Influence of the multiple scattering of relativistic electrons on the line width of the backward Parametric X-ray Radiation in the absence of photo absorption

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by  Mehdi Tabrizi

Influence of the multiple scattering of relativistic electrons on the line width of the
backward Parametric X-ray Radiation in the absence of photo absorption
Tabrizi Mehdi
Parallel Computing Laboratory (2216), Department of Physics, Faculty of Science, Razi University of Kermanshah, 67149-67346, Iran
Abstract
5The multiple scattering effect on the line width of the backward Parametric X-ray Radiation (PXR) in the extremely
1Bragg geometry, produced by low energy relativistic electrons traversing a single crystal, is discussed. It is shown that
0
there exist conditions, when the influence of photo absorption on the line width can be neglected, and the only multiple
2
scatteringprocessofrelativisticelectronsincrystalleadstothePXRlines’broadening. Basedontheobtainedtheoretical

band numerical results for the line width broadening, caused by the multiple scattering of 30 and 50 MeV relativistic
e
electrons in a Si crystal of varying thicknesses, an experiment could be performed to help to reveal the scattering effect
F
on the PXR lines in the absence of photo absorption. This leads to a more accurate understanding of the influence of

5scattering phenomenon on the line width of the backward PXR and helps to a better construction of a table-top narrow

bandwidth X-ray source for scientific and industrial applications.
]
h
pKeywords: parametric X-ray radiation, line width, relativistic electrons, multiple scattering
-
c
c
1. Introduction angular radiation density as a result of the interference of
a
. reflected waves from crystallographic atomic planes, ori-
s Parametric X-ray Radiation (PXR) is produced when
c ented perpendicular to the z-axis (Fig. 2). The natural
irelativistic electrons fall at a small incidence angle with width of these lines is determined by the number of crys-
s
respect to one of the crystallographic atomic planes (see
y tallographicplaneswhichtherelativisticelectronsinteract
h[1, 2, 3] and references therein). Recently, production of with. However,theresultsofexperiment[7]showthatthe
pPXR by ultra-relativistic protons was observed in a bent
line width of the backward PXR is much larger than its
[crystals [4]. This radiation is mainly concentrated in di-
natural line.
4 rectionsclosetotheBragg’sanglesofparticle’sfieldreflec- There are two kinds of effects which destructively influ-
vtionfromsuchplanes. ThecaseofPXRinthe”extremely
ence on PXR lines. The ”instrumental” effects such as fi-
6Bragg geometry” or ”backward” PXR is of special inter-
nite detector opening [11], finiteness of the collimator [12]
3
est,whenrelativisticelectronsfallontoacrystalatasmall
8 andangularspreadoftheelectronbeam[13]areconnected
angle ψ > ψ with respect to one of the crystallographic
0 c with the instruments for measurement of PXR lines.
0axes (z-axis in Fig. 1), since in this case, the contribution The ”non-instrumental” effects on PXR lines, caused by
.of bremsstrahlung and channeling radiation are then con-
1 (cid:112) such processes as absorption of radiated photons and the
siderablysuppressed. Hereψ = 4Ze2/Edisthecritical
0 c multiple scattering of relativistic electrons in crystal are
5angle of axial channeling [5], d - the inter atomic distance connected with the physical phenomena while producing
1along the z-axis, E - particle energy.
PXR. In the subsequent discussion, the two aforemen-
:The backward PXR in extremely Bragg geometry was ob-
v tioned ”non-instrumental” effects on the line width of the
Xiserved at low [6] and high [7] energies of relativistic elec- backward PXR will be considered .
trons, but the line width was measured in [7]. Measure-
For thick crystals, such that their thickness L is larger
rmentsofthelinewidthofthebackwardPXRathighener-
a than the absorption length L of radiated photons, the
a
gies of electrons [7] show that the PXR has a very narrow
absorptioneffectplaystheleadingroleintheformationof
line width of a few meV (milli-electron-Volt) which points
PXR lines [7, 14]. Although, at L (cid:29) L , PXR intensity
a
to the fact that such a quasi-monochromatic and narrow
ultimately no longer increases with L [15]. The multiple
bandwidth X-ray radiation can be used in many applica-
scattering process of relativistic electrons in crystal is the
tions [8, 9, 10]. Such narrow lines appear in the spectral
anotherphysicalphenomenawhichmakesdestructivecon-
tribution to the line width of PXR [7, 11, 13] (see Fig. 1).
