- Email: [email protected]
A study of alpha decay
Chaos, Solitons and Fractals 12 (2001) 1157±1166
www.elsevier.nl/locate/chaos
A study of alpha decay Leon Katz Physics Department, University of Saskatchewan, Saskatoon, Sask., Canada S7N 0W0 Accepted 3 April 2000
Abstract The process of alpha decay in nuclei has traditionally been assumed to be governed by the many characteristic properties (Coulomb barrier, spin, parity, etc.) of the parent isotope and the alpha particle. This implies that each pair of parent±daughter transition is only marginally related to other pairs of the same element. In this paper, the author shows that the experimental data suggests there is a strong scale-independent relationship between the isotopes of an alpha emitting element. In most cases, the relationship between the energy and half-life values for the isotopes of an element can be represented by a single power law with mathematical precision. The presence of this relationship can be used to deduce information on the nuclear structure of some nuclei. Also it suggests that alpha decay properties may play a less restrictive role in the process than traditional models have previously assumed. The exponent of the power law is negative and slightly decreases in value linearly with increasing mass number. There is also a large negative peak in the vicinity of mass 82. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction It has been known for some time [1] that in elements with even±even nuclei the experimental values of the partial half-life for alpha decay and the energy released in a ground-state transitions of the elementÕs isotopes can be represented by a semi-log equation with a fair degree of accuracy. The author has been investigating non-linear dynamic systems in the social, physical and life sciences, where Levy ¯ight statistics [2,3] apply and often the relationship between pairs of variables can be represented by a power law equation. In view of the recent developments in our understanding about non-linear systems a re-analysis of alpha decay using a power law analysis technique (Eq. (1)) and the latest experimental values was in order. n Q MeV ct1=2 ;
1
where c and n are constants and Q M parent M daughter M 4 He and t1=2 is the partial half-life of that transition. The ability to represent the relation between energy and half-life in alpha decay of an element's isotopes by a power law would indicate that the process is a scale-independent. Another way of expressing this is to say that the process is self-similar. On the other hand, assuming the relationship is exponential does not tell us anything about the process involved. The graphs in Fig. 1(a) illustrate log±log and semi-log plots of the even±even lead isotopes. Note that the ®ve of the six isotopes (number A186±A210) lie along a straight line with mathematical precision (Table 1) while isotope, A192, is o the line. It would be reasonable to assume that all six isotopes with transitions of zero spin and no parity change would behave similarly. Yet ®ve of them lie along a straight line while the E-mail address: [email protected] (L. Katz). 0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 0 8 7 - 4
1158 L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Fig. 1. Examples of power law graphs of log(Q) versus log(partial half-life).
Table 1 Partial half-life analysis Element
At. no.
Transition
Slope
S.D.
Intercept
S.D.
R2
F
d. o. f. (q2 )
Spin Parent
Daughter
Fig. 1(a)
Pb
E±E
Fig. 1(b) Fig. 1(c)
Cm Fr
82 Semi-log 96 87
E±E O±E
)0.0148 )0.1698 )0.0140 )0.0150
0.0001 0.0076 0.0002 0.0001
0.8234 6.5336 0.8945 0.8577
0.0008 0.0684 0.0015 0.0002
0.9999 0.9940 0.9994 0.9997
28547 496 6689 18674
3 3 4 5
Fr
87
O±E
)0.0152
0.0002
0.8467
0.0007
0.9999
9128
1
Pu
94
E±O
)0.0133
0.8872
)0.0133
0.8776
0+ 0+ 0+ 9/2) 9/2) 9/2) (9/2)) (9/2)) (9/2)) 9/2) 3/2()) 5/2) 9/2) 5/2+ ? 1/2+ (5/2+) ? 9/2) 9/2) 3/2) (5/2+) (5/2)) 0+ 0+
0+ 0+ 0+ 9/2) 9/2) 9/2) (9/2)) (9/2)) ? 9/2) ? 9/2) 9/2) 1/2+ (3/2+) 7/2) (5/2)) (3/2)) 9/2) 9/2) 3/2) (3/2)) (5/2)) 0+ 0+
Fig. 1(d)
Fig. 1(e)
Fig. 1(f)
Pa
91
O±E
)0.0179
0.0001
0.8728
0.0006
1.0000
21020
1
Pa
91
O±E
)0.0124
0.0003
0.8598
0.0028
0.9993
1343
1
U
92
E±E E±E
)0.0132 )0.0164
0.0000 0.0001
0.8567 0.8793
0.0006 0.0006
1.0000 0.9999
110421 15955
2 2
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Figure
Note: 0.1% F-distribution values for systems of two variables n F 1; n 1 4053 2 998 3 167 4 74 5 47
1159
1160
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
sixth diering only by two neutrons from those on either side is far o this line (there is a seventh alpha emitter, A202, but its BR is not known). It would need a fourfold decrease in BR or an equal increase in half-life or a change in Q by +107 keV to move these values on the line, well beyond the experimental errors assigned to them. This kind of information cannot be deduced from the exponential graph since there is no scienti®c basis for the values to lie on the graph. If on the other hand, we can show in this analysis of alpha decay data, that the process is scale-independent, then the position of the A192 becomes highly signi®cant, indicating perhaps some interesting change in the nuclear structure of the isotope. This analysis found that with very few exceptions most of the isotopes of alpha emitting elements had partial half-lives and Q values that fell along linear regression lines in log±log plots. Surely this signals a dynamic in alpha decay not previously noticed or predicted. In many cases, some of the values are o the lines formed by the others, as A192 in the above example. A study of these exceptional cases could throw new light on alpha decay dynamics. The analysis also included a study of the half-life for all the alphas emitted by an isotope, this will be called the ``total half-life'' to avoid confusion, and its connection to the energy of the alpha from that isotope to the ground state. Surprisingly, as will be shown later this hybrid analysis gave the same results as the partial half-life and ground-state transition when the dependence of slopes and the intercepts on Z were compared. In addition, this analysis indicated a sharp change in slope and intercept at the magic number of Z 82. In the interval between Z 74 and 102, the region covered by this analysis, there are a few cases, where essentially all the isotopes of an element have branching ratios (BRs) near 100%, where this agreement would be expected but only 40% of all the alpha emitters have such high ratios. Since less than half of the alphas emitting isotopes have BRs known to three-®gure accuracy, this hybrid analysis had more than twice as many useful data values. 