COMPARISON OF THE SPECTRUM LINE WIDTHS OF Dy1 6 2 AND "NATURAL" Dy
S141961
Tony C. Hill
Submitted in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY May, 1961
Signature redacted
Signature of Author--- --- ---- --- - ---epart nt of Physics, May 20, 1961
Signature redacted
Certified by-f - - --- K- -
---Thesis Supervisor
Signature redacted
Accepted
by---Chairman, Departmental Committee onTheses by
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Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available.
Thank you.
The following pages were not included in the original document submitted to the MIT Libraries.
Page 9
-2-TABLE OF CONTENTS TITLE---TABLE OF CONTENTS--- ACKNOWLEDGEMENTS---I. INTRODUCTION---II. THEORY---III. APPARATUS---IV. RESULTS---V. CONCLUSION---TABULATION---- ---DATA SHEETS --- BIBLIOGRAPHY---Page 1 2 3 4 5 10 16 17 18 19 - 28 29
-3-ACKNOWLEDGEMENTS
The author wishes to thank Dr. K. Vander Sluis for expo-sing the photographic plate and for guidence in takeng the data.
The author also wishes to thank Professor L. C. Bradley for helping in organizing the theory and for advice in preparing this paper.
-4-I. INTRODUCTION
"Natural' Dy containing 6 isotopes of Dy is inexpensive compared to a single isotope of Dy. It is desired to study the
spectrum lines of Dy in a Zeeman size, magnetic field using an echelle spectrometer. My thesis is then to find out what kind
of differences do appear between "natural" Dy and, in this instance, Dy
-5-II. THEORY
Each isotope of Dy will have radiation specturm lines which may be broader or shifd from each of the other isotope lines, (if the shift is less than the width of a line, it will appear as a broadening of the combined patterns). The broadening, due to nucleus differences, will take place through one or more of the following mechanisms.
Each nucleus has a potential associated with it, and it is this potential which appears in the Schroedinger equation, so that solutions (eigen values) of the equation will be different due to a different potential. Since each isotope of Dy has a slightly different shape and wieight due to extra neutrons, the energy levels (solutions eigen values) will be blightly different from isotope to isotope. The effect of the difference in potential is reduced even further by the fact that for most paths of the elec-trons the nucleus will be screened from the outer elecelec-trons by the $sed shells of inner electrons. Electrons whose paths have great eccenticities will "pass through" the completed shells and will notice the difference in nuclear potential. Consequently, electrons moving close to the nucleus will have their energy
levels changed. However, the selection rules governing the exci-tation of electrons are such that electrons may only jump to orbits which are fairly similar to the original orbit. But if both orbits are alterged by a change in potential by a similar amount then
the energy radiated (difference in energy level between the two orbits) will be the same for a similar but slightly different sets of potentials.
Now that we have associated with each isotope a set of energy levels Ei, we see by the uncertainty principle that there is still
an uncertainty of "knowing" these levels which is induced by the finite time we have to measure than
h
AT-where -r
=
finite measurement time. Therefore we get an uncertainty in the emission frequencies since the frequency is proportional to the difference in two energy levels. Fortunately, this natural uncertainty gives a frequency distribution whose half width is two places smaller than the half widths from other factors.One of the other fractors which predicts a half width to the distribution function on the order of those half widths observed is the well known Doppler broadening. If a moving object gives off radiation of a certain frequency, a stationary observer will see a different frequency.
u vobserved - vobject
=
v=
where u is the velocity of emitting particle towards or away from the observer. If the gas distributions function is Maxwellian, then
-Mu -A V Ijk
f(u) = Ae =Ae with a half width of 1. 67 kT
-7-Still another phenomena giving broadening is collision damping. If two molecules collide while one is radiating energy, then the phases and amplitudes of the various components in the radiated wave will be changed. By Fourier analysis we can derive a distribution function for v if we assume the collision time is negligible compared to the mean time between collisions.
Io
-r = mean flight timehalf width=-0
= 4 Np 2 d R p = mole, cross section
m
Also connected with the collisions is another broadening phenomena. During the collisions the potentials of the excited particles are changed due to the presence of the other excited particles. Consequently the energy levels are changed and the emitted radiation is shifted. Thus the collision damping
distribu-tion becomes asymmetrical.
