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The Constant Linear Force Model

 

Summary

 

This model is based on the assumption that all charged elementary particles have the same quantity of Linear Force (Fl). We show how this assumption leads to an alternative interpretation of particle experiments.

 

Linear Force

 

Linear force is found by the following equation:

 

……………………………………………….………….………..1

 

Fl = linear force

m = mass (MeV)

r = radius (fermi)

 

Using the Classical Electron Radius and electron mass to find the Linear Force constant we have:

 

 0.5109989MeV x 2.817941fermi = 1.4399648......………………………….…2

 

Radius

 

Using Fl and m we can calculate the radii off other particles as show in Table 1

 

Charge

 

Using the formula for finding the Classical Radius to find the radii of other particles produces the same result as shown in Table 1 only if all particles have the same charge value.

 

de Broglie Wavelength

 

Calculations show that:

 

Circumference divided by de Broglie wavelength = constant……………….3

 

 

Confirming that a mathematical relationship exists between particle and particle wavelength.

 

Energy

 

Energy multiplied by radius = constant………………………….……………….4

 

 


 

 

FQHE

 

Robert B Laughlin, Horst L StΦrmer and Daniel C Tsui1,  were awarded the 1998 Nobel Prize in Physics for their discovery of Fractionally Charged Electrons; the fractional sequence is 1/3, 2/5, 3/7, 4/9 etc.

 

 

In astrophysics the sequence found by Tsui et al, is the measurement of waves within a field for example; measuring the distance between the centres of the dust bands around comet Hale-Bop on two radii (maximum and minimum length) gives the following table:

 

Band

 

A

B

C

D

4 (outer band)

Actual

38

39

60

48

 

less 1/3 =

25.3

26

40

32

3

Actual

25

27

40

32

 

less 2/5 =

15

16.2

24

19.2

2

actual

16.5

16

23

19

 

less 3/7 =

9.5

9.1

13

10.9

1 (inner band)

Actual

9.4

9

12

10

 

The cause of these dust bands is believed to be the sun’s magnetic field that I take to mean - magnetic compaction of charged (dust) particles.

 

We are unable to match the Tsui sequence to particle compaction and find that the magnetic compaction of single particles can be better explained using the Laughlin and Jain sequences.

 

H Heiselberg’s2 quotes the Laughlin sequence (1/2, 1/3, 1/4, 1/5 etc) as being the most important sequence and the Jain sequence (1/2, 2/3, 3/4, 4/5 etc) as being the next most important sequence; no comment is made on the fact that, placed in order these two sequences add up to a constant value of 1.

 

One possible cause for the constant is that Laughlin and Jain sequences are two measurements of the same particle in different compaction states as shown in Fig. 1.

 

Fig.1

 

This compaction between two parallel force fields will be referred to as single plane compaction. It was then necessary to match the fractional sequences to the particle mass values reported by the Particle Data Group (2004 tables).

 

Mass

 

Two points immediately become obvious. The first is that if we are to give a cause for stopping compaction, then the practice of measuring wavelength at mid-point between peak and trough; has to be abandoned as it does not provide a cause. But, at the peak and trough the wave changes direction to start a new wave and this provides a cause because at both points, there must be sufficient wave force to commence the building of a new wave.

 

Secondly it was realised that each fraction is a fraction of the previous wave and not a fraction of the wave at which compaction started.

 

This lead to the construction of Table 1 as illustrated by Fig.2


 

 

 


TABLE 1

(Extracted from full table)

Fraction
deducted

radius

theoretical

mass:
CLF
formula

mass

from PDG

2004 tables

Margin
of
error

PDG reference

a

b

c

d

e

f

(1)   

14.089705

0.1021998

 

 

Graviton?

 1/2

7.0448525

0.2043996

0.26

±48 

56 CHETYRKIN 98 THEO MS scheme

 1/3

4.6965684

0.3065994

0.3

±10 

57 CUCCHIERI 98 LATT MS scheme

 1/4

3.5224263

0.4087991

0.43

± 8

52 MALTMAN 99 THEO MS scheme

 1/5

2.817941

0.5109989

0.5109989

electron

 

