Consider this: Coulomb's Law says that the electrical attraction between two oppositely charged particles is DIRECTLY proportional to the charge of each of the two (as well as a dependence on the inverse square of the distance between them.) But in 1913, a scientist named Moseley discovered that the radiation created by different elements is proportional to the SQUARE of the electrical charge of the nucleus. Consider the simplest possible situation inside an atom, a single negativelly-charged electron orbiting a (very tiny) nucleus which has a single positive charge in it. This is a standard hydrogen atom. There is obviously some energy of attraction between the opposite charges, and we will call that 13.59 electron-Volts. Now, let's consider the same situation, but where the nucleus contains TWO positive charges, but there is still one electron orbiting it. This is an atom of helium which has been once ionized. Coulomb's Law indicates that the attraction between the electron and nucleus should be TWICE as great, right? But when this is actually measured, the energy turns out it is actually FOUR times as great, extremely accurately (54.42 electron-volts).

We might think that was just a coincidence, because we KNOW that Coulomb's Law is true! So we now examine a twice-ionized Lithium atom, where a single electron orbits a nucleus with THREE charges in it. Is the attraction three times? No. You probably already guess that it is experimentally shown to be NINE times, again, extremely accurately (122.45 eV). And a triply-ionized Beryllium atom (one electron circling a nucleus with four charges in it) is experimentally shown to have 16 times the binding energy (217.72 eV). And a quadruply-ionized Boron atom has 25 times (340.23 eV). These are all remarkably accurate multiples of the hydrogen binding energy.

We have not considered the radius of the orbits of these electrons, simply the SQUARE of the positive nuclear charge of these atoms. In 1913, Moseley did massive research of the radiation given off by various atoms when an electron was removed, and Moseley's data charts show very impressive exact 'electric charge squared' dependency. Yes, thed orbital radii vary in a way that Coulomb's Law appears to be preserved, but the evidence shown here suggest a real importance of Moseley's research.

This is NOT an accident! Researchers have managed to check 32 different elements
where there is a single electron orbiting the nucleus, and they
all very accurately follow this same pattern, where even a 31-times-ionized
Germanium atom has 32^{2} or 1024 times as great a binding
energy of that single electron (14,119.4 eV).

With all variables in the atoms being identical, with the exception that the ionized Germanium nucleus has 32 times the electrical charge, there is no logical way of explaining why the Binding energy would be 32^2 or 1024 times as great.

This all absolutely contradicts the basic claim of Coulomb's Law! And not just in random ways but in very precise ways.

These findings also seem to provide an actual physical meaning behind the equations used for the Balmer series of lines of Hydrogen and the Rydberg equation, as well as suggesting how Rydberg could be extended to essentially any element in any ionized state!

They also conclusively prove that the (alleged) Quantum Defect, a number that has always believed to be a random correction, is actually a specific and precise number, accurately specified by equations! No one else has ever noticed that before!

This presentation was first placed on the Internet in June 2007.

This analysis shows that the concepts can also be extended to describing atoms with multiple electrons, with all the equations still accurately applying. And further, the entire structure of the atomic shells and sub-shells, are all clearly predicted by surprisingly simple mathematical equations.

It certainly appears that Coulomb was wrong regarding the electrical attractions inside atoms!

Each neutral element has equal numbers of protons in the nucleus and electrons orbiting it. We are going to look at this situation in the old-fashioned traditional way, where discrete electrons were actually seen as orbiting the nucleus.

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The second distinct way is where the electron is sent to an infinite distance from the nucleus, in other words, entirely removed from the atom, and which is the greatest possible amount of energy which can be inserted or removed from that electron. Effectively, this is the common situation when chemical reactions occur.

The exact wavelengths of the resulting radiation from all elements have been extremely accurately determined. This means that the exact amount of energy transfer regarding any of these ionizations is also known extremely accurately.

