40-6 X Rays and the Ordering of the Elements#

Prompts

Where do x rays sit in the electromagnetic spectrum? How are they produced when high-energy electrons strike a target?
Distinguish the continuous x-ray spectrum from the characteristic x-ray spectrum. What causes each? What determines the cutoff wavelength \(\lambda_{\min}\)?
Describe the two-step process for characteristic x rays: knock out a K-shell electron, then an L- or M-shell electron fills the hole. What are the K\(\alpha\) and K\(\beta\) lines?
What is a Moseley plot? Why does \(\sqrt{f}\) vs. \(Z\) give a straight line? What did Moseley conclude about the ordering of elements?
Explain the screening effect: why does a K-shell electron “see” an effective nuclear charge \((Z-1)e\)? How does this lead to \(f \propto (Z-1)^2\) for the K\(\alpha\) line?

Lecture Notes#

Overview#

X rays are short-wavelength EM radiation produced when high-energy electrons strike a target. The spectrum has two parts: a continuous component and characteristic peaks.
The continuous spectrum arises from electrons losing energy in collisions (bremsstrahlung); the cutoff wavelength \(\lambda_{\min} = hc/K_0\) occurs when an electron loses all its kinetic energy in one collision.
Characteristic x rays are produced when an incident electron knocks out an inner-shell electron and an outer electron fills the hole, emitting a photon. The K\(\alpha\) line comes from L\(\to\)K transitions; K\(\beta\) from M\(\to\)K.
Moseley’s work (1913) showed that \(\sqrt{f}\) vs. atomic number \(Z\) is a straight line — proving that nuclear charge \(Z\), not atomic weight, orders the elements in the periodic table.

X-Ray Production#

When a solid target (e.g., copper, tungsten, molybdenum) is bombarded with electrons of kinetic energy \(K_0\) in the kiloelectron-volt range, x rays are emitted. X rays lie in the electromagnetic spectrum between ultraviolet and gamma rays (wavelengths roughly 0.01–10 nm).

The Continuous X-Ray Spectrum#

An incident electron can collide with a target atom and lose part of its kinetic energy \(\Delta K\); that energy is radiated as an x-ray photon. The electron may undergo multiple collisions, each producing a photon of different energy. The collection of such photons forms the continuous x-ray spectrum (bremsstrahlung).

Cutoff wavelength: The shortest possible wavelength occurs when an electron loses all its kinetic energy \(K_0\) in a single collision. The photon then carries energy \(hf = K_0\):

(423)#\[ K_0 = hf = \frac{hc}{\lambda_{\min}} \quad \Rightarrow \quad \lambda_{\min} = \frac{hc}{K_0} \]

\(\lambda_{\min}\) depends only on \(K_0\) — it is independent of the target material.
Increasing \(K_0\) decreases \(\lambda_{\min}\).

The Characteristic X-Ray Spectrum#

Superimposed on the continuous spectrum are sharp peaks at specific wavelengths — the characteristic x-ray spectrum of the target element.

Two-step process:

An incident electron knocks out a deep-lying (low \(n\)) electron from the target atom, creating a vacancy (hole) in that shell.
An electron from a higher shell jumps down to fill the hole, emitting an x-ray photon.

Shell labels (historical): K (\(n=1\)), L (\(n=2\)), M (\(n=3\)), N (\(n=4\)), …

Line	Transition	Hole moves
K\(\alpha\)	L \(\to\) K (\(n=2 \to n=1\))	K \(\to\) L
K\(\beta\)	M \(\to\) K (\(n=3 \to n=1\))	K \(\to\) M

Hole picture

It is often convenient to describe characteristic x rays in terms of holes rather than electrons. An energy-level diagram for holes shows the energy of the atom with a hole in each shell. A transition that fills a K-shell hole (electron L\(\to\)K) is drawn as the hole moving from K to L.

