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Field curvature as a source of systematic errors in nuclear emulsion measurements
NUCLEAR
INSTRUMENTS
AND METHODS
21 (1963) 202--208; N O R T H - H O L L A N D
PUBLISHING
CO.
FIELD CURVATURE AS A SOURCE OF SYSTEMATIC ERRORS IN NUCLEAR EMULSION MEASUREMENTS D. A N G H E L E S C U and J. S. AUSL&NDElZ
Received 4 August 1962
Polytechnieal Institute, Bucharest
I t is shown t h a t curvature of the microscopic field of view m a y give rise to systematic e r r o r s i n n u c l e a r emulsion measure-
merits, especially if dip or polar angles are concerned. Simple correction procedures are outlined.
1. Introduction Purstang other investigations we noticed the surprising fact that in a nuclear emulsion soaked with Th(NOa)4 the e tracks from ThC' directed towards the glass surface seem to outnumber significantly the same kind of tracks directed towards the air surface. Controls showed that the same situation arises for other e prongs of Th stars, too, if their horizontal projection x exceeds N 28 #m, A discussion given in sec. 3, as wel! as control measurements described there and in sec, 4 prove that bias as a possible explanation is out of question. A satisfactory explanation of the up-down asymmetry is obtained by taking into account the curvature of the field of view in the microscope (sees. 4 and 5), which has been measured also independently (sec. 6).
wards all pertinent measurements were made with the same optical equipment. Control measurements were performed also with "homogeneous" equipment, i.e. with objective and eye-pieces of same manufacture (Meopta; C. Zeiss, Jena, etc.); the results are practically the same. Details concerning microscopes and optical equipments used, can be found in sec. 6. The main measurements refer to all prongs of Th stars, with origin more than ~ 32 #m away from either surface of the unprocessed emulsion (fiducial planes). Only stars with three 1) or four prongs were accepted, one of which had to be due toThC', showing ahorizontalproiection x >~ 41 #In (i.e. their vertical projections in the unprocessed emulsion were surely N 32 t~m, the distance oi the fiducial planes mentioned above). For each prong of the accepted stars its apparent vertical proj ection z' in the processed emulsion was measured, z' is considered positive (negative) if the track is directed upwards (downwards) ; the same applies as well to z = kz' (where k is the shrinkage factor), i.e. to the apparent vertical projection in the unprocessed emulsion. For ~ 90% of the accepted stars, chosen at random, the distance in the processed emulsion between star origin and air surface was measured, too. Furthermore, the thickness of the processed emulsion was measured in 30 points distributed at
2. Processing of the Plates and Scanning Conditions The plates used were of type NIKFI-R, with a nominal thickness of 200/~m. They were soaked for 30minutesin a 0.1% Th(NOa) * solution, dried with aicohol and processed after 48 hours in the usual manner. Initially other features of the stars had been under investigation with a binocular microscope (manufactured by Meopta, Prague), equipped incidentally with Meopta oil objective, 100 x ,' and with I.O.R. (Bucharest) eye-pieces, 10 x. After-
x) H. J. Taylor, V. 13. Dabholkar, Proc. Ind. Ac. Sci, A3,
(1936) 265; D. Anghelescu, Bul. Inst. Politehnic Bucuresti 19 (1957) 99. 202
FIELD
CURVATURE
AS A SOURCE
OF SYSTEMATIC
ERRORS
203
random over its surface. This thickness, transformed into unprocessed emulsion, turned out to be (134.0 + 4,4)/zm, the standard error being that of the individual value. The absolute value of the shrinkage factor seems to be of little importance, as it does not affect tile relative error of z determinations. We therefore admitted for it the rough value of 2, obtained in other experiments with plates of the same type, under similar processing2). For the same reason, we made no effort to determine the conversion factor of the microscope's depth screw from scale divisions into microns, but accepted the nominal value given by the manufacturer.
for h > 98/~m uniformity is obviously perturbed. Now, the lower fiducial plane is situated at ~ 32/~m from the even glass surface, i.e. at (~ 102.0 + 4.4) /an from the uneven, ondulated air surface. The loss of events for h > 98 #m is easily explained by the existence of such. undulations of the free emulsion surface, which are characterized numerically by the standard error of the unprocessed emulsion thickness (4.4/zm). It will be shown in the next section that elimination of the stars with h > 98/zm does not introduce any essential change in the detected up-down asymmetry. Thus, beyond any reasonable doubt this asymmetry eannot be ascribed to bias.
3. Discussion of Possible Bias
4. Distribution o f Vertical Projections o f ThC' a Tracks
Higher detection efficiency for stars nearer to the air surface would be a possible bias, able to produce the observed asymmetry, only if i~o fiducial planes (~ 32 #m from either emulsion surface) were used. Homogeneity of the detection efficiency with respect to the distance h between air surface and origin of a star in the unprocessed emulsion has been checked nevertheless in a simple manner, by means of the frequency distribution of h. This distribution would be nonuniform in the case of inhomogeneous detection efficiency.
The length z of the vertical projection of ThC' c~ tracks in the unprocessed emulsion has been determined, by direct measurement of z' in the processed emulsion. The origin of each star was placed in the centre ot the field of view, which was considered to be coincident with the centre of an eye-piece micrometer. The horizontal projection of the tracks was measured by means of this latter aN~ ¢ az
a~(M
20
[ ] ~'ev,,t~
$0 $
I -30
]Fig, I. D e p t h d i s t r i b u t i o n of T h C ' s t a r s .