 1/6

2.3482842

0.6131987

0.553

±12 

55 BECIREVIC 98 LATT MS scheme

 1/7

2.012815

0.7153985

0.66

±19 

58 DOMINGUEZ 98 THEO MS scheme

1/13   

1.0838235

1.3285972

1.3

±0.3

ASTIER 00D NOMD 

1/17   

0.8288062

1.7373963

1.7

±0.3

1 AUBIN 04A LATT MS scheme 

1/18   

0.7827614

1.8395961

1.79

±0.38

VILAIN 99 THEO MS scheme 

1/23   

0.6125959

2.350595

2.3

±0.4

3 NARISON 99 THEO MS scheme 

1/27   

0.521841

2.7593942

2.7

 ±0.06 4.72

3 AUBERT 04X THEO

1/28   

0.5032038

2.8615939

2.9

±0.6

2 JAMIN 02 THEO MS scheme 

1/30   

0.4696569

3.0659935

3

±0.7

5 NARISON 95C THEO MS scheme 

1/33   

0.4269608

3.3725929

3.4

±0.11

5 HOANG 04 THEO 

1/35   

0.402563

3.5769924

3.6

±0.03 4.68

4 BAUER 04 THEO 

1/37   

0.3808029

3.781392

3.8

±0.2 

27 EICKER 97 LATT MS scheme 

1/38   

0.3707817

3.8835918

3.9

±0.5

6 AUBIN 04A LATT MS scheme 

1/39   

0.3612745

3.9857916

3.95

±0.3 

17 CHIU 02 LATT MS scheme 

1/40   

0.3522427

4.0879913

4.05

±0.6 

19 MALTMAN 01 THEO MS scheme 

1/41   

0.3436514

4.1901911

4.19

±0.9

7 JAMIN 02 THEO MS scheme 

1/42   

0.3354692

4.2923909

4.25

±0.7

5 NARISON 95C THEO MS scheme 

1/43   

0.3276676

4.3945907

4.4

±0.1 ±0.4

14 BECIREVIC 03 LATT MS scheme 

1/44   

0.3202206

4.4967905

4.5

±0.11

6 MCNEILE 04 LATT 

1/45   

0.3131046

4.5989903

4.57

 

20 AOKI 00 LATT MS scheme 

1/46   

0.306298

4.70119

4.7

±2 

21 GOECKELER 00 LATT MS scheme 

1/51   

0.2762688

5.212189

5.2

±0.9

7 JAMIN 02 THEO MS scheme 

1/63   

0.2236461

6.4385864

6.4

±1.1 8

NARISON 99 THEO MS scheme 

1/69   

0.2041986

7.0517851

7

±1.1

9 JAMIN 95 THEO MS scheme 

1/72   

0.1956904

7.3583844

7.4

±0.7

10 NARISON 95C THEO MS scheme 

1/223   

0.0631826

22.790552

22.7

±20 

61 EICKER 97 LATT MS scheme 

1/224   

0.0629005

22.892752

22.8

±14.1 

59 CHETYRKIN 97 THEO MS scheme

1/744   

0.0189378

76.036639

76

±0.09 

9 CORCELLA 03 THEO

1/793   

0.0177676

81.044428

81

±0.09

7 BAUER 03 THEO

1/827   

0.0170372

84.519221

84.5

±0.10

11 EIDEMULLER 03 THEO

1/861   

0.0163644

87.994014

88

±0.05

16 KUHN 01 THEO

1/900   

0.0156553

91.979805

92

±0.06

13 MAHMOOD 03 THEO

1/910   

0.0154832

93.001803

93

±0.05

8 BORDES 03 THEO 

1/930   

0.0151502

95.045799

95

± 4 

49 GOECKELER 00 LATT MS scheme

1/969   

0.0145405

99.03159

99

±0.090 ±0.025

18 PINEDA 01 THEO

1/978   

0.0144067

99.951388

100

±0.82

19 BARATE 00V ALEP

1/979   

0.014392

100.05359

100

±14 

50 AOKI 99 LATT MS scheme 

1/1008   

0.0139779

103.01738

103

±0.57 

15 ABBIENDI 01S OPAL

1/1027   

0.0137193

104.95918

105

±17 

40 GAMIZ 03 THEO MS scheme

1/1037   

0.013587

105.98118

106

±0.031

12 ERLER 03 THEO

1/1086   

0.012974

110.98896

111

±12 

55 BECIREVIC 98 LATT MS scheme

1/1115   

0.0126365

113.95276

114

±12 

45 MALTMAN 02 THEO MS scheme

1/1125   

0.0125242

114.97476

115

±0.06

17 NARISON 01B THEO 

1/1135   

0.0124139

115.99675

116

±0.10

10 DEDIVITIIS 03 LATT 

1/1145   

0.0123054

117.01875

117

±0.070

14 PENIN 02 THEO

1/1155   

0.0121989

118.04075

118

±17 

41 GAMIZ 03 THEO MS scheme

1/1223   

0.0115206

124.99034

125

± 2 ± 8 38 

BECIREVIC 03 LATT MS scheme

1/1262   

0.0111646

128.97613

129

±16

44 JAMIN 02 THEO MS scheme

1/1272   

0.0110768

129.99812

130

± 9 ±16 

39 CHIU 03 LATT MS scheme

1/1370   

0.0102845

140.0137

140

±15 

48 AOKI 00 LATT MS scheme

1/1448   

0.0097305

147.98529

148

±48 

56 CHETYRKIN 98 THEO MS scheme

1/1487   

0.0094753

151.97108

152

±27 

47 KOERNER 01 THEO MS scheme

1/1663   

0.0084725

169.95824

170

-3

42 ALIKHAN 02 LATT MS scheme 

 

 

 

 

 

 

 

 

 

Fig.2

 

A section of Table I illustrating how particles are compacted by the
external wave system; five particles found by experiment are shown
 set in waves with wavelengths drawn to scale (amplitude is not to
scale).


