In the Classroom edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
Positive and Negative Temperatures in a Two-Level System: Thermodynamic and Statistical-Mechanical Perspectives Mark B. Masthay* and Harry B. Fannin Department of Chemistry, Murray State University, Murray, KY 42071-3346; *
[email protected] Negative temperatures were first reported in the early 1950s for nuclear spin–lattices having a finite number of energy levels (1–8). Since that time, negative temperatures have been reported for lasing media (9), information systems (10), two-dimensional vortex fluids (11–20), plasmas (21–23), and additional spin–lattice systems (24–27), although their legitimacy has been questioned on theoretical grounds (28).1 As noted in a recent article in this Journal (29), negative temperatures occur when one of the degrees of freedom in a system (i) has a finite number of energy levels, (ii) has an inverted Boltzmann population, and (iii) is in equilibrium with itself but not with the other degrees of freedom. A representative case in point is a nuclear spin–lattice immediately after a 180⬚ rotation of an external magnetic field. Because the spins cannot instantaneously follow the field as its polarization is rotated, the low-energy (spins parallel to field) and high-energy (spins antiparallel to field) states switch identity, giving rise to a population inversion that can exist for a significant period of time when the coupling between the spins and the thermal vibrations of the lattice is weak. While negative temperatures occur in only a small number of systems and seem to contradict the common notion that thermodynamic (i.e., Kelvin) temperatures are always positive, they provide an effective platform for illustrating the relationship between the thermodynamic and statisticalmechanical formulations of temperature. In this article we present a set of calculations for a two-level system that graphically illustrate the statistical nature of temperature and the fundamental equivalence of its thermodynamic and statistical-mechanical formulations. These calculations, which we have applied in our undergraduate- and graduate-level physical chemistry courses, provide useful insights into the physical meaning of a variety of thermodynamic and statistical-mechanical concepts (e.g., entropy, temperature, and partition functions) that students frequently have difficulty understanding. We derive the statistical-mechanical expression for the temperature of a two-level system (eq 12) by combining the thermodynamic temperature (eq 3) with the Boltzman (entropy) equation (eq 7). We then utilize eq 12 to calculate the temperatures of the states of a two-level system containing various numbers of particles. We present the results of our calculations in both tabular (Tables 1–3) and graphical (Figure 3) form. In addition to providing a pedagogically simple illustration of the thermodynamic and statistical-mechanical temperature expressions, our results clearly demonstrate the statistical nature of temperature as well as the physical origin of positive and negative temperatures. www.JCE.DivCHED.org
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Thermodynamic and Statistical-Mechanical Definitions of Temperature The change in the internal energy of a closed system is given by the well-known expression (1)
dU = TdS − PdV
in which U, T, S, P, and V represent internal energy, temperature, entropy, pressure, and volume, respectively (30, 31). Because U is a state function, its total differential dU is equal to a sum of partial differentials with respect to its natural variables S and V: ∂ U ∂ S
dU =
∂ U ∂ V
dS + V
dV
(2)
S
By combining eqs 1 and 2 we obtain (by identity): T =
∂ U ∂ S
(3) V
Likewise, the total differential for the enthalpy H is equal to (30) dH = T dS + V dP =
∂ H ∂ S
P
∂ H ∂ S
P
dS +
∂ H ∂ P
S
dP (4)
so that (by identity): T =
(5)
Hence, temperature is defined thermodynamically as the rate of change of internal energy with respect to entropy under constant volume conditions (eq 3) and as the rate of change of enthalpy with respect to entropy under constant pressure conditions (eq 5) (29, 30). For brevity, we derive all of our results below using eq 3, though eq 5 could also be used. While the derivation of the thermodynamic temperature expressions is straightforward, their physical interpretation is somewhat obscure, particularly to students encountering them for the first time. Students’ understanding of eqs 2 and 3 can be significantly enhanced, however, by utilizing the statistical-mechanical definition of temperature
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T =
•
1 kB
∂ U ∂ lnΩ
