CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
ANALYSIS OF THE SLOWING OF A HIGH ENERGY PROTON SHOT
THROUGH A TARGET IN THE FRAME OF THE FOKKER-PLANCK
EQUATION
V. Molinari and F. Teodori
Istituto Nazionale di Fisica della Materia (INFM) and
Laboratorio di Montecuccolino dell'Universita di Bologna
via dei Colli,16 40136 Bologna Italy
Abstract. When a high energy proton shot hits a target, energy and momentum transfer is distributed non
uniformly within a certain volume, whose depth depends on the proton energy and on the composition of the
target.
By using the Fokker-Planck equation it has been obtained a full description of the energy and space
distribution of the protons as a function of time. This is essential to compute energy deposition and momentum
transfer laws and it is useful to predict the response of the target.
INTRODUCTION
is the diffusion velocity tensor, where cr(g,%) is the
differential cross section, Al^ is the change in proton
velocity, ~£ is the relative velocity of two colliding
particles and % is the deflection angle in the center of
mass system.
In this paper we consider the slowing down and diffusion of a pulsed beam of high energy protons, shot at
the surface of a homogeneous target. The aim of this
work is to investigate the evolution of the ion beam
parameters in time and in space. For this purpose the
Fokker-Planck equation has been used, which is a
powerful analysis tool for studying the diffusion of
neutral and charged particles through matter [1, 2, 3]:
SOLUTION OF THE FOKKER-PLANCK
EQUATION
As a consequence of the symmetry of the problem,
denoting by VQ the velocity of the incoming protons
and by ju the cosine of the angle between the direction
of the protons and VQ, Eq. (1) becomes
(D
In it,
is the ion distribution function,
is the dynamical frictional coefficient and
The boundary conditions are
/(r,v,^0)=JV 0 8(vlim,W(r,v,Ai,0)=0
(3)
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(5)
(6)
Equation (15) has been solved with the boundary conditions (12) and (13) obtaining the result
When Coulomb interactions are considered, the dynamical frictional coefficient a, and the coefficients
P and j of the diffusion velocity tensor are expressed
by
(7)
(8)
with
(16)
A=
— A2 in A
-
where jp-\,n is the n — th zero of the Bessel function
of order p — 1. The parameter p depends on the ratio
between the dynamical friction and the diffusion in
velocity coefficient
(9)
(10)
where mf is the field particle mass, mp the proton
mass and %m/-w the lower limit of the deflection angle % in the center of mass system of two colliding
particles.
An approach to the problem by the use of Eq.s (4),
(5) and (6) is very difficult. In this work, the attention
is focused on the space and energy distribution of the
protons. Therefore both sides of Eq. (4) can be integrated over the domain of the variable ju obtaining the
balance equation
P=^
(17)
Integrating the distribution function and expressing
the streaming term in the diffusion approximation any
information on the initial strong anisotropy of the incident beam has been lost. As a consequence, at very
short distance from the surface of the target, r < ^,
where ^ is the mean free path at the initial energy,
a first order transport correction is needed. For this
purpose a source term can be added to Eq. (4)
(11)
(18)
This must be solved with the boundary conditions
and new boundary conditions must be considered
(12)
(13)
/(0,v,0=0
lim,^oo/(r,v,0=0
/(r,v,0)=0.
obtained integrating over p the Eq.s (5) and (6). The
streaming term can be expressed in the diffusion approximation
dp
;/(r,v,o = - D j / ( r , v , r ) ,
(19)
(20)
(21)
Only after interacting with matter and acquiring some
angular spreading, the protons are studied via the
Fokker-Planck equation. The solution of the problem
is
(14)
4-1
where D is the diffusion coefficient. This assumption
leads to the new equation
(22)
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estimation of the thickness of the energy deposition zone. After interacting with matter, the protons
undergo a redistribution both in physical and momentum space. The packet moves forward, the initial anisotropy is progressively forgotten and the diffusion theory gives a good description of the phenomenon. This can be seen in Fig (2), where the spatial spreading and the momentum straggling of the
proton beam are sensibly increased.
