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General solution for the distribution function in the continuum model of planar channeling
Volume 85A, number 1
PHYSICS LETTERS
7 September 1981
GENERAL SOLUTION FOR THE DISTRIBUTION FUNCTION IN THE CONTINUUM MODEL OF PLANAR CHANNELING V.G. NOVITZKY Institute of Metal Physics, 252142 Kiev, USSR Received 15 August 1980 Revised manuscript received 15 June 1981
The time-dependent phase-space distribution function in the classical continuum model of planar channeling is obtained explicitly for arbitrary interplanar potential and arbitrary initial distribution.
The quantum-mechanical discussion of the channeling effect [1] leads, neglecting inelastic scattering, to the problem of determining the evolution of an initial plane wave in a periodic one-dimensional (planar channeling) effective potential. However, the quantum solution of this oversimplified problem can be constructed only in the simplest case of the Kronig—Penney model potential (or the same potential with two steps in a unit cell for binary alloys [2]). As a first attempt to deal with this problem, one must discuss the classical approximation, which is known as the continuum model. Significant progress in this discussion was achieved in the paper of Ellison [31, who obtained the classical phase-space distribution function (in implicit form), taking into account only particles below the barrier (see below). The phase-space distribution function of planar channeling in the continuum model satisfies the following onedimensional Liouville equation [3]: af/at+(p/m)af/ax
—
V’(x)af/ap =0,
(1)
where the planar continuum potential V(x) is such that V(x) = V(~—x)= V(x + 2a), V(0) = Vmin = 0, V(a) Vmax 2t,and E~is the = V 0, p is theenergy momentum the x-axis, the distance z from the channel equals z = (2E~/m)’/ longitudinal of the along particle. The present work reports on the construction of the Green’s function for eq. (1), which gives the solution of the Gauchy problem for this equation with arbitrary initial value f(p,x, 0) f 0(p,x)
(2)
in the following way: f(P,x,t)ffdKd~G(P,~x,~t)fo(K,~).
(3)
The Green’s function is here obviously given by G(p,g;x,~t)=&(p—P(K,~,t))6(x—X(K,~,t)), provided the functions P(K, 1——V’(X), 38
~
~,
t)
and X(ic, XP/m,
~,
t)
(4)
satisfy the following set of equations:
X(K,~,0)~.
(5)
0 031- ~9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
Volume 85A, number 1
PHYSICS LETTERS
7 September 1981
Thus defined, the function (4) represents the partial phase-space density due to a single point-like particle which at t = 0 has momentum p = g and coordinate x- = Explicit solutions to the set of equations (5) can be constructed only for the simplest potentials, but in fact one does not need them as the identity 6(x—X(K,~,t))=~(t—T(K,~,x))/IXI(XO) is a standard result if integration over t is considered. Its validity can also be shown in the case of integration over K and following e.g. ref. [4]. This identity can easily be applied to particles which move without reflections, i.e. if E,~= K212m + V(~) >V 0. In this case sign P = sign K and ~,
G(p,K;x,~t)=~(p—mu)~(t—T(K,~,x))/IuI,
(6)
where 2signK,
T(K,~,x)fdz/u(K,~,z).
(7,8)
u=u(k,~,x){(2/m)[E~_ V(x)]}V These particles correspond to above-the-barrier quantum states [1], and they must be taken into account as long as one considers a uniformly distributed incident flux. Below-the-barrier particles (Ext < V 0) move with period 2r, r=r(K~~)=fdz0(E~ V(z))/Iu(K,~,z)I.
(9)
—
(Introduction of the step function 0(x) = 1 if x > 0,0(x) = 0 if x <0, enables one to avoid writing explicitly the limits of integration, whose dependence on K, ~is determined by the form of the potential function.) All the time these particles stay in the same unit cell so that in the time interval between the 2nth and (2n + l)th reflections (10)
tT(K,~,x)+2nr, G~—~(p—mu)t5(t—T(~,~,x)—2nr)/IuI. In the remaining time intervals sign P
=
—sign ~ and if x and ~ belong to the Nth unit cell (2Na
—
a ~x, ~ ~ 2Na
+ a), then
t——T(K,4Na—~,x)+(2n+l)r, G~=6(p+mu)t5(t+T(~,4Na—~,x)—(2n+l)r)/luI.