Fig. i shows a plot A N (h)/zlh vs. h for 1527 stars located inside the fiducial planes. As can be seen, up to h --- 98 #tm (1428 stars) the h distribution is uniform (Pearson probability p22 ~ 27 %), while 2} M. Nicolae, p r i v a t e c o m m u n i c a t i o n .
-20
-[a
9
,40
*zo
1,.
*~70 z[jz.m)
Fig. 2. L e n g t h d i s t r i b u t i o n of v e r t i c a l projections of • t r a c k s from ThC'.
and the vertical projection by aid of the microscope's depth screw. All z' measurements were performed so as to eliminate the backlash of this screw. Fig. 2 is a plot of the frequency density zlN/Az vs.
204
D. A N G H E L E S C U AND J. S. A U S L A N D E R
z for a sample of 1727 events. T h e a s y m m e t r y coefficient b, defined as b = 2 ( N + -- N - ) / ( N
+ + N-),
(1)
where N + (respectively N - ) is t h e n u m b e r of e v e n t s w i t h z > 0 (respectively z < 0), t u r n s out to be b =
-
0.194
4- 0 . 0 4 9 ,
(2)
i.e. highly significant. F o r t h e c o m p u t a t i o n of t h i s b-value all events recorded as z = O, h a v e n o t b e e n t a k e n into account. If t h e tracks w i t h h > 98 # m are eliminated, t h e corresponding value of t h e a s y m m e t r y coefficient becomes b(h < 98~m) = - 0.174 + 0.051, (2') i n excellent a g r e e m e n t w i t h t h e v a l u e given b y eq. (2). I t seems v e r y s o u n d to a s s u m e t h a t t h e difference Ab --- - 0.018 + 0.071 is due to fluctuations a n d not to a s y s t e m a t i c displacement, since for t h e deeper lying stars one would r a t h e r expect a positive excess t h a n a n e n h a n c e d n e g a t i v e one. Consider now a plane i n t h e object space (fig. 3) ; i n t h e case of field c u r v a t u r e w i t h r o t a t i o n a l symm e t r y it will be r e p r e s e n t e d b y a c u r v e d surface in t h e image space (dashed i n fig. 3). L e t t h e origin O of a s t a r be placed i n C, i.e. i n t h e centre of t h e field of view. Then, all u p w a r d t r a c k OA will be seen as OA', i.e. its vertical projection will be recorded as z < Z, where Z m e a n s t h e t r u e v a l u e of t h i s projeetion; t h e same i n e q u a l i t y is v a l i d also for a d o w n w a r d t r a c k OB a n d its image OB'. A
.B'
/
Fig..3. Curvature of field of view (schematically).
I n absolute value, however, ] z ] < ] Z [ for upw a r d tracks (OA) a n d I z I > f Z ! for d o w n w a r d tracks (OB) a n d for strictly h o r i z o n t a l ones. There is one exception to this rule: Tracks w i t h small positive values of Z (actually upward) m i g h t be
recorded as cases w i t h small negative values of z, i.e. as tracks which seem t o b e directed downward. T h e general t r e n d is t h u s clearly a loss of u p w a r d tracks a n d a s i m u l a t e d excess of d o w n w a r d tracks. If this i n t e r p r e t a t i o n is correct, t h e n t h e distrib u t i o n of z will be s p r e a d on t h e negative side u p to higher absolute values of z t h a n o n t h e positive side. A c t u a l l y this features appears i n fig. 2. F u r t h e r m o r e , we m a y displace t h e axis of t h e figure to negative values, u n t i l a m i n i m u m of t h e difference b e t w e e n t h e n u m b e r of events to t h e r i g h t of t h i s new axis a n d t h e n m n b e r of e v e n t s t o its left is reached, i.e. practically u n t i l t h e new a s y m m e t r y coefficient b' becomes a p p r o x i m a t e l y equal to zero. I n fig. 2 t h e d a s h e d vertical line a t z = - 2.5 ~ m represents this new axis, for w h i c h b' =
-
0.006 +_ 0.048
;
(3)
this s t a n d a r d deviation is slightly smaller t h a n the one i n eq. (2), since here events w i t h z = 0 are not eliminated. Obviously, t h e small v a l u e of b' is n o t yet a n a r g u m e n t in favour of actual s y m m e t r y , since t h e new axis has b e e n chosen so, as to ensure a practically v a n i s h i n g b' - value. If t h e Z - values are actually s y m m e t r i c a l l y distributed, while t h e zvalues are n o t because of t h e c u r v a t u r e outlined i n fig. 3, t h e n t h e r i g h t h a n d p a r t of t h e h i s t o g r a m in fig. 2, a n d t h e left h a n d p a r t (right a n d Ieft u n d e r stood w i t h respect to t h e dashed, i.e. displaced, axis) m u s t show m i r r o r s y m m e t r y b e t w e e n each other, obviously w i t h allowance for fluctuations. This mirror hypothesis can be checked b y a X2 test 3) for consistency b e t w e e n t h e two halves of t h e histogram, which yields P X 2 ~ 25%. T h e same test, b u t r i g h t a n d left w i t h respect to t h e undisplaced axis, yields PX 2 m 0.03%. T h e e x p l a n a t i o n of t h e up-down a s y m m e t r y b y field c u r v a t u r e is definitely confirmed b y t h e i n d e p e n d e n t a n d direct m e a s u r e m e n t of this curvature, described i n sec. 6; f u r t h e r m o r e a decisive control experfinent with ~ tracks of ThC' is related i n sec. 7. Likewise the z m e a s u r e m e n t s of o t h e r prongs of T h stars, described in sec. 5, provide us w i t h f u r t h e r evidence in t h e same sense. ~) A. ~Iald, Statisticai Theory with Engineering Applications; quoted after Russian translation, Moscow 1956, p. 633.
