NOISE ASSOCIATED WITH ELECTRON STATISTICS
IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF aPARTICLES
By
JENG GONG
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1983
ACKNOWLEDGMENTS
I am deeply indebted to Dr. Carolyn Van Vliet, Dr. Aldert
van der Ziel, and Dr. Peter Handel for their most generous and valuable guidance and assistance during the preparation of this work. I thank Dr. Alan Sutherland, Dr. Eugene Chenette, and Dr. Gys Bosman for their advice and encouragement. Dr. Wiliam Ellis has been most helpful during many phases of this work and his help, especially in discussions of radiation detection techniques, is greatly appreciated.
I also wish to express my appreciation to Dr. R. J. McIntyre and Dr. Dean Schoenfeld for providing RCA diodes and the Argonion Laser, respectively. I am particularly indebted to Mrs. S. L. Wang for the preparation of the figures presented in this work.
My deepest gratitude goes to my parents, my wife, and my son who have always had faith, encouragement, and understanding when it was most needed.
This research was supported by AFOSR contract, #820226.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ......... ......................... . i.. ii
ABSTRACT ............. ............................. v
CHAPTER I: INTRODUCTION ....... ....................... 1
PART A
NOISE FROM HOLEINITIATED
PHOTO AVALANCHE PROCESSES
CHAPTER II: THEORY OF NOISE IN AVALANCHE DIODES .... .........5
CHAPTER III: DEVICE DESCRIPTION AND EXPERIMENTAL SETUP
FOR MEASURING PHOTODIODE NOISE ... ........... ...20
CHAPTER IV: RESULTS FOR HOLEINITIATED AVALANCHE CURRENTS ..... 29 PART B
EMISSION STATISTICS OF PARTICLES CHAPTER V: THEORETICAL PERSPECTIVES ..... .............. ... 49
5.1 Handel's Theory ..... .............. ... 49
5.2 The Allan Variance Theorem ............... 52
5.3 Application of the Allan Variance
Theorem to Counting Statistics ...... 54 CHAPTER VI: EXPERIMENTAL SETUP FOR LONGTIME COUNTING ..... 57 CHAPTER VII: COUNTING RESULTS ......... .................. 62
7.1 Convergence of the Allan Variance
for N  . ................62
7.2 Results of the Relative Allan
Variance vs. l/T ..... ............. 62
7.3 Results of the Allan Variance vs. T . . .. 66 CHAPTER VIII: DISCUSSION OF RESULTS ..... ................ ...76
CHAPTER IX: APPENDIX:
REFERENCES
BIOGRAPHICAL
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK ..... .................
EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS
SKETCH . . . . . . . . . . . . . . . . . .
Page
87 91 98 100
Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOISE ASSOCIATED WITH ELECTRON STATISTICS IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF aPARTICLES
By
Jeng Gong
August 1983
Chairperson: C. M. Van Vliet
Major Department: Electrical Engineering
It is known that the standard theories of avalanche statistics for twocarrier impact ionization by Tager, McIntyre, and Personick only deal with processes for which the number of possible ionizations is very large. On the contrary, it is believed that in many modern devices the number N of possible ionizations is finite and perhaps even very small (N = 1  5).
In 1978, van Vliet and Rucker developed a new theory forthis
small N case, which has been fully confirmed for the electroninitiated ionization process. Part A of this dissertation presents the results of a detailed noise study with holeinitiated avalanche currents which corroborate van Vliet and Rucker's theory for the second time.
Data obtained from extensive measurements on counting techniques for aparticles radioactive decay from 95Am241 are presented in Part B
of this dissertation. These data have shown that the statistics are nonPoissonian for large counting times (order 1,000 minutes) in contrast with the fact that many textbooks cite adecay as an example for Poisson statistics.
Detailed measurements of the Allan variance indicated a
"flicker floor" due to the presence of 1/f noise in the decay, of 10 This result is in agreement with Handel's quantum 1/f noise theory. If upheld by further measurements, then this would be the first quantitative indication that 1/f noise is caused by emission of long wavelength infraquanta, such as soft photons causing minute inelastic losses in the scattered wave packet.
CHAPTER I
INTRODUCTION
Avalanche multiplication in devices occurs when the free carriers gain enough energy in an electric field so that they can ionize bound carriers upon impact. This phenomenon has been observed in a number of devices, such as reverse biased pn junctions, impact diodes, Read diodes, FET's, etc. Most authors consider the ionization process as a continuous process [16], which means that either the region over which the avalanching occurs is very long, or the number of ionizing collisions per primary carrier transit N is infinite or at least very large.
Ionizations can occur due to electron or hole impact. The
primary carriers responsible for the ionization can be either thermally generated, or arise from tunneling (as in Zener diodes), or stem from light absorption; this is the case in avalanche photodiodes, as considered in these studies.
For presentday small dimension devices, the number of ionizations per carrier transit N is small. In 1978 van Vliet and Rucker used a new statistical approach, referred to as the "method of recurrent generation functions," to develop a new theory which is valid for an arbitrary number of N [7,8]. This theory has been corroborated for electroninitiated ionization process [9].
Measurements with holeinitiated avalanche currentaremade and investigated; the results will be discussed in PartAof this dissertation.
Whereas most of the noise phenomena, like shot noise, thermal noise, and generationrecombination noise, are well understood, 1/f noise remains an enigma. This noise has been observed almost everywhere: semiconductor devices, music, traffic flow, hourglass flow, the frequency of sunspots, the light output of quasars, etc. Because of its universality, some investigators believe that there must be some universal phenomenon operative in all these manifestations [1012].
One of the theories of 1/f noise is the quantum theory, based on the infrared divergence phenomenon and developed by Handel [10]. This theory is fundamental in the sense that it derives the 1/f spectrum from basic quantum physics at the level of a single charged particle subject to scattering, although the final result depends essentially on the presence of many carriers, making 1/f noise similar to diffraction patterns which are oneparticle effects but can be seen only if many particles are diffracted. In addition, the theory is universal in the sense that any infraquanta with infrareddivergent coupling to the current carriers will give a contribution to the observed 1/f noise proportional to their coupling constant. Such infraquanta are, for example, very low frequency photons, various types of phonons, shallow electronhole pairs on the Fermi surface of a metal, spin waves, correlated states [11], etc.
Recently, this theory has been reformulated with quantumoptical terminology and compoundPoisson statistics in a paper written by Van Vliet, van der Ziel, and Handel, which led to the idea of
verifying the theory on a "clean" system outside the domain of solidstate physicsradioactive adecay [13].
The purpose of this study is to examine the existence of 1/f
noise in radioactive decay from 95Am241 cparticle source. The results are presented in Part B of this dissertation.
PART A
NOISE FROM HOLEINITIATED PHOTO AVALANCHE PROCESSES
CHAPTER II
THEORY OF NOISE IN AVALANCHE DIODES The first theories for noise in avalanche diodes were given by Tager (1965) [1] and by McIntyre (1966) [2]. Tager assumed that the ionization coefficients for ionization by electrons and holes are equal. Under these conditions the noise spectral density is given by
Sld = 2q prM3 (21) where Id is the avalanche current, Ipr is the primary photocurrent, q is the electronic charge, M is the average gain, and where it is assumed that the incoming particles show full shot noise.