(11)
After the introduction of the corresponding step functions and summation over n, eqs. (6), (10) and (11) give for the Green’s function the following expression: G(p,K;x,~t)0(E,~—V0)6(p—mu)~(t—T(K,~,x))/IuI +0(a— Ix!)0(a— I~J)0(V0_E,~)0(E~ V(x)) —
(12)
XE ~ n0 This result is valid for lxi ~a. Generalization to arbitraryx is obvious. To describe the experimental situation in channeling, the initial flux must be chosen uniform alongx, so that f0(p,x) =f0(p). In this case solution (3) takes the form
39
Volume 85A, number 1
PHYSICS LETTERS
7 September 1981
Fig. 1. Depth-averaged density of incident particles on atomic planes in the classical (solid line) and the quantum (dashed line) cases for the mterplanar Kronig—Penney model potential. The relative width of the barrier b/a = 0.04, height V 0 pro2/2m. These parameters correspond to 1 MeV
0
(300h/a) ton channeling in Ge. =
1
f(p,x,t)=0(E~~ V —
0)
f d~6(t+T)fo(mu)/Iul (13)
+0(Vo_Epx)fd~0(Epx_ V(~))E6(t+T—nr)f0((—ly~mu)/lul, where lxi
f dpf(p,x,ti)=fd~fdp a
0(E ~ ~
—
V(x))f0(p)
(14)
From a theoretical point of view it is interesting to compare the classical and quantum results. The latter are obtainable so far only for a monochromatic initial flux in the Kronig—Penney model potential. Fig. 1 shows the angular dependence of p(a). This quantity determines essentially such observables as the nuclear reaction yield. The classical curve is given by eq. (14) with f0(p) = 6 (p p0), where p0 is proportional to the incident angle. The quantum curve is taken from ref. [2J (see also ref. [1]). This comparison shows that the classical result is a very good approximation to the exact quantum calculations (at least in the case of proton channeling) and can be widely used owing to its generality. The usefulness of the Green’s function (12) will also be proved later in the discussion of perturbation theory taking into account inelastic effects and structural corrections to the continuum model. —
References [1] Yu.M. Kagan and Yu.V. Kononetz, Zh. Eksp. Teor. Fiz. 58 (1970) 226. [21 V.B. Molodkin and V.G. Novitzky, preprint IMP 79.8, Kiev (1979) (in Russian). [31 J.A. Ellison, Phys. Rev. B18 (1978) 5948. [4] P. Antosik, J. Mikusinski and R. Sikorski, Theory of distributions (Elsevier—PWN, Amsterdam—Warsaw, 1973).
40
PHYSICS LETTERS
7 September 1981
GENERAL SOLUTION FOR THE DISTRIBUTION FUNCTION IN THE CONTINUUM MODEL OF PLANAR CHANNELING V.G. NOVITZKY Institute of Metal Physics, 252142 Kiev, USSR Received 15 August 1980 Revised manuscript received 15 June 1981
The time-dependent phase-space distribution function in the classical continuum model of planar channeling is obtained explicitly for arbitrary interplanar potential and arbitrary initial distribution.
The quantum-mechanical discussion of the channeling effect [1] leads, neglecting inelastic scattering, to the problem of determining the evolution of an initial plane wave in a periodic one-dimensional (planar channeling) effective potential. However, the quantum solution of this oversimplified problem can be constructed only in the simplest case of the Kronig—Penney model potential (or the same potential with two steps in a unit cell for binary alloys [2]). As a first attempt to deal with this problem, one must discuss the classical approximation, which is known as the continuum model. Significant progress in this discussion was achieved in the paper of Ellison [31, who obtained the classical phase-space distribution function (in implicit form), taking into account only particles below the barrier (see below). The phase-space distribution function of planar channeling in the continuum model satisfies the following onedimensional Liouville equation [3]: af/at+(p/m)af/ax
—
V’(x)af/ap =0,
(1)
where the planar continuum potential V(x) is such that V(x) = V(~—x)= V(x + 2a), V(0) = Vmin = 0, V(a) Vmax 2t,and E~is the = V 0, p is theenergy momentum the x-axis, the distance z from the channel equals z = (2E~/m)’/ longitudinal of the along particle. The present work reports on the construction of the Green’s function for eq. (1), which gives the solution of the Gauchy problem for this equation with arbitrary initial value f(p,x, 0) f 0(p,x)
(2)
in the following way: f(P,x,t)ffdKd~G(P,~x,~t)fo(K,~).
(3)
The Green’s function is here obviously given by G(p,g;x,~t)=&(p—P(K,~,t))6(x—X(K,~,t)), provided the functions P(K, 1——V’(X), 38
~
~,
t)
and X(ic, XP/m,
~,
t)
(4)
satisfy the following set of equations:
X(K,~,0)~.