F I E L D C U R V A T U R E AS A S O U R C E OF S Y S T E M A T I C E R R O R S
5. Distribution of Vertical Projections of Other ct Tracks of the Th Family I n a three- (or four-) prong star with ThC' the other c~tracks belong to ThA, Tn (and ThX) ; their ranges ( ~ 24 pm to ~ 32/am) show considerable overlapping, due to straggling. In the light of the preceding section it seems reasonable to investigate the up-down a s y m m e t r y of these tracks by classifying them with respect to the lengths x of their horizontal projections. Column 1 of table 1 lists three classes of horizontal projections x of ~ tracks from ThA, Tn and T h X , taken together; the fourth class, ThC', is included for comparison. Column 2 shows the updown a s y m m e t r y coefficients b. T h e y are not significant for the first two classes. For the last two classes, for which b is significantly different from zero, column 3 gives the axis displacement A z, for which a m i n i m u m of the up-down asymm e t r y coefficient b' is reached; the value of b' is given in column 4. The differences between a~ and ~ arise from elimination and non-elimination of the interval with z = 0. The figures of table 1 show clearly that tracks with large horizontal projections x are affected by curvature of the field of view (see also sec. 6), while the shortest ones are not; for intermediate values of x the effect seems to be present, without reaching in this experiment the usual significance level of 3 standard errors. 6. Direct Measurement of Field Curvature In N I K F I - R emulsions irradiated in Dubna with l0 GeV protons, several relativistic tracks were chosen, with dip angles less t h a n ~ 7 °. Consider two grains A and B of such a track, with a projected horizontal distance x between them, measured by means of all eyepiece micrometer (or in the case of the Zeiss KSM microscope, by the so called S-device for horizontal displacement of the field of view). If A is placed ill the centre of the field of view, a certain apparent vertical projection Z'A~ of the distance AB will be recorded by means of the microscope's depth screw; if B is placed in the centre the apparent vertical projection recorded will be z~A; generally s p e a k i n g Z~A # "AB*'"Both ZAB'and z~A' will be ex-
205
pressed in microns of processed emulsion and only their absolute values will be considered here. If [ Z ' [ is in absolute value the actual vertical distance between A and B in the processed emulsion, then
[ Z' [ = I I Z'AB t + a(X) [ = I I z~A J -T- A(X) I ,
(4)
where A (x) is the positive correction term due to field curvature. I t is supposed that A (x) does not depend on the azimuthal angle, or at least that it remains unchanged by an increase or decrease of 180 ° of the latter. This latter assumption has been checked experimentally for several pairs of distances + x and - x, approximately equal in absolute value. Only if
is practically equal to zero for all values of x, the field of view is to be considered as flat. A (x) has been measured with several combinations of microscope and optical equipment (listed in table 2) for different values of x. This was done by chosing a fixed grain A and different grains B1, B2 etc. at variable distances x; sometimes instead of A another, neighbouring grain A ' had to be chosen, in order to obtain a set of x-values reasonably spread over the whole diameter of the field of view. Figs. 4a and b are plots of A (x) vs. x for different microscopes and optical equipments; the symbols used in these figures are explained in table 2. Fig. 4a, concerning the Meopta microscope, contains also data obtained on a dipping ( ~ 14 °) track; evidently the dip is of n O importance for the value of A, which depends only on x. As can be seen from figs. 4a and b, the field curvature is often large enough, so that it must not be neglected. 7. Discussion 1. F r o m a qualitative point of v i e w it is not without interest to compare fig. 4 with the displacements A z given in table 1 and fig. 2. Transforming these Az (unprocessed emulsion) into Az" (processed emulsion) we have Az' = 0.75 pm and 1.25 a m for tracks with x /> 28.5/~m (say, on the a v e r a g e ~ ~ 31/an) and w i t h x ~>41 /am
206
D.
A N G H E L E $ C U AND
J.
S.
AUSLANDER
T~LE 1 A s y m m e t r y coefficients b and b' for c¢tracks due to ThX, Tn, ThA, and ThC" Horizontal projection x
tracks
ThX / Tn
x <~ 24.3/~m •
24.3/~m ~ x <
ThA j
+
28.5/~m
--
28.5/~m ~ . x
ThC"
Az
b X 10a
x ~ 41/2m
b~ × 10:l
32 -4- 39 127 4-
62
- - 2 6 2 4- 60
1.5/~m
+ 34 4- 57
--
2.5/~ra
--
Dip angle of track used*
Observation
194 4- 49
6±48
T~L~ 2 Signs used in figs. 4a and 4b
Sign
Manufacturer
1V~eopta (Prague) idem idem
×
o
e
+
C. Zeiss (Jena) idem E. Leitz (Wetzlar) Fratelli Koristka (Milano)
Binocular factor
1.25 X
Objective Oil immersion
Num. ap.
