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Properties of Potassium
T.G. Tiecke
v1.03, (Jun, 2019)
1 Introduction
This document is a stand-alone version of Appendix A of my PhD thesis [1]. It is meant to provide an overview of the properties of atomic potassium useful for experiments on ultracold gases. A thorough review of the properties of lithium has been given in the thesis of Michael Gehm [2, 3]. For the other alkali atoms extended reviews have been given for Na, Rb and Cs by Daniel Steck [4].
version date changes 1.0 2009 first standalone version 1.02 2011 fixed typos 1.03 2019 fixed units in captions of figure 1 and 2
2 General Properties
Potassium is an alkali-metal denoted by the chemical symbol K and atomic number Z = 19. It has been discovered in 1807 by deriving it from potassium hydroxide KOH. Being an alkali atom it has only one electron in the outermost shell and the charge of the nucleus is being shielded by the core electrons. This
3 OPTICAL PROPERTIES
Mass number A Neutrons N Abundance (%) [6] m (u) [8] τ [9] I [9] 39 20 93.2581(44) 38.96370668(20) stable 3/ 40 21 0.0117(1) 39.96399848(21) 1. 28 × 109 y 4 41 22 6.7302(44) 40.96182576(21) stable 3/
Table 1: Naturally occurring isotopes of potassium. The atomic number of potassium is Z = 19. The given properties are the atomic number A, the number of neutrons in the nucleus N , the abundance, the atomic mass m, the lifetime τ and the nuclear spin I.
Melting point 63. 65 ◦C (336.8 K) [10] Boiling point 774. 0 ◦C (1047.15 K) [10] Density at 293 K 0 .862 g/cm^3 [10] Ionization energy 418 .8 kJ mol−^1 [10] 4 .34066345 eV [11] Vapor pressure at 293 K 1. 3 × 10 −^8 mbar [5] Electronic structure 1 s^22 s^2 p^63 s^2 p^64 s^1
Table 2: General properties of potassium
makes the element very chemically reactive due to the relatively low ionization energy of the outermost electron. The basic physical properties of potassium are listed in Table 2. Potassium has a vapor pressure given in mbar by [5]:
(solid) log p = 7. 9667 −
T
298 K < T < Tm. (1)
(liquid) log p = 7. 4077 −
T
Tm < T < 600 K
Figure 1 depicts the vapor pressure over the valid range of Eq. 1.
Potassium has a chemical weight of 39.0983(1) [6] and appears naturally in three isotopes, 39 K, 40 K and 41 K which are listed in Table 1. The fermionic isotope 40 K has two radioactive decay channels. In 89% of the cases it decays through a β−^ decay of 1.311MeV resulting in the stable 40 Ar. In the remaining 11% it decays through electron capture (K-capture) to 40 Ca [7]. The former decay channel is commonly used for dating of rocks.
3 Optical properties
The strongest spectral lines of the ground state potassium atom are the D1 (^2 S → 2 P 1 / 2 ) and D (^2 S → 2 P 3 / 2 ) lines. The most recent high precision measurements of the optical transition frequencies of potassium have been published by Falke et al. [12]. Tables 3 to 8 list the properties of the D1 and D lines for the various isotopes. The natural lifetime τ of an excited state is related to the linewidth of the associated transition by
Γ =
τ
where Γ is the natural linewidth. A temperature can be related to this linewidth, which is referred to as the Doppler temperature
kB TD =
where kB is the Boltzmann constant. The wavenumber k and frequency ν of a transition are related to the wavelength λ by
k = 2 π λ
, ν = c λ
3 OPTICAL PROPERTIES
Property symbol value reference Frequency ν 389 .286058716(62) THz [12] Wavelength λ 770 .108385049(123) nm Wavenumber k/ 2 π 12985 .1851928(21) cm−^1 Lifetime τ 26 .72(5) ns [13] Natural linewidth Γ/ 2 π 5 .956(11) MHz Recoil velocity vrec 1 .329825973(7) cm/s Recoil Temperature Trec 0. 41436702 μK Doppler Temperature TD 145 μK
Table 3: Optical properties of the 39 K D1-line.
Property symbol value reference Frequency ν 391 .01617003(12) THz [12] Wavelength λ 766 .700921822(24) nm Wavenumber k/ 2 π 13042 .8954964(4) cm−^1 Lifetime τ 26 .37(5) ns [13] Natural linewidth Γ/ 2 π 6 .035(11) MHz Recoil velocity vrec 1 .335736144(7) cm/s Recoil Temperature Trec 0. 41805837 μK Doppler Temperature TD 145 μK Saturation intensity Is 1 .75 mW/cm^2
Table 4: Optical properties of the 39 K D2-line.
When an atom emits or absorbs a photon the momentum of the photon is transferred to the atom by the simple relation mvrec = ℏk (4)
where m is the mass of the atom, vrec is the recoil velocity obtained (lost) by the absorption (emission) process and ℏ = h/ 2 π is the reduced Planck constant. A temperature can be associated to this velocity, which is referred to as the recoil temperature
kB Trec =
mv^2 rec (5)
Finally, we can define a saturation intensity for a transition. This intensity is defined as the intensity where the optical Rabi-frequency equals the spontaneous decay rate. The optical Rabi-frequency depends on the properties of the transition, here we only give the expression for a cycling transition
Is =
πhc 3 λ^3 τ
3 OPTICAL PROPERTIES
Property symbol value reference Frequency ν 389 .286184353(73) THz [12] Wavelength λ 770 .108136507(144) nm Wavenumber k/ 2 π 12985 .1893857(24) cm−^1 Lifetime τ 26 .72(5) ns [13] Natural linewidth Γ/ 2 π 5 .956(11) MHz Recoil velocity vrec 1 .296541083(7) cm/s Recoil Temperature Trec 0. 40399576 μK Doppler Temperature TD 145 μK
Table 5: Optical properties of the 40 K D1-line.
Property symbol value reference Frequency ν 391 .016296050(88) THz [12] Wavelength λ 766 .700674872(173) nm Wavenumber k/ 2 π 13042 .8997000(29) cm−^1 Lifetime τ 26 .37(5) ns [13] Natural linewidth Γ/ 2 π 6 .035(11) MHz Recoil velocity vrec 1 .302303324(7) cm/s Recoil Temperature Trec 0. 40399576 μK Doppler Temperature TD 145 μK Saturation intensity Is 1 .75 mW/cm^2
Table 6: Optical properties of the 40 K D2-line.
Property symbol value reference Frequency ν 389 .286294205(62) THz [12] Wavelength λ 770 .107919192(123) nm Wavenumber k/ 2 π 12985 .1930500(21) cm−^1 Lifetime τ 26 .72(5) ns [13] Natural linewidth Γ/ 2 π 5 .956(11) MHz Recoil velocity vrec 1 .264957788(6) cm/s Recoil Temperature Trec 0. 41408279 μK Doppler Temperature TD 145 μK
Table 7: Optical properties of the 41 K D1-line.
Property symbol value reference Frequency ν 391 .01640621(12) THz [12] Wavelength λ 766 .70045870(2) nm Wavenumber k/ 2 π 13042 .903375(1) cm−^1 Lifetime τ 26 .37(5) ns [13] Natural linewidth Γ/ 2 π 6 .035(11) MHz Recoil velocity vrec 1 .2070579662(7) cm/s Recoil Temperature Trec 0. 41408279 μK Doppler Temperature TD 145 μK Saturation intensity Is 1 .75 mW/cm^2
Table 8: Optical properties of the 41 K D2-line.
4 FINE STRUCTURE, HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT
(^2) P 3/
(^2) S 1/
(^2) P 1/
766.701 nm
770.108 nm
F=9/2 (-571.5)
F=7/2 (714.3)
F=9/2 (-69.0)
F=7/2 (86.3)
F'=11/2 (-46.4)
F'=9/2 (-2.3)
F'=7/2 (31.0)
F'=5/2 (55.2)
(1285.8)
40 K
39 K
41 K
(^2) P 3/
(^2) S 1/
(^2) P 1/
F'=0 (-19.4)
F'=1 (-16.1)
F'=2 (-6.7)
F'=3 (14.4)
(461.7)
(^2) P 3/
(^2) S 1/
(^2) P 1/
(254.0) F=1 (-158.8)
F=2 (95.3)
F'=0 (-8.4)
F'=1 (-8.4)
F'=2 (-5.0)
F'=3 (8.4)
F=1 (-19.1)
F=2 (11.4)
F'=1 (-34.7)
F'=1 (-288.6)
F'=2 (20.8)
F'=2 (173.1)
(125.6)
(126.0)
(235.5)
(236.2)
Figure 2: Optical transitions of the D1 and D2-lines of 39 K, 40 K and 41 K. A similar plot including 37 K, (^38) K can be found in [9]. The energy levels of the hyperfine structure are taken from [12] and [14] and
given in units of MHz. Note the inverted hyperfine structure for 40 K.
4 FINE STRUCTURE, HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT
B (^) hf,K=357G
F=9/
F=7/
m =-9/2F
m =+9/2F
m =+7/2F
m =-7/2F
m =+7/2F
Figure 3: The hyperfine structure of the 2 S 1 / 2 groundstate of 40 K. The states are labeled with their low-field quantum numbers |F, mF 〉. Note the inverted hyperfine structure.
2
P1/
2
P3/
m =-1/2J
m =+1/2J
m =+3/2J
m =-3/2J
Figure 4: Hyperfine structure of the 2 P 1 / 2 (D1) and the 2 P 3 / 2 (D2) levels of 40 K.
Hint^ = Hhf^ + HZ^ (6)
In the absence of orbital angular momentum, L = 0, and for S = 1/2, the eigenvalues of Eq. 6 correspond to the Breit-Rabi formula [16]
Ehf^ (B) = −
ahf 4
ahf^ (I + 1/2) 2
4 mf x 2 I + 1
x =
(gJ − gI )μB ahf^ (I + 1/2)
B
where μB = 9. 27400915 × 10 −^24 JT−^1 is the Bohr-magneton and the sign corresponds to the manifolds with F = I ± S. Figures 3 and 4 show the eigenvalues of Eq. 6 for the 2 S 1 / 2 ground state and the 2 P 1 / 2 and (^2) P 3 / 2 excited states of 40 K respectively.
5 SCATTERING PROPERTIES
-7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/
-5/2 -3/2 -1/2 1/2 3/2 5/
-9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/
-9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/
-9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/
-7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/
-11/2 11/
(^1458243036455103680487481093513365) 729
243
277221561617115577046223177 144025603360384040003840336025601440
(^2314627701155161721562772) 77
1120192024002560240019201120 5103364524301458729243
Figure 5: Transition probabilities for 40 K (I = 4) on σ+^ transitions, normalized to integer values. Similar diagrams for 39 K and 41 K can be found in Ref. [18].
-7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/
-5/2 -3/2 -1/2 1/2 3/2 5/
-9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/
-9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/
-9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/
-7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/
-11/2 11/
(^218729163402364536453402291621871215)
308539693770770693539308 1960100036040 40 360100019603240
(^539693770770693539) 308
100036040 40 36010001960 7291215145814581215729
1960
308
3240
1215
Figure 6: Transition probabilities for 40 K (I = 4) on π transitions, normalized to integer values. Similar diagrams for 39 K and 41 K can be found in Ref. [18].
5 SCATTERING PROPERTIES
isotope as at 39/39 138 .49(12) − 33 .48(18) 39/40 − 2 .84(10) −1985(69) 39/41 113 .07(12) 177 .10(27) 40/40 104 .41(9) 169 .67(24) 40/41 − 54 .28(21) 97 .39(9) 41/41 85 .53(6) 60 .54(6)
Table 11: s-wave scattering lengths for the various isotope-combinations of potassium, values are taken from Ref. [20]
value units C 6 3925. 9 Eha^60 C 8 4. 224 × 105 Eha^80 C 10 4. 938 × 107 Eha^100 r 0 (^39 K) 64. 61 a 0 r 0 (^40 K) 65. 02 a 0 r 0 (^41 K) 65. 42 a 0
Table 12: Van der Waals properties of the scattering potential of potassium. Vvdw (r) = −C 6 /r^6 −C 8 /r^8 − C 8 /r^8.
[
ℏ^2
2 μ
∂^2
∂r^2
r
∂r
l(l + 1) r^2
]
R(r) = ǫR(r), (8)
where R(r) is the radial wavefunction, l is the angular momentum quantum number and V (r) is the scattering potential. Many ultracold scattering properties can be obtained with sufficient accuracy for general use in the lab by only using the accumulated phase method [19] and V (r) = −C 6 /r^6. However, for Potassium accurate potentials have been published by Falke, et al. [20]. Because potassium has S = 1/2 the total spin of the potassium dimer can be either singlet (S = 0) or triplet (S = 1). Figure 7 shows the Born-Oppenheimer potentials for the singlet X^1 Σ and triplet a^3 Σ potentials. Solving Eq. 8 for ǫ ↓ 0 one can obtain the scattering length. Table 11 lists the s-wave scattering lengths of the various potassium isotopes [20].
To qualitatively describe the scattering for 40 K we compare the scattering lengths to the the van der Waals range. The van der Waals range is a measure for the typical range of the potential for an atomic species. It is defined as the range where the kinetic energy of confinement in the potential equals the potential energy and is given by [21]
r 0 =
2 μC 6 ℏ^2
Using the van der Waals coefficient of C 6 = 3925.9 Eha^60 [20] for 40 K we obtain a van der Waals range of r 0 ≃ 65 a 0. The scattering lengths of both the singlet and triplet potentials are much larger than r 0 indicating resonant scattering due to the presence of a weakly bound state in both the singlet and triplet scattering potentials. Figure 8 shows the wavefunctions of the least bound states in the singlet and triplet potentials for 40 K. Note the horizontal logarithmic scale. The wavefunctions extend far into the asymptotic van der Waals tail of the potentials.
5.1 Feshbach resonances
The use of Feshbach resonances are essential for the study of ultracold gases, in particular for fermionic isotopes. A Feshbach resonance occurs due to a resonant coupling of a scattering pair of atoms with
REFERENCES
mf 1 , mf 2 s/p B 0 (G) ∆B (G) Ref. -9/2 + -7/2 s 202. 10 ± 0. 07 7. 8 ± 0. 6 [23, 24, 25] -9/2 + -5/2 s 224. 21 ± 0. 05 9. 7 ± 0. 6 [23, 26] -7/2 + -7/2 p ∼ 198. 8 [23, 24, 27]
Table 14: Feshbach resonances for 40 K. All resonances are between spin states in the F = 9/2 manifold. This table has been adapted from Ref. [23]
an energetically closed molecular state. The s-wave scattering length a in the vicinity of a Feshbach resonance is parameterized by
a(B) = abg
∆B
B − B 0
where abg is the background scattering length in absence of coupling to the molecular state, B 0 is the resonance position and ∆B is the magnetic field width of the resonance. Due to the resonant scattering in the open channels (i.e. a large background scattering length) the Feshbach resonances of 40 K have a broad character. For 39 K eight resonances have been experimentally obtained and are listed together with some theoretical predictions in table 13. For 40 K two experimentally characterized s-wave Feshbach resonances and one p-wave resonance have been published. The resonances are summarized in Table 14.
Acknowledgements
I would like to thank Antje Ludewig and Jook Walraven for suggestions and corrections.
References
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