2. Data analysis All data on alpha decay of nuclei were taken from the Lawrence Berkeley National Laboratory Isotopes Project in collaboration with LUNDS Universitet web site. 1 Altogether there are 29 elements listed in that site with eight or more alpha emitting isotopes, 25 of which had enough isotopes and BR data for isotopes with even or odd numbers of neutrons to be analyzed separately. Q values vary between 3 and 10 MeV only, usually having less than half that range in any one element but are known predominantly to four signi®cant ®gures. Some of the high Z elements are known only to three signi®cant ®gures and others have Q values listed to six ®gures. All Q values are given in MeV. Half-life values are mostly known to two or three signi®cant ®gures, a few with greater accuracy. However, in any one element the isotope half-lives range over many orders of magnitude, from a minimum of three orders in some of the low Z elements to a maximum of 24 in thorium. The wide range over which the values are spread allows the calculation of a regression line with good accuracy in a log(partial halflife))log(Q) plot. In fact the R2 (goodness of ®t) values obtained in such an analysis may be a bit misleading since the wide range of the Q values in¯ates the calculated values to some extent. A few elements with less than four odd or four even alpha emitting isotopes often do not have enough data points to de®ne the regression lines. All half-life values are given in seconds. The partial half-life±Q analysis yielded a total of 47 graphs (E±E, E±O, O±E, and O±O) from 29 elements: 25 with single lines, 13 with two parallel lines, 6 cases of two lines at an angle (a single line followed by one of less steep slope) and 3 more complex cases. In the case of multiple lines the average slope and intercept of the parallel lines were used in further analysis. Some results of a power law analysis are listed in Table 1 and a few illustrated in Fig. 1(a)±(f). The following is a short discussion of each of these ®gures: Fig. 1(a). There are 11 even±even (E±E) alpha emitting isotopes of lead, only the BRs of mass numbers 194 and 210 have been measured to three ®gure accuracy, those of mass 188, 190, 192 and 196 to two ®gures 1
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L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
1161
only and are not known for the remaining ®ve. In view of this limited BR accuracy the degree to which ®ve of them fall along a straight line in a log±log plot remains to be explained. There is no question however that they do follow a power law distribution and that if this observed property of alpha decay is correct, then any deviation from the line can be estimated quantitatively and may lead to a better understanding of the alpha emission process. The author made use of a similar analysis of world running records by men and identi®ed variations in the aerobic and anaerobic energies of the athlete as a function of exertion time. The analysis involved examining the distribution of the data values about a regression power law line through the running data [4]. Fig. 1(b). Of the seven alpha emitting E±E curium isotopes six have BR values of 100% or close to it and are known with great accuracy, the seventh, A250, has a BR value of 8%, not accurate enough to include in this analysis. There is no question that in this case they have a power law distribution. Fig. 1(c). In many cases the alpha decay values de®ne two parallel lines in a log±log plot. This ®gure shows one such case. There are 12 odd±even alpha decay isotopes of francium. All BRs have values greater than 85% except A223 which has a value of 0.006%. The lines are parallel to better than 1% in slope. If the BR of A223 were 0.005%, then the lines would have identical slopes of )0.01505. R2 and F-distribution values are listed in Table 1. Fig. 1(d). Another interesting case of parallel lines is illustrated in this graph. Only two values de®ne each line, but they have slopes of )0.01328 and )0.01329. These are the values of the E±O isotopes of plutonium. There are only ®ve values and the BRs of all of them have been measured to three-®gure accuracy. Only A239 has 100% BR, the other values are all much less than one. The transition characteristic of the two lines may turn out to be very interesting except the parity and spin of A233 are not known. Fig. 1(e). Many graphs showed sudden changes in slope. A good example of this behaviour is the two lines with dierent slope in this diagram. There are 10 odd±even alpha emitting isotopes of protactinium, only eight of which have measured BRs, six with 100%, A229 with 0.48% and A227 with 85%. The BRs of A213 and A217 are not known. As can be seen in Table 1, the probability that power laws can represent the distribution of these values is very high in one case and reasonable in the other. The spin and parity of the transitions are listed in Table 1 but they are not known for the two isotopes that are o the line. Fig. 1(f). All 10 of the even±even isotopes of uranium have well-known BR value, 9 at 100% and A228 at 95%. Eight of these de®ne two lines of dierent slope. The two o-lines values are: A218 which would require a reduction in the BR by a factor of 12.9 or a lower Q value by 361 keV and A224 which needs a reduction in the BR by a factor of 2.4 or a 124 keV reduction in its Q value. As mentioned previously the analysis was repeated for all alpha-emitting elements by plotting the hybrid combination log(Q) against log(total half-life). In this analysis there were 41 graphs: 11 with single lines through the data, 8 with two lines of dierent slope, 18 with at least two lines (some with parallel lines and a third line of dierent slope) and 4 more complex cases. Fig. 2(a)±(f) are some example graphs. Data for these are summarised in Table 2. The following is a short discussion of the Fig. 2(a±f): Fig. 2(a). There are