Atomic Transitions

Electrons in atoms do not have just any possible energy. Instead, they occupy specific quantized energy levels.

An atomic transition occurs when an electron changes from one allowed energy state to another.

There are two main possibilities:

Electron moves upward → energy is absorbed

Electron moves downward → energy is released

These transitions explain why atoms absorb and emit only particular wavelengths of electromagnetic radiation and why every element has a characteristic spectrum.

Quantized Energy Levels

Electrons bound within atoms can occupy only particular allowed energy states.

These are called energy levels.

A simplified diagram might look like this:

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────────────

Each horizontal line represents an allowed energy state.

An electron can occupy:

E₁, E₂, E₃ or E₄

but it cannot occupy an arbitrary energy between these levels.

This is what is meant by quantization.

---

Energy Level Diagrams

An energy level diagram is a convenient way of representing the allowed energies of electrons.

The vertical direction represents energy, not physical height or distance from the nucleus.

Higher lines represent higher energies.

Lower lines represent lower energies.

For example:

higher energy

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────────────

lower energy

The electron's position on this diagram tells us its energy state.

It does not show the literal location of the electron within the atom.

The Ground State

The ground state is the lowest available energy state of an atom.

For a simple diagram:

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────●───────

the electron is in E₁.

If E₁ is the lowest allowed level, the electron is in its ground state.

The ground state is generally the most stable state.

---

Excited States

An excited state is an allowed energy state above the ground state.

For example:

E₄ ─────────────

E₃ ─────●───────

E₂ ─────────────

E₁ ─────────────

The electron is now in an excited state.

For this to happen, the atom must have gained energy.

Possible sources include:

• electromagnetic radiation
• electrical energy
• thermal energy
• collisions with particles

Excited states are generally less stable than the ground state.

---

Absorbing Energy

An electron can move from a lower energy state to a higher energy state by absorbing energy.

For example:

E₁ → E₃

is an upward transition.

The electron must gain an amount of energy equal to the difference between the two levels.

The energy change is:

ΔE = E_final − E_initial

For an upward transition:

ΔE > 0

because the electron gains energy.

Worked Example: Energy Absorbed

Suppose:

E₁ = −10 eV

E₂ = −6 eV

An electron moves:

E₁ → E₂

Calculate the energy change.

ΔE = E_final − E_initial

ΔE = −6 − (−10)

ΔE = +4 eV

The positive sign tells us that the electron has gained energy.

Therefore:

4 eV must be absorbed.

---

Absorption of Photons

One way an electron can gain energy is by absorbing a photon.

The photon carries electromagnetic energy.

Photon energy is:

E = hf

where:

E = photon energy

h = Planck's constant

f = photon frequency

For an atomic transition to occur:

E_photon = ΔE

The photon must supply the appropriate amount of energy for the allowed transition.

---

Why the Energy Must Match

Suppose an atom has:

E₁ = −10 eV

E₂ = −6 eV

The energy difference is:

4 eV

A photon with the appropriate 4 eV energy can cause:

E₁ → E₂

But a 2 eV photon cannot simply move the electron halfway between E₁ and E₂.

There is no allowed bound state there.

Likewise, in a simplified energy-level model, a 3 eV photon cannot produce this particular 4 eV transition.

This is a direct consequence of quantized energy levels.

---

Electrons Do Not Stay Excited Forever

After absorbing energy, an electron may occupy an excited state.

However, many excited states are unstable.

Eventually, the electron can return to a lower-energy state.

For example:

E₃ → E₂

or:

E₃ → E₁

When this happens, energy is released.

The energy released corresponds to the difference between the two energy states.

Emission of a Photon

When an electron makes a downward transition, energy can be emitted as a photon.

For example:

E₃ → E₁

The photon energy is equal to the magnitude of the energy difference:

E_photon = |E_final − E_initial|

If:

E₃ = −2 eV

and:

E₁ = −10 eV

then:

E_photon = |−10 − (−2)|

E_photon = 8 eV

Therefore, an 8 eV photon is emitted.

