoNc ~ c N ~ m , e~.) so l h ~ even one~imenNonN peneUation of a w a v e ~ k e t through a b ~ r i ~ m u ~ be sN~ed numericN~ [ 1]. N tNs ~ t u ~ n one must ~ o k for some app r o ~ m a ~ anNytic sNutiom A promiNng poss~flit~ appearing with the ~ o d u c f i o n of the Wigner d~tr~ufiom ~nction approach [2], win be conNde~d here with the application to the continuum model of N a n ~ channeling [3]. This m o d ~ ~ads to the proNem of de~rmining the evNufion of the initiN flux ~ an exactly perio~c on~ ~ m e n ~ o n ~ effective po~nti~. Quantum-mechaNcNly tNs m o d d w ~ derived ~ ~ [4], and the quantummechaNc~ sNution for a ~N~tic modal po~ntiN w ~ Nven ~ ~ [5] for the c ~ e cf a m o n o c h r o m ~ inifiN flux of channebd protons. Later an exact d ~ c ~ sdution ~ r ~ N t r g y ~ r p l a n ~ potentiN and nonmonochr~ matic initiN flux was constructed ~ ~ ~ ] . AccorSng to re£ [5], the effect of proton channefing d e m o n ~ s egenfia~y d ~ c N ~ u ~ s . This means that the ~miclassicN reset, if R N obtNnaNe, win pro~de a good a ~ procreation to the exact quantum-mechan~N sNution. Such r ~ t may p ~ s ~ s ~ n t i ~ l y quantum-mecha~cN properti~ in the case of the channe~ng of hght p ~ f i ~ . ~ the p N ~ n t p a p ~ the ~miclassic~ cor~ction to the ~ N c N sNution w ~ be obtNned ~ terms of the Wigner ~ t i b u t i o n function [2,7], wNch estab~sh~ the most n~urN ~nk between ~as~cN and quantum mec h a ~ . A ~cent ~ e w of Rs properti~ may be ~ u n d ~ ~ £ [8-10]. The sinNe~article one-~mensionN Wigner function ~ defined by f ( ~ ~ 0 = (gh)-I hs ~ u f i o n
~
f @ ~p@N~)ff*~
~ ~m~
by the ~ e r
_N 05~
(1)
+Y, O .
e ~
+ (p/m)Sff~ = L ~ = 0 ~ 2 ~ 1 f f
~ ~ ~p[N~
- p~]
[V(x +y) - V ~ - y ~ f ~ ~ O.
(2)
In the fimit ~ = 0 t~s equation c ~ n d d ~ with the Houfine equation for the d ~ f i c ~ phase~pace d~tribution function of an en~mNe of n o ~ n t ~ t ~ g particles in an extern~ p o ~ n t i ~ V~):
bfc~t + (P/m)bfc/bX = L c f e = V ' ~ ) b f e ~ P .
O)
Fo~owing the m ~ h o d used ~ re~ [6], the Green function G c ~ r thg equation can be con~ructed ~ r an ~ b R ~ r y on~dimengonN p ~ e n t i ~ . Th~ e n a N ~ one to ~ansform the Cauchy proNem for eq. (2) with lhe i~tiN v~ue 0 031 ~ 163/83/0000-0000/$ 03.00 © 1983 North-Holland
437
Vo~me 94A, numb~ 9
PHYSICS LETTERS
4 April 1983
f ( ~ a o) = ~ ( ~ x)
~)
~ t o the following ~ N
equation
t
f=fc + f dtl Gc(t- tl)(Lq - L c ) f Q 1 ) ,
(5)
0 when fc( N ~ 0) = f 0 ( ~ x). A ~micl~NcN approximation ~ now obtNned by iterat~g eq. (5) OMs idea bdonN to SNmkov [9]) t ~ =fc + f l ,
f l = f dtlGcQ - t l ) ( t q - L c ) f c ~ l ) " 0
(6)
~ the case of Nan~ channeling V~) = Voo@),
f 0 ( ~ x) = / 0 ( P ) ,
o~) = ~
~ exp(~nx~),
where v(0) = Omin = 0, o ~ ) = Vm~ = 1, ~ = V_n, and 2a is the ~ r ~ a n ~
x ~,
p~m~)~p,
(7) spacing. In ~men~oN~s variables
t~/2~)~2at,
(8)
the das~cN ~ncaons are Nven by [6] ~
1
dq f dyGc( N q ; ~ y ; t ) ~ ) ,
~ ( N x, Q= f --~
O)
--1
Gc( p, q;x, y; t) = ~ {O(Eqy - 1)8(p - u)8(t - T(q, y, x) - nr) n
+O(1-~)[6(p-u)b(t-~x)-2nr)+6(p+u)b~+T~,-y,x)-(2n+
1)r)l)/lul,
(10)
where 0 g the stop ~ncfion, ~ y = q2 + o(y), u =u~G
x) = M Y - o ~ V 2
~gn~),
(11)
x
1
f eo%y y
(12, 13)
1 1
L(gat)
=fd~Sq+~-nrl)[O H -1
~ -
1)~O1)+0(1-~)0~-o~-l~u~ull.