V
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Equation 6 is obtained by combining eq 3 with the wellknown Boltzmann equation (7)
S = kB ln Ω
in which Boltzmann’s constant kB = 1.38 × 10᎑23 J K᎑1 particle᎑1 and the microcanonical partition function Ω(N, V, U ) is equal to the degeneracy of an energy level U (i.e., Ω is equal to the number of ways a set of N distinguishable particles in a volume V can be distributed over a set of one-particle energy states to yield a multiparticle state with total internal energy U ). With a few notable exceptions (27, 32–34), the thermodynamic properties of two-level systems have been derived using the canonical ensemble Q(N, V, U ) (3, 4, 8, 28, 29, 35), as is typical for closed systems in general. Our derivation of temperature using the microcanonical ensemble in eq 6 is thus somewhat unconventional. Nevertheless, it is consistent with a recent and growing trend originating largely from the pioneering studies by Rugh (36, 37), in which thermodynamic properties are derived from the microcanonical ensemble (36–59). This approach has been applied to a wide range of interesting systems, including two-level systems (27, 32–34), lattice gases (33, 46), Ising lattices (54, 55), molecular dynamics simulations of systems of identical harmonic oscillators (38), particles in a one-dimensional box (38), molecular clusters (47–49), Lennard–Jones chains (59), systems far from equilibrium (42, 50, 51, 53), Bose–Einstein condensates (45), and to various problems in nuclear (43, 52) and elementary particle (44, 53) physics. In a number of these recent studies, temperatures have been derived from the microcanonical partition function (36–42, 45, 47–49, 51, 53, 55, 56, 59).2 While such “microcanonical temperatures” are not in general numerically identical to the conventional “canonical temperatures” for finite systems, they are expected to be of the same order of magnitude, follow similar trends, and become identical to the canonical temperature in the thermodynamic (large N) limit (32, 33, 36–42, 44, 46–49, 51, 53–58, 60). We use microcanonical temperatures in our classes because they are easier to both visualize and calculate than canonical temperatures in two-level systems. For brevity, we refer to the microcanonical temperature simply as “temperature” in the derivations and discussion below.
Temperature in a Two-Level System Containing N Particles The statistical-mechanical (eq 6) and thermodynamic (eq 3) formulations of temperature can both be clearly illustrated for any system with a finite number of energy levels provided the occupancy of states obeys Boltzmann statistics (i.e., provided the system is internally at thermal equilibrium) (35). Two-level systems, such as those generated when spin 1/2 nuclei are placed in static magnetic fields, are the simplest of all such systems. Under such conditions, the system splits into a “spin–up” (↑) ground state in which the z component of the spin points parallel to the magnetic field H (which also points up) and a “spin-down” (↓) excited state in which the z component of the spin is directed antiparallel to the field, as shown in Figure 1. When heat flows into a collection of independent, distinguishable particles—each of which has U↑ and U↓ energy levels (as occurs when a collection of spin 1/2 nuclei, each in a separate atom in a regularly ordered lattice, are placed in a magnetic field)3—the spin-down levels are populated at the expense of the spin-up levels. Conversely, the spin-up levels are populated at the expense of the spin-down levels when the thermal energy is removed. We assume for simplicity that the energy of the ground state is equal to zero Joules ( J) and that the energy of the excited state lies ε J above the ground state (i.e., U↑ = 0 J and U↓ = ε J). The energy U[n↑,n↓] of a multiparticle state [n↑, n↓] with n↑ spin-up particles and n↓ spin-down particles is then equal to U n
↑ ,n↓
= n↓ ε
(8a)
and the transition energy between any two multiparticle states [n↑initial, n↓initial] and [n↑final, n↓final] is equal to
(
)
∆U = n↓ final − n↓ initial ε = ∆n↓ ε
(8b)
According to eq 7, the entropy of any state [n↑, n↓] is equal to
S n
↑ ,n↓
= kB ln Ω n
↑ ,n↓
(9a)
so that the difference in the entropies of any two states is spin-down (U2 = ε)
∆S = kB ln H
Ω final n ,n
↑ ↓ initial Ω n↑ ,n↓
(9b)
∆U22 = h(U2 − U2) = ε
Noting that the degeneracy of [n↑, n↓] is equal to Ω n
↑ ,n↓
spin-up (U2 = 0) Figure 1. A two-level system generated when spin 1/2 nuclei are placed in a static magnetic field H (h is Planck’s constant).
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=
N! N! = n↑ ! n↓ ! n↑ ! ( N − n↑ ) !