Tumns
aTCOI-a~~~ts~~o'at + - AE&_i
As the pressure goes up, the number of collisions per second goes up, and the asymmetry becomes more pronounced.
Comparing the above effects we see that the half widths of the line do not vary faster than m3/2 so that since m 160 the widths of the lines of the various isotopes will start to vary in the
-8-3rd or 4th place. However, the finite widths of the lines become more important if there is a shift in some of the isotope lines because the overall change in width will be proportional to the widths and to the frequency shift (see below).
There is another cause of broadening which appears in some isotope lines and not in others - hyperfine structure. There is an interaction of the spin of the nucleus with the magnetic field caused by the rotating electrons. The result is that with each energy level field of an isotope in a mag. field characterized by
I
quantum numbers, there is also another set of energy levelsassociated with the energy of the spin of the nucleus and charac-terized by quantum numbers I and m. The actual splitting due to the nuclear moments is too small to be seen in this experiment but shows up as broadening. Even number isotopes have no spin, odd number isotopes will have a spin which varies from isotope to isotope, so that there will definitely be a difference between the
hyperfine broadening of Dy 16 2 (no spin, no broadening) and "natural" Dy (various spins, various broadenings.)
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MISSING PAGE(S)APPARATUS
A tube containing powdered Dy162 or "natural" Dy is excited by some means in a Zeeman sized magnetic field. The radiation from this tube perpendicular to the field is passed through a small slit (see Fig. 1 and 2)
1P
40N , AMD
/ Fig. I - ~~ G-c ' &iDA ~1 ~ACP~oAJ -10~ III. P4v 1A,
T16-M 'Fig. 2
-11-The slit is at the focal point of a large mirror so that when the waves from the slit reach the mirror, they are reflec-ted as a plane wave. The resultant plane wave is then directed onto the echelle at verticM angle i . The echelle is designed so that each step of the echelle acts alnost like a'mirror in that it re-flects back most of the light at an angle -i 0 to the horizontal
(see Fig. 3)
-12-The reflected wave has an intensity distribution as in Fig. 4. The result, of course, is that in front of the diffrac-tion grating the waves from each of the steps of the echelle interfere to give an interference pattern for each wave-length according to C'>g3;' +x y for max. intensity.
A
Since the intensity distribution from the individual steps of the echelle is closely grouped around anglei 01, only the max-imum corresponding to m = 1 will have a significant intensity. The result is that at the diffraction grating we see but
a single horizontal line at angle i - i .
When this pattero consisting of one line, hits the grating, it is likewise diffracted according to a similar formula
Q ci WLLFb
7
(21
for max.
where a and b are constants of the grating. At the same time the rays of light are now focused at the plate by the
curvature of the grating.
nos7 Z0 rrTY 0 F
-13-We can see from Fig. 1 and 2 that X = A sin (i" - i0)
Y=Bsin(i' - i
so by making i"- i 0 0 and i I- 10 small we find that
for any X
which gives us for white light a pattern like this on the plate
Also by use of Taylor series we can also get:
where m X C Iand C2-- etc. are constant and can be found from
frequencies which are easily identifiable.
The above analysis was done for a point source. The slit dimensions are small enough so that it can be considered a number of points side by side. Thus an image of the slit shows up on the photo plate. In the experiment half the slit was covered while the tube of Dyl 6 2 was used and the other half covered while natural
Dy was used so that the two spectrum were displayed side by side on the plate.
-14-Error.
The X on the data sheet are within 1 mm. of the actual distance on the plate and are used only to locate the lines rather than to calculate the actual wavelengths.
Errors in the echelle plate and the diffraction grating may have produced errors in the line spectra, however, since the
wavelength differences between Dy 6 2 and natural Dy are so small) there would be no noticable difference in the final pattern as it
falls on the plate. There was a noticable shift of one side of the spectra. This can be explained in that there was no way to con-trol the pressure in the tube other than to try to concon-trol the energy supplied by the exciting apparatus, thus producing a collision
damping difference. A difference in pressure outside would change the optical length and also the diffraction pattern.