∗Fax: [phone] However,thisprocessmakesitselfmoreevidentinthefor-
Email address: [email protected], [email protected](Tabrizi mation of PXR lines, if low energy relativistic electrons
Mehdi)
traverse a crystal of the thickness L less than L .
URL: http://fm.razi.ac.ir/tabrizi(TabriziMehdi) a
Preprint for arXiv February 9, 2015
x y
k
g
ϕ
ψ
z
v x
0
②
v
d x
①
ψ v
y
N
z
v
0
Figure 1: 2D Schematic representation of the multiple scattering
of relativistic electrons in a crystal producing the backward PXR
Figure2: 3Dschematicrepresentationofasinglescatteringofelec-
in extremely Bragg geometry (view from the y-axis): v0 - velocity tronontheatom1ofthetopplanegoingtotheatom2ofthemiddle
vectoroftheincomingelectron,g-reciprocallatticevector,k-wave
planealongthex-andy-axes.
vectoroftheradiatedphotons,ψ -incidenceangleofelectronswith
respecttothez-axis,φ-anglebetweenkandg,N -numberofthe
crystalplanes,d-distancebetweenlatticeplanes.
in Fig. 2, v (t) has two components v (t)=(v ,v ).
⊥ ⊥ x y
The deviation of trajectory of relativistic electrons from
Inthispaper,theinfluenceofthesmallanglemultiplescat- the straight forward direction in a crystal, is mainly con-
tering of relativistic electrons of different energies on the nected with the multiple scattering process [17] and the
linewidthofthebackwardPXRinextremelyBragggeom- channeling phenomenon [5]. In the following, it will be
etry in a single crystal of varying thicknesses L (L < L ) assumed that the relativistic electron impinges to one of
a
undertheconditionsofabsenceofphotoabsorption, isin- the crystallographic axes of a crystal (z-axis in Fig. 1) at
vestigated. The conditions, when the absorption effect of an angle ψ, which is larger than the critical angle of ax-
radiated photons is negligible, are found. In what follows, ial channeling ψc. Therefore, the channeling of relativistic
we shall use the system of units, in which c=h¯ =1. electrons in the crystal does not take here place.
As it is shown in Fig. 1, the small angle scattering of
relativistic electrons during the passage through the crys-
2. Spectral angular density of the backward PXR
talleadstothesmallchangeofmeanfreepathofparticles
includingthesmallanglemultiplescatteringpro-
between sequential collisions with crystallographic atomic
cess
planes. This, in its turn, leads to the destruction of in-
terference of reflected waves from the various planes. One
Analysis of the experimental data on the backward
oftheimportantfeaturesofscatteringprocessofrelativis-
PXRshows[7]that,apartfromtheabsorption,theimpor-
tic electrons at ψ > ψ , is that the process is asymmetric
tant contribution to the characteristics of the PXR lines, c
in (x,y)-plane (Fig. 2), namely, the scattering along the
suchaslinewidth,ismadebydeflectionofrelativisticelec-
x-axis in (v,z)-plane and along the y-axis in (x,y)-plane
trons from initial straight forward direction in crystal. In
essentiallyaredifferent. Thereasonisthatthecorrelations
other words, the assumption of the particle’s straight tra-
betweensubsequentcollisionsofrelativisticelectronswith
jectory and its no-effect on the line shape is inconsistent
atomicchains,paralleltothez-axisareofsubstantial. Due
with the experimental data.
to these correlations, the asymmetric scattering occurs.
For a relativistic electron (E (cid:29) m ), the variation of its
e
The correlated scattering takes place mainly along the
velocity|v˙|duetoscatteringissmall,since|v˙|∼1/E [16].
azimuthal angle in (x,y)-plane, perpendicular to the z-
Therefore, for relativistic electrons, falling onto a crystal
axis[16]. A redistribution of relativistic electrons occurs
at a small angle ψ > ψ , one can represent the velocity
c
over this angle due to the multiple scattering by different
v(t) in the form [16]
atomic chains. At ψ > ψ , the scattering process can be
c
1 approximated by a Gaussian one [17]. The mean square
v(t)≈v (1− v2)+v (t), (1)
0 2v02 ⊥ ⊥ of the multiple scattering angle is then given by θc2 =qcL,
where q is the mean square of the correlated scattering
c
where v0 - velocity of the incident electrons, v⊥(t) - com- angle per unit length. The quantity qc differs from the
ponent of v(t) which is perpendicular to v0 and v⊥ (cid:28)v0. corresponding one in an amorphous medium by a factor
The latter inequality allows us to use the so-called small of the order of R/4ψa [17], where R is the Thomas-Fermi
anglescatteringapproximationθ ∼(v⊥/v). Asitisshown radius of screening of the potential of a separate crystal
2
atom. 1/L to within the order of magnitude. At σ (cid:29) 1 and
x
The non-correlated scattering of relativistic electrons oc- α(cid:29)1, the scattering process leads to the line broadening
curs on thermal vibrations of atoms in the crystal, and of the backward PXR and ∆ω is defined to within the
the mean square scattering angle in this case is given by order of magnitude as
θ2 =q L of that of an amorphous medium [16].
a a
Thus, the deviation of the particle’s trajectory along the ∆ω ∼ασx/L, (4)
y-axis,whichisorientedperpendiculartothe(v,z)-plane,
which is much larger than the natural line width for the
is caused by both correlated and non-correlated scatter-
same crystal thickness.
ing in the crystal, and the mean square scattering angle is
givenbyθ2 =q L,whereq =q +q /2. Thedeviationof
y y y c a
particle’s trajectory along the x-axis is mainly caused by 3. Necessary conditions for neglecting the photo
non-correlated scattering with mean square scattering an- absorption effect on the line width
gleofθ2 =qL,whereq =q /2. Theq valuediffersbyless
x x x
than 10÷15% from the corresponding value for an amor- In experiment [7], the backward PXR with the photon
phousmediumqa [16]. Inthemultiplescatteringtheoryof energies of several KeV was produced by relativistic elec-
x
high energy electrons in an amorphous medium, the value trons of the energy of E ≈855 MeV in a Si crystal of the
qxa is defined by the relation qxa = η(13.6MeV/E)2/LR thickness of 525µm. For such a thick Si crystal (L > La)
[18], where L is the radiation length of relativistic elec- besidesthescatteringprocess, theanotherimportantone,
R
trons and η =(1+0.038ln L )2. In section 4, we will use causing the line broadening of the backward PXR, is the
qa instead of q for numericLaRl calculations. photo absorption. Both processes exert a parasitic influ-
x x
In [19], the spectral angular density of the backward PXR ence on the line width and were experimentally investi-
in extremely Bragg geometry, averaged on the multiple gated in [7]. The interpretation of the experimental data
scatteringprocessbythemethodoffunctionalintegration forthemultiplescatteringeffectofelectronsinthecrystal
[20] was obtained. Thus, the appropriate formula with on the line width of the backward PXR was explained in
somemodificationsforthespectralangularradiationden- [7] on the basis of [21]. In the work [21], the PXR ampli-
sity near the lines with the energy ω ≈ g/2 reads as the tude was first obtained without any directional changes of
n
following electrons in the crystal, and then, was modified for small
angle multiple scattering. At the end, the corresponding
d2W amplitudewascalculatedusingtheprobabilisticmethods.
=CΦ(φ,ψ)L2F(L,∆ω), (2)
dωdΩ But there is the another point of view. On one hand, the
multiplescatteringofelectronsduringthepassagethrough
where C = e2ωn2|ε(cid:48)ωn|2, Φ(φ,ψ)= (φ−ψ)2 , e - charge the crystal is a usual parasitic process and occurs along
4π2d2 [γ−2+(φ−ψ)2]2
of particle, ε(cid:48) - Fourier component of material part of the whole path of charged particles traversing the crystal.
ωn
the medium permittivity ε = 1+ε(cid:48) , g = 2πn/a, n - Fromtheotherhand,PXRisproducedbyreflectionofrel-
ωn ωn
integers, γ - Lorentz factor of particle, and F(L,∆ω) is ativisticelectron’sfieldfromallofthecrystalplanes,even
defined as from the last layers. Consequently, the scattering process
of electrons destructively influences on the line width of
(cid:90) 1 (cid:90) z
F(L,∆ω)=2 dz dycos[2yL∆ω] thebackwardPXRfromthefirstplanesofthecrystaltra-
0 0 (3) versed by electrons till the last points, when the particles
×exp(−α2σ2y2z), leave it. Therefore to obtain (3), the directional changes
x
ofelectronsincrystalwerefromtheoutsetincludedinthe
√
where∆ω =ω−ωn,α=φr/ 2qxL,φr =φ−ψ andσx = PXR spectral angular density, and afterwards, the latter
ωnqxL2. The parameters α, σx and function F(L,∆ω) was averaged on the random scattering process.
aredimensionless. Otherrelevantquantitiesaredefinedin However, Eq. (3) has been obtained without taking into
Fig. 1. accounttheabsorptioneffectonthelinewidthoftheback-
In Eq. (2), C and Φ(φ,ψ) do not depend on the multiple wardPXR.Inordertofindouttheconditions,underwhich
scatteringparameterqx, butF(L,∆ω)doesanddescribes it is possible to neglect the influence of the photo absorp-
thebehaviouroflinewidthofthebackwardPXR.Formula tion on the line width and to put the formula (3) into
(3) has been obtained in the approximation in which the practice, we consider the following.
main contribution to the line width originates from the Fromtheonehand,thefactorασ in(3)isproportionalto
√ x
non-correlated multiple scattering of relativistic electrons Lasασ ∝L L. Therefore,accordingtotheformula(4),
x √
in the crystal, i.e. when α(cid:29)1 and σx (cid:29)1. thelinewidthofthebackwardPXRdecreasesas Lwhen
The factor ασx in F(L,∆ω) stands for influence of the decreasingthethickness. Fromtheotherhand,ασx ispro-
multiplescatteringofelectronsonPXRlines. Ifασx →0, portionaltotheenergyofchargedparticlesasασx ∝1/E.
which means qx →0, then the influence of non-correlated Hence, if we choose thin crystals and decrease the energy
multiple scattering process on line width of the backward of charged particles, but retaining the latter in relativistic
PXRcanbeignoredand∆ω inthiscaseisgivenby∆ω ∼
3
domain, then it is possible to fit such values for L and E,
Table 1: Approximated and exact values of the line width of the
so that the following conditions can be fulfilled:
backwardPXR(fourthandfifthcolumns)forelectronsoftheenergy
of E =30 MeV traversing a Si crystals of varying thicknesses (first
L<La, ασx (cid:29)1. (5) column). Theconditionασx(cid:29)1isfulfilled.
L[µm] α σ ∆ω [eV](4) ∆ω [eV]
x app ex
In other words, at the fulfilment of the conditions (5), the
20 22.24 8.14 1.81 1.52
influenceofthephotoabsorptioninthecrystalontheline
25 19.64 13.05 2.05 1.68
width can be ignored and the main contribution to the
30 17.75 19.17 2.26 1.83
line broadening of the backward PXR in extremely Bragg
geometry is made by the multiple scattering process of
relativistic electrons in crystal.
Table2: SameasinTable1butforelectronsoftheenergyofE=50
MeVtraversingSicrystalofdifferentthicknesses(firstcolumn).
4. Numerical results and discussion
L [µm] α σ ∆ω [eV](4) ∆ω [eV]
x app ex
Formula (3) is appropriate for a parametric consider- 35 27.17 9.55 1.48 1.21
ation of the multiple scattering effect of relativistic elec- 40 25.23 12.66 1.59 1.29
tronsonthelinewidthofthebackwardPXR.Butitisless 45 23.64 16.22 1.70 1.36
suitable for numerical analysis from the wall time point of 50 22.30 20.26 1.80 1.44
view. Therefore, to investigate numerically the influence
of the multiple scattering on the line width, one has to
modify formula (3) as in the following.