100 X
1.25 idem idem
idem idem
Eye-pieces
10 ×
N
idem 10 ×
N
14°
N
7°
4°
1×
90 X
1.30
15
2×
50 X monochromat 100 × apochromat 100 X
1.33
12,5 ×
1.32
10 ×
N 6°
1.25
12X
N
1.25 X IX
×
Eye-pieces 1.0.R. (Bucharest) idem
3°
Microscope mod. Ng.
4°
Nuclear emulsion microscope K.S.M. Microscope mod. Ortholux Koristka optics mounted on a Zeiss microscope, mod. Ng.
4°
* Dip angle in processed emulsion.
(ThC'; say, on the average x ~ 44 #m). F r o m fig. 4a we read the corresponding values A (31 pro) 0.65/~m and A (44/~m) ~ 1.3 pro. This agreement is a qualitative justification a posteriori of the axis displacements chosen in sees. 4 and 5. 2. As a final check for the explanation of the updown a s y m m e t r y as a field curvature effect, we have performed z' measurements on !000 ~ tracks of ThC', placing the end A of the track instead of its origin O (fig. 3) in the centre C of the field of view. If z' is considered positive for upward tracks, irrespective whether the origin O of the star or the end A of the prong is placed in C, t h a n we should
expect (in selfevident notations), bC__-O ~ -- bC~A,
(6)
with due allowance for slight differences in the absolute values, caused by reading errors. Experimentally we obtain for these 1 0 0 0 , tracks be= o = - 0.174 + 0.065 ; be_=A = + 0.163 + 0.065,
(7) in perfectly satisfactory agreement -with the expectation given b y eq. (6). 3. A numerical example concerning polar ang'te measurements of jet tracks in nuclear emulsion might be of illustrative value.
FIELD
CURVATURE
AS A SouRcE
Consider a jet with horizontal axis and denote the apparent polar angle of a jet track b y 8, the actual polar angle by 0, their projection onto the emulsion plane by 9, the apparent and the actual dip angle by $ and ~g; then evidently
(8)
cos ~ = cos q~ cos @ = cos q~ [1 + (kz'/l) 2]-~,
OF SYSTEMATIC
207
refer to the apparent direction of a track, as it is seen in the microscope before applying any corrections. As can be seen from fig. 5, the difference (8 - 8) between the true polar angle and the apparent one can reach non-negligibl e values, especially for small angles 8. Larger values of k increase this difference still more.
cos 0 = cosq~ cos T = cOS@ [1 + (k(z' + A)/1)2] -~ ,
(9)
ERRORS
, (0- 4-) ° 3 •~
_ _
* ~p# d','ttc~:s
where k is the shrinkage factor and l the projected distance at which z' is measured. ~0
(a)
20
30
40
~0
60
i
I/"
I
\
(/m~°
"
\
t,0
I tt
.2L
}
-3
$0
30
2O
40
~0
~o
\
.//
-f
I"1
J ~
Fig. 5. Difference b e t w e e n a c t u a l (0) a n d a p p a r e n t (8) polar a n g l e s of j e t t r a c k s .
8. Conclusions
i 0.~"
\
}
T+
(b) L
Fig. 4. ~ i e l d c u r v a t u r e m e a s u r e d on r e l a t i v i s t i c t r a c k s . (a) M e o p t a microscope; (b) O t h e r microscopes. Signs u s e d a r e e x n l a i n e d in t a b l e 2.
Considering e.g. a Zeiss KSM microscope with its optical equipment and taking l = 40/~m, i.e. A = 0.88 pm (see fig. 4b), k = 2.5 and different values of 9, we obtain the differences (0 - 0) as plotted in fig. 5. I n this figure " u p " and " d o w n "
1. Though manufacturers usually pay little attention to correction of field curvature*), this latter m a y become under certain circumstances of importance, giving rise to seriously misleading results. 2. For strongly dipping tracks the consequences of field curvature are practically negligible. 3. Most affected are flat or nearly flat tracks, which actually are directed horizontally or slightly upwards, but seem to be directed downwards. 4. There are several ways to avoid or to correct for the systematic errors introduced by field curvature, e.g. : a) to place the segment whose z coordinates are measured in such a manner in the field of view, that its extremities be located symmetrically with respect to the field centre; ~} see e . g . C . P . Sbillaber, P h o t o m i c r o g r a p h y in T h e o r y a n d P r a c t i c e ( N e w Y o r k a n d London, 1949) p. 212.
208
D. & N G H E L E S C U AND J. S. A U S L A N D E R
b) to measure the z coordinates twice, viz. once with one extremity and then with the other ex~ tremity placed in the centre of the field of view (recommendable only if the vertical noise of the stage is of little importance) ; c) to measure the curvature oi the field of view previously with sufficient accuracy, to establish an analytical expresion of the correction term, e.g. by the method of least squares for a parabolic dependence, or to plot it graphically, and to apply the corresponding corrections whenever necessary.
5. Since even high quality optical equipment is often uncompletely corrected for field curvature, this feature deserves more attention in emulsion work than it is usually given.
Acknowledgements We feel most indebted to M. Nicolae for her wholehearted assistance and for soaking and processing the plates, to E. NI. Friedliinder and Dr. R. Grigorovici for valuable discussions and to our scanning aid M. ~erban-Costescu for her able and patient work in scanning and measurements
INSTRUMENTS
AND METHODS
21 (1963) 202--208; N O R T H - H O L L A N D
PUBLISHING
CO.