The theory by McIntyre allowed for the possibility that the ionization coefficients are unequal. Let c(x)dx be the probability for ionization by an electron in the interval (x,x+dx) and let (x) be the ionization probability by a hole. For simplicity, McIntyre assumed (x) = kc(x). The noise spectral density is then given by
Sd = 2qIprM3 [ (1  k)[M ) (22)
for the case that the primary carriers injected into the avalanche region are electrons; in case these carriers are holes, replace k by 1/k; if both occur, add the two partial results.
It is not necessary to commit oneself to Poisson statistics of the primary particles; so, one can consider immediately the statistics of the avalanche process. Thus let X denote the offspring (plus original carrier) due to one incoming primary carrier. Then, according to the variance theorem,
SId = M2S + 2q1 var X (23)
where M = . From this we find that (21) and (22) are tantamount to
var X = M2(M 1) (Tager) (24) and
var X = M(M ) + k(M 1)2M (McIntyre) (25)
The above theories assumed that the region over which avalanche occurs is very long, so that the number N of possible ionizations per carrier transmit is very large. Therefore, these theories can be referred to as asymptotic theories. Lukaszek et al. at the University of Florida (1974) were the first authors to show that this assumption is unrealistic [14], in particular for the region of onset (low gain) of avalanche multiplication. They developed a theory forthe case that N = 1 and N = 2. Besides for short avalanche regions, these formulas should always be applicable at the onset of avalanche ionization. For N = 1, their results give
var X = M(M 1) + 2i (Lukaszek) (26)
where X is the a priori chance for ionization by an electron after the electron has gained enough energy to ionize, whereas 11 pertains to ionization by a hole.
In 1978 van Vliet and Rucker reinvestigated the problem. A new theoretical method was developed, named the "method of recurrent generating functions" [7,8]. By this method they were able to solve the complete problem, in which N possible ionizations per carrier transit, or per traveling holeelectron pair (the electron and hole going in opposite directions) are possible. In this case the basic parameters are the a priori ionization probabilities by electron impact (X) or by hole impact (p), once these carriers have covered a path which is long enough to gain the necessary ionization energy from the electrical
field.
For the case that the primary carriers injected into
the avalanche region are electrons, the following results are found
N
e ,e> ( + X)(l  k)
MN N (1 + kX)N I  k(l + X)N+l
MeM )1 k) 1
e <(AXe)2> = N )(I k) + 1  kA2 var X  2 + + k 1 + k
,[Mek 1 1 (28)
The asymptotic limit for N . is not trivial; in the paper of van Vliet and Rucker it is shown that (27) and (28) lead to McIntyre's results.
For hole initiated avalanche case, replace k by I/k, X by i, and 1i by X. The results are
h. h h (1 k)[k(l + ji)]N
MN
k (l + _ (k+ )N+l (29)
M h(Mh r k (1 2
var X h <(AX )2> N N ï¿½ k )  k)  2 k + p N N ~2k + pk +
x + I _] (210)
We now consider Equations (29) and (210) in more detail. At the onset of ionization the field is just high enough to sustain one possible ionization in the avalanche region; thus N = 1. With increasing field, the value of p, denoted as 1(l), increases and so the gain increases according to Equation (29). The length 1 necessary to gain the ionizing energy simultaneously decreases, until two ionizations per carrier transit are possible. The value of 1(l), just prior to this is denoted as (l) max' When two ionizations per carrier transit are possible, the diode switches over from the regime N = 1 to the regime N = 2. To realize the gain M of that operating point, .the value of i for the new regime, denoted as p(2),is considerably less than the value of i(l)max prior to the switchover, as is found by inverting (29) for 1 with M fixed, taking N = 1 and N = 2,
respectively. See Figure 21. At the M value for which the switchover occurs, the variance jumps from var X to var X2. Equation (210) indicates that the decrease in p and the increase in N cause var X to make a positive jump at this particular M, which we call a breakpoint value.
If now the field is further increased, the paths Z1 + k2
decreases, p(2) increases, and M increases, until at p = p(2)max three ionizations per carrier transit are possible. Once the regime N = 3 is possible, the p(3) for these processes is much smaller than 11(2) in max
order to realize the same M. The switchover marks another breakpoint value for M, at which the variance jumps due to the reduction of 11, from
(2) max to 11(3), etc. We thus obtain an overall curve of var X versus M, which shows discontinuities at the breakpoint values of M, for which the regime switches N N + 1. In order to construct this overall curve we need only plot Equation (210) with p expressed as li(M,N) according to the inversion of (29), where N = 1 and a = i(l) up to
(1)max ; next N = 2 and 11 = p(2) up to p(2) max, etc.
We conjecture that p(N)max is related to the pole value,
1p(N), for which the denominator of (29) is zero. These pole values are listed in Table 21. On general ground we assumed that the p max is related to 1p by
_ N p(N) (211) (Nmax N + 5.
The reason for this presumed relationship is that for very large N we must have p(N)max + (N) in order thatM can rise without saturation.
Figure 21. p vs. M
Table 21. hpole(N) and max(N) for different values of k, all values x 106
N
k 1 2 3 4 5 6 7 8 9 10
Ppol e
0.02 141,500 57,960 35,050 24,820 19,130 15,520 13,040 11,240 9,868 8,793 0.016 126,500 50,720 30,360 21,370 16,400 13,270 11,130 9,575 8,398 7,476 0.014 118,322 46,807 27,833 19,520 14,944 12,073 10,112 8,692 7,618 6,778 1imax
0.02 23,580 16,560 13,140 11,030 9,565 8,465 7,607 6,917 6,343 5,862 0.016 21,080 14,490 11,390 9,498 8,200 7,238 6,493 5,892 5,399 4,984 0.014 19,720 13,373 10,437 8,676 7,472 6,585 5,899 5,349 4,897 4,519
Converting Equation (23) to terms of equivalent saturated diode current, we obtain
Sld = M2slpr + 2qlpr var X = 2qleqd (212)
When the primary current shows full shot noise, we have
I Ir =M2 + var X (213)
while
Id/Ipr = M (214)
Thus, suppose we plot theoretically log (M2 + var X) versus log (M). Such a curve is identical to a plot of log (Ieqd) versus log (Id); the origins of the two curves are displaced by a line under 45', the Xaxis and Yaxis displacements being log (I pr). Therefore, by comparing the experimental curve of leqd versus Id on loglog paper with the theoretical curves for M2 + var X versus M on loglog paper, we find immediately I from the displacement of the origin. A special favorable feature pr
is that the theoretical curves of M2 + var X on loglog paper are very close for different values of k. Hence k needs only be approximately known for the determination of I pr. Once Ipr has been found, we then plot leq/I pr vs. Id/I pr on linear paper, and compare it with curves for M2 + var X versus M on linear paper. From the magnitude as well as from the breakpoints in the curve, k is obtained for the best fit. The
13
computer plots for M2 + var X versus M are obtained from (29) and (210). The results are given in loglog form in Figures 22 to 24 and on linear paper in Figures 25 to 27.
k 0.0 1 4
1 10
Var X + M2 for k = 0.014 in loglog scale
1001
Figure 22.
k= 0.0 1 6
C4
x
010
1
1 10
Var X + M2 vs. M for k = 0.016 in loglog scale
Figure 23.