Ordering the Elements: Moseley’s Work#

In 1913, H. G. J. Moseley measured the characteristic x-ray wavelengths of many elements. He found that for a given spectral line (e.g., K\(\alpha\)), a plot of \(\sqrt{f}\) vs. atomic number \(Z\) yields a straight line — a Moseley plot.

Conclusion: The position of an element in the periodic table is set by \(Z\) (nuclear charge), not by atomic weight. Before Moseley, ordering by mass required inversions for some pairs; his work showed that \(Z\) is the fundamental quantity.

The characteristic x-ray spectrum became the signature of an element, resolving disputes about new elements and properly ordering the lanthanides (rare earths).

Accounting for the Moseley Plot: Screening#

The K-shell electrons lie very close to the nucleus and are sensitive to its charge. Because of the screening effect — the other K-shell electron partially shields the nucleus — each K electron effectively “sees” a nuclear charge of about \((Z-1)e\) rather than \(Ze\).

Using a hydrogen-like model with effective charge \((Z-1)e\), the energy of a level \(n\) is

(424)#\[ E_n \approx -\frac{(13.6\ \text{eV})(Z-1)^2}{n^2} \]

For the K\(\alpha\) transition (L\(\to\)K, \(n=2 \to n=1\)):

(425)#\[ \Delta E = E_2 - E_1 = (13.6\ \text{eV})(Z-1)^2\left(\frac{1}{1^2} - \frac{1}{2^2}\right) = (10.2\ \text{eV})(Z-1)^2 \]

Since \(hf = \Delta E\):

(426)#\[ f = \frac{(10.2\ \text{eV})(Z-1)^2}{h} \propto (Z-1)^2 \]

Taking the square root:

(427)#\[ \sqrt{f} = C(Z-1) = CZ - C \]

where \(C\) is a constant. This is the equation of a straight line — matching Moseley’s data.

Why x rays, not optical?

Optical spectra involve outer electrons, which are heavily screened by inner electrons and are poor probes of nuclear charge. The K electrons are close to the nucleus and directly reflect \(Z\), so characteristic x rays show clean, element-specific regularities.

Using the Moseley Plot#

From \(\sqrt{f} = C(Z-1)\) and \(f = c/\lambda\), we have \(\sqrt{c/\lambda} \propto Z-1\). For two elements with atomic numbers \(Z_1\) and \(Z_2\):

(428)#\[ \sqrt{\frac{\lambda_1}{\lambda_2}} = \frac{Z_2 - 1}{Z_1 - 1} \]

Worked example: Identifying an impurity

A cobalt target (\(Z_{\text{Co}} = 27\)) is bombarded with electrons. The K\(\alpha\) wavelengths are 178.9 pm (cobalt) and 143.5 pm (impurity). Identify the impurity.

Solution: Using \(\sqrt{\lambda_{\text{Co}}/\lambda_X} = (Z_X - 1)/(Z_{\text{Co}} - 1)\):

\[ \sqrt{\frac{178.9}{143.5}} = \frac{Z_X - 1}{26} \quad \Rightarrow \quad Z_X - 1 = 26 \sqrt{\frac{178.9}{143.5}} \approx 29.0 \quad \Rightarrow \quad Z_X \approx 30 \]

The impurity is zinc (\(Z = 30\)). The shorter wavelength for zinc means a larger energy jump (higher \(Z\) → stronger nuclear attraction).

Summary#

Continuous spectrum: Electrons lose energy in collisions; \(\lambda_{\min} = hc/K_0\) is independent of the target.
Characteristic spectrum: Inner-shell electron knocked out; outer electron fills hole and emits x ray. K\(\alpha\) (L\(\to\)K), K\(\beta\) (M\(\to\)K).
Moseley plot: \(\sqrt{f}\) vs. \(Z\) is linear; elements are ordered by nuclear charge \(Z\).
Screening: K electrons see effective charge \((Z-1)e\); leads to \(f \propto (Z-1)^2\) for K\(\alpha\).