The differences which we would like to measure are differences in size and shape of the distribution of intensities versus frequencies curve. Because of the characteristics of the photographic plate and the crude method used to take the data from
it, we are limited to results of the kind which give an apparent change in line width or increase of background (see diagram below).
-15-Consequently, it would be rather easy to get different results from side to side merely by increasing the intensity of light or
exposure time on one side. What this means is that in judging the width of the lines, we can not use the results to varify the
original prediction that the "natural" Dy side (right side) be broader, but we must assume that our predictions are well founded, and
the theory, at least, is varified by previous experiments. What we can realize is that the exposure times and intensities were kept as close as possible during the experiment. Also, from the
data we find several lines which are darker on the left side Dy1 6 2 than1thi right, so that according to our theory this can be due only to an overexposure of the left side. Then, if anything, the right side lines should be wider. However, there are so compara-tively few lines which show a broadening of the left side, we are justified in thinking that the exposures were approximately equal.
The lines on the plate were almost always grouped together so that the lines formed a triplet of small groups of lines.
-16-IV. RESULTS
As can be seen from the data sheets a pattern of blurring appeared.When the lines were well spaced and narrow, no differ-ence was seen from side to side. The broader the line, the more difference could be seen. This was reasonalbe from the two
points of view. First, wide lines and correspondingly wide differ-ences are easier to see with the naked eye, and secondly, wide lines usually indicate large J and m quantum numbers which afford more lines to give blurring. When there appeared a great many narrow lines closely spaced, the differences from side to side became quite noticeable. The backgrounds almost always increased and there was usually blurring on the darker parts of the natural Dy pattern. So, as we might expect in this best of all possible worlds, the more interesting patterns become more obscured.
-17-V. CONCLUSION
As an example of what I classified as no difference, I refer to any of the lines so classified in the data sheet and on the photographic plate. The most blurred pattern was that aroundXA X1z0&0,5 As for the use of Dy162 or natural Dy
that is left up to the future experimenters who can refer to the data or the photographic plate. However, a 'quick glance at the plate or data will show that practically no lines are "lost" from side to side, so that in experiments of moderate accuracy
-18-TABULATION
Total lines 273 orN usable lines 20
Lines definately darker on left side 10 Lines with no difference -154
Percent lines of no difference to usable lines 62 Lines of light backgrounds 46
Lines of heavy background or flurring 40
Percent lines of heavy background increased or
m k 0 607856 C1 C 2 No No No 26.2 Comment difference difference difference
Same background towards top lines Background on outside edges
Same background on bottom lines 3393. 3393. 3393. 3395. 3396. 3396. 3407. 3407. 3407. 3408. 3413. 3413. 3413. 3414. 3433. 3434. 3434. 179 179 179 179 179 179 178 178 178 178 179 178 178 177 177 3395. 3395. 3395. 3414. 3414. 3414. 3414. 3414. 3414. 3414. 3414. 3434. 3434.
