ELECTRON ENERGY LEVELS IN ATOMS

For almost 30 years the Balmer formula was a small oasis of regularity in the field of spectroscopy—the science of measuring and cataloging the wavelengths of radiation emitted and absorbed by different elements and compounds. Unfortunately, the Balmer formula could not be explained, or applied to any other element, or even applied to other known wavelengths emitted by hydrogen atoms. It might well have been a mere coincidence, without any significance. Bohr’s model of the hydrogen atom not only explained the Balmer formula, but also gave scientists their first glimpse of atomic structure. It still serves as the basis for most scientists’ working picture of an atom.

Bohr adopted Rutherford’s nuclear model that had been successful in explaining scattering experiments with alpha particles between 1910 and 1912. In other words, Bohr assumed that almost all the mass of a hydrogen atom is concentrated in a positively charged nucleus, allowing most of the atomic volume free for the motion of the much lighter electron. The electron was assumed attracted to the nucleus by the Coulomb force law governing opposite charges (Fig. 2.1). In magnitude this force is

F¼ 1 4pe₀

e²

r²: (2:2:2)

Bohr also assumed that the electron travels in a circular orbit about the massive nucleus. Moreover, he assumed the validity of Newton’s laws of motion for the orbit.

Thus, in common with every planetary body in a circular orbit, the electron was assumed to experience an inward (centripetal) acceleration of magnitude

a¼v²

r : (2:2:3)

Newton’s second law of motion,F¼ma, then gives mv²

r ¼ 1 4pe₀

e²

r², (2:2:4)

which is the same as saying that the electron’s kinetic energy,T¼¹₂mv², is half as great as the magnitude of its potential energy,V¼2e²/4pe0r. In the Coulomb field of the

e

p r

Figure 2.1 The electron in the Bohr model is attracted to the nucleus with a force of magnitude F¼e²/4pe₀r².

nucleus the electron’s total energy is therefore E¼TþV¼ 1

4pe₀ e²

2r: (2:2:5)

These results are familiar consequences of Newton’s laws. Bohr then introduced a single, radical, unexplained restriction on the electron’s motion. He asserted that only certain circles are actually used by electrons as orbits. These orbits are the ones that permit the electron’s angular momentumLto have one of the values

L¼n h

2p^{, (2:2:6)}

wherenis an integer (n¼1, 2, 3,. . .) andhis the constant of Planck’s radiation formula:

h6:62510³⁴ J-s: (2:2:7) With the definition of angular momentum for a circular orbit,

L¼mvr, (2:2:8)

it is easy to eliminaterbetween (2.2.4) and (2.2.8) and find v¼ 1

4pe₀ e²

L ¼ 1 4pe₀

2pe²

nh : (2:2:9)

Then the combination of (2.2.4) and (2.2.5), namely E¼mv²

2 , (2:2:10)

together with (2.2.9), gives the famous Bohr formula for the allowed energies of a hydrogen electron:

E_n¼ 1 4pe₀ 2

me⁴

2n²h²: (2:2:11)

This can be seen, by comparison with (2.2.5), to be the same as a formula for the allowed values of electron orbital radius:

r_n¼4pe₀ⁿ

2h²

me²: (2:2:12)

In both (2.2.11) and (2.2.12) we have adopted the modern notation for Planck’s constant:

h ¼ h

2p^1:05410

34 J-s: (2:2:13) The first thing to be said about Bohr’s model and his unsupported assertion (2.2.6) is that they were not contradicted by known facts about atoms, and, for small values ofn,

2.2 ELECTRON ENERGY LEVELS IN ATOMS 19

the allowed radii defined by (2.2.12) are numerically about right.²For example, the smallest of these radii (conventionally called “the Bohr radius” and denoteda0) is

a₀¼r₁¼4pe₀ ^h

2

me²0:53 A: (2:2:14) This might have been an accident without further consequences. Since no way existed to measure such small distances with any precision, Bohr needed a connection between (2.2.12) and a possible laboratory experiment. A second unsupported assertion supplied the connection.