100
+
x a10
k= 0.02
Var X + M2 vs. M for k = 0.02 in loglog scale
1 10
Figure 24.
k= 0.0 14
N: 9
McIntyre
N:8
N: 7
Figure 25. Var X + M2 vs. M for k = 0.014 in linear scale
18
1 k: 0.0 1 6
9
N= 9 7
McIntyre
+
x N: 8
5
N 7 N 6
3
N 5 N4
1 1.1 1.2 1.3 M
Figure 26. Var X + M2 vs. M for k = 0.016 in linear scale
k =0.02
N:9
McIntyre
N: 7
N:6
N :5
Var X + M2 vs. M for k = 0.02 in linear scale
Figure 27.
CHAPTER III
DEVICE DESCRIPTION AND EXPERIMENTAL SETUP FOR MEASURING PHOTO DIODE NOISE
Two specially designed RCA "reachthrough" diodes are used
in these holeinitiated avalanche process studies. Figure 31 shows the diode structure and impurity concentration profile. This structure is designed such that the depletion region in the ptype material reaches through to the lightly doped iregion when the reversebias voltage is about 200 volts. Further increase in the reversebias voltage quickly depletes the Trregion.
+
A window is opened on the n p junction side of the diode so that
+
light can be shone on the n region and then generate electronhole pairs. Because the diode is reverse biased, holes will drift into the highfield region and initiate avalanche processes. The wavelength of the light is chosen at 502.8 nm so that the absorption depth is shallow, no light can penetrate to the ptype region, which excludes the possibility of electroninitiated avalanche multiplication.
The block diagram of the whole system is shown in Figure 32. The light source is a Spectra Physics 17108 Argonion Laser. The TEMOI mode is used. This laser gives a doughnutshaped light which is focused by a plainconvex lens. Figure 33 shows the biasing circuit of the device under test (DUT). High Q capacitors are used, so as to reduce the thermal noise as much as possible. Six 70volt batteries and
4X 1012
120 /im
15
3.4X 10
n  T1/2 LD
5 X 1012
1/2
13"' LA
P+= 1 Pm
/4 LD
LD= 1.7 pm LA 7.6 jFm
deep
Figure 31. Diode structure and impurity concentration profile
Device Under Test See Fig. 33
Standard Noise Source
Low Noise Amplifier See Fig. 35
Figure 32. Block diagram of the noise measurement system
500Kfl.
20V
O.,pF O.IF 1oo2 SO2N4220A
NOISE 0 OUTPUT TO LNA
Figure 33. Biasing circuit of the DUT
a 5.5 M potentiometer give the correct bias voltage needed. The diode current Id is monitored by a Keithley 602 solidstate electrometer which is capable of measuring current accurately to less than 1 picoampere. An Nchannel FET source follower is set in between the biasing circuit and the lownoise amplifier (LNA) to match the impedances.
The 5722 vacuum noise diode is chosen as the standard noise
source because its known level of Gaussian white noise and proven performance. The complete circuit is shown in Figure 34. The anode current is adjusted by varying the heater voltage, VHH, with a car battery to supply heater current. The anode load is a 100 millihenry high Q miniature inductor.
The lownoise amplifier as designed by L. M. Rucker of the University of Florida is shown in Figure 35; it uses a high transconductance JFET, type 2N6451, as the input of a cascade input stage. The output is a bipolar transistor, type 2N3904. The input stage has an overall gain of 30 dB. The second and third stages are identical 20 dB gain cascades using 2N4220A FET inputs and 2N3904 bipolar outputs. Because the gain of the first stage, no special effect was made to reduce the noise in the later stages. The final stage is a unity gain buffer having an output impedance of approximately 50 ohms. The buffer is connected to one of the earlier stages through a rotary switch having a low capacitance between the contacts. The overall gain of the amplifier may then be set to 30, 50, or 70 dB. The bandwidth depends on the gain setting and is 3.5, 2.5, and 2.2 MHz, respectively.
5722
Noise Diode
12V
Circuit of a noise standard using a vacuum diode
Noise Output
Figure 34.
+20V
Circuit for a lownoise amplifier (LNA)
INPUT
Figure 35.
Immediately following the LNA is the HP310A Wave Analyzer,
which is used essentially as a narrowband, highgain, tunable filter. It is used in the AM mode with a bandwidth of 3 KHz.
The output of the wave analyzer is connected, through an RC
lowpass filter (or an integrator), to the Yaxis input of the MFE 815 XY plotter with an MFE 7T time base on the plotter's Xaxis input terminals. The Ydirection displacement is proportional to the output voltage of the HP310 Wave Analyzer and its time average over a certain period of time can be obtained from this plotter. This timeaveraged voltage reading gives more accurate results than that obtained directly from an rms volt meter.
The whole system is first operated with both the light and
calibration noise source off. The variable capacitor (see Figure 33) is tuned at the minimum capacitance position and the receiver (HP310 Wave Analyzer) is set to the resonant frequency of the biasing circuit, 203 KHz, such that the maximum noise output is obtained. A voltage meter reading M1 is recorded and the noise power at the output of the aave analyzer in this case is
M= BGIS bkgJ (31)
where B is the bandwidth of the whole system, G the power gain of the amplifier, and Sbkg stands for the spectral density of background noise, which includes the amplifier's noise and noise due to the dark current of the diode.
Next, we turn the calibration noise source on and record the meter reading M2 and the average anode current of the 5722 saturated
vacuum tube diode, IA. Now,
M 2 GB(Sbkg + Scal) (32)
where G, B, and Sbkg are the same as in the previous case, Scal = 2qlA is the spectral density of the calibration noise source where q is electronic charge.
Finally, we turn on the light and turn off the calibration noise source and obtain a new reading M3,
M3 = GB(Sbkg + Sld + Slaser) (33)
where Sld is the noise spectral density due to avalanche processes, and Slaser is the noise introduced by the laser light source which is considered as constant background noise. Here we assume SId and S laser are independent.
The equivalent saturated diode current for Sld + Slaser, Ieq, can then be obtained by calculating
Ieq = [r 3 M IjIA (34)
The procedure is repeated for various bias voltages of the
DUT. Experimental results are plotted and compared with the theory given in Chapter II.
