16 X – Ray Spectroscopy
Contents:
1. The Origin of X – Rays
2. Emission Spectra of X – Ray
3. Atomic Number and the Position of Emission Lines
4. X – ray Emission (Doublet) Spectra
5. Satellites
6. Continuous X – Ray Emission
7. X – Ray Absorption Spectra
8. Assignment
The students will be able to learn about X-Rays origin and X-Ray Spectroscopy
1. The Origin of X – Rays
X – Rays are radiations in the electromagnetic spectrum of wavelength between ~.01 to ~10 nm. These are characterized by index of refraction very near to unity for all materials. Electrons are emitted thermally from the heated cathode C and are accelerated towards the anode target A (a metal) by the applied potential V. When accelerated electrons strike the target, X – rays are produced. The face of the target is at an angle relative to the direction of electron beam and the X – rays that leave the anode target pass through the side of the tube. The geometry of the tube is such that it evacuates to allow electrons to reach the target without any interruption.
The X – rays are produced both by deceleration of electrons in the metal target and by the excitation of the core electrons in the atom of the target. The first process gives a broad continuous spectrum and the second gives sharp lines.
When a moving electron is stopped suddenly, all its energy appears as photon of frequency υ of X – rays. The energy of an electron of charge ‘e’ in dropping through a potential difference V is eV and
An electron does not lose all its energy in this way; it will have a number of glancing collisions with the atoms that it collides and causing them to vibrate. The temperature of the target increases due to this process. The minimum value of is possible that accounts for the short wavelength cut off. However, the larger wavelengths are more probable and thus the rapid increase in the intensity. The figure presents the X – ray spectrum that result when Molybdenum target is bombarded by electron at 35 keV. The electron beam on striking the target gets decelerated and also a small fraction of electron of the beam strikes the target and ejects the inner shells electrons. The atom becomes unstable and outer shell electrons in the same atom drops into the hole/vacancy caused by the ejection of the electron. In this process it loses energy and a photon is emitted.
Fig. x – ray spectra from Molybdenum target at 35kV accelerating potential
As E is a definite quantity associated with the electron energy change in the atom, the wavelength achieved is specific. Several wavelengths are possible and they constitute the characteristic X – ray line spectrum shown as peaks in the figure. The energy of the characteristic X – ray produced is very weakly dependent on the chemical structure in which the atom is bound indicating that non- bonding shells of atoms are the characteristic X – ray source. The resulting characteristic spectrum is superimposed on the continuum. An atom remains ionized for a very short time of the order of ~10-14s. Thus the incident electrons that arrive about every ~10-17 s can repeatedly ionize an atom as not all outer incident electrons fall into holes to produce X – rays.
Assuming that an electron from K shell is ejected and a hole is created in the shell, then an electron from the higher shell L, M, N,…. makes a radiative transition filling the hole. The energy of the photon emitted is in the range from a few keV to a few hundred keV and thus lie in the X – ray region. The emission spectrum that results in a line spectrum forms a simple series. The lines originating from a K shell vacancy are called Kα , Kβ, K lines and corresponding to L→K, M→K, N→K transition. Kα is the strongest and the emission of K series is accompanied by other series. The vacancies are created in L, M, N, … shells as a result of lines of K series, those are filled by M,N, … shell electrons. Thus K series may be accompanied by L, M, N, … and so on. These transitions can be shown on an energy level diagram.
This is different from that used in the atomic spectra of valence electrons in the sense that X – ray diagram gives the energy of the atom when one electron of the quantum number n, l, j is missing . Thus the diagram depicts the energy levels of a hole, with quantum number n, l, j that jumps from one subshell to the next when the atom emits X-ray line spectrum. The absence of an electron of negative energy (representation of a hole) , the energy associated with the hole is positive. In the figure, the levels are also represented by capital letters K, L, M, N, corresponding to n = 1, 2, 3, 4, respectively.
X – rays can be classified as hard or soft depending upon the wavelength,. The K X – rays are called hard and L radiation is soft and M, N, O X – rays are very soft.
M and L radiation arising from heavy elements are harder and are of shorter wavelengths than K radiation of lighter elements.
K radiation of an element is more penetrating than that of L, M, N, O etc.
2. Emission Spectra of X – Ray
Spin, orbital and total angular momenta for every filled subshell or shell are zero. Suppose that one electron is removed from the closed subshell, then the allowed values of spin, orbital and total angular momenta for the rest of the electrons in the subshell are identical with those of the single electron that can complete the subshell. This infers that the states of np1, nd1……. are same as that of np5, np9, .. , respectively, that is, the quantum number n, l and j of subshell with one missing electron are identical with those of an electron that completes the subshell. As only one electron is missing from the subshell, therefore s = S = ½ and multiplicity is 2. The figure depicts the energy levels corresponding to various values of n. There are n-1 values of l for each value of n, and for each value of l, there are two j values corresponding to j = 1+1/2 and j =1 – ½.