Particle jets

 

In a paper on particle jet experiments J M Campbell, M A Cullen and E W N Glover3 reported on the difficulties of extracting useful information. By measuring the width of the jets shown in figure 3 of their report and comparing the result with CLF radii predictions we show how the particle jets match CLF predictions. Six of the jet particles have already been discovered and reported by the PDG while two appear to be new discoveries

TABLE 2

fraction of

remainder

CLF

2r

particle jet

width

PDG mass

PDG ref.

a

b

c

d

e

 

 

 

 

 

1/10

2.817941013

2.75

 

 

1/11

2.561764557

 

 

 

1/12

2.348284177

 

 

 

1/13

2.167646933

 

2.1676469

13

1/14

2.012815009

 

 

 

1/15

1.878627342

 

 

 

1/16

1.761213133

 

 

 

1/17

1.65761236

 

1.6576124

17

1/18

1.565522785

1.6

1.5655228

18

1/19

1.483126849

 

 

 

1/20

1.408970506

 

 

 

1/21

1.341876673

 

 

 

1/22

1.280882278

 

 

 

1/23

1.225191745

1.21

1.2251917

23

1/24

1.174142089

 

 

 

1/25

1.127176405

 

 

 

1/26

1.083823466

1.07

 

 

1/27

1.043681856

 

1.0436819

27

1/28

1.006407504

 

1.0064075

28

1/29

0.971703797

 

 

 

1/30

0.939313671

 

 

 

1/31

0.90901323

 

 

 

1/32

0.880606566

 

 

 

1/33

0.853921519

 

0.8539215

33

1/34

0.82880618

 

 

 

1/35

0.805126004

0.81

0.805126

35

1/36

0.782761392

 

 

 

1/37

0.761605679

 

0.7616057

37

1/38

0.741563424

0.75

0.7415634

38

1/39

0.722548978

 

0.722549

39

1/40

0.704485253

 

0.7044853

40

1/41

0.687302686

 

0.6873027

41

1/42

0.670938336

 

0.6709383

42

1/43

0.655335119

 

0.6553351

43

1/44

0.640441139

0.64

0.6404411

44

1/45

0.626209114

 

0.6262091

45

1/46

0.612595872

 

0.6125959

46

1/47

0.599561918

 

 

 

1/48

0.587071044

 

 

 

1/49

0.575090003

 

 

 

1/50

0.563588203

 

 

 

1/51

0.552537453

0.55

0.5525375

51

 

 

A graph of Table 2 is shown in Fig. 3 (below).

 

Structural constants

 

Einstein showed that:

.................................................5
       c2

 

The CLF model formula shows that:

…………………………….    6
       r

 

Then:

………………………………7
c2    r

 

c2 and Fl are constants therefore:

Constant…………..……………..8

 

Because the linear force is the same for all elementary charged particles and because speed alters the position of the field force centre, the density forward of the force centre increases with speed; causing an increase in the apparent mass of the particle, when, in fact; the total matter content of the particle remains unchanged. Energy increases with increasing speed because of the increase in density forward of the field force centre, and is at it maximum on the shortest radial,.


 

Circular waves

 

Using the Laughlin sequence to visualize the internal wave structure we show how wave rotation creates the prism effect that leads to the ring pattern observed in experiments.

 

 

Returning to atomic structure, the wave pattern shows how the wave structure contains the nuclear particles and why nuclear particles undergo a reduction in volume. Particles compacted between the rings are subject to compression in one plane only and do not undergo a change in volume because they can expand in the non-compaction (concentric) plane, but, Particles compacted in the nucleus are subject to compaction in all planes and consequently are forced to undergo a reduction in volume.

 

The circumference shows a possible cause for de Broglie waves (dotted line) in that there is a variation in wave force on the particle/field surface.

 

 

Electron binding energies5

 

The average value for each sub-shell for elements 1 to 925 is used to construct table 3. Limiting the fractions to those with single digit denominators, reveals the Jain sequence; indicating that the internal wave structure is the probable cause of the Jain sequence. These fractions occur on the transverse radial of electrons as shown in Fig. 1.

 

The space between groups of sub-shells can be seen clearly in a graph of average values. It does not agree with current grouping in the two outermost shells. The current grouping is, of course; present in individual atoms, but disappears in the table of averages.

 

(Table 3 [produces the fractions used to construct the circular wave diagram).

 

Table 3

Shell

Average

value

Middle

shell value

Fraction

Of

Higher

value

1s

35923.376

35923.376

 

 

 

 

 

2s

6607.7259

 

 

2p(1/2)

6415.4193

6415.4193

 1/6

2p(3/2)

5660.2928

 

 

 

 

 

 

3s

1710.7432

 

 

3p(1/2)

1547.9213

 

 

3p(3/2)

1363.292

1363.292

 1/5

3d(3/2)

1302.9508

 

 

3d(5/2)

1257.5032

 

 

 

 

 

 

4s

481.01455

 

 

4p(1/2)

396.22807

 

 

4p(3/2)

338.29123

338.29123

 1/4

4d(3/2)

275.32955

 

 

4d(5/2)

258.94318

 

 

 

 

 

 

4f(5/2)

112.325

 

 

4f(7/2)

104.26765

104.26765

 1/3

5s

110.00811

 

 

 

 

 

 

5p(1/2)

78.442105

 

 

5p(3/2)

63.705405

 

 

5d(3/2)

52.784615

52.784615

 1/2

5d(5/2)

50.830769

 

 

6s

37.86

 

 

6p(1/2)

21.325

 

 

 

 

By extending the EBE energy investigation to cover each sub-shell we find that the following fractions are (Heiselberg) n1 sequence fractions:

 

2p(1/2)   and    2p(3/2)
   1s                   1s

 

Shells 3,4,5, and sub-shell 2s divided by 1s produce fractions in the sequence:

 

1    1    1    1    1     1     etc

2    3    4    5    6     7

Laughlin sequence fractions are shown in bold type, the complete sequence will be referred to as the 'modified Laughlin sequence'. The complete table is shown in appendix A.