CHAPTER IV
RESULTS FOR HOLEINITIATED AVALANCHE CURRENTS
Table 41 lists the experimental data for an RCA diode. The photocurrent Id generally rises monotonically with the bias voltage Vb. Figure 41 shows the measured data of Id/I dVb = 30 V versus Vb. They are in accord with the manufacturer's data which indicates that no electroninitiated avalanche occurred in these experiments. From Chapter II we have
SId = Slaser = 2qIeq (41)
Now, let SId = 2 qIeqd, Slaser = 2eqleql, then
'eq = Ieqd + Ieq. (42)
If the multiplication factor M equals unity, and the primary photocurrent shows full shot noise, then from Equations (210) and (212) the following result is obtained:
SId = SIpr = 2ql pr (for M = 1) (43)
In this case, leqd = Ipr. The photocurrent Id can hardly be multiplied at the bias voltage of Vb = 30 V; therefore, we can assume Ipr = Id Vb=30V with little error. The constant Ieqz can now be decided by
Table 41. Experimental
data for I = 412 nA
pr
V Id leq leqd(leq  3,733) Id/412 Ieqd/412
30 100
150 170 185 200
240 260 275
300 320 340
412 417 422 432 442 449 458
469 480 492
512 532
4,145 4,159 4,180 4,268 4,480 4,637 4,722 4,792 5,058 5,352 5,887 6,855
412 426 447 535 747
904 989
1,059 1,325
1,619 2,154 3,122
1 .000 1.012 1.024 1 .049 1 .073 1.090
1.112 1.138 1.165
1.194 1.243 1.291
1.000 1.030 1.085 1.299 1.810 2.190 2.400 2.570 3.220 3.930 5.230 7.580
0
II .03
Figure 4I.
Manufacturer's Data
0 Eep. Data
Id/Id Vb=30V
vs. Vb for Ipr = 412 nA
leq = eq Vb:.30V  I' Vb=30V (44)
For the data presented in Table 41, leqk = 3,733 nA; Ieqd can then be found by using Equation (42).
After obtaining leqd' we plot log (Ieqd) versus log (Id). As discussed in Chapter II, the primary current can then be decided by comparing this plot with theoretical curves. The Ipr so determined in Figure 42 is exactly equal to what has been assumed, namely Ipr = IdlVb=3OV. The results of leqd, Id/Ipr, and Ieqd/Ipr are also given in Table 41.
In the data given here, we show the following types of measurements: (1) the avalanche photocurrent Id as a function of bias voltage;
(2) the noise, as expressed in the equivalent saturated diode current leqd, as a function of bias voltage; (3) leqd as a function of diode current Id. The measurements are typical of many others. We determined the diode current Id and the noise Ieqd for different light intensities. The data for different light intensities are quite similar and generally consistent with each other as predicted by the theory.
Figures 43 to 45 show the experimental data obtained for a
primary current Ipr of 412 nA. The Id vs. Vb graph shows a slight bend which marks the reachthrough voltage. The noise is plotted in Figures 44 and 45 as a function of Vb and Id, respectively.
Table 42 lists the data obtained for a different light intensity. The leqk for these measurements is obtained by Equation (44).
I = 3,962 nA  180 nA = 3,782 nA (45)
eq
Theoretical Curve for k:0.014
* Exp. Data +
x
4
Figure 42.
I d(nA)
Log(I eq d vs. log(Id)
for I = 412 nA
550
500
Reach Through Point
450
_o
4001
0 100 200 300
Vb( V) Figure 43. Id vs. Vb for Ipr = 412 nA
I I
vb(v)
100
200
300
400
vs. Vb for Ipr
eq d
4000 V
3000 I
2000 F
1000h
= 412 nA
Figure 44.
3500 3000 2500
4
2000
15001000
500
200
400 450 500 'd (A)
Figure 45. leq vs. I d for IDr = 412 nA
Table 42. Experimental data for I
pr
V Id leqd(leq  3,782) Id/180 leqd/180
30 100
135 150 175 200 240 260 280 300 315 330 340 350
180 181 185 190 195
199 204 208 214 220 226 231 237 244
3,962 4,039 4,050 4,104 4,172 4,234 4,309 4,407 4,506 4,608 4,793 4,967 5,217 5,526
180 257 268 322 390 452 527 625 724 826 1,011
1,185 1,435 1,744
= 180 nA
1.000 1.006 1.028 1.056 1.083 1.106 1.133
1.156 1.189 1.222 1.256 1.283 1.317 1.356
1.00 1.43 1.49 1.79 2.17 2.51 2.93 3.47 4.02 4.59 5.62 6.58 7.97 9.69
which is very close to what we found for previous measurements, 3,733 nA,atadifferent light intensity. This is in accord with our assumptions that Slaser is a constant background noise and independent of SId.
By comparing the curve of log (Ieqd) versus log (Id) with the theoretical plots, we note that the curves coincide if the origins are displaced under 450 by an amount log Ipr, with Ipr = 180 nA, see Figure 46. Figures 47 to 49 show the results of Id versus Vb, leqd versus Vb, and leqd versus Id,
Dividing now both axes of Figure 45 by the primary current, we can make a plot of the experimental values
leq Ipr M2 + var X versus Id/ I pr = M (46)
This result is given in Figure 410.
The breakpoint structure is not well pronounced, which is due
to the fact that the current gain, or multiplication factor M, for holeinitiated avalanche current is too small. Therefore, largevoltage increments are necessary in order to observe current increase while increasing the bias voltage, and the breakpoint structure could be easily glossed over. However, we compare the results of Figure 410 with the theoretical curves of M2 + var X versus M. The best fit, with respect to magnitude, is obtained for k = 0.014. The experimental data plus the theoretical curve is given in Figure 411. Comparison of these experimental results with the asymptotic theory (McIntyre's) would result in a k value of 0.03, which is too large to believe for a modern device.
I 000
"0
100
/
/
45o
I I I I I I
I I I I I I I I I
10 100 1000
Id (nA)
Figure 46. Log(Ieq ) vs. log(Id) for Ipr = 180 nA dp
240
220
Reach Through Point
200h
180 l
160
100
1.50
200
250
300
350
Vb( v )
Figure 47. Id vs. Vb for Ipr = 180 nA
2000 1500
500
0 I I
0 100 200 300 400
v
Fb
Figure 48. Ie vs. Vb for Ip = 180 nA
2000 1500
C
1000 500
0
150 200 250 1d1 (nA) Figure 49. 1eq vs. I for I pr = 180 nA
Figure 410.
1.1
Var X + M2 vs. M for Ipr
1.2
1.3
= 412 nA
44
9
Theoretical Curve for k: 0.0 14
/
/
Exp. Results
f
7 /
N: 8
C14
> 5 V N:7
/
N:6
/
/
3 ,/NN:
/
N 5
N:4
1 1.1 1.2 1.3
Fiqure 411. Var X + M2 vs. M with theoretical curve for I = 412 nA pr
The experimental M 2 + var X versus M curve obtained from lower light intensity measurements is presented in Figure 412. Comparison with the theoretical curve for k = 0.014 is made in Figure 413. This diode shows consistent noise performance while operated at different light intensities.