Bohr’s second assertion was that the atom was stable when the electron was in one of the permitted orbits, but that jumps from one orbit to another were possible if accompanied by light emission or absorption. To be specific, Bohr combined earlier ideas of Planck and Einstein and stated that a jump from a higher to a lower orbit would find the decrease of the electron’s energy transformed into a quantum of radiation that would be emitted in the process. In other words, Bohr postulated that

(DE)_n,n⁰ ¼hn¼energy of emitted photon: (2:2:15) Here (DE)_n,n⁰denotes the energy lost by the electron in switching orbits fromrntorn⁰and nis the frequency of the photon emitted in the process (Fig. 2.2).

The relation (2.2.15) led immediately to a connection between Bohr’s theory and all the spectroscopic data known for atomic hydrogen. By using (2.2.11) for two different orbits, that is, for two different values of the integern, we easily find for the energy decrement (DE)_n,n⁰the expression

EnEn⁰ ¼ 1 4pe_{0 2}

me⁴ 2h² 1

n⁰² 1 n²

: (2:2:16)

Furthermore, the connection between the frequency and wavelength of a light wave is l¼c

n: (2:2:17)

+ E_n

E_n¢

hn

Figure 2.2 A radiative transition of an atomic electron in the Bohr model.

2Lord Rayleigh (1890) was able to estimate molecular dimensions by dropping olive oil onto a water surface.

Assuming that an oil drop spreads until it forms a layer one molecule thick (amolecular monolayer), he could give a reasonable estimate of a molecular diameter from the area of the layer and the volume of the original drop. A century earlier, Benjamin Franklin tried the same oil-on-water experiment on a pond in London while on diplomatic assignment there.

Thus, Bohr’s statement (2.2.15) and his energy formula (2.2.11) are actually equivalent to the postulate that all spectroscopic wavelengths of light associated with atomic hydrogen fit the formula

l¼ hc

DE_n,n⁰ ¼(4pe₀)² 4ph³c me⁴

n²n⁰²

n²n⁰², (2:2:18) wherenandn⁰are integers to be chosen, but where all the other parameters are fixed.

It is obvious that ifn⁰¼2 andn.2, then (2.2.18) becomes

l¼(4pe₀)²16ph³c me⁴

n²

n²4, (2:2:19) which is exactly the Balmer formula (2.2.1). The numerical value of the product of the coefficients in (2.2.19) is 3645.6 A˚ , just what Balmer had said the constant b was, 28 years earlier. Bohr’s expression (2.2.18) was quickly found, for values of n⁰ not equal to 2, to agree with other wavelengths associated with hydrogen but which had not fitted the Balmer formula (Problem 2.1).

Bohr’s theory opened a new viewpoint on atomic spectroscopy. All observed spectro- scopic wavelengths could be interpreted as evidence for the existence of certain allowed electron orbits in all atoms, even if formulas corresponding to Bohr’s (2.2.18) were not known for any atom but hydrogen.

In Chapter 3 we will see that, for many aspects of the interaction of light and matter, atoms can be regarded as a set of electrons acting as harmonic oscillators.

It might be supposed that the oscillation frequencies of the electrons in this classical electron oscillator model of an atom, which was used with considerable success before quantum theory, are associated somehow with the “transition frequencies” n in Bohr’s formula (2.2.15). In this way the most useful features of the classical oscillator model of an atom survived the quantum revolution unchanged. How this is possible, in view of the obvious fact that the assumed Coulomb force (2.2.2) between electron and nucleus is not a harmonic oscillator force, will be explained in Chapter 3.

It is easy to have second thoughts about Bohr’s model, no matter how successful it is. For example, one can ask why (2.2.6) does not include the possibility n¼0.

