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{{Short description|Mass effects on spectroscopy}}

{{Quantum book backlink|Atomic and spectroscopy}}
The '''isotopic shift''' (also called isotope shift) is the shift in various forms of [[Physics:Spectroscopy|spectroscopy]] that occurs when one nuclear [[Physics:Isotope|isotope]] is replaced by another.
[[File:Isotopic shift.jpg|thumb|450px|Isotopic shift: variations in nuclear mass and size cause small shifts in atomic energy levels, leading to observable differences in spectral lines (Δλ) between isotopes.]]
==NMR spectroscopy==
[[File:H2&HDlowRes (2).png|thumb|[[Wikipedia:H NMR spectrum|[H NMR spectrum]] of a solution of HD (labeled with red bars) and H2 (blue bar). The 1:1:1 triplet arises from the coupling of the 1H nucleus ([[Wikipedia:nuclear spin|nuclear spin|''I'']] = 1/2) to the 2H nucleus (''I'' = 1).]] 
In NMR spectroscopy, isotopic effects on chemical shifts are typically small, far less than 1 ppm, the typical unit for measuring shifts. The {{chem|1|H}} NMR signals for {{chem|1|H|2}} and {{chem|1|H}}{{chem|2|H}} ("HD") are readily distinguished in terms of their chemical shifts. The asymmetry of the signal for the "protio" impurity in {{chem|CD|2|Cl|2}} arises from the differing chemical shifts of {{chem|CDHCl|2}} and {{chem|CH|2|Cl|2}}.

==Vibrational spectra==
Isotopic shifts are best known and most widely used in vibration spectroscopy, where the shifts are large, being proportional to the ratio of the square root of the isotopic masses. In the case of hydrogen, the "H-D shift" is (1/2)1/2 ≈ 1/1.41. Thus, the (totally symmetric) C−H and C−D vibrations for {{chem|CH|4}} and {{chem|CD|4}} occur at 2917 cm−1 and 2109 cm−1 respectively.<ref>{{cite web |author=Takehiko Shimanouchi |title=Tables of Molecular Vibrational Frequencies Consolidated |volume=I |date=1972 |id=NSRDS-NBS-39 |publisher=National Bureau of Standards |url=https://www.nist.gov/data/nsrds/NSRDS-NBS-39.pdf |access-date=2017-07-13 |archive-date=2016-08-04 |archive-url=https://web.archive.org/web/20160804010334/http://www.nist.gov/data/nsrds/NSRDS-NBS-39.pdf |url-status=dead }}</ref> This shift reflects the differing [[Physics:Reduced mass|reduced mass]] for the affected bonds.

==Atomic spectra==
Isotope shifts in atomic spectra are minute differences between the electronic energy levels of isotopes of the same element. They are the focus of a multitude of theoretical and experimental efforts due to their importance for atomic and nuclear physics. If atomic spectra also have [[Physics:Hyperfine structure|hyperfine structure]], the shift refers to the [[Centre wavelength|center of gravity]] of the spectra.

From a nuclear physics perspective, isotope shifts combine different precise atomic physics probes for studying [[Physics:Nuclear structure|nuclear structure]], and their main use is nuclear-model-independent determination of charge-radii differences.

Two effects contribute to this shift:

===Mass effects===
The mass difference (mass shift), which dominates the isotope shift of light elements.<ref>{{Citation |last=King |first=W. H. |chapter=Isotope Shifts in X-Ray Spectra |date=1984 |pages=55–61 |publisher=Springer US |isbn=9781489917881 |doi=10.1007/978-1-4899-1786-7_5 |title=Isotope Shifts in Atomic Spectra}}.</ref> It is traditionally divided to a '''normal mass shift''' (NMS) resulting from the change in the reduced electronic mass, and a '''specific mass shift''' (SMS), which is present in multi-electron atoms and ions.

The NMS is a purely kinematical effect, studied theoretically by Hughes and Eckart.<ref>{{cite journal |first=D. J. |last=Hughes |first2=C. |last2=Eckart |journal=Phys. Rev. |volume=36 |issue=4 |date=1930 |pages=694–698 |title=The Effect of the Motion of the Nucleus on the Spectra of Li I and Li II |doi=10.1103/PhysRev.36.694 |bibcode=1930PhRv...36..694H }}</ref> It can be formulated as follows:

In a theoretical model of atom, which has a infinitely massive nucleus, the energy (in [[Physics:Wavenumber|wavenumber]]s) of a transition can be calculated from [[Physics:Rydberg formula|Rydberg formula]]:
<math display="block">
\tilde{\nu}_\infty = R_\infty \left(\frac{1}{n^2} - \frac{1}{n'^2} \right),
</math>
where <math>n</math> and <math>n'</math> are principal quantum numbers, and <math>R_\infty</math> is [[Physics:Rydberg constant|Rydberg constant]].