FIELD CURVATURE AS A SOURCE OF SYSTEMATIC ERRORS IN NUCLEAR EMULSION MEASUREMENTS D. A N G H E L E S C U and J. S. AUSL&NDElZ
Received 4 August 1962
Polytechnieal Institute, Bucharest
I t is shown t h a t curvature of the microscopic field of view m a y give rise to systematic e r r o r s i n n u c l e a r emulsion measure-
merits, especially if dip or polar angles are concerned. Simple correction procedures are outlined.
1. Introduction Purstang other investigations we noticed the surprising fact that in a nuclear emulsion soaked with Th(NOa)4 the e tracks from ThC' directed towards the glass surface seem to outnumber significantly the same kind of tracks directed towards the air surface. Controls showed that the same situation arises for other e prongs of Th stars, too, if their horizontal projection x exceeds N 28 #m, A discussion given in sec. 3, as wel! as control measurements described there and in sec, 4 prove that bias as a possible explanation is out of question. A satisfactory explanation of the up-down asymmetry is obtained by taking into account the curvature of the field of view in the microscope (sees. 4 and 5), which has been measured also independently (sec. 6).
wards all pertinent measurements were made with the same optical equipment. Control measurements were performed also with "homogeneous" equipment, i.e. with objective and eye-pieces of same manufacture (Meopta; C. Zeiss, Jena, etc.); the results are practically the same. Details concerning microscopes and optical equipments used, can be found in sec. 6. The main measurements refer to all prongs of Th stars, with origin more than ~ 32 #m away from either surface of the unprocessed emulsion (fiducial planes). Only stars with three 1) or four prongs were accepted, one of which had to be due toThC', showing ahorizontalproiection x >~ 41 #In (i.e. their vertical projections in the unprocessed emulsion were surely N 32 t~m, the distance oi the fiducial planes mentioned above). For each prong of the accepted stars its apparent vertical proj ection z' in the processed emulsion was measured, z' is considered positive (negative) if the track is directed upwards (downwards) ; the same applies as well to z = kz' (where k is the shrinkage factor), i.e. to the apparent vertical projection in the unprocessed emulsion. For ~ 90% of the accepted stars, chosen at random, the distance in the processed emulsion between star origin and air surface was measured, too. Furthermore, the thickness of the processed emulsion was measured in 30 points distributed at
2. Processing of the Plates and Scanning Conditions The plates used were of type NIKFI-R, with a nominal thickness of 200/~m. They were soaked for 30minutesin a 0.1% Th(NOa) * solution, dried with aicohol and processed after 48 hours in the usual manner. Initially other features of the stars had been under investigation with a binocular microscope (manufactured by Meopta, Prague), equipped incidentally with Meopta oil objective, 100 x ,' and with I.O.R. (Bucharest) eye-pieces, 10 x. After-
x) H. J. Taylor, V. 13. Dabholkar, Proc. Ind. Ac. Sci, A3,
(1936) 265; D. Anghelescu, Bul. Inst. Politehnic Bucuresti 19 (1957) 99. 202
FIELD
CURVATURE
AS A SOURCE
OF SYSTEMATIC
ERRORS
203
random over its surface. This thickness, transformed into unprocessed emulsion, turned out to be (134.0 + 4,4)/zm, the standard error being that of the individual value. The absolute value of the shrinkage factor seems to be of little importance, as it does not affect tile relative error of z determinations. We therefore admitted for it the rough value of 2, obtained in other experiments with plates of the same type, under similar processing2). For the same reason, we made no effort to determine the conversion factor of the microscope's depth screw from scale divisions into microns, but accepted the nominal value given by the manufacturer.
for h > 98/~m uniformity is obviously perturbed. Now, the lower fiducial plane is situated at ~ 32/~m from the even glass surface, i.e. at (~ 102.0 + 4.4) /an from the uneven, ondulated air surface. The loss of events for h > 98 #m is easily explained by the existence of such. undulations of the free emulsion surface, which are characterized numerically by the standard error of the unprocessed emulsion thickness (4.4/zm). It will be shown in the next section that elimination of the stars with h > 98/zm does not introduce any essential change in the detected up-down asymmetry. Thus, beyond any reasonable doubt this asymmetry eannot be ascribed to bias.
3. Discussion of Possible Bias
4. Distribution o f Vertical Projections o f ThC' a Tracks
Higher detection efficiency for stars nearer to the air surface would be a possible bias, able to produce the observed asymmetry, only if i~o fiducial planes (~ 32 #m from either emulsion surface) were used. Homogeneity of the detection efficiency with respect to the distance h between air surface and origin of a star in the unprocessed emulsion has been checked nevertheless in a simple manner, by means of the frequency distribution of h. This distribution would be nonuniform in the case of inhomogeneous detection efficiency.
The length z of the vertical projection of ThC' c~ tracks in the unprocessed emulsion has been determined, by direct measurement of z' in the processed emulsion. The origin of each star was placed in the centre ot the field of view, which was considered to be coincident with the centre of an eye-piece micrometer. The horizontal projection of the tracks was measured by means of this latter aN~ ¢ az
a~(M
20
[ ] ~'ev,,t~
$0 $
I -30
]Fig, I. D e p t h d i s t r i b u t i o n of T h C ' s t a r s .