seven even±even alpha emitting isotopes in curium. Five of the isotopes, namely mass numbers 240, 242, 244, 246 and 248 line up and give the single line seen in Fig. 1(b). They all have BRs which are close to 100%. A238 with a BR of 3.84% that was on the line in Fig. 1(b) is now o the line and A250 which has a poorly measured BR of 8 is very far o the line. It would need a half-life of 9.2E + 12 s instead of 2.84 s or a BR much smaller in order for it to be on the line. Alternately, the Q value would have to be 2.562 MeV larger. Fig. 2(b). Seven of the eight E±E alpha emitting isotopes of plutonium follow a power law distribution with great accuracy (Table 2). All except mass numbers A232 and A234 have 100% BR. The two exceptions have BR values of 23% and 6%, respectively. No obvious reason could be found for the position of the A234 value in this ®gure being so far o the regression line. For this point to fall on the line it would need 165 keV greater energy or a half-life that is 6.6-fold longer, both of which are much larger than the errors assigned to these values. Mass calculations of the Q value con®rmed that this is a ground-state transition. In the partial half-life analysis these transitions are slightly above the line but the line slopes and intercepts are the same in both cases. Fig. 2(c). This is an example of a hybrid graph with two lines of the same slope (within 1 part in 160). Ten of the 12 E±E radium alpha emitting isotopes de®ne two lines in the log±log plot with great precision. The
1162 L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Fig. 2. Examples of power law graphs of log(Q) versus log(total half-life).
Table 2 Total half-life analysis Element
At. no.
Transition
Slope
S.D.
Intercept
S.D.
R2
F
d. o. f.
Spin Parent
Daughter
Fig. 2(a) Fig. 2(b) Fig. 2(c)
Cm Pu Ra
Fig. 2(d)
Pt
96 94 88 88 78
E±E E±E E±E E±E E±O
)0.0140 )0.0131 0.0167 0.0166 )0.0325
0.0001 0.0001 0.0003 0.0002
0.895 0.870 0.853 0.865 0.775
0.001 0.001 0.001 0.001
0.9997 0.9997 0.9993 0.9995
12849 15253 2674 9515
4 5 2 5
78
E±O
)0.0327
0.00001
0.768
0.00001
1.0000
2.64E+07
1
78
E±O
)0.0605
0.0005
0.815
0.001
0.9999
16505
1
Hg
80
E±O
)0.025
0.0003
0.805
0.001
1
5265
2
Hg
80
E±O
)0.122
0.0014
0.969
0.003
1
7160
1
Bi
83
O±E
)0.0735
0.0006
0.933
0.002
0.9999
14010
2
83
O±E
)0.0234
0.0003
0.8572
0.0006
0.9998
6321
1
0+ 0+ 0+ 0+ 9/2+ 1/2) 1/2) 1/2) ? 1/2) (5/2)) ? 1/2) 1/2) ? ? 3/2) 13/2+ 1/2) 9/2) (9/2)) (9/2)) (9/2)) (9/2)) (9/2)) (9/2))
0+ 0+ 0+ 0+ 1/2) (5/2)) (1/2)) (1/2)) ? (1/2)) (5/2)) (5/2)) 1/2) 1/2) ? ? 9/2+ 1/2) 1/2) 1/2+ 1/2(+) (1/2+) (1/2+) (1/2+) (1/2+) (1/2+)
Fig. 2(e)
Fig. 2(f)
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Figure
1163
1164
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
isotope that is o the lines has mass number A206. Its position from the line is much further than the experimental error assigned to it. Fig. 2(d). The eight even±odd alpha emitting isotopes of platinum de®ne two parallel line and a third line crossing them with a much steeper slope. This and the next two diagrams are examples of the 17 cases, where the data follows two lines with dierent slopes. The few such cases of dierent sloped lines in the partial half-life analysis (e.g., Fig. 1(e) and (f)) are always a steep line followed by one of lesser slope with increasing half-life time, while in this case the reverse is true. Fig. 2(e). Six of the eight E±O isotopes of mercury de®ne two lines, with one value common to both lines. The values A179 and A181 are slightly above one of the line. Seven of these isotopes in the partial half-life plot lie along a single line with A187 missing since its BR is not known. Fig. 2(f). An interesting example of the distribution of total half-life and Q values for the 10 O±E nuclei of bismuth is shown in this ®gure. The best estimate of the spins in these transitions is from 9/2) to 1/2+. Slopes and goodness of ®t are again listed in Table 2. The fact that one line extends into the region of the other indicates that whatever caused the lower line to change slope had not aected the top line. The dynamics of the decay process are indeed very complex. Mass numbers 187, 203 and 211 are o the lines by fairly large factors. For example, to move A211 from above the top line onto it would require a half-life of 15.5 s instead of 128 s or a reduction in the Q value by 327 keV. 2.1. Parallel line separation It is interesting to determine whether the ``parallel'' lines are truly parallel or dier by some DQ value. If they are actually parallel, then the Q values in the two lines dier by a constant multiple, if the dierence is some ®xed DQ, then the lines would not be quite parallel. Taking the average slope of the top 33 parallel lines and the average slope of the bottom lines gave slopes of )0.01945 and )0.01944 with an average separation between the lines of 0.24 MeV. If the dierence between the lines was a ®xed value DQ, then for DQ 0:24 MeV the dierence in slope would be 100 times greater than 1 part in 1900. 2.2. Slopes The low sloped lines in all the panels (partial and total half-life) were analysed by averaging the slopes of those with even and odd numbers of neutrons in each element since there did not seem to be any systematic dierence between them. Then a further running average was made over three adjacent values. The steeply sloped lines in the 17 hybrid panels were averaged in the same way though the six cases in the partial halflife panels did not oer enough data for such further treatment. The results are shown in Fig. 3. Note that far from atomic number 82 all lines have essentially the same slope, low slope, while in the vicinity of Z 82 only the hybrid analysis shows a sharp increase in steepness of the line slopes. Total half-life: Low slope triangles slope
0:0387 0:000242Z;
Steep slope diamonds slope
0:0502 0:000309Z:
2 3
Partial half-life: Low slope circles slope
0:0270 0:000129Z:
4
2.3. Intercepts The intercepts have an interesting physical interpretation. In Eq. (1), the intercept c is equal to Q when t 1 s. In other words, this would be the Q value if that element had an isotope of 1-s half-life. Fig. 4 is a graph of these Q values for the ``low'' and ``high'' sloped lines. Regression lines through the downward sloping data in these ®gures gave Eqs. (2) and (3).