 

 

The modified Laughlin sequence agrees with the wave structure shown in fig. 2 and the tables shown in fig.1 explain the appearance of the n1 sequence in the following manner:

 

Within an atom, electrons are compacted in the transverse plane (modified Laughlin sequence), allowing the longitudinal plane to spread on the atomic field concentric. When the atomic field concentric sphere is no longer large enough to allow for transverse plane electron compaction, the electrons are forced to compact on the longitudinal plane (Heiselberg n1 sequence). When there is insufficient space for both transverse and longitudinal compaction; the electron is forced to compact in all planes (spherical compaction). The particle is now small enough to allow for further compaction in the transverse plane. When there is insufficient space for plane or spherical compaction, further (spherical) compaction is only possible by the creation of composite particles.

 

 

Summary

 

We have shown that compaction of a single elementary particle is responsible for the creation of all observed charged elementary particles. The fractional sequence:

1    1    1    1    1     1     etc

2    3    4    5    6     7

used to determine compaction changes is similar to that found in the compaction of particles within an atomic field, as shown by an examination of the Electron Binding Force values. There is however one obvious difference, collision experiment particle compaction occurs in steps that are fractions of the remainder (remaining radius) while atomic compaction occurs in fractions that are fractions of the whole (1s); this occurs because atomic compaction is compaction within a field; collision compaction is compaction within a vortex.

 

 

References 

1 http://nobelprize.org/nobel_prizes/physics/laureates/1998/press.html

2 arXiv:cond-mat/0510688 v2 20 Dec 2005

3arXiv:hep-ph/9809429v2 17 Nov 1998

4 ‘The structure of Spiral Galaxies' by Berlini and Linn.

5 The Elements, John Emsley, OUP, Third edition, 2000

 

 

 

Appendix A

 

             
    Laughlin     n1  
    fractions     sequence  
             
  2s/1s 2s/1s    2p(1/2)/1s  2p(1/2)/1s  
Element   True sequence   True sequence  
 No. EBE fraction fraction error fraction fraction error
(1s) a b (a-b) a b (a-b)
             
