Does Neutron Star Collapse Violate Quantum Theory?

With the death of a star meaning that radiation pressure can no longer support the gravitational crush due to its large mass, Pauli’s exclusion principle maintains an equilibrium in what has now become a neutron star. Heisenberg uncertainty allows for a neutron star to undergo ultimate collapse, as a star becomes more massive momentum space expands and allows for many more plausible wavefunctions, for a given spatial location. This therefore allows for a neutron star to form a black hole without violating the exclusion principle that at first keeps it in equilibrium. However, when a black hole is formed and the matter crashes down into a single point, a singularity, the real question of violation arises. In this paper I intend to explain how Pauli exclusion is not violated by the collapse of a neutron star initially, and explore whether any of the laws of quantum mechanics are violated once all the mass has fallen beneath the event horizon.

I. Introduction– The Neutron star

Neutron stars contain some of the most bizarre matter in the universe, and are one of the rare examples of quantum mechanics apparent on a macroscopic scale. Neutron stars are quantum entities, their collapse and stability governed by uncertainty and exclusion. This makes them fundamentally complex, both their stability and ultimate collapse are made possible by certain quantum phenomena. Before we understood their weird nature, Neutron stars were first theorised in 1934 by Baade and Zwicky [1], a year after the discovery of the neutron. Since then we have seen them mostly as Pulsars (Neutron stars with large magnetic fields that emit pulses of radio waves) [16]. Mostly because they are very small on a cosmic scale, and tend to be rather far away.

Stars have huge masses. As gravitational field strength is proportional to mass, it is clear that stars have massive gravitational field strengths. When a star is on the main sequence it is actively fusing elements in its core. For the majority of its life this consists of fusing hydrogen atoms and forming helium, releasing large amounts of energy. This causes a radiation pressure outward, balancing the inwards crush of the star’s gravitational field, maintaining stellar equilibrium. However, towards the end of a high mass stars’ life (typically greater than 8 M [3]), the temperatures and pressures become so high that heavier elements are fused, Helium, Carbon, Neon, Magnesium… [22]. However, when the stellar core contracts and heats up enough to fuse silicon into iron, the process can no longer go on. This is because when iron fuses, it doesn’t release energy like the other reactions, it takes in more energy- its endothermic. This means that there is no longer the radiation pressure supporting the star- the core collapses.

In the last moments of a massive star’s life [3], the collapsing core smashes electrons and protons together, forming neutrons and neutrinos- as shown by the Feynman diagram in figure 1.1. The collapse in the core is only stopped by a phenomenon named Neutron degeneracy, a concept explained in more depth later, (see chapter 2). Briefly, it happens because neutrons in the core have nowhere else to go, they cannot be compacted further. The rest of the imploding star bounces off this core causing the spectacular explosion we see as a type II supernova. What’s left behind is the Neutron star. A celestial body comprised (according to our understanding) mostly of neutrons, with a density comparable to an atomic nucleus.

Using data from Heidelberg University [4] on nearby Neutron star masses and radii, a simple density calculation can be made, for example on pulsar “PSR B1257+12” *:

 \rho _{B1257+12} = \frac{1.5(m_{sol})}{\frac{4}{3}\pi(2\times 10^{-5})(r_{sol})}= 2.6\times 10^{17}kgm^{-3}

Figure 1.2 shows the other calculations made using some other data from Heidelberg University [4]. The data shows that the density of a Neutron star is in the order of magnitude 10^{17} kgm^{-3}, which is incomprehensibly large, especially compared with matter we are used to (lead has a density of 1.13 \time 10^{4} kgm^{-3}). When put into perspective it is much easier to understand the huge densities of neutron stars. Neutron stars have masses greater than the sun, with radii around 20km. This is comparable to shrinking the sun to the size of Manhattan. Atoms in regular matter are comprised of 99.9999999999996% empty space [20]. When all that empty space is filled with neutrons, it is possible to comprehend how neutron stars are as dense as they are.