Fig. i shows a plot A N (h)/zlh vs. h for 1527 stars located inside the fiducial planes. As can be seen, up to h --- 98 #tm (1428 stars) the h distribution is uniform (Pearson probability p22 ~ 27 %), while 2} M. Nicolae, p r i v a t e c o m m u n i c a t i o n .
-20
-[a
9
,40
*zo
1,.
*~70 z[jz.m)
Fig. 2. L e n g t h d i s t r i b u t i o n of v e r t i c a l projections of • t r a c k s from ThC'.
and the vertical projection by aid of the microscope's depth screw. All z' measurements were performed so as to eliminate the backlash of this screw. Fig. 2 is a plot of the frequency density zlN/Az vs.
204
D. A N G H E L E S C U AND J. S. A U S L A N D E R
z for a sample of 1727 events. T h e a s y m m e t r y coefficient b, defined as b = 2 ( N + -- N - ) / ( N
+ + N-),
(1)
where N + (respectively N - ) is t h e n u m b e r of e v e n t s w i t h z > 0 (respectively z < 0), t u r n s out to be b =
-
0.194
4- 0 . 0 4 9 ,
(2)
i.e. highly significant. F o r t h e c o m p u t a t i o n of t h i s b-value all events recorded as z = O, h a v e n o t b e e n t a k e n into account. If t h e tracks w i t h h > 98 # m are eliminated, t h e corresponding value of t h e a s y m m e t r y coefficient becomes b(h < 98~m) = - 0.174 + 0.051, (2') i n excellent a g r e e m e n t w i t h t h e v a l u e given b y eq. (2). I t seems v e r y s o u n d to a s s u m e t h a t t h e difference Ab --- - 0.018 + 0.071 is due to fluctuations a n d not to a s y s t e m a t i c displacement, since for t h e deeper lying stars one would r a t h e r expect a positive excess t h a n a n e n h a n c e d n e g a t i v e one. Consider now a plane i n t h e object space (fig. 3) ; i n t h e case of field c u r v a t u r e w i t h r o t a t i o n a l symm e t r y it will be r e p r e s e n t e d b y a c u r v e d surface in t h e image space (dashed i n fig. 3). L e t t h e origin O of a s t a r be placed i n C, i.e. i n t h e centre of t h e field of view. Then, all u p w a r d t r a c k OA will be seen as OA', i.e. its vertical projection will be recorded as z < Z, where Z m e a n s t h e t r u e v a l u e of t h i s projeetion; t h e same i n e q u a l i t y is v a l i d also for a d o w n w a r d t r a c k OB a n d its image OB'. A
.B'
/
Fig..3. Curvature of field of view (schematically).
I n absolute value, however, ] z ] < ] Z [ for upw a r d tracks (OA) a n d I z I > f Z ! for d o w n w a r d tracks (OB) a n d for strictly h o r i z o n t a l ones. There is one exception to this rule: Tracks w i t h small positive values of Z (actually upward) m i g h t be
recorded as cases w i t h small negative values of z, i.e. as tracks which seem t o b e directed downward. T h e general t r e n d is t h u s clearly a loss of u p w a r d tracks a n d a s i m u l a t e d excess of d o w n w a r d tracks. If this i n t e r p r e t a t i o n is correct, t h e n t h e distrib u t i o n of z will be s p r e a d on t h e negative side u p to higher absolute values of z t h a n o n t h e positive side. A c t u a l l y this features appears i n fig. 2. F u r t h e r m o r e , we m a y displace t h e axis of t h e figure to negative values, u n t i l a m i n i m u m of t h e difference b e t w e e n t h e n u m b e r of events to t h e r i g h t of t h i s new axis a n d t h e n m n b e r of e v e n t s t o its left is reached, i.e. practically u n t i l t h e new a s y m m e t r y coefficient b' becomes a p p r o x i m a t e l y equal to zero. I n fig. 2 t h e d a s h e d vertical line a t z = - 2.5 ~ m represents this new axis, for w h i c h b' =
-
0.006 +_ 0.048
;
(3)
this s t a n d a r d deviation is slightly smaller t h a n the one i n eq. (2), since here events w i t h z = 0 are not eliminated. Obviously, t h e small v a l u e of b' is n o t yet a n a r g u m e n t in favour of actual s y m m e t r y , since t h e new axis has b e e n chosen so, as to ensure a practically v a n i s h i n g b' - value. If t h e Z - values are actually s y m m e t r i c a l l y distributed, while t h e zvalues are n o t because of t h e c u r v a t u r e outlined i n fig. 3, t h e n t h e r i g h t h a n d p a r t of t h e h i s t o g r a m in fig. 2, a n d t h e left h a n d p a r t (right a n d Ieft u n d e r stood w i t h respect to t h e dashed, i.e. displaced, axis) m u s t show m i r r o r s y m m e t r y b e t w e e n each other, obviously w i t h allowance for fluctuations. This mirror hypothesis can be checked b y a X2 test 3) for consistency b e t w e e n t h e two halves of t h e histogram, which yields P X 2 ~ 25%. T h e same test, b u t r i g h t a n d left w i t h respect to t h e undisplaced axis, yields PX 2 m 0.03%. T h e e x p l a n a t i o n of t h e up-down a s y m m e t r y b y field c u r v a t u r e is definitely confirmed b y t h e i n d e p e n d e n t a n d direct m e a s u r e m e n t of this curvature, described i n sec. 6; f u r t h e r m o r e a decisive control experfinent with ~ tracks of ThC' is related i n sec. 7. Likewise the z m e a s u r e m e n t s of o t h e r prongs of T h stars, described in sec. 5, provide us w i t h f u r t h e r evidence in t h e same sense. ~) A. ~Iald, Statisticai Theory with Engineering Applications; quoted after Russian translation, Moscow 1956, p. 633.