---

Positive and Negative Energy Changes

Using:

ΔE = E_final − E_initial

the sign of ΔE tells us whether the electron gains or loses energy.

Positive ΔE

If:

ΔE > 0

the electron has moved to a higher-energy state.

Energy has been absorbed.

Negative ΔE

If:

ΔE < 0

the electron has moved to a lower-energy state.

Energy has been released.

For example:

E_initial = −2 eV

E_final = −10 eV

ΔE = −10 − (−2)

ΔE = −8 eV

The electron loses 8 eV.

The emitted photon therefore has energy:

8 eV

Photon energy itself is normally reported as a positive magnitude.

---

A Simple Rule to Remember

Moving Up

electron gains energy

ΔE > 0

energy absorbed

Moving Down

electron loses energy

ΔE < 0

energy released

For photon emission, use the magnitude of the energy difference:

E_photon = |ΔE|

---

One Excited State, Several Possible Transitions

Suppose an atom has:

E₄ = −1 eV

E₃ = −3 eV

E₂ = −6 eV

E₁ = −10 eV

An electron in E₄ could potentially return through several transitions.

For example:

E₄ → E₃

Energy released:

|−3 − (−1)| = 2 eV

E₄ → E₂

Energy released:

|−6 − (−1)| = 5 eV

E₄ → E₁

Energy released:

|−10 − (−1)| = 9 eV

Each transition produces a different photon energy.

Direct Transitions and Cascades

An excited electron does not necessarily return directly to the ground state.

Suppose an electron is at E₄.

It might transition directly:

E₄ → E₁

Or it could return in stages:

E₄ → E₃ → E₂ → E₁

This sequence of transitions is sometimes called a cascade.

Each downward transition can release a photon with an energy corresponding to that particular energy gap.

---

Worked Example: A Cascade

Suppose:

E₄ = −1 eV

E₃ = −3 eV

E₂ = −6 eV

E₁ = −10 eV

An electron follows:

E₄ → E₃ → E₁

First transition:

E₄ → E₃

Energy released:

|−3 − (−1)| = 2 eV

Second transition:

E₃ → E₁

Energy released:

|−10 − (−3)| = 7 eV

So two photons can be emitted:

2 eV photon

and:

7 eV photon

The total energy released is:

2 + 7 = 9 eV

Notice that this equals the overall difference:

E₄ → E₁

|−10 − (−1)| = 9 eV

Energy is conserved.

---

Atomic Transitions and Conservation of Energy

Atomic transitions obey the law of conservation of energy.

If an electron gains 5 eV during an upward transition, that energy must come from somewhere.

If an electron loses 5 eV during a downward transition, that energy must go somewhere.

For photon interactions:

energy gained by electron = energy lost by photon

or:

energy lost by electron = energy carried by emitted photon

Energy is transferred, not created or destroyed.

---

Photon Frequency and Atomic Transitions

The energy of an emitted photon is related to frequency:

E = hf

Therefore:

f = E/h

A larger atomic energy difference produces a higher-energy photon.

A higher-energy photon has a higher frequency.

So:

larger ΔE → higher photon energy → higher frequency

---

Photon Wavelength and Atomic Transitions

Frequency and wavelength are related by:

c = fλ

Therefore:

λ = c/f

Combining this with:

E = hf

gives:

E = hc/λ

This means:

larger energy gap → shorter wavelength

and:

smaller energy gap → longer wavelength

This connection explains the colours and wavelengths found in atomic spectra.