From the data presented above for low M values and various
light intensities, we conclude that Van Vliet and Rucker's theory is applicable for holeinitiated avalanche currents.
9 7
x
5 3
1 1.1 1.2 1.3 M Figure 412. Var X + M2 vs. M for I = 180 nA pr
Theoretical curve for k: 0.014 Exp. Results
/ N:6
4 N:5
N 5
Figure 413. Var X + M2 vs. M with theoretical curve for I
I.1 1.2 1.3
= 180 nA
PART B
EMISSION STATISTICS OF cxPARTICLES
CHAPTER V
THEORETICAL PERSPECTIVES
5.1 Handel's Theory
It is known that upon scattering a beam of electrons will
emit bremsstrahlung. The power spectrum W(f) of the emitted radiation is independent of frequency (W = constant) at low frequencies and decreases to zero at an upper frequency limit fm which is approximated by E/h, where E is the kinetic energy of the electrons, h the Planck constant. Consequently, the rate of photon emission per unit frequency interval is N(f) = W/Hf, i.e., proportional to 1/f (see Figure 51). Therefore, we conclude that the fraction of electrons scattered with energy loss c is proportional to I/:, i.e., the relative squared matrix element for scattering with energy loss E is IbT()12 % 1/6.
If the incoming beam of electrons is described by a wave function exp[(i/f)(Pi  r Et)], the scattered beam will contain a large nonbremsstrahlung part of amplitude a, and an incoherent mixture of waves of amplitude abT(E) with bremsstrahlung energy loss E ranging from some resolution threshold c0 to an upper limit A < E, of the order of the kinetic energy E of the electrons,
r = exp r  Et]a 1 + f bT()eit/dE] (51)
0
where bT()z IbT(E)eiYe has a random phase Y which implies incoherence of all bremsstrahlung parts, and IbT(E)2 is proportional to I/c.
49
W W) N(f) :W hf
W(f)actua I
fmf
Figure 51. Spectral density of bremsstrahlung in power W and
photon emission rate N
In Equation (51) the frequencyshifted components present in the integral interfere with the elastic term, yielding beats of frequency E/h. The particle density given by Equation (51) is
IT12 =a2{1 + 2fA 1bT(E)icos t .YJdE
+ r fA b*(E)bT(e )eiE ' )t/'rdEd61 (52) C0 E0
the second term in large parentheses describes the particle density beats.
If the particle concentration fluctuation is defined by 61 12 = 1 12  <1 12>, its autocorrelation function will be
22 >_ <2212>2
 1a14A Ib( )12cos ]dE (53)
which is proportional to jb(6)j2 and hence proportional to 1/f. Therefore, the spectral density of the particle concentration fluctuation [the Fourier transform of Equation (53)] is proportional to 1/f.
The relative bremsstrahlung rate lb(E)j2 can be derived as follows. The constant spectral energy density can be written as w(f) = 4e2(AV)2/3c3K, where e is electronic charge, c is the velocity of light, K is the dielectric constant of the medium, and AV is the
 I
velocity change in the scattering process. The relative scattering rate density with energy loss ï¿½, lb( )l2, is obtained by dividing w(f) by the energy of a photon E = hf:
Ib(f)2  4e2(AV)2
3c32 IrtfK
b(s) 12  4e2(AV)2
3c hsK
e2 2(A02 _ cLA hc 3rrf K fK
aA
K
A  2 (Z)2
3 'r
e2
and a =
Tic
where K0 is
sity of the
is the fine structure constant. In the MKS system, a the dielectric constant of vacuum. The spectral denrelative fluctuations is
S 2(f)2 = 2[l + cA kn (A/e0)2ctA/fK 2'p
= 2aA/f K
5.2 The Allan Variance Theorem
The main link between counting statistics and particle current noise is provided by MacDonald's theorem,
d  2 Sm (W)W sin wTdw
(58)
where
(54) (55)
AV
c
(56)
e2 TWcK0
(57)
with inversion
Sm(w) = 2w sin T[ M dT (59) where is the variance of the total number of particles detected in
T
a time interval (t, t+T) and Sm(w) the noise spectral density of the flux fluctuations Am(t). This theorem is useful for Poissonian statistics. Unfortunately, for 1/f noise, Equation (58) is not applicable, since the integral diverges. However, a useful concept in this case is the "two sample variance" or "Allan variance." Let m(l) be the average
T
counting rate in (t, t+T) and M(2) the counting rate in (t + T, t + 2T).
T
Then the Allan variance is defined by
A2 = 1 () T2 (510)
GmT = <(mT T
and
A2 = 1  M(2))2> = T2A2 (511)
MT 2 T T mT
The variance [A2 which means ( A ) 2] turns out to be finite for 1/f noise.
The theorem reads [15]
aA2 (T) = 4 Sm ( ) sin d (512)
with inversion
.1 iM Cos 1P~
S() dP o P 2P r(A2P(T)
m 2rif 1  2 P OTP MT
(513)
For Poissonian shot noise Sm(W) = 2mo, where m is the average counting A2
rate. Substituting Sm(w) into Equation (512), one has aMT(T) = moT. For 1/f noise with a spectrum of S0 2C
, = Tw , where C is a constant, the Allan variance cA2 = 2CT2 Zn 2. The various results are summarized in Table 51.
5.3 Application of the Allan Variance Theorem to Counting Statistics
The presence of 1/f noise in counting statistics can now be
determined from a measurement of the Allan variance as a function of T. For suppose that the noise is composed of shot noise and 1/f noise, i.e.,
a(T) = m T + 2CT2 Zn 2 (514)
M T 0
recall that = m0T, then a measurement of the relative Allan variance R(T) = M(T)/2 yields
R(T) T 2C' kn 2 (515)
0
I/m oT is dominant, hence R(T) is proportional to I/T. When T is long enough, 2C' Zn 2 becomes dominant; R(T) is, therefore, a constant, called the "flicker floor."
Table 51. Various results of Allan variance theorem
S (W)
Poissonian
shot noise General
shot noise 1/f noise
Lorentzian
flicker noise
Pathological
noise
2m0
2T 2rC/ I wI
4B 2
a +2
= mOT
2
= KmoT 2CT2 kn 2
3 [4e T  e2aT a2
L/IwI X1
0 < X < 4; X # 2
+ 2cT  3]
LTX(l  2X2)
sin (7TX/2)(X + 1)
crA2 (T) MT
/
According to Handel's theory, the constant C' in the 1/f noise term in the rela4ve Allan variance is
C' = 2A /K (516) where is a coherence factor; for aparticles it is expected to be close to one; a is the fine structure constant 1/137, and A = 2(Av)2/3c c2 where Av is the velocity change of the particles in the emission process, c is the velocity of light. For aparticles one finds
aA = 8.32 x 107 x E (517)
K K
where E is energy in MeV and K is the dielectric constant of the radioactive material.