However, for a nucleus with finite mass <math>M_N</math>, reduced mass is used in the expression of Rydberg constant instead of mass of electron:
<math display="block">
\tilde{\nu} = \tilde{\nu}_\infty \frac{M_N}{m_e + M_N}.
</math>

For two isotopes with [[Physics:Atomic mass|atomic mass]] approximately <math>A' M_p</math> and <math>A'' M_p</math>, the difference in the energies of the same transition is
<math display="block">
\Delta\tilde{\nu} =
\tilde{\nu}_\infty \left( \frac{1}{1 + \frac{m_e}{A'' M_p}} - \frac{1}{1 + \frac{m_e}{A' M_p}} \right) \approx
\tilde{\nu}_\infty \left[ 1 - \frac{m_e}{A'' M_p} \left( 1 - \frac{m_e}{A' M_p} \right) \right] \approx
\frac{m_e}{M_p} \frac{A'' - A'}{A' A''} \tilde{\nu}_\infty.
</math>
The above equations imply that such mass shift is greatest for hydrogen and deuterium, since their mass ratio is the largest, <math>A'' = 2A'</math>.

The effect of the specific mass shift was first observed in the spectrum of neon isotopes by [[Biography:Hantaro Nagaoka|Nagaoka]] and Mishima.<ref>H. Nagaoka and T. Mishima, Sci. Pap. Inst. Phys. Chem. Res. (Tokyo) '''13''', 293 (1930).</ref>

Consider the [[Physics:Kinetic energy|kinetic energy]] operator in [[Schrödinger equation]] of multi-electron atoms:
<math display="block">
T = \frac{p_n^2}{2M_N} + \sum_i \frac{p_i^2}{2m_e},
</math>
For a stationary atom, the conservation of momentum gives
<math display="block">
p_n = -\sum_i p_i.
</math>
Therefore, the kinetic energy operator becomes
<math display="block">
T =
\frac{\left( \sum_i p_i \right)^2}{2M_N} + \frac{\sum_i p_i^2}{2m_e} =
\frac{\sum_i p_i^2}{2M_N} + \frac{1}{M_N} \sum_{i > j} p_i \cdot p_j + \frac{\sum_i p_i^2}{2m_e}.
</math>

Ignoring the second term, the rest two terms in equation can be combined, and original mass term need to be replaced by the reduced mass <math>\mu = \frac{m_e M_N}{m_e + M_N}</math>, which gives the normal mass shift formulated above.

The second term in the kinetic term gives an additional isotope shift in spectral lines known as specific mass shift, giving
<math display="block">
\frac{1}{M_N} \sum_{i > j} p_i \cdot p_j = -\frac{\hbar^2}{M_N} \sum_{i > j} \nabla_i \cdot \nabla_j.
</math>
Using perturbation theory, the first-order energy shift can be calculated as
<math display="block">
\Delta E = -\frac{\hbar^2}{M} \sum_{i > j} \int \psi^* \nabla_i \cdot \nabla_j \psi \,d^3 r,
</math>
which requires the knowledge of accurate many-electron [[Wave function|wave function]]. Due to the <math>1/M_N</math> term in the expression, the specific mass shift also decrease as <math>1/M_N^2</math> as mass of nucleus increase, same as normal mass shift.

===Volume effects===
The volume difference (field shift) dominates the isotope shift of heavy elements. This difference induces a change in the electric charge distribution of the nucleus. The phenomenon was described theoretically by Pauli and Peierls.<ref>W. Pauli, R. E. Peierls, Phys. Z. 32 (1931) 670.</ref><ref>{{cite book |first=P. |last=Brix |first2=H. |last2=Kopfermann |chapter=Neuere Ergebnisse zum Isotopieverschiebungseffekt in den Atomspektren |title=Festschrift zur Feier des Zweihundertjährigen Bestehens der Akademie der Wissenschaften in Göttingen |lang=de |publisher=Springer |year=1951 |isbn=978-3-540-01540-6 |doi=10.1007/978-3-642-86703-3_2 |pages=17–49 }}</ref><ref>{{cite book |first=H. |last=Kopfermann |title=Nuclear Moments |publisher=[[Company:Academic Press|Academic Press]] |year=1958 |url=https://archive.org/details/nuclearmoments0000kopf |url-access=registration }}</ref> Adopting a simplified picture, the change in an energy level resulting from the volume difference is proportional to the change in total electron probability density at the origin times the mean-square charge radius difference.