Worked Example: Frequency of an Emitted Photon

Suppose an atomic transition releases:

E = 3.98 × 10⁻¹⁹ J

Use:

E = hf

Rearrange:

f = E/h

Using:

h = 6.63 × 10⁻³⁴ J·s

f = (3.98 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 6.00 × 10¹⁴ Hz

Therefore, the emitted photon has a frequency of approximately:

6.00 × 10¹⁴ Hz

---

Worked Example: Wavelength of an Emitted Photon

Using:

f = 6.00 × 10¹⁴ Hz

and:

c = 3.00 × 10⁸ m/s

Use:

λ = c/f

λ = (3.00 × 10⁸)/(6.00 × 10¹⁴)

λ = 5.00 × 10⁻⁷ m

Convert to nanometres:

λ = 500 nm

So this atomic transition produces a photon with a wavelength of approximately 500 nm.

---

Energy in Joules and Electron Volts

Atomic energies are commonly expressed in electron volts (eV).

The conversion is:

1 eV = 1.60 × 10⁻¹⁹ J

Suppose a transition releases:

3.0 eV

Convert to joules:

E = (3.0)(1.60 × 10⁻¹⁹)

E = 4.80 × 10⁻¹⁹ J

This value can then be used in:

E = hf

to calculate the photon's frequency.

---

Worked Example: From Energy Levels to Wavelength

Suppose:

E_initial = −2.0 eV

E_final = −5.0 eV

Calculate the energy change

ΔE = −5.0 − (−2.0)

ΔE = −3.0 eV

Therefore, a photon with energy:

3.0 eV

is emitted.

Convert to joules

E = (3.0)(1.60 × 10⁻¹⁹)

E = 4.80 × 10⁻¹⁹ J

Calculate frequency

f = E/h

f = (4.80 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 7.24 × 10¹⁴ Hz

Calculate wavelength

λ = c/f

λ = (3.00 × 10⁸)/(7.24 × 10¹⁴)

λ ≈ 4.14 × 10⁻⁷ m

or:

414 nm

Therefore, the transition produces a photon with wavelength approximately:

414 nm

---

Reading an Energy Level Diagram

When given an energy-level diagram, use a systematic approach.

Identify the Initial Level

Where does the electron start?

Identify the Final Level

Where does the electron finish?

Determine the Direction

Upward arrow:

absorption

Downward arrow:

emission

Calculate the Energy Change

Use:

ΔE = E_final − E_initial

Interpret the Sign

Positive:

energy absorbed

Negative:

energy released

If a Photon Is Emitted

Use:

E_photon = |ΔE|

Then, if required, calculate:

f = E/h

and:

λ = c/f

---

Interpreting Arrow Lengths

On a properly scaled energy-level diagram, a longer vertical transition represents a larger energy difference.

Therefore:

larger vertical transition → larger |ΔE|

For emitted photons:

larger |ΔE| → higher photon energy → higher frequency → shorter wavelength

For smaller transitions:

smaller |ΔE| → lower photon energy → lower frequency → longer wavelength

This allows many questions to be answered without performing calculations.

---

Example: Comparing Three Transitions

Suppose three downward transitions are shown:

A: E₄ → E₃

B: E₄ → E₂

C: E₄ → E₁

If E₁ is the lowest state, Transition C has the largest energy difference.

Therefore, C produces:

• the highest-energy photon
• the highest-frequency photon
• the shortest-wavelength photon

Transition A has the smallest energy difference.

Therefore, A produces:

• the lowest-energy photon
• the lowest frequency
• the longest wavelength

---

Atomic Transitions and Emission Spectra

Each possible downward atomic transition can produce a photon with a particular energy.

Since:

E = hf

each photon has a particular frequency.

Since:

c = fλ

each has a particular wavelength.

When these emitted wavelengths are separated, they form an emission spectrum.

Why Emission Spectra Have Lines

Atomic energy levels are discrete.

Therefore, the differences between those levels are also specific.

This means atoms cannot emit every possible photon energy during bound-bound transitions.

Instead, they emit particular photon energies.

Therefore:

quantized energy levels

↓

specific energy differences

↓

specific photon energies

↓

specific frequencies

↓

specific wavelengths

↓

spectral lines

This is why atomic emission spectra contain lines rather than a continuous rainbow.

