Isotopic shift explained

The isotopic shift (also called isotope shift) is the shift in various forms of spectroscopy that occurs when one nuclear isotope is replaced by another.

NMR spectroscopy

In NMR spectroscopy, isotopic effects on chemical shifts are typically small, far less than 1 ppm, the typical unit for measuring shifts. The NMR signals for and ("HD") are readily distinguished in terms of their chemical shifts. The asymmetry of the signal for the "protio" impurity in arises from the differing chemical shifts of and .

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 and occur at 2917 cm⁻¹ and 2109 cm⁻¹ respectively.^[1] This shift reflects the differing 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 hyperfine structure, the shift refers to the center of gravity of the spectra.

From a nuclear physics perspective, isotope shifts combine different precise atomic physics probes for studying 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.^[2] 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.^[3] It can be formulated as follows:

In a theoretical model of atom, which has a infinitely massive nucleus, the energy (in wavenumbers) of a transition can be calculated from Rydberg formula:$$\backslash tilde\_\backslash infty\; =\; R\_\backslash infty\; \backslash left(\backslash frac\; -\; \backslash frac\; \backslash right),$$where

n and n' are principal quantum numbers, and R _{infty} is Rydberg constant.

However, for a nucleus with finite mass

M _{N} , reduced mass is used in the expression of Rydberg constant instead of mass of electron:$$\backslash tilde\; =\; \backslash tilde\_\backslash infty\; \backslash frac.$$

For two isotopes with atomic mass approximately

A' M _{p} and A'' M _{p} , the difference in the energies of the same transition is$$\backslash Delta\backslash tilde\; =\; \backslash tilde\_\backslash infty\; \backslash left(\backslash frac\; -\; \backslash frac\; \backslash right)\; \backslash approx\; \backslash tilde\_\backslash infty\; \backslash left[1\; -\; \backslash frac\{m\_e\}\{A\text{'}\text{'}\; M\_p\}\; \backslash left(1\; -\; \backslash frac\{m\_e\}\{A\text{'}\; M\_p\}\; \backslash right)\; \backslash right]\; \backslash approx\; \backslash frac\; \backslash frac\; \backslash tilde\_\backslash infty.$$The above equations imply that such mass shift is greatest for hydrogen and deuterium, since their mass ratio is the largest, A'' = 2 A' .

The effect of the specific mass shift was first observed in the spectrum of neon isotopes by Nagaoka and Mishima.^[4]

Consider the kinetic energy operator in Schrödinger equation of multi-electron atoms:$$T\; =\; \backslash frac\; +\; \backslash sum\_i\; \backslash frac,$$For a stationary atom, the conservation of momentum gives$$p\_n\; =\; -\backslash sum\_i\; p\_i.$$Therefore, the kinetic energy operator becomes$$T\; =\; \backslash frac\; +\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; \backslash sum\_\; p\_i\; \backslash cdot\; p\_j\; +\; \backslash frac.$$

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

\mu = (m _{e} M _{N})/(m _{e} + M _{N}) , 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$$\backslash frac\; \backslash sum\_\; p\_i\; \backslash cdot\; p\_j\; =\; -\backslash frac\; \backslash sum\_\; \backslash nabla\_i\; \backslash cdot\; \backslash nabla\_j.$$Using perturbation theory, the first-order energy shift can be calculated as $$\backslash Delta\; E\; =\; -\backslash frac\; \backslash sum\_\; \backslash int\; \backslash psi^*\; \backslash nabla\_i\; \backslash cdot\; \backslash nabla\_j\; \backslash psi\; \backslash ,d^3\; r,$$which requires the knowledge of accurate many-electron wave function. Due to the

1 /M _{N} term in the expression, the specific mass shift also decrease as 1 /M ^{2} _{N} 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.^{[5] [6] [7]} 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

R = r _{0} A ^{1/3} , where A is the atomic mass number, and r _{0} ≈ 1 . 2 x 10 ^{-15} m is a constant.

Similarly, calculating the electrostatic potential of an ideal charge density uniformly distributed in a sphere, the nuclear electrostatic potential isWhen the unperturbed Hamiltonian is subtracted, the perturbation is the difference of the potential in the above equation and Coulomb potential

- (Ze ^{2})/(( 4 \pi\epsilon _{0)r}) :

Such a perturbation of the atomic system neglects all other potential effect, like relativistic corrections. Using the perturbation theory (quantum mechanics), the first-order energy shift due to such perturbation is$$\backslash Delta\; E\; =\; \backslash langle\; \backslash psi\_\; |\; H\text{'}\; |\; \backslash psi\_\; \backslash rangle.$$The wave function

\psi _{nlm} = R _{nl} (r)Y _{lm} (\theta, \phi) has radial and angular parts, but the perturbation has no angular dependence, so the spherical harmonic normalize integral over the unit sphere:$$\backslash Delta\; E\; =\; \backslash frac\; \backslash int\_0^R\; |R\_(r)|^2\; \backslash left(\backslash frac\; +\; \backslash frac\; -\; 3\; \backslash right)\; r^2\; \backslash ,dr.$$Since the radius of nuclues R is small, and within such a small region r \leq R , the approximation R _{nl} (r) ≈ R _{nl} ( 0 ) is valid. And at r ≈ 0 , only the s sublevel remains, so l = 0 . Integration gives$$\backslash Delta\; E\; \backslash approx\; \backslash frac\; \backslash frac\; |R\_(0)|^2\; =\; \backslash frac\; \backslash frac\; R^2\; |\backslash psi\_(0)|^2.$$

The explicit form for hydrogenic wave function,

|\psi _{n00} ( 0 )| ^{2} = (Z ^{3})/(\pi a ^{3} _{\mu} n ^{3}) , gives$$\backslash Delta\; E\; \backslash approx\; \backslash frac\; \backslash frac\; R^2\; \backslash frac.$$

In an real experiment, the difference of this energy shift of different isotopes

\delta E is measured. These isotopes have nuclear radius difference \delta R . Differentiation of the above equation gives the first order in \delta R :$$\backslash delta\; E\; \backslash approx\; \backslash frac\; \backslash frac\; R^2\; \backslash frac\; \backslash frac.$$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

• Kinetic isotope effect
• Magnetic isotope effect

Notes and References

1. Web site: Takehiko Shimanouchi . Tables of Molecular Vibrational Frequencies Consolidated . I . 1972 . NSRDS-NBS-39 . . 2017-07-13 . 2016-08-04 . https://web.archive.org/web/20160804010334/http://www.nist.gov/data/nsrds/NSRDS-NBS-39.pdf . dead .
2. .
3. D. J. . Hughes . C. . Eckart . Carl Eckart . Phys. Rev. . 36 . 4 . 1930 . 694–698 . The Effect of the Motion of the Nucleus on the Spectra of Li I and Li II . 10.1103/PhysRev.36.694 . 1930PhRv...36..694H .
4. H. Nagaoka and T. Mishima, Sci. Pap. Inst. Phys. Chem. Res. (Tokyo) 13, 293 (1930).
5. W. Pauli, R. E. Peierls, Phys. Z. 32 (1931) 670.
6. Book: Brix, P. . H. . Kopfermann . Hans Kopfermann . Neuere Ergebnisse zum Isotopieverschiebungseffekt in den Atomspektren . Festschrift zur Feier des Zweihundertjährigen Bestehens der Akademie der Wissenschaften in Göttingen . de . Springer . 1951 . 978-3-540-01540-6 . 10.1007/978-3-642-86703-3_2 . 17–49 .
7. Book: Kopfermann, H. . Nuclear Moments . . 1958 . registration .