As an example, for L shell: n = 2, the 1 values are 0 and 1 and corresponding j values are ½ corresponding to 1 = 0 and ½, 3/2 for l=1 . Thus each n level splits up into 2n-1 components.
In X – rays, shells are designated with n from 1 to 7 with letters K, L, M, N, O, P, Q, respectively and levels in each shell with Roman numerals in indices
The Nomenclature of X – ray states is shown in the table for ready reference
A part of energy level diagram for X – ray emission spectrumIndicating only K and L levels is shown below:
There is a transition from one excited state to another for emission of an X – ray line and it is notable that, both the initial and final states are excited. The emission of K line is due to transition of an electron level L to that of K accompanied by emission of a photon whose energy is equal to the difference in energy between these levels, so the selection rules for the transitions are
∆1 = ±1, ∆ = 0, ±1
All the transitions between K and L levels are not allowed as per the selection rules. As L1→K is not allowed since∆1=0 is forbidden. One of the two allowed transitions is shown in the figure. The transition has been are denoted by an arrow pointing in the direction of electron transition with emission of X – ray photon i.e. from level Llll to level K.
The two allowed transitions between L and K levels are further given the name 1 and 2 . The 1 is the intense line while weaker is 2. Similarly, 1 and 2 are designated. A similar system is used for lines. The 1 line arises from a K-Llll transition(1 1/2 → 2 3/2) for the atom corresponding to the electron from Llll filling the vacancy in the K shell. The 2 line corresponds to electron from Lll filling the vacancy in the K shell. The statistical factor of Llll is 2j+1=4 while that of Lll is 2, therefore 1 is to be twice as intense as the 2 line. Similarly, 1 (K-Mlll transition) is twice as strong as 2 (K-Mll transition).
It is noteworthy that, that 1 and 1 are not equally intense. The rule of proportionality to the statistical weight is good only for the intensity of lines of very nearly the same energy.
3. Atomic Number and the Position of Emission Lines
Moseley, in 1913, observed that the wavelength of characteristic X – ray lines shift continuously with the change in the element atomic number Z. He observed that with increasing Z, wavelength of X – ray decreases and hence the frequency of X – ray increased.
A plot of square root of frequency versus Z gives a straight line. This increase in frequency with increase of Z is understood to be due to increasing binding energy of the electron with increase of number of protons in the nucleus.
The electron energy Enl for a given n and 1 is given by
This equation is known as Mosley Law. This provided the information that where elements were missing from the periodic table and has led to the discovery of some of the elements.
For Kα line, nf=1 and ni=2 and thus σ~1 and 7.4 for K and L shells, respectively.
From the above equation wavelength for K series is λ=1/ v
for L series it can also be obtained by putting the suitable values z∗and a.
4. X – ray Emission (Doublet) Spectra
An X – ray doublet is referred to the separation between adjacent X- ray terms of the same principal quantum number. The intervals that correspond to differences between levels with the same n, 1 and s but different j are called spin doublets also called regular doublets. The frequency interval remains constant all through the spectrum of the same element but increases very slowly with rising value of Z. Examples of spin doublets are Lll – Llll, Mll or Mlv – Mv etc.
The irregular doublets or the screening doublets are pair of energy levels having equal n and j but quantum number 1differs by 1. The separation between lines remains constant for all the elements.
The examples of screening doublets are:
Ll – Lll (2s1/2 – 2p1/2);
Mlll – Mlv (3p3/2 – 3d3/2);
Ml – Mll (3s1/2 – 3p1/2) ……..
The explanation of the doublet separation is the following:
The energy of a given level due to the Coulomb interaction of an electron with the nucleus and the energy of a level (in a first approximation) is
σ1 being the screening constant depending on the value of 1. The position of the energy level is given by a similar expression to that of hydrogen therby taking into account the correction due to relativistic effect and spin-orbit interaction, given by
As the screening constant represents an average over ranges of effective Z value, therefore different screening constants for expression with different power of Z are expected. The screening constant σ2 < σ1.
The term value is
And the doublet separation
Thus the energy difference ∆T ( cm −1) between regular doublets is proportional to the fourth power of (? − ?2).
The value of ?2 is independent of Z but depend on the orbit.
?2 increases with the subshells going farther from the nucleus. It is reported that ?2 = 3.5 for Lll ,Llll ; 8.5 for Mll ,Mlll ; 13 for Mlv and Mv etc.
Ignoring the term in ?2
5. Satellites
The lines in the X – ray spectra are often found to be accompanied by faint satellites. As these lines do not fit into conventional energy level diagram and are therefore, called non-diagram lines. They occur usually on the short wavelength side of the diagram lines. The origin of satellite is related to transition between the states of double and multiple ionization.
Considering that the incident electron beam striking the target of X – ray tube has sufficiently high energy and it ejects an electron from K shell as well as from L shell that results in holes in K and L shells and such a state undergoes a radiative transition into any one of a number of other double ionization like KL→LL (an electron dropping from L shell into K shell). The loss of energy due to emission of line in the transition KL→LL is greater than the normal K→L transition that gives rise toKα line.