For a simple nuclear model of an atom, the nuclear charge is distributed uniformly in a sphere with radius <math>R = r_0 A^{1/3}</math>, where ''A'' is the atomic mass number, and <math>r_0 \approx 1.2 \times 10^{-15}\ \text{m}</math> is a constant.

Similarly, calculating the electrostatic potential of an ideal charge density uniformly distributed in a sphere, the nuclear electrostatic potential is
<math display="block">
V(r) = \begin{cases}
\dfrac{Ze^2}{(4\pi\epsilon_0)2R} \left( \dfrac{r^2}{R^2} - 3 \right), & r \leq R, \\
-\dfrac{Ze^2}{(4\pi\epsilon_0)r}, & r \geq R.
\end{cases}
</math>
When the unperturbed Hamiltonian is subtracted, the perturbation is the difference of the potential in the above equation and Coulomb potential <math>-\frac{Ze^2}{(4\pi\epsilon_0)r}</math>:
<math display="block">
H' = \begin{cases}
\dfrac{Ze^2}{(4\pi\epsilon_0)2R} \left( \dfrac{r^2}{R^2} + \dfrac{2R}{r} - 3 \right), & r \leq R, \\
0, & r \geq R.
\end{cases}
</math>

Such a perturbation of the atomic system neglects all other potential effect, like relativistic corrections. Using the [[Perturbation theory (quantum mechanics)|perturbation theory (quantum mechanics)]], the first-order energy shift due to such perturbation is
<math display="block">
\Delta E = \langle \psi_{nlm} | H' | \psi_{nlm} \rangle.
</math>
The wave function <math>\psi_{nlm} = R_{nl}(r)Y_{lm}(\theta, \phi)</math> has radial and angular parts, but the perturbation has no angular dependence, so the spherical harmonic normalize integral over the unit sphere:
<math display="block">
\Delta E = \frac{Ze^2}{(4\pi\epsilon_0)2R} \int_0^R |R_{nl}(r)|^2 \left( \frac{r^2}{R^2} + \frac{2R}{r} - 3 \right) r^2 \,dr.
</math>
Since the radius of nuclues <math>R</math> is small, and within such a small region <math>r \leq R</math>, the approximation <math>R_{nl}(r) \approx R_{nl}(0)</math> is valid. And at <math>r \approx 0</math>, only the ''s'' sublevel remains, so <math>l = 0</math>. Integration gives
<math display="block">
\Delta E \approx
\frac{Ze^2}{(4\pi\epsilon_0)} \frac{R^2}{10} |R_{n0}(0)|^2 =
\frac{Ze^2}{(4\pi\epsilon_{0})} \frac{2\pi}{5} R^2 |\psi_{n00}(0)|^2.
</math>

The explicit form for [[Hydrogen-like atom|hydrogenic]] wave function, <math>|\psi_{n00}(0)|^2 = \frac{Z^3}{\pi a_\mu^3 n^3}</math>, gives
<math display="block">
\Delta E \approx \frac{e^2}{(4\pi\epsilon_0)} \frac{2}{5} R^2 \frac{Z^4}{a_\mu^3 n^3}.
</math>

In a real experiment, the difference of this energy shift of different isotopes <math>\delta E</math> is measured. These isotopes have nuclear radius difference <math>\delta R</math>. Differentiation of the above equation gives the first order in <math>\delta R</math>:
<math display="block">
\delta E \approx \frac{e^2}{(4\pi\epsilon_0)} \frac{4}{5} R^2 \frac{Z^4}{a_\mu^3 n^3} \frac{\delta R}{R}.
</math>
This equation confirms that the volume effect is more significant for hydrogenic atoms with larger ''Z'', which explains why volume effects dominates the isotope shift of heavy elements.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
*[[Chemistry:Kinetic isotope effect|Kinetic isotope effect]]
*[[Physics:Magnetic isotope effect|Magnetic isotope effect]]

= References =
{{reflist|3}}

{{Author|Harold Foppele}}
[[Category:Emission spectroscopy]]

{{Sourceattribution|Physics:Isotopic shift|1}}

Physics:Quantum Isotopic shift - Revision history

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