(14)
Here u 1 = u ( ~ ~ ~, ~ = ~ ~ ~L rl = r ( ~ x) as defined ~ eqs. ( 1 1 ) ~ 1 3 ~ It shouN be noted ~ a eq. (10) and r~e O~ by wNch tNs Green ~ncfion can be ~ e d , are v ~ d oNy ~ r Ix l ~ 1 and ~ r Nn~ions pefio~c ~ coordinate ~ace (wMch g the case in the present d ~ u s ~ o n ~ ~ t h tMs in ~ n d and ~Mng ~ m account that ~ r the pommM (7) and variaNm ~ the owrator Lq (2) takes the ~ r m L q [ ( N x) = ( g / 2 ~ ~ ~ s i n ~ x ~ ( p n
438
- ng, x),
(15)
V~ume 94A, numb~ 9 ~e
g=( ~ m
PHYSICSLETTERS ~ ,
the ~m~hsfic~ correction to the d a t u m c o o ~ v s p ~ e t ~ 1 dtl 0
+ 0(1 - ~
~
4 April 1983
f
f dY [uIM ~m
-~
-1
denfi~ b ~ o m ~
{O~y-1)6Q-tl-T~x)-mr)
[~ Q - t 1 - T ~ ~ x ) - 2mr) + 6 Q - t 1 + T ~ , ~ , x ) - (2m + 1)~] }
- o~
X ~ ~ fin~)~n
~
tl)~ +nO~
~ tl)~q ] .
(16)
~ ~ncfion, ~ ~h ~ ~ ~ s the ~ d a ~ c ~ ~ ~ ~ ~ ~ ~ t ~g pm~om ~ to the ~ a n c e ~om ~ ~ sur~ce. It ~ ~so ~ n g m d ~ m ~ e the ~ ~ ~ n ~ y , whi~ ~ n be done by m~ng the ~ a c e ~ e [I1] ofeq. (16): d t P l ~ , 0 = s~ m 0 s S dt e- a P l ~ 0 .
0
(17)
0
(~e c~n~g d~fic~ ~ncfion is ~ven ~ re~ [6].) E ~ n g necessary now to de~rm~e a ~ ~rm S=s
Sdt e-~t ft dtl 0
0
~
6 Q - tl _
T1 _
the ~
dependence from eq. (16), R is
mZl)6Q1 _ T2 - n z 2 ) ,
m,~
where I~1 ~ ~ ~ = 1,2~ ~ r ~ r m ~ g the ~ e g r ~ , S =s H ~ p ( ~ + n ~ + n ~ -i=1,2 n
~ obt~n the pmdu~ ~ ~ o ~ o m e ~ c ~ series:
=s H ~ p ( ~ ( ~ + ~ x p ~ u ) - l ] - l } . i=1,2
For s ~ 0 this ~ncfion b e h ~ l~e S = ~-1 +~rl f i ~ ( ~ ) _
~ +½r2 f i ~ ( ~ ) _
~ +O~1~
"
(18)
The ~ # ~ term ~ n g ~ m v ~ ~ f i c ~ y after ~ e ~ t ~egr~s over ~ and y. ~Mng ~ o accoum the ~ m m e t ~ p m ~ of the ~egrand and the demema~ rdafion 0~) + 0 ( ~ ) = 1, we obtain ~om eqs. ( 1 6 ~ 1 ~ the finM reset ~ r the ~ c M c ~ t ~ u ~ n to the d e c l a i m e d d e n n y ~
1
=
-
--~
X~
sin~)~(p
oe
O
y
-
-
÷
-
--1
- q +ng)~ 4 ~
- p)]~(u2)~2Zl~
[q ,
(19)
n
where u 1 = u(q, y, x ~ u 2 = u ( p , y, ~), 71 = r(q, y ~ r 2 = r ( p , y ~ T 1 = T(q, y, x ~ T 2 = T(p, y, ~). The term with the 6-functions in eq. (19) ori#n~es from the operator Lq - L c. ~ w~ ~ff unintegrated for convenience. Owing to complexity of the prob~m (periodic potenti~), the qu~itafive an~y~s of the semiclas~c~ solutions (16) and (19) ~ a difficult task. Th~ task may be fimp~fied in the d~cus~on of problems with a fine,humped p~ tenfi~ [1,12]. In condufion ff mum be emphafized that the proposed method can be applied to any on~dimension~ prob~m, or to any problem for which the dasfic~ sdufion ~ ava~able. The numeric~ ~sul~ concerning the channeling eft ~ct will be #yen hte~ 439
V o ~ m e 94A, n u m b ~ 9 R [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
440
PHYSICS LETTERS
~ R.A. Sacks and J.E. R~hardson, Phy~ Rev. B21 (1980) 1449. E. Wigner, Phys. Rev. 40 (1932) 749. J. Lindhard, K. Dan. Vid. Sdsk. Mat. Fys. Medd. 34, no. 14 (1965). Yu. Kagan and Yu.V. Konone~, Zh. Eksp. Teor. F~. 58 (1970) 226. Yu. Kagan, Yu.V. Kononets and A.A. Mamonto~ Phys. LetL 72A (1979) 247. V.G. Nofi~ky, Phys. Lett. 85A (1981) 38. J.E. Moy~, Pro~ Cambridge Philos. Soc. 45 (1949) 99. C.R. de Groot and L.G. Suttorp, Foundations of electrodynamics (North-Holland, Amsmrdam, 1972). Yu.M. Shirokov, Part. Nucl. 10 (1979) 5. G.J. Ia~ate, H.L. Grubin and D.K. Ferry, Phys. Lett. 87A (1982) 145. W. Kohn and J.M. Lutfinger, Phys. Rev. 108 (1957) 590. A. Har~ein, Z.A. Wdnberg and D.J. DiMaria, Phys. Rev. B25 (1982) 7174.
4 Apdl 1983