(10)
in which N = n↓ + n↑ and combining eqs 3, 8b, 9b, and 10 we obtain a general expression for the temperature registered
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In the Classroom Table 1. Energies U [ n↑ , n ↓ ], Microcanonical Partition Function Values Ω [ n ↑ , n ↓ ], and Entropies S[n↑, n↓] of the 11 Possible States of a Two-Level (U↑ = 0 J and U↓ = ε J) System Containing 10 Particles
State
Internal Energy/J
U n
n↑ , n↓
Ω n
= n↓ ε
↑ ,n↓
Entropy/(J K᎑1)
Degeneracy ↑ ,n↓
=
10 ! n↑ ! (10 − n↑ ) !
S n
↑ ,n↓
= kB ln Ω n
↑ ,n↓
[10, 0]
00ε
001
0
[9, 1]
01ε
010
2.302kB
[8, 2]
02ε
045
3.807kB
[7, 3]
03ε
120
4.787kB
[6, 4]
04ε
210
5.347kB
[5, 5]
05ε
252
5.529kB
[4, 6]
06ε
210
5.347kB
[3, 7]
07ε
120
4.787kB
[2, 8]
08ε
045
3.807kB
[1, 9]
09ε
010
2.302kB
[0, 10]
10ε
001
0
during a transition from [n↑initial, n↓initial] to [n↑final, n↓final]: T =
∆U ∆S
∆n↓ ε
= V
kB ln
Ω final n ,n
↑ ↓ initial Ω n ,n ↑ ↓
Trend 1
∆n↓ ε
= kB ln
8a, 10, and 9a, respectively. We specify the temperatures calculated for the ∆n↓ = +1 (i.e., [n↑, n↓] → [(n↑ − 1), (n↓ + 1)]) transitions using eq 12 in Table 2, and illustrate these temperatures graphically in the plot of U versus S shown in Figure 3. The calculations presented in Table 2 and Figure 3 illustrate three trends, which we detail below.
(11)
n↑ initial ! n↓ initial ! n↑ final ! n↓ final !
For brevity and simplicity, we will restrict our discussion to transitions between adjacent levels (i.e., ∆n↓ = ±1 transitions), for which eq 11 reduces to
The temperature is positive for states in which n↑ > n↓, which appear on the ascending (positive slope) portion of the curve in Figure 3. Conversely, the temperatures are negative for states in which n↑ ≤ n↓, which appear on the descending (negative slope) portion of the curve in Figure 3.
Although this latter result contradicts common experience (since the minimum temperature possible in the absolute
⫾ε
T∆n↓ = ± 1 = kB ln
(n
n↑ initial ! n↓ initial !
initial ↑
)(
)
⫿ 1 ! n↓ initial ⫾1) !
(12)
spin-down
spin-up
Application to Two-Level Systems Containing 10, 50, 100, 104, NA, and ∞ Particles Eleven states are possible in a two-level system containing 10 distinguishable particles: [10, 0], [9, 1], [8, 2], [7, 3], [6, 4], [5, 5], [4, 6], [3, 7], [2, 8], [1, 9], and [0, 10] (in order of increasing energy). The minimum energy, minimum entropy state [10, 0]; maximum energy, minimum entropy state [0, 10]; and the intermediate energy, maximum entropy state [5, 5] are illustrated schematically in Figure 2. In Table 1, we specify U[n↑,n↓], Ω[n↑,n↓], and S[n↑,n↓] for each of the states in a two-level, 10-particle system using eqs
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minimum energy state [10, 0]
maximum energy state [0, 10]
intermediate energy state [5, 5]
U[10,0] = 0ε
U[0,10] = 10ε
U[5,5] = 5ε
Figure 2. The maximum, minimum, and intermediate energy states for a collection of 10 particles in a two-level system. The entropies of the maximum and minimum energy states are zero, whereas the entropy is maximized in the intermediate energy state (see eqs 7 and 10 and Table 1 in the text).
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through an entropy maximum at states of intermediate energy (29, 35).
12
Trend 2
10
[1, 9]
[0, 10]
The temperature attains its minimum magnitude as the system approaches the minimum entropy ([N, 0] and [0, N ]) states.