---

Atomic Transitions and Absorption Spectra

The same atomic energy levels also explain absorption spectra.

Suppose an atom can make:

E₂ → E₄

if it absorbs the correct photon.

The required photon energy is:

E₄ − E₂

The reverse transition:

E₄ → E₂

can emit a photon corresponding to the same energy difference.

Therefore, absorption and emission lines are closely related.

---

The Hydrogen Atom

Hydrogen is particularly useful for studying atomic transitions because it contains only one electron.

Its energy levels are commonly labelled:

n = 1, 2, 3, 4...

The ground state is:

n = 1

Higher values represent excited states.

Transitions between these states produce hydrogen's characteristic spectrum.

Hydrogen's Balmer Transitions

Some of hydrogen's most familiar visible emission lines occur when electrons fall to:

n = 2

Examples include:

n = 3 → 2

n = 4 → 2

n = 5 → 2

n = 6 → 2

These transitions form part of the Balmer series.

Because each transition has a different energy difference, each produces a different wavelength.

---

Why Different Elements Produce Different Spectra

Different elements have different:

• numbers of protons
• nuclear charges
• numbers of electrons
• electron arrangements
• allowed energy states

Therefore, their possible atomic transitions are different.

Different transitions produce different photon energies.

As a result, each element has a characteristic spectral pattern.

This allows scientists to identify elements using spectroscopy.

---

Atomic Transitions Are Not Literal Jumps Through Space

Energy-level diagrams can create a misleading mental picture.

When an electron moves from:

E₁ → E₂

it should not be imagined as a tiny ball physically travelling upward through the diagram.

The vertical axis represents energy.

An atomic transition is a change in the electron's quantum state.

The electron does not occupy all of the intermediate energies during the transition.

---

Excitation vs Ionization

It is important to distinguish between excitation and ionization.

Excitation

An electron moves from one bound energy state to a higher bound energy state.

The electron remains associated with the atom.

Ionization

Enough energy is supplied to remove the electron from the atom.

The atom becomes an ion.

The Ionization Limit

Energy-level diagrams often show an ionization limit.

For many diagrams, the reference:

E = 0

represents a free electron separated from the atom.

Bound electron states are shown with negative energies.

For example:

E = 0 eV — ionization limit

E₃ = −1.5 eV

E₂ = −3.4 eV

E₁ = −13.6 eV

To ionize an electron in E₁:

ΔE = 0 − (−13.6)

ΔE = 13.6 eV

So at least 13.6 eV is required in this simplified hydrogen example.

---

Real-World Importance of Atomic Transitions

Atomic transitions are important in many areas of science and technology.

Spectroscopy

Spectral lines reveal which elements are present in a substance.

Astronomy

Scientists identify elements in stars and gas clouds by analysing spectral lines.

Lasers

Controlled atomic or molecular transitions produce photons.

Fluorescent Lamps

Electrical energy excites atoms, which then emit radiation as they return to lower-energy states.

Neon Lighting

Excited gas atoms emit characteristic wavelengths.

LEDs

Although semiconductor physics differs from isolated atomic transitions, photons are also produced through transitions between allowed electronic energy states.

Atomic Clocks

Extremely precise transition frequencies are used to measure time.

---

A Complete Transition Problem

Suppose an atom has:

E₄ = −1 eV

E₃ = −3 eV

E₂ = −6 eV

E₁ = −10 eV

An electron begins in E₄ and moves directly to E₁.

Determine the Energy Change

ΔE = E_final − E_initial

ΔE = −10 − (−1)

ΔE = −9 eV

Interpret the Sign

The negative sign means the electron loses energy.

Therefore, energy is emitted.