In these experiments E = 5.48 MeV, so that the value of 2C' kn 2 (the value of the flicker floor, F) is
F = 4C 8.32 x 107 x 5.48 x kn 2 = 126.4 x 10x7 x
K (518)
CHAPTER VI
EXPERIMENTAL SETUP FOR LONGTIME COUNTING
The block diagram of the counting system being used to investigate 1/f fluctuations in the aparticle emission rate is shown in Figure 61. The source is 95Am241, which decays with a halflife time of T1/2 = 458 years with the emission of 5.48 MeV aparticles into 93NP237 The detector, a silicon surfacebarrier detector, is reverse biased at 80 volts, and the dead times of the ND575 AnalogtoDigital Converter (ADC) and ND66 MultiChannel Analyzer are 60 nseconds and
6 pseconds, respectively. Therefore, no deadtime correction is necessary, as long as the counting rate is kept lower than 1,000 counts per second [16] (or the averaged time elapse between two counts is higher than 1,000 pseconds).
A typical fullenergy spectrum measured in these experiments is shown in Figure 62 in semilogarithmic scale. The spectrum is shown on a display screen while accumulating counts and the final result, after a chosen time T, is stored in the memory units of the ND66. The Full Width Half Maximum (FWHF) of the spectrum can be found by moving the cursor on the display, which indicates the number of counts under each energy channel. The systematic range used in these experiments was from peak channel FWHM x 6 to peak channel +FWHM x 2. Therefore, the total number of counts MT, which will be analyzed later, is always under a fixed portion of the full spectrum.
Ortec 210 Detector
Control
Unit
Detector Ortec 109
I.   PreAmp.
Vacuum
Chamber
SouIrce:
241
95 Am
Ortec 410
Multimode Amplifier
ND 575
Analog
to
Digital Convertor
ND 66
MultiChannel Analyzer
Block diagram counting system
&1 54
Figure 61.
* t.. . .~..
50 100
250
ENERGY CHANNEL
Figure 62.
Typical fullenergy spectrum
The MT'S of adjacent time intervals can be read directly from the memory units of the ND66 MultiChannel Analyzer; thus the Allan variance can be calculated by
A2_ 1 [ 1 N I ) _ M.i+l)]2 (61)
aM T N 7J T T
since
1 M.i) (62)
the relative Allan variance, R(T), defined by Equation (515), can be found by using Equations (61) and (62).
Part of the data, mostly the total number of counts for T longer than 1,000 minutes, was not read directly from the ND66 MultiChannel Analyzer. An "addup" method was adopted; namely, MO) was obtained by
Analyzer. ~1000waobindy
adding up M() and M(2), and M(2)000 equals the sum of M(500 and M(4) etc. Physically, since M500s were measured in adjacent time intervals, of course the first two can be added up as the total number of counts for the first 1,000minute interval, and the third and the fourth can be summed up as M(2) However, in order to check the validity of this
1000
method, the following experiment has. been done.
By making use of a Tconnector, the output signal of the Ortec 410 amplifier was fed simultaneously into two ND575 ADCs (see Figure 61). The first one (ADC #1) counted 100minute measuremenst for 310 times, and the second one (ADC #2) accumulated 500minute counts for 62
times; hence, both ADCs covered exactly the same time span. The "addup" method was applied to Ml00s, obtained from ADC #1, to find out the calculated M500S.
Table 61 lists the results calculated from both ADCs for
500minute measurements. The difference between them, in each category, is <1%. This shows the validity of the "addup" method.
Table 61.
Comparison of the results obtained from the "addup" method (ADC #1) and the realtime measurements (ADC #2)
ADC #1 ADC #2
9,062,464.36 9,067,009.84 0A2
aM5007,862,171.3 7,943,285.5 R(500) 9.573 x 108 9.662 x 108
CHAPTER VII
COUNTING RESULTS
7.1 Convergence of the Allan Variance for N
Because the existence of a "variance of the variance" (or, a noise of the noise), the relative Allan variance itself is a fluctuating parameter. In order to obtain an accurate value of R(T), a sufficient number of measurements must be made especially when T is short. To determine the minimum number of measurements needed, we plot R(T) versus N for different T's. These figures show that when N is small, R(T) is spread over a wide range. When T is increased the spread in R(T) diminishes and finally R(T) converts to a stable value. For example, Figure 71 shows that for T = 100 minutes, 21 measurements of M(1O0) yield an R(1O0) = (7.09 ï¿½ 3.94) x 107; for N = 24, R(T) = (7.23 ï¿½ 2.14) x 107; for N = 27, R(T) = (7.13 ï¿½ 1.85) x 107. Here R(T) is given in the form of (mean value ï¿½ standard deviation). From Figure 72 we know that for T = 1 minute at least 70 measurements are necessary for a reliable value of R(T). Figure 73 shows that for T = 3 minutes we need N > 50.
7.2 Results of the Relative Allan Variance versus l/T
From Equation (515) we have
R(T) = 1_ + 2C' n 2 (7) moT
40 1
30 F
20 I
10 I
Figure 71. R(T) vs. N for T = 100 min.
6 5 In
0
x 4
3 2 1 0
Figure 72 R(T) vs. N vor T = 1 min.
I
x 3
2
Ir
2ï¿½a
00
0 I I60
0 20 40 60
N
Figure 73. R(T) vs. N for T = 3 min.
where C' is a constant, m0 is the averaged counting rate, and T is the counting time.
Figures 74 to 710 show seven series of experimental results of R(T) versus I/T. Figures 74 to 76 show clearly that for T small (I/T large), R(T) is equal to the value of Poissonian shot noise, the first term of Equation 71. But for T large (l/T small), the values of R(T) is larger than the value for shot noise. Figures 77 to 710 show the results of some experiments which were concentrated on longtime measurements. They all show that for T long enough the experimental values of R(T) are higher than 1/m0T and the curves start to level off, which implies the existence of 1/f noise. The second, constant term in Equation (71) becomes dominant.
7.3 Results of the Allan Variance versus T
Figures 711 and 712 give the plot of the Allan variance, TA2 versus T corresponding to the measurements of Figures 74 and 75, respectively. Recall that for shot noiseMT = mT, which is proportional to T, but for 1/f noise cA2 = 2CT2 kn 2, which is proportional OMT
to T2.
Since these figures (Figures 711 and 712) are plotted on loglog scale, the slope of log (aA) versus log(T) is really the exponent of T in the expression of MT see Equation (514). Figures 711 and 712 show clearly the dependence of A2 on T2 and T while dMT
different noise terms dominate the noise spectrum.
67
Poisson Noise .104.0 Exp. Result
a
106
70
10
3 2 1
10 10 10
T (min.) Figure 74. R ep(T) vs. 1/T for m = 27,700/min.
 Poisson Noise
4 0 Exp. Result 10
100
x
: 6 10
1
0
I . . ... ... . . .....I . . ..
3 2 1
10 10 10 1
! mi."
Figure 75. Rexp (T) vs. I/T for mo = 17,980/min.
4
10
5
10
0.