An object so massive has a very large gravitational field strength. Using the example of the pulsar “PSR B1257+12” a calculation can be made to determine just how large acceleration due to gravity is near its surface. Using a combination of Newton’s second law, and his equation for universal gravitation the acceleration due to gravity at“PSR B1257+12”’s surface can be determined [23]:

 g=\frac{GM}{r^2}

 g=\frac{G(1.5)M_{sol}}{((2.5\times10^{-5})R_{sol})^{2}}=1.03\times10^{12} ms^{-2}

This gravitational field strength is a factor of 1011 stronger than Earth’s gravitational field. With a gravitational field strength so large (thus with such an extreme space-time curvature) effects of gravitational time dilation are apparent. With clocks running noticeably slower than in an inertial frame of reference (such as a frame in deep space) [6]. With this crushing field strength, it is difficult to imagine what resists it, for the neutron star to stay in equilibrium. Quantum mechanics allows for neutron stars not to collapse initially, however quantum mechanics also allows for them to ultimately collapse. So does neutron star collapse violate quantum theory?

*Table values stated and calculations made in terms of solar masses (M), and solar radii (R). In the calculation numerical values for these numbers substituted in. [5]

II. Pauli’s Exclusion Principle and Degenerate matter

With such a large mass the gravitational crush of the Neutron star on itself is significant. Before it’s death the star actively fused elements in its core for most of its life (on the main sequence), this was the fusion of hydrogen atoms into helium, creating a lot of energy, as mentioned in the previous chapter. This “radiation pressure” caused an outwards push, helping to resist the inwards crush of the star’s gravitational field. Once the star has got to the neutron star phase, its days of nuclear fusion have long gone. Therefore, there must be something else resisting the huge gravitational inwards force. This ends up being entirely described by quantum mechanics. It is quite rare to see the work of quantum mechanics on the macroscopic scale, but the neutron star is an example of exactly this. To explain how the neutron star is in equilibrium it is first important to understand Wolfgang Pauli’s exclusion principle.

The Pauli exclusion principle states that no two fermions can be in the same quantum state [8]. This holds true for neutrons (hadrons), not only fundamental particles such as the electron [9]. In essence any particle with a non-integer spin (a quantum property) is subject to Pauli’s exclusion principle. Only particles like bosons aren’t as they have integer spins. Thus, no two neutrons can have the same position, and the same momentum (and any other quantum state) [10].

The exclusion principle can be proved in terms of the wave function of the fermion (\Psi), and in particular the wave function of a system comprising two fermions. The wave function of a fermion describes mathematically its position and other such quantum properties [11]. For example, the probability of locating a particle, described by wavefunction , in a given place is equal to the square of the wavefunction (\begin{vmatrix}\Psi^{2} \end{vmatrix}) at that given place [19].

Quantum mechanics states the following is true about the wave function, \Psi, of a composite system of positions of two neutrons in phase space. This is because neutrons are said to have asymmetric wave functions- the same isn’t true for bosons who have symmetric wave functions thus aren’t affected by the exclusion principle [12]. This can be explained by the fact that fermions have non-integer spins (a quantum property and form of angular momentum):

 \Psi_{n_{1},n_{2}} = -\Psi_{n_{2},n_{1}}

However if both neutrons have identical quantum properties, and location, they must have identical wave functions, thus in this case:

 \Psi_{n_{1},n_{2}} = \Psi_{n_{2},n_{1}}

The only way for both the above to be satisfied \Psi=-\Psi, Which must mean that either one or both of the wave functions of our neutrons must equal 0 (meaning this system can’t exist). Therefore, showing two particles can never have the same wave function, thus never have symmetric quantum properties.

As mentioned in the introduction chapter the space the neutron star occupies is entirely filled with neutrons**. But what does filled mean? When talking about the space the neutron star occupies, due to the nature of the matter, it is insufficient to talk just about 3 dimensional space. Due to its quantum nature it is more appropriate to look at the entity in terms of quantum phase space [7]. Phase space represents all the possible states of a dynamic system, in this case the neutron star, with “all states” simplified to just position and momentum [7]. This simplification is aided by the fact that the neutron has no charge. In the case of our neutron star we have to consider not only the 3 spatial dimensions, but also a momentum dimension associated with all the spatial ones. For this reason, the space the neutron star occupies is more accurate described as 6 dimensional quantum phase space. With the six dimensions coming from the 3 spatial and 3 momentum dimensions. In the case of the neutron star phase space is full of neutrons. Thus every spatial position and every possible momentum linked to each position is full of neutrons. This state where phase space is full is called degeneracy, and in this case neutron degeneracy. The reason why neutrons cannot be further compacted is explained by the Pauli exclusion principle, mentioned above.