The ions undergo the first interaction with matter at
a time after the pulse is generate, which depends on
the free path before the collision. For this reason, at
very short time, t < ^-U, a second order transport correction is needed. The source term used in Eq. (18) is
not appropriate and must be substituted by
CONCLUSIONS
(23)
Via the Fokker-Planck equation, the evolution in time
and space of the parameters of a pulsed heavy ion
beam has been fully analyzed. Solving the equation,
without introducing some approximations, was out of
question. We have chosen to express the streaming
term in the diffusion approximation, after averaging
the distribution function over the angle. A closed analytical solution has been found, that gives a detailed
description of the spreading of the ions both in physical and momentum space. A first and a second order
transport correction have been applied to take into account the effects of the initial beam strong anisotropy.
The knowledge of the energy spectra is important in
those fields, such as microbeam analysis [6], radioprotection [7] and Boron Neutron Capture Therapy
[8], where quantifying and identifying the reactions
produced in the medium is of fundamental interest,
depending the induced phenomena on the energy of
the interacting particles. The aim of this work was to
suggest a theoretical approach to the problem based
on the transport theory. When applied to microbeam
analysis, the model gave predictions in good agreement with experimental results.
Equation (23) solved with the boundary conditions
(19), (20) and (21), gives
, (24)
where
v = Lv
;
L
\i -Jp-i
n
(25)
^
Equations (16), (22) and (24) give a description of the
ion slowing down with improved detail and precision,
in spite of increasing complexity. Which result must
be used depends on the particular problem is considered: the thickness of the specimen, the resolution desired, the observation time.
REFERENCES
RESULTS
1. Shkarofsky, I.P., Johnston, T.W., Bachhynski, M.P.,
The Particle Kinetics of Plasmas, Addison-Wesley,
Reading, USA 1996
2. Klimontovich, Y.L., Statistical Theory of Open Systems,
Vol. 1, Kluwer Academic Publishers, Dordrecht,
Netherlands 1995
3. Risken, H., The Fokker-Planck Equation, SpringerVerlag, New York 1984
4. Manservisi, S., Molinari, V., Nespoli, A., // Nuovo
Cimento D 18, 435-448, (1996).
5. Utkin, A.V., Kanel, G.I., Baumung, K., "Experimental
Calculations have been performed, assuming an aluminum target and a pulsed proton beam of 1.53 MeV,
whose projected range p, in the CSDA approximation, is 2.66 jam [5]. In Fig. (1) the distribution function is shown at short time from the emission of the
pulse, t c± ~£ ' calculated both with the diffusion approximation and applying the transport correction.
Due to the strong anisotropy of the beam, at short
time, the diffusion approximation leads to a wrong
273
Figure 1. Proton distribution function at short time from the emission of the pulse, calculated in the diffusion approximation
(on the left) and by applying the the transport correction (on the right).
Figure 2. Proton distribution function after they have interacted with matter, calculated in the diffusion approximation (on
the left) and by applying the the transport correction (on the right). The collisions determine a spreading in physical space and
a momentum straggling. We see that the effects of the anisotropy are weaker than before, but non completely negligible yet.
Observation of State of Matter heated by the Ion
Beam", in Shock Compression of Condensed Matter7995, edited by S.C, Schmidt et aL, AIP Conference
"The Effect of Straggling on the Slowing Down of
Neutrons in Radiation Protection", in Proceedings of
the SIEN'99, Bucarest, 1999, pp. 141-145.
Teodori, E, Mostacci, D., Molinari, V.G., Sumini,
ML, "A Simple Model for Neutral and Charged
Particle Diffusion and its Application to BNCT", in
Proceedings of the M&C '99, Madrid, 1999.
Proceedings 370, New York, 1996, pp. 1243-1246.
Fernandez, J., Molinari, V., Teodori, E, Mikrochimica
AcM 132, 219-224, (2000)
7. Mostacci, D., Molinari, V., Teodori, E, Pesic, ML,
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