[2, 8] 8
[3, 7]
U ε
[4, 6] 6
[5, 5] 4
[6, 4]
This effect occurs because ∆S increases in magnitude as the system approaches the minimum entropy states, whereas the magnitude ε of ∆U is identical for all ∆n↓ = ±1 transitions. It is noteworthy that T[10, 0] and T[0, 10] are not equal to zero. This interesting result is due to the inverse dependence of the temperature of the minimum entropy states on lnN:
[7, 3]
ε T[ N ,0] = −T[0,N ] = kB ln N
2
[8, 2] [10, 0]
[9, 1]
0 0
1
2
3
4
5
6
S kB Figure 3. Internal energy plotted as a function of entropy for a twolevel system containing 10 particles. The temperature for any [ n↑, n↓] → [(n↑ − 1), (n↓ + 1)] transition equals the slope of this plot between the points (S[n↑, n↓], U[n↑, n↓],) and (S[(n↑ − 1)],(n↓ + 1)], U[(n↑ − 1),(n↓ + 1)]).
[Kelvin] temperature scale is zero), negative temperatures occur in two-level systems because promotion of a particle from the spin-up to the spin-down level in states with n↑ ≤ n↓ leads to states of higher energy but lower entropy. For the same reason, negative temperatures occur in any system in which the entropy is not a monotonically increasing function of temperature. This property is characteristic for all systems having a finite number of levels, since such systems pass
(13)
Equation 13, which is obtained by applying eq 12 to the [N, 0] → [(N − 1), 1] and [0, N] → [1, (N − 1)] transitions, indicates that the temperatures of the minimum entropy states approach zero only as N → ∞ (the thermodynamic limit) or as ε → 0 (the degenerate states limit). We demonstrate how the temperature approaches the thermodynamic limit in Table 3, in which we tabulate the values of T[N,0] and T[0,N] for systems containing 10, 50, 100, 104, and NA particles. In accord with eq 13, T[N,0] = ᎑T[0,N] becomes smaller as the number of particles increases, demonstrating that temperature is an inherently statistical property. Notably however, the temperature of the minimum and maximum entropy states remains finite (±0.018ε兾kB) even in systems of typical “chemical” size (N ∼ NA = 6.02 × 1023). Hence, the temperatures of the minimum and maximum energy (minimum entropy) states approach zero only when N becomes extremely large (57). This behavior is consistent with the inverse dependence of T[N,0] = ᎑T[0,N] on the logarithm of N (since lnN increases more slowly than N itself ).
Table 2. Changes in Internal Energy ∆U, Changes in Entropy ∆S, and Temperatures T for ∆n↓ = +1 Transitions in a Two-Level (U↑ = 0 J and U↓ = ε J) System Containing 10 Particles
Transition
n↑ , n↓
Transition Energy/J
n↑ − 1 , n↓ + 1
∆U = ∆n↓ ε
Transition Entropy/(J K᎑1)
∆S = kB ln
Ω final n↑ ,n↓ Ω initial n↑ ,n↓
Temperature/K
T =
[10, 0] → [9, 1]
+ε
+2.302kB
+0.434ε/kB
[9, 1] → [8, 2]
+ε
+1.504kB
+0.665ε/kB
[8, 2] → [7, 3]
+ε
+0.981kB
+1.02ε/kB
[7, 3] → [6, 4]
+ε
+0.560kB
+1.79ε/kB
[6, 4] → [5, 5]
+ε
+0.182kB
+5.48ε/kB
[5, 5] → [4, 6]
+ε
᎑0.182kB
᎑5.48ε/kB
[4, 6] → [3, 7]
+ε
᎑0.560kB
᎑1.79ε/kB
[3, 7] → [2, 8]
+ε
᎑0.981kB
᎑1.02ε/kB
[2, 8] → [1, 9]
+ε
᎑1.504kB
᎑0.665ε/kB
[1, 9] → [0, 10]
+ε
᎑2.302kB
᎑0.434ε/kB
NOTE: ∆U, ∆S, and T calculated using eqs 8b, 9b, and 12, respectively.
870
∆U ∆S
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10 particles are present the maximum magnitude temperatures are obtained for the [6, 4] → [5, 5] transition (T = +5.48ε兾k B ) and the [5, 5] → [4, 6] transition ((T = ᎑5.48ε兾kB). Applying eq 12 to the [(N兾2 + 1), (N兾2 − 1)] → [N兾2, N兾2] and the [(N兾2 − 1), (N兾2 + 1)] → [N兾2, N兾2] transitions, it is easy to show that the maximum temperature is equal to
Since the energetic splitting between spin-up and spindown levels is proportional to the magnetic field strength, the degenerate states limit applies to spin systems only in the absence of magnetic fields. In contrast, the degenerate states limit applies under all conditions to the translational states of gas molecules on which the Kelvin scale is based, because translational states form an effective energetic continuum (31). Hence, the Kelvin temperature of the minimum entropy state for an ideal gas is equal to zero even when a finite number of particles are present, consistent with eq 13.