There are an immense number of ways that the NIST data might be
studied. The transitions from one energy state to a lower energy state
must certainly include many valuable insights, but there are a number
of variables involved, including for example a very large number of
possible target energy states. The single electron in a hydrogen
atom is known to have at least 187 raised energy levels available
to it (which is known by the fact that there are at least 187
different wavelengths of radiation emitted when the electron
later falls back to its neutral state). This also enables a
hydrogen atom to emit many thousands of different wavelengths
of light, as the electron falls from one of those states to another
of them. **For now, we choose to
avoid this complexity and only consider electrons that are
ionized to an infinite distance.**

**The electrons in atoms are in many different quantum orbitals and
sub-orbitals, and the total effect of many of them on each other
figures to be quite complex, so we wish to temporarily eliminate
this complexity as well.** It turns out that we can do that!
It turns out that researchers have discovered and experimentally
analyzed 32 different atoms which each have a single electron
orbiting the nucleus.

The hydrogen atom has one electron. NEUTRAL helium has two, but if the helium is singly-ionized, it also has one electron. NEUTRAL lithium has three electrons, but doubly-ionized lithium has one. NEUTRAL beryllium has four electrons but triply-ionized beryllium has one. And so on. The discussion here will stop at element 32, Germanium, which is 31-times-ionized, such that it too only has one remaining electron.

**Therefore, we have simplified the situation into one where there
only is a single electron** (we will not now specify any shell or
sub-shell) **orbiting around various nuclei which have (attractive)
(integer) positive charges of from one to thirty-two.**

If we look at Coulomb's Law regarding these 32 atoms, which are essentially identical except for the electrical charge of the nucleus, we might then assume that the attractive force acting should be in the ratio of 1:2:3: _ :32. That is NOT the case!

The EXPERIMENTALLY MEASURED NIST data on these atoms has an amazing pattern in it! Consider for a moment only the H I and He II atoms. The NIST ionization potentials, in electron-Volts, are 13.5984340 and 54.4177630, meaning that these values represent the actual binding energy of the single electron for these two cases, as compared to zero, the potential at infinite distance.

This is quite close to being a 1:4 proportion, within 1/20 of one percent! We might them look at the Lithium II value of 122.454353 and see that it is remarkably close to NINE times the hydrogen value, again around a match of around 1/20 of one percent. Could this be some sort of fluke, where the universally accepted experimental data seems to have a remarkable match with the square of the electric charge of the nucleus? No! The match continues, consistently amazingly accurate, throughout the entire sequence of these 32 elements.

Here is the NIST data (in electron-volts) and the nucleus-charge-squared multiple of hydrogen's.

nuclear charge | NIST ionization value | multiple of hydrogen |
---|---|---|

1 | 13.5984340 | 13.5984340 |

2 | 54.4177630 | 54.393736 |

3 | 122.454353 | 122.385906 |

4 | 217.718572 | 217.574944 |

5 | 340.225993 | 339.96085 |

6 | 489.99312 | 489.543624 |

7 | 667.04602 | 666.323266 |

8 | 871.40969 | 870.299776 |

9 | 1103.1171 | 1101.473154 |

10 | 1362.1986 | 1359.843 |

11 | 1648.70105 | 1645.4105 |

12 | 1962.6642 | 1958.1745 |

13 | 2304.1401 | 2298.1353 |

14 | 2673.1807 | 2665.2931 |

15 | 3069.84143 | 3059.648 |

16 | 3494.1877 | 3481.199 |

17 | 3946.2907 | 3929.947 |

18 | 4426.2226 | 4405.8926 |

19 | 4934.0439 | 4909.0347 |

20 | 5469.8614 | 5439.3736 |

21 | 6033.7551 | 5996.9094 |

22 | 6625.81 | 6581.6421 |

23 | 7246.1196 | 7193.5716 |

24 | 7894.80 | 7832.70 |

25 | 8571.94 | 8499.02 |

26 | 9277.6874 | 9192.54 |

27 | 10012.1 | 9913.26 |

28 | 10775.4 | 10661.17 |

29 | 11567.612 | 11436.28 |

30 | 12388.928 | 12238.59 |

31 | 13239.4881 | 13068.10 |

32 | 14119.4287 | 13924.80 |

**This is very troublesome! By doubling the charge in the nucleus,
and being careful not to change any other variables, standard
understandings in Physics do NOT allow any conclusion where the
Binding Energy becomes four times as great!**

However, this information is in exact accordance with the Balmer:

and Rydberg:

with the special case for Hydrogen Balmer being:

mathematical equations for spectral lines. (These equations are presented in their traditional form, for solving for the wave number nu. By multiplying that value by Planck's constant, the energy is determined, so we could have presented these to solve for energy in electron-Volts by that simple multiplication.)

