Ultramicroscopy 1 (1975) 167-169 © North-Holland Publishing Company
LETTER TO THE EDITOR THE INFLUENCE OF PLURAL SCATTERING ON THE LIMIT OF RESOLUTION IN ELECTRON MICROSCOPY* H. ROSE
lnstitut flit theoretischePhysik, TechnicalUniversityof Darmstadt, Darmstadt, Fed. Rep. Germany Received 28 October 1975
Recently the influence of plural scattering on the resolution limit of point scatterers embedded in thick amorphous layers was treated by Groves for the CEM and the STEM respectively [1 ]. The spatial distribution of fast electrons in thick amorphous objects was also investigated both theoretically and experimentally by Jost and Kessler [2] and Reimer et al. [3,4]. The latter authors have evaluated the top bottom effect in the STEM by means of Monte-Carlo calculations, but did not perform the convolution over the incident electrons. Groves started from the Boltzmann transport equation to achieve a theoretical description of the beam broadening. A thorough study of his paper has shown that it contains some errors which may severely influence the numerical results presented. In the following we shall restrict ourself to discuss these errors and deduce the correct solution for the STEM mode. According to [1] the spatial and angular distribution F(r,6, z) of the electrons within the amorphous foil can be obtained from the small angle approximation of the Boltzmann transport equation:
DFa__+_zOx -~aF+ Oy ayaF- F +~ fF(r,O,,z)f.(lO_O,i)d20,
(I)
Here f(0) and k are the angular distribution for single scattering and tile mean free path of the electrons, respectively. For solving this equation it is useful to take the double Fourier transform with respect to the variables r and 0.The solutions of the resulting partial differential equation**
ag
X--
aO x
r
~Oy
-
~(I- f) A
(2)
--
of the transformed distribunon flanction /~ =/7(R,O, z) = ~
ffF(r,O, z) exp [--i(Rr + 00)] d2r d20
(3)
can be found easier than those o f ( l ) ;
f =f(®) =fJ'(O)
(4)
exp (iO0) d 20.
Unfortunately the three-dimensional solution of (2) proposed by Groves does not fulfill this equation (not even in the case of rotational symmetry), a fact which can easily be proved by differentiation. Without going into the details of the calculation we present the exact solution and prove its validity. The general solution of (2) is given * Comment on a paper by Groves [1]. ** It should be noted that the signs of the second and third term are positive in [1]. 167
H. Rose / Huml scattering and the limit of resolution
168
by
where a is an arbitrary function, and the upper surface of the foil is located in the plane z = 0. in order to prove that (5) indeed is a solution of (2) we calculate the following derivatives: aF
aOR +(z-- zo}R2"-
l'"(IO+zRI)-l'=,l.t IF
(6)
~f(OX+z'X)X+((~y+z'Y)YdZ'IO+Z'RI
X~x~)/l~ +~O~ I'c~f _aa X[Ox+(Z-IO+(2z0)X]+-Y[Ov+(Z-z0)Y]Fz0)R[ +--~0
aF- OR + (z, =;
IO+(-z
z0)R2
z0)RI
+FI=(IO+ zRI)--.t
-.t'(®)]
x
(7)
The dots indicate differentiation with respect to the variables IO+ (z - z0)RI and 10+ z'RI respectively. Subtracting the relation (7) from (6) yields the equation1 (2). Hence, the relation (5) is the general solution of the transformed Boltzmann equation. The function a = a(R, IO + (z - z0)RI ) has to be determined from the boundary conditions. For the STEM the solution F(r, O, z)
=ffP(R,O,z) exp [i(rR + 00)] d2R d 2 0
(8)
of the Boltzmann equation (1) must fulfill the boundary condition
_L1
(9)
F(r,O, z = Zo, X = oo) = 7tO2 j(r) T(O/O0) ,
which means that in the absence of the foil (X = o~) the minimal scanning spot is located in the plane z = z 0 of optimum focus as can be seen from fig. I. In the case of an ideal microscope the radial current density distribution /(r) is the well known Airy distribution and z 0 the gaussian focal plane. The step function
T(O/Oo)= {
1
f o r 0 < 00,
0
for 0
(1o)
> 0o,
~, ~0 .}foil *z Fig. l. Spreadingof an electron beam within an amorphous foil due to plural scattering.
H. Rose /Plural scattering and the limit of resolution
169
where 00 is tile limiting angle of tile objective, guarantees that all electrons are uniformly distributed within the cone of illumination. Tile function a(R, IO+ (z - zo)R I) is entirely determined when it is known in the plane z = z 0. The resulting function a(R,O) can be determined from (9) in addition with the relations (5), (8) and (10) by taking the double Fourier transform with respect to r and 0. As a result we lind a(R,O) =
Jl(O00) - ~ ( R ) - (2n) 3 00® '
(11)
where ~ ( R ) = (1/lo)fj(r) exp(irR)d2r is tile contrast transfer function for completely incoherent illumination and 10 the total current of the electron beam. The radial current density distribution ](r, z) in a plane z within the amorphous foil is found by integrating (8) over all angles 0:
](r,z)=
I--~OS JI[OoR(z- zo)] ( l Z[" -f(z'R)]dz') 0 ~(R) OoR(Z - Zo) Jo(rR) exp - kOJ [l
R dR .