F I E L D C U R V A T U R E AS A S O U R C E OF S Y S T E M A T I C E R R O R S
5. Distribution of Vertical Projections of Other ct Tracks of the Th Family I n a three- (or four-) prong star with ThC' the other c~tracks belong to ThA, Tn (and ThX) ; their ranges ( ~ 24 pm to ~ 32/am) show considerable overlapping, due to straggling. In the light of the preceding section it seems reasonable to investigate the up-down a s y m m e t r y of these tracks by classifying them with respect to the lengths x of their horizontal projections. Column 1 of table 1 lists three classes of horizontal projections x of ~ tracks from ThA, Tn and T h X , taken together; the fourth class, ThC', is included for comparison. Column 2 shows the updown a s y m m e t r y coefficients b. T h e y are not significant for the first two classes. For the last two classes, for which b is significantly different from zero, column 3 gives the axis displacement A z, for which a m i n i m u m of the up-down asymm e t r y coefficient b' is reached; the value of b' is given in column 4. The differences between a~ and ~ arise from elimination and non-elimination of the interval with z = 0. The figures of table 1 show clearly that tracks with large horizontal projections x are affected by curvature of the field of view (see also sec. 6), while the shortest ones are not; for intermediate values of x the effect seems to be present, without reaching in this experiment the usual significance level of 3 standard errors. 6. Direct Measurement of Field Curvature In N I K F I - R emulsions irradiated in Dubna with l0 GeV protons, several relativistic tracks were chosen, with dip angles less t h a n ~ 7 °. Consider two grains A and B of such a track, with a projected horizontal distance x between them, measured by means of all eyepiece micrometer (or in the case of the Zeiss KSM microscope, by the so called S-device for horizontal displacement of the field of view). If A is placed ill the centre of the field of view, a certain apparent vertical projection Z'A~ of the distance AB will be recorded by means of the microscope's depth screw; if B is placed in the centre the apparent vertical projection recorded will be z~A; generally s p e a k i n g Z~A # "AB*'"Both ZAB'and z~A' will be ex-
205
pressed in microns of processed emulsion and only their absolute values will be considered here. If [ Z ' [ is in absolute value the actual vertical distance between A and B in the processed emulsion, then
[ Z' [ = I I Z'AB t + a(X) [ = I I z~A J -T- A(X) I ,
(4)
where A (x) is the positive correction term due to field curvature. I t is supposed that A (x) does not depend on the azimuthal angle, or at least that it remains unchanged by an increase or decrease of 180 ° of the latter. This latter assumption has been checked experimentally for several pairs of distances + x and - x, approximately equal in absolute value. Only if
is practically equal to zero for all values of x, the field of view is to be considered as flat. A (x) has been measured with several combinations of microscope and optical equipment (listed in table 2) for different values of x. This was done by chosing a fixed grain A and different grains B1, B2 etc. at variable distances x; sometimes instead of A another, neighbouring grain A ' had to be chosen, in order to obtain a set of x-values reasonably spread over the whole diameter of the field of view. Figs. 4a and b are plots of A (x) vs. x for different microscopes and optical equipments; the symbols used in these figures are explained in table 2. Fig. 4a, concerning the Meopta microscope, contains also data obtained on a dipping ( ~ 14 °) track; evidently the dip is of n O importance for the value of A, which depends only on x. As can be seen from figs. 4a and b, the field curvature is often large enough, so that it must not be neglected. 7. Discussion 1. F r o m a qualitative point of v i e w it is not without interest to compare fig. 4 with the displacements A z given in table 1 and fig. 2. Transforming these Az (unprocessed emulsion) into Az" (processed emulsion) we have Az' = 0.75 pm and 1.25 a m for tracks with x /> 28.5/~m (say, on the a v e r a g e ~ ~ 31/an) and w i t h x ~>41 /am
206
D.
A N G H E L E $ C U AND
J.
S.
AUSLANDER
T~LE 1 A s y m m e t r y coefficients b and b' for c¢tracks due to ThX, Tn, ThA, and ThC" Horizontal projection x
tracks
ThX / Tn
x <~ 24.3/~m •
24.3/~m ~ x <
ThA j
+
28.5/~m
--
28.5/~m ~ . x
ThC"
Az
b X 10a
x ~ 41/2m
b~ × 10:l
32 -4- 39 127 4-
62
- - 2 6 2 4- 60
1.5/~m
+ 34 4- 57
--
2.5/~ra
--
Dip angle of track used*
Observation
194 4- 49
6±48
T~L~ 2 Signs used in figs. 4a and 4b
Sign
Manufacturer
1V~eopta (Prague) idem idem
×
o
e
+
C. Zeiss (Jena) idem E. Leitz (Wetzlar) Fratelli Koristka (Milano)
Binocular factor
1.25 X
Objective Oil immersion
Num. ap.