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
1165
Fig. 3. All slopes of partial and total half-life lines. Fig. 4. Intercepts of partial and total half-life lines.
Total half-life: Low slope triangles Q MeV
1:922 0:1043Z;
Steep slope diamonds Q MeV
5
2:265 0:1093Z:
6
1:586 0:101Z:
7
Partial half-life: Low slope open circles
Q MeV
Q values calculated from these equations in the range 70 6 Z 6 100 dier by less than 4%. The peak at 82 is 1.8 MeV (81.7 in the graph) above these lines. The slopes as a function of Z for the low and high sloped lines in Fig. 3 were expected to show an anomaly in the vicinity of the magic number Z 82. But why this anomaly appeared in the hybrid curves only and why the dip in the slopes for the high sloped lines is about ®ve times as deep as that for the low sloped lines remains to be explained.
3. Discussion To summarize, in the Q±partial half-life ground-state transitions of all the elements examined, only those in the range Z72±Z102 had sucient alpha emitting isotopes for analysis with at least four isotopes per graph. These 29 elements have the potential of giving 58 graphs when the even±even, even±odd, odd±even and odd±odd isotopes are analyzed separately. Actually 27 were found to give single log±log lines with R2 P 0:999; 12 were found to have two parallel lines with R2 P 0:999 and less then 1% dierence in slope, many with much less dierence. In 11 cases the BRs of the alpha transitions were poorly known. Thus out of a possible 47 graphs 39 yielded good log±log plots. The frequency with which the data points line up on log±log graphs should be a sucient indication that this is not a random occurrence but the result of some inner dynamic of these systems. The alpha decay process is generally assumed to depend on the spin and parity of the transition, the odd±even characteristics of the parent nucleus, formation probability of ®nding an alpha in the parent nucleus and the Coulomb barrier penetration as well as the presence of closed shells. It is dicult to comprehend how these combine to follow such a simple equation as the power law, which is the signature of a scale-independent system. It is even more dicult to understand how power law distributions can develop in the isotopes of a single element, whose isotopes dier so greatly in the spin and parity change for alpha emission as illustrated in Tables 3 and 4. From a more general point of view these alpha emission dynamic seem to indicate that the parameters usually assumed to govern such processes may play a less restrictive role than previously thought. For example, the even±odd isotopes of Radon, whose transition spin and parity are listed in Table 3, fall along a
1166
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Table 3 Radon even±odd A
Parent
Daughter
221 207 205 203 201 219 215
7/2(+) 5/2) 5/2) (3/2,7/2)) (3/2)) 5/2+ 9/2+
? 5/2) 3/2) (3/2)) (3/2)) 9/2+ 9/2+
A
Parent
Daughter
231 225 215 221
5/2+ (3/2)+ (1/2)) (7/2+)
3/2+ 5/2+ 3/2) 1/2)
Table 4 Thorium even±odd
single line with a goodness of ®t R2 0:9999 or F-distribution of 68,802 with 5 degrees of freedom. Thorium (see Table 4) is another example that has a goodness of ®t R2 0:9999 or F-distribution of 30,125 with two degrees of freedom. Both these cases are for partial half-life and ground-state transitions. It is tempting to assume that isotopes of an element that display a power law distribution, though diering in the number of neutrons they possess, act as if they were dierent phases of some more general dynamic system for alpha emission. In that case alpha emission though limited by the above conditions (Coulomb barrier penetration, etc.) could still be a statistical process subject to Levy ¯ight dynamics. From this point of view a sudden change in slope implies that the statistical characteristics of the nuclei involved are somehow dierent from the others, that they constitute a dierent scale-independent system. References [1] [2] [3] [4]
See for example, Table of Isotopes, 7th ed. p. 23 [Appendices]. Shlesinger MF, Zaslavsky GM, Klafter J. Strange kinetics. Nature 1993;363:31. Shlesinger MF, Zaslavsky GM, Frisch U, Levy ¯ights and related topics in physics. New York: Springer, 1995. Sylvan KJ, Katz L. Power laws and athletic performance. J Sports Sci. 1999;17:467±76.
