Hyperfine structure

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In atomic physics, the hyperfine structure refers to a small shift and separation in the energy levels of atoms, molecules and ions, due to the interaction between the state of the nucleus and the state of the electron cloud.

In atoms, the hyperfine structure arises from the energy of the nuclear magnetic dipole moment interacting with the magnetic field generated by the electrons and energy of the nuclear power quadrupole moment in the electric field of the gradient due to the charge distribution within the atom. The molecular hyperfine structure is generally dominated by these two effects, but also includes the energy associated with the interaction between the magnetic moments associated with different magnetic cores in a molecule, as well as between the nuclear magnetic moment and the magnetic field generated by its molecular rotation.

The Hyperfine structure contrasts with the fine structure , which results from the interaction between magnetic moments associated with electron spin and the angular momentum of the electron orbital. The structure of hyperfine, with energy shifts is usually a smaller sequence than a subtle shift in structure, resulting from the interaction of nuclei (or nuclei, in molecules) with internally generated electrical and magnetic fields.

Video Hyperfine structure

Histori

The structure of optical hyperfine was observed in 1881 by Albert Abraham Michelson. However, it can be explained only in terms of quantum mechanics when Wolfgang Pauli proposed the existence of a small nuclear magnetic moment in 1924.

In 1935, H. SchÃÆ'¼ler and Theodor Schmidt proposed the existence of a nuclear quadrupole moment to explain the anomalies in the hyperfine structure.

Maps Hyperfine structure

Theory

The theory of hyperfine structure comes directly from electromagnetism, which consists of interactions of nuclear multipole moments (excluding electrical monopole) with internally generated fields. This theory is derived first for the case of atoms, but can be applied to each core in a molecule. After this there is a discussion about the additional effects unique to the molecular case.

The atomic hyperfine structure

Magnetic Dipol

Istilah dominan dalam Hamiltonian hyperfine biasanya adalah istilah dipol magnetik. Inti atom dengan spin nuklir non-nol ${\ displaystyle \ mathbf {I}}  ÂÂ$ memiliki momen dipol magnetik, yang diberikan oleh:

${\ displaystyle {\ boldsymbol {\ mu}} _ {\ text {I}} = g _ {\ text {I}} \ mu_ {\ text {N}} \ mathbf {I},}  ÂÂ$

di mana ${\ displaystyle g _ {\ text {I}}}  ÂÂ$ adalah g -factor dan ${\ displaystyle \ mu_ {\ text {N}}}  ÂÂ$ adalah magneton nuklir.

Ada energi yang terkait dengan momen dipol magnetik di hadapan medan magnet. Untuk momen dipol magnetik nuklir, ? _I , ditempatkan dalam medan magnet, B , istilah yang relevan di Hamiltonian diberikan oleh:

${\ displaystyle {\ hat {H}} _ {\ text {D}} = - {\ boldsymbol {\ mu}} _ {\ text {I}} \ cdot \ mathbf {B}.}  ÂÂ$

Dengan tidak adanya medan yang diaplikasikan secara eksternal, medan magnet yang dialami oleh nukleus adalah yang terkait dengan orbital ( l ) dan spin ( s ) momentum sudut elektron:

${\ displaystyle \ mathbf {B} \ equiv \ mathbf {B} _ {\ text {el}} = \ mathbf {B} _ {\ text {el}} ^ {l} \ mathbf {B} _ {\ text {el}} ^ {s}.}  ÂÂ$

The electron momentum angle of the orbital results from an electron motion about some fixed external point that we will take to be the nucleus location. The magnetic field in the nucleus because of the movement of one electron, with the charge - e at position r relative to the nucleus, given by:

${\ displaystyle \ mathbf {B} _ {\ text {el}} ^ {l} = {\ dfrac {\ mu_ {0}} {4 \ pi}} {\ dfrac {-e \ mathbf {v} \ times - \ mathbf {r}} {r ^ {3}}},} Â Â$

dimana - r memberikan posisi inti relatif terhadap elektron. Ditulis dalam istilah magneton Bohr, ini memberikan:

${\ displaystyle \ mathbf {B} _ {\ text {el}} ^ {l} = - 2 \ mu_ {\ text {B}} {\ dfrac {\ mu _ {0}} {4 \ pi}} {\ dfrac {1} {r ^ {3}}} {\ dfrac {\ mathbf {r} \ kali m _ {\ text {e}} \ mathbf {v} } {\ hbar}}.}  ÂÂ$

Mengakui bahwa m _e v adalah momentum elektron, p , dan itu r ÃÆ'— p /? adalah momentum sudut orbital dalam satuan ? , l , kita dapat menulis:

${\ displaystyle \ mathbf {B} _ {\ text {el}} ^ {l} = - 2 \ mu_ {\ text {B}} {\ dfrac {\ mu _ {0}} {4 \ pi}} {\ dfrac {1} {r ^ {3}}} \ mathbf {l}.}  ÂÂ$

The electron spin angle momentum is a fundamentally different property that is intrinsic to the particles and therefore does not depend on electron motion. However it is the angular momentum and every angular momentum associated with charged particles produces magnetic dipole moment, which is the source of the magnetic field. An electron with a rotational angle momentum, s , has a magnetic moment, ? _s , provided by:

${\ displaystyle {\ boldsymbol {\ mu}} _ {\ text {s}} = - g_ {s} \ mu_ {\ text {B}} \ mathbf {s},} Â Â$

where g _s is the spin electron g -factor and the negative sign because the electron is negatively charged (assuming that the particles are negatively charged and positive with identical masses, running on an equivalent path, will have the same angular momentum, but will produce a current in the opposite direction).

Medan magnet momen dipol, ? _s , diberikan oleh:

${\ displaystyle \ mathbf {B} _ {\ text {el}} ^ {s} = {\ dfrac {\ mu_ {0}} {4 \ pi r ^ {3}}} \ kiri (3 ({\ boldsymbol {\ mu}} _ {\ text {s}} \ cdot {\ hat {\ mathbf {r}}}) {\ hat {\ mathbf {r}} } - {\ boldsymbol {\ mu}} _ {\ text {s}} \ right) {\ dfrac {2 \ mu_ {0}} {3}} {\ boldsymbol {\ mu}} _ {\ text {s}} \ delta ^ {3} (\ mathbf {r}).}  ÂÂ$

The first term gives nuclear dipole energy in the field due to the electronic orbital angular momentum. The second term energizes the "limited distance" interaction of the nuclear dipole with the field because of the magnetic moment of the electron spin. The latter term, often known as the term "Fermi contact" relates to the direct interaction of a nuclear dipole with a rotating dipole and is only non-zero for a state with an electron spin density confined to an unpaired (unpaired) core position. electrons in s -subshells). It has been argued that one can obtain different expressions when calculating the detailed nuclear magnetic moment distribution.

Untuk negara dengan l ? 0 ini dapat diekspresikan dalam formulir

${\ displaystyle {\ hat {H}} _ {D} = 2g_ {I} \ mu_ {\ text {B}} \ mu_ {\ text {N} } {\ dfrac {\ mu_ {0}} {4 \ pi}} {\ dfrac {\ mathbf {I} \ cdot \ mathbf {N}} {r ^ {3}}},}  ÂÂ$

dimana:

${\ displaystyle \ mathbf {N} = \ mathbf {l} - (g_ {s}/2) \ kiri [\ mathbf {s} -3 (\ mathbf {s } \ cdot {\ hat {\ mathbf {r}}}) {\ hat {\ mathbf {r}}} \ right].}  ÂÂ$

Ini biasanya ditulis sebagai

${\ displaystyle {\ hat {H}} _ {\ text {D}} = {\ hat {A}} \ mathbf {I} \ cdot \ mathbf {J} ,}  ÂÂ$

dengan ${\ displaystyle \ scriptstyle {\ langle {\ hat {A}} \ rangle}}  ÂÂ$ adalah konstanta struktur hyperfine yang ditentukan oleh eksperimen. Sejak I . J = ½ { F . F - Saya . Saya - J . J } (di mana F = Saya J adalah momentum sudut total), ini memberikan energi

${\ displaystyle \ Delta E _ {\ text {D}} = {\ frac {1} {2}} \ langle {\ hat {A}} \ rangle [F ( F 1) -I (I 1) -J (J 1)].}  ÂÂ$

In this case the hyperfine interaction meets the interval rules of Landà ©.

Quadrupole power

Inti atom dengan spin ${\ displaystyle \ scriptstyle {I \ geq 1}}  ÂÂ$ memiliki momen quadrupole listrik. Dalam kasus umum ini diwakili oleh peringkat-2 tensor, ${\ displaystyle \ scriptstyle {\ garis bawah {\ garis bawah {Q}}}}  ÂÂ$ , dengan komponen yang diberikan oleh:

${\ displaystyle Q_ {ij} = {\ dfrac {1} {e}} \ int \ kiri (3x_ {i} ^ {\ prime} x_ {j} ^ { \ prime} - (r ^ {\ prime}) ^ {2} \ delta _ {ij} \ right) \ rho (\ mathbf {r} ^ {\ prime}) d ^ {3} r ^ {\ prime} ,}  ÂÂ$

where i and j are tensor indexes running from 1 to 3, x _i and x << su

Source of the article : Wikipedia