There is no apparent reason why zero angular momentum must be excluded, except that the energy formula (2.2.11) is not defined forn¼0. This point, whether physical significance can be assigned to the orbit with zero angular momentum, cannot be clarified within the Bohr theory, and it proved puzzling to physicists for more than a decade until quantum mechanics was developed. In a similar fashion, one can ask how Bohr’s results are modified by relativity. The kinetic energy formula used above, T¼¹₂mv², is certainly nonrelativistic. Again, this point was not fully answered until the development of quantum mechanics.

† Relativistic corrections to Newtonian physics become important when particle velocities approach the velocity of light. Ifvis the velocity of a particle, then typically the first correction terms are found to be proportional to (v/c)², wherec is the velocity of light. The

2.2 ELECTRON ENERGY LEVELS IN ATOMS 21

value of (v/c)²can easily be estimated within the Bohr theory. It follows from (2.2.4) that v

c

2¼ 1 4pe0

e²

rmc²: (2:2:20)

By insertingrfrom (2.2.12) and taking the square root, the ratiov/ccan be found for any of the allowed orbits:

v c¼ 1

4pe₀ e²

nhc: (2:2:21)

Equation (2.2.21) shows that the largest velocity to be expected in the Bohr atom is associated with the lowest orbit, n¼1. The ratio of this maximum velocity to the velocity of light is given by the remarkable (dimensionless) combination of electromagnetic and quantum mechanical constants,e²=4pe0hc. The numerical value of this parameter is easily found:

1 4pe0

e²

hc¼ (1:60210¹⁹C)²

(4p)(8:85410¹²C=V-m)(1:05410³⁴J-s)(2:99810⁸m=s)

¼0:007297 (¼1=137:04): (2:2:22)

The value found in (2.2.22) is small enough that corrections to the Bohr model from relati- vistic effects are of the relative order of magnitude 10²⁴ or smaller, and thus negligible in most circumstances. Spectroscopic measurements, however, are commonly accurate to five sig- nificant figures. Arnold Sommerfeld, in the period 1915 – 1920, studied the relativistic corrections to Bohr’s formulas and showed that they accounted accurately for some of the fine details orfine structurein observed spectra. For this reason the parametere²=4pe₀hcis calledSommerfeld’s fine-structure constant.

The fine-structure constant appears so frequently in expressions of atomic radiation physics that it is very useful to remember its numerical value. Because the value given in (2.2.22) is very nearly equal to 1/137, it is in this form that its value is memorized

by physicists. †

Quantum States and Degeneracy

In the Bohr model a state of the electron is characterized by thequantum number n.

Everything the model can say about the allowed states of the electron is given in terms ofn.

The full quantum theory of the hydrogen atom also yields the allowed energies (2.2.11). However, in the quantum theory a state of the electron is characterized by other quantum numbers in addition to the principal quantum number n appearing in (2.2.11). The results of the quantum theory for the hydrogen atom, in addition to (2.2.11), are mainly the following:

(i) For each principal quantum numbern(¼1, 2, 3,. . .) there arenpossible values of theorbital angular momentum quantum number‘. The allowed values of‘are 0, 1, 2,. . .,n21. Thus, forn¼1 we can have only‘¼0, whereas forn¼2 we can have‘¼0 or 1, and so on.

(ii) For each‘ there are 2‘þ1 possible values of themagnetic quantum number m.

The possible values of magnetic quantum number m are ‘,‘þ1,. . ., 1, 0, 1,. . .,‘1,‘.

(iii) In addition to orbital angular momentum, an electron also carries an intrinsic angular momentum, which is called simplyspin. The spin of an electron always has magnitude¹₂(in units ofh). But in any given direction the electron spin can be either “up” or “down”; that is, quantum theory says that when the component of electron spin along any direction is measured, we will always find it to have one of two possible values.³Because of this, an electron state must also be labeled by an additional quantum numberms, called thespin magnetic quantum number, whose only possible values are+¹₂.

