Diamagnetic EDMs and Nuclear Structure J. Engel University of North Carolina February 15, 2013
One Way Things Get EDMs Starting at fundamental level and working up: N Underlying fundamental g ¯ theory generates three T -violating π NN vertices: π ? New physics
One Way Things Get EDMs Starting at fundamental level and working up: N Underlying fundamental g ¯ theory generates three T -violating π NN vertices: π ? New physics Then neutron gets EDM, γ e.g., from chiral-PT diagrams like this: π − ¯ g g n p n
How Diamagnetic Atoms Get EDMs γ Nucleus can get one from ¯ g nucleon EDM or π T -violating NN interaction:
How Diamagnetic Atoms Get EDMs γ Nucleus can get one from ¯ g nucleon EDM or π T -violating NN interaction: �� � g 0 τ 1 · τ 2 − ¯ g 1 2 ( τ z 1 + τ z g 2 ( 3 τ z 1 τ z V PT ∝ ¯ 1 ) + ¯ 2 − τ 1 · τ 2 ) ( σ 1 − σ 2 ) � · ( ∇ 1 − ∇ 2 ) exp ( − m π | r 1 − r 2 | ) − ¯ g 1 2 ( τ z 1 − τ z 2 ) ( σ 1 + σ 2 ) m π | r 1 − r 2 |
How Diamagnetic Atoms Get EDMs γ Nucleus can get one from ¯ g nucleon EDM or π T -violating NN interaction: �� � g 0 τ 1 · τ 2 − ¯ g 1 2 ( τ z 1 + τ z g 2 ( 3 τ z 1 τ z V PT ∝ ¯ 1 ) + ¯ 2 − τ 1 · τ 2 ) ( σ 1 − σ 2 ) � · ( ∇ 1 − ∇ 2 ) exp ( − m π | r 1 − r 2 | ) − ¯ g 1 2 ( τ z 1 − τ z 2 ) ( σ 1 + σ 2 ) m π | r 1 − r 2 | Finally, atom gets one from nucleus. Electronic shielding makes the relevant nuclear object the “Schiff moment” � S � ≈ � � p r 2 p z p + . . . � rather than the dipole moment � D z � .
How Diamagnetic Atoms Get EDMs γ Nucleus can get one from ¯ g nucleon EDM or π T -violating NN interaction: �� � g 0 τ 1 · τ 2 − ¯ g 1 2 ( τ z 1 + τ z g 2 ( 3 τ z 1 τ z V PT ∝ ¯ 1 ) + ¯ 2 − τ 1 · τ 2 ) ( σ 1 − σ 2 ) � · ( ∇ 1 − ∇ 2 ) exp ( − m π | r 1 − r 2 | ) − ¯ g 1 2 ( τ z 1 − τ z 2 ) ( σ 1 + σ 2 ) m π | r 1 − r 2 | Finally, atom gets one from nucleus. Electronic shielding makes the relevant nuclear object the “Schiff moment” � S � ≈ � � p r 2 p z p + . . . � rather than the dipole moment � D z � . Job of nuclear theory: calculate dependence of � S � on the ¯ g’s.
How Does Shielding Work? Theorem (Schiff) T he nuclear dipole moment causes the atomic electrons to rearrange themselves so that they develop a dipole moment opposite that of the nucleus. In the limit of nonrelativistic electrons and a point nucleus the electrons’ dipole moment exactly cancels the nuclear moment, so that the net atomic dipole moment vanishes.
How Does Shielding Work? Proof Consider atom with nonrelativistic constituents (with dipole moments � d k ) held together by electrostatic forces. The atom has a “bare” edm � d ≡ � k � d k and a Hamiltonian p 2 � � � d k · � � k H = + V ( � r k ) − E k 2 m k k k k
How Does Shielding Work? Proof Consider atom with nonrelativistic constituents (with dipole moments � d k ) held together by electrostatic forces. The atom has a “bare” edm � d ≡ � k � d k and a Hamiltonian p 2 � � � d k · � � k H = + V ( � r k ) − E k 2 m k k k k + � k ( 1 / e k ) � d k · � = ∇ V ( � r k ) H 0 dipole perturbation K.E. + Coulomb
How Does Shielding Work? Proof Consider atom with nonrelativistic constituents (with dipole moments � d k ) held together by electrostatic forces. The atom has a “bare” edm � d ≡ � k � d k and a Hamiltonian p 2 � � � d k · � � k H = + V ( � r k ) − E k 2 m k k k k + � k ( 1 / e k ) � d k · � = ∇ V ( � r k ) H 0 � � � � ( 1 / e k ) d k · � p k , H 0 = H 0 + i k dipole perturbation K.E. + Coulomb
How Does Shielding Work? The perturbing Hamiltonian � � � � H d = i ( 1 / e k ) d k · � p k , H 0 k shifts the ground state | 0 � to
How Does Shielding Work? The perturbing Hamiltonian � � � � H d = i ( 1 / e k ) d k · � p k , H 0 k shifts the ground state | 0 � to | m � � m | H d | 0 � � | ˜ 0 � = | 0 � + E 0 − E m m | m � � m | i � k ( 1 / e k ) � d k · � p k | 0 � ( E 0 − E m ) � = | 0 � + E 0 − E m m � � � ( 1 / e k ) � = 1 + i d k · � p k | 0 � k
How Does Shielding Work? d ′ is The induced dipole moment � � � d ′ � ˜ r j | ˜ = 0 | e j � 0 � j
How Does Shielding Work? d ′ is The induced dipole moment � � � d ′ � ˜ r j | ˜ = 0 | e j � 0 � j � � �� � 1 − i � k ( 1 / e k ) � = � 0 | d k · � p k j e j � r j � � 1 + i � k ( 1 / e k ) � × d k · � p k | 0 � �� � r j , � k ( 1 / e k ) � = i � 0 | j e j � d k · � p k | 0 � � � � � = − � 0 | d k | 0 � = − d k k k − � = d
How Does Shielding Work? d ′ is The induced dipole moment � � � d ′ � ˜ r j | ˜ = 0 | e j � 0 � j � � �� � 1 − i � k ( 1 / e k ) � = � 0 | d k · � p k j e j � r j � � 1 + i � k ( 1 / e k ) � × d k · � p k | 0 � �� � r j , � k ( 1 / e k ) � = i � 0 | j e j � d k · � p k | 0 � � � � � = − � 0 | d k | 0 � = − d k k k − � = d So the net EDM is zero!
All is Not Lost, Though. . . Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM � d . Post-screening nucleus-electron interaction proportional to Schiff moment: � � � p − 5 � r 2 3 � R 2 S ≡ e p ch � r p + . . . � p
All is Not Lost, Though. . . Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM � d . Post-screening nucleus-electron interaction proportional to Schiff moment: � � � p − 5 � r 2 3 � R 2 S ≡ e p ch � � r p + . . . p If, as you’d expect, �� N � � D � , then � d is down from � � S � ≈ R 2 D � by � � ≈ 10 − 8 , R 2 N / R 2 O A
All is Not Lost, Though. . . Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM � d . Post-screening nucleus-electron interaction proportional to Schiff moment: � � � p − 5 � r 2 3 � R 2 S ≡ e p ch � r p + . . . � p If, as you’d expect, �� N � � D � , then � d is down from � � S � ≈ R 2 D � by � � ≈ 10 − 8 , R 2 N / R 2 O A Ughh! Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10 Z 2 ≈ 10 5 .
All is Not Lost, Though. . . Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM � d . Post-screening nucleus-electron interaction proportional to Schiff moment: � � � p − 5 � r 2 3 � R 2 S ≡ e p ch � r p + . . . � p If, as you’d expect, �� N � � D � , then � d is down from � � S � ≈ R 2 D � by � � ≈ 10 − 8 , R 2 N / R 2 O A Ughh! Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10 Z 2 ≈ 10 5 . Overall suppression of � � D � is only about 10 − 3 .
Theory for Heavy Nuclei S ∝ Z 2 , so experiments are in heavy nuclei but can’t solve Schr¨ odinger eq’n for A > 40 . Usually apply approximation scheme, then account for omitted physics by modifying operators.
Theory for Heavy Nuclei S ∝ Z 2 , so experiments are in heavy nuclei but can’t solve Schr¨ odinger eq’n for A > 40 . Usually apply approximation scheme, then account for omitted physics by modifying operators. Paradigm: Density functional Theory H¨ ohenberg-Kohn-Sham: Can get exact density from Hartree calculation with appropriate effective interaction (density functional).
Theory for Heavy Nuclei S ∝ Z 2 , so experiments are in heavy nuclei but can’t solve Schr¨ odinger eq’n for A > 40 . Usually apply approximation scheme, then account for omitted physics by modifying operators. Paradigm: Density functional Theory H¨ ohenberg-Kohn-Sham: Can get exact density from Hartree calculation with appropriate effective interaction (density functional). Nuclear version: Mean-field theory with density-dependent interactions (called Skyrme interactions) built from delta functions and deriviatives of delta functions plus whatever corrections one can manage, e.g. projection of deformed wave functions onto states with good angular momentum mixing of several mean fields . . .
Theory for Heavy Nuclei S ∝ Z 2 , so experiments are in heavy nuclei but can’t solve Schr¨ odinger eq’n for A > 40 . Usually apply approximation scheme, then account for omitted physics by modifying operators. Paradigm: Density functional Theory H¨ ohenberg-Kohn-Sham: Can get exact density from Hartree calculation with appropriate effective interaction (density functional). Nuclear version: Mean-field theory with density-dependent interactions (called Skyrme interactions) built from delta functions and deriviatives of delta functions plus whatever corrections one can manage, e.g. projection of deformed wave functions onto states with good angular momentum mixing of several mean fields . . . Density functional still obtained largely through phenomenology.
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