This investigation seems to give a physical reality to the mathematics
of those equations. **A critical point is that it has generally been
assumed that all the numbers in the denominators were DISTANCES**
(because of the well-known inverse-square distance rule of
electromagnetic phenomena), but this investigation seems to
definitely prove that **the a and b parameters are rather the
charge in the nucleus!** The fact that these quantities of
charge need to be squared for these equations appears to
contradict Coulomb's Law. (The Rydberg constant is 109,677.8/cm
or 13.5978 eV equivalent, the ionization potential of neutral
hydrogen.).

For clarification, the first term of the Rydberg equation describes
the initial energy state of the electron and the second term describes
the final energy state. The initial state is therefore described by
R / (n + a)^{2}. For Hydrogen, we are describing here that
a is the reciprocal of the nuclear charge of one (which is still one),
and the first n is zero, such that
we have the energy become 13.6 eV / (1^{2}) or 13.6 eV.
Regarding the final state being at infinite radius, we have m is
infinite and the energy is 0 eV.

Here are these values graphed, which shows the parabolic shape of the curve.

And with our calculated parabola of EXACT INTEGERS in yellow:

Traditional thought would say that with double the central electrostatic attraction, then the attractive force, that is, the centripetal force must also be exactly doubled. But then all the traditional analysis of energy and angular momentum do NOT result in the energy going up as the square of that centripetal force! But it clearly must, per this parabola!

Here is the general reasoning that has always been applied, for each of planets in a gravitational field and electrons in an electrostatic field.

Coulomb's Law says that the force of attraction between two charged
objects is given by F = k * q_{1} * q_{2} / r^{2},
where the q's are the amounts of charge and r is the distance separating
them (with k a constant depending on the system of measurements).
Newton's Law similarly says that the force of attraction between two
massive objects is given by F = G * m_{1} * m_{2} /
r^{2}, where the m's are the amounts of mass and r is the
distance separating them (with G, the Gravitational constant, depending
on the system of measurements).

Newton's Law F = m * a for central force for circular motion is F =
m_{1} * v^{2} / r. This can also be written F = I *
omega^{2} or I * v^{2} / r^{2}.

It might also be speculated that the radius of the orbit of the electron would adapt itself to account for this different force. However, calculations show that that is not a viable explanation. Yes, the orbital radius WOULD change to some extent, but it is physically impossible that it could change enough to account for this full effect. An easy proof of that is that the Binding Energy is known to be in the inverse square proportion to the orbital radius. We have just been discussing a Germanium XXXII and a Hydrogen I, which each have one electron. The NIST data shows that the Ge atom has a binding energy of over 1,000 times that of the Hydrogen, meaning that its orbital radius would necessarily be 1/32nd the radius. That electron would be extremely close to being inside the nucleus!

But the standard calculations do not even result in this solution! There is NO conventional way where multiplying the charge of the nucleus by some integer can possibly result in the Binding Energy increasing by the square of that integer! At least, not if Coulomb's Law is assumed to still apply.