100 X
1.25 idem idem
idem idem
Eye-pieces
10 ×
N
idem 10 ×
N
14°
N
7°
4°
1×
90 X
1.30
15
2×
50 X monochromat 100 × apochromat 100 X
1.33
12,5 ×
1.32
10 ×
N 6°
1.25
12X
N
1.25 X IX
×
Eye-pieces 1.0.R. (Bucharest) idem
3°
Microscope mod. Ng.
4°
Nuclear emulsion microscope K.S.M. Microscope mod. Ortholux Koristka optics mounted on a Zeiss microscope, mod. Ng.
4°
* Dip angle in processed emulsion.
(ThC'; say, on the average x ~ 44 #m). F r o m fig. 4a we read the corresponding values A (31 pro) 0.65/~m and A (44/~m) ~ 1.3 pro. This agreement is a qualitative justification a posteriori of the axis displacements chosen in sees. 4 and 5. 2. As a final check for the explanation of the updown a s y m m e t r y as a field curvature effect, we have performed z' measurements on !000 ~ tracks of ThC', placing the end A of the track instead of its origin O (fig. 3) in the centre C of the field of view. If z' is considered positive for upward tracks, irrespective whether the origin O of the star or the end A of the prong is placed in C, t h a n we should
expect (in selfevident notations), bC__-O ~ -- bC~A,
(6)
with due allowance for slight differences in the absolute values, caused by reading errors. Experimentally we obtain for these 1 0 0 0 , tracks be= o = - 0.174 + 0.065 ; be_=A = + 0.163 + 0.065,
(7) in perfectly satisfactory agreement -with the expectation given b y eq. (6). 3. A numerical example concerning polar ang'te measurements of jet tracks in nuclear emulsion might be of illustrative value.
FIELD
CURVATURE
AS A SouRcE
Consider a jet with horizontal axis and denote the apparent polar angle of a jet track b y 8, the actual polar angle by 0, their projection onto the emulsion plane by 9, the apparent and the actual dip angle by $ and ~g; then evidently
(8)
cos ~ = cos q~ cos @ = cos q~ [1 + (kz'/l) 2]-~,
OF SYSTEMATIC
207
refer to the apparent direction of a track, as it is seen in the microscope before applying any corrections. As can be seen from fig. 5, the difference (8 - 8) between the true polar angle and the apparent one can reach non-negligibl e values, especially for small angles 8. Larger values of k increase this difference still more.
cos 0 = cosq~ cos T = cOS@ [1 + (k(z' + A)/1)2] -~ ,
(9)
ERRORS
, (0- 4-) ° 3 •~
_ _
* ~p# d','ttc~:s
where k is the shrinkage factor and l the projected distance at which z' is measured. ~0
(a)
20
30
40
~0
60
i
I/"
I
\
(/m~°
"
\
t,0
I tt
.2L
}
-3
$0
30
2O
40
~0
~o
\
.//
-f
I"1
J ~
Fig. 5. Difference b e t w e e n a c t u a l (0) a n d a p p a r e n t (8) polar a n g l e s of j e t t r a c k s .
8. Conclusions
i 0.~"
\
}
T+
(b) L
Fig. 4. ~ i e l d c u r v a t u r e m e a s u r e d on r e l a t i v i s t i c t r a c k s . (a) M e o p t a microscope; (b) O t h e r microscopes. Signs u s e d a r e e x n l a i n e d in t a b l e 2.
Considering e.g. a Zeiss KSM microscope with its optical equipment and taking l = 40/~m, i.e. A = 0.88 pm (see fig. 4b), k = 2.5 and different values of 9, we obtain the differences (0 - 0) as plotted in fig. 5. I n this figure " u p " and " d o w n "
1. Though manufacturers usually pay little attention to correction of field curvature*), this latter m a y become under certain circumstances of importance, giving rise to seriously misleading results. 2. For strongly dipping tracks the consequences of field curvature are practically negligible. 3. Most affected are flat or nearly flat tracks, which actually are directed horizontally or slightly upwards, but seem to be directed downwards. 4. There are several ways to avoid or to correct for the systematic errors introduced by field curvature, e.g. : a) to place the segment whose z coordinates are measured in such a manner in the field of view, that its extremities be located symmetrically with respect to the field centre; ~} see e . g . C . P . Sbillaber, P h o t o m i c r o g r a p h y in T h e o r y a n d P r a c t i c e ( N e w Y o r k a n d London, 1949) p. 212.
208
D. & N G H E L E S C U AND J. S. A U S L A N D E R
b) to measure the z coordinates twice, viz. once with one extremity and then with the other ex~ tremity placed in the centre of the field of view (recommendable only if the vertical noise of the stage is of little importance) ; c) to measure the curvature oi the field of view previously with sufficient accuracy, to establish an analytical expresion of the correction term, e.g. by the method of least squares for a parabolic dependence, or to plot it graphically, and to apply the corresponding corrections whenever necessary.
5. Since even high quality optical equipment is often uncompletely corrected for field curvature, this feature deserves more attention in emulsion work than it is usually given.
Acknowledgements We feel most indebted to M. Nicolae for her wholehearted assistance and for soaking and processing the plates, to E. NI. Friedliinder and Dr. R. Grigorovici for valuable discussions and to our scanning aid M. ~erban-Costescu for her able and patient work in scanning and measurements