T ( N
),(N 2 − 1) = −T ( N 2 − 1),( N 2 + 1)
2 +1
Trend 3
=
The absolute magnitude of the temperature increases as the system approaches the maximum entropy ([N兾2, N兾2]) configuration.
ε kB ln (1 +
2
(14)
N)
As eq 14 and the data presented in Table 3 indicate, the temperatures of these transitions are infinite in the limit of large N. Significantly, these results apply for all values of the transition energy ε, indicating that ε need not be large for the temperature to be large. Hence, the large temperatures of the
This effect occurs because ∆S decreases in magnitude as the system approaches this configuration, whereas the magnitude ε of ∆U is identical for all ∆n↓ = ±1 transitions. Hence, when
Table 3. Temperatures of the Minimum Entropy ([N, 0] and [0, N]) and Maximum Entropy ([N/2, N/2]) States of a Two-Level System Containing Various Numbers of Particles
Transition
N
Temperature/K
T = ⫾
+: [N, 0] → [(N – 1), 1] –: [1, (N – 1)] → [0, N]
ε kB ln N
[10, 0] → [9, 1]
10
+0.434ε/kB
[1, 9] → [0, 10]
10
᎑0.434ε/kB
[50, 0] → [49, 1]
50
+0.256ε/kB
[1, 49] → [0, 50]
50
᎑0.256ε/kB
[100, 0] → [99, 1]
100
+0.217ε/kB
[1, 99] → [0, 100]
100
᎑0.217ε/kB
[10000, 0] → [9999, 1]
10,000
+0.109ε/kB
[1, 9999] → [0, 10000]
10,000
᎑0.109ε/kB
[NA, 0] → [(NA – 1), 1]
NA = 6.02 × 1023
+0.018ε/kB
[1, (NA – 1)] → [0, NA]
NA = 6.02 × 1023
᎑0.018ε/kB
[∞, 0] → [(∞ – 1), 1]
∞
+0ε/kB
[1, (∞ – 1)] → [0, ∞ ]
∞
᎑0ε/kB
Transition
N
Temperature/K
ε
T = ⫾
+: [(N/2 + 1), (N/2 – 1)] → [N/2, N/2] –: [N/2, N/2] → [(N/2 – 1), (N/2 + 1)]
kB ln 1 +
[6, 4] → [5, 5]
10
+5.48ε/kB
[5, 5] → [4, 6]
10
᎑5.48ε/kB
[26, 24] → [25, 25]
50
+25.5ε/kB
[25, 25] → [24, 26]
50
᎑25.5ε/kB
[51, 49] → [50, 50]
100
+50.5ε/kB
[50, 50] → [49, 51]
100
᎑50.5ε/kB
[5001, 4999] → [5000, 5000]
10,000
+5000.5ε/kB
[5000, 5000] → [4999, 5001]
10,000
᎑5000.5ε/kB
[(NA/2 + 1),(NA/2 - 1)] → [NA/2, NA/2]
NA = 6.02 × 1023
[NA/2, NA/2] → [(NA/2 - 1),(NA/2 + 1)]
23
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maximum entropy states that occur in the thermodynamic limit originate exclusively from large N; that is, they are a purely statistical effect. Finally, it is noteworthy that the temperature of the intermediate energy, maximum entropy states approach infinity more rapidly with increasing N than the temperatures of the minimum and maximum energy, minimum entropy states approach zero because ln(1 + 2兾N) → 0 more rapidly than lnN → ∞ as N becomes large (eqs 13 and 14). Summary By combining the thermodynamic expression for temperature T = (∂U兾∂S )V with the well-known statistical-mechanical definition of entropy S = k lnΩ, we arrive at a simple method for illustrating the physical meaning of temperature both numerically and graphically. We apply this technique to a two-level system containing between 10 and 6.02 × 1023 independent, distinguishable particles to illustrate the statistical nature of temperature and the physical origin of negative temperatures. We have employed exercises that utilize this technique in our undergraduate and graduate level physical chemistry courses, as they provide pedagogically useful insights into the meaning of a variety of thermodynamic and statistical-mechanical concepts that students frequently have difficulty grasping. Additional information, including specific classroom exercises, are available.61
= U, and hence “collapses” into a single microcanonical ensemble Ω(N, V, U *) = Ω(N, V, U ), under these conditions (see refs 32, 33, 36–42, 44, 46–49, 51, 53–58, and 60, and, especially, pp 33– 35 of ref 8 and pp 63–64 of ref 31). In contrast, in small systems the energy of the canonical ensemble is not sharply peaked at U, but rather is broadly distributed over a number of microstates with energies ≠ U. As a result, Ω(N, V, U ) and Ω(N, V, T ) yield similar but not rigorously identical thermodynamic values for small systems. For example, the internal energies of a lattice gas containing 12 particles calculated using Ω are of the same order of magnitude but are not identical to those calculated using Q (see pp 70–72 of ref 8 and pp 57–72 of Part II of ref 32 ). Assuming the two-level spin system employed in this article obeys Boltzmann statistics, we obtain the relationship