7 409.9 0.090998    1/11 0.000089      
8 543.1 0.076597    1/13 -0.000326      
9 696.7     0.000000      
10 870.2 0.055734    1/18 0.000179 0.447423  4/9 0.002978
11 1070.8 0.059301    1/17 0.000478 0.478740  1/2 -0.021260
12 1303 0.067997    1/15 0.001330 0.559819  5/9 0.004264
13 1559 0.075561    1/13 -0.001362 0.618846  5/8 -0.006154
14 1839 0.081403    1/12 -0.001930 0.666667  2/3 0.000000
15 2145.5 0.088091    1/11 -0.002818 0.719577  5/7 0.005291
16 2472 0.093406    1/11 0.002497 0.708532  5/7 -0.005754
17 2822 0.095677    1/10 -0.004323 0.748148  3/4 -0.001852
18 3205.9 0.101781    1/9  -0.009330 0.768005  3/4 0.018005
19 3608.4 0.104922    1/9  -0.006189 0.785261  7/9 0.007484
20 4038.5 0.108555    1/9  -0.002556 0.797673  4/5 -0.002327
21 4492 0.110864    1/9  -0.000247 0.810442  4/5 0.010442
22 4966 0.112948    1/9  0.001837 0.820467  5/6 -0.012866
23 5465 0.114675    1/9  0.003564 0.829424  5/6 -0.003909
24 5989 0.116213    1/9  0.005102 0.838793  5/6 0.005460
25 6539 0.117617    1/9  0.006506 0.845014  5/6 0.011680
26 7112 0.118757    1/8  -0.006243 0.852356  6/7 -0.004787
27 7709 0.120003    1/8  -0.004997 0.857421  6/7 0.000278
28 8333 0.121037    1/8  -0.003963 0.862582  6/7 0.005439
29 8979 0.122141    1/8  -0.002859 0.868332    7/8  -0.006668
30 9659 0.123843    1/8  -0.001157 0.873516    7/8  -0.001484
31 10367 0.125301    1/8  0.000301 0.880062    7/8  0.005062
32 11103 0.127407    1/8  0.002407 0.882299    7/8  0.007299
33 11867 0.128676    1/8  0.003676 0.890046    8/9  0.001157
34 12658 0.130510    1/8  0.005510 0.892433    8/9  0.003545
35 13474 0.132255    1/8  0.007255 0.895623    8/9  0.006734
36 14326 0.134092    1/7  -0.008765 0.901041    8/9  0.012152
37 15200 0.135855    1/7  -0.007002 0.902663    8/9  0.013775
38 16105 0.137597    1/7  -0.005260 0.905686    8/9  0.016797
39 17038 0.139277    1/7  -0.003580 0.908555    8/9  0.019666
40 17993 0.140721    1/7  -0.002136 0.911137    8/9  0.022249
41 18986 0.142105    1/7  -0.000752 0.913640    8/9  0.024751
42 20000 0.143300    1/7  0.000443 0.925331    8/9  0.036443
43 21044 0.144602    1/7  0.001745 0.917844    8/9  0.028955
44 22117 0.145770    1/7  0.002913 0.920285    8/9  0.031396
45 23220 0.146942    1/7  0.004085 0.922040    8/9  0.033151
46 24350 0.148008    1/7  0.005151 0.923973    8/9  0.035084
47 25514 0.149173    1/7  0.006316 0.925906    8/9  0.037018
48 26711 0.150425    1/7  0.007568 0.927576    8/9  0.038687
49 27940 0.151682    1/7  0.008825 0.929212    8/9  0.040323
50 29200 0.152911    1/7  0.010054 0.930795    8/9  0.041906
51 30491 0.153914    1/6  -0.012752 0.933305    8/9  0.044416
52 31814 0.155246    1/6  -0.011421 0.933792    8/9  0.044903
53 33169 0.156411    1/6  -0.010256 0.935235    8/9  0.046346
54 34561 0.157779    1/6  -0.008888 0.936549    8/9  0.047660
55 35985 0.158788    1/6  -0.007878 0.937872    8/9  0.048983
56 37441 0.159958    1/6  -0.006708 0.939055    8/9  0.050166
57 38925 0.160976    1/6  -0.005690 0.940153    8/9  0.051264
58 40443 0.161907    1/6  -0.004760 0.941356    8/9  0.052467
59 41991 0.162773    1/6  -0.003894 0.942209    8/9  0.053320
60 43569 0.163557    1/6  -0.003110 0.943306    8/9  0.054417
61 45184 0.164394    1/6  -0.002272 0.944130    8/9  0.055241
62 46834 0.165200    1/6  -0.001466 0.945069    8/9  0.056180
63 48519 0.165956    1/6  -0.000711 0.945976    8/9  0.057087
64 50239 0.166723    1/6  0.000056 0.946753    8/9  0.057864
65 51996 0.167474    1/6  0.000808 0.947634    8/9  0.058745
66 53789 0.168176    1/6  0.001509 0.948596    8/9  0.059707
67 55618 0.168902    1/6  0.002235 0.949329    8/9  0.060440
68 57486 0.169624    1/6  0.002957 0.950056    8/9  0.061168
69 59390 0.170332    1/6  0.003665 0.950672    8/9  0.061783
70 61332 0.170971    1/6  0.004304 0.951554    8/9  0.062666
71 63314 0.171684    1/6  0.005017 0.952070    8/9  0.063181
72 65351 0.172469    1/6  0.005802 0.952799    8/9  0.063910
73 67416 0.173282    1/6  0.006616 0.953261    8/9  0.064373
74 69525 0.174038    1/6  0.007371 0.954050    8/9  0.065161
75 71676 0.174773    1/6  0.008106 0.954658    8/9  0.065769
76 73871 0.175793    1/6  0.009126 0.953719    8/9  0.064831
77 76111 0.176308    1/6  0.009642 0.955660    8/9  0.066771
78 78395 0.177052    1/6  0.010385 0.956268    8/9  0.067379
79 80725 0.177801    1/6  0.011135 0.956873    8/9  0.067984
80 83102 0.178564    1/6  0.011897 0.957544    8/9  0.068655
81 85530 0.179434    1/6  0.012767 0.957712    8/9  0.068823
82 88005 0.180228    1/6  0.013562 0.958325    8/9  0.069437
83 90526 0.181031    1/6  0.014364 0.958689    8/9  0.069800
84 93105 0.181934    1/5  -0.018066 0.958970    8/9  0.070082
85 95730 0.182733    1/5  -0.017267 0.959527    8/9  0.070638
86 98404 0.183417    1/5  -0.016583 0.960552    8/9  0.071663
87 101137 0.184295    1/5  -0.015705 0.960728    8/9  0.071839
88 103922 0.185110    1/5  -0.014890 0.960857    8/9  0.071968
89 106755 0.185846    1/5  -0.014154 0.961845    8/9  0.072956
90 109651 0.186701    1/5  -0.013299 0.961948    8/9  0.073059
91 112601 0.187432    1/5  -0.012568 0.962521    8/9  0.073632
92 115606 0.188200    1/5  -0.011800 0.964471    8/9  0.075582

 

             
    Laughlin     n1  
    fractions     sequence  
             
  2s/1s 2s/1s    2p(1/2)/1s  2p(1/2)/1s  
Element   True sequence   True sequence  
 No. EBE fraction fraction error fraction fraction error
(1s) a b (a-b) a b (a-b)
             
