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Improved energy calculation for high l states (l>5) and quantum defect updates for Cs and K. #173

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Improved energy calculation for high l states (l>5) and quantum defect updates for Cs and K. #173

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8fef945
Added Polarisation Energy for high l states
johnny-sa Jul 3, 2024
f55756e
Updated qds for K and Cs and Core polarisabilities
johnny-sa Jul 3, 2024
e4a2e49
Added polarisation energy for high l (l>5) states
johnny-sa Jul 3, 2024
d54d900
Formatting and References for high L energies
johnny-sa Aug 7, 2024
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174 changes: 132 additions & 42 deletions arc/alkali_atom_data.py
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Original file line number Diff line number Diff line change
Expand Up @@ -127,7 +127,6 @@
"Potassium41",
]


class Hydrogen(AlkaliAtom):
"""
Properties of hydrogen atoms
Expand Down Expand Up @@ -192,6 +191,15 @@ class Caesium(AlkaliAtom):
model potential parameters from [#c1]_

"""

a_d_eff = 15.79

a_q_eff = 38.7

"""
Effective core polarisabilities from https://doi.org/10.1364/JOSA.71.000423
"""

#
a1 = [3.49546309, 4.69366096, 4.32466196, 3.01048361]
"""
Expand Down Expand Up @@ -219,6 +227,8 @@ class Caesium(AlkaliAtom):

I = 3.5 # 7/2



#: (eV), Ref. [#jd2016]_.
ionisationEnergy = (
31406.4677325
Expand All @@ -234,24 +244,24 @@ class Caesium(AlkaliAtom):

quantumDefect = [
[
[4.04935665, 0.2377037, 0.255401, 0.00378, 0.25486, 0.0],
[3.59158950, 0.360926, 0.41905, 0.64388, 1.45035, 0.0],
[4.0493532, 0.2391, 0.06, 11, -209, 0.0],
[3.5915871, 0.36273, 0.0, 0.0, 0.0, 0.0],
[2.4754562, 0.009320, -0.43498, -0.76358, -18.0061, 0.0],
[0.03341424, -0.198674, 0.28953, -0.2601, 0.0, 0.0],
[0.00703865, -0.049252, 0.01291, 0.0, 0.0, 0.0],
],
[
[4.04935665, 0.2377037, 0.255401, 0.00378, 0.25486, 0.0],
[3.5589599, 0.392469, -0.67431, 22.3531, -92.289, 0.0],
[2.46631524, 0.013577, -0.37457, -2.1867, -1.5532, -56.6739],
[4.0493532, 0.2391, 0.06, 11, -209, 0.0],
[3.5590676, 0.37469, 0.0, 0.0, 0.0, 0.0],
[2.4663144, 0.01381, -0.392, -1.9, 0.0, 0.0],
[0.03341424, -0.198674, 0.28953, -0.2601, 0.0, 0.0],
[0.00703865, -0.049252, 0.01291, 0.0, 0.0, 0.0],
],
]
"""
quantum defects for :math:`S_{1/2}`, :math:`nP_{1/2}`, :math:`D_{5/2}`,
quantum defects for :math:`S_{1/2}`, :math:`nP_{1/2}`, :math:`nP_{3/2}`, :math:`D_{5/2}` are from https://doi.org/10.1103/PhysRevA.93.013424
:math:`F_{5/2}` and :math:`G_{7/2}` are from [#Weber1987]_, while
quantum defects for :math:`nP_{3/2}`,:math:`D_{3/2}` are from [#Lorenzen1984]_,
:math:`D_{3/2}` are from [#Lorenzen1984]_,

Note:
f_7/2 quantum defects are PUT TO BE EXACTLY the same as f_5/2 (~10MHz difference?!)
Expand Down Expand Up @@ -390,7 +400,6 @@ class Cesium(Caesium):

pass


class Rubidium85(AlkaliAtom):
"""
Properites of rubidium 85 atoms
Expand All @@ -403,6 +412,15 @@ class Rubidium85(AlkaliAtom):

"""

a_d_eff = 9.089

a_q_eff = 16.8

"""
Effective core polarisabilities from https://doi.org/10.1103/PhysRevA.102.062818

"""

a1 = [3.69628474, 4.44088978, 3.78717363, 2.39848933]
"""
model potential parameters from [#c1]_
Expand Down Expand Up @@ -561,7 +579,15 @@ class Rubidium87(AlkaliAtom):
model potential parameters from [#c1]_

"""
a_d_eff = 9.089

a_q_eff = 16.8

"""
Effective core polarisabilities from https://doi.org/10.1103/PhysRevA.102.062818

"""

a1 = [3.69628474, 4.44088978, 3.78717363, 2.39848933]
"""
model potential parameters from [#c1]_
Expand Down Expand Up @@ -711,7 +737,17 @@ class Lithium6(AlkaliAtom): # Li
"""
model potential parameters from [#c1]_


"""

a_d_eff = 0.1883

a_q_eff = 0.04579

"""
(Effective) core plarisabiltiesm from https://doi.org/10.1103/PhysRevA.16.1141
"""

# model potential parameters from Marinescu et.al, PRA 49:982 (1994)
a1 = [2.47718079, 3.45414648, 2.51909839, 2.51909839]
"""
Expand Down Expand Up @@ -867,6 +903,15 @@ class Lithium7(AlkaliAtom): # Li
model potential parameters from [#c1]_

"""

a_d_eff = 0.1883

a_q_eff = 0.04579

"""
(Effective) core plarisabiltiesm from https://doi.org/10.1103/PhysRevA.16.1141
"""

a1 = [2.47718079, 3.45414648, 2.51909839, 2.51909839]
"""
model potential parameters from [#c1]_
Expand Down Expand Up @@ -1012,6 +1057,16 @@ class Sodium(AlkaliAtom): # Na23
model potential parameters from [#c1]_

"""

a_d_eff = 0.9980

a_q_eff = 0.351
"""
(Effective) core plarisabiltiesm from https://doi.org/10.1103/PhysRevA.38.4985
"""



a1 = [4.82223117, 5.08382502, 3.53324124, 1.11056646]
"""
model potential parameters from [#c1]_
Expand Down Expand Up @@ -1155,6 +1210,18 @@ class Potassium39(AlkaliAtom):
model potential parameters from [#c1]_

"""


a_d_eff = 5.49

a_q_eff = 18


"""
(Effective) core plarisabiltiesm from https://doi.org/10.1103/PhysRevA.100.012501
"""


a1 = [3.56079437, 3.65670429, 4.12713694, 1.42310446]
"""
model potential parameters from [#c1]_
Expand Down Expand Up @@ -1197,22 +1264,22 @@ class Potassium39(AlkaliAtom):
# quantum defects from Physica Scripta 27:300 (1983)
quantumDefect = [
[
[2.1801985, 0.13558, 0.0759, 0.117, -0.206, 0.0],
[1.713892, 0.233294, 0.16137, 0.5345, -0.234, 0.0],
[0.27697, -1.024911, -0.709174, 11.839, -26.689, 0.0],
[0.010098, -0.100224, 1.56334, -12.6851, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[2.18020826, 0.134534, 0.0952, 0.0021, 0.0, 0.0],
[1.71392626, 0.23114, 0.1948, 0.3683, 0.0, 0.0],
[0.27698453, -1.02691, -0.665, 10.9, 0.0, 0.0],
[0.0094576, -0.0446, 0.0, 0.0, 0.0, 0.0],
[0.0024080, -0.0209, 0.0, 0.0, 0.0, 0.0],
],
[
[2.1801985, 0.13558, 0.0759, 0.117, -0.206, 0.0],
[1.710848, 0.235437, 0.11551, 1.1015, -2.0356, 0.0],
[0.2771580, -1.025635, -0.59201, 10.0053, -19.0244, 0.0],
[0.010098, -0.100224, 1.56334, -12.6851, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[2.18020826, 0.134534, 0.0952, 0.0021, 0.0, 0.0],
[1.71087854, 0.23233, 0.1961, 0.3716, 0.0, 0.0],
[0.27715665, -1.02493, -0.640, 10.0, 0.0, 0.0],
[0.0094576, -0.0446, 0.0, 0.0, 0.0, 0.0],
[0.0024080, -0.0209, 0.0, 0.0, 0.0, 0.0],
],
]
"""
quantum defects from Ref. [#c7]_.
p1/2 and p3/2, s1/2, d3/2, d5/2 and f and g (centre of manifold) from https://doi.org/10.1103/PhysRevA.100.012501
"""

levelDataFromNIST = "k_NIST_level_data.ascii"
Expand Down Expand Up @@ -1305,6 +1372,18 @@ class Potassium40(AlkaliAtom):
model potential parameters from [#c1]_

"""

a_d_eff = 5.49

a_q_eff = 18


"""
(Effective) core plarisabiltiesm from https://doi.org/10.1103/PhysRevA.100.012501
"""



a1 = [3.56079437, 3.65670429, 4.12713694, 1.42310446]
"""
model potential parameters from [#c1]_
Expand Down Expand Up @@ -1347,22 +1426,22 @@ class Potassium40(AlkaliAtom):
# quantum defects from Physica Scripta 27:300 (1983)
quantumDefect = [
[
[2.1801985, 0.13558, 0.0759, 0.117, -0.206, 0.0],
[1.713892, 0.233294, 0.16137, 0.5345, -0.234, 0.0],
[0.27697, -1.024911, -0.709174, 11.839, -26.689, 0.0],
[0.010098, -0.100224, 1.56334, -12.6851, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[2.18020826, 0.134534, 0.0952, 0.0021, 0.0, 0.0],
[1.71392626, 0.23114, 0.1948, 0.3683, 0.0, 0.0],
[0.27698453, -1.02691, -0.665, 10.9, 0.0, 0.0],
[0.0094576, -0.0446, 0.0, 0.0, 0.0, 0.0],
[0.0024080, -0.0209, 0.0, 0.0, 0.0, 0.0],
],
[
[2.1801985, 0.13558, 0.0759, 0.117, -0.206, 0.0],
[1.710848, 0.235437, 0.11551, 1.1015, -2.0356, 0.0],
[0.2771580, -1.025635, -0.59201, 10.0053, -19.0244, 0.0],
[0.010098, -0.100224, 1.56334, -12.6851, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[2.18020826, 0.134534, 0.0952, 0.0021, 0.0, 0.0],
[1.71087854, 0.23233, 0.1961, 0.3716, 0.0, 0.0],
[0.27715665, -1.02493, -0.640, 10.0, 0.0, 0.0],
[0.0094576, -0.0446, 0.0, 0.0, 0.0, 0.0],
[0.0024080, -0.0209, 0.0, 0.0, 0.0, 0.0],
],
]
"""
quantum defects from Ref. [#c7]_.
p1/2 and p3/2, s1/2, d3/2, d5/2 and f and g (centre of manifold) from https://doi.org/10.1103/PhysRevA.100.012501
"""

levelDataFromNIST = "k_NIST_level_data.ascii"
Expand Down Expand Up @@ -1445,6 +1524,17 @@ class Potassium41(AlkaliAtom):
model potential parameters from [#c1]_

"""

a_d_eff = 5.49

a_q_eff = 18


"""
(Effective) core plarisabiltiesm from https://doi.org/10.1103/PhysRevA.100.012501
"""


a1 = [3.56079437, 3.65670429, 4.12713694, 1.42310446]
"""
model potential parameters from [#c1]_
Expand Down Expand Up @@ -1487,22 +1577,22 @@ class Potassium41(AlkaliAtom):
# quantum defects from Physica Scripta 27:300 (1983)
quantumDefect = [
[
[2.1801985, 0.13558, 0.0759, 0.117, -0.206, 0.0],
[1.713892, 0.233294, 0.16137, 0.5345, -0.234, 0.0],
[0.27697, -1.024911, -0.709174, 11.839, -26.689, 0.0],
[0.010098, -0.100224, 1.56334, -12.6851, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[2.18020826, 0.134534, 0.0952, 0.0021, 0.0, 0.0],
[1.71392626, 0.23114, 0.1948, 0.3683, 0.0, 0.0],
[0.27698453, -1.02691, -0.665, 10.9, 0.0, 0.0],
[0.0094576, -0.0446, 0.0, 0.0, 0.0, 0.0],
[0.0024080, -0.0209, 0.0, 0.0, 0.0, 0.0],
],
[
[2.1801985, 0.13558, 0.0759, 0.117, -0.206, 0.0],
[1.710848, 0.235437, 0.11551, 1.1015, -2.0356, 0.0],
[0.2771580, -1.025635, -0.59201, 10.0053, -19.0244, 0.0],
[0.010098, -0.100224, 1.56334, -12.6851, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[2.18020826, 0.134534, 0.0952, 0.0021, 0.0, 0.0],
[1.71087854, 0.23233, 0.1961, 0.3716, 0.0, 0.0],
[0.27715665, -1.02493, -0.640, 10.0, 0.0, 0.0],
[0.0094576, -0.0446, 0.0, 0.0, 0.0, 0.0],
[0.0024080, -0.0209, 0.0, 0.0, 0.0, 0.0],
],
]
"""
quantum defects from Ref. [#c7]_.
p1/2 and p3/2, s1/2, d3/2, d5/2 and f and g (centre of manifold) from https://doi.org/10.1103/PhysRevA.100.012501
"""

levelDataFromNIST = "k_NIST_level_data.ascii"
Expand Down
66 changes: 62 additions & 4 deletions arc/alkali_atom_functions.py
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Original file line number Diff line number Diff line change
Expand Up @@ -787,7 +787,33 @@ def getEnergy(self, n, l, j, s=0.5):

# else, use quantum defects
defect = self.getQuantumDefect(n, l, j, s=s)
return -self.scaledRydbergConstant / ((n - defect) ** 2)



if l <= 3:
return -self.scaledRydbergConstant / ((n - defect) ** 2)
elif l == 4:
fineStructSplit = -(2*self.scaledRydbergConstant)*pow(scipy.constants.alpha,2)/(2*l*(l+0.5)*(l+1)*pow(n,3))

if j == l + 0.5:
fineStructSplit *= 0.5*l

elif j == l - 0.5:
fineStructSplit *= -1*0.5*(l+1)

return -self.scaledRydbergConstant / ((n - defect) ** 2) - fineStructSplit

else:
fineStructSplit = -(2*self.scaledRydbergConstant)*pow(scipy.constants.alpha,2)/(2*l*(l+0.5)*(l+1)*pow(n,3))

if j == l + 0.5:
fineStructSplit *= 0.5*l

elif j == l - 0.5:
fineStructSplit *= -1*0.5*(l+1)

return -self.scaledRydbergConstant / ((n) ** 2) - fineStructSplit - 2*self.scaledRydbergConstant*defect


def _getSavedEnergy(self, n, l, j, s=0.5):
if abs(j - (l - 0.5)) < 0.001:
Expand Down Expand Up @@ -841,9 +867,41 @@ def getQuantumDefect(self, n, l, j, s=0.5):
+ modifiedRRcoef[5] / ((n - modifiedRRcoef[0]) ** 10)
)
else:
# use \delta_\ell = \delta_g * (4/\ell)**5
# from https://journals.aps.org/pra/abstract/10.1103/PhysRevA.74.062712
defect = self.quantumDefect[0][4][0] * (4 / l) ** 5
n = int(n)
l = int(l)

# Calculate r4 and r6 coefficients from https://journals.aps.org/pra/abstract/10.1103/PhysRevA.9.1087
top_r4 = 3 * pow(n, 2) - (l * (l + 1))
bottom_r4 = (
2 * pow(n, 5) * (l + 1.5) * (l + 1) * (l + 0.5) * l * (l - 0.5)
)
r4 = top_r4 / bottom_r4

top_r6 = (
35 * pow(n, 4)
- 5 * pow(n, 2) * (6 * l * (l + 1) - 5)
+ 3 * (l + 2) * (l + 1) * l * (l - 1)
)
bottom_r6 = (
8
* pow(n, 7)
* (l + 2.5)
* (l + 2)
* (l + 1.5)
* (l + 1)
* (l + 0.5)
* l
* (l - 0.5)
* (l - 1)
* (l - 1.5)
)
r6 = top_r6 / bottom_r6

# Calculate the energy difference from polarisation energy from https://journals.aps.org/pra/abstract/10.1103/PhysRevA.14.1614
defect = (
0.5 * (self.a_d_eff * r4 + self.a_q_eff * r6)
)

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I believe above is correct based on your numerical results, however, is there some reference where these equations have been originally derived, that we can reference in commend directly in code?

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I believe the original is from J. E. Mayer and M. G. Mayer, Phys. Bev. 43, 605
(1933).

The Freeman and Kleppner Physical Review A 14.5 (1976): 1614 I reference cites this work.

On overview is also given in pgs 340-364 `Spectroscopy_of_alkali_Rydberg_states' in Gallagher's Rydberg atoms (which they themselves cite Mayer and Mayer as the origin)

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if you are referring to the expectation values see https://journals.aps.org/pra/abstract/10.1103/PhysRevA.42.1123 or more appropriately https://journals.aps.org/pra/abstract/10.1103/PhysRevA.9.1087

return defect

def getRadialMatrixElement(
Expand Down