1 i → f Tµcan
1 i → f Tcan
N − n↓ initiall kB ln ε n↓ initial + 1
=
+
The authors thank C. J. Cairns, J. R. Cox, T. W. McCreary, and P. G. Nelson for helpful discussions and R. E. Jones, M. E. Kelleher, J. M. Mabrouk, J. B. McGregor, R. J. Provost, R. G. Raspberry, J. A. Schatz, M. R. Short, T. S. Sirls, and L. K. White for editorial assistance. MBM thanks the National Science Foundation for partial funding of this project by grant number NSF-EPS-0132295. This article is respectfully and affectionately dedicated to Hartland H. Schmidt.
∑ Ω ( N , V, E j ) e
−E j
(k B T )
j
which is dominated by states having the most probable energy U*
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kB ln ε
n
(N − n↓ initial −1),(n↓ initial +1)
n
(N − n↓ initial ),n↓ initial
for ∆n↓ = +1 (i.e., ∆U = +ε) transitions and to
N − n↓ initial + 1 kB ln ε n↓ initiall
−
Q (N, V, T ) =
N − n↓ initial kB ln ε n↓ initial
Notes 1. In ref 28 the idea of negative temperatures is obviated by redefining the Second Law for the upper branch of the U versus S curve shown in Figure 3. 2. We calculate temperature using Ω(N, V, U ) rather than Q(N, V, U ) because of the greater simplicity and pedagogical clarity afforded by Ω-based temperature expressions for two-level systems. Though the microcanonical (µcan) ensemble applies most naturally to isolated (constant N, V, U ) systems, it yields thermodynamic values that are similar to those obtained using the canonical (can) ensemble when applied to finite, closed (constant N, V, T ) systems. Microcanonical and canonical values converge in the thermodynamic (large N ) limit because the canonical ensemble is actually a Boltzmann distribution of microcanonical ensembles, that is,
n kB ln f ε ni
+
for transitions from an initial state, i, to a final state, f, in which ni and nf are the Boltzmann populations of the initial and final states. This expression reduces to
Acknowledgments
872
=
=
kB ln ε
N − n↓ initial + 1
kB ln ε
n↓ initial − 1
n
(N − n↓ initial + 1),(n↓ initial −1)
n
(N − n↓ initial ),n↓ initial
for ∆n↓ = ᎑1 (i.e., ∆U = ᎑ε) transitions, in which Tµcani→f is given by eq 12 and Tcani→f is given by the equation in the caption of Figure 7, p 94 of ref 35, which we have amended to apply to a system of total energy Nε. As we demonstrate in a forthcoming publication, this expression indicates that Tµcani→f ≠ Tcani→f for all i → f transitions, [(N − n↓initial), n↓initial] → [(N − n↓initial ⫿ 1), (n↓initial ⫾ 1)], when N is finite, but that the microcanonical and canonical temperatures converge for all initial and final states in the thermodynamic (large N) limit. 3. In this system, more than two spin 1/2 nuclei are allowed to occupy the spin-up or spin-down state because the nuclei are distinguishable. The nuclei are distinguishable because they are localized in separate quantum systems (i.e., separate atoms), and hence have independent Hamiltonians so that they are not subject exchange antisymmetry and Pauli exclusion principle constraints. The Pauli exclusion principle applies only to indistinguishable particles
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In the Classroom (i.e., identical particles that are subject to exchange antisymmetry constraints because they are in the same quantum system [atom]).
Literature Cited 1. 2. 3. 4.
5. 6. 7. 8. 9. 10.
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