Poisson Noise (D Exp. Result
3 2
10 10
i
10
(min'")
Figure 76. R exp(T) vs. 1/T for mo = 18,160/min.
I 
5
10
6
10
3 2
10 0 10 1I
~1.(mm:")
Figure 77. R exp(T) vs. I/T for mï¿½ = 17,180/min.
71
Poisson Noise
) Exp. Result
0 5
10.
I.
x
,v 6
10
0
7
10
3 2 1
10 10 10 1(..i) r mn'
Figure 78. R (T) vs. 1/T for m = 18,356/min.
exp o
5
10
7
10
Poisson Noise
0 Exp. Result
2
10
Figure 79. R exp(T) vs. 1/T for mo = 17,807/min.
10 ( i)
10
17
8
10
Poisson Noise
0 Exp. Result
0
~ /
3
10
2
10
Figure 710. Rexp (T) vs. I/T for mo = 18,180/min.
T. (min.')
74
109
108
10 0
w Sop=2
4
QCQ
Z 6
10
0
10
Slope 10 2 3 T (min.)
1 10 10 10
Figure 711. A (T) vs. T for m= 27,700/min.
M T
75
10 9
7
10
z slopes Z 0
z
6
10 0
0
10
4
10
2 3 4 1 10 10 10 10
T (m in.)
Figure 712. aA2 (T) vs. T for mo = 17,960/min.
'"T
CHAPTER VIII
DISCUSSION OF RESULTS
The curves shown in Chapter VII are not smooth, which is due to the nature of the noise fluctuations. In order to obtain a universal curve, the following procedure has been carried out.
Assume the experimental results of R(T) contain a fluctuation term AR, due to the presence of "noise of noise," i.e.,
R exp(T) = mo +2C' kn 2 + AR (81)
where the average value of AR, , should be zero. The experiment was then repeated for several times, and the average of R exp(T),
= Lo + 2C' kn 2 + (82)
exp m 0T
is obtained. Although the value of in Equation (82) can hardly be exactly zero for a finite number of measurements, it should be reduced by a great deal compared with the value of AR for a single measurement. Now, the value of gives the best estimation to the true value of R(T).
The distance between the radioactive source and the detector was adjusted such that very close counting rates were obtained while repeating the measurements. However, it is difficult to obtain identical
counting rates. Therefore, the shot noise term in the relative Allan variance, R(T), is slightly different for each series of measurements.
Before the average of R (T) is taken, the shot noise term, exp
1
, should be normalized to the same counting rate. Here a rate of
0
18,000 counts per minute was chosen. The normalized relative Allan variance Rn(T) is then
Rn(T) = R (T) 1+ 1 1 + 2C' kn 2 nmexp oT 18,000 x T 18,000 x T
(83)
Tables 81 and 82 show the values of R exp(T) and R n(T). Instead of , is now used to estimate the true value of R(T). The value of is calculated as follows:
ERni (T) x DF.
1DF (84) n EDF.
1
where R ni(T) is the value of R n(T) obtained from the ith series and DFi stands for the degrees of freedom of that particular value, which equals the number of measurements minus one.
Figure 81 shows the comparison of and (1/18,000 x T) + 1 x 107 versus l/T, which suggests that the value of 2C' Zn 2 is about 1 x 107.
Figure 82 shows the comparison of the average value of the normalized Allan variance, ,w
Table 81. The values of Rexp(T) and Rn(T) for T = 150, all values x 107, the numbers in parentheses
being the number of measurements
Time (min.)
m 1 2 5 10 20 50
0
27,700/min.
17,980/min. 18,160/min.
17,807/min.
R (T) exp R n(T)
R (T) exp R n(T)
R (T) exp Rn (T)
R (T)
exp Rn (T)
348.7 544.1(96) 572.9 572.5(177) 504.8 509.8(248)
205.3 303.0(48) 232.1 231.9(88) 263.1
265.6(124)
84.27 123.30(19)
149.6 149.6(93)
71.85 72.85(49)
39.07 58.58(38) 41.16 42,78(83)
66,03 66.55(88)
11.91 21.66(18) 22.68 23,49(41) 34.11 34.37(44)
4.957
8.844(7)
11.65 11.970(16) 11.350 11.450(149) 12.150 12.030(56)
Table 82. The values of Rexg(T) and Rn(T) for T
ses being the number of measurements
= 1003,000, all values x 107, the numbers in parenthe
Time (min.)
m 100 200 500 1,000 2,000 3,000
27,770/min.
17,980/min.
18,160/min.
17,180/min.
18,356/min.
R n(T) exp(T) R (T)
n
Rexp (T) R (T)
Rexp(T)
R (T)
n
R n(T) Rexp(T
R (T)
n
4.126
6.075(18)
6.359
6.348(72)
7.398
7.437(310)
7.192
6.927(26)
6.003
6.111(30)
3.281
4.255(9)
4.093
4.087(36)
3.505
3.524(155)
8.390
8.257(13)
1.970
2.024(15)
6.105
6.494(3)
3.404
3.401(14)
0.9662
0.9742(62)
8.9990
8.9460(5)
2.3430
2.3640(6)
6. 8300
6.8290(7)
0.8133
0.8175(31)
1.3830
1.3850(15)
3.360
2.3610(10)
4.4280
4.4390(3)
Table 82Continued
Time (min.)
mo 100 200 500 1,000 2,000 5,000
R exp(T) 8.147 4.263 2.3630 11,807/min. ex
R n(T) 8.087(28) 4.233(14) 2.3510(5)
Rexp (T) 3.808 1.3200 0.8042 0.6299 0.5865 18,180/mi n.0
Rn(T) 3.836(40) 1.3310(30) 0.8099(15) 0.6327(7) 0.5884(5)
Page
Missing
or
Unavailable
Theoretical Value
0 Exp. Result
Figure 82.
< 2T)> vs. T for mo
MT
18,000/min.
109
8
10
7
10
z
4
4
z
6 S106
0
z
105
T (min.)
= x (18,000 x T)2 (85)
Very good agreement is obtained as shown in this figure.
The value of and 1/18,000 x T (shot noise level) are
n
listed in Table 83. It shows clearly that at this counting rate, 18,000 counts per minute, for T longer than 100 minutes, 1/f noise becomes noticeable. For T longer than 1,000 minutes, 1/f noise totally dominates the noise spectrum.
Besides the fluctuation of scattering cross section described in Handel's theory, two other possibilities may contribute to the fluctuation in the total number of counts. First, the pulse height, which is produced when an aparticle is absorbed by the detector, is a strong function of the bias voltage applied to the detector [17]. If the bias voltage was unstable, then aparticles with the same energy would have been registered in different energy channels. Fortunately, the detector control unit used in these experiments, Ortec 210, has a very good stability: bias voltage variation with line voltage is <ï¿½0.005% for 105125 V AC input, and thestability is ï¿½0.01%[18]. Secondly, since a fixed portion of the full spectrum is counted, whether particles with energies around boundaries fall in or out of boundaries contributes fluctuations to the total number of counts.
