Physics:Quantum Isotopic shift - Revision history

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	{{Quantum book backlink\|Atomic and spectroscopy}}		{{Quantum book backlink\|Atomic and spectroscopy}}
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	{{Short description\|Mass effects on spectroscopy}}~~Book I~~		{{Short description\|Mass effects on spectroscopy}}

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	'''Isotopic shift''' ~~is a Book I topic in~~ the ~~Quantum Collection. 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. 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 . 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.		'''Isotopic shift''' 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. 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 . 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.
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	==NMR spectroscopy==		==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 H<sub>2</sub> (blue bar). The 1:1:1 triplet arises from the coupling of the <sup>1</sup>H nucleus (~~[[Wikipedia:nuclear spin\|~~nuclear spin\|''I'']] = 1/2) to the <sup>2</sup>H nucleus (''I'' = 1).]]<br>		[[File:H2&HDlowRes (2).png\|thumb\|[[Wikipedia:H NMR spectrum\|[H NMR spectrum]] of a solution of HD (labeled with red bars) and H<sub>2</sub> (blue bar). The 1:1:1 triplet arises from the coupling of the <sup>1</sup>H nucleus (nuclear spin\|''I'' = 1/2) to the <sup>2</sup>H nucleus (''I'' = 1).]]<br>
	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<!-- (as well as the H-D coupling)-->. 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}}.		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<!-- (as well as the H-D coupling)-->. 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}}.

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	~~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.		'''Isotopic shift''' is a Book I topic in the Quantum Collection. 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. 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 . 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.
	</div>		</div>

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	==Vibrational spectra==		==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)<sup>1/2</sup> ≈ 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<sup>−1</sup> and 2109 cm<sup>−1</sup> 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.		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)<sup>1/2</sup> ≈ 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<sup>−1</sup> and 2109 cm<sup>−1</sup> 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 reduced mass for the affected bonds.

	==Atomic spectra==		==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.		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 ~~[[Physics:Nuclear structure\|~~nuclear structure]], and their main use is nuclear-model-independent determination of charge-radii differences.		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:		Two effects contribute to this shift:
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	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:		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]]~~:		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:
	<math display="block">		<math display="block">
	\tilde{\nu}_\infty = R_\infty \left(\frac{1}{n^2} - \frac{1}{n'^2} \right),		\tilde{\nu}_\infty = R_\infty \left(\frac{1}{n^2} - \frac{1}{n'^2} \right),
	</math>		</math>
	where <math>n</math> and <math>n'</math> are principal quantum numbers, and <math>R_\infty</math> is ~~[[Physics:~~Rydberg constant~~\|Rydberg constant]]~~.		where <math>n</math> and <math>n'</math> are principal quantum numbers, and <math>R_\infty</math> is 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:		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:
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	</math>		</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		For two isotopes with 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">		<math display="block">
	\Delta\tilde{\nu} =		\Delta\tilde{\nu} =
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	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 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>		The effect of the specific mass shift was first observed in the spectrum of neon isotopes by 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:		Consider the kinetic energy operator in Schrödinger equation of multi-electron atoms:
	<math display="block">		<math display="block">
	T = \frac{p_n^2}{2M_N} + \sum_i \frac{p_i^2}{2m_e},		T = \frac{p_n^2}{2M_N} + \sum_i \frac{p_i^2}{2m_e},
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	\Delta E = -\frac{\hbar^2}{M} \sum_{i > j} \int \psi^* \nabla_i \cdot \nabla_j \psi \,d^3 r,		\Delta E = -\frac{\hbar^2}{M} \sum_{i > j} \int \psi^* \nabla_i \cdot \nabla_j \psi \,d^3 r,
	</math>		</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.		which requires the knowledge of accurate many-electron 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===		===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.		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=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.		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.
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	</math>		</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		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
	<math display="block">		<math display="block">
	\Delta E = \langle \psi_{nlm} \| H' \| \psi_{nlm} \rangle.		\Delta E = \langle \psi_{nlm} \| H' \| \psi_{nlm} \rangle.
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	</math>		</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		The explicit form for hydrogenic wave function, <math>\|\psi_{n00}(0)\|^2 = \frac{Z^3}{\pi a_\mu^3 n^3}</math>, gives
	<math display="block">		<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}.		\Delta E \approx \frac{e^2}{(4\pi\epsilon_0)} \frac{2}{5} R^2 \frac{Z^4}{a_\mu^3 n^3}.
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	=See also=		=See also=
	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	*~~[[Chemistry:~~Kinetic isotope effect~~\|Kinetic isotope effect]]~~		*Kinetic isotope effect
	*~~[[Physics:Magnetic isotope effect\|~~Magnetic isotope effect]]		*Magnetic isotope effect

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 {{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.]]
+</div>
 ==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 H<sub>2</sub> (blue bar). The 1:1:1 triplet arises from the coupling of the <sup>1</sup>H nucleus ([[Wikipedia:nuclear spin|nuclear spin|''I'']] = 1/2) to the <sup>2</sup>H nucleus (''I'' = 1).]]<br>