The exclusion principle means that it is important to look at the space occupied by the neutron star in terms of phase space. This is because it is possible for more than one neutron to be in the same position as another in 3 dimensional space, provided their momenta are different (or any other quantum property). As a neutron occupies each position and every momentum associated with each position in the star, every possible wave function is defined. This is why the neutrons in the neutron star cannot be compacted any further. No force in the universe can cause them to be further compacted, as it isn’t a question of force- no two fermions can occupy the same quantum state and that’s that. Mathematics simply doesn’t allow it without one of the wave functions being 0.

The packed neutrons cause something know as degeneracy pressure. This is an outwards “force” which is solely responsible for resisting the inwards gravitational crush, and keeping the neutron star stable. It is important to remember however that in the case of neutron stars with high rotation rates the added centrifugal force also resists gravity. The force isn’t a fundamental force like the strong nuclear force is, but “a consequence of the limitations the exclusion principle puts on allowed wave functions”[13]. The fact that all wave functions are defined makes this degeneracy pressure rather strong, resisting that huge gravitational crush mentioned before.

In our neutron star system for a neutron to be further crushed by the force of gravity it must take a wave function which has already been defined by another neutron. This is of course impossible, as shown above. This causes an outwards pressure [14][15].

This degeneracy pressure can be mathematically modelled, as Dr S. Chandrasekhar did this in his paper “the highly collapsed configurations of a stellar mass”[25]. The matter in the neutron star is modelled as a “Fermi gas”. This is the relativistic and quantum equivalent of the ideal gas treatment in classical thermodynamics. We assume that all the neutrons have momentum smaller than a maximum value, which we will call p_{0}. The volume of phase space occupied by the star is thus given by:

 \frac{4}{3}\pi p_{0}^{3}V

This can be derived by thinking about an element of phase space. A small unit of phase space is given by the following:

 du=dVdp_{x}dp_{y}dp_{z}

To find the volume of the whole phase space the star occupies we must sum all the small units of phase space, defined above. Clearly the physical volume of the star spatially will be the volume that the star occupies, we will call this V, however thinking about the star’s momentum “volume” is less obvious. Above we defined the maximum neutron momentum to be a constant (we called \rho_0). Since this is the largest momentum a neutron can have, the momentum “volume” occupied by the neutron star must be equal to this value in each of the 3 spatial dimensions. The constant \frac{4}{3}\pi arises because we are dealing with a spherical space.

The smallest possible volume in phase space is given by h^{3} (Planck’s constant cubed) [25]. In this space two neutrons can exist, each with opposite spin, for the Pauli exclusion principle to hold true. To work out the number of neutrons in the whole space we can multiply the number of neutrons per cubic Planck constant by the volume we derived above:

 N= \frac{2}{h^{3}} \cdot \frac{4}{3}\pi p_{0}^{3}V

 N= \frac{8}{3h^{3}} \pi p_{0}^{3}V

Given that the “number density (n)” is defined by the number of neutrons per unit volume, it is simple to find the number density from the expression above:

 n= \frac{8}{3h^{3}} \pi p_{0}^{3}

Some arbitrary re-arangement allows us to see an interesting patter, we see the form of an integral- which is useful in deriving the degeneracy pressure from the number density:

 n= \frac{8}{h^{3}} \pi \frac{1}{3}p_{0}^{3}

 n= \frac{8}{h^{3}} \pi \int_{0}^{p_{0}}p_{0}^{2}dp

From this we can work out the total energy of the system, given out number density and volume of the space (ξ=total energy):

\xi= \frac{8\pi V}{h^{3}}\int_{0}^{p_{0}}E_{kin}p^{2}dp

As pressure can be proved to be the total energy of our system per unit volume of phase space, to get an equation for the degeneracy pressure we can divide the energy by the volume of the spatial part, and take the derivative with respect to momentum for the momentum part. We therefore have an equation for the pressure exerted by the degenerate neutrons due to limitations put on them by the available wavefunctions and the exclusion principle:

P=\frac{8\pi}{3h^{3}}\int_{0}^{p_{0}}\frac{dE_{kin}}{dp}p^{3}dp

This ends up being the degeneracy pressure when relativistic effects can be neglected, however to be precise for the neutron star these cannot be neglected. By using the famous mass-energy equivalence equation, it is possible to derive a relativistic formula for the degeneracy pressure [25], after some re-arranging and “minor transformations” as described by Chandrasekhar in his paper (this is relatively straight forward):

P= \frac{8\pi m^{4}c^{5}}{3h^{3}}\int_{0}^{\theta_{0}}sinh\theta d\theta

Where \theta= \frac{p}{mc} and \theta_{0}= \frac{p_{0}}{mc}.