at 3440 there are strong Dy lines e even stronger Fe lines 3445.3 3434.2 177 3445. 3445. 3446. 3445. 3445. 3445. 3447. 6 7 0 4 7 75 0 3434.2 3434.2 3434.2 3453.7 3453.7 3453.7 3453.7 Around to 177 177 177 176 176 176 176 176- + analyze. 75 77 77. 79. .148 .148 .148 .148 .149 .149 .149 .149
To many other lines to make judgement
No difference No difference
Bottom of plate, Bad right side 5 5 56. 54. 54.5 51. 5 Other lines Slight blurring No difference
0 - 10 there are some lines which are dark but hard
x m C1 m AX 18.5 15.0 No No No No No No No No difference difference difference difference difference difference difference difference 5 5 5 0 8 7 5 .146 .146 .146 .147 .147 . 147 . 147 .147 .147 148 148 Increased background
Background on outside edges Increased background 1. 4. 5. 4. 4. 4. 9. 7. 7. 5. 6 8 8 2 9 3 7 5 5 48 00 2 .8 .3
I m m AX Comments 0 mn 3456. 1 3453.7 176 . 149 16 No difference 3456.4 3453.7 176 . 149 18 No difference 3456.8 3453.7 176 .149 21 3460.5 3453.7 176 . 149 45.5 No difference 3460.9 3453.7 176 . 149 48.5 No difference 3461-.4 3453.7 176 . 149 51. 5 Extra Line
3468 3473.5 175 .149 10=30 Many, many lines of which is
same background over most of
lines Rt. side 3473.4 3473.5 175 .149 .5 No difference 3473.65 3473.5 175 . 149 1 No difference 3473.9 3473.5 3 No difference 3473.6 3473.5 175 . 149 21. 5 No difference 3477. 1 3473.5 175 . 149 24 No difference 3477.4 3473.5 175 . 149 26 No difference 3494.0 3493.4 174 .150 4 No difference 3494.2 3493.4 174 . 150 5.5 No difference 3494.7 3493.4 174 .150 8.5 No difference 3498.2 3493.4 174 . 150 32 Extra Line 3498.7 3493.4 174 . 150 32 No difference
3499.0 3493.4 174 . 150 37 Possible slight blurr
3504.1 3493.4 174 . 150 71 No difference
3504.4 3493.4 174 .. 150 74 No difference
3504.9 3493.4 174 . 150 76 No difference
3504.2 3513.6 173 .151 61
3506.5 3513.6 173 . 151 47 black on top
3507.0 3513.6 173 . 151 44 Black on outside edge 3507.4 3513.6 173 .151 41 Black on bottom
x 0 Comments 3519. 1 3513.6 3523.7 3513.6 3523.9 3513.6 3524.3 3513.6 3531.4 3534.0 3531.7 3534. 0 3532.0 3534.0 3532.5 3534.0 3532.8 3535.1 173 173 173 173 172 172 172 151 151 .151 .151 152 .152 .152 24 67 68- 69 71 17 15 13 3 5. 5 7. 5 3535.5 3535.9 3536. 1 3538. 0 3538.3 3538.6 3538.8 3546.4 3554.7 3546.9 3554.7 3547.4 3554.7 3549.8 3554.7 3550.0 3554.7 3550.4~ 3554.7 3550.7 3554.7 3551.3 3554.7 3551.6 3554.7 3551.9 3554.7 171 171 171 171 171 171 171 171 .153 . 153 . 153 . 153 .153 . 153 .153 10 12.5 14 26.5 28 30 31. 5 54 51 48 32 30 28 26 22 20 .153 18 No difference No difference
Too many other lines
No difference No difference No difference No difference No difference Other line s
Smearing, increased background Blurring on inside edge
Increased background
Smearing, increased background Blurring, increased background
I 'II
I t
I t t I
Slightly increased
Background, prac. no blurring
t t t f I t I
No difference No difference No difference
Cl
k m AX Comments
3562.1 55 Here 2 Dy patterns overlap
--The stronger appears as a triplet whose top line right side is made up of many lines and whose top line left side is blurred together. The second pattern has many line equally spaced and unchanging from side to side.
3573.4 3575.6 170 .154 14
3573.9 3575.6 ,170 .154 11 No difference
3574.2 3575.6 170 .154 9
3573.8 3575.6 170 .154 12 Heavy background
3574. 1 3575.6 170 .154 10 Blurring on light lines
3574.4 3575.6 170 .154 8 Triplet of many lines
3575.7 3575.6 170 .154 .5 Heavy background