(5)
0 031- ~9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
Volume 85A, number 1
PHYSICS LETTERS
7 September 1981
Thus defined, the function (4) represents the partial phase-space density due to a single point-like particle which at t = 0 has momentum p = g and coordinate x- = Explicit solutions to the set of equations (5) can be constructed only for the simplest potentials, but in fact one does not need them as the identity 6(x—X(K,~,t))=~(t—T(K,~,x))/IXI(XO) is a standard result if integration over t is considered. Its validity can also be shown in the case of integration over K and following e.g. ref. [4]. This identity can easily be applied to particles which move without reflections, i.e. if E,~= K212m + V(~) >V 0. In this case sign P = sign K and ~,
G(p,K;x,~t)=~(p—mu)~(t—T(K,~,x))/IuI,
(6)
where 2signK,
T(K,~,x)fdz/u(K,~,z).
(7,8)
u=u(k,~,x){(2/m)[E~_ V(x)]}V These particles correspond to above-the-barrier quantum states [1], and they must be taken into account as long as one considers a uniformly distributed incident flux. Below-the-barrier particles (Ext < V 0) move with period 2r, r=r(K~~)=fdz0(E~ V(z))/Iu(K,~,z)I.
(9)
—
(Introduction of the step function 0(x) = 1 if x > 0,0(x) = 0 if x <0, enables one to avoid writing explicitly the limits of integration, whose dependence on K, ~is determined by the form of the potential function.) All the time these particles stay in the same unit cell so that in the time interval between the 2nth and (2n + l)th reflections (10)
tT(K,~,x)+2nr, G~—~(p—mu)t5(t—T(~,~,x)—2nr)/IuI. In the remaining time intervals sign P
=
—sign ~ and if x and ~ belong to the Nth unit cell (2Na
—
a ~x, ~ ~ 2Na
+ a), then
t——T(K,4Na—~,x)+(2n+l)r, G~=6(p+mu)t5(t+T(~,4Na—~,x)—(2n+l)r)/luI.
(11)
After the introduction of the corresponding step functions and summation over n, eqs. (6), (10) and (11) give for the Green’s function the following expression: G(p,K;x,~t)0(E,~—V0)6(p—mu)~(t—T(K,~,x))/IuI +0(a— Ix!)0(a— I~J)0(V0_E,~)0(E~ V(x)) —
(12)
XE ~ n0 This result is valid for lxi ~a. Generalization to arbitraryx is obvious. To describe the experimental situation in channeling, the initial flux must be chosen uniform alongx, so that f0(p,x) =f0(p). In this case solution (3) takes the form
39
Volume 85A, number 1
PHYSICS LETTERS
7 September 1981
Fig. 1. Depth-averaged density of incident particles on atomic planes in the classical (solid line) and the quantum (dashed line) cases for the mterplanar Kronig—Penney model potential. The relative width of the barrier b/a = 0.04, height V 0 pro2/2m. These parameters correspond to 1 MeV
0
(300h/a) ton channeling in Ge. =
1
f(p,x,t)=0(E~~ V —
0)
f d~6(t+T)fo(mu)/Iul (13)
+0(Vo_Epx)fd~0(Epx_ V(~))E6(t+T—nr)f0((—ly~mu)/lul, where lxi
f dpf(p,x,ti)=fd~fdp a
0(E ~ ~
—
V(x))f0(p)
(14)
From a theoretical point of view it is interesting to compare the classical and quantum results. The latter are obtainable so far only for a monochromatic initial flux in the Kronig—Penney model potential. Fig. 1 shows the angular dependence of p(a). This quantity determines essentially such observables as the nuclear reaction yield. The classical curve is given by eq. (14) with f0(p) = 6 (p p0), where p0 is proportional to the incident angle. The quantum curve is taken from ref. [2J (see also ref. [1]). This comparison shows that the classical result is a very good approximation to the exact quantum calculations (at least in the case of proton channeling) and can be widely used owing to its generality. The usefulness of the Green’s function (12) will also be proved later in the discussion of perturbation theory taking into account inelastic effects and structural corrections to the continuum model. —
References [1] Yu.M. Kagan and Yu.V. Kononetz, Zh. Eksp. Teor. Fiz. 58 (1970) 226. [21 V.B. Molodkin and V.G. Novitzky, preprint IMP 79.8, Kiev (1979) (in Russian). [31 J.A. Ellison, Phys. Rev. B18 (1978) 5948. [4] P. Antosik, J. Mikusinski and R. Sikorski, Theory of distributions (Elsevier—PWN, Amsterdam—Warsaw, 1973).
40