www.elsevier.nl/locate/chaos
A study of alpha decay Leon Katz Physics Department, University of Saskatchewan, Saskatoon, Sask., Canada S7N 0W0 Accepted 3 April 2000
Abstract The process of alpha decay in nuclei has traditionally been assumed to be governed by the many characteristic properties (Coulomb barrier, spin, parity, etc.) of the parent isotope and the alpha particle. This implies that each pair of parent±daughter transition is only marginally related to other pairs of the same element. In this paper, the author shows that the experimental data suggests there is a strong scale-independent relationship between the isotopes of an alpha emitting element. In most cases, the relationship between the energy and half-life values for the isotopes of an element can be represented by a single power law with mathematical precision. The presence of this relationship can be used to deduce information on the nuclear structure of some nuclei. Also it suggests that alpha decay properties may play a less restrictive role in the process than traditional models have previously assumed. The exponent of the power law is negative and slightly decreases in value linearly with increasing mass number. There is also a large negative peak in the vicinity of mass 82. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction It has been known for some time [1] that in elements with even±even nuclei the experimental values of the partial half-life for alpha decay and the energy released in a ground-state transitions of the elementÕs isotopes can be represented by a semi-log equation with a fair degree of accuracy. The author has been investigating non-linear dynamic systems in the social, physical and life sciences, where Levy ¯ight statistics [2,3] apply and often the relationship between pairs of variables can be represented by a power law equation. In view of the recent developments in our understanding about non-linear systems a re-analysis of alpha decay using a power law analysis technique (Eq. (1)) and the latest experimental values was in order. n Q MeV ct1=2 ;
1
where c and n are constants and Q M parent M daughter M 4 He and t1=2 is the partial half-life of that transition. The ability to represent the relation between energy and half-life in alpha decay of an element's isotopes by a power law would indicate that the process is a scale-independent. Another way of expressing this is to say that the process is self-similar. On the other hand, assuming the relationship is exponential does not tell us anything about the process involved. The graphs in Fig. 1(a) illustrate log±log and semi-log plots of the even±even lead isotopes. Note that the ®ve of the six isotopes (number A186±A210) lie along a straight line with mathematical precision (Table 1) while isotope, A192, is o the line. It would be reasonable to assume that all six isotopes with transitions of zero spin and no parity change would behave similarly. Yet ®ve of them lie along a straight line while the E-mail address: [email protected] (L. Katz). 0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 0 8 7 - 4
1158 L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Fig. 1. Examples of power law graphs of log(Q) versus log(partial half-life).
Table 1 Partial half-life analysis Element
At. no.
Transition
Slope
S.D.
Intercept
S.D.
R2
F
d. o. f. (q2 )
Spin Parent
Daughter
Fig. 1(a)
Pb
E±E
Fig. 1(b) Fig. 1(c)
Cm Fr
82 Semi-log 96 87
E±E O±E
)0.0148 )0.1698 )0.0140 )0.0150
0.0001 0.0076 0.0002 0.0001
0.8234 6.5336 0.8945 0.8577
0.0008 0.0684 0.0015 0.0002
0.9999 0.9940 0.9994 0.9997
28547 496 6689 18674
3 3 4 5
Fr
87
O±E
)0.0152
0.0002
0.8467
0.0007
0.9999
9128
1
Pu
94
E±O
)0.0133
0.8872
)0.0133
0.8776
0+ 0+ 0+ 9/2) 9/2) 9/2) (9/2)) (9/2)) (9/2)) 9/2) 3/2()) 5/2) 9/2) 5/2+ ? 1/2+ (5/2+) ? 9/2) 9/2) 3/2) (5/2+) (5/2)) 0+ 0+
0+ 0+ 0+ 9/2) 9/2) 9/2) (9/2)) (9/2)) ? 9/2) ? 9/2) 9/2) 1/2+ (3/2+) 7/2) (5/2)) (3/2)) 9/2) 9/2) 3/2) (3/2)) (5/2)) 0+ 0+
Fig. 1(d)
Fig. 1(e)
Fig. 1(f)
Pa
91
O±E
)0.0179
0.0001
0.8728
0.0006
1.0000
21020
1
Pa
91
O±E
)0.0124
0.0003
0.8598
0.0028
0.9993
1343
1
U
92
E±E E±E
)0.0132 )0.0164
0.0000 0.0001
0.8567 0.8793
0.0006 0.0006
1.0000 0.9999
110421 15955
2 2
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Figure
Note: 0.1% F-distribution values for systems of two variables n F 1; n 1 4053 2 998 3 167 4 74 5 47
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L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
sixth diering only by two neutrons from those on either side is far o this line (there is a seventh alpha emitter, A202, but its BR is not known). It would need a fourfold decrease in BR or an equal increase in half-life or a change in Q by +107 keV to move these values on the line, well beyond the experimental errors assigned to them. This kind of information cannot be deduced from the exponential graph since there is no scienti®c basis for the values to lie on the graph. If on the other hand, we can show in this analysis of alpha decay data, that the process is scale-independent, then the position of the A192 becomes highly signi®cant, indicating perhaps some interesting change in the nuclear structure of the isotope. This analysis found that with very few exceptions most of the isotopes of alpha emitting elements had partial half-lives and Q values that fell along linear regression lines in log±log plots. Surely this signals a dynamic in alpha decay not previously noticed or predicted. In many cases, some of the values are o the lines formed by the others, as A192 in the above example. A study of these exceptional cases could throw new light on alpha decay dynamics. The analysis also included a study of the half-life for all the alphas emitted by an isotope, this will be called the ``total half-life'' to avoid confusion, and its connection to the energy of the alpha from that isotope to the ground state. Surprisingly, as will be shown later this hybrid analysis gave the same results as the partial half-life and ground-state transition when the dependence of slopes and the intercepts on Z were compared. In addition, this analysis indicated a sharp change in slope and intercept at the magic number of Z 82. In the interval between Z 74 and 102, the region covered by this analysis, there are a few cases, where essentially all the isotopes of an element have branching ratios (BRs) near 100%, where this agreement would be expected but only 40% of all the alpha emitters have such high ratios. Since less than half of the alphas emitting isotopes have BRs known to three-®gure accuracy, this hybrid analysis had more than twice as many useful data values. 