In conclusion to this derivation, this is the pressure which prevents the large inward pointing gravitational crush felt by the matter in the neutron star, from collapsing the star. This creates an equilibrium which is very difficult to overcome, as discussed no force in the universe can overcome the exclusion principle. It is also very interesting to see how different the two solutions are between the one that takes into consideration relativistic effects, and the one that doesn’t. It shows why neutron stars and black holes are so complex, as quantum mechanics and relativity are both significant and cannot be neglected.

So no force in the universe can crush the neutrons into a smaller space, but black holes exist and are proof of this exact thing happening. So is the exclusion principle violated as the neutron star collapses to form a blackhole?…

III. Heisenberg’s Uncertainty Principle and Ultimate Collapse

As mentioned in chapter two, the neutron star has reached a stage where it is impossible for the matter to be further compacted. However Black holes exist, and are proof that the neutron star undergoes ultimate collapse. So how does the star do this? And does this ultimate collapse violate the Pauli exclusion principle? The answer lies in another principle of quantum mechanics.

The Heisenberg Uncertainty principle states that for complementary variables, such as position and momentum, or energy and time, the more precisely you measure (or know about, or confine) one of these variable- the less you know about the other [17]. This is expressed as an inequality, for example:

\Delta x \Delta p \geq \frac{\hbar}{2}

The inequality above states that there is a “minimum for the product of uncertainties” [17] for the two complementary variables. This shows that the more defined the position of a particle, the more uncertain the momentum of the particle. This is, of course, because as the uncertainty of the position tends towards 0, the momentum uncertainty must get much larger, so that the product of the two remain greater than the minimum. This has huge implications when thinking about the neutron star. Quantum mechanics tells us that properties of particles are fundamentally uncertain. A neutron, for example, doesn’t have a position – it just exists as a cloud of possible positions, which may be tightly bound, or the opposite, until it interacts with another particle and its wave function collapses (according to the Copenhagen interpretation) [18]. This uncertainty in any quantum property is what the Heisenberg uncertainty principle describes.

The degenerate state of the matter in the neutron star, as discussed in chapter 2, means that the neutrons are as bound as they can possibly be in position. This means that their uncertainties in position are extremely small. Figure 3.1 shows how the Heisenberg uncertainty principle describes bounds for uncertainty in momentum and position. The arrows show the range of possible positions and momenta for a given particle. Figure 3.2 shows that when position is tightly bound, the uncertainty in position is small, therefore the uncertainty in momentum must be large, because the minimum product of the uncertainties is constant.

This principle applies to the neutron star. It is the constituent neutrons that are affected by the limits the principle puts on possible positions and momenta. As discussed before the neutrons are as tightly packed in position space as they could possibly be, due to the limits Pauli’s exclusion principle puts on their wave functions. The inequalities governed by the Heisenberg uncertainty principle show that for the neutrons to be so tightly constrained, their momenta must be highly undefined. Applying this so the 6 dimensional quantum phase space model (established in chapter 2) shows that when position space shrinks, momentum space becomes very large, therefore phase space expands. This is very significant, as it shows that the tighter the position space is confined, the large the phase space as a whole becomes:

 \Delta x \rightarrow 0

 \Delta p \rightarrow \infty

Figure 3.3 shows that as the more massive the neutron star, the more likely the neutrons which it comprises will have high momenta, and the larger the possible momentum space. As the “momentum space” grows, and hence the probability space for possible neutron momenta also grows more neutrons can occupy the same position. This is because there are so many more “positions” in momentum space linked to every location in position space. Since “momentum space” expands more mass can occupy the same position, Pauli’s exclusion principle allows for fermions to have the same position if their momenta are different. This is significant for the neutron star, the more massive it gets, the smaller the uncertainty in position, hence the greater the uncertainty in momentum- and the larger the momentum probability space becomes. This means that the greater the neutron star’s mass, and the more particles that make it up, the smaller the radius.