nuclear charge | NIST ionization value | Second NIST value |
---|---|---|

2 | 24.5873876 | |

3 | 5.3917191 | |

4 | 18.211153 | 136.80273 |

5 | 37.930620 | 205.99 |

6 | 64.49390 | |

7 | 97.89013 | |

8 | 138.1196 | |

9 | 185.1868 | |

10 | 239.0969 | |

11 | 299.864 | 1407.705 |

12 | 367.497 | 1698.61 |

13 | 441.999 | 2017.0 |

14 | 523.4203 | 2362.8411 |

15 | 611.74 | 2736.31 |

16 | 707.01 | 3137.35 |

17 | 809.2129 | |

18 | 918.3861 | |

19 | 1034.5 | |

20 | 1158 | |

21 | 1288.0 | |

22 | 1425.4 | |

23 | 1569.6399 | |

24 | 1721.1 | |

25 | 1879.8 | |

26 | 2045.7391 | |

27 | 2218.9 |

nuclear charge | NIST ionization value |
---|---|

6 | 11.26030 |

7 | 14.53413 |

8 | 35.12111 |

9 | 62.7084 |

10 | 97.1168 |

11 | 138.40 |

12 | 186.76 |

13 | 241.76 |

14 | 303.5381 |

15 | 372.13 |

16 | 447.46 |

17 | 529.2761 |

18 | 618.73 |

19 | 714.6 |

20 | 817.7 |

21 | 927.5 |

22 | 1043.947 |

23 | 1167.931 |

24 | 1299 |

25 | 1437 |

26 | 1575.5911 |

27 | 1735 |

28 | 1894 |

29 | 2060.6172 |

30 | 2234.1951 |

31 | 2418.9316 |

32 | 2665.6601 |

When an r-squared analysis was done of these various data sets, it seems that a "pure" parabolic shape was very close to the absolute best curve-fit. Of around a hundred common simple equations, the y = a + b * (x-c)

Here are the three graphs above repeated, and some other numbers of electrons, but with the statistical data and the parameters now included and the parabolic curve-fit included:

And here are the Residuals for each of these curve-fits, which each show interesting patterns:

It seems appropriate to show the data for six-electrons again. You probably noted that the Residuals for six- had a large excursion outlier at a nuclear charge of 31. Look at the red dot on the following graph of six- again, to see that even that extreme outlier is actually quite close to the parabolic curve. This demonstrates just how good these parabolas are.

Look at element 26 and element 32.

Look at element 26.

Look at element 20.

The parabolic equation y = a + b * (x-c)

These graph analyses present some amazing results! We will collect numbers
from these parabolic graphs. For non-statisticians, the r^{2}
value is an indication of how well a mathematical curve fits the
data, with 1.000 meaning a perfect fit. These are mighty close,
meaning that the data is truly extremely close to parabolic!

electrons | a | b | c | r^{2} |
---|---|---|---|---|

one | 19.3 | 13.938338 | 0.21 | 0.99999693 |

. | ||||

two | 6.3 | 3.5312093 | 1.94 | 0.99999095 |

three | -1.5 | -3.4442806 | 2.22 | 0.99999944 |

four | -2.3 | -3.4642755 | 3.21 | 0.99999939 |

five | -4.4 | -3.4478263 | 3.84 | 0.99999941 |

six | -12.8 | 3.3879534 | 4.31 | 0.99996958 |

nine | -18.1 | 3.4165106 | 6.64 | 0.99999867 |

. | ||||

ten | 4.1 | 1.601446 | 8.60 | 0.99999826 |

eleven | 8.4 | 1.5728455 | 8.81 | 0.99998215 |

twelve | -10.0 | 1.5421735 | 9.86 | 0.9999791 |

seventeen | -24.9 | 1.5293853 | 12.96 | 0.99997388 |

. | ||||

eighteen | -15.7 | -1.6134294 | 15.79 | 0.99955473 |

nineteen | -9.0 | -1.7198515 | 17.20 | 0.99981609 |

twenty | -1.6 | -2.1111569 | 19.14 | 0.99991352 |

All of the electron numbers from two through nine have a b parameter very close to 3.45. These match up to the Quantum level 2 electrons, and it is surprising that they share that same parameter. Starting with ten electrons, that b parameter suddenly and consistently drops to around 1.57, which seems to be true for the Quantum level three electrons. The single electron situation with the b parameter of 13.9 matches up to the Quantum level 1 of Hydrogen. Note that TWO electrons seems to belong with the next quantum level, implying that the Periodic Chart may be wrong! It could be that each shell starts out with an EMPTY state as the first entry, meaning the Noble Gases might belong on the LEFT side of the Periodic Table rather than the right where they have always been shown (as being FILLED shells). In the same way, TEN electrons appears to belong with the next shell!

These results show parabolas that are NOT tangent to the x-axis, having a vertical differential equal to the a parameter. The graphs were curve-fitted with a requirement that there was that tangent requirement, but every such curve was substantially worse than the ones presented above. Here is one example, for 12-electrons.

The r^{2} value is good, but not nearly as good as the parabola
presented well above which would tangent around the -10 eV line.
That seems to suggest that it is not appropriate to force the
parabolas to be tangent to the x-axis.