2. Data analysis All data on alpha decay of nuclei were taken from the Lawrence Berkeley National Laboratory Isotopes Project in collaboration with LUNDS Universitet web site. 1 Altogether there are 29 elements listed in that site with eight or more alpha emitting isotopes, 25 of which had enough isotopes and BR data for isotopes with even or odd numbers of neutrons to be analyzed separately. Q values vary between 3 and 10 MeV only, usually having less than half that range in any one element but are known predominantly to four signi®cant ®gures. Some of the high Z elements are known only to three signi®cant ®gures and others have Q values listed to six ®gures. All Q values are given in MeV. Half-life values are mostly known to two or three signi®cant ®gures, a few with greater accuracy. However, in any one element the isotope half-lives range over many orders of magnitude, from a minimum of three orders in some of the low Z elements to a maximum of 24 in thorium. The wide range over which the values are spread allows the calculation of a regression line with good accuracy in a log(partial halflife))log(Q) plot. In fact the R2 (goodness of ®t) values obtained in such an analysis may be a bit misleading since the wide range of the Q values in¯ates the calculated values to some extent. A few elements with less than four odd or four even alpha emitting isotopes often do not have enough data points to de®ne the regression lines. All half-life values are given in seconds. The partial half-life±Q analysis yielded a total of 47 graphs (E±E, E±O, O±E, and O±O) from 29 elements: 25 with single lines, 13 with two parallel lines, 6 cases of two lines at an angle (a single line followed by one of less steep slope) and 3 more complex cases. In the case of multiple lines the average slope and intercept of the parallel lines were used in further analysis. Some results of a power law analysis are listed in Table 1 and a few illustrated in Fig. 1(a)±(f). The following is a short discussion of each of these ®gures: Fig. 1(a). There are 11 even±even (E±E) alpha emitting isotopes of lead, only the BRs of mass numbers 194 and 210 have been measured to three ®gure accuracy, those of mass 188, 190, 192 and 196 to two ®gures 1
http://ie.lbl.gov/toi.html
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
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only and are not known for the remaining ®ve. In view of this limited BR accuracy the degree to which ®ve of them fall along a straight line in a log±log plot remains to be explained. There is no question however that they do follow a power law distribution and that if this observed property of alpha decay is correct, then any deviation from the line can be estimated quantitatively and may lead to a better understanding of the alpha emission process. The author made use of a similar analysis of world running records by men and identi®ed variations in the aerobic and anaerobic energies of the athlete as a function of exertion time. The analysis involved examining the distribution of the data values about a regression power law line through the running data [4]. Fig. 1(b). Of the seven alpha emitting E±E curium isotopes six have BR values of 100% or close to it and are known with great accuracy, the seventh, A250, has a BR value of 8%, not accurate enough to include in this analysis. There is no question that in this case they have a power law distribution. Fig. 1(c). In many cases the alpha decay values de®ne two parallel lines in a log±log plot. This ®gure shows one such case. There are 12 odd±even alpha decay isotopes of francium. All BRs have values greater than 85% except A223 which has a value of 0.006%. The lines are parallel to better than 1% in slope. If the BR of A223 were 0.005%, then the lines would have identical slopes of )0.01505. R2 and F-distribution values are listed in Table 1. Fig. 1(d). Another interesting case of parallel lines is illustrated in this graph. Only two values de®ne each line, but they have slopes of )0.01328 and )0.01329. These are the values of the E±O isotopes of plutonium. There are only ®ve values and the BRs of all of them have been measured to three-®gure accuracy. Only A239 has 100% BR, the other values are all much less than one. The transition characteristic of the two lines may turn out to be very interesting except the parity and spin of A233 are not known. Fig. 1(e). Many graphs showed sudden changes in slope. A good example of this behaviour is the two lines with dierent slope in this diagram. There are 10 odd±even alpha emitting isotopes of protactinium, only eight of which have measured BRs, six with 100%, A229 with 0.48% and A227 with 85%. The BRs of A213 and A217 are not known. As can be seen in Table 1, the probability that power laws can represent the distribution of these values is very high in one case and reasonable in the other. The spin and parity of the transitions are listed in Table 1 but they are not known for the two isotopes that are o the line. Fig. 1(f). All 10 of the even±even isotopes of uranium have well-known BR value, 9 at 100% and A228 at 95%. Eight of these de®ne two lines of dierent slope. The two o-lines values are: A218 which would require a reduction in the BR by a factor of 12.9 or a lower Q value by 361 keV and A224 which needs a reduction in the BR by a factor of 2.4 or a 124 keV reduction in its Q value. As mentioned previously the analysis was repeated for all alpha-emitting elements by plotting the hybrid combination log(Q) against log(total half-life). In this analysis there were 41 graphs: 11 with single lines through the data, 8 with two lines of dierent slope, 18 with at least two lines (some with parallel lines and a third line of dierent slope) and 4 more complex cases. Fig. 2(a)±(f) are some example graphs. Data for these are summarised in Table 2. The following is a short discussion of the Fig. 2(a±f): Fig. 2(a). There are seven even±even alpha emitting isotopes in