7 409.9 0.090998    1/11 0.000089      
8 543.1 0.076597    1/13 -0.000326      
9 696.7     0.000000      
10 870.2 0.055734    1/18 0.000179 0.447423  4/9 0.002978
11 1070.8 0.059301    1/17 0.000478 0.478740  1/2 -0.021260
12 1303 0.067997    1/15 0.001330 0.559819  5/9 0.004264
13 1559 0.075561    1/13 -0.001362 0.618846  5/8 -0.006154
14 1839 0.081403    1/12 -0.001930 0.666667  2/3 0.000000
15 2145.5 0.088091    1/11 -0.002818 0.719577  5/7 0.005291
16 2472 0.093406    1/11 0.002497 0.708532  5/7 -0.005754
17 2822 0.095677    1/10 -0.004323 0.748148  3/4 -0.001852
18 3205.9 0.101781    1/9  -0.009330 0.768005  3/4 0.018005
19 3608.4 0.104922    1/9  -0.006189 0.785261  7/9 0.007484
20 4038.5 0.108555    1/9  -0.002556 0.797673  4/5 -0.002327
21 4492 0.110864    1/9  -0.000247 0.810442  4/5 0.010442
22 4966 0.112948    1/9  0.001837 0.820467  5/6 -0.012866
23 5465 0.114675    1/9  0.003564 0.829424  5/6 -0.003909
24 5989 0.116213    1/9  0.005102 0.838793  5/6 0.005460
25 6539 0.117617    1/9  0.006506 0.845014  5/6 0.011680
26 7112 0.118757    1/8  -0.006243 0.852356  6/7 -0.004787
27 7709 0.120003    1/8  -0.004997 0.857421  6/7 0.000278
28 8333 0.121037    1/8  -0.003963 0.862582  6/7 0.005439
29 8979 0.122141    1/8  -0.002859 0.868332    7/8  -0.006668
30 9659 0.123843    1/8  -0.001157 0.873516    7/8  -0.001484
31 10367 0.125301    1/8  0.000301 0.880062    7/8  0.005062
32 11103 0.127407    1/8  0.002407 0.882299    7/8  0.007299
33 11867 0.128676    1/8  0.003676 0.890046    8/9  0.001157
34 12658 0.130510    1/8  0.005510 0.892433    8/9  0.003545
35 13474 0.132255    1/8  0.007255 0.895623    8/9  0.006734
36 14326 0.134092    1/7  -0.008765 0.901041    8/9  0.012152
37 15200 0.135855    1/7  -0.007002 0.902663    8/9  0.013775
38 16105 0.137597    1/7  -0.005260 0.905686    8/9  0.016797
39 17038 0.139277    1/7  -0.003580 0.908555    8/9  0.019666
40 17993 0.140721    1/7  -0.002136 0.911137    8/9  0.022249
41 18986 0.142105    1/7  -0.000752 0.913640    8/9  0.024751
42 20000 0.143300    1/7  0.000443 0.925331    8/9  0.036443
43 21044 0.144602    1/7  0.001745 0.917844    8/9  0.028955
44 22117 0.145770    1/7  0.002913 0.920285    8/9  0.031396
45 23220 0.146942    1/7  0.004085 0.922040    8/9  0.033151
46 24350 0.148008    1/7  0.005151 0.923973    8/9  0.035084
47 25514 0.149173    1/7  0.006316 0.925906    8/9  0.037018
48 26711 0.150425    1/7  0.007568 0.927576    8/9  0.038687
49 27940 0.151682    1/7  0.008825 0.929212    8/9  0.040323
50 29200 0.152911    1/7  0.010054 0.930795    8/9  0.041906
51 30491 0.153914    1/6  -0.012752 0.933305    8/9  0.044416
52 31814 0.155246    1/6  -0.011421 0.933792    8/9  0.044903
53 33169 0.156411    1/6  -0.010256 0.935235    8/9  0.046346
54 34561 0.157779    1/6  -0.008888 0.936549    8/9  0.047660
55 35985 0.158788    1/6  -0.007878 0.937872    8/9  0.048983
56 37441 0.159958    1/6  -0.006708 0.939055    8/9  0.050166
57 38925 0.160976    1/6  -0.005690 0.940153    8/9  0.051264
58 40443 0.161907    1/6  -0.004760 0.941356    8/9  0.052467
59 41991 0.162773    1/6  -0.003894 0.942209    8/9  0.053320
60 43569 0.163557    1/6  -0.003110 0.943306    8/9  0.054417
61 45184 0.164394    1/6  -0.002272 0.944130    8/9  0.055241
62 46834 0.165200    1/6  -0.001466 0.945069    8/9  0.056180
63 48519 0.165956    1/6  -0.000711 0.945976    8/9  0.057087
64 50239 0.166723    1/6  0.000056 0.946753    8/9  0.057864
65 51996 0.167474    1/6  0.000808 0.947634    8/9  0.058745
66 53789 0.168176    1/6  0.001509 0.948596    8/9  0.059707
67 55618 0.168902    1/6  0.002235 0.949329    8/9  0.060440
68 57486 0.169624    1/6  0.002957 0.950056    8/9  0.061168
69 59390 0.170332    1/6  0.003665 0.950672    8/9  0.061783
70 61332 0.170971    1/6  0.004304 0.951554    8/9  0.062666
71 63314 0.171684    1/6  0.005017 0.952070    8/9  0.063181
72 65351 0.172469    1/6  0.005802 0.952799    8/9  0.063910
73 67416 0.173282    1/6  0.006616 0.953261    8/9  0.064373
74 69525 0.174038    1/6  0.007371 0.954050    8/9  0.065161
75 71676 0.174773    1/6  0.008106 0.954658    8/9  0.065769
76 73871 0.175793    1/6  0.009126 0.953719    8/9  0.064831
77 76111 0.176308    1/6  0.009642 0.955660    8/9  0.066771
78 78395 0.177052    1/6  0.010385 0.956268    8/9  0.067379
79 80725 0.177801    1/6  0.011135 0.956873    8/9  0.067984
80 83102 0.178564    1/6  0.011897 0.957544    8/9  0.068655
81 85530 0.179434    1/6  0.012767 0.957712    8/9  0.068823
82 88005 0.180228    1/6  0.013562 0.958325    8/9  0.069437
83 90526 0.181031    1/6  0.014364 0.958689    8/9  0.069800
84 93105 0.181934    1/5  -0.018066 0.958970    8/9  0.070082
85 95730 0.182733    1/5  -0.017267 0.959527    8/9  0.070638
86 98404 0.183417    1/5  -0.016583 0.960552    8/9  0.071663
87 101137 0.184295    1/5  -0.015705 0.960728    8/9  0.071839
88 103922 0.185110    1/5  -0.014890 0.960857    8/9  0.071968
89 106755 0.185846    1/5  -0.014154 0.961845    8/9  0.072956
90 109651 0.186701    1/5  -0.013299 0.961948    8/9  0.073059
91 112601 0.187432    1/5  -0.012568 0.962521    8/9  0.073632
92 115606 0.188200    1/5  -0.011800 0.964471    8/9  0.075582
 