Thus, for a givenn, there arenpossible values of‘, and for each‘there are 2‘þ1 possible values ofm, for a total of

Xⁿ¹

‘¼0

(2‘þ1)¼n², (2:2:23)

states. And each of these states is characterized further byms, which may beþ¹₂or¹₂. Therefore, there are 2n²states associated with each principal quantum numbern. In con- trast to the Bohr model, in which an allowed state of the electron in the hydrogen atom is characterized byn, quantum theory characterizes each allowed state by the four quantum numbers n, ‘, m, and ms; and since the electron energy depends only on n [recall (2.2.11)], there are 2n²states with the same energy for every value of n. These 2n² states are calleddegenerate statesor are said to be degenerate in energy.

Historical designations for the orbital angular momentum quantum numbers are still in use:

‘¼0

‘¼1

‘¼2

‘¼3

‘¼4

designates the so-calledsorbital porbital

dorbital forbital gorbital

The first three letters came from the wordssharp,principal, anddiffuse, which described the character of atomic emission spectra in a qualitative way long before quantum theory showed that they could be associated systematically with different orbital angular momentum values for an electron in the atom.

The Periodic Table

Although hydrogen is the only atom for which explicit expressions such as (2.2.11) can be written down, we can nevertheless understand the gross features of the periodic table of the elements. That is, we can understand the chemical regularity, or periodicity, that occurs as the atomic numberZincreases. The key to this understanding is theexclusion principleof Wolfgang Pauli (1925), which forbids two electrons from occupying the

3The magnitude of the spin angular momentum vector is (Section 2.4) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(sþ1)

p h ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2

1 2þ1

q

h ¼ ffiffi

3 4

q h, so that its two allowed components (spin “up” and spin “down”) make angles cos¹(+¹2)= ffiffi

3 4

h q i

¼54:748and (18054:74)¼125:268with the axis along which the spin is measured.

2.2 ELECTRON ENERGY LEVELS IN ATOMS 23

same quantum mechanical state. The Pauli exclusion principle may be proved only at an advanced level that is well beyond the scope of this book. We will simply accept it as a fundamental truth.

But the Pauli principle alone is not sufficient for an understanding of the periodic table. We must also deal with the electron – electron interactions in a multielectron atom. These interactions present us with an extremely complicated many-body problem that has never been solved. A useful approximation, however, is to assume that each electron moves independently of all the others; each electron is thought of as being in a spherically symmetric potentialV(r) due to the Coulomb field of the nucleus plus theZ21 other electrons. In this independent-particle approximation an electron state is still characterized by the four quantum numbers (n,‘,m,ms), as in the case of hydro- gen. However, in this case the simple energy formula (2.2.11) does not apply, and in particular the energy depends on bothnand‘(but notmorms) as sketched in Fig. 2.3.

The simplest multielectron atom, of course, is helium, in which there areZ¼2 elec- trons. The lowest energy state for each electron is characterized in the independent- particle approximation by the quantum numbers n¼1,‘¼0,m¼0, and ms¼+¹₂. Since the energy depends now on bothn and‘, we can label this particularelectron configurationas 1s, a shorthand notation meaning n¼1 and ‘¼0. Both electrons are in theshell n¼1, one having spin up m_s ¼¹₂

, the other spin downm_s ¼ ¹₂ . Since 2 is the maximum number of electrons allowed by the Pauli exclusion principle for the 1sconfiguration, we say that the 1sshell is completely filled in the helium atom.

Energy

Bohr’s hydrogen

Generic atom Energy

3d (10) 3p (6)

2p (6) 3s (2)

2s (2)

1s (2) n = 3

n = 2

n = 1

Figure 2.3 The main differences between Bohr’s model for hydrogen and a generic many-electron atom arise from the Pauli exclusion principle and the dependence of level energies on bothnand‘. In parentheses we show the number of different states, (2‘þ1)2, permitted by assignment ofmandms

values for each‘. Carbon’s 6 electrons, for example, occupy the 2 states in each of the 1sand 2slevels and 2 of the 2pstates. For clarity the energy separations are not properly scaled.