curium. Five of the isotopes, namely mass numbers 240, 242, 244, 246 and 248 line up and give the single line seen in Fig. 1(b). They all have BRs which are close to 100%. A238 with a BR of 3.84% that was on the line in Fig. 1(b) is now o the line and A250 which has a poorly measured BR of 8 is very far o the line. It would need a half-life of 9.2E + 12 s instead of 2.84 s or a BR much smaller in order for it to be on the line. Alternately, the Q value would have to be 2.562 MeV larger. Fig. 2(b). Seven of the eight E±E alpha emitting isotopes of plutonium follow a power law distribution with great accuracy (Table 2). All except mass numbers A232 and A234 have 100% BR. The two exceptions have BR values of 23% and 6%, respectively. No obvious reason could be found for the position of the A234 value in this ®gure being so far o the regression line. For this point to fall on the line it would need 165 keV greater energy or a half-life that is 6.6-fold longer, both of which are much larger than the errors assigned to these values. Mass calculations of the Q value con®rmed that this is a ground-state transition. In the partial half-life analysis these transitions are slightly above the line but the line slopes and intercepts are the same in both cases. Fig. 2(c). This is an example of a hybrid graph with two lines of the same slope (within 1 part in 160). Ten of the 12 E±E radium alpha emitting isotopes de®ne two lines in the log±log plot with great precision. The
1162 L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Fig. 2. Examples of power law graphs of log(Q) versus log(total half-life).
Table 2 Total half-life analysis Element
At. no.
Transition
Slope
S.D.
Intercept
S.D.
R2
F
d. o. f.
Spin Parent
Daughter
Fig. 2(a) Fig. 2(b) Fig. 2(c)
Cm Pu Ra
Fig. 2(d)
Pt
96 94 88 88 78
E±E E±E E±E E±E E±O
)0.0140 )0.0131 0.0167 0.0166 )0.0325
0.0001 0.0001 0.0003 0.0002
0.895 0.870 0.853 0.865 0.775
0.001 0.001 0.001 0.001
0.9997 0.9997 0.9993 0.9995
12849 15253 2674 9515
4 5 2 5
78
E±O
)0.0327
0.00001
0.768
0.00001
1.0000
2.64E+07
1
78
E±O
)0.0605
0.0005
0.815
0.001
0.9999
16505
1
Hg
80
E±O
)0.025
0.0003
0.805
0.001
1
5265
2
Hg
80
E±O
)0.122
0.0014
0.969
0.003
1
7160
1
Bi
83
O±E
)0.0735
0.0006
0.933
0.002
0.9999
14010
2
83
O±E
)0.0234
0.0003
0.8572
0.0006
0.9998
6321
1
0+ 0+ 0+ 0+ 9/2+ 1/2) 1/2) 1/2) ? 1/2) (5/2)) ? 1/2) 1/2) ? ? 3/2) 13/2+ 1/2) 9/2) (9/2)) (9/2)) (9/2)) (9/2)) (9/2)) (9/2))
0+ 0+ 0+ 0+ 1/2) (5/2)) (1/2)) (1/2)) ? (1/2)) (5/2)) (5/2)) 1/2) 1/2) ? ? 9/2+ 1/2) 1/2) 1/2+ 1/2(+) (1/2+) (1/2+) (1/2+) (1/2+) (1/2+)
Fig. 2(e)
Fig. 2(f)
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Figure
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L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
isotope that is o the lines has mass number A206. Its position from the line is much further than the experimental error assigned to it. Fig. 2(d). The eight even±odd alpha emitting isotopes of platinum de®ne two parallel line and a third line crossing them with a much steeper slope. This and the next two diagrams are examples of the 17 cases, where the data follows two lines with dierent slopes. The few such cases of dierent sloped lines in the partial half-life analysis (e.g., Fig. 1(e) and (f)) are always a steep line followed by one of lesser slope with increasing half-life time, while in this case the reverse is true. Fig. 2(e). Six of the eight E±O isotopes of mercury de®ne two lines, with one value common to both lines. The values A179 and A181 are slightly above one of the line. Seven of these isotopes in the partial half-life plot lie along a single line with A187 missing since its BR is not known. Fig. 2(f). An interesting example of the distribution of total half-life and Q values for the 10 O±E nuclei of bismuth is shown in this ®gure. The best estimate of the spins in these transitions is from 9/2) to 1/2+. Slopes and goodness of ®t are again listed in Table 2. The fact that one line extends into the region of the other indicates that whatever caused the lower line to change slope had not aected the top line. The dynamics of the decay process are indeed very complex. Mass numbers 187, 203 and 211 are o the lines by fairly large factors. For example, to move A211 from above the top line onto it would require a half-life of 15.5 s instead of 128 s or a reduction in the Q value by 327 keV. 2.1. Parallel line separation It is interesting to determine whether the ``parallel'' lines are truly parallel or dier by some DQ value. If they are actually parallel, then the Q values in the two lines dier by a constant multiple, if the dierence is some ®xed DQ, then the lines would not be quite parallel. Taking the average slope of the top 33 parallel lines and the average slope of the bottom lines gave slopes of )0.01945 and )0.01944 with an average separation between the lines of 0.24 MeV. If the dierence between the lines was a ®xed value DQ, then for DQ 0:24 MeV the dierence in slope would be 100 times greater than 1 part in 1900. 2.2. Slopes The low sloped lines in all the panels (partial and total half-life) were analysed by averaging the slopes of those with even and odd numbers of neutrons in each element since there did not seem to be any systematic dierence between them. Then a further running average was made over three adjacent values. The steeply sloped lines in the 17 hybrid panels were averaged in the same way though the six cases in the partial halflife panels did not oer enough data for such further treatment. The results are shown in Fig. 3. Note that far from atomic number 82 all lines have essentially the same slope, low slope, while in the vicinity of Z 82 only the hybrid analysis shows a sharp increase in steepness of the line slopes. Total half-life: Low slope triangles slope
0:0387 0:000242Z;
Steep slope diamonds slope
0:0502 0:000309Z:
2 3
Partial half-life: Low slope circles slope
0:0270 0:000129Z:
4
2.3. Intercepts The intercepts have an interesting physical interpretation. In Eq. (1), the intercept c is equal to Q when t 1 s. In other words, this would be the Q value if that element had an isotope of 1-s half-life. Fig. 4 is a graph of these Q values for the ``low'' and ``high'' sloped lines. Regression lines through the downward sloping data in these ®gures gave Eqs. (2) and (3).