             
    n1     Laughlin  
    sequence     fractions  
             
  2p(3/2)/1s 2p(3/2)/1s   3s/1s 3s/1s  
Element   True sequence   True sequence  
 No. EBE fraction fraction error fraction fraction error
(1s) a b (a-b) a b (a-b)
             
10 870.2 0.445361  4/9 0.000916      
11 1070.8 0.480315  1/2 -0.019685      
12 1303 0.555305  5/9 -0.000251      
13 1559 0.615450  5/8 -0.009550      
14 1839 0.662659  2/3 -0.004008      
15 2145.5 0.714286  5/7 0.000000      
16 2472 0.703768  5/7 -0.010518      
17 2822 0.740741  3/4 -0.009259      
18 3205.9 0.761263  3/4 0.011263 0.009139 1/109  
19 3608.4 0.778130  7/9 0.000352 0.009644    1/104 0.000029
20 4038.5 0.789690  4/5 -0.010310 0.010969    1/91 -0.000020
21 4492 0.800602  4/5 0.000602 0.011376    1/88 0.000012
22 4966 0.809057  4/5 0.009057 0.011820    1/85 0.000056
23 5465 0.817137  4/5 0.017137 0.012132    1/82 -0.000063
24 5989 0.824856  5/6 -0.008477 0.012373    1/81 0.000027
25 6539 0.830451  5/6 -0.002882 0.012586    1/79 -0.000072
26 7112 0.836846  5/6 0.003513 0.012837    1/78 0.000017
27 7709 0.841098  5/6 0.007765 0.013102    1/76 -0.000056
28 8333 0.845429  5/6 0.012096 0.013297    1/75 -0.000037
29 8979 0.850278  6/7 -0.006865 0.013643    1/73 -0.000056
30 9659 0.854205  6/7 -0.002938 0.014474    1/69 -0.000019
31 10367 0.859430  6/7 0.002287 0.015385    1/65 0.000001
32 11103 0.860314  6/7 0.003171 0.016221    1/62 0.000092
33 11867 0.866798  7/8 -0.008202 0.017250    1/58 0.000008
34 12658 0.867978  7/8 -0.007022 0.018139    1/55 -0.000043
35 13474 0.869809  7/8 -0.005191 0.019074    1/52 -0.000157
36 14326 0.873712  7/8 -0.001288 0.020438    1/49 0.000030
37 15200 0.873608  7/8 -0.001392 0.021493    1/47 0.000217
38 16105 0.875451  7/8 0.000451 0.022273    1/45 0.000050
39 17038 0.876528  7/8 0.001528 0.023007    1/43 -0.000248
40 17993 0.877962  7/8 0.002962 0.023915    1/42 0.000105
41 18986 0.878799  7/8 0.003799 0.024576    1/41 0.000186
42 20000 0.879274  7/8 0.004274 0.025315    1/40 0.000315
43 21044 0.879724  7/8 0.004724 0.025851    1/39 0.000210
44 22117 0.880273  7/8 0.005273 0.026500    1/38 0.000184
45 23220 0.880422  7/8 0.005422 0.027050    1/37 0.000023
46 24350 0.880411  7/8 0.005411 0.027581    1/36 -0.000197
47 25514 0.880452  7/8 0.005452 0.028181    1/35 -0.000391
48 26711 0.880538  7/8 0.005538 0.028902    1/35 0.000331
49 27940 0.880132  7/8 0.005132 0.029606    1/34 0.000195
50 29200 0.879955  7/8 0.004955 0.030298    1/33 -0.000005
51 30491 0.880460  7/8 0.005460 0.031026    1/32 -0.000224
52 31814 0.878923  7/8 0.003923 0.031621    1/32 0.000371
53 33169 0.878373  7/8 0.003373 0.032319    1/31 0.000061
54 34561 0.877682  7/8 0.002682 0.033237    1/30 -0.000096
55 35985 0.877144  7/8 0.002144 0.033653    1/30 0.000320
56 37441 0.876106  7/8 0.001106 0.034534    1/29 0.000052
57 38925 0.875040  7/8 0.000040 0.034990    1/29 0.000508
58 40443 0.874007  7/8 -0.000993 0.035507    1/28 -0.000208
59 41991 0.872568  7/8 -0.002432 0.035984    1/28 0.000270
60 43569 0.871176  7/8 -0.003824 0.036150    1/28 0.000435
61 45184 0.869548  7/8 -0.005452     0.000000
62 46834 0.868037  7/8 -0.006963 0.036790    1/27 -0.000248
63 48519 0.866493   6/7 0.009350 0.037099    1/27 0.000062
64 50239 0.864733   6/7 0.007590 0.037441    1/27 0.000404
65 51996 0.862885   6/7 0.005742 0.037849    1/26 -0.000612
66 53789 0.861154   6/7 0.004011 0.038056    1/26 -0.000405
67 55618 0.859165   6/7 0.002023 0.038261    1/26 -0.000201
68 57486 0.857143   6/7 0.000000 0.038375    1/26 -0.000087
69 59390 0.854883   6/7 -0.002260 0.038845    3/77 -0.000116
70 61332 0.852947   6/7 -0.004196 0.039099    2/51 -0.000117
71 63314 0.850414   6/7 -0.006729 0.039344    2/51 0.000128
72 65351 0.848283   6/7 -0.008860 0.039800    1/25 -0.000200
73 67416 0.845831   5/6 0.012498 0.040169    1/25 0.000169
74 69525 0.843554   5/6 0.010220 0.040561    3/74 0.000020
75 71676 0.840983   5/6 0.007650 0.040906    2/49 0.000090
76 73871 0.837132   5/6 0.003799 0.041275    4/97 0.000038
77 76111 0.835755   5/6 0.002422 0.041702    1/24 0.000036
78 78395 0.833141   5/6 -0.000192 0.042043    1/24 0.000377
79 80725 0.830419   5/6 -0.002915 0.042428   0.042428
80 83102 0.827819   5/6 -0.005515 0.042863    1/23 -0.000615
81 85530 0.824787   5/6 -0.008547 0.043306    1/23 -0.000172
82 88005 0.823088   5/6 -0.010245 0.043759    1/23 0.000281
83 90526 0.818831   5/6 -0.014502 0.044175    1/23 0.000697
84 93105 0.815514   4/5 0.015514 0.044563   0.044563
85 95730 0.812554   4/5 0.012554 0.045096    1/22 -0.000359
86 98404 0.809962   4/5 0.009962 0.045547    1/22 0.000092
87 101137 0.806427   4/5 0.006427 0.045997    1/22 0.000542
88 103922 0.802828   4/5 0.002828 0.046400   0.046400
89 106755 0.799950   4/5 -0.000050 0.046855    1/21 -0.000764
90 109651 0.796209   4/5 -0.003791 0.047259    1/21 -0.000360
91 112601 0.792845   4/5 -0.007155 0.047664    1/21 0.000045
92 115606 0.788987   4/5 -0.011013 0.047991    1/21 0.000372
 