(12)
In tile case of a parallel incident electron beam (0 0 = 0,./2 (R) = 1) this equation becomes identical to the formula given by Jost and Kessler [2]. With a homogeneous film having no discrete scattering center, there will be no contrast and hence no resolution. When tile homogeneous foil contains an additional point scatterer, that point acts as a probe for the beam. That is, recognizable contrast can occur only when the point is inside the broadened scanning spot resulting from plural electron scattering inside the foil above the point scatterer. Thus the instrumental resolution, determined from point scattering, depends r ,imarily on only tile electron beam size in tile plane of the scatterer and not on detector parameters such as the collector geometry. In the case of a real scatterer, for example a havy atom, the recorded image does depend on the detector geometry, but then no linear contrast transfer does exist *, which means that the Fourier transform of the image cannot be described as a mu~iplication of the transformed scattering distribution.fA of the atom with the transformed distribution function F, contrary to what was assumed in [1]. It should be noted that the angular limits of tile integration cannot depend on the detector geometry because the beam size inside the foil cannot be influenced by file physical parameters of any apparatus following the film. It appears that in the derivation o f e q . (2) in [I] tile limiting detector acceptance angle has been used as the upper limit of the 0 integration and thus the result cannot be assumed to describe correctly the resolution measured. This could also be the cause of the substantial difference between the predicted resolutions in the STEM and CEM. The "top-bottom effect" is obtained by comparing the distribution ](r, z = z0, k = oo) without the foil with tile distribution ](r, z = z 0, k) resulting from (12). In the latter case the illuminating beam is focused onto the bottom of tile foil located in the plane z = z 0. A detailed quantitative investigation of the beam broadening and a complete theoretical description of image formation of single objects embedded in thick amorphous material or in a wet specimen chamber will be given in forthcoming papers on a wave-mechanical basis [5].
References [1] [2] [3] [4] [5]
T. Groves, Ultramicroscopy 1 (1975) 15. K. Jost and J. Kessler, Zeit. Phys. 176 (1963) 126. L. Reimer, H. Gilde and K. Sommer, Optik 30 (1970) 590. R. Gentsch, H. Gilde and L. Reimer, J. Microscopy 100 (1974) 81. J. Fertig and H. Rose, Optik to be published.
~ A detailed proof of this statement will be given in [5 ].
LETTER TO THE EDITOR THE INFLUENCE OF PLURAL SCATTERING ON THE LIMIT OF RESOLUTION IN ELECTRON MICROSCOPY* H. ROSE
lnstitut flit theoretischePhysik, TechnicalUniversityof Darmstadt, Darmstadt, Fed. Rep. Germany Received 28 October 1975
Recently the influence of plural scattering on the resolution limit of point scatterers embedded in thick amorphous layers was treated by Groves for the CEM and the STEM respectively [1 ]. The spatial distribution of fast electrons in thick amorphous objects was also investigated both theoretically and experimentally by Jost and Kessler [2] and Reimer et al. [3,4]. The latter authors have evaluated the top bottom effect in the STEM by means of Monte-Carlo calculations, but did not perform the convolution over the incident electrons. Groves started from the Boltzmann transport equation to achieve a theoretical description of the beam broadening. A thorough study of his paper has shown that it contains some errors which may severely influence the numerical results presented. In the following we shall restrict ourself to discuss these errors and deduce the correct solution for the STEM mode. According to [1] the spatial and angular distribution F(r,6, z) of the electrons within the amorphous foil can be obtained from the small angle approximation of the Boltzmann transport equation:
DFa__+_zOx -~aF+ Oy ayaF- F +~ fF(r,O,,z)f.(lO_O,i)d20,
(I)
Here f(0) and k are the angular distribution for single scattering and tile mean free path of the electrons, respectively. For solving this equation it is useful to take the double Fourier transform with respect to the variables r and 0.The solutions of the resulting partial differential equation**
ag
X--
aO x
r
~Oy
-
~(I- f) A
(2)
--
of the transformed distribunon flanction /~ =/7(R,O, z) = ~
ffF(r,O, z) exp [--i(Rr + 00)] d2r d20
(3)
can be found easier than those o f ( l ) ;
f =f(®) =fJ'(O)
(4)
exp (iO0) d 20.