This might suggest that there is some physical importance in the fact that the a parameter is not zero. In fact, in progressing from 2-electrons to 9-electrons, it is clear from the chart above that the a value constantly becomes more negative. The same trending toward negative a seems to exist starting with 10-electrons. There may be a physical meaning for this.

It is probably silly to note a potential relationship between these
b parameters. If we say that the first value is 14, then the second shell
parameter is pretty close to 14/2^{2} or 3.5 and the third
is pretty close to 14/3^{2} or 1.56. It will be interesting
to see if the next shell (possibly beginning with 18 electrons) might have
a b parameter of 14/4^{2} or 0.875, and the shell after that
(beginning with 36 electrons) having 14/5^{2} or 0.56

There does not seem to be any logical reason for this to be the case, and I am merely noticing a possible pattern in the data numbers.

Certain errors in the NIST
data for specific ions probably cause some variation and the
consistency may even be better than that. The amazingly high
values for r^{2} for each graph show the very good curve-fit.

Therefore, it seems that a simple equation certainly exists which takes the (number of electrons) and (number of nuclear charges) and gives the Ionization Potential in electron-Volts!

For real ions, the number of electrons cannot be greater than the number of nuclear charges, although this equation seems to allow certain limited exceptions. (notice the Five- and Four-electron parabolas which clearly extend to the left of the physically possible numbers). This might aid in explaining how and why some chemical molecules can form. The upward curve of the graphs at the left might be describing the REAL situation when an atom is in a compound. In salt, a neutral Chlorine has 17 electrons, but Chemistry has taught us that the single electron that a Sodium atom has handy can "complete the shell" for the Chlorine atom. That is, the Chlorine atom technically has an extra electron in that compound. These graphs might provide some insight into which compounds are possible and what energy amounts are required for each bonding.

I don't know if this is credible reasoning or not, but it seems an interesting possibility. It might help explain a lot about electrochemical processes.

The 10- data actually includes three separate curves! They have very different b parameters. This might indicate the distinction of whether the electron emitted is a 1s electron, or a 2s or a 2p electron.

Keep in mind that most of these atoms have already been multiply ionized, meaning that some electrons are already gone. The point being made here is that it seems possible to know the SEQUENCE of where the electrons are emitted from! A Ca XI ion has already given up ten electrons, suggesting that all the 1s, 2s, and 2p electrons are already gone. Therefore, the only electrons that should still remain would be 3-level electrons, and the graph (10 C) seems to show that.

There are clearly complex changes that occur within the electron movements, which apparently are needed to try to describe how S VI would be able to expel a 1s electron which would normally be expected to have been long gone.

Looking again at the form of the Rydberg Equation, the first term was described as E = 13.6 / (n + a)

The present analysis has shown the physical reality of the a parameter in the Rydberg Equation, being the nuclear charge. The n and m parameters appear to still be distance/radius, which brings up an interesting question regarding how an equation could mix and match charge and distance! There appears to be a level of understanding still beyond us all!

This presentation was first placed on the Internet in June 2007.

Conservation of Angular Momentum - An Exception or Violation (Sept 2006)

**Galaxy Spiral Arms** **Stability and Dynamics** A purely Newtonian gravitational explanation (Nov 1997, Aug 1998)

**Twins Paradox of Relativity Is Absolutely Wrong** (research 1997-2004, published Aug 2004)

Perturbation Theory. Gravitational Theory and Resonance (Aug 2001, Dec 2001)

Origin of the Earth. **Planetary Gravitational Resonances** (Dec 2001)

Rotation of the Sun (Jan 2000)

**Origin of the Universe**. Cosmogony - Cosmology (more logical than the Big Bang) (devised 1960, internet 1998)

**Time Passes Faster Here on Earth than on the Moon** (but only a fraction of a second per year!) (Jan 2009)

Globular Clusters. All Globulars Must Regularly Pass Through the cluttered Galaxy Plane, which would be very disruptive to their pristine form. (Nov 1997, Aug 1998)

Existence of Photons. A Hubble Experiment to Confirm the Existence of Individual Photons (experimental proof of quanta) (Feb 2000)

**Origin of the Moon - A New Theory** (June 2000)

**Planetary Rotation of Jupiter, Saturn, and the Earth** (Jupiter has a lot of gaseous turbulence which should have slowed down its rapid rotation over billions of years) (March 1998)

