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Comparison of calculated third-order aberrations of a magnetic quadrupole lens
NUCLEAR
INSTRUMENTS
AND
METHODS
99
(1972) 6o9-6Io;
©
NORTH-HOLLAND
PUBLISHING
CO.
C O M P A R I S O N OF CALCULATED THIRD-ORDER A B E R R A T I O N S
OF A M A G N E T I C Q U A D R U P O L E LENS G. E. LEE-WHITING
Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada Received 17 November 197l D,iscrepancies between expressions for third-order aberration coefficients of a magnetic quadrupole lens of uniform gradient calculated by Smith t) and by Lee-Whitingz) are traced to an algebraic error in the work of the former.
D. L. Smith (ref. 1) has given expressions for thirdorder focusing coefficients applicable to the interior regions (transverse field) of electric and magnetic quadrupole lenses. The formulae for the magnetic case do not agree completely with results given by myself 2) in a paper submitted not long before the journal containing Smith's work became available in North America. In particular Smith's eqs. (133)to (142) ought to agree with table 3 of ref. 2. There are discrepancies in four cases: Smith's (x] X oXo12"), (X] X ot3) , (XrXoY'o~) and (xfxdy'o ~) fail to agree with my terms 3, 4, 7 and 10. Similarly Smith's eqs. (145) to (154) ought to agree with my table 4; there are discrepancies in (x'lxoX'o~), (x'fx~), (x'lxoY'o ~) and (x'lx~y'o ~) or my terms 3, 4, 7 and 10. Note that the differences in the x- and in the x'-coefficients occur for the same four third-order combinations of initial parameters. Discrepancies occur again in the (yl...) and (Y' I...) coefficients in analogous terms; in both papers x is used for the plane in which the lens is convergent and y for the plane in which it is divergent. Smith does not make explicit mention of the transformation 2) whereby y-coefficients may be obtained from x-coefficients - interchange of x and y and replacement of/3 by i/3, in Smith's notation -, but all his results in eqs. (155) to (178) conform to it; obvious misprints involving the dropping of a/3 in eqs. (169) and (1173) have been corrected. Hence we may confine our discussion to the x motion. The fact that in all four cases the discrepancies occur for the same (or analogous) combinations of initial parameters suggests that they are caused by the same flaw in the analysis, not by a collection of errors in the detailed algebra. We have verified Smith's section 3 from eq. (111) to eq. (129). However, we fail to agree with the expression for ofg given a few lines below eq. (130), finding instead - ~ =/3(1
+ X'o2+Y'oZ)+.
Smith has ½ in place of the exponent ¼. This difference turns out not to be a misprint, but an error responsible for the discrepancies between our two sets of coefficients. In working out aberration coefficients one makes use of an approximate third-order differential equation for trajectories. Following Smith's development one would obtain
X"+FI2X= /32(x'yy '-3xx'2--2 )Ix"2 Y 1
2
x (x + y )-v(x0 +yo 2) rz
, 2
t 2
+ flZf(xZ-- yZ)-(xZ-- y2)]}.
(2)
With the form of ~ appearing in Smith's paper one gets eq. (2) with v = 2; correcting the alleged error gives v = 1. The third-order equation used in ref. 2 and derived in Steffen's 3) book lacks all the terms below the first in line eq. (2). The difference occurs because instead of using the integrated form of the z-equation to get Vo/V= for use on the right in eq. (120) as Smith does, we employ
v0/v= = [1 +x'2+y'2]
(3)
which is essentially Smith's eq. (29). Regardless of the reason for the difference, the two forms must agree when we substitute first-order expressions for x, x', y and y' in the third-order terms; i.e. the expression in the {...} on the right ofeq. (2) must vanish. It does for v = 1, but not for v = 2. Now let ¢ be the difference between the corrected x and x as calculated by Smith. Using eq. (2) one sees that ~ satisfies ~,,+/32~ =
I 2 ~/3 X(Xo,2 + Yo,2);
(4)
the x on the right is to be replaced by the first-order approximation
(1)
X=XoCOSdp+fi-lxosin4), ~=flz. 609
(5)
610
(;. E. LEE-WHITING
i l
work and that it is not a gross misconception oll our part, the coefficient (x]x~) was computed numerically. Since the alleged error is in the reduction of the equations of motion to trajectory equations we went right back to the former. Smith's eqs. (117) and (I 18) were integrated by a R u n g e - K u t t a method starting from initial values z=O, x=O, v,=V:o~-Vo(I +x~)2) -~ and vx=x~)V~o up to a value of t for which z=4/[J. The coefficients C(~b) in eq. (7) was extracted from the numerical results for a set o f values of Xo.
I
x = I1 '[x,;sin(~+xo3C(qS) + ...].
,ee-Whi,io0 f \\
-.2}1
×"~ \~×
-.3 i
i
\\
c(#i
×~ \
-.4! i
\
Smith
X \
-51
iJ
\\ 1
-'7 0
,2
.4
.6
I .8
1.0
1,2
114
'r
1,6
Fig. I. The points shown are the results of numerical calculations of the coefficient C(~) defined by eq. (7). The curve labelled LeeWhiting is calculated from line 4 of table 3 of ref. 2. The curve labelled Smith is from eq. (139) of ref. l. Clearly 4 will contain terms in "o~o, " ,,,z ~o, ,,3 XoYo,2 and x6y'o2 exactly the terms in which the Smith and Lee-Whiting coefficients differ. The solution o f eq. (4) obeying the initial conditions 4 = 0 , 4 ' = 0 at z = 0 is
-
I
t2
t2
"
--
=~(Xo +yo)[XodpSlnO+[ ~ 'Xo(sinq)-qScosq~)].
(6)
The addition of the terms in eq. (6) to Smith's (xl...) coefficients brings them into complete agreement with Lee-Whiting's work; the same is true for ~' and the (x'l...) coefficients. This agreement serves as a verification of the results in tables 3 and 4 of ref. 2, especially since rather different calculational methods were used in the two papers. As a check that there really is an error in Smith's
(7)
The calculations were carried out f o r / ~ = 1.5, but the value offi is immaterial. The results are shown in fig. I. The discrepancy between the numerical results and the Lee-Whiting curve is too small to show in the figure. In conclusion we wish to stress that the two sets of formula being compared apply to a region of uniform magnetic gradient. Results applicable to the rectangular model - i.e. including the effects of the finite jumps in gradient at the ends of the lens - appear in tables 5 and 6 of ref. 2, but were not given by Smith. The foregoing paper has been examined by Dr. Smith and is published with his assent. It is his wish that two typographical errors also be corrected: (1) a multiplicative p2 should be inserted before the entire expression in eq. (80), and (2) Bx in eq. ( l l 7 ) should be By. References
t) D. L. Smith, Nucl. Instr. and Meth. 79 (1970) 144. 2) G. E. Lee-Whiting, Nucl. Instr. and Meth. 83 (1970) 232. :3) K. G. Steffen, High energy beam optics (lnterscience, New York, 1964).
INSTRUMENTS
AND
METHODS
99
(1972) 6o9-6Io;
©
NORTH-HOLLAND
PUBLISHING
CO.
C O M P A R I S O N OF CALCULATED THIRD-ORDER A B E R R A T I O N S
OF A M A G N E T I C Q U A D R U P O L E LENS G. E. LEE-WHITING
Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada Received 17 November 197l D,iscrepancies between expressions for third-order aberration coefficients of a magnetic quadrupole lens of uniform gradient calculated by Smith t) and by Lee-Whitingz) are traced to an algebraic error in the work of the former.
D. L. Smith (ref. 1) has given expressions for thirdorder focusing coefficients applicable to the interior regions (transverse field) of electric and magnetic quadrupole lenses. The formulae for the magnetic case do not agree completely with results given by myself 2) in a paper submitted not long before the journal containing Smith's work became available in North America. In particular Smith's eqs. (133)to (142) ought to agree with table 3 of ref. 2. There are discrepancies in four cases: Smith's (x] X oXo12"), (X] X ot3) , (XrXoY'o~) and (xfxdy'o ~) fail to agree with my terms 3, 4, 7 and 10. Similarly Smith's eqs. (145) to (154) ought to agree with my table 4; there are discrepancies in (x'lxoX'o~), (x'fx~), (x'lxoY'o ~) and (x'lx~y'o ~) or my terms 3, 4, 7 and 10. Note that the differences in the x- and in the x'-coefficients occur for the same four third-order combinations of initial parameters. Discrepancies occur again in the (yl...) and (Y' I...) coefficients in analogous terms; in both papers x is used for the plane in which the lens is convergent and y for the plane in which it is divergent. Smith does not make explicit mention of the transformation 2) whereby y-coefficients may be obtained from x-coefficients - interchange of x and y and replacement of/3 by i/3, in Smith's notation -, but all his results in eqs. (155) to (178) conform to it; obvious misprints involving the dropping of a/3 in eqs. (169) and (1173) have been corrected. Hence we may confine our discussion to the x motion. The fact that in all four cases the discrepancies occur for the same (or analogous) combinations of initial parameters suggests that they are caused by the same flaw in the analysis, not by a collection of errors in the detailed algebra. We have verified Smith's section 3 from eq. (111) to eq. (129). However, we fail to agree with the expression for ofg given a few lines below eq. (130), finding instead - ~ =/3(1
+ X'o2+Y'oZ)+.
Smith has ½ in place of the exponent ¼. This difference turns out not to be a misprint, but an error responsible for the discrepancies between our two sets of coefficients. In working out aberration coefficients one makes use of an approximate third-order differential equation for trajectories. Following Smith's development one would obtain
X"+FI2X= /32(x'yy '-3xx'2--2 )Ix"2 Y 1
2
x (x + y )-v(x0 +yo 2) rz
, 2
t 2
+ flZf(xZ-- yZ)-(xZ-- y2)]}.
(2)
With the form of ~ appearing in Smith's paper one gets eq. (2) with v = 2; correcting the alleged error gives v = 1. The third-order equation used in ref. 2 and derived in Steffen's 3) book lacks all the terms below the first in line eq. (2). The difference occurs because instead of using the integrated form of the z-equation to get Vo/V= for use on the right in eq. (120) as Smith does, we employ
v0/v= = [1 +x'2+y'2]
(3)
which is essentially Smith's eq. (29). Regardless of the reason for the difference, the two forms must agree when we substitute first-order expressions for x, x', y and y' in the third-order terms; i.e. the expression in the {...} on the right ofeq. (2) must vanish. It does for v = 1, but not for v = 2. Now let ¢ be the difference between the corrected x and x as calculated by Smith. Using eq. (2) one sees that ~ satisfies ~,,+/32~ =
I 2 ~/3 X(Xo,2 + Yo,2);
(4)
the x on the right is to be replaced by the first-order approximation
(1)
X=XoCOSdp+fi-lxosin4), ~=flz. 609
(5)
610
(;. E. LEE-WHITING
i l
work and that it is not a gross misconception oll our part, the coefficient (x]x~) was computed numerically. Since the alleged error is in the reduction of the equations of motion to trajectory equations we went right back to the former. Smith's eqs. (117) and (I 18) were integrated by a R u n g e - K u t t a method starting from initial values z=O, x=O, v,=V:o~-Vo(I +x~)2) -~ and vx=x~)V~o up to a value of t for which z=4/[J. The coefficients C(~b) in eq. (7) was extracted from the numerical results for a set o f values of Xo.
I
x = I1 '[x,;sin(~+xo3C(qS) + ...].
,ee-Whi,io0 f \\
-.2}1
×"~ \~×
-.3 i
i
\\
c(#i
×~ \
-.4! i
\
Smith
X \
-51
iJ
\\ 1
-'7 0
,2
.4
.6
I .8
1.0
1,2
114
'r
1,6
Fig. I. The points shown are the results of numerical calculations of the coefficient C(~) defined by eq. (7). The curve labelled LeeWhiting is calculated from line 4 of table 3 of ref. 2. The curve labelled Smith is from eq. (139) of ref. l. Clearly 4 will contain terms in "o~o, " ,,,z ~o, ,,3 XoYo,2 and x6y'o2 exactly the terms in which the Smith and Lee-Whiting coefficients differ. The solution o f eq. (4) obeying the initial conditions 4 = 0 , 4 ' = 0 at z = 0 is
-
I
t2
t2
"
--
=~(Xo +yo)[XodpSlnO+[ ~ 'Xo(sinq)-qScosq~)].
(6)
The addition of the terms in eq. (6) to Smith's (xl...) coefficients brings them into complete agreement with Lee-Whiting's work; the same is true for ~' and the (x'l...) coefficients. This agreement serves as a verification of the results in tables 3 and 4 of ref. 2, especially since rather different calculational methods were used in the two papers. As a check that there really is an error in Smith's
(7)
The calculations were carried out f o r / ~ = 1.5, but the value offi is immaterial. The results are shown in fig. I. The discrepancy between the numerical results and the Lee-Whiting curve is too small to show in the figure. In conclusion we wish to stress that the two sets of formula being compared apply to a region of uniform magnetic gradient. Results applicable to the rectangular model - i.e. including the effects of the finite jumps in gradient at the ends of the lens - appear in tables 5 and 6 of ref. 2, but were not given by Smith. The foregoing paper has been examined by Dr. Smith and is published with his assent. It is his wish that two typographical errors also be corrected: (1) a multiplicative p2 should be inserted before the entire expression in eq. (80), and (2) Bx in eq. ( l l 7 ) should be By. References
t) D. L. Smith, Nucl. Instr. and Meth. 79 (1970) 144. 2) G. E. Lee-Whiting, Nucl. Instr. and Meth. 83 (1970) 232. :3) K. G. Steffen, High energy beam optics (lnterscience, New York, 1964).