Determine Photon Energy

E_photon = 9 eV

Convert to Joules

E = (9)(1.60 × 10⁻¹⁹)

E = 1.44 × 10⁻¹⁸ J

Calculate Frequency

f = E/h

f = (1.44 × 10⁻¹⁸)/(6.63 × 10⁻³⁴)

f ≈ 2.17 × 10¹⁵ Hz

Calculate Wavelength

λ = c/f

λ = (3.00 × 10⁸)/(2.17 × 10¹⁵)

λ ≈ 1.38 × 10⁻⁷ m

or:

138 nm

This wavelength lies in the ultraviolet region.

---

Common Mistakes

Thinking Upward Transitions Release Energy

Upward transitions require energy to be absorbed.

Thinking Downward Transitions Require Energy

Downward transitions release energy.

Ignoring Negative Energy Values

Remember to subtract carefully:

−3 − (−10) = +7

Reporting Negative Photon Energy

ΔE for the electron may be negative during emission, but the photon energy is reported as a positive magnitude.

Thinking Electrons Can Have Any Energy

Bound electrons occupy allowed quantized states.

Thinking an Electron Can Stop Between Levels

The spaces between levels do not represent allowed bound electron energies.

Assuming Every Excited Electron Goes Directly to the Ground State

Electrons may return through several transitions.

Forgetting Conservation of Energy in a Cascade

The total energy released through several transitions equals the overall energy difference between the starting and ending states.

Thinking Energy-Level Diagrams Show Physical Height

They show energy, not physical position.

Confusing Excitation with Ionization

Excitation keeps the electron bound.

Ionization removes the electron.

Forgetting the Frequency Relationship

Larger energy gap means higher photon frequency.

Forgetting the Wavelength Relationship

Larger energy gap means shorter photon wavelength.

---

Key Terms

Atomic transition — A change of an electron from one allowed atomic energy state to another.

Energy level — An allowed energy state of an electron.

Quantized — Restricted to particular discrete values.

Ground state — The lowest allowed energy state.

Excited state — An allowed energy state above the ground state.

Excitation — The process of moving an electron to a higher bound energy state.

De-excitation — A transition from a higher energy state to a lower one.

Absorption — The transfer of energy to an atom, potentially causing an upward transition.

Emission — The release of energy from an atom, often as a photon.

Photon — A discrete packet of electromagnetic energy.

Energy gap — The energy difference between two allowed states.

Energy-level diagram — A representation showing the allowed electron energies in an atom.

Initial state — The energy state occupied before a transition.

Final state — The energy state occupied after a transition.

Energy change (ΔE) — The difference between final and initial energies.

Planck's constant — The constant relating photon energy to frequency.

Frequency (f) — The number of wave cycles passing a point each second.

Wavelength (λ) — The distance between equivalent points on successive waves.

Electron volt (eV) — A unit of energy commonly used for atomic-scale processes.

Cascade — A sequence of transitions through intermediate energy states.

Ionization — The removal of an electron from an atom.

Ionization energy — The minimum energy required to remove an electron from a specified atomic state.

Ionization limit — The energy at which an electron is no longer bound to the atom.

Emission spectrum — The characteristic wavelengths emitted by excited matter.

Absorption spectrum — A spectrum showing wavelengths reduced by absorption.

Spectral line — A feature corresponding to a particular photon wavelength and atomic transition.

Conservation of energy — The principle that energy cannot be created or destroyed, only transferred or transformed.

---

Key Takeaways

• Electrons in atoms occupy quantized energy levels.
• An atomic transition occurs when an electron changes from one allowed energy state to another.
• The ground state is the lowest allowed energy state.
• Higher allowed states are excited states.
• Electrons move upward by absorbing energy.
• Electrons move downward by releasing energy.
• The electron energy change is:

ΔE = E_final − E_initial

• Positive ΔE means the electron gained energy.
• Negative ΔE means the electron lost energy.
• During photon emission, the photon carries the magnitude of the lost energy:

E_photon = |ΔE|

• Photon energy and frequency are related by:

E = hf

• Frequency and wavelength are related by:

c = fλ

• Larger energy gaps produce higher-energy photons.
• Higher-energy photons have higher frequencies.
• Higher-frequency photons have shorter wavelengths.
• Electrons cannot occupy arbitrary energies between allowed bound states.
• A photon must have an appropriate energy to cause a particular upward transition.
• Excited electrons can return directly to a lower level or through a cascade of intermediate levels.
• Energy is conserved during atomic transitions.
• Different transitions produce different photon energies.
• Different photon energies produce different spectral lines.
• Atomic emission spectra therefore provide evidence for quantized energy levels.
• Absorption spectra are produced when photons cause allowed upward transitions.
• Energy-level diagrams show energy states, not literal electron positions.
• Excitation and ionization are different: excitation leaves the electron bound, while ionization removes it.
• Different elements have different energy-level structures and therefore different characteristic spectra.

The simplest rules to remember are:

Electron up → energy absorbed

Electron down → energy emitted

Bigger energy gap → higher frequency → shorter wavelength

---

Check Your Understanding

1. What is an atomic transition?

2. What does it mean for atomic energy levels to be quantized?

3. Define the ground state.

4. Define an excited state.

5. What must happen for an electron to move to a higher energy level?

6. What happens when an excited electron returns to a lower energy level?

7. What does an upward arrow represent on an energy-level diagram?

8. What does a downward arrow represent?

9. State the equation for electron energy change.

10. An electron moves from −10 eV to −6 eV. Calculate ΔE and state whether energy is absorbed or released.

11. An electron moves from −2 eV to −8 eV. Calculate ΔE.

12. How much photon energy is emitted in Question 11?

13. Why is the emitted photon's energy reported as positive even though the electron's ΔE is negative?

14. Explain why a 3 eV photon cannot cause a transition requiring 5 eV in a simplified atomic model.

15. What is meant by a cascade?

16. An electron drops 2 eV and then 5 eV in two transitions. How much total energy has it lost?

17. How does this demonstrate conservation of energy?

18. What happens to photon frequency as the energy difference between levels increases?

19. What happens to photon wavelength as the energy difference increases?

20. A transition releases 4.80 × 10⁻¹⁹ J. Calculate the photon frequency using h = 6.63 × 10⁻³⁴ J·s.

21. Calculate the wavelength of the photon in Question 20.

22. Explain how atomic transitions produce an emission spectrum.

23. Explain how atomic transitions produce an absorption spectrum.

24. Why does each element have a characteristic spectral pattern?

25. Explain why an energy-level diagram should not be interpreted as showing an electron's physical height.

26. Explain the difference between excitation and ionization.

27. Why can an excited electron produce more than one photon while returning to the ground state?

28. Convert 5.0 eV into joules.

29. An electron falls through an energy difference of 6.0 eV. Would its photon have a longer or shorter wavelength than a photon produced by a 3.0 eV transition? Explain.

30. Challenge: An atom has the following energy levels:

E₄ = −1 eV
E₃ = −3 eV
E₂ = −6 eV
E₁ = −10 eV

a. Identify the ground state.
b. Identify the excited states shown.
c. Calculate ΔE for E₁ → E₂.
d. Is energy absorbed or emitted in part c?
e. Calculate ΔE for E₂ → E₄.
f. Calculate ΔE for E₄ → E₁.
g. What is the energy of the photon emitted in part f?
h. Calculate the energy released during E₄ → E₃.
i. Calculate the energy released during E₃ → E₁.
j. Show that the total energy released in E₄ → E₃ → E₁ equals the energy released in E₄ → E₁.
k. Which possible downward transition produces the highest-energy photon?
l. Which produces the highest-frequency photon?
m. Which produces the shortest-wavelength photon?
n. Could a 4 eV photon be absorbed by an electron initially in E₁? Identify the resulting state if possible.
o. Could a 5 eV photon produce an upward transition from E₁ using the levels shown? Explain.
p. Explain how these energy levels could produce several different spectral lines.