Figure 3.4 shows how radius varies with mass for neutron stars. The discrete lines show how other factors can affect the mass-radius relationship. These include rapid rotations, which help in supporting the neutron star with the added centripetal force. It is clear to say that there is a strong relationship between the two, and that the radius gets rapidly smaller as mass increases, for the neutron star.

So does the neutron star contracting violate the exclusion principle? Well, the fundamental uncertainty in the behaviour of particles at the quantum scale gives rise to the exclusion principle, as discussed in this chapter. The fact that as the star gains mass the radius gets smaller, doesn’t violate quantum theory, but in fact proves that it must be correct. At no point during the initial collapse of the star does more than one neutron occupy the same quantum state, hence having the same wavefunction, the uncertainty principle just allows for phase space to expand in momentum terms, making the number of possible wavefunctions larger. Thus this process actually validates our theories of quantum mechanics instead of violating them.

However, once the neutron star gains enough mass to collapse into a black hole, it isn’t so simple…

IV. Black Holes- The Singularity and possible exclusion violation

For a neutron star to become a black hole and ultimately collapse it must gain more mass. As discussed before, as the neutron star gets more massive, hence has more matter comprising it doesn’t get bigger in position space, in fact it contracts. There is a point at which the neutron star has so much mass that it undergoes ultimate collapse.

Everything in the universe (with mass) potentially can become a black hole, if compacted into a space smaller than its Schwarzschild radius. If an object were to be compressed to its Schwarzschild radius its escape velocity would be equal to the speed of light. It is possible to find the Schwarzschild radius r_{s} for any given body by first starting with finding its escape velocity equation. To do so the first step is to begin with Newton’s law of gravitation and take an object at the surface of body for which we are trying to find the Schwarzschild radius. The gravitational potential energy at this point, hence the work done to bring the object from infinity to its current position, is the Force multiplied by distance from centre of mass. The kinetic energy required to overcome this potential energy is equal to the potential energy at the surface. Solving for velocity gives the escape velocity for that given body. The Schwarzschild radius is then found by allowing the escape velocity to be equal to the speed of light (there also exists a proper derivation from the Schwarzschild metric):

r_{s}= \frac{2GM}{c^{2}}

As mentioned before the more massive the Neutron star, the smaller the radius. There will be a point where, as the star gains mass and contracts, that the star’s radius becomes smaller than its Schwarzschild radius. This happens because not only is the star getting smaller in radius, but it is gaining mass- meaning the Schwarzschild radius is getting bigger. These will inevitably meet if the neutron star continues to gain mass. The maximum mass at which the Neutron can theoretically exist is called the “Tolman–Oppenheimer–Volkoff limit”. This is said to be the upper bound mass for an object composed of neutron degenerate-matter. The “Tolman–Oppenheimer–Volkoff limit” is actually a bit smaller than some neutron stars observed. Therefore, when working out maximum masses for cold stellar bodies supported by neutron degeneracy the neutron-neutron repulsion at very small distances, mediated by the strong force, must be taken into consideration. This gives an upper mass limit in the region of 3 solar masses for a neutron star, which agrees with experimental evidence.

The neutron star shirks as it gains mass, this was established in the previous chapter. As the neutron star’s radius meets its Schwarzschild radius the fate of the neutron star is sealed, it has become a black hole. As mentioned in the previous chapter, for the neutron star to overcome the exclusion principle it must gain mass. How does it do this?

The neutron star can gain mass in a number of different ways. The merger of two neutron stars often means what is left behind is a black hole, as the product of the merger is so massive that its radius is smaller than the Schwarzschild radius for that combined mass. We now see neutron star mergers using laser interferometers to detect the gravitational waves given off when neutron stars’ orbits decay before they merge. Another way in which a neutron star can gain enough mass to collapse into a black hole is to be in a binary system with another star. This star, if close enough, will have matter ripped away from it due to the great tidal forces exerted on it by the neutron star. The neutron star “eats” away at the star before becoming massive enough to collapse into a black hole. When the star collapses past the Schwarzschild radius, hence becoming a black hole, the real problem with our current theories arise.

