Lecture 9Lecture 9

Atoms, atomic transitions, & neon lighting

Absorption in Solar SpectrumAtomic transitions

Neon LightingLasers

Recap from last week: Electron Wavefunctions

iEtSteady-state total wavefunction:

iEtxtx, )exp()(

E energy of the electron E=energy of the electron

t=time

ψ(x) = electron wavefunction that describes only the spatially ψ(x) electron wavefunction that describes only the spatially behavior

Experimentally, we measure the probability of finding an electron in a given position at time t (like an intensity):

22 |)(||)(| zy,x,tz,y,x,

S h di ’ i f di i

Time independent Schrodinger equation

022

VEmd

Schrodinger’s equation for one dimension

022 VEdx

Schrondinger’s equation for three dimensions

0)(222

2

2

2

2

2

VEm

zyx zyx

A mathematical “crank”: we input the potential V of the electron (i.e., the ‘force’ it experiences, F=-dV/dx), and can obtain the

electron energies E and their wavefunctions / probability

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)

electron energies E and their wavefunctions / probability distributions.

Example 1: electrons in a 1D box

axnAxn sin)(

2222

Wavefunction:

2

22

2

22

82)(

manh

manEn

Electron energy in an infinite PE well:

2 )12( nh

Energy separation in an infinite PE well:

21 8)12(

manhEEE nn

Example 1: electrons in a 1D box

Wavefunction: Probability:

Atoms: A 3D Quantum well

Electron confined in three dimensions by a three-dimensional infinite PE box. yEverywhere inside the box, V = 0, but outside, V = . The electron cannot escape

from the box.

Quantum mechanics in 3D: 3 quantum numbers

Electron wavefunction in infinite PE well

znynxn

czn

byn

axnAz y, x,nnn

321 sinsinsin)(321

Electron energy in infinite PE box

2

22

2

23

22

21

2

88321

NhnnnhE nnn

22 88321 mamannn

23

22

21

2 nnnN 321 nnnN

Atoms: electrons in a 3D spherical box

2m 022

2 VEm

The electron in the hydrogenic atom is atom is attracted by a central force that is always directed toward the positive nucleusis always directed toward the positive nucleus.

Spherical coordinates centered at the nucleus are used to describe the positionof the electron. The PE of the electron depends only on r.

Atoms: electrons in a 3D spherical box

The electron’s potential energy V(r) in a hydrogenic atom (i.e., just one electron) is used in the Schrödinger equation.just one electron) is used in the Schrödinger equation.

Recall: r2=x2+y2+z2

2m

Solutions to Schrodinger’s equation in an atom

022

2 VEm

rZerV4

)(2

ro4

:2 f

)()()( ),()(),,( YrRr

R: Radial wavefunction – depends on two quantum numbers, “n” and “l”d w d p d w q b , dY: Angular wavefunction – depends on another quantum number, “ml”

(A fourth quantum number, also in Y, arises from relativity: “ms”)

Radial wavefunctions, R(r)

Radial wavefunctions of the electron in a hydrogenic atom for various n and values.

1 2 3 (j t lik ti l i b )n=1,2,3,…(just like particle in a box)=0,1,2,3,…(n-1) =0: “s” =1: “p” =2: “d” =3: “f”

Radial Probability

r2 |Rn,2| gives the radial probability density., i.e., where are we most

likely to find the electron around the nucleuslikely to find the electron around the nucleus

Radial ProbabilityBohr radius (a0): 0.0529nmBohr radius (a0): 0.0529nm

r2 |Rn,2| gives the radial probability density., i.e., where are we most

likely to find the electron around the nucleuslikely to find the electron around the nucleus

Angular Wavefunctions, Y(θ,φ)

The polar plots of Y(, ) for 1s and 2p states.

Angular Probability Distribution

The angular dependence of the probability distribution, which is proportional to | Y(, )|2.

Total Wavefunction: ψ(r, θ, φ)=R(r)Y(θ,φ)

n=1,2,3,…(just like particle in a box)(j p )=0,1,2,3,…(n-1) =0: “s” =1: “p” =2: “d” =3: “f”m =- …,0,…

Knowing ψ we can use the Schrodinger equation to find the electron

Electron energies

Knowing ψ, we can use the Schrodinger equation to find the electron energies.

Just like electrons in a 1D or 3D quantum well, the electron energy in the hydrogenic atom is quantized.

24Z222

24

8 nhZmeEn

8 nho

Ionization energy of hydrogen: energy required to remove the

(Z is atomic number, n is the quantum number, 1,2,3,…)

184me

Ionization energy of hydrogen: energy required to remove the electron from the ground state in the H-atom

eV13.6J1018.28

1822

hmeE

oI

Electron energies in 1-electron atoms

Energies are more closely spaced for

higher n

Radial electron ‘position’ & ionization energies in a hydrogenic atom (Z≥1)

an 2

y g ( )

Zanr o

max Z

2

2effective eV)6.13(ZE nI 2, nnI

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)

Example of Zeff: Li as a ‘hydrogenic atom’

The Li atom has a nucleus with charge +3e, 2 electrons in the K shell , which is closed, and one electron in the 2s orbital. (b) A simple view of (a) would be one ( ) p ( )

electron in the 2s orbital that sees a single positive charge, Z = 1

The simple view Z = 1 is not a satisfactory description for the outer electron because it has a probability distribution that penetrates the inner shell We can because it has a probability distribution that penetrates the inner shell. We can

instead use an effective Z, Zeffective = 1.26, to calculate the energy of the outer electron in the Li atom.

Energy transitions can occur via photons

Absorption of a photon

Energy transitions can occur via photons

Absorption of a photon

Emission of a photon

Example: Solar Spectrum

1829: Josef von Fraunhofer1829: Josef von Fraunhoferλdark

1=656.3 nmλdark

2=486.1 nm

eVEn

Enh

meZEo

n 6.13)1(8 121222

42

Convenient conversion: λ [eV]= 1241.341/λ [nm]

Example: Solar Spectrum

1829: Josef von Fraunhoferλdark

1=656.3 nm = 1.89eVdark(n=3 to n=2)

λd k2=486 1 nm = 2 55 eVλdark =486.1 nm = 2.55 eV

(n=4 to n=2)

eVEn

Enh

meZEo

n 6.13)1(8 121222

42

Energy transitions can also occur via collisions

An atom can become excited by a collision with another atom.An atom can become excited by a collision with another atom.

Energy transitions can also occur via collisions

An atom can become excited by a collision with another atomAn atom can become excited by a collision with another atom.When it returns to its ground energy state, the atom emits a photon.

Neon lighting occurs via quantum transitions!