In the caseZ¼3 (lithium), there is one electron left over after the 1sshell is filled.

The next allowed electron configuration is 2s(n¼2,‘¼0), and one of the electrons in lithium is assigned to this configuration. Since the 2sconfiguration can accommodate two electrons, the 2s subshellin lithium is only partially filled. The next element is beryllium, withZ¼4 electrons, and in this case the 2ssubshell is completely filled, there being two electrons in this “slot.”

ForZ¼5 (boron), the added electron goes into the 2pconfiguration (n¼2,‘¼1).

This configuration can accommodate 2(2‘þ1)¼6 electrons. Thus, there are five other elements (C, N, O, F, and Ne) in which the outer subshell of electrons corresponds to the configuration 2p. The eight elements lithium through neon, for which the outermost electrons belong to then¼2 shell, constitute the first full row of the periodic table.

Inside the back cover of this book we list the first 36 elements and their electron configurations. The configurations are assigned in a similar manner as done above for Z¼1 – 10. Also listed is the ionization energy, defined as the energy required to remove one electron from the atom. For hydrogen the ionization energyWImay be cal- culated from Eq. (2.2.11) withn¼1, that is,WIis just the binding energy of the electron in ground-state hydrogen:

WI¼ jE1j ¼ 1 4pe_{0 2}

me⁴

2h² ¼2:1710¹⁸ J: (2:2:24) We already pointed out, in connection with photon energy in Chapter 1, that such small energies are usually expressed in units of electron volts, an electron volt being the energy acquired by an electron accelerated through a potential difference of 1 volt:

1 eV¼(1:60210¹⁹ C)(1 V)¼1:60210¹⁹ J: (2:2:25) The ionization energy of hydrogen is therefore

W_I¼ 2:1710¹⁸ J

1:60210¹⁹ J=eV¼13:6 eV: (2:2:26) The ionization energy of a hydrogen atom in any state (n, ‘, m, ms) is likewise (13.6 eV)/n².

The elements He, Ne, Ar, and Kr are chemically inactive. We note that each of these atoms has a completely filled outer shell. Evidently, an atom with a filled outer shell of electrons tends to be “satisfied” with itself, having very little proclivity to share its electrons with other atoms (i.e., to join in chemical bonds). However, a filled outer subshell does not necessarily mean chemical inertness. Beryllium, for instance, has a filled 2s subshell, but it is not inert. Furthermore, even some of the noble gases are not entirely inert.

The alkali metals Li, Na, and K have only one electron in an outer subshell, and their outer electrons are weakly bound, leading to low ionization energies of these elements.

These elements are highly reactive; they will readily give up their “extra” electron. On the other hand, the halogens F, Cl, and Br are one electron short of a filled outer subshell.

These atoms will readily take another electron, and so they too are quite reactive chemi- cally and the halogens are sufficiently “eager” to combine with elements that can easily

2.2 ELECTRON ENERGY LEVELS IN ATOMS 25

contribute an electron that they can formnegativeions, stably but weakly binding an extra electron. This even includes H², the negative hydrogen ion. Hydrogen is in the odd position of having some properties in common with the alkali metals and some in common with the halogens.

The characterization of atomic electron states in terms of the four quantum numbers n,‘,m, andms, together with the Pauli exclusion principle, thus allows us to understand why Na is chemically similar to K, Mg is chemically similar to Ca, and so forth. These chemical periodicities, according to which the periodic table is arranged, are con- sequences of the way electrons fill in the allowed “slots” when they combine with nuclei to form atoms.

Of course, there is a great deal more that can be said about the periodic table. For a rigorous treatment of atomic structure, we must refer the reader to textbooks on atomic physics. As mentioned earlier, however, we can understand lasers without a more detailed understanding of atomic and molecular physics.

In document Laser Physics by Milonni P.W., Eberly J.H. .pdf (Page 36-45)