Taking p≡2L∆ω and s≡α2σ2 as new numerical dimen-
Table 1).
sionless quantities and using Euler representation for co-
Theoptimalupperlimitforenergyofelectronsis50MeV,
sine function, one gets the following formula for F(L,∆ω)
since at the chosen thicknesses (first column of Table 2),
the conditions (5) (second and third columns of Table 2)
√ (cid:34) (cid:32) (cid:33) arestillfulfilledforphotonsinthe7−8KeVenergyrange.
F(L,∆ω)=2(cid:90) 1 √π e−4ps2z erf ip+√2sz2 For100MeVelectronstraversingSicrystalofthethickness
2 sz 2 sz
0 10µm ≤ L ≤ 60µm, the values of σ are in the range of
(6) x
(cid:32)ip−2sz2(cid:33)(cid:35) 0.17≤σx ≤7.4. Therefore, the condition σx (cid:29)1 can not
−erf √ dz, be fulfilled. It seems that more thicker crystal L > 60µm
2 sz
could be used to increase σ . But such range of crystal
x
thickness is not suitable, since the absorption of radiated
where the line width ∆ω is included in the arguments of
photons parasitically influences on the line width of the
complex Error and exponential functions (In Eq.(6), i is
backwardPXRandthefirstpartofconditions(5)cannot
the imaginary unit).
be fulfilled. Thus, the upper limit of energy for electrons
The Eq. (6) depends on the parameters such as thickness
is set to 50 MeV.
Lofcrystal,energyE ofrelativisticelectrons,incidentan-
Based on the used ω together with the selected ranges
gle ψ of electrons to the crystallographic axis (ψ must be n
for energy of electrons and the corresponding values of in-
more than ψ ) and ω . For concrete practical application
c n cident angle ψ of electrons and appropriate thickness of
of formula (6) under the conditions (5), we consider the
Si, the numerical calculations of the influence of multiple
line width of the backward PXR produced by relativistic
scattering of E = 30 MeV and E = 50 MeV relativistic
electrons of different E = 30 MeV and E = 50 MeV en-
electrons on the line width of the backward PXR in ex-
ergies in a single Si crystal of varying thicknesses. The
tremely Bragg geometry are shown in Figs. 3 and 4.
radiation length for Si is L = 9.37 cm[22] and we use
R Let us discuss the obtained curves in both figures by the
ωn ≈ √34dπSin with n=3, where dSi =5.43×10−8cm [23] example of L=20µm for E =30 MeV electrons (Fig. 3).
(for absorption length of radiated photons in the energy
The maximum value of F(L,∆ω) for L=20µm at chosen
range of 7−8 KeV in Si, see X-Ray Attenuation Length
energy of electron in Fig. 3 is (cid:39) 3.3 × 10−2. Dividing
[24]).
it by two and finding the corresponding value of ∆ω for
For above ω , the thickness of Si crystal and the values of
n it on the abscissa of Fig. 3 and multiplying the obtained
the energy of relativistic electrons are chosen so that the
∆ω by two, we get ∆ω = 1.52eV for the line width of
ex
condition α(cid:29)1 and σ (cid:29)1 of application of Eq. (2) to-
x the backward PXR produced by E = 30 MeV relativistic
gether with (5) could be fulfilled. Then the optimal value
electrons traversing Si crystal of thickness L=20µm tak-
for the energy of relativistic electrons is started from 30
ing into account the scattering process in the absence of
MeV to retain particles in relativistic domain. The crys-
photoabsorption, where∆ω meanstheexactcalculated
ex
tal thickness (first column of Table 1) is in the range of
line width by Eq.6 . This value is shown in the fifth col-
20µm ≤ L ≤ 30µm (L < L ) in order to fulfil both con-
a umn of Table 1. The same value of ∆ω for L=20µm,
app
ditions α (cid:29) 1 and σ (cid:29) 1 (second and third columns of
x
4
0.035 0.045
20µm 0.04 35µm
0.03 25µm 40µm
30µm 0.035 45µm
50µm
0.025
0.03
)ω 0.02 )ω 0.025
∆ ∆
L, L,
F( 0.015 F( 0.02
0.015
0.01
0.01
0.005
0.005
0 0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
∆ω [eV] ∆ω [eV]
Figure 3: Numerical results of the multiple scattering effect on the Figure4: SameasinFig. 3butforelectronsoftheenergyofE=50
line width ∆ω of the backward PXR according to (6) produced by MeVtraversingaSicrystalofthevaryingthicknesses35µm≤L≤
relativistic electrons of the energy of E = 30 MeV traversing a Si 50µm(seeTable2).