Therefore, a wider energy range should have less fluctuations
in the total number of counts in a certain time period. This is indeed the case; see Table 84. However, for T = 100 minutes, the difference between R(T)'s for wider range and narrower range is negligible. For
Table 83.
The values of and 1/18,000xT, all values x 107
Time (min.) 1 2 5 10 20 50 537.4 261.0 123.3 55.66 27.86 11.56 1 555.6 277.8 111.1 55.56 27.78 11.11
18,000 x T
Time (min.) 100 200 500 1,000 2,000 3,000 7.156 3.825 1.799 1.6480 1.1590 1.8160
1
1 5.556 2.778 1.111 0.5556 0.2778 0.1852
18,000 x T
Table 84. Comparison of experimental results for narrower energy
range (peak channelFWHM x 6 to peak channel + FWHM x 2)
and wider energy range (peak channelFWHM x 12 to peak
channel + FWHM x 6)
Time (min.) Channels 127147a Channels 112157a
A2
R(100)
A2
11500
R(500)
A2
M3000
R(3000)
1,812,492.94 2,430,286.7
7
7.398 x 10
9,067,009.8 7,943,285.5 9.662 x 108
54,398,712.1 698,293,362 2.360 x 107
apeak channel: 142; FWHM:
2.5 channels
500
1,828,630.89 2,472,416.52 7.394 x 107
9,147,489.02 7,762,255.74 9.277 x 108
54,881 ,014.1 634,854,514 2.108 x 107
3,000
T = 50 minutes, the difference is 4%, but the value of is
n
62% higher than the shot noise level (see Table 83). For T = 3,000 minutes, the difference is 11%, and the value of is ten times the value for the shot noise level. Hence the energy range chosen in these experiments affected little the final results.
In certain R(T) versus l/T figures in Chapter VII and in
Figure 81, when T becomes large R(T) shows the tendency to be proportional to T. This phenomenon is not exclusive, since according to the Allan variance theorem R(T) is indeed proportional toT for very slow Lorentzian flicker noise [15].
CHAPTER IX
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
From the experimental data presented in the Appendix, we see that the ratio for the averaged value of the total number of counts, , to the variance of M is very close to one when T is less than 100 minutes, which is an indication for Poissonian statistics. However, when T is greater than 100 minutes this ratio becomes greater than unity and generally increases monotonically with T, which indicates superPoisonian statistics. With the help of the newly devised Allan variance theorem, we found that this excess noise in the radioactive decay rate has a 1/f spectrum.
Quantitatively, from Equation (518), we know the flicker floor F has a value of
F = 126.4 x 10 x (91)
To the author's knowledge, nobody has ever measured the dielectric constant of AmO2, the ctparticle source used in these experiments; thus the value of K in Equation (91) is unknown. However, since the atomic structure of AmO2 is similar to that for U02, the dielectric constant of UO2 (20.4 ï¿½ 1.5 [19], 21.7 ï¿½ 0.5 [20], and 21.0 ï¿½ 1 [21]) can then be used as a reference.
If one considers the following factors that (1) the coherence factor, , is always less than unity; (2) the dielectric constant of
AmO2 may be larger than 21, then the measured value of F = 1 x 107 is in the right ballpark to verify Handel's theory. In particular, for C = 0.17, K = 21, one obtains F = 1.02 x lO, in accord with the observed value of Figures 81 and 82. We believe, therefore, that these experiments constitute the first experimental verification of Handel's quantum 1/f noise theory.
The data presented here all referred to 95Am241. If an asource with different energy peaks is used, the counting could then be done in each of these energy peaks. This should result in an R(T) versus l/T as given in Figure 91.
If Handel's theory is correct, the flicker floor should be
proportional to v /c , i.e., to the energy E. Thus, the flicker floors shown in Figure 91 should have the ratios of these energy peaks. This experiment will therefore be a very strong test to verify the applicability and correctness of Handel's theory.
6
10
/
E1 > E2 > E3
1o3 102 I01
Figure 91. R(T) vs. 1/T for different energy peaks
APPENDIX
EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS
Experimental data for m0 = 27,700
var A2
T N var. V MT Rexp (T)
27,761 55,525 138,797 277,369 554,689
1,386,606 2,772,026 5,544,053
13,855,521
29,759 64,254 174,979 271,146 407,757
1,556,691 13,322,534 49,414,157 202,833,876
1.07
1.16 1.26
0.98 0.74 0.83
4.81 8.91
14.64
26,874 63,300 162,334 300,625 366,675 950,741
3,170,123 10,084,511 117,194,752
1
2
5
10
20 50 100
200 500
3.487
2.053 8.427
3.907 1.191
4.945 4.126 3.281 6.105
x lo5 x 105 x lo6 x lo6 x 106 x lo7 x 107 x lo7 x 107
Table A1.
Table A2. Experimental data for m0 = 17,980
TNvr var A29 TQq N N>vr R (T) . MT exp
17,987 35,970 89,984 185,379 370,752
926,836 1,796,155 3,592,309
8,980,092 17,960,183
17,155 29,055 108,003 154,777 324,936 868,989
15,133,266 55,659,746 324,379,609 1,311,457,070
0.95 0.81 1.20 0.84 0.88
0.94 8.43
15.49 36.12 73.02
18,534 30,030
121,148 141,464 311,757 1,000,977 2,051,505 5,282,260 27,448,241 220,310,187
1
2 5 10
20 50 100
200 500
1,000
5.729 2.321 1.496 4.116
2.268 1.165
6.359 4.093
3.404 6.830
x 105 x 105 x 105 x lo6 x 1o6
x lo6 x lo7 x 107 x lo7 x 107
Table A3. Experimental data for m0 = 18,160
yar. A2 R (T) T N var y OMT exp
18,160 36,321
90,795 181,684 363,367 908,647 1,812,493 3,625,004
9,067,010 18,134,020 36,265,809 54,398,714
16,829 34,706 64,814 214,732 436,054 1,274,359 9,116,358 31,272,439
168,503,103 668,501,571 2,626,291 ,950 5,991,195,130
0.93 0.96 0.71 1.18 1.20 1.40
5.03 8.63 16.58 36.86 72.42 110.14
16,647
34,714 59,228 217,969 450,422 936,880
2,430,287 4,606,285 7,943,286 26,745,242 181,946,661 698,293,362
5.048 x 105 2.631 x 105
7.185 x 106 6.603 x 106 3.411 x 106 1.135 x 106 7.398 x 107 3.505 x 107 9.662 x 108 8.133 x 108 1.383 x 107 2.360 x 107
1 2 5 10
20 50 100
200 500 1,000
2,000 3,000
Table A4. Experimental data for m 0 17,180
var A2
>var < TRMT R (T)
100 26 1,717,908 10,533,131 6.13 2,122,404 7.192 x 107 200 13 3,435,815 38,770,910 11.28 9,904,640 8.390 x 107
7
500 58,589,200 186,528,449 21.72 66,392,064 8.999 xI0