Black holes were inferred by solving the Einstein field equations [27]. Peculiar mathematical solutions gave rise to an object which distorts space-time so greatly, that not even electromagnetic radiation can escape [26]. They were thought to be just bizarre solutions to Einstein’s equations, but we eventually observed black holes, and now know them to be real objects in space and time. The problem with quantum theory arises when the neutron star collapses past the Schwarzschild radius, and all the matter cascades towards the “singularity”. The singularity of a black hole is defined as a region of infinite space-time curvature [28]. This is, according to the field equations and current theory, where all the matter of whatever collapses into the black hole, and any matter engulfed after its creation, lies. This is a problem, because unless momentum space is infinitely large, it will be impossible to have all the particles in different quantum states, hence violating the exclusion principle. Furthermore, the position of the matter will be 100% certain, as it has all collapsed to one point. The fundamental uncertainty woven in the skin of quantum mechanics has issues with this, what will happen to the uncertainty in momentum, if the uncertainty in position is 0?

Since neutron stars often have large rotation rates it is possible that all the matter collapse into a “Kerr singularity” [29] also known as a ring singularity. A point cannot have angular momentum, if the mass of a rotating body where to collapse into a point the law of conservation of angular momentum would be violated. This singularity has a ring shape, with 0 thickness. However once again the same problem with exclusion arises, because the ring has no thickness, there simply isn’t enough space to support multiple solar masses of fermionic matter, without violating the exclusion principle.

The idea of a singularity is largely scrutinised by physicists. Our current understanding on how general relativity and quantum mechanics fit together is not good enough to model events at highly distorted space-time on a quantum scale. A theory of “quantum gravity”, or alternatives such as string theory are needed to fill in the gaps. String theorists are trying to bridge the gap between quantum mechanics and general relativity, explaining everything in the universe as being made up of strings of length on Planck length. Perhaps the answer to what exactly all the matter collapses down into lays within the complex mathematics which describes string theory, perhaps it does not. So does the neutron star collapsing beneath its event horizon violate quantum theory?…

V. Does Neutron Star Collapse Violate Quantum Theory?

The Pauli exclusion principle doesn’t allow for the neutron star to collapse. As explained in chapter two, there is no force in the universe that could further compact the constituent neutrons of the star. However black holes exist and are proof that matter must collapse further than the neutron degenerate stage. As explained in chapter three, Heisenberg uncertainty means that neutron stars are able to be compressed further, into radii smaller than their Schwarzschild radii, creating a black hole. But does this violate the exclusion principle?

The collapse of a neutron star itself does not violate the exclusion principle. The principle states that no two fermions can occupy the same quantum state, and at no stage during the collapse does this ever change.

As explained before, for the inequalities given by the uncertainty principle to hold true (shown above), the momentum uncertainty must be large, as the position uncertainty is so small. Large neutron momentum space is a consequence of this, allowing for many more possible momentum “positions” to be associated with each spatial one. As the neutron star gains mass, as mentioned in chapter three, the neutrons have a larger uncertainty in momentum, therefore more neutrons can exist within the same spatial position without violating the exclusion principle, thanks to the uncertainty principle. At no point do two neutrons have the same wavefunction, thus the exclusion principle holds true. No violation of the principle is therefore observed.

Once the initial collapse of the neutron star has been verified not to violate the exclusion principle, it is important to consider what happens one it has collapsed past the Schwarzschild radius, and is now hidden behind its event horizon. As discussed in chapter four, there are different theories of what happens once a black hole is formed, do these seem to violate the exclusion principle?

The major problem with the violation of the principle comes once the neutron star has collapsed past its Schwarzschild radius. The most mainstream theory at the moment is that all the mass collapses towards a “singularity”, mentioned in the previous chapter. As also discussed, there are major problems with thinking of solar masses of matter in one spatial location, mostly as this seems to violate the exclusion principle, unless there is an infinite size of momentum space.

On the other hand, it is possible to argue that once the neutrons have collapsed past the Schwarzschild radius they somehow lose their spin, or someone gain spin, to have integer spin and act as bosons. Bosons are not governed by the exclusion principle; therefore, it is possible to have infinitely many in one quantum state. This matter can all exist in one spatial location without violating the exclusion principle.

Not enough is known about the black hole within its event horizon, and the nature of this location in our universe, means that no information can ever escape it, so we may never know whether neutron star collapse violates quantum theory.

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