Cepheid Variable Stars. Velocity Graph Analysis (Feb 2003)

Compton Effect of Astrophysics. A Possible New Compton Effect (Mar 2003)

Olbers Paradox Regarding Neutrinos (Oct 2004)

Kepler and Newton. Calculations (2006)

Pulsars. Pulsars May Be Quite Different than we have Assumed (June 2008)

Sun and Stars - How the Sun Works - Nuclear Fusion in Creating Light and Heat (Aug 2006)

Stars - How They Work - Nuclear Fusion. Lives of Stars and You (Aug 2004)

Sundial Time Correction - Equation of Time. Sundial to Clock-Time Correction Factor (Jan 2009)

General Relativity - A Moon Experiment to Confirm It. Confirming General Relativity with a simple experiment. (Jan 2009)

General Relativity and Time Dilation. Does Time Dilation Result? (Jan 2009)

Geysers on Io. Source of Driving Energy (June 1998)

Mass Extinction, a New Explanation. A New Explanation for Apparent Periodicity of Mass Extinctions (May 1998, August 2001)

Precession of Gyroscopes and of the Earth. **Gyroscope Precession and Precession of the Earth's Equinoxes** (Apr 1998)

Ocean Tides - The Physics and Logic. Mathematical Explanation of Tides (Jan 2002)

Earth's Spinning - Perfect Energy Source (1990, Dec. 2009)

**Earth's Magnetic Field - Source and Logic**. Complex nature of the magnetic field and its source (March 1996)

**Earth Spinning Energy - Perfect Energy Source** From the Earth's Spinning (1990, Nov. 2002)

**Nuclear or Atomic Physics Related Subjects:**

**Nuclear Physics - Statistical Analysis of Isotope Masses** Nuclear Structure. (research 1996-2003, published Nov 2003)

Quantum Defect is NOT a Mathematical Defect- It Can Be Calculated **The Quantum Defect is a Physical Quantity and not a Fudge Factor**(July 2007)

Atomic Physics - NIST Atomic Ionization Data Patterns **Surprising Patterns in the NIST Data Regarding Atomic Ionization** (June 2007)

Nuclear Physics - Logical Inconsistencies (August 2007)

Neutrinos - Where Did they all Come From? (August 2004)

Neutrinos - Olbers Paradox Means Neutrinos from Everywhere (Oct 2004)

Quantum Nuclear Physics. A Possible Alternative (Aug 2001, Dec 2001, Jan 2004)

Quantum Physics - Quantum Dynamics. A Potential Improvement (2006)

Quantum Physics is Compatible with the Standard Model (2002, Sept 2006, Oct 2010)

Quantum Dynamics (March 2008)

Ionization Potential - NIST Data Patterns. Surprising patterns among different elements (March 2003)

Mass Defect Chart. (calculation, formula) (research 1996-2003, published Nov 2003)

**Assorted other Physics Subjects:**

Precession of Gyroscopes and of the Earth. **Gyroscope Precession and Precession of the Earth's Equinoxes** (Apr 1998)

**Earth's Magnetic Field - Source and Logic**. Complex nature of the magnetic field and its source (March 1996)

**Earth Spinning Energy - Perfect Energy Source** (1990, Nov. 2002)

Earth Energy Flow Rates due to Precessional Effects (63,000 MegaWatts) (Sept 2006)

Accurate Mass of the Earth. Gravitational Constant - An Important Gravitation Experiment. (Feb 2004)

Tornadoes - The Physics of How They Operate, including How they Form. **Solar Energy, an Immense Source of Energy, Far Greater than all Fossil Fuels** (Feb 2000, Feb 2006, May 2009)

Radiometric Age Dating - Carbon-14 Age Determination. Carbon-14, C-14 (Dec 1998)

Mass Extinction, an Old Explanation. An Old Explanation for Apparent Periodicity of Mass Extinctions (Aug 2003)

**Hurricanes, the Physics and Analysis** **A Credible Approach to Hurricane Reduction** (Feb 2001)

Sundial Time Correction - Equation of Time. Sundial to Clock-Time Correction Factor (Jan 2009)

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Carl W. Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago