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Properties of Potassium
T.G. Tiecke
van der Waals-Zeeman institute University of Amsterdam The Netherlands v1.0 (February, 2010)
1 Introduction
This document is a stand-alone version of the appendix on potasssium properties of my 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].
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 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]:
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
(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
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
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)
4 FINE STRUCTURE, HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT
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 .37(5) ns [13] Natural linewidth Γ/ 2 π 6 .03(1) 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.
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
4 Fine structure, Hyperfine structure and the Zeeman effect
The fine structure interaction originates from the coupling of the orbital angular momentum L of the valence electron and its spin S with corresponding quantum numbers L and S respectively. The total electronic angular momentum is given by:
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 .37(5) ns [13] Natural linewidth Γ/ 2 π 6 .035(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.
4 FINE STRUCTURE, HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT
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 .37(5) ns [13] Natural linewidth Γ/ 2 π 6 .035(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
J = L + S
and the quantum number J associated with the operator J is in the range of |L − S| ≤ J ≤ L + S. The electronic ground state of 40 K is the 4^2 S 1 / 2 level, with L = 0 and S = 1/2, therefore J = 1/2. For the first excited state L = 1 and S = 1/2 therefore J = 1/2 or J = 3/2 corresponding to the states 4^2 P 1 / 2 and 4^2 P 3 / 2 respectively. The fine structure interaction lifts the degeneracy of the 4^2 P 1 / 2 and 4^2 P 3 / 2 levels, splitting the spectral lines in the D 1 line (4^2 S 1 / 2 → 42 P 1 / 2 ) and the D 2 line (4^2 S 1 / 2 → 42 P 3 / 2 ). The hyperfine interaction originates from the coupling of the nuclear spin I with the total electronic angular momentum
F = J + I
where the quantum number F associated with the operator F is in the range of |J − I| ≤ F ≤ J + I, where I is the quantum number corresponding to the operator I. For 40 K the fine-structure splitting is ∆EF S ≃ h × 1 .7 THz, therefore the two excited states can be considered separately when considering smaller perturbations like the hyperfine or Zeeman interaction which are on the order of a few GHz or less. The Hamiltonian describing the hyperfine structure for the two excited states described above is given by [14, 15]
Hhf^ =
ahf ℏ^2
I · J+
bhf ℏ^2
3(I · J)^2 + 32 (I · J) − I^2 J^2
2 I(2I − 1)J(2J − 1)
where ahf and bhf are the magnetic dipole and electric quadrupole constants respectively. The dot product is given by
I · J =
(F^2 − I^2 − J^2 )
This hyperfine interaction lifts the spin degeneracy due to the different values of the total angular mo- mentum F. The energy shift of the manifolds are given by
δEhf =
ahf 2
[F (F + 1) − I(I + 1) − J(J + 1)]
For a S = 1/2 system in the electronic grounstate, J = 1/2, the energy splitting due to the hyperfine interaction in zero field is given by
∆Ehf = ahf 2
I +
In the presence of an external magnetic field the Zeeman interaction has to be taken into account
HZ^ = (μB /ℏ)(gJ J + gI I) · B,
where gJ is the Land´e g-factor of the electron and gI the nuclear gyromagnetic factor. Note that different sign conventions for gI are used in literature, here we take the convention consistent with the common references in this context [14, 4, 3], such that μ = −gI μB I. The factor gJ can be written as
gJ = gL
J(J + 1) − S(S + 1) + L(L + 1)
2 J(J + 1)
J(J + 1) + S(S + 1) − L(L + 1)
2 J(J + 1)
where gS is the electron g-factor, gL is the gyromagnetic factor of the orbital, given by gL = 1 − me/mn, where me is the electron mass and mn is the nuclear mass. The total hyperfine interaction in the presence of an external magnetic field is now given by the internal hamiltonian
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]
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.
Ehf^ (B) = − ahf 4
- gI μB mf B ± ahf^ (I + 1/2) 2
4 mf x 2 I + 1
x =
(gS − 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.
4.1 Transition strengths
In this section we present the transition strengths for 40 K. We do not elaborate on the physics behind the transition dipole matrix elements. For a more thorough description and the transition strengths for (^39) K and 41 K we refer to Ref. [18]. The transition matrix element coupling a ground state defined by the
quantum numbers J, F, mF to an excited state with quantum numbers J′, F ′, m′ F is given by
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.
-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 40360100019603240
(^539693770770693539) 308
100036040 4036010001960 7291215145814581215729
1960
308
3240
1215
Figure 6: Transition probabilities for 40 K (I = 4) on π transitions, normalized to integer values.
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.
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 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.
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: All resonances are between spin states in the F = 9/2 manifold. This table has been adapted from Ref. [23]
References
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REFERENCES
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