Unfortunately the three-dimensional solution of (2) proposed by Groves does not fulfill this equation (not even in the case of rotational symmetry), a fact which can easily be proved by differentiation. Without going into the details of the calculation we present the exact solution and prove its validity. The general solution of (2) is given * Comment on a paper by Groves [1]. ** It should be noted that the signs of the second and third term are positive in [1]. 167
H. Rose / Huml scattering and the limit of resolution
168
by
where a is an arbitrary function, and the upper surface of the foil is located in the plane z = 0. in order to prove that (5) indeed is a solution of (2) we calculate the following derivatives: aF
aOR +(z-- zo}R2"-
l'"(IO+zRI)-l'=,l.t IF
(6)
~f(OX+z'X)X+((~y+z'Y)YdZ'IO+Z'RI
X~x~)/l~ +~O~ I'c~f _aa X[Ox+(Z-IO+(2z0)X]+-Y[Ov+(Z-z0)Y]Fz0)R[ +--~0
aF- OR + (z, =;
IO+(-z
z0)R2
z0)RI
+FI=(IO+ zRI)--.t
-.t'(®)]
x
(7)
The dots indicate differentiation with respect to the variables IO+ (z - z0)RI and 10+ z'RI respectively. Subtracting the relation (7) from (6) yields the equation1 (2). Hence, the relation (5) is the general solution of the transformed Boltzmann equation. The function a = a(R, IO + (z - z0)RI ) has to be determined from the boundary conditions. For the STEM the solution F(r, O, z)
=ffP(R,O,z) exp [i(rR + 00)] d2R d 2 0
(8)
of the Boltzmann equation (1) must fulfill the boundary condition
_L1
(9)
F(r,O, z = Zo, X = oo) = 7tO2 j(r) T(O/O0) ,
which means that in the absence of the foil (X = o~) the minimal scanning spot is located in the plane z = z 0 of optimum focus as can be seen from fig. I. In the case of an ideal microscope the radial current density distribution /(r) is the well known Airy distribution and z 0 the gaussian focal plane. The step function
T(O/Oo)= {
1
f o r 0 < 00,
0
for 0
(1o)
> 0o,
~, ~0 .}foil *z Fig. l. Spreadingof an electron beam within an amorphous foil due to plural scattering.
H. Rose /Plural scattering and the limit of resolution
169
where 00 is tile limiting angle of tile objective, guarantees that all electrons are uniformly distributed within the cone of illumination. Tile function a(R, IO+ (z - zo)R I) is entirely determined when it is known in the plane z = z 0. The resulting function a(R,O) can be determined from (9) in addition with the relations (5), (8) and (10) by taking the double Fourier transform with respect to r and 0. As a result we lind a(R,O) =
Jl(O00) - ~ ( R ) - (2n) 3 00® '
(11)
where ~ ( R ) = (1/lo)fj(r) exp(irR)d2r is tile contrast transfer function for completely incoherent illumination and 10 the total current of the electron beam. The radial current density distribution ](r, z) in a plane z within the amorphous foil is found by integrating (8) over all angles 0:
](r,z)=
I--~OS JI[OoR(z- zo)] ( l Z[" -f(z'R)]dz') 0 ~(R) OoR(Z - Zo) Jo(rR) exp - kOJ [l
R dR .
(12)
In tile case of a parallel incident electron beam (0 0 = 0,./2 (R) = 1) this equation becomes identical to the formula given by Jost and Kessler [2]. With a homogeneous film having no discrete scattering center, there will be no contrast and hence no resolution. When tile homogeneous foil contains an additional point scatterer, that point acts as a probe for the beam. That is, recognizable contrast can occur only when the point is inside the broadened scanning spot resulting from plural electron scattering inside the foil above the point scatterer. Thus the instrumental resolution, determined from point scattering, depends r ,imarily on only tile electron beam size in tile plane of the scatterer and not on detector parameters such as the collector geometry. In the case of a real scatterer, for example a havy atom, the recorded image does depend on the detector geometry, but then no linear contrast transfer does exist *, which means that the Fourier transform of the image cannot be described as a mu~iplication of the transformed scattering distribution.fA of the atom with the transformed distribution function F, contrary to what was assumed in [1]. It should be noted that the angular limits of tile integration cannot depend on the detector geometry because the beam size inside the foil cannot be influenced by file physical parameters of any apparatus following the film. It appears that in the derivation o f e q . (2) in [I] tile limiting detector acceptance angle has been used as the upper limit of the 0 integration and thus the result cannot be assumed to describe correctly the resolution measured. This could also be the cause of the substantial difference between the predicted resolutions in the STEM and CEM. The "top-bottom effect" is obtained by comparing the distribution ](r, z = z0, k = oo) without the foil with tile distribution ](r, z = z 0, k) resulting from (12). In the latter case the illuminating beam is focused onto the bottom of tile foil located in the plane z = z 0. A detailed quantitative investigation of the beam broadening and a complete theoretical description of image formation of single objects embedded in thick amorphous material or in a wet specimen chamber will be given in forthcoming papers on a wave-mechanical basis [5].
References [1] [2] [3] [4] [5]
T. Groves, Ultramicroscopy 1 (1975) 15. K. Jost and J. Kessler, Zeit. Phys. 176 (1963) 126. L. Reimer, H. Gilde and K. Sommer, Optik 30 (1970) 590. R. Gentsch, H. Gilde and L. Reimer, J. Microscopy 100 (1974) 81. J. Fertig and H. Rose, Optik to be published.
~ A detailed proof of this statement will be given in [5 ].