 

 

             
    Laughlin     Laughlin  
    fractions     fractions  
             
  3p(1/2)/1s 3p(1/2)/1s   3p(3/2)/1s 3p(3/2)/1s  
Element   True sequence   True sequence  
 No. EBE fraction fraction error fraction fraction error
(1s) a b (a-b) a b (a-b)
             
18 3205.9 0.004960    1/202 0.000009 0.004897    1/204 -0.000005
19 3608.4 0.005071    1/197 -0.000005 0.005071    1/197 -0.000005
20 4038.5 0.006289    1/159 0.000000 0.006289    1/159 0.000000
21 4492 0.006300    1/159 0.000011 0.006300    1/159 0.000011
22 4966 0.006565    1/152 -0.000014 0.006565    1/152 -0.000014
23 5465 0.006807    1/147 0.000004 0.006807    1/147 0.000004
24 5989 0.007046    1/142 0.000004 0.007046    1/142 0.000004
25 6539 0.007218    1/139 0.000024 0.007218    1/139 0.000024
26 7112 0.007410    1/135 0.000003 0.007410    1/135 0.000003
27 7709 0.007640    1/131 0.000007 0.007640    1/131 0.000007
28 8333 0.008160    1/123 0.000030 0.007944    1/126 0.000008
29 8979 0.008609    1/116 -0.000012 0.008364    1/120 0.000031
30 9659 0.009463    1/108 0.000203 0.009173    1/109 -0.000002
31 10367 0.009984    1/100 -0.000016 0.009646    1/104 0.000031
32 11103 0.011249    1/89 0.000013 0.010880    1/92 0.000010
33 11867 0.012320    1/81 -0.000026 0.011899    1/84 -0.000006
34 12658 0.013154    1/76 -0.000004 0.012696    1/79 0.000037
35 13474 0.014027    1/71 -0.000057 0.013507    1/74 -0.000006
36 14326 0.015510    1/64 -0.000115 0.014966    1/67 0.000040
37 15200 0.016362    1/61 -0.000032 0.015730    1/64 0.000105
38 16105 0.017405    1/57 -0.000139 0.016765    1/60 0.000098
39 17038 0.018230    1/55 0.000048 0.017537    1/57 -0.000007
40 17993 0.019091    1/52 -0.000140 0.018329    1/55 0.000148
41 18986 0.019809    1/50 -0.000191 0.018993    1/53 0.000125
42 20000 0.020580    1/49 0.000172 0.019700    1/51 0.000092
43 21044 0.021270    1/47 -0.000007 0.019849    1/50 -0.000151
44 22117 0.021870    1/46 0.000131 0.020866    1/48 0.000033
45 23220 0.022450    1/45 0.000228 0.021382    1/47 0.000106
46 24350 0.022994    1/43 -0.000262 0.021860    1/46 0.000121
47 25514 0.023665    1/42 -0.000144 0.022458    1/46 0.000719
48 26711 0.024432    1/41 0.000042 0.023152    1/43 -0.000104
49 27940 0.025168