3576. 0 3575.6 170 .154 2.5 Blurring on light lines 3576.5 3575.6 170 . 154 6 Triplet of many lines
3576.5 3575.6 170 .154 6
3576.8 3575.6 170 .154 7.5 No difference
3577.1 3575.6 170 .154 9.5
2 overlapping Dy patterns with many
3585.6 65 Fe lines, one Dy seems unchanged
3589.7 3596.7 169 155 45 Sharp 3590. 0 3596.7 169 155 43 No difference Dark 3590.3 3596.7 169 155 41 Lines 3590.5 3596.7 169 155 40 3590.7 3596.7 169 155 39 3591. 0 3596.7 169 155 37 Light background
3591.4 3596.7 169 155 34 Light background - blurring top edge 3591.8 3596.7 169 155 31.5 Heavy background
3592.0 3596.7 169 155 30 No difference
Comments
Heavy background, blurring Heavy background, blurring Heavy background, blurring Heavy background, blurring 3594.5 3594.8 3595. 1 3595.5 3605.7 3606.0 3606.3 3612..6 3612.9 3613.2 3623.8 3624.0 3624.6 3629.0 3629.4 3629.9 3630.2 3630.5 3630.7 3632.5 3633.0 3633.3 3632.8 3633..3 3633.5 3643.4 3643.8 3644.3 3596.7 3596.7 3596.7 3596.7 3596.7 3596.7 3596.7 3618.2 3618.2 3618.2 3618.2 3618.2 3618.2 3618.2 3618.2 3618.2 3639.9 3639.9 3639.9 3639.9 .3639.9 3639.9 3639.9 3639.9 3639.9 3639.9 3639.9 3639.9 169 169 169 169 169 169 169 168 168 168 168 168 168 168 168 168 167 167 167 167 167 167 167 167 167 167 167 167 x 0 .155 .155 .155 155 155 .155 .155 .156 . 156 .156 .156 .156 .156 .156 . 156 .156 . 157 . 157 . 157 .157 .157 .157 . 157 . 157 157 157 157 157 m 14 12 10 8 58 60 62 36 34 22 36 38.5 41 69 72 75 61.5 60. 0 58.5 47 44 42 45 43 41 22 25 28 No difference No difference No difference
Light background, blurring Light background, blurring Light background, blurring
No difference
No difference
Very light background No difference No difference No difference 2 Dy lines together CI m Sharp Dark Line s AX No difference
Comments 3643.5 3643.9 3644.3 3645.0 3645.4 3645.8 3649.9 3650.2 3650.6 3671.8 3672.4 3672.9 3676.4 3676.7 3687.0 3694.6 3694.9 3695.0 3695.3 3697.8 3698.2 3698.4 3648.*7 3716.4 3716.9 3717.3 3730.2 3730.6 3731.0 X 0 C 1 I AX 3639.9 3639.9 3639.9 3639.9 3639.9 3639.9 3639.9 3639.9 3639.9 3661.8 3661.8 3661.8 3684.0 3684.0 3684.0 3684.0 3684.0 3684.0 3684.0 3706.4 3706.4 3706.4 3706.4 3706.4 3706.4 3706.4 3729.2 3729.2 3729.2 m 167 167 167 167 167 167 167 167 167 160 160 160 165 165 165 165 165 165 165 164 164 164 164 164 164 164 163 163 163 157 157 157 157 .157 .157 .157 .157 . 157 .158 .158 .158 .159 .159 * 159 .159 .159 .159 .159 .160 .160 .160 . 160 .160 .160 .160 .161 .161 . 161 22 25.5 28 32.5 35.0 37.5 64 66 68 63 67 70 48 46 44 66.5 68.5 69.0 71.0 54 51.5 50 48 63 65.5 68 6 8.5 11 No difference No difference 2 Dy lines together No difference Dark background Broadening of lines
Possibly due to extra lines Very slight increase
Background, slight Blurry of middle set Slight background Increase in dark Sections
No difference
Light background , blurring Light background , blurring Light background, blurring Light background , blurring
No difference
Comments 3739.0 .3729.2 3739.3 3729.2 3739.7 3729.2 3740.1 3729.2 3740.5 3729.2 3753.0 3752.2 3753.5 3752.2 3753.8 3752.2 3767.8 3775.5 3768.2 3775.5 3770.8 3775.5 3771.1 3775.5 3771.4 3775.5 163 163 163 163 163 162 162 162 161 161 161 161 161 161 161 .161 .161 .161 .162 .162 .162 .163 .163 .163 .163 . 163 3781.0 3781.4 3781.9 3785.6 3786. 1 3786.6 3805.7 3799.1 3806.2 3806.6 3816.8 3823.0 3817.1 3817.4 3836.2 3847.2 3836.7 3837.1 3847.2 160 159 158 .164 .165 .166 61 63 65 68 70 5 8 10 47.5 48 29 27 25 34 36.5 39 62 65 68 40.5 43 46 39 37 35 66 63.5 61 No difference No difference
left side wider and darker No difference
No difference Left side darker No difference No difference No difference No difference No difference Extra Fe line No difference No difference 0 m C m AX
Cly x m AX Comments 3852.5 3847.2 32 3852.7 3847.2 34.5 No difference 3853.2 3847.2 36 3871.6 3871.7 157 .167 .5 No difference
3872.0 2 Slight broadening - especially
of lower line
3872.5 4.5 No difference
3898.0 3896.5 156 . 168 9 Many lines in triplet, heavy 3898.4 3896.5 156 .168 11.5 Increase in background 3898.9 3896.5 156 .168 14.6 Increase in background
3930.5 3921.7 155 .169 56.5
3930.7 58 No difference
3931.1 60.5
3944.1 3947. 1 154 . 170 17 Although the re are (2) two
3944.5 15 overlapping lines from others
3944.7 14 it looks like
3945.2 11 no difference
3967.9 3972.9 153 . 171 29 Right side Broadened
3968.5 26 No difference
3969.0 3972.9 153 . 171 23 No difference
3978.0 3972.9 153 . 171 30 2 Dy triplet patterns overlapped
3978.5 3972.9 153 .. 171 33 One has many lines
3979.1 3972.9 153 .171 36 Light background increase
3981.5 3972.9 153 .171 50 3981.8 3972.9 153 .171 52 No difference 3982.0 3972.9 153 .171 53 3982.3 3972.9 153 .171 55 3983.2 3972.9 153 .171 60 3983.6. 3972.9 153 . 171 62.5 No difference 3984.0 3972.9. 153 .171 65
Comments 3999. 4000. 4001. 4045. 4046. 4046. 4050. 4050. 4051. 4072. 4073. 4073. 4077. 4078. 4078. 3999. 1 8 5 2 6 1 6 1 6 2 7 3 8 5 0 5 4052. 4052. 4052. 4052. 4052. 4052. 4079. 4079. 4079. 4079. 4079. 4079. 4 4 4 4 4 4 6 6 6 6 6 6 4102. 7 4107. 1 4104.4 4107.1 4110. 4111. 4111. 4111. 8 0 3 7 4135.8 4136.1 4136.5 4167.4 4167.9 4168.3 4107. 4107. 4107. 4107. 152 150 150 150 150 150 150 149 149 149 149 149 149 148 148 148 148 148 148 147 1 1 1 1 4135.1 4163.4 146 .172 .175 .175 . 175 .175 .175 .175 .176 .176 .176 .176 .176 .176 .177 .177 .177 .177 .177 .177 .178 .179 4 8 12 39 36 33 13 0 7 39 36 33 12 9 6 25 25 21 22 24 26 60 62 64 22.5 25 27.5
Many lined triplet
Heavy background and blurring on upper 2 lines, middle left is darker and sharper than middle right.
Although there are some extraneous lines, the left side definitely is broader Many lined triplet,
Left side lines are sharper
Light increase in right backgroun
No difference overlapping
Many lined, left side sharper Right side slightly blurred
Manyg, many lines left side sharper, darker, right side Blurred, light background Many lines
Left side sharper, right side blurred No difference No difference x0 m CI Thi AX
Comments 4181. 4182. 4185. 4186. 4186. 4187. 4194. 4194. 4195. 4195. 4211. 4211. 4212. 4213. 4213. 4214. 4215. 4213. 4217. 4218. 4218. 4220. 4221. 4221. 4224. 4225. 4225. 6 5 6 3 9 4 3 5 0 4 2 8 3 6 9 6 2 7 6 1 7 5 0 6 7 0 7 x0 C Im 4192. 4192. 4192. 4192. 4192. 4192. 4192. 4192. 4192. 4192. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. 4221. AX 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 145 145 145 145 145 145 145 145 145 145 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 . 181 .181 .181 . 181 .181 .181 . 181 .181 .181 .181 m 58 53 36 32 29 26 12 13.5 16 18
Many lines, Left side sharper
Left side
slightly darker - both very black
Many lined, quad. Left side sharper and
darker
Very dark No difference
Left side sharper Left side sharper Many lined triplet, No difference No difference
Left side darker, right side blurred
No difference Left side darker No difference .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 .182 0 5 55. 51. 49 47 40 No difference 36 33 30 20 17 14 4 1 2 19 21 23
-29.-BIBLIOGRAPHY
The theory of braodening was taken from