The other important source of satellite line is the auger process. In this process when an electron from an outer orbit make a transition to a hole in the core orbital, a photon is emitted that may be passed to another electron of outer orbit leading to its ionization. Thus a doubly ionized state is produced. For example, an electron from one of the higher shells. This infers that Auger effect produces a doubly ionized state, that give rise to the emission of satellite of L series.
6. Continuous X – Ray Emission
In addition to characteristic spectrum a continuous spectrum of a different nature arises, when X – rays are excited by electron. The continuous part of the spectrum is due to slowed down movement of an electron striking the target in the Coulomb field of the nucleus.
The continuous spectrum has a sharp limit 0 on the lower wavelength side whose value depends on the X – ray tube voltage and is independent of atomic number Z of the target element. The relationship between wavelength and X – ray tube voltage is already established as 
Below this short wavelength limit no X – ray radiation is observed. Wavelength of maximum intensity ?m is approximately one and half time 0 and therefore
???1/2 = ???????t
Intensity of continuous X – ray is nearly proportional to the square of voltage and to the first power of atomic number that is
I = k Z V2
Here k is constant of proportionality.
The position of characteristic spectrum is independent of voltage of X – ray tube. It is to note that by raising voltage the limit of continuous X – ray spectrum shifts towards shorter wavelength side.
7. X – Ray Absorption Spectra
The intensity of rays become weaker because of absorption and scattering, when a beam of X – ray passes through a sample. The absorption of X – ray emission energy occurs as a result of a single process. The X – ray photon knocks out electron of a shell and the energy of the absorbed photon is thus transformed into kinetic energy of the electron plus the potential energy of the excited atom that equals binding energy of the electro.
The X – ray emission line yields information on the difference in binding energies between two electronic states. However, to determine absolute binding energies, absorption of X – ray has to be studied. For such a purpose an X – ray continuum is used and sample absorption as a function of wavelength is measured.
The X – rays of the longest wavelength force out electrons from the outer shell. With increasing energy of X – rays (decreasing wavelength) a small part of it is required to knock out the electron from the given shell. This causes reduced absorption. This decrease continue till the X – ray energy is sufficient to knock out electron from the next deeper lying inner shell and give rise to a sharp increase in absorption as shown in the figure.
It causes a discontinuous behavior of absorption with λ. The discontinuous behavior of absorption and wavelength or frequency is called absorption edge and corresponds to photon critical energy. The absorption edges are labeled as K edge, L edge, M edge etc. from the observed absorption edge an approximate binding energy is obtained.
For increasing energy of X – ray, the threshold for photoemission of deeper shells are reached and additional contributions to the total absorption are obtained resulting in a number of edges. Apart from the edge there is a general fall off in absorption due to λ3 dependence of the absorption coefficient given by ? =? ? ?3?3. The figure depicts the X –ray absorption spectrum with absorption edges. Here k is a constant and ? is the density.
It is worth to learn that the soft X – rays are absorbed strongly in comparison with hard X – rays. The wavelength of the absorption edge is defined in terms of the orbital energy
The absorption edge exhibits fine structure corresponding to the fine structure of the core state. However, K edge also exhibits a structure that is due to discrete levels close to the series limit.
Assignments:
1. Find the shortest wavelength fine structure structure of the core state. Potential is 50 keV.
2. What energy is needed to excite Cd atoms (Z =48) so that all series are observed?
To observed all series, it is necessary to have a vacancy in K shell. The energy required to remove a K electron is
3. What energy is required to excite Cd atoms (Z=48) so that L-series X –rays are observed (? ?? 7.4 ??? ? ?????s)
For L series to be observed, there should be a vacancy in L shell. Therefore, energy required to remove the electron from the L –shell by
Using l eV=8065.48 cm-1
4. Which element hasX – ray line whose wavelength is 1541.23 XU? 1000 XU=1.00202 Å. The wavelength = 1.54123 1.00202 Å = 1.5443 Å
The element with atomic number Z = 29 is Cu.
5. If K and L energy levels of an element lie at roughly 78 keV and 12keV, respectively, compute the approximate wavelength of ?a line. What minimum potential difference across an X – ray tube is required to excite this line? At approximately what wavelength is the K absorption edge?
For observing ?a line, a vacancy has to be created in the K shell. The K level lie at 78 keV therefore, this much potential difference has to be applied to remove the electron. The line is due to transition between K and L shells. The energy difference between these two is (78-12) keV=66keV. The wavelength corresponding to this much of potential difference from
6. The Lll – Llll regular doublet separation for Ag (Z=47) is 0.173 keV. Evaluate the Lll – Llll regular doublet separation in In (Z = 49)?2 = 3.5 for Lll , Llll .
| you can view video on X – Ray Spectroscopy | |