L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
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Fig. 3. All slopes of partial and total half-life lines. Fig. 4. Intercepts of partial and total half-life lines.
Total half-life: Low slope triangles Q MeV
1:922 0:1043Z;
Steep slope diamonds Q MeV
5
2:265 0:1093Z:
6
1:586 0:101Z:
7
Partial half-life: Low slope open circles
Q MeV
Q values calculated from these equations in the range 70 6 Z 6 100 dier by less than 4%. The peak at 82 is 1.8 MeV (81.7 in the graph) above these lines. The slopes as a function of Z for the low and high sloped lines in Fig. 3 were expected to show an anomaly in the vicinity of the magic number Z 82. But why this anomaly appeared in the hybrid curves only and why the dip in the slopes for the high sloped lines is about ®ve times as deep as that for the low sloped lines remains to be explained.
3. Discussion To summarize, in the Q±partial half-life ground-state transitions of all the elements examined, only those in the range Z72±Z102 had sucient alpha emitting isotopes for analysis with at least four isotopes per graph. These 29 elements have the potential of giving 58 graphs when the even±even, even±odd, odd±even and odd±odd isotopes are analyzed separately. Actually 27 were found to give single log±log lines with R2 P 0:999; 12 were found to have two parallel lines with R2 P 0:999 and less then 1% dierence in slope, many with much less dierence. In 11 cases the BRs of the alpha transitions were poorly known. Thus out of a possible 47 graphs 39 yielded good log±log plots. The frequency with which the data points line up on log±log graphs should be a sucient indication that this is not a random occurrence but the result of some inner dynamic of these systems. The alpha decay process is generally assumed to depend on the spin and parity of the transition, the odd±even characteristics of the parent nucleus, formation probability of ®nding an alpha in the parent nucleus and the Coulomb barrier penetration as well as the presence of closed shells. It is dicult to comprehend how these combine to follow such a simple equation as the power law, which is the signature of a scale-independent system. It is even more dicult to understand how power law distributions can develop in the isotopes of a single element, whose isotopes dier so greatly in the spin and parity change for alpha emission as illustrated in Tables 3 and 4. From a more general point of view these alpha emission dynamic seem to indicate that the parameters usually assumed to govern such processes may play a less restrictive role than previously thought. For example, the even±odd isotopes of Radon, whose transition spin and parity are listed in Table 3, fall along a
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L. Katz / Chaos, Solitons and Fractals 12 (2001) 1157±1166
Table 3 Radon even±odd A
Parent
Daughter
221 207 205 203 201 219 215
7/2(+) 5/2) 5/2) (3/2,7/2)) (3/2)) 5/2+ 9/2+
? 5/2) 3/2) (3/2)) (3/2)) 9/2+ 9/2+
A
Parent
Daughter
231 225 215 221
5/2+ (3/2)+ (1/2)) (7/2+)
3/2+ 5/2+ 3/2) 1/2)
Table 4 Thorium even±odd
single line with a goodness of ®t R2 0:9999 or F-distribution of 68,802 with 5 degrees of freedom. Thorium (see Table 4) is another example that has a goodness of ®t R2 0:9999 or F-distribution of 30,125 with two degrees of freedom. Both these cases are for partial half-life and ground-state transitions. It is tempting to assume that isotopes of an element that display a power law distribution, though diering in the number of neutrons they possess, act as if they were dierent phases of some more general dynamic system for alpha emission. In that case alpha emission though limited by the above conditions (Coulomb barrier penetration, etc.) could still be a statistical process subject to Levy ¯ight dynamics. From this point of view a sudden change in slope implies that the statistical characteristics of the nuclei involved are somehow dierent from the others, that they constitute a dierent scale-independent system. References [1] [2] [3] [4]
See for example, Table of Isotopes, 7th ed. p. 23 [Appendices]. Shlesinger MF, Zaslavsky GM, Klafter J. Strange kinetics. Nature 1993;363:31. Shlesinger MF, Zaslavsky GM, Frisch U, Levy ¯ights and related topics in physics. New York: Springer, 1995. Sylvan KJ, Katz L. Power laws and athletic performance. J Sports Sci. 1999;17:467±76.