crystalofthevaryingthicknesses20µm≤L≤30µm(seeTable1).
tering process makes the important contribution to the
but according to the formula (4) is shown in the fourth line width of the backward PXR in the absence of photo
column of Table 1. absorption.
There is a small discrepancy ∆ between the values of the The line width of the backward PXR in extremely Bragg
linewidthforL=20µminthefourthandfifthcolumnsof geometryatlowenergyrelativisticelectronsinexperiment
Table 1, where ∆ = |∆ωapp−∆ωex|. This is because the [6] was not measured, although a boost of intensity by a
line width in the fourth column ∆ωapp is obtained by the factor of two was observed. As far as we know, apart
approximate formula (4), whereas ∆ωex in fifth column is from the measurement of the line width of the backward
extracted from the exact calculation of (6) in Fig. 3. PXR at high energies of electrons[7], no any experiment
The above discussion concerning the numerical result of has been performed yet on the measurement of the line
the line width at L = 20µm in Fig. 3 can be given for width of the backward PXR in extremely Bragg geome-
otherthicknessesinFig. 3andTable1, asforcorrespond- try, produced by low energy relativistic electrons in thin
ing values in Fig. 4 and Table 2. crystals. The experiment could make the influence of the
This is very important to note that the natural line width multiplescatteringeffectonthelinewidthofthebackward
of the backward PXR produced by E =30 MeV relativis- PXR more clear, since this is a still open question.
tic electrons at L = 20µm in extremely Bragg geometry
under the condition of the absence of photo absorption
6. Acknowledgements
is about ∆ω ∼ 10 meV to within the order of magni-
tude. This value is much smaller than the line width of
TheauthorisgratefultoN.F.Shul’gafromtheKharkov
the backward PXR of the same thickness, but affected by
Institute of Physics and Technology (Ukraine) for discus-
the multiple scattering effect ∆ω = 1.52 (fifth column
ex sion concerning the work and reading the manuscript.
in Table 1). This difference is of significant for all thick-
The author would like in particular to express his grat-
nesses at both E =30 MeV and E =50 MeV energies. It
itude to Tanaji Sen from the Fermi National Lab (USA)
means that the multiple scattering of relativistic electrons
forfruitfuldiscussionsconcerningthisworkandusefulrec-
traversingasinglecrystalmakesasignificantcontribution
ommendations during reading the manuscript.
to the line width of the backward PXR in the absence of
Figs. 1 and 2 have been prepared in the free and open-
photo absorption.
source software ”LibreOffice Draw”[25].
The numerical results in Figs. 3 and 4 have been calcu-
5. Conclusions lated in the free and open-source software Sage[26] at the
Parallel Computing Laboratory (2216) of the Department
The influence of the multiple scattering of low energy of Physics at the Razi University of Kermanshah using
30 and 50 MeV relativistic electrons on the line width ∆ω GSL (GNU Scientific Library) and NumPy packages..
of the backward PXR in extremely Bragg geometry in a Active administrative support from Dr. Ardeshir Rabeie
Sicrystalatvaryingthicknesseshasbeenconsidered. The together with the financial support from the Department
conditions, when the photo absorption effect on the line of Physics for installing and establishment of the Parallel
width is negligible, has been obtained. Computing Laboratory, are gratefully acknowledged.
The obtained numerical values of ∆ω show that the scat-
5
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