### Nucleus Kinetic Energy

The following equations are for the ground state of the single-electron ion.  Beta represents the electron velocity as a fraction of the speed of light and is defined in this model as:

$\beta =v/c=\alpha Z$

The electron's kinetic energy is, therefore, given by:

$E_{k}=(\gamma -1)m_{e}c^{2}$

Where;

$\gamma =\frac{1}{\sqrt{1-\beta ^{2}}}=\frac{1}{\sqrt{1-\alpha ^{2}Z^{2}}}$

It will also be useful to express beta as a function of gamma:

$\beta =\frac{\sqrt{\gamma ^{2}-1}}{\gamma }$

So that;

$\gamma \beta =\sqrt{\gamma ^{2}-1}$

The momentum of the electron is:

$p=\gamma m_{e}v=\gamma \beta m_{e}c=\sqrt{\gamma ^{2}-1}m_{e}c$

Any impulse (momentum change) that the nucleus imparts to the electron is always matched by an equal and opposite impulse on the nucleus.  It is, therefore, assumed that the momentum of the nucleus is identical to the momentum of the electron.

For a nucleus of finite mass, this momentum represents a kinetic energy of:

$E_{nuc}=\frac{p^{2}}{2m_{n}}=\frac{1}{2}(\gamma ^{2}-1)\frac{m_{e}}{m_{n}}m_{e}c^{2}$

(Since the velocity of the nucleus is so much smaller than the speed of light, relativistic effects are ignored.)

This kinetic energy is the energy stored in the wobble of the nucleus.  Since kinetic energy is always positive, it is dissociative. (i.e., it will help the ionization process.)

The energy which must be added to remove the electron is, therefore, less than it would have been for an infinite nuclear mass.

$E_{escape}=E_{\infty }-E_{nuc}$

The proposed model defines the the theoretical ionization energy for an infinite nuclear mass:

$E_{\infty }=\left ( 1-\frac{1}{\gamma } \right )m_{e}c^{2}$

The ionizing particle must increase the electron's energy by:

$E_{escape}=\left ( 1-\frac{1}{\gamma }-\frac{1}{2}(\gamma ^{2}-1)\frac{m_{e}}{m_{n}} \right )m_{e}c^{2}$

By equating the escape energy to the kinetic energy of an electron, an equivalent escape-gamma can be inferred.

$(\gamma _{escape}-1)m_{e}c^{2}=\left ( 1-\frac{1}{\gamma }-\frac{1}{2}(\gamma ^{2}-1)\frac{m_{e}}{m_{n}} \right )m_{e}c^{2}$

$\gamma _{escape}= 2-\frac{1}{\gamma }-\frac{1}{2}(\gamma ^{2}-1)\frac{m_{e}}{m_{n}}$

The ionizing particle must transfer an impulse to the electron of:

$p_{escape} =\sqrt{\gamma _{escape}^{2}-1}m_{e}c^{2}$

As before, any impulse delivered to the electron will also be delivered to  the nucleus.  This ionizing impulse will give the nucleus a residual translational energy of:

$E_{residual}=\frac{1}{2}(\gamma _{escape}^{2}-1)\frac{m_{e}}{m_{n}}m_{e}c^{2}$

The ionizing particle must provide both the escape energy and the residual kinetic energy of the nucleus.  The observed ionization energy is, therefore:

$E_{obs}=E_{escape}+E_{residual}=E_{\infty }-E_{nuc}+E_{residual}$

A B C D E F
Z
W
W
Element
Nucleus
kinetic
energy
eV

Observed
+nucleus
(D+E)
eV

1

1.008
2.0141
H Hydrogen
D Deuterium
0.00740
0.00371
13.59844
13.60000
13.60584
13.60584
2
4.003
He Helium54.4182154.417780.000430.0000079
3
6.941
Li Lithium122.45622122.454290.001930.0000158
4
9.012
Be Beryllium217.72421217.718650.005560.0000255
5
10.811
B Boron340.23832340.225800.012520.0000368
6
12.01
C Carbon490.01750489.993340.024160.0000493
7
14
N Nitrogen667.08823667.046000.042230.0000633
8
16
O Oxygen871.47758871.410100.067480.0000774
9
19
F Fluorine1103.220161103.117600.102560.0000930
10
20
Ne Neon1362.347971362.199500.148470.0001090
11
23
Na Sodium1648.909901648.702000.207900.0001261
12
24.3
Mg Magnesium1962.945711962.665000.280710.0001430
13
27
Al Aluminium2304.511792304.141000.370790.0001609
14
28
Si Silicon2673.658092673.182000.476090.0001781
15
31
P Phosphorus3070.451433069.842000.609430.0001985
16
32
S Sulfur3494.949393494.189200.760190.0002175
17
35.5
Cl Chlorine3947.229713946.296000.933710.0002365
18
40
Ar Argon4427.363634426.229601.134030.0002561
19
39
K Potassium4935.420114934.046001.374110.0002784
20
40
Ca Calcium5471.495425469.864001.631420.0002982
21
45
Sc Scandium6035.682786033.712001.970780.0003265
22
48
Ti Titanium6628.065806625.820002.245800.0003388
23
51
24
52
Cr Chromium7897.829647894.810003.019640.0003823
25
55
Mn Manganese8575.428488571.940003.488480.0004068
26
56
Fe Iron9281.653269277.690003.963260.0004270
27
59
Co Cobalt10016.6314210012.120004.511420.0004504
28
59
Ni Nickel10780.4816110775.400005.081610.0004714
29
64
Cu Copper11573.3493611567.617005.732360.0004953

A B C D E F
Z
W
Element Calculated
Bonding
Energy
eV
Ionization
Potential
+Z-1 to +Z
(CRC handbook)
eV
Absolute
Error
(C-D)
eV
Relative
Error
(E/C)
1
1.008
H HydrogenH Hydrogen13.598440.000030.0000024
2
4.003
He Helium54.4182154.417780.000430.0000079
3
6.941
Li Lithium122.45622122.454290.001930.0000158
4
9.012
Be Beryllium217.72421217.718650.005560.0000255
5
10.811
B Boron340.23832340.225800.012520.0000368
6
12.01
C Carbon490.01750489.993340.024160.0000493
7
14
N Nitrogen667.08823667.046000.042230.0000633
8
16
O Oxygen871.47758871.410100.067480.0000774
9
19
F Fluorine1103.220161103.117600.102560.0000930
10
20
Ne Neon1362.347971362.199500.148470.0001090
11
23
Na Sodium1648.909901648.702000.207900.0001261
12
24.3
Mg Magnesium1962.945711962.665000.280710.0001430
13
27
Al Aluminium2304.511792304.141000.370790.0001609
14
28
Si Silicon2673.658092673.182000.476090.0001781
15
31
P Phosphorus3070.451433069.842000.609430.0001985
16
32
S Sulfur3494.949393494.189200.760190.0002175
17
35.5
Cl Chlorine3947.229713946.296000.933710.0002365
18
40
Ar Argon4427.363634426.229601.134030.0002561
19
39
K Potassium4935.420114934.046001.374110.0002784
20
40
Ca Calcium5471.495425469.864001.631420.0002982
21
45
Sc Scandium6035.682786033.712001.970780.0003265
22
48
Ti Titanium6628.065806625.820002.245800.0003388
23
51
24
52
Cr Chromium7897.829647894.810003.019640.0003823
25
55
Mn Manganese8575.428488571.940003.488480.0004068
26
56
Fe Iron9281.653269277.690003.963260.0004270
27
59
Co Cobalt10016.6314210012.120004.511420.0004504
28
59
Ni Nickel10780.4816110775.400005.081610.0004714
29
64
Cu Copper11573.3493611567.617005.732360.0004953

A B C D E F
Z
W
ElementNucleus
kinetic energy
eV
Observed
eV
Observed
(corrected)
D+E
1
1.008H Hydrogen13.5984413.598440.0000024
2
4.003He Helium54.4177854.417780.0000079
3
6.941122.45429122.454290.0000158
4
9.012
Be Beryllium217.72421217.718650.005560.0000255
5
10.811
B Boron340.23832340.225800.012520.0000368
6
12.01
C Carbon490.01750489.993340.024160.0000493
7
14
N Nitrogen667.08823667.046000.042230.0000633
8
16
O Oxygen871.47758871.410100.067480.0000774
9
19
F Fluorine1103.220161103.117600.102560.0000930
10
20
Ne Neon1362.347971362.199500.148470.0001090
11
23
Na Sodium1648.909901648.702000.207900.0001261
12
24.3
Mg Magnesium1962.945711962.665000.280710.0001430
13
27
Al Aluminium2304.511792304.141000.370790.0001609
14
28
Si Silicon2673.658092673.182000.476090.0001781
15
31
P Phosphorus3070.451433069.842000.609430.0001985
16
32
S Sulfur3494.949393494.189200.760190.0002175
17
35.5
Cl Chlorine3947.229713946.296000.933710.0002365
18
40
Ar Argon4427.363634426.229601.134030.0002561
19
39
K Potassium4935.420114934.046001.374110.0002784
20
40
Ca Calcium5471.495425469.864001.631420.0002982
21
45
Sc Scandium6035.682786033.712001.970780.0003265
22
48
Ti Titanium6628.065806625.820002.245800.0003388
23
51
24
52
Cr Chromium7897.829647894.810003.019640.0003823
25
55
Mn Manganese8575.428488571.940003.488480.0004068
26
56
Fe Iron9281.653269277.690003.963260.0004270
27
59
Co Cobalt10016.6314210012.120004.511420.0004504
28
59
Ni Nickel10780.4816110775.400005.081610.0004714
29
64